diff --git a/thys/Virtual_Substitution/ExecutiblePolyProps.thy b/thys/Virtual_Substitution/ExecutiblePolyProps.thy
--- a/thys/Virtual_Substitution/ExecutiblePolyProps.thy
+++ b/thys/Virtual_Substitution/ExecutiblePolyProps.thy
@@ -1,1083 +1,1084 @@
 text "Executable Polynomial Properties"
 theory ExecutiblePolyProps
   imports
     Polynomials.MPoly_Type_Univariate
     MPolyExtension
 begin
 
 text \<open>(Univariate) Polynomial hiding\<close>
 
 lifting_update poly.lifting
 lifting_forget poly.lifting
 
 text \<open>\<close>
 
 no_notation MPoly_Type.div (infixl "div" 70)
 no_notation MPoly_Type.mod (infixl "mod" 70)
 
 subsection "Lemmas with Monomial and Monomials"
 
 lemma of_nat_monomial: "of_nat p = monomial p 0"
   by (auto simp: poly_mapping_eq_iff lookup_of_nat fun_eq_iff lookup_single)
 
 lemma of_nat_times_monomial: "of_nat p * monomial c i = monomial (p*c) i"
   by (auto simp: poly_mapping_eq_iff prod_fun_def fun_eq_iff of_nat_monomial
       lookup_single mult_single)
 
 lemma monomial_adds_nat_iff: "monomial p i adds c \<longleftrightarrow> lookup c i \<ge> p" for p::"nat"
   apply (auto simp: adds_def lookup_add)
   by (metis add.left_commute nat_le_iff_add remove_key_sum single_add)
 
 lemma update_minus_monomial: "Poly_Mapping.update k i (m - monomial i k) = Poly_Mapping.update k i m"
   by (auto simp: poly_mapping_eq_iff lookup_update update.rep_eq fun_eq_iff lookup_minus
       lookup_single)
 
 lemma monomials_Var: "monomials (Var x::'a::zero_neq_one mpoly) = {Poly_Mapping.single x 1}"
   by transfer (auto simp: Var\<^sub>0_def)
 
 lemma monomials_Const: "monomials (Const x) = (if x = 0 then {} else {0})"
   by transfer' (auto simp: Const\<^sub>0_def)
 
 lemma coeff_eq_zero_iff: "MPoly_Type.coeff c p = 0 \<longleftrightarrow> p \<notin> monomials c"
   by transfer (simp add: not_in_keys_iff_lookup_eq_zero)
 
 lemma monomials_1[simp]: "monomials 1 = {0}"
   by transfer auto
 
 lemma monomials_and_monoms: 
   shows "(k \<in> monomials m) = (\<exists> (a::nat). a \<noteq> 0 \<and> (monomials (MPoly_Type.monom k a)) \<subseteq> monomials m)"
 proof - 
   show ?thesis using monomials_monom by auto
 qed
 
 lemma mult_monomials_dir_one:
   shows "monomials (p*q) \<subseteq> {a+b | a b . a \<in> monomials p \<and> b \<in> monomials q}"
   using monomials_and_monoms mult_monom
   by (simp add: keys_mult monomials.rep_eq times_mpoly.rep_eq) 
 
 lemma monom_eq_zero_iff[simp]: "MPoly_Type.monom a b = 0 \<longleftrightarrow> b = 0"
   by (metis MPolyExtension.coeff_monom MPolyExtension.monom_zero)
 
 lemma update_eq_plus_monomial:
   "v \<ge> lookup m k \<Longrightarrow> Poly_Mapping.update k v m = m + monomial (v - lookup m k) k"
   for v n::nat
   by transfer auto
 
 lemma coeff_monom_mult':
   "MPoly_Type.coeff ((MPoly_Type.monom m' a) * q) (m'm)  = a * MPoly_Type.coeff q (m'm - m')"
   if *: "m'm = m' + (m'm - m')"
   by (subst *) (rule More_MPoly_Type.coeff_monom_mult)
 
 lemma monomials_zero[simp]: "monomials 0 = {}"
   by transfer auto
 
 lemma in_monomials_iff: "x \<in> monomials m \<longleftrightarrow> MPoly_Type.coeff m x \<noteq> 0"
   using coeff_eq_zero_iff[of m x] by auto
 
 lemma monomials_monom_mult: "monomials (MPoly_Type.monom mon a * q) = (if a = 0 then {} else (+) mon ` monomials q)"
   for q::"'a::semiring_no_zero_divisors mpoly"
   apply auto
   subgoal by transfer' (auto elim!: in_keys_timesE)
   subgoal by (simp add: in_monomials_iff More_MPoly_Type.coeff_monom_mult)
   done
 
 subsection "Simplification Lemmas for Const 0 and Const 1"
 lemma add_zero : "P + Const 0 = P"
 proof -
   have h:"P + 0 = P" using add_0_right by auto
   show ?thesis unfolding Const_def using h by (simp add: Const\<^sub>0_zero zero_mpoly.abs_eq)
 qed
 
 (* example *)
 lemma add_zero_example : "((Var 0)^2 - (Const 1)) + Const 0 = ((Var 0)^2 - (Const 1))"
 proof -
   show ?thesis by (simp add : add_zero)
 qed
 
 lemma mult_zero_left : "Const 0 * P = Const 0"
 proof -
   have h:"0*P = 0" by simp
   show ?thesis unfolding Const_def using h by (simp add: Const\<^sub>0_zero zero_mpoly_def)
 qed
 
 lemma mult_zero_right : "P * Const 0 = Const 0"
   by (metis mult_zero_left mult_zero_right)
 
 lemma mult_one_left : "Const 1 * (P :: real mpoly) = P"
   by (simp add: Const.abs_eq Const\<^sub>0_one one_mpoly_def)
 
 lemma mult_one_right : "(P::real mpoly) * Const 1 = P"
   by (simp add: Const.abs_eq Const\<^sub>0_one one_mpoly_def)
 
 
 subsection "Coefficient Lemmas"
 lemma coeff_zero[simp]: "MPoly_Type.coeff 0 x = 0"
   by transfer auto
 
 lemma coeff_sum: "MPoly_Type.coeff (sum f S) x = sum (\<lambda>i. MPoly_Type.coeff (f i) x) S"
   apply (induction S rule: infinite_finite_induct) 
     apply (auto)
   by (metis More_MPoly_Type.coeff_add)
 
 lemma coeff_mult_Var: "MPoly_Type.coeff (x * Var i ^ p) c = (MPoly_Type.coeff x (c - monomial p i) when lookup c i \<ge> p)"
   by transfer'
     (auto simp: Var\<^sub>0_def pprod.monomial_power lookup_times_monomial_right
       of_nat_times_monomial monomial_adds_nat_iff)
 
 lemma lookup_update_self[simp]: "Poly_Mapping.update i (lookup m i) m = m"
   by (auto simp: lookup_update intro!: poly_mapping_eqI)
 
 lemma coeff_Const: "MPoly_Type.coeff (Const p) m = (p when m = 0)"
   by transfer' (auto simp: Const\<^sub>0_def lookup_single)
 
 lemma coeff_Var: "MPoly_Type.coeff (Var p) m = (1 when m = monomial 1 p)"
   by transfer' (auto simp: Var\<^sub>0_def lookup_single when_def)
 
 subsection "Miscellaneous"
 lemma update_0_0[simp]: "Poly_Mapping.update x 0 0 = 0"
   by (metis lookup_update_self lookup_zero)
 
 lemma mpoly_eq_iff: "p = q \<longleftrightarrow> (\<forall>m. MPoly_Type.coeff p m = MPoly_Type.coeff q m)"
   by transfer (auto simp: poly_mapping_eqI)
 
 lemma power_both_sides :
   assumes Ah : "(A::real) \<ge>0"
   assumes Bh : "(B::real) \<ge>0"
   shows "(A\<le>B) = (A^2\<le>B^2)"
   using Ah Bh by (meson power2_le_imp_le power_mono)
 
 lemma in_list_induct_helper: 
   assumes "set(xs)\<subseteq>X"
   assumes  "P []"
   assumes "(\<And>x. x\<in>X \<Longrightarrow> ( \<And>xs. P xs \<Longrightarrow> P (x # xs)))"
   shows "P xs" using assms(1)
 proof(induction xs)
   case Nil
   then show ?case using assms by auto
 next
   case (Cons a xs)
   then show ?case using assms(3) by auto
 qed
 
 theorem in_list_induct [case_names Nil Cons]: 
   assumes  "P []"
   assumes "(\<And>x. x\<in>set(xs) \<Longrightarrow> ( \<And>xs. P xs \<Longrightarrow> P (x # xs)))"
   shows "P xs"
   using in_list_induct_helper[of xs "set(xs)" P] using assms by auto
 
 subsubsection "Keys and vars"
 
 lemma inKeys_inVars : "a\<noteq>0 \<Longrightarrow> x \<in> keys m \<Longrightarrow> x \<in> vars(MPoly_Type.monom m a)"
   by(simp add: vars_def)
 
 lemma notInKeys_notInVars : "x \<notin> keys m \<Longrightarrow> x \<notin> vars(MPoly_Type.monom m a)"
   by(simp add: vars_def)
 
 lemma lookupNotIn : "x \<notin> keys m \<Longrightarrow> lookup m x = 0"
   by (simp add: in_keys_iff)
 
 subsection "Degree Lemmas"
 
 lemma degree_le_iff: "MPoly_Type.degree p x \<le> k \<longleftrightarrow> (\<forall>m\<in>monomials p. lookup m x \<le> k)"
   by transfer simp
 
 lemma degree_less_iff: "MPoly_Type.degree p x < k \<longleftrightarrow> (k>0 \<and> (\<forall>m\<in>monomials p. lookup m x < k))"
   by (transfer fixing: k) (cases "k = 0"; simp)  
 
 lemma degree_ge_iff: "k > 0 \<Longrightarrow> MPoly_Type.degree p x \<ge> k \<longleftrightarrow> (\<exists>m\<in>monomials p. lookup m x \<ge> k)"
   using Max_ge_iff by (meson degree_less_iff not_less) 
 
 lemma degree_greater_iff: "MPoly_Type.degree p x > k \<longleftrightarrow> (\<exists>m\<in>monomials p. lookup m x > k)"
   by transfer' (auto simp: Max_gr_iff)
 
 lemma degree_eq_iff:
   "MPoly_Type.degree p x = k \<longleftrightarrow> (if k = 0
     then (\<forall>m\<in>monomials p. lookup m x = 0)
     else (\<exists>m\<in>monomials p. lookup m x = k) \<and> (\<forall>m\<in>monomials p. lookup m x \<le> k))"
   by (subst eq_iff) (force simp: degree_le_iff degree_ge_iff intro!: antisym)
 
 declare poly_mapping.lookup_inject[simp del]
 
 lemma lookup_eq_and_update_lemma: "lookup m var = deg \<and> Poly_Mapping.update var 0 m = x \<longleftrightarrow>
   m = Poly_Mapping.update var deg x \<and> lookup x var = 0"
   unfolding poly_mapping_eq_iff
   by (force simp: update.rep_eq fun_eq_iff)
 
 
 lemma degree_const : "MPoly_Type.degree (Const (z::real)) (x::nat) = 0"
   by (simp add: degree_eq_iff monomials_Const)
 
 lemma degree_one : "MPoly_Type.degree (Var x :: real mpoly) x = 1"
   by(simp add: degree_eq_iff monomials_Var)
 
 subsection "Lemmas on treating a multivariate polynomial as univariate "
 lemma coeff_isolate_variable_sparse:
   "MPoly_Type.coeff (isolate_variable_sparse p var deg) x =
   (if lookup x var = 0
   then MPoly_Type.coeff p (Poly_Mapping.update var deg x)
   else 0)"
   apply (transfer fixing: x var deg p)
   unfolding lookup_sum
   unfolding lookup_single
   apply (auto simp: when_def)
    apply (subst sum.inter_filter[symmetric])
   subgoal by simp
   subgoal by (simp add: lookup_eq_and_update_lemma Collect_conv_if)
   by (auto intro!: sum.neutral simp add: lookup_update)
 
 lemma isovarspar_sum: 
   "isolate_variable_sparse (p+q) var deg = 
   isolate_variable_sparse (p) var deg
   + isolate_variable_sparse (q) var deg"
   apply (auto simp add: mpoly_eq_iff coeff_isolate_variable_sparse )
    apply (metis More_MPoly_Type.coeff_add coeff_isolate_variable_sparse)
   by (metis More_MPoly_Type.coeff_add add.comm_neutral coeff_isolate_variable_sparse less_numeral_extra(3))
 
 lemma isolate_zero[simp]: "isolate_variable_sparse 0 var n = 0"
   by transfer' (auto simp: mpoly_eq_iff)
 
 lemma coeff_isolate_variable_sparse_minus_monomial:
   "MPoly_Type.coeff (isolate_variable_sparse mp var i) (m - monomial i var) =
   (if lookup m var \<le> i then MPoly_Type.coeff mp (Poly_Mapping.update var i m) else 0)"
   by (simp add: coeff_isolate_variable_sparse lookup_minus update_minus_monomial)
 
 lemma sum_over_zero: "(mp::real mpoly) = (\<Sum>i::nat \<le>MPoly_Type.degree mp x. isolate_variable_sparse mp x i * Var x^i)"
   by (auto simp add: mpoly_eq_iff coeff_sum coeff_mult_Var if_if_eq_conj not_le
       coeff_isolate_variable_sparse_minus_monomial when_def lookup_update degree_less_iff
       simp flip: eq_iff
       intro!: coeff_not_in_monomials)
 
 lemma const_lookup_zero : "isolate_variable_sparse (Const p ::real mpoly) (x::nat) (0::nat) = Const p"
   by (auto simp: mpoly_eq_iff coeff_isolate_variable_sparse coeff_Const when_def)
     (metis lookup_update_self)
 
 lemma const_lookup_suc : "isolate_variable_sparse (Const p :: real mpoly) x (Suc i) = 0"
   apply(auto simp add: mpoly_eq_iff coeff_isolate_variable_sparse coeff_Const when_def)
   by (metis lookup_update lookup_zero nat.distinct(1))
 
 lemma isovar_greater_degree : "\<forall>i > MPoly_Type.degree p var. isolate_variable_sparse p var i = 0"
   apply(auto simp add: mpoly_eq_iff coeff_isolate_variable_sparse degree_less_iff)
   by (metis coeff_not_in_monomials less_irrefl_nat lookup_update)
 
 lemma isolate_var_0 : "isolate_variable_sparse (Var x::real mpoly) x 0 = 0"
   apply(auto simp add: mpoly_eq_iff coeff_isolate_variable_sparse coeff_Var when_def)
   by (metis gr0I lookup_single_eq lookup_update_self n_not_Suc_n)
 
 lemma isolate_var_one : "isolate_variable_sparse (Var x :: real mpoly) x 1 = 1"
   by (auto simp add: mpoly_eq_iff coeff_isolate_variable_sparse coeff_Var coeff_eq_zero_iff)
     (smt More_MPoly_Type.coeff_monom One_nat_def add_diff_cancel_left' lookup_eq_and_update_lemma
       lookup_single_eq lookup_update_self monom_one plus_1_eq_Suc single_diff single_zero update_minus_monomial)
 
 lemma isovarspase_monom :
   assumes notInKeys : "x \<notin> keys m"
   assumes notZero : "a \<noteq> 0"
   shows "isolate_variable_sparse (MPoly_Type.monom m a) x 0 = (MPoly_Type.monom m a :: real mpoly)"
   using assms
   by (auto simp add: mpoly_eq_iff coeff_isolate_variable_sparse coeff_eq_zero_iff in_keys_iff)
     (metis lookup_update_self)
 
 lemma isolate_variable_spase_zero : "lookup m x = 0 \<Longrightarrow>
     insertion (nth L) ((MPoly_Type.monom m a)::real mpoly) = 0 \<Longrightarrow>
     a \<noteq> 0 \<Longrightarrow> insertion (nth L) (isolate_variable_sparse (MPoly_Type.monom m a) x 0) = 0"
   by (simp add: isovarspase_monom lookup_eq_zero_in_keys_contradict notInKeys_notInVars)
 
 lemma isovarsparNotIn : "x \<notin> vars (p::real mpoly) \<Longrightarrow> isolate_variable_sparse p x 0 = p"
 proof(induction p rule: mpoly_induct)
   case (monom m a)
   then show ?case
     apply(cases "a=0")
     by (simp_all add: isovarspase_monom vars_monom_keys)
 next
   case (sum p1 p2 m a)
   then show ?case 
     by (simp add: monomials.rep_eq vars_add_monom isovarspar_sum)
 qed
 
 
 lemma degree0isovarspar :
   assumes deg0 : "MPoly_Type.degree (p::real mpoly) x = 0"
   shows "isolate_variable_sparse p x 0 = p"
 proof -
   have h1 : "p = (\<Sum>i::nat \<le>MPoly_Type.degree p x. isolate_variable_sparse p x i * Var x ^ i)"
     using sum_over_zero by auto
   have h2a : "\<forall>f. (\<Sum>i::nat \<le>0. f i) = f 0"
     apply(simp add: sum_def)
     by (metis add.right_neutral comm_monoid_add_class.sum_def finite.emptyI insert_absorb insert_not_empty sum.empty sum.insert)
   have h2 : "p = isolate_variable_sparse p x 0 * Var x ^ 0"
     using deg0 h1 h2a by auto
   show ?thesis using h2
     by auto 
 qed
 
 
 subsection "Summation Lemmas"
 
 lemma summation_normalized :
   assumes nonzero : "(B ::real) \<noteq>0"
   shows "(\<Sum>i = 0..<((n::nat)+1). (f i :: real) * B ^ (n - i)) = (\<Sum>i = 0..<(n+1). (f i) / (B ^ i)) * (B^n)"
 proof -
   have h1a : "\<forall>x::real. ((\<Sum>i = 0..<(n+1). (f i) / (B ^ i)) * x = (\<Sum>i = 0..<(n+1). ((f i) / (B ^ i)) * x))"  
     apply(induction n )
      apply(auto)
     by (simp add: distrib_right)
   have h1 : "(\<Sum>i = 0..<(n+1). (f i) / (B ^ i)) * (B^n) = (\<Sum>i = 0..<(n+1). ((f i) / (B ^ i)) * (B^n))"
     using h1a by auto
   have h2 : "(\<Sum>i = 0..<(n+1). ((f i) / (B ^ i)) * (B^n)) = (\<Sum>i = 0..<(n+1). (f i) * ((B^n) / (B ^ i)))"
     by auto
   have h3 : "(\<Sum>i = 0..<(n+1). (f i) * ((B^n) / (B ^ i))) = (\<Sum>i = 0..<(n+1). (f i) * B ^ (n - i))"
     using add.right_neutral nonzero power_diff by fastforce
   show ?thesis using h1 h2 h3 by auto
 qed
 
 lemma normalize_summation :
   assumes nonzero : "(B::real)\<noteq>0"
   shows "(\<Sum>i = 0..<n+1. f i * B ^ (n - i))= 0
           \<longleftrightarrow>
   (\<Sum>i = 0..<(n::nat)+1. (f i::real) / (B ^ i)) = 0"
 proof - 
   have pow_zero : "\<forall>i. B^(i :: nat)\<noteq>0" using nonzero by(auto)
   have single_division_zero : "\<forall>X. B*(X::real)=0 \<longleftrightarrow> X=0" using nonzero by(auto)
   have h1: "(\<Sum>i = 0..<n+1. (f i) / (B ^ i)) = 0 \<longleftrightarrow> ((\<Sum>i = 0..<n+1. (f i) / (B ^ i)))*B^n = 0"
     using nonzero single_division_zero by(auto)
   have h2: "((\<Sum>i = 0..<n+1. (f i) / (B ^ i))*(B^n)) = ((\<Sum>i = 0..<n+1. (f i) * (B ^ (n- i))))"
     using summation_normalized nonzero by auto
   show ?thesis using h1 h2 by auto
 qed
 
 
 lemma normalize_summation_less :
   assumes nonzero : "(B::real)\<noteq>0"
   shows "(\<Sum>i = 0..<(n+1). (f i) * B ^ (n - i)) * B ^ (n mod 2) < 0
           \<longleftrightarrow>
   (\<Sum>i = 0..<((n::nat)+1). (f i::real) / (B ^ i)) < 0"
 proof - 
   have h1 : "(\<Sum>i = 0..<(n+1). (f i) * B ^ (n - i)) * B ^ (n mod 2) < 0
         \<longleftrightarrow>  (\<Sum>i = 0..<(n+1). (f i) / (B ^ i)) * (B^n) * B ^ (n mod 2) < 0"
     using summation_normalized nonzero by auto
   have h2a : "n mod 2 = 0 \<or> n mod 2 = 1" by auto
   have h2b : "n mod 2 = 1 \<Longrightarrow> odd n" by auto
   have h2c : "(B^n) * B ^ (n mod 2) > 0"
     using h2a h2b apply auto
     using nonzero apply presburger
     by (metis even_Suc mult.commute nonzero power_Suc zero_less_power_eq)    
   have h2 : "\<forall>x. ((x * (B^n) * B ^ (n mod 2) < 0) = (x<0))"
     using h2c using mult.assoc by (metis mult_less_0_iff not_square_less_zero) 
   show ?thesis using h1 h2 by auto
 qed
 
 subsection "Additional Lemmas with Vars"
 
 lemma not_in_isovarspar : "isolate_variable_sparse (p::real mpoly) var x = (q::real mpoly) \<Longrightarrow> var\<notin>(vars q)"
   apply(simp add: isolate_variable_sparse_def vars_def)
 proof -
   assume a1: "MPoly (\<Sum>m | m \<in> monomials p \<and> lookup m var = x. monomial (MPoly_Type.coeff p m) (Poly_Mapping.update var 0 m)) = q"
   { fix pp :: "nat \<Rightarrow>\<^sub>0 nat"
     have "isolate_variable_sparse p var x = q"
       using a1 isolate_variable_sparse.abs_eq by blast
     then have "var \<notin> keys pp \<or> pp \<notin> keys (mapping_of q)"
       by (metis (no_types) coeff_def coeff_isolate_variable_sparse in_keys_iff) }
   then show "\<forall>p\<in>keys (mapping_of q). var \<notin> keys p"
     by meson
 qed 
 
 lemma not_in_add : "var\<notin>(vars (p::real mpoly)) \<Longrightarrow> var\<notin>(vars (q::real mpoly)) \<Longrightarrow> var\<notin>(vars (p+q))"
   by (meson UnE in_mono vars_add)
 
 lemma not_in_mult : "var\<notin>(vars (p::real mpoly)) \<Longrightarrow> var\<notin>(vars (q::real mpoly)) \<Longrightarrow> var\<notin>(vars (p*q))"
   by (meson UnE in_mono vars_mult)
 
 lemma not_in_neg : "var\<notin>(vars(p::real mpoly)) \<longleftrightarrow> var\<notin>(vars(-p))"
 proof -
   have h: "var \<notin> (vars (-1::real mpoly))" by (metis add_diff_cancel_right' add_neg_numeral_special(8) isolate_var_one isolate_zero isovarsparNotIn isovarspar_sum not_in_isovarspar)
   show ?thesis using not_in_mult using h by fastforce
 qed
 
 lemma not_in_sub : "var\<notin>(vars (p::real mpoly)) \<Longrightarrow> var\<notin>(vars (q::real mpoly)) \<Longrightarrow> var\<notin>(vars (p-q))"
   using not_in_add not_in_neg by fastforce
 
 
 lemma not_in_pow : "var\<notin>(vars(p::real mpoly)) \<Longrightarrow> var\<notin>(vars(p^i))"
 proof (induction i)
   case 0
   then show ?case using isolate_var_one not_in_isovarspar
     by (metis power_0) 
 next 
   case (Suc i)
   then show ?case using not_in_mult by simp
 qed
 
 lemma not_in_sum_var: "(\<forall>i. var\<notin>(vars(f(i)::real mpoly))) \<Longrightarrow> var\<notin>(vars(\<Sum>i\<in>{0..<(n::nat)}.f(i)))"
 proof (induction n)
   case 0
   then show ?case using isolate_zero not_in_isovarspar by fastforce
 next
   case (Suc n)
   have h1: "(sum f {0..<Suc n}) = (sum f {0..< n}) + (f n)" using sum.atLeast0_lessThan_Suc by blast
   have h2: "var \<notin> vars (f n)" by (simp add: Suc.prems)
   then show ?case using h1 vars_add by (simp add: Suc.IH Suc.prems not_in_add)
 qed
 
 lemma not_in_sum : "(\<forall>i. var\<notin>(vars(f(i)::real mpoly))) \<Longrightarrow> \<forall>(n::nat). var\<notin>(vars(\<Sum>i\<in>{0..<n}.f(i)))"
   using not_in_sum_var by blast
 
 lemma not_contains_insertion_helper : 
   "\<forall>x\<in>keys (mapping_of p). var \<notin> keys x \<Longrightarrow>
          (\<And>k f. (k \<notin> keys f) = (lookup f k = 0)) \<Longrightarrow>
          lookup (mapping_of p) a = 0 \<or>
          (\<Prod>aa. (if aa < length L then L[var := y] ! aa else 0) ^ lookup a aa) =
          (\<Prod>aa. (if aa < length L then L[var := x] ! aa else 0) ^ lookup a aa)"
   apply(cases "lookup (mapping_of p) a = 0") 
    apply auto
   apply(rule Prod_any.cong) 
   apply auto
   using lookupNotIn nth_list_update_neq power_0 by smt
 
 lemma not_contains_insertion : 
   assumes notIn : "var \<notin> vars (p:: real mpoly)"
   assumes exists : "insertion (nth_default 0 (list_update L var x)) p = val"
   shows "insertion (nth_default 0 (list_update L var y)) p = val"
   using notIn exists
   apply(simp add: insertion_def insertion_aux_def insertion_fun_def)
   unfolding vars_def nth_default_def
   using not_in_keys_iff_lookup_eq_zero  
   apply auto
   apply(rule Sum_any.cong) 
   apply simp
   using not_contains_insertion_helper[of p var _ L y x]
 proof -
   fix a :: "nat \<Rightarrow>\<^sub>0 nat"
   assume a1: "\<forall>x\<in>keys (mapping_of p). var \<notin> keys x"
   assume "\<And>k f. ((k::'a) \<notin> keys f) = (lookup f k = 0)"
   then show "lookup (mapping_of p) a = 0 \<or> (\<Prod>n. (if n < length L then L[var := y] ! n else 0) ^ lookup a n) = (\<Prod>n. (if n < length L then L[var := x] ! n else 0) ^ lookup a n)"
     using a1 \<open>\<And>a. \<lbrakk>\<forall>x\<in>keys (mapping_of p). var \<notin> keys x; \<And>k f. (k \<notin> keys f) = (lookup f k = 0)\<rbrakk> \<Longrightarrow> lookup (mapping_of p) a = 0 \<or> (\<Prod>aa. (if aa < length L then L[var := y] ! aa else 0) ^ lookup a aa) = (\<Prod>aa. (if aa < length L then L[var := x] ! aa else 0) ^ lookup a aa)\<close> by blast
 qed
 
 
 subsection "Insertion Lemmas"
 lemma insertion_sum_var : "((insertion f (\<Sum>i\<in>{0..<(n::nat)}.g(i))) = (\<Sum>i\<in>{0..<n}. insertion f (g i)))"
 proof (induction n)
   case 0
   then show ?case by auto
 next
   case (Suc n)
   then show ?case by (simp add: insertion_add)
 qed
 
 (* changed to explicitly typecast n as a nat *)
 lemma insertion_sum : "\<forall>(n::nat). ((insertion f (\<Sum>i\<in>{0..<n}.g(i))) = (\<Sum>i\<in>{0..<n}. insertion f (g i)))"
   using insertion_sum_var by blast
 
 
 lemma insertion_sum' : "\<And>(n::nat). ((insertion f (\<Sum>i\<le>n. g(i))) = (\<Sum>i\<le>n. insertion f (g i)))"
   by (metis (no_types, lifting) fun_sum_commute insertion_add insertion_zero sum.cong) 
 
 lemma insertion_pow : "insertion f (p^i) = (insertion f p)^i"
 proof (induction i)
   case 0
   then show ?case by auto
 next
   case (Suc n)
   then show ?case by (simp add: insertion_mult)
 qed
 
 lemma insertion_neg : "insertion f (-p) = -insertion f p"
   by (metis add.inverse_inverse insertionNegative)
 
 lemma insertion_var : 
   "length L > var \<Longrightarrow> insertion (nth_default 0 (list_update L var x)) (Var var) = x"
   by (auto simp: monomials_Var coeff_Var insertion_code nth_default_def)
 
 lemma insertion_var_zero : "insertion (nth_default 0 (x#xs)) (Var 0) = x" using insertion_var
   by fastforce
 
 lemma notIn_insertion_sub : "x\<notin>vars(p::real mpoly) \<Longrightarrow> x\<notin>vars(q::real mpoly)
                              \<Longrightarrow> insertion f (p-q) = insertion f p - insertion f q"
   by (metis ab_group_add_class.ab_diff_conv_add_uminus insertion_add insertion_neg)
 
 lemma insertion_sub : "insertion f (A-B :: real mpoly) = insertion f A - insertion f B"
   using insertion_neg insertion_add
   by (metis uminus_add_conv_diff)
 
 lemma insertion_four : "insertion ((nth_default 0) xs) 4 = 4"
   by (metis (no_types, lifting) insertion_add insertion_one numeral_plus_numeral one_add_one semiring_norm(2) semiring_norm(6))
 
 lemma insertion_add_ind_basecase:
   "insertion (nth (list_update L var x)) ((\<Sum>i::nat \<le> 0. isolate_variable_sparse p var i * (Var var)^i))
   = (\<Sum>i = 0..<(0+1).  insertion (nth (list_update L var x)) (isolate_variable_sparse p var i * (Var var)^i))"
 proof - 
   have h1: "((\<Sum>i::nat \<le> 0. isolate_variable_sparse p var i * (Var var)^i))
    = (isolate_variable_sparse p var 0 * (Var var)^0)"
     by auto
   show ?thesis using h1
     by auto 
 qed
 
 lemma insertion_add_ind:
   "insertion (nth_default 0 (list_update L var x)) ((\<Sum>i::nat \<le> d. isolate_variable_sparse p var i * (Var var)^i))
   = (\<Sum>i = 0..<(d+1).  insertion (nth_default 0 (list_update L var x)) (isolate_variable_sparse p var i * (Var var)^i))"
 proof (induction d)
   case 0
   then show ?case using insertion_add_ind_basecase nth_default_def
     by auto
 next
   case (Suc n)
   then show ?case using insertion_add apply auto
     by (simp add: insertion_add)
 qed
 
 lemma sum_over_degree_insertion :
   assumes lLength : "length L > var"
   assumes deg : "MPoly_Type.degree (p::real mpoly) var = d"
   shows "(\<Sum>i = 0..<(d+1). insertion (nth_default 0 (list_update L var x)) (isolate_variable_sparse p var i) * (x^i))
           = insertion (nth_default 0 (list_update L var x)) p"
 proof -
   have h1: "(p::real mpoly) = (\<Sum>i::nat \<le>(MPoly_Type.degree p var). isolate_variable_sparse p var i * (Var var)^i)" using sum_over_zero by auto
   have h2: "insertion (nth_default 0 (list_update L var x)) p = 
     insertion (nth_default 0 (list_update L var x)) ((\<Sum>i::nat \<le> d. isolate_variable_sparse p var i * (Var var)^i))" using h1 deg by auto
   have h3:  "insertion (nth_default 0 (list_update L var x)) p = (\<Sum>i = 0..<(d+1).  insertion (nth_default 0 (list_update L var x)) (isolate_variable_sparse p var i * (Var var)^i))"
     using h2 insertion_add_ind nth_default_def
     by (simp add: )
   show ?thesis using h3 insertion_var insertion_pow
     by (metis (no_types, lifting) insertion_mult lLength sum.cong)
 qed
 
 
 
 lemma insertion_isovarspars_free :
   "insertion (nth_default 0 (list_update L var x)) (isolate_variable_sparse (p::real mpoly) var (i::nat))
   =insertion (nth_default 0 (list_update L var y)) (isolate_variable_sparse (p::real mpoly) var (i::nat))"
 proof -
   have h : "var \<notin> vars(isolate_variable_sparse (p::real mpoly) var (i::nat))"
     by (simp add: not_in_isovarspar)
   then show ?thesis using not_contains_insertion
     by blast 
 qed
 lemma insertion_zero : "insertion f (Const 0 ::real mpoly) = 0"
   by (metis add_cancel_right_right add_zero insertion_zero)
 
 lemma insertion_one : "insertion f (Const 1 ::real mpoly) = 1"
   by (metis insertion_one mult.right_neutral mult_one_left)
 
 lemma insertion_const : "insertion f (Const c::real mpoly) = (c::real)"
   by (auto simp: monomials_Const coeff_Const insertion_code)
 
 
 subsection "Putting Things Together"
 subsubsection "More Degree Lemmas"
 lemma degree_add_leq : 
   assumes h1 : "MPoly_Type.degree a var \<le> x"
   assumes h2 : "MPoly_Type.degree b var \<le> x"
   shows "MPoly_Type.degree (a+b) var \<le> x"
   using degree_eq_iff coeff_add coeff_not_in_monomials
   by (smt (z3) More_MPoly_Type.coeff_add add.left_neutral coeff_eq_zero_iff degree_le_iff h1 h2)
 
 lemma degree_add_less : 
   assumes h1 : "MPoly_Type.degree a var < x"
   assumes h2 : "MPoly_Type.degree b var < x"
   shows "MPoly_Type.degree (a+b) var < x"
 proof -
   obtain pp :: "nat \<Rightarrow> nat \<Rightarrow> 'a mpoly \<Rightarrow> nat \<Rightarrow>\<^sub>0 nat" where
     "\<forall>x0 x1 x2. (\<exists>v3. v3 \<in> monomials x2 \<and> \<not> lookup v3 x1 < x0) = (pp x0 x1 x2 \<in> monomials x2 \<and> \<not> lookup (pp x0 x1 x2) x1 < x0)"
     by moura
   then have f1: "\<forall>m n na. (\<not> MPoly_Type.degree m n < na \<or> 0 < na \<and> (\<forall>p. p \<notin> monomials m \<or> lookup p n < na)) \<and> (MPoly_Type.degree m n < na \<or> \<not> 0 < na \<or> pp na n m \<in> monomials m \<and> \<not> lookup (pp na n m) n < na)"
     by (metis (no_types) degree_less_iff)
   then have "0 < x \<and> (\<forall>p. p \<notin> monomials a \<or> lookup p var < x)"
     using assms(1) by presburger
   then show ?thesis
     using f1 by (metis MPolyExtension.coeff_add add.left_neutral assms(2) coeff_eq_zero_iff)
 qed
 
 lemma degree_sum : "(\<forall>i\<in>{0..n::nat}. MPoly_Type.degree (f i :: real mpoly) var \<le> x) \<Longrightarrow> (MPoly_Type.degree (\<Sum>x\<in>{0..n}. f x) var) \<le> x"
 proof(induction n)
   case 0
   then show ?case by auto
 next
   case (Suc n)
   then show ?case by(simp add: degree_add_leq)
 qed
 
 lemma degree_sum_less : "(\<forall>i\<in>{0..n::nat}. MPoly_Type.degree (f i :: real mpoly) var < x) \<Longrightarrow> (MPoly_Type.degree (\<Sum>x\<in>{0..n}. f x) var) < x"
 proof(induction n)
   case 0
   then show ?case by auto
 next
   case (Suc n)
   then show ?case by(simp add: degree_add_less)
 qed
 
 
 lemma varNotIn_degree : 
   assumes "var \<notin> vars p"
   shows "MPoly_Type.degree p var = 0"
 proof-
   have "\<forall>m\<in>monomials p. lookup m var = 0"
     using assms unfolding vars_def keys_def
     using monomials.rep_eq by fastforce
   then show ?thesis
     using degree_less_iff by blast
 qed
 
 lemma degree_mult_leq : 
   assumes pnonzero: "(p::real mpoly)\<noteq>0"
   assumes qnonzero: "(q::real mpoly)\<noteq>0"
   shows "MPoly_Type.degree ((p::real mpoly) * (q::real mpoly)) var \<le> (MPoly_Type.degree p var) + (MPoly_Type.degree q var)"
 proof(cases "MPoly_Type.degree (p*q) var =0")
   case True
   then show ?thesis by simp
 next
   case False
   have hp: "\<forall>m\<in>monomials p. lookup m var \<le> MPoly_Type.degree p var" using degree_eq_iff
     by (metis zero_le) 
   have hq: "\<forall>m\<in>monomials q. lookup m var \<le> MPoly_Type.degree q var" using degree_eq_iff
     by (metis zero_le) 
   have hpq: "\<forall>m\<in>{a+b | a b . a \<in> monomials p \<and> b \<in> monomials q}.
       lookup m var \<le> (MPoly_Type.degree p var) + (MPoly_Type.degree q var)"
     by (smt add_le_mono hp hq mem_Collect_eq plus_poly_mapping.rep_eq)
   have h1: "(\<forall>m\<in>monomials (p*q). lookup m var \<le> (MPoly_Type.degree p var) + (MPoly_Type.degree q var))" 
     using mult_monomials_dir_one hpq
     by blast 
   then show ?thesis using h1 degree_eq_iff False
     by (simp add: degree_le_iff)
 qed
 
 lemma degree_exists_monom: 
   assumes "p\<noteq>0"
   shows  "\<exists>m\<in>monomials p. lookup m var = MPoly_Type.degree p var"
 proof(cases "MPoly_Type.degree p var =0")
   case True
   have h1: "\<exists>m\<in>monomials p. lookup m var = 0" unfolding monomials_def
     by (metis True assms(1) aux degree_eq_iff in_keys_iff mapping_of_inject monomials.rep_eq monomials_def zero_mpoly.rep_eq)
   then show ?thesis using h1
     using True by simp
 next
   case False
   then show ?thesis using degree_eq_iff assms(1) apply(auto)
     by (metis degree_eq_iff dual_order.strict_iff_order) 
 qed
 
 lemma degree_not_exists_monom: 
   assumes "p\<noteq>0"
   shows  "\<not> (\<exists> m\<in>monomials p. lookup m var > MPoly_Type.degree p var)"
 proof - 
   show ?thesis using degree_less_iff by blast 
 qed
 
 lemma degree_monom: "MPoly_Type.degree (MPoly_Type.monom x y) v = (if y = 0 then 0 else lookup x v)"
   by (auto simp: degree_eq_iff)
 
 lemma degree_plus_disjoint:
   "MPoly_Type.degree (p + MPoly_Type.monom m c) v = max (MPoly_Type.degree p v) (MPoly_Type.degree (MPoly_Type.monom m c) v)"
   if "m \<notin> monomials p"
   for p::"real mpoly"
   using that
   apply (subst degree_eq_iff)
   apply (auto simp: monomials_add_disjoint)
             apply (auto simp: degree_eq_iff degree_monom)
        apply (metis MPoly_Type.degree_zero degree_exists_monom less_numeral_extra(3))
   using degree_le_iff apply blast
   using degree_eq_iff
      apply (metis max_def neq0_conv)
     apply (metis degree_eq_iff max.coboundedI1 neq0_conv)
    apply (metis MPoly_Type.degree_zero degree_exists_monom max_def zero_le)
   using degree_le_iff max.cobounded1 by blast
 
 subsubsection "More isolate\\_variable\\_sparse lemmas"
 
 lemma isolate_variable_sparse_ne_zeroD:
   "isolate_variable_sparse mp v x \<noteq> 0 \<Longrightarrow> x \<le> MPoly_Type.degree mp v"
   using isovar_greater_degree leI by blast
 
 context includes poly.lifting begin
 lift_definition mpoly_to_nested_poly::"'a::comm_monoid_add mpoly \<Rightarrow> nat \<Rightarrow> 'a mpoly Polynomial.poly" is
   "\<lambda>(mp::'a mpoly) (v::nat) (p::nat). isolate_variable_sparse mp v p"
   \<comment> \<open>note \<^const>\<open>extract_var\<close> nests the other way around\<close>
   unfolding MOST_iff_cofinite
 proof -
   fix mp::"'a mpoly" and v::nat
   have "{p. isolate_variable_sparse mp v p \<noteq> 0} \<subseteq> {0..MPoly_Type.degree mp v}"
     (is "?s \<subseteq> _")
     by (auto dest!: isolate_variable_sparse_ne_zeroD)
   also have "finite \<dots>" by simp
   finally (finite_subset) show "finite ?s" .
 qed
 
 lemma degree_eq_0_mpoly_to_nested_polyI:
   "mpoly_to_nested_poly mp v = 0 \<Longrightarrow> MPoly_Type.degree mp v = 0"
   apply transfer'
   apply (simp add: degree_eq_iff)
   apply transfer'
   apply (auto simp: fun_eq_iff)
 proof -
   fix mpa :: "'a mpoly" and va :: nat and m :: "nat \<Rightarrow>\<^sub>0 nat"
   assume a1: "\<forall>x. (\<Sum>m | m \<in> monomials mpa \<and> lookup m va = x. monomial (MPoly_Type.coeff mpa m) (Poly_Mapping.update va 0 m)) = 0"
   assume a2: "m \<in> monomials mpa"
   have f3: "\<forall>m p. lookup (mapping_of m) p \<noteq> (0::'a) \<or> p \<notin> monomials m"
     by (metis (full_types) coeff_def coeff_eq_zero_iff)
   have f4: "\<forall>n. mapping_of (isolate_variable_sparse mpa va n) = 0"
     using a1 by (simp add: isolate_variable_sparse.rep_eq)
   have "\<forall>p n. lookup 0 (p::nat \<Rightarrow>\<^sub>0 nat) = (0::'a) \<or> (0::nat) = n"
     by simp
   then show "lookup m va = 0"
     using f4 f3 a2 by (metis coeff_def coeff_isolate_variable_sparse lookup_eq_and_update_lemma)
 qed
 
 lemma coeff_eq_zero_mpoly_to_nested_polyD: "mpoly_to_nested_poly mp v = 0 \<Longrightarrow> MPoly_Type.coeff mp 0 = 0"
   apply transfer'
   apply transfer'
   by (metis (no_types) coeff_def coeff_isolate_variable_sparse isolate_variable_sparse.rep_eq lookup_zero update_0_0)
 
 lemma mpoly_to_nested_poly_eq_zero_iff[simp]:
   "mpoly_to_nested_poly mp v = 0 \<longleftrightarrow> mp = 0"
   apply (auto simp: coeff_eq_zero_mpoly_to_nested_polyD degree_eq_0_mpoly_to_nested_polyI)
 proof -
   show "mpoly_to_nested_poly mp v = 0 \<Longrightarrow> mp = 0" 
     apply (frule degree_eq_0_mpoly_to_nested_polyI)
     apply (frule coeff_eq_zero_mpoly_to_nested_polyD)
     apply (transfer' fixing: mp v)
     apply (transfer' fixing: mp v)
     apply (auto simp: fun_eq_iff mpoly_eq_iff intro!: sum.neutral)
   proof -
     fix m :: "nat \<Rightarrow>\<^sub>0 nat"
     assume a1: "\<forall>x. (\<Sum>m | m \<in> monomials mp \<and> lookup m v = x. monomial (MPoly_Type.coeff mp m) (Poly_Mapping.update v 0 m)) = 0"
     assume a2: "MPoly_Type.degree mp v = 0"
     have "\<forall>n. isolate_variable_sparse mp v n = 0"
       using a1 by (simp add: isolate_variable_sparse.abs_eq zero_mpoly.abs_eq)
     then have f3: "\<forall>p. MPoly_Type.coeff mp p = MPoly_Type.coeff 0 p \<or> lookup p v \<noteq> 0"
       by (metis (no_types) coeff_isolate_variable_sparse lookup_update_self)
     then show "MPoly_Type.coeff mp m = 0"
       using a2 coeff_zero
       by (metis coeff_not_in_monomials degree_eq_iff)
   qed
   show "mp = 0 \<Longrightarrow> mpoly_to_nested_poly 0 v = 0" 
     subgoal
       apply transfer'
       apply transfer'
       by (auto simp: fun_eq_iff intro!: sum.neutral)
     done
 qed
 
 lemma isolate_variable_sparse_degree_eq_zero_iff: "isolate_variable_sparse p v (MPoly_Type.degree p v) = 0 \<longleftrightarrow> p = 0"
   apply (transfer')
   apply auto
 proof -
   fix pa :: "'a mpoly" and va :: nat
   assume "(\<Sum>m | m \<in> monomials pa \<and> lookup m va = MPoly_Type.degree pa va. monomial (MPoly_Type.coeff pa m) (Poly_Mapping.update va 0 m)) = 0"
   then have "mapping_of (isolate_variable_sparse pa va (MPoly_Type.degree pa va)) = 0"
     by (simp add: isolate_variable_sparse.rep_eq)
   then show "pa = 0"
     by (metis (no_types) coeff_def coeff_eq_zero_iff coeff_isolate_variable_sparse degree_exists_monom lookup_eq_and_update_lemma lookup_zero)
 qed
 
 lemma degree_eq_univariate_degree: "MPoly_Type.degree p v =
     (if p = 0 then 0 else Polynomial.degree (mpoly_to_nested_poly p v))"
   apply auto
   apply (rule antisym)
   subgoal
     apply (rule Polynomial.le_degree)
     apply (auto simp: )
     apply transfer'
     by (simp add: isolate_variable_sparse_degree_eq_zero_iff)
   subgoal apply (rule Polynomial.degree_le)
     apply (auto simp: elim!: degree_eq_zeroE)
     apply transfer'
     by (simp add: isovar_greater_degree)
   done
 
 lemma compute_mpoly_to_nested_poly[code]:
   "coeffs (mpoly_to_nested_poly mp v) =
     (if mp = 0 then []
     else map (isolate_variable_sparse mp v) [0..<Suc(MPoly_Type.degree mp v)])"
   unfolding coeffs_def unfolding mpoly_to_nested_poly_eq_zero_iff degree_eq_univariate_degree apply auto
   subgoal by transfer' (rule refl)
   by transfer' (rule refl)
 
 end
 
 lemma isolate_variable_sparse_monom: "isolate_variable_sparse (MPoly_Type.monom m a) v i =
   (if a = 0 \<or> lookup m v \<noteq> i then 0 else MPoly_Type.monom (Poly_Mapping.update v 0 m) a)"
 proof -
   have *: "{x. x = m \<and> lookup x v = i} = (if lookup m v = i then {m} else {})"
     by auto
   show ?thesis
     by (transfer' fixing: m a v i) (simp add:*)
 qed
 
 
 
 lemma isolate_variable_sparse_monom_mult:
   "isolate_variable_sparse (MPoly_Type.monom m a * q) v n =
     (if n \<ge> lookup m v
     then MPoly_Type.monom (Poly_Mapping.update v 0 m) a * isolate_variable_sparse q v (n - lookup m v)
     else 0)"
   for q::"'a::semiring_no_zero_divisors mpoly"
   apply (auto simp: MPoly_Type.mult_monom)
   subgoal
     apply transfer'
     subgoal for mon v i a q
       apply (auto simp add: monomials_monom_mult sum_distrib_left)
       apply (rule sum.reindex_bij_witness_not_neutral[where
             j="\<lambda>a. a - mon"
             and i="\<lambda>a. mon + a"
             and S'="{}"
             and T'="{}"
             ])
               apply (auto simp: lookup_add)
       apply (auto simp: poly_mapping_eq_iff fun_eq_iff single.rep_eq Poly_Mapping.mult_single)
       apply (auto simp: when_def More_MPoly_Type.coeff_monom_mult)
       by (auto simp: lookup_update lookup_add split: if_splits)
     done
   subgoal
     apply transfer'
     apply (auto intro!: sum.neutral simp: monomials_monom_mult )
     apply (rule poly_mapping_eqI)
     apply (auto simp: lookup_single when_def)
     by (simp add: lookup_add)
   done
 
 lemma isolate_variable_sparse_mult:
   "isolate_variable_sparse (p * q) v n = (\<Sum>i\<le>n. isolate_variable_sparse p v i * isolate_variable_sparse q v (n - i))"
   for p q::"'a::semiring_no_zero_divisors mpoly"
 proof (induction p rule: mpoly_induct)
   case (monom m a)
   then show ?case
     by (cases "a = 0")
       (auto simp add: mpoly_eq_iff coeff_sum coeff_mult if_conn if_distrib if_distribR
         isolate_variable_sparse_monom isolate_variable_sparse_monom_mult
         cong: if_cong)
 next
   case (sum p1 p2 m a)
   then show ?case
     by (simp add: distrib_right isovarspar_sum sum.distrib)
 qed
 
 subsubsection "More Miscellaneous"
 lemma var_not_in_Const : "var\<notin>vars (Const x :: real mpoly)"
   unfolding vars_def keys_def
   by (smt UN_iff coeff_def coeff_isolate_variable_sparse const_lookup_zero keys_def lookup_eq_zero_in_keys_contradict) 
 
 lemma mpoly_to_nested_poly_mult:
   "mpoly_to_nested_poly (p * q) v = mpoly_to_nested_poly p v * mpoly_to_nested_poly q v"
   for p q::"'a::{comm_semiring_0, semiring_no_zero_divisors} mpoly"
   by (auto simp: poly_eq_iff coeff_mult mpoly_to_nested_poly.rep_eq isolate_variable_sparse_mult)
 
 lemma get_if_const_insertion : 
   assumes "get_if_const (p::real mpoly) = Some r"
   shows "insertion f p = r"
 proof-
   have "Set.is_empty (vars p)"
     apply(cases "Set.is_empty (vars p)")
      apply(simp) using assms by(simp)
   then have "(MPoly_Type.coeff p 0) = r"
     using assms by simp
   then show ?thesis
     by (metis Set.is_empty_def \<open>Set.is_empty (vars p)\<close> empty_iff insertion_irrelevant_vars insertion_trivial)
 qed
 
 subsubsection "Yet more Degree Lemmas"
 lemma degree_mult:
   fixes p q::"'a::{comm_semiring_0, ring_1_no_zero_divisors} mpoly"
   assumes "p \<noteq> 0"
   assumes "q \<noteq> 0"
   shows "MPoly_Type.degree (p * q) v = MPoly_Type.degree p v + MPoly_Type.degree q v"
   using assms
   by (auto simp add: degree_eq_univariate_degree mpoly_to_nested_poly_mult Polynomial.degree_mult_eq)
 
 lemma degree_eq:
   assumes "(p::real mpoly) = (q:: real mpoly)"
   shows "MPoly_Type.degree p x = MPoly_Type.degree q x"
   by (simp add: assms)
 
 lemma degree_var_i : "MPoly_Type.degree (((Var x)^i:: real mpoly)) x = i"
 proof (induct i)
   case 0
   then show ?case using degree_const
     by simp
 next
   case (Suc i)
   have multh: "(Var x)^(Suc i) = ((Var x)^i::real mpoly) * (Var x:: real mpoly)"
     using power_Suc2 by blast
   have deg0h: "MPoly_Type.degree 0 x = 0"
     by simp
   have deg1h: "MPoly_Type.degree (Var x::real mpoly) x = 1"
     using degree_one
     by blast 
   have nonzeroh1: "(Var x:: real mpoly) \<noteq> 0" 
     using degree_eq deg0h deg1h by auto 
   have nonzeroh2: "((Var x)^i:: real mpoly) \<noteq> 0" 
     using degree_eq deg0h Suc.hyps
     by (metis one_neq_zero power_0) 
   have sumh: "(MPoly_Type.degree (((Var x)^i:: real mpoly) * (Var x:: real mpoly)) x) = (MPoly_Type.degree ((Var x)^i::real mpoly) x) + (MPoly_Type.degree (Var x::real mpoly) x)"
     using degree_mult[where p = "(Var x)^i::real mpoly", where q = "Var x::real mpoly"]  nonzeroh1 nonzeroh2
     by blast 
   then show ?case using sumh deg1h
     by (metis Suc.hyps Suc_eq_plus1 multh) 
 qed
 
 
 lemma degree_less_sum: 
   assumes "MPoly_Type.degree (p::real mpoly) var = n"
   assumes "MPoly_Type.degree (q::real mpoly) var = m"
   assumes "m < n"
   shows "MPoly_Type.degree (p + q) var = n"
 proof - 
   have h1: "n > 0"
     using assms(3) neq0_conv by blast
   have h2: "(\<exists>m\<in>monomials p. lookup m var = n) \<and> (\<forall>m\<in>monomials p. lookup m var \<le> n)"
     using degree_eq_iff assms(1)
     by (metis degree_ge_iff h1 order_refl)
   have h3: "(\<exists>m\<in>monomials (p + q). lookup m var = n)"
     using h2
     by (metis MPolyExtension.coeff_add add.right_neutral assms(2) assms(3) coeff_eq_zero_iff degree_not_exists_monom) 
   have h4: "(\<forall>m\<in>monomials (p + q). lookup m var \<le> n)"
     using h2 assms(3) assms(2)
     using degree_add_leq degree_le_iff dual_order.strict_iff_order by blast
   show ?thesis using degree_eq_iff h3 h4
     by (metis assms(3) gr_implies_not0) 
 qed
 
 lemma degree_less_sum': 
   assumes "MPoly_Type.degree (p::real mpoly) var = n"
   assumes "MPoly_Type.degree (q::real mpoly) var = m"
   assumes "n < m"
   shows "MPoly_Type.degree (p + q) var = m" using degree_less_sum[OF assms(2) assms(1) assms(3)]
   by (simp add: add.commute) 
 
 (* Result on the degree of the derivative  *)
 
 lemma nonzero_const_is_nonzero: 
   assumes "(k::real) \<noteq> 0"
   shows "((Const k)::real mpoly) \<noteq> 0"
   by (metis MPoly_Type.insertion_zero assms insertion_const)
 
 lemma degree_derivative : 
   assumes "MPoly_Type.degree p x > 0"
   shows "MPoly_Type.degree p x = MPoly_Type.degree (derivative x p) x + 1"
 proof -
   define f where "f i = (isolate_variable_sparse p x (i+1) * (Var x)^(i) * (Const (i+1)))" for i
   define n where "n = MPoly_Type.degree p x-1"
   have d : "\<exists>m\<in>monomials p. lookup m x = n+1"
     using n_def degree_eq_iff assms
     by (metis add.commute less_not_refl2 less_one linordered_semidom_class.add_diff_inverse)    
   have h1a : "\<forall>i. MPoly_Type.degree (isolate_variable_sparse p x i) x = 0"
     by (simp add: not_in_isovarspar varNotIn_degree)
   have h1b : "\<forall>i::nat. MPoly_Type.degree ((Var x)^i:: real mpoly) x = i"
     using degree_var_i by auto
   have h1c1 : "\<forall>i. (Var(x)^(i)::real mpoly)\<noteq>0"
     by (metis MPoly_Type.degree_zero h1b power_0 zero_neq_one)
+  fix i
   have h1c1var: "((Var x)^i:: real mpoly) \<noteq> 0" using h1c1 by blast
   have h1c2 : "((Const ((i::nat) + 1))::real mpoly)\<noteq>0"
     using nonzero_const_is_nonzero
     by auto 
   have h1c3: "(isolate_variable_sparse p x (n + 1)) \<noteq> 0" using d
     by (metis One_nat_def Suc_pred add.commute assms isolate_variable_sparse_degree_eq_zero_iff n_def not_gr_zero not_in_isovarspar plus_1_eq_Suc varNotIn_degree)
   have h3: "(isolate_variable_sparse p x (i+1) = 0) \<longrightarrow> (MPoly_Type.degree (f i) x) = 0"
     by (simp add: f_def)
   have degh1: "(MPoly_Type.degree (((Const ((i::nat)+1))::real mpoly)*(Var x)^i) x) = 
     ((MPoly_Type.degree ((Const (i+1))::real mpoly) x) + (MPoly_Type.degree ((Var x)^i:: real mpoly) x))"
     using h1c2 h1c1var degree_mult[where p="((Const ((i::nat)+1))::real mpoly)", where q="((Var x)^i:: real mpoly)"]
     by blast
   then have degh2: "(MPoly_Type.degree (((Const ((i::nat)+1))::real mpoly)*(Var x)^i) x) = i" 
     using degree_var_i degree_const
     by simp 
   have nonzerohyp: "(((Const ((i::nat)+1))::real mpoly)*(Var x)^i) \<noteq> 0"
   proof (induct i)
     case 0
     then show ?case
       by (simp add: nonzero_const_is_nonzero) 
   next
     case (Suc i)
     then show ?case using degree_eq degh2
       by (metis Suc_eq_plus1 h1c1 mult_eq_0_iff nat.simps(3) nonzero_const_is_nonzero of_nat_eq_0_iff)
   qed
   have h4a1: "(isolate_variable_sparse p x (i+1) \<noteq> 0) \<longrightarrow> (MPoly_Type.degree (isolate_variable_sparse p x (i+1) * ((Var x)^(i) * (Const (i+1)))::real mpoly) x = 
       (MPoly_Type.degree (isolate_variable_sparse p x (i+1):: real mpoly) x) + (MPoly_Type.degree (((Const (i+1)) *  (Var x)^(i))::real mpoly) x))"
     using nonzerohyp degree_mult[where p = "(isolate_variable_sparse p x (i+1))::real mpoly", where q = "((Const (i+1)) *  (Var x)^(i)):: real mpoly", where v = "x"]
     by (simp add: mult.commute)
   have h4: "(isolate_variable_sparse p x (i+1) \<noteq> 0) \<longrightarrow> (MPoly_Type.degree (f i) x) = i"
     using h4a1 f_def degh2 h1a
-    by (metis (no_types, hide_lams) add.left_neutral mult.commute mult.left_commute of_nat_1 of_nat_add) 
+    by (metis (no_types, lifting) degh1 degree_const h1b mult.commute mult.left_commute of_nat_1 of_nat_add)
   have h5: "(MPoly_Type.degree (f i) x) \<le> i" using h3 h4 by auto
   have h6: "(MPoly_Type.degree (f n) x) = n" using h1c3 h4
     by (smt One_nat_def add.right_neutral add_Suc_right degree_const degree_mult divisors_zero f_def h1a h1b h1c1 mult.commute nonzero_const_is_nonzero of_nat_1 of_nat_add of_nat_neq_0) 
   have h7a: "derivative x p = (\<Sum>i\<in>{0..MPoly_Type.degree p x-1}. isolate_variable_sparse p x (i+1) * (Var x)^i * (Const (i+1)))" using derivative_def by auto
   have h7b: "(\<Sum>i\<in>{0..MPoly_Type.degree p x-1}. isolate_variable_sparse p x (i+1) * (Var x)^i * (Const (i+1))) = (\<Sum>i\<in>{0..MPoly_Type.degree p x-1}. (f i))" using f_def
     by (metis Suc_eq_plus1 add.commute semiring_1_class.of_nat_simps(2)) 
   have h7c: "derivative x p = (\<Sum>i\<in>{0..MPoly_Type.degree p x-1}. (f i))" using h7a h7b by auto
   have h7: "derivative x p = (\<Sum>i\<in>{0..n}. (f i))" using n_def h7c
     by blast 
   have h8: "n > 0 \<longrightarrow> ((MPoly_Type.degree (\<Sum>i\<in>{0..(n-1)}. (f i)) x) < n)"
   proof (induct n)
     case 0
     then show ?case
       by blast 
   next
     case (Suc n)
     then show ?case using h5 degree_less_sum
       by (smt add_cancel_right_right atLeastAtMost_iff degree_const degree_mult degree_sum_less degree_var_i diff_Suc_1 f_def h1a le_imp_less_Suc mult.commute mult_eq_0_iff)
   qed 
   have h9a: "n = 0 \<longrightarrow> MPoly_Type.degree (\<Sum>i\<in>{0..n}. (f i)) x = n" using h6
     by auto   
   have h9b: "n > 0 \<longrightarrow> MPoly_Type.degree (\<Sum>i\<in>{0..n}. (f i)) x = n" 
   proof - 
     have h9bhyp: "n > 0 \<longrightarrow> (MPoly_Type.degree (\<Sum>i\<in>{0..n}. (f i)) x = MPoly_Type.degree ((\<Sum>i\<in>{0..(n-1)}. (f i)) + (f n)) x)"
       by (metis Suc_diff_1 sum.atLeast0_atMost_Suc)
     have h9bhyp2: "n > 0 \<longrightarrow> ((MPoly_Type.degree ((\<Sum>i\<in>{0..(n-1)}. (f i)) + (f n)) x) = n)" 
       using h6 h8 degree_less_sum
       by (simp add: add.commute) 
     then show ?thesis using h9bhyp2 h9bhyp
       by linarith
   qed
   have h9:  "MPoly_Type.degree (\<Sum>i\<in>{0..n}. (f i)) x = n" using h9a h9b
     by blast 
   have h10: "MPoly_Type.degree (derivative x p) x = n" using h9 h7
     by simp
   show ?thesis using h10 n_def
     using assms by linarith
 qed
 
 
 lemma express_poly :
   assumes h : "MPoly_Type.degree (p::real mpoly) var = 1 \<or> MPoly_Type.degree p var = 2"
   shows "p =
      (isolate_variable_sparse p var 2)*(Var var)^2
     +(isolate_variable_sparse p var 1)*(Var var)
     +(isolate_variable_sparse p var 0)"
 proof-
   have h1a: "MPoly_Type.degree p var = 1 \<longrightarrow> p =
     isolate_variable_sparse p var 0 + 
     isolate_variable_sparse p var 1 * Var var"
     using sum_over_zero[where mp="p",where x="var"]
     by auto
   have h1b: "MPoly_Type.degree p var = 1 \<longrightarrow> isolate_variable_sparse p var 2 = 0"
     using isovar_greater_degree
     by (simp add: isovar_greater_degree)
   have h1: "MPoly_Type.degree p var = 1 \<longrightarrow> p =
     isolate_variable_sparse p var 0 + 
     isolate_variable_sparse p var 1 * Var var
     + isolate_variable_sparse p var 2 * (Var var)^2" using h1a h1b by auto
   have h2a: "MPoly_Type.degree p var = 2 \<longrightarrow> p = (\<Sum>i::nat \<le> 2. isolate_variable_sparse p var i * Var var^i)"
     using sum_over_zero[where mp="p", where x="var"] by auto
   have h2b: "(\<Sum>i::nat \<le> 2. isolate_variable_sparse p var i * Var var^i) =
    (\<Sum>i::nat \<le> 1. isolate_variable_sparse p var i * Var var^i) +
    isolate_variable_sparse p var 2 * (Var var)^2" apply auto
     by (simp add: numerals(2))
   have h2:  "MPoly_Type.degree p var = 2 \<longrightarrow> p =
     isolate_variable_sparse p var 0 + 
     isolate_variable_sparse p var 1 * Var var + 
     isolate_variable_sparse p var 2 * (Var var)^2"
     using h2a h2b by auto
   have h3: "isolate_variable_sparse p var 0 + 
     isolate_variable_sparse p var 1 * Var var + 
     isolate_variable_sparse p var 2 * (Var var)^2 = 
     isolate_variable_sparse p var 2 * (Var var)^2 +
     isolate_variable_sparse p var 1 * Var var + 
     isolate_variable_sparse p var 0" by (simp add: add.commute)
 
   show ?thesis using h h1 h2 h3 by presburger
 qed
 
 lemma degree_isovarspar : "MPoly_Type.degree (isolate_variable_sparse (p::real mpoly) x i) x = 0"
   using not_in_isovarspar varNotIn_degree by blast 
 
 
 end                           
diff --git a/thys/Virtual_Substitution/GeneralVSProofs.thy b/thys/Virtual_Substitution/GeneralVSProofs.thy
--- a/thys/Virtual_Substitution/GeneralVSProofs.thy
+++ b/thys/Virtual_Substitution/GeneralVSProofs.thy
@@ -1,2390 +1,2390 @@
 theory GeneralVSProofs
   imports  DNFUni EqualityVS VSAlgos
 begin
 
 
 fun separateAtoms :: "atomUni list \<Rightarrow> (real * real * real) list * (real * real * real) list * (real * real * real) list * (real * real * real) list" where
   "separateAtoms [] = ([],[],[],[])"|
   "separateAtoms (EqUni p # L) = (let (a,b,c,d) = separateAtoms(L) in (p#a,b,c,d))"|
   "separateAtoms (LessUni p # L) = (let (a,b,c,d) = separateAtoms(L) in (a,p#b,c,d))"|
   "separateAtoms (LeqUni p # L) = (let (a,b,c,d) = separateAtoms(L) in (a,b,p#c,d))"|
   "separateAtoms (NeqUni p # L) = (let (a,b,c,d) = separateAtoms(L) in (a,b,c,p#d))"
 
 
 lemma separate_aEval :
   assumes "separateAtoms L = (a,b,c,d)"
   shows "(\<forall>l\<in>set L. aEvalUni l x) = 
       ((\<forall>(a,b,c)\<in>set a. a*x^2+b*x+c=0) \<and> (\<forall>(a,b,c)\<in>set b. a*x^2+b*x+c<0) \<and>
       (\<forall>(a,b,c)\<in>set c. a*x^2+b*x+c\<le>0) \<and> (\<forall>(a,b,c)\<in>set d. a*x^2+b*x+c\<noteq>0))"
   using assms proof(induction L arbitrary :a b c d)
   case Nil
   then show ?case by auto
 next
   case (Cons At L)
   then have Cons1 : "\<And>a b c d. separateAtoms L = (a, b, c, d) \<Longrightarrow>
     (\<forall>l\<in>set L. aEvalUni l x) =
     ((\<forall>a\<in>set a. case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c = 0) \<and>
      (\<forall>a\<in>set b. case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c < 0)\<and>
      (\<forall>a\<in>set c. case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c \<le> 0) \<and>
      (\<forall>a\<in>set d. case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c \<noteq> 0))" "
     separateAtoms (At # L) = (a, b,c,d)" by auto
   then show ?case proof(cases At)
     case (LessUni p)
     show ?thesis proof(cases b)
       case Nil
       show ?thesis using Cons(2) unfolding LessUni separateAtoms.simps Nil
         apply(cases "separateAtoms L") by simp
     next
       case (Cons p' b')
       then have p_def : "p' = p" using Cons1(2) unfolding LessUni separateAtoms.simps
         apply(cases "separateAtoms L") by simp
       have h1 :  "separateAtoms L = (a,b',c,d)" using Cons Cons1(2) unfolding LessUni separateAtoms.simps
         apply(cases "separateAtoms L") by simp
       have h2 : "(\<forall>a\<in>set (p # b'). case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c < 0) = (
           (\<forall>a\<in>set (b'). case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c < 0)\<and> (case p of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c < 0))"
         by auto
       have h3 : "(\<forall>l\<in>set (LessUni p # L). aEvalUni l x) = ((\<forall>l\<in>set (L). aEvalUni l x)\<and>(case p of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c < 0))"
         by auto
       show ?thesis unfolding Cons LessUni p_def h2 h3 using Cons1(1)[OF h1]
         by auto
     qed
   next
     case (EqUni p)
     show ?thesis proof(cases a)
       case Nil
       show ?thesis using Cons(2) unfolding EqUni separateAtoms.simps Nil
         apply(cases "separateAtoms L") by simp
     next
       case (Cons p' a')
       then have p_def : "p' = p" using Cons1(2) unfolding EqUni separateAtoms.simps
         apply(cases "separateAtoms L") by simp
       have h1 :  "separateAtoms L = (a',b,c,d)" using Cons Cons1(2) unfolding EqUni separateAtoms.simps
         apply(cases "separateAtoms L") by simp
       have h2 : "(\<forall>a\<in>set (p # a'). case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c = 0) = (
           (\<forall>a\<in>set (a'). case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c = 0)\<and> (case p of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c = 0))"
         by auto
       have h3 : "(\<forall>l\<in>set (EqUni p # L). aEvalUni l x) = ((\<forall>l\<in>set (L). aEvalUni l x)\<and>(case p of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c = 0))"
         by auto
       show ?thesis unfolding Cons EqUni p_def h2 h3 using Cons1(1)[OF h1]
         by auto
     qed
   next
     case (LeqUni p)
     then show ?thesis proof(cases c)
       case Nil
       show ?thesis using Cons(2) unfolding LeqUni separateAtoms.simps Nil
         apply(cases "separateAtoms L") by simp
     next
       case (Cons p' a')
       then have p_def : "p' = p" using Cons1(2) unfolding LeqUni separateAtoms.simps
         apply(cases "separateAtoms L") by simp
       have h1 :  "separateAtoms L = (a,b,a',d)" using Cons Cons1(2) unfolding LeqUni separateAtoms.simps
         apply(cases "separateAtoms L") by simp
       have h2 : "(\<forall>a\<in>set (p # a'). case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c \<le> 0) = (
           (\<forall>a\<in>set (a'). case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c \<le> 0)\<and> (case p of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c \<le> 0))"
         by auto
       have h3 : "(\<forall>l\<in>set (LeqUni p # L). aEvalUni l x) = ((\<forall>l\<in>set (L). aEvalUni l x)\<and>(case p of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c \<le> 0))"
         by auto
       show ?thesis unfolding Cons LeqUni p_def h2 h3 using Cons1(1)[OF h1]
         by auto
     qed
   next
     case (NeqUni p)
     then show ?thesis proof(cases d)
       case Nil
       show ?thesis using Cons(2) unfolding NeqUni separateAtoms.simps Nil
         apply(cases "separateAtoms L") by simp
     next
       case (Cons p' a')
       then have p_def : "p' = p" using Cons1(2) unfolding NeqUni separateAtoms.simps
         apply(cases "separateAtoms L") by simp
       have h1 :  "separateAtoms L = (a,b,c,a')" using Cons Cons1(2) unfolding NeqUni separateAtoms.simps
         apply(cases "separateAtoms L") by simp
       have h2 : "(\<forall>a\<in>set (p # a'). case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c \<noteq> 0) = (
           (\<forall>a\<in>set (a'). case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c \<noteq> 0)\<and> (case p of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c \<noteq> 0))"
         by auto
       have h3 : "(\<forall>l\<in>set (NeqUni p # L). aEvalUni l x) = ((\<forall>l\<in>set (L). aEvalUni l x)\<and>(case p of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c \<noteq> 0))"
         by auto
       show ?thesis unfolding Cons NeqUni p_def h2 h3 using Cons1(1)[OF h1]
         by auto
     qed
   qed
 qed
 
 lemma splitAtoms_negInfinity :
   assumes "separateAtoms L = (a,b,c,d)"
   shows "(\<forall>l\<in>set L. evalUni (substNegInfinityUni l) x) = (
   (\<forall>(a,b,c)\<in>set a.(\<exists>x. \<forall>y<x. a*y^2+b*y+c=0))\<and>
   (\<forall>(a,b,c)\<in>set b.(\<exists>x. \<forall>y<x. a*y^2+b*y+c<0))\<and>
   (\<forall>(a,b,c)\<in>set c.(\<exists>x. \<forall>y<x. a*y^2+b*y+c\<le>0))\<and>
   (\<forall>(a,b,c)\<in>set d.(\<exists>x. \<forall>y<x. a*y^2+b*y+c\<noteq>0)))"
   using assms proof(induction L arbitrary :a b c d)
   case Nil
   then show ?case by auto
 next
   case (Cons At L)
   then have Cons1 : "\<And>a b c d. separateAtoms L = (a, b, c, d) \<Longrightarrow>
     (\<forall>l\<in>set L. evalUni (substNegInfinityUni l) x) =
     ((\<forall>a\<in>set a. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c = 0) \<and>
      (\<forall>a\<in>set b. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0)\<and>
      (\<forall>a\<in>set c. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<le> 0)\<and>
      (\<forall>a\<in>set d. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<noteq> 0))" "separateAtoms (At # L) = (a, b, c, d)" by auto
   then show ?case proof(cases At)
     case (LessUni p)
     show ?thesis using LessUni Cons proof(induction b rule : list.induct)
       case Nil
       then have Nil : "b = []"
         using Cons.prems by auto
       show ?case using Cons(2) unfolding LessUni separateAtoms.simps Nil
         apply(cases "separateAtoms L") by simp
     next
       case (Cons p' b')
       then have p_def : "p' = p" using Cons1(2) unfolding LessUni separateAtoms.simps
         apply(cases "separateAtoms L") by simp
       have h1 :  "separateAtoms L = (a,b',c,d)" using Cons Cons1(2) unfolding LessUni separateAtoms.simps
         apply(cases "separateAtoms L") by simp
       have h2 : "(\<forall>a\<in>set (p # b'). case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0) = (
           (\<forall>a\<in>set ( b'). case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0)\<and> (case p of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0))"
         by auto
       have one: "(\<exists>x. \<forall>y<x. aEvalUni (LessUni p) y) = (case p of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0)"
         apply(cases p) by simp
       have "(\<forall>l\<in>set (LessUni p # L). evalUni (substNegInfinityUni l) x) = ((evalUni (substNegInfinityUni (LessUni p)) x)\<and>(\<forall>l\<in>set ( L). evalUni (substNegInfinityUni l) x))"
         by auto
       also have "... = (
       (case p of (a,ba,c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0)\<and>
       (\<forall>a\<in>set a. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c = 0) \<and>
      (\<forall>a\<in>set b'. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0)\<and>
      (\<forall>a\<in>set c. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<le> 0)\<and>
      (\<forall>a\<in>set d. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<noteq> 0))"
         unfolding infinity_evalUni[of "LessUni p" x, symmetric] Cons(3)[OF h1]  LessUni one by simp
       finally have h3 : "(\<forall>l\<in>set (LessUni p # L). evalUni (substNegInfinityUni l) x) = (
       (case p of (a,ba,c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0)\<and>
       (\<forall>a\<in>set a. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c = 0) \<and>
      (\<forall>a\<in>set b'. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0)\<and>
      (\<forall>a\<in>set c. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<le> 0)\<and>
      (\<forall>a\<in>set d. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<noteq> 0) )"
         by auto
       show ?case unfolding Cons LessUni p_def h2 h3 using Cons1(1)[OF h1]
         by auto
     qed
   next
     case (EqUni p)
     show ?thesis using EqUni Cons proof(induction a rule : list.induct)
       case Nil
       then have Nil : "a = []"
         using Cons.prems by auto
       show ?case using Cons(2) unfolding EqUni separateAtoms.simps Nil
         apply(cases "separateAtoms L") by simp
     next
       case (Cons p' a')
       then have p_def : "p' = p" using Cons1(2) unfolding EqUni separateAtoms.simps
         apply(cases "separateAtoms L") by simp
       have h1 :  "separateAtoms L = (a',b,c,d)" using Cons Cons1(2) unfolding EqUni separateAtoms.simps
         apply(cases "separateAtoms L") by simp
       have h2 : "(\<forall>a\<in>set (p # a'). case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c = 0) = (
           (\<forall>a\<in>set ( a'). case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c = 0)\<and> (case p of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c = 0))"
         by auto
       have one: "(\<exists>x. \<forall>y<x. aEvalUni (EqUni p) y) = (case p of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c = 0)"
         apply(cases p) by simp
       have "(\<forall>l\<in>set (EqUni p # L). evalUni (substNegInfinityUni l) x) = ((evalUni (substNegInfinityUni (EqUni p)) x)\<and>(\<forall>l\<in>set ( L). evalUni (substNegInfinityUni l) x))"
         by auto
       also have "... = (
       (case p of (a,ba,c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c = 0)\<and>
       (\<forall>a\<in>set a'. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c = 0) \<and>
      (\<forall>a\<in>set b. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0)\<and>
      (\<forall>a\<in>set c. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<le> 0)\<and>
      (\<forall>a\<in>set d. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<noteq> 0))"
         unfolding infinity_evalUni[of "EqUni p" x, symmetric] Cons(3)[OF h1] EqUni one by simp
       finally have h3 : "(\<forall>l\<in>set (EqUni p # L). evalUni (substNegInfinityUni l) x) = (
       (case p of (a,ba,c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c = 0)\<and>
       (\<forall>a\<in>set a'. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c = 0) \<and>
      (\<forall>a\<in>set b. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0)\<and>
      (\<forall>a\<in>set c. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<le> 0)\<and>
      (\<forall>a\<in>set d. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<noteq> 0))"
         by auto
       show ?case unfolding Cons EqUni p_def h2 h3 using Cons1(1)[OF h1]
         by auto
     qed
   next
     case (LeqUni p)
     show ?thesis using LeqUni Cons proof(induction c rule : list.induct)
       case Nil
       then have Nil : "c = []"
         using Cons.prems by auto
       show ?case using Cons(2) unfolding LeqUni separateAtoms.simps Nil
         apply(cases "separateAtoms L") by simp
     next
       case (Cons p' c')
       then have p_def : "p' = p" using Cons1(2) unfolding LeqUni separateAtoms.simps
         apply(cases "separateAtoms L") by simp
       have h1 :  "separateAtoms L = (a,b,c',d)" using Cons Cons1(2) unfolding LeqUni separateAtoms.simps
         apply(cases "separateAtoms L") by simp
       have h2 : "(\<forall>a\<in>set (p # c'). case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<le> 0) = (
           (\<forall>a\<in>set ( c'). case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<le> 0)\<and> (case p of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<le> 0))"
         by auto
       have one: "(\<exists>x. \<forall>y<x. aEvalUni (LeqUni p) y) = (case p of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<le> 0)"
         apply(cases p) by simp
       have "(\<forall>l\<in>set (LeqUni p # L). evalUni (substNegInfinityUni l) x) = ((evalUni (substNegInfinityUni (LeqUni p)) x)\<and>(\<forall>l\<in>set ( L). evalUni (substNegInfinityUni l) x))"
         by auto
       also have "... = (
       (case p of (a,ba,c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<le> 0)\<and>
       (\<forall>a\<in>set a. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c = 0) \<and>
      (\<forall>a\<in>set b. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0)\<and>
      (\<forall>a\<in>set c'. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<le> 0)\<and>
      (\<forall>a\<in>set d. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<noteq> 0))"
         unfolding infinity_evalUni[of "LeqUni p" x, symmetric] Cons(3)[OF h1]  LeqUni one 
         by simp
       finally have h3 : "(\<forall>l\<in>set (LeqUni p # L). evalUni (substNegInfinityUni l) x) = (
       (case p of (a,ba,c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<le> 0)\<and>
       (\<forall>a\<in>set a. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c = 0) \<and>
      (\<forall>a\<in>set b. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0)\<and>
      (\<forall>a\<in>set c'. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<le> 0)\<and>
      (\<forall>a\<in>set d. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<noteq> 0) )"
         by auto
       show ?case unfolding Cons LeqUni p_def h2 h3 using Cons1(1)[OF h1]
         by auto
     qed
   next
     case (NeqUni p)
     show ?thesis using NeqUni Cons proof(induction d rule : list.induct)
       case Nil
       then have Nil : "d = []"
         using Cons.prems by auto
       show ?case using Cons(2) unfolding NeqUni separateAtoms.simps Nil
         apply(cases "separateAtoms L") by simp
     next
       case (Cons p' d')
       then have p_def : "p' = p" using Cons1(2) unfolding NeqUni separateAtoms.simps
         apply(cases "separateAtoms L") by simp
       have h1 :  "separateAtoms L = (a,b,c,d')" using Cons Cons1(2) unfolding NeqUni separateAtoms.simps
         apply(cases "separateAtoms L") by simp
       have h2 : "(\<forall>a\<in>set (p # d'). case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<noteq> 0) = (
           (\<forall>a\<in>set ( d'). case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<noteq> 0)\<and> (case p of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<noteq> 0))"
         by auto
       have one: "(\<exists>x. \<forall>y<x. aEvalUni (NeqUni p) y) = (case p of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<noteq> 0)"
         apply(cases p) by simp
       have "(\<forall>l\<in>set (NeqUni p # L). evalUni (substNegInfinityUni l) x) = ((evalUni (substNegInfinityUni (NeqUni p)) x)\<and>(\<forall>l\<in>set ( L). evalUni (substNegInfinityUni l) x))"
         by auto
       also have "... = (
       (case p of (a,ba,c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<noteq> 0)\<and>
       (\<forall>a\<in>set a. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c = 0) \<and>
      (\<forall>a\<in>set b. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0)\<and>
      (\<forall>a\<in>set c. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<le> 0)\<and>
      (\<forall>a\<in>set d'. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<noteq> 0))"
         unfolding infinity_evalUni[of "NeqUni p" x, symmetric] Cons(3)[OF h1]  NeqUni one 
         by simp
       finally have h3 : "(\<forall>l\<in>set (NeqUni p # L). evalUni (substNegInfinityUni l) x) = (
       (case p of (a,ba,c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<noteq> 0)\<and>
       (\<forall>a\<in>set a. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c = 0) \<and>
      (\<forall>a\<in>set b. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0)\<and>
      (\<forall>a\<in>set c. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<le> 0)\<and>
      (\<forall>a\<in>set d'. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<noteq> 0) )"
         by auto
       show ?case unfolding Cons NeqUni p_def h2 h3 using Cons1(1)[OF h1]
         by auto
     qed
   qed
 qed
 
 lemma set_split : 
   assumes "separateAtoms L = (eq,les,leq,neq)"
   shows "set L = set (map EqUni eq @ map LessUni les @ map LeqUni leq @ map NeqUni neq)"
   using assms proof(induction L arbitrary :eq les leq neq)
   case Nil
   then show ?case by auto
 next
   case (Cons At L)
   then show ?case proof(cases At)
     case (LessUni p)
     have "\<exists>les'. p#les' = les \<and> separateAtoms L = (eq, les', leq, neq)" using Cons(2) unfolding LessUni apply (cases "separateAtoms L") by auto
     then obtain les' where les' : "p#les' = les" "separateAtoms L = (eq, les', leq, neq)" by auto
     show ?thesis unfolding LessUni les'(1)[symmetric] using Cons(1)[OF les'(2)] by auto
   next
     case (EqUni p)
     have "\<exists>eq'. p#eq' = eq \<and> separateAtoms L = (eq', les, leq, neq)" using Cons(2) unfolding EqUni apply (cases "separateAtoms L") by auto
     then obtain eq' where eq' : "p#eq' = eq" "separateAtoms L = (eq', les, leq, neq)" by auto
     show ?thesis unfolding EqUni eq'(1)[symmetric] using Cons(1)[OF eq'(2)] by auto
   next
     case (LeqUni p)
     have "\<exists>leq'. p#leq' = leq \<and> separateAtoms L = (eq, les, leq', neq)" using Cons(2) unfolding LeqUni apply (cases "separateAtoms L")
       by auto
     then obtain leq' where leq' : "p#leq' = leq" "separateAtoms L = (eq, les, leq', neq)" by auto
     show ?thesis unfolding LeqUni leq'(1)[symmetric] using Cons(1)[OF leq'(2)] by auto
   next
     case (NeqUni p)
     have "\<exists>neq'. p#neq' = neq \<and> separateAtoms L = (eq, les, leq, neq')" using Cons(2) unfolding NeqUni apply (cases "separateAtoms L")
       by auto
     then obtain neq' where neq' : "p#neq' = neq" "separateAtoms L = (eq, les, leq, neq')" by auto
     show ?thesis unfolding NeqUni neq'(1)[symmetric] using Cons(1)[OF neq'(2)] by auto
   qed
 qed
 
 lemma set_split' : assumes "separateAtoms L = (eq,les,leq,neq)"
   shows "set (map P L) = set (map (P o EqUni) eq @ map (P o LessUni) les @ map (P o LeqUni) leq @ map (P o NeqUni) neq)"
   unfolding image_set[symmetric] set_split[OF assms]
   unfolding image_set map_append map_map by auto
 
 lemma split_elimVar :
   assumes "separateAtoms L = (eq,les,leq,neq)"
   shows "(\<exists>l\<in>set L. evalUni (elimVarUni_atom L' l) x) = 
   ((\<exists>(a',b',c')\<in>set eq. (evalUni (elimVarUni_atom L' (EqUni(a',b',c'))) x))
   \<or> (\<exists>(a',b',c')\<in>set les. 
     (evalUni (elimVarUni_atom L' (LessUni(a',b',c'))) x))
 \<or> (\<exists>(a',b',c')\<in>set leq. 
     (evalUni (elimVarUni_atom L' (LeqUni(a',b',c'))) x))
 \<or> (\<exists>(a',b',c')\<in>set neq. 
     (evalUni (elimVarUni_atom L' (NeqUni(a',b',c'))) x)))"
 proof-
   have c1: "(\<exists>l\<in>set eq. evalUni (elimVarUni_atom L' (EqUni l)) x) = (\<exists>(a', b', c')\<in>set eq. evalUni (elimVarUni_atom L' (EqUni (a', b', c'))) x)"
     by (metis (no_types, lifting) case_prodE case_prodI2)
   have c2: "(\<exists>l\<in>set les. evalUni (elimVarUni_atom L' (LessUni l)) x) = (\<exists>(a', b', c')\<in>set les. evalUni (elimVarUni_atom L' (LessUni (a', b', c'))) x)"
     by (metis (no_types, lifting) case_prodE case_prodI2)
   have c3: "(\<exists>l\<in>set leq. evalUni (elimVarUni_atom L' (LeqUni l)) x) = (\<exists>(a', b', c')\<in>set leq. evalUni (elimVarUni_atom L' (LeqUni (a', b', c'))) x)"
     by (metis (no_types, lifting) case_prodE case_prodI2)
   have c4: "(\<exists>l\<in>set neq. evalUni (elimVarUni_atom L' (NeqUni l)) x) = (\<exists>(a', b', c')\<in>set neq. evalUni (elimVarUni_atom L' (NeqUni (a', b', c'))) x)"
     by (metis (no_types, lifting) case_prodE case_prodI2)
   have h :  "((\<exists>l\<in>EqUni ` set eq. evalUni (elimVarUni_atom L' l) x) \<or>
          (\<exists>l\<in>LessUni ` set les. evalUni (elimVarUni_atom L' l) x) \<or>
     (\<exists>l\<in>LeqUni ` set leq. evalUni (elimVarUni_atom L' l) x) \<or>
     (\<exists>l\<in>NeqUni ` set neq. evalUni (elimVarUni_atom L' l) x)
     ) = 
         ((\<exists>l\<in>set eq. evalUni (elimVarUni_atom L' (EqUni l)) x) \<or>
          (\<exists>l\<in>set les. evalUni (elimVarUni_atom L' (LessUni l)) x) \<or>
     (\<exists>l\<in>set leq. evalUni (elimVarUni_atom L' (LeqUni l)) x) \<or>
     (\<exists>l\<in>set neq. evalUni (elimVarUni_atom L' (NeqUni l)) x)
     )"
     by auto
   then have "... = ((\<exists>(a', b', c')\<in>set eq. evalUni (elimVarUni_atom L' (EqUni (a', b', c'))) x) \<or>
      (\<exists>(a', b', c')\<in>set les. evalUni (elimVarUni_atom L' (LessUni (a', b', c'))) x) \<or>
      (\<exists>(a', b', c')\<in>set leq. evalUni (elimVarUni_atom L' (LeqUni (a', b', c'))) x) \<or>
      (\<exists>(a', b', c')\<in>set neq. evalUni (elimVarUni_atom L' (NeqUni (a', b', c'))) x))"
     using c1 c2 c3 c4 by auto
   then show ?thesis 
     unfolding set_split[OF assms] set_append bex_Un image_set[symmetric]
     using case_prodE case_prodI2  by auto
 qed
 
 lemma split_elimvar : 
   assumes "separateAtoms L = (eq,les,leq,neq)"
   shows "evalUni (elimVarUni_atom L At) x = evalUni (elimVarUni_atom ((map EqUni eq)@(map LessUni les) @ map LeqUni leq @ map NeqUni neq) At) x"
 proof(cases At)
   case (LessUni p)
   then show ?thesis apply(cases p) apply simp unfolding eval_list_conj_Uni set_split'[OF assms] by simp
 next
   case (EqUni p)
   then show ?thesis apply(cases p) apply simp unfolding eval_list_conj_Uni set_split'[OF assms] by simp
 next
   case (LeqUni p)
   then show ?thesis apply(cases p) apply simp unfolding eval_list_conj_Uni set_split'[OF assms] by simp
 next
   case (NeqUni p)
   then show ?thesis apply(cases p) apply simp unfolding eval_list_conj_Uni set_split'[OF assms] by simp
 qed
 
 
 
 
 lemma less : "
          ((a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. evalUni (substInfinitesimalLinearUni b' c' (EqUni (d, e, f))) x) \<and>
          (\<forall>(d, e, f)\<in>set b. evalUni (substInfinitesimalLinearUni b' c' (LessUni (d, e, f))) x) \<and>
          (\<forall>(d, e, f)\<in>set c. evalUni (substInfinitesimalLinearUni b' c' (LeqUni (d, e, f))) x) \<and>
          (\<forall>(d, e, f)\<in>set d. evalUni (substInfinitesimalLinearUni b' c' (NeqUni (d, e, f))) x) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               evalUni
                (substInfinitesimalQuadraticUni (- b') 1 (b'\<^sup>2 - 4 * a' * c') (2 * a')
                  (EqUni (d, e, f)))
                x) \<and>
           (\<forall>(d, e, f)\<in>set b.
               evalUni
                (substInfinitesimalQuadraticUni (- b') 1 (b'\<^sup>2 - 4 * a' * c') (2 * a')
                  (LessUni (d, e, f)))
                x) \<and>
           (\<forall>(d, e, f)\<in>set c.
               evalUni
                (substInfinitesimalQuadraticUni (- b') 1 (b'\<^sup>2 - 4 * a' * c') (2 * a')
                  (LeqUni (d, e, f)))
                x) \<and>
           (\<forall>(d, e, f)\<in>set d.
               evalUni
                (substInfinitesimalQuadraticUni (- b') 1 (b'\<^sup>2 - 4 * a' * c') (2 * a')
                  (NeqUni (d, e, f)))
                x) \<or>
           (\<forall>(d, e, f)\<in>set a.
               evalUni
                (substInfinitesimalQuadraticUni (- b') (- 1) (b'\<^sup>2 - 4 * a' * c') (2 * a')
                  (EqUni (d, e, f)))
                x) \<and>
           (\<forall>(d, e, f)\<in>set b.
               evalUni
                (substInfinitesimalQuadraticUni (- b') (- 1) (b'\<^sup>2 - 4 * a' * c') (2 * a')
                  (LessUni (d, e, f)))
                x) \<and>
           (\<forall>(d, e, f)\<in>set c.
               evalUni
                (substInfinitesimalQuadraticUni (- b') (- 1) (b'\<^sup>2 - 4 * a' * c') (2 * a')
                  (LeqUni (d, e, f)))
                x) \<and>
           (\<forall>(d, e, f)\<in>set d.
               evalUni
                (substInfinitesimalQuadraticUni (- b') (- 1) (b'\<^sup>2 - 4 * a' * c') (2 * a')
                  (NeqUni (d, e, f)))
                x))) = 
 
           ((a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              (\<exists>y'::real>-c'/b'. \<forall>x::real \<in>{-c'/b'<..y'}. aEvalUni (EqUni (d, e, f)) x)) \<and>
          (\<forall>(d, e, f)\<in>set b.
             (\<exists>y'::real>-c'/b'. \<forall>x::real \<in>{-c'/b'<..y'}. aEvalUni (LessUni (d, e, f)) x))\<and>
          (\<forall>(d, e, f)\<in>set c.
              (\<exists>y'::real>-c'/b'. \<forall>x::real \<in>{-c'/b'<..y'}. aEvalUni (LeqUni (d, e, f)) x)) \<and>
          (\<forall>(d, e, f)\<in>set d.
             (\<exists>y'::real>-c'/b'. \<forall>x::real \<in>{-c'/b'<..y'}. aEvalUni (NeqUni (d, e, f)) x)) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               (\<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
         \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
            aEvalUni (EqUni (d,e,f)) x)) \<and>
           (\<forall>(d, e, f)\<in>set b.
               (\<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
         \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
            aEvalUni (LessUni (d,e,f)) x)) \<and>
           (\<forall>(d, e, f)\<in>set c.
               (\<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
         \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
            aEvalUni (LeqUni (d,e,f)) x)) \<and>
           (\<forall>(d, e, f)\<in>set d.
               (\<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
         \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
            aEvalUni (NeqUni (d,e,f)) x)) \<or>
           (\<forall>(d, e, f)\<in>set a.
               (\<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
         \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
            aEvalUni (EqUni (d,e,f)) x)) \<and>
           (\<forall>(d, e, f)\<in>set b.
               (\<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
         \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
            aEvalUni (LessUni (d,e,f)) x)) \<and> 
           (\<forall>(d, e, f)\<in>set c.
               (\<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
         \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
            aEvalUni (LeqUni (d,e,f)) x)) \<and>
           (\<forall>(d, e, f)\<in>set d.
               (\<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
         \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
            aEvalUni (NeqUni (d,e,f)) x))))"
 proof(cases "a'=0")
   case True
   then have a' :  "a'=0" by auto
   then show ?thesis proof(cases "b'=0")
     case True
     then show ?thesis using a' by auto
   next
     case False
     then show ?thesis using True unfolding infinitesimal_linear'[of b' c' _ x, symmetric, OF False] by auto
   qed 
 next
   case False
   then have a' : "a' \<noteq> 0" by auto
   then have d : "2 * a' \<noteq> 0" by auto
   show ?thesis proof(cases "0 \<le> b'\<^sup>2 - 4 * a' * c'")
     case True
     then show ?thesis using False
       unfolding infinitesimal_quad[OF d True, of "-b'", symmetric] by auto
   next
     case False
     then show ?thesis using a' by auto
   qed 
 qed
 
 lemma eq_inf : "(\<forall>(a, b, c)\<in>set (a::(real*real*real) list). \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c = 0) = (\<forall>(a, b, c)\<in>set a. a=0\<and>b=0\<and>c=0)"
   using infinity_evalUni_EqUni[of _ x] by auto
 
 
 
 text "This is the main quantifier elimination lemma, in the simplified framework. We are located directly underneath 
 the most internal existential quantifier so F is completely free in quantifier and consists only of conjunction and disjunction.
 The variable we are evaluating on, x, is removed in the generalVS\\_DNF converted formula as expanding out the evalUni function
 determines that x doesn't play a role in the computation of it. It would be okay to replace the x in the second half with any variable,
 but it is simplier this way
 
 This conversion is defined as a \"veritcal\" translation as we remain within the univariate framework in both sides of the expression"
 
 lemma eval_generalVS'' : "(\<exists>x. evalUni (list_conj_Uni (map AtomUni L)) x) =
                                evalUni (generalVS_DNF L) x"
 proof(cases "separateAtoms L")
   case (fields a b c d)
   have a : "\<And> P. (\<forall>l\<in>set (map EqUni a) \<union> (set (map LessUni b) \<union> (set (map LeqUni c) \<union> set (map NeqUni d))).P l) = 
             ((\<forall>(d,e,f)\<in>set a. P (EqUni (d,e,f))) \<and> (\<forall>(d,e,f)\<in>set b. P (LessUni (d,e,f))) \<and> (\<forall>(d,e,f)\<in>set c. P (LeqUni (d,e,f))) \<and> (\<forall>(d,e,f)\<in>set d. P (NeqUni (d,e,f))))"
     by auto
   show ?thesis apply(simp add: eval_list_conj_Uni separate_aEval[OF fields]
         splitAtoms_negInfinity[OF fields] eval_list_disj_Uni 
         del:elimVar.simps)
 
     unfolding eval_conj_atom generalVS_DNF.simps 
       split_elimVar[OF fields ] 
 
 split_elimvar[OF fields]
 
     unfolding elimVarUni_atom.simps evalUni.simps aEvalUni.simps
       Rings.mult_zero_class.mult_zero_left Groups.add_0 eval_list_conj_Uni Groups.group_add_class.minus_zero 
       eval_map_all linearSubstitutionUni.simps quadraticSubUni.simps evalUni_if aEvalUni.simps
       set_append a less eq_inf
     using qe  by auto
 qed
 
 
 lemma forallx_substNegInf : "(\<not>evalUni (map_atomUni substNegInfinityUni F) x) = (\<forall>x. \<not> evalUni (map_atomUni substNegInfinityUni F) x)"
 proof(induction F)
   case TrueFUni
   then show ?case by simp
 next
   case FalseFUni
   then show ?case  by simp
 next
   case (AtomUni At)
   then show ?case apply(cases At) by auto  
 next
   case (AndUni F1 F2)
   then show ?case  by auto
 next
   case (OrUni F1 F2)
   then show ?case  by auto
 qed
 
 lemma linear_subst_map: "evalUni (map_atomUni (linearSubstitutionUni b c) F) x = evalUni F (-c/b)"
   apply(induction F)by auto
 
 lemma quadratic_subst_map : "evalUni (map_atomUni (quadraticSubUni a b c d) F) x = evalUni F ((a+b*sqrt(c))/d)"
   apply(induction F)by auto
 
 
 
 
 fun convert_atom_list :: "nat \<Rightarrow> atom list \<Rightarrow> real list \<Rightarrow> (atomUni list) option" where
   "convert_atom_list var [] xs = Some []"|
   "convert_atom_list var (a#as) xs = (
   case convert_atom var a xs of Some(a) \<Rightarrow> 
   (case convert_atom_list var as xs of Some(as) \<Rightarrow> Some(a#as) | None \<Rightarrow> None)
    | None \<Rightarrow> None
 )"
 
 
 
 
 
 lemma convert_atom_list_change :
   assumes "length xs' = var"
   shows "convert_atom_list var L (xs' @ x # \<Gamma>) = convert_atom_list var L (xs' @ x' # \<Gamma>)"
   apply(induction L)using convert_atom_change[OF assms] apply simp_all
   by (metis)
 
 lemma negInf_convert :
   assumes "convert_atom_list var L (xs' @ x # xs) = Some L'"
   assumes "length xs' = var"
   shows "(\<forall>f\<in>set L. eval (substNegInfinity var f) (xs' @ x # xs))
          = (\<forall>f\<in>set L'. evalUni (substNegInfinityUni f) x)"
   using assms
 proof(induction L arbitrary : L')
   case Nil
   then show ?case by auto
 next
   case (Cons a L)
   then show ?case proof(cases a)
     case (Less p)
     have h:  "MPoly_Type.degree p var < 3 \<Longrightarrow>
           eval (substNegInfinity var (Less p)) (xs' @ x # xs) = evalUni
            (substNegInfinityUni
              (LessUni
                (insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 2),
                 insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var (Suc 0)),
                 insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 0))))
            x"
       using convert_substNegInfinity[of var "Less p" xs' x xs, OF _ assms(2)] by simp
     show ?thesis using Cons(2)[symmetric] Cons(1) unfolding Less apply(cases " MPoly_Type.degree p var < 3")
       defer apply simp apply(cases "convert_atom_list var L (xs' @ x # xs)") apply (simp_all del: substNegInfinity.simps substNegInfinityUni.simps)
       unfolding h
       using assms(2) by presburger
   next
     case (Eq p)
     have h:  "MPoly_Type.degree p var < 3 \<Longrightarrow>
           eval (substNegInfinity var (Eq p)) (xs' @ x # xs) = evalUni
            (substNegInfinityUni
              (EqUni
                (insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 2),
                 insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var (Suc 0)),
                 insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 0))))
            x"
       using convert_substNegInfinity[of var "Eq p", OF _ assms(2)] by simp
     show ?thesis using Cons(2)[symmetric] Cons(1) unfolding Eq apply(cases " MPoly_Type.degree p var < 3")
       defer apply simp apply(cases "convert_atom_list var L (xs' @ x # xs)") apply (simp_all del: substNegInfinity.simps substNegInfinityUni.simps)
       unfolding h assms by auto
   next
     case (Leq p)
     have h:  "MPoly_Type.degree p var < 3 \<Longrightarrow>
           eval (substNegInfinity var (Leq p)) (xs' @ x # xs) = evalUni
            (substNegInfinityUni
              (LeqUni
                (insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 2),
                 insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var (Suc 0)),
                 insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 0))))
            x"
       using convert_substNegInfinity[of var "Leq p", OF _ assms(2)] by simp
     show ?thesis using Cons(2) unfolding Leq apply(cases " MPoly_Type.degree p var < 3") 
       defer apply simp 
       apply(cases "convert_atom_list var L (xs' @ x # xs)") 
       apply (simp_all del: substNegInfinity.simps substNegInfinityUni.simps)
       unfolding h using Cons.IH assms by auto 
   next
     case (Neq p)
     have h:  "MPoly_Type.degree p var < 3 \<Longrightarrow>
           eval (substNegInfinity var (Neq p)) (xs' @ x # xs) = evalUni
            (substNegInfinityUni
              (NeqUni
                (insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 2),
                 insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var (Suc 0)),
                 insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 0))))
            x"
       using convert_substNegInfinity[of var "Neq p", OF _ assms(2)] by simp
     show ?thesis using Cons(2) unfolding Neq apply(cases " MPoly_Type.degree p var < 3") defer apply simp 
       apply(cases "convert_atom_list var L (xs' @ x # xs)") 
       apply (simp_all del: substNegInfinity.simps substNegInfinityUni.simps)
       unfolding h using Cons.IH assms by auto
   qed
 qed
 
 lemma elimVar_atom_single :
   assumes "convert_atom var A (xs' @ x # xs) = Some A'"
   assumes "convert_atom_list var L2 (xs' @ x # xs) = Some L2'"
   assumes "length xs' = var"
   shows "eval (elimVar var L2 [] A) (xs' @ x # xs) = evalUni (elimVarUni_atom L2' A') x"
 proof(cases A)
   case (Less p)
   define a where "a = insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 2)"
   have a_def' : "a = insertion (nth_default 0 (xs' @ 0 # xs)) (isolate_variable_sparse p var 2)" unfolding a_def
     using insertion_isovarspars_free[of "(xs' @ x # xs)" var x p 2 0] assms by auto
   define b where "b = insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var (Suc 0))"
   have b_def' : "b = insertion (nth_default 0 (xs' @ 0 # xs)) (isolate_variable_sparse p var (Suc 0))" unfolding b_def
     using insertion_isovarspars_free[of "(xs' @ x # xs)" var x p "(Suc 0)" 0] assms by auto
   define c where "c = insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 0)"
   have c_def' : "c = insertion (nth_default 0 (xs' @ 0 # xs)) (isolate_variable_sparse p var 0)" unfolding c_def
     using insertion_isovarspars_free[of "(xs' @ x # xs)" var x p 0 0] assms by auto
   have linear : "b\<noteq>0 \<Longrightarrow> (\<forall>f\<in>set L2.
          eval
           (substInfinitesimalLinear var (-isolate_variable_sparse p var 0) (isolate_variable_sparse p var (Suc 0)) f)
           (xs' @ x # xs)) = (\<forall>l\<in>set L2'. evalUni (substInfinitesimalLinearUni b c l) x)"
     using assms(2) proof(induction L2 arbitrary : L2')
     case Nil
     then show ?case by auto
   next
     case (Cons At L2)
     have "\<exists>At'. convert_atom var At (xs' @ x # xs) = Some At'" proof(cases At)
       case (Less p)
       then show ?thesis using Cons(3) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Eq p)
       then show ?thesis using Cons(3) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Leq p)
       then show ?thesis using Cons(3) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     next
       case (Neq p)
       then show ?thesis using Cons(3) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     qed 
     then obtain At' where At' : "convert_atom var At (xs' @ x # xs) = Some At'" by auto
     have "\<exists>L2's. convert_atom_list var L2 (xs' @ x # xs) = Some L2's"
       using Cons(3) At'
       apply(cases "convert_atom_list var L2 (xs' @ x # xs)") by auto
     then obtain L2's where L2's : "convert_atom_list var L2 (xs' @ x # xs) = Some L2's" by auto
     have L2' : "L2' = At' # L2's"
       using Cons(3) At'
       by (simp_all add: L2's)
     have h : "eval
          (substInfinitesimalLinear var (-isolate_variable_sparse p var 0) (isolate_variable_sparse p var (Suc 0))
             At)
          (xs' @ x # xs) = evalUni (substInfinitesimalLinearUni b c At') x"
     proof(cases "convert_atom var At (xs' @ x # xs)")
       case None
       then show ?thesis using At' apply(cases At)  by simp_all
     next
       case (Some a)
       have h1 : "var \<notin> vars (isolate_variable_sparse p var (Suc 0))"
         by (simp add: not_in_isovarspar) 
       have h2 : "var \<notin> vars (isolate_variable_sparse p var 0)"by (simp add: not_in_isovarspar) 
       have h :  "evalUni (substInfinitesimalLinearUni b c a) x =
     evalUni (substInfinitesimalLinearUni b c At') x"
       proof(cases At)
         case (Less p)
         then show ?thesis using At'[symmetric] Some[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Eq p)
         then show ?thesis using At'[symmetric] Some[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Leq x3)
         then show ?thesis using At' Some by auto 
       next
         case (Neq x4)
         then show ?thesis using At' Some by auto 
       qed
       show ?thesis unfolding convert_substInfinitesimalLinear[OF Some b_def[symmetric] c_def[symmetric] Cons(2) h1 h2 assms(3)]
         using h .
     qed
     show ?case unfolding L2' using h Cons(1)[OF Cons(2) L2's] by auto
   qed
   have quadratic_1 : "(a \<noteq> 0) \<Longrightarrow>
      (4 * a * c \<le> b\<^sup>2) \<Longrightarrow> (\<forall>f\<in>set L2.
           eval
            (substInfinitesimalQuadratic var
              (- isolate_variable_sparse p var (Suc 0)) 1
              ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
               4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)
              (2 * isolate_variable_sparse p var 2) f)
            (xs' @ x # xs)) = (\<forall>l\<in>set L2'.
           evalUni
            (substInfinitesimalQuadraticUni (- b) 1 (b\<^sup>2 - 4 * a * c) (2 * a) l)
            x)"
     using assms(2) proof(induction L2 arbitrary: L2')
     case Nil
     then show ?case by auto
   next
     case (Cons At L2)
     have "\<exists>At'. convert_atom var At (xs' @ x # xs) = Some At'" proof(cases At)
       case (Less p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Eq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Leq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     next
       case (Neq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     qed 
     then obtain At' where At' : "convert_atom var At (xs' @ x # xs) = Some At'" by auto
     have "\<exists>L2's. convert_atom_list var L2 (xs' @ x # xs) = Some L2's"
       using Cons(4) At'
       apply(cases "convert_atom_list var L2 (xs' @ x # xs)") by auto
     then obtain L2's where L2's : "convert_atom_list var L2 (xs' @ x # xs) = Some L2's" by auto
     have L2' : "L2' = At' # L2's"
       using Cons(4) At' apply(cases At) apply auto
       by (simp_all add: L2's)
     have h1 :  "var < length (xs' @ x # xs)" using assms by auto
     have h2 : "2*a \<noteq>0" using Cons by auto
     have h3 : "0\<le>b^2-4*a*c" using Cons(3) by auto
     have h4 : "var\<notin>vars ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
             4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)"
       by (metis add_uminus_conv_diff not_in_add not_in_isovarspar not_in_mult not_in_neg not_in_pow num_double numeral_times_numeral one_add_one power_0)
     have h5 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (- isolate_variable_sparse p var (Suc 0)) = -b"
       unfolding insertion_neg b_def
       by (metis insertion_isovarspars_free list_update_id) 
     have h6 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) 1 = 1" by auto
     have h7 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa]))
         ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
          4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0) =
        b\<^sup>2 - 4 * a * c" apply(simp add: insertion_four insertion_mult insertion_sub insertion_pow b_def a_def c_def)
       by (metis insertion_isovarspars_free list_update_id)
     have "\<And>xa. insertion (nth_default 0 (xs' @ xa # xs)) (2::real mpoly)  = (2::real)"
       by (metis MPoly_Type.insertion_one insertion_add one_add_one)  
     then have h8 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (2 * isolate_variable_sparse p var 2) = 2 * a"
       unfolding insertion_mult a_def apply auto
       by (metis assms(3) insertion_lowerPoly1 list_update_length not_in_isovarspar)
     have h9 : "var\<notin>vars(- isolate_variable_sparse p var (Suc 0))"
       by (simp add: not_in_isovarspar not_in_neg)
     have h10 : "var\<notin>vars(1::real mpoly)"
       by (metis h9 not_in_pow power.simps(1))
     have h11 : "var\<notin>vars(2 * isolate_variable_sparse p var 2)"
       by (metis isovarspar_sum mult_2 not_in_isovarspar)
     have h :  "eval
       (substInfinitesimalQuadratic var (- isolate_variable_sparse p var (Suc 0)) 1
         ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
          4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)
         (2 * isolate_variable_sparse p var 2) At)
       (xs' @ x # xs) =  evalUni
       (substInfinitesimalQuadraticUni (- b) 1 (b\<^sup>2 - 4 * a * c) (2 * a) At') x"
     proof (cases "convert_atom var At (xs' @ x # xs)")
       case None
       then show ?thesis using At' apply(cases At) by auto
     next
       case (Some aT)
       have h1 : "insertion (nth_default 0 (xs' @ x # xs)) (- isolate_variable_sparse p var (Suc 0)) = (-b)" unfolding b_def insertion_neg by auto
       have h2 : "insertion (nth_default 0 (xs' @ x # xs)) 1 = 1" by auto
       have h3 : "insertion (nth_default 0 (xs' @ x # xs)) (((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
         4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)) = (b\<^sup>2 - 4 * a * c)"
         unfolding insertion_mult insertion_pow insertion_four insertion_neg insertion_sub a_def b_def c_def
         by auto
       have h4 : "insertion (nth_default 0 (xs' @ x # xs)) (2 * isolate_variable_sparse p var 2) = 2 * a"
         unfolding insertion_mult a_def
         by (metis insertion_add insertion_mult mult_2)
       have h5 : "2 * a \<noteq> 0" using Cons by auto
       have h6 : "0 \<le> b\<^sup>2 - 4 * a * c" using Cons by auto
       have h7 : "var\<notin>vars(- isolate_variable_sparse p var (Suc 0))"
         by (simp add: not_in_isovarspar not_in_neg)
       have h8 : "var\<notin>vars(1::real mpoly)"
         by (metis h9 not_in_pow power.simps(1))
       have h9 : "var \<notin> vars ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
              4 * isolate_variable_sparse p var 2 *
              isolate_variable_sparse p var 0)"
         by (metis add_uminus_conv_diff not_in_add not_in_isovarspar not_in_mult not_in_neg not_in_pow num_double numeral_times_numeral one_add_one power_0)
       have h10 : "var\<notin>vars(2 * isolate_variable_sparse p var 2)"
         by (metis isovarspar_sum mult_2 not_in_isovarspar)
       have h : "evalUni (substInfinitesimalQuadraticUni (- b) 1 (b\<^sup>2 - 4 * a * c) (2 * a) aT)
      x =
     evalUni (substInfinitesimalQuadraticUni (- b) 1 (b\<^sup>2 - 4 * a * c) (2 * a) At')
      x"proof(cases At)
         case (Less p)
         then show ?thesis using At'[symmetric] Some[symmetric] apply(cases "MPoly_Type.degree p var < 3") by auto
       next
         case (Eq p)
         then show ?thesis using At'[symmetric] Some[symmetric] apply(cases "MPoly_Type.degree p var < 3") by auto
       next
         case (Leq x3)
         then show ?thesis using At' using Some by auto
       next
         case (Neq x4)
         then show ?thesis using At' using Some by auto 
       qed
       show ?thesis unfolding convert_substInfinitesimalQuadratic[OF Some h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 assms(3)]
         using h .
     qed
 
 
     show ?case
       unfolding L2' apply(simp del : substInfinitesimalQuadratic.simps substInfinitesimalQuadraticUni.simps)
       unfolding 
         Cons(1)[OF Cons(2) Cons(3) L2's]
       unfolding h
       by auto
   qed
   have quadratic_2 : "(a \<noteq> 0) \<Longrightarrow>
      (4 * a * c \<le> b\<^sup>2) \<Longrightarrow> (\<forall>f\<in>set L2.
           eval
            (substInfinitesimalQuadratic var
              (- isolate_variable_sparse p var (Suc 0)) (- 1)
              ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
               4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)
              (2 * isolate_variable_sparse p var 2) f)
            (xs' @ x # xs)) = (\<forall>l\<in>set L2'.
           evalUni
            (substInfinitesimalQuadraticUni (- b) (- 1) (b\<^sup>2 - 4 * a * c) (2 * a)
              l)
            x)" 
     using assms(2) proof(induction L2 arbitrary: L2')
     case Nil
     then show ?case by auto
   next
     case (Cons At L2)
     have "\<exists>At'. convert_atom var At (xs' @ x # xs) = Some At'" proof(cases At)
       case (Less p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Eq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Leq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     next
       case (Neq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     qed 
     then obtain At' where At' : "convert_atom var At (xs' @ x # xs) = Some At'" by auto
     have "\<exists>L2's. convert_atom_list var L2 (xs' @ x # xs) = Some L2's"
       using Cons(4) At'
       apply(cases "convert_atom_list var L2 (xs' @ x # xs)") by auto
     then obtain L2's where L2's : "convert_atom_list var L2 (xs' @ x # xs) = Some L2's" by auto
     have L2' : "L2' = At' # L2's"
       using Cons(4) At' apply(cases At) apply auto
       by (simp_all add: L2's)
     have h1 :  "var < length (xs' @ x # xs)" using assms by auto
     have h2 : "2*a \<noteq>0" using Cons by auto
     have h3 : "0\<le>b^2-4*a*c" using Cons(3) by auto
     have h4 : "var\<notin>vars ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
             4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)"
       by (metis add_uminus_conv_diff not_in_add not_in_isovarspar not_in_mult not_in_neg not_in_pow num_double numeral_times_numeral one_add_one power_0)
     have h5 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (- isolate_variable_sparse p var (Suc 0)) = -b"
       unfolding insertion_neg b_def
       by (metis insertion_isovarspars_free list_update_id) 
     have h6 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (-1) = (-1)" unfolding insertion_neg by auto
     have h7 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa]))
         ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
          4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0) =
        b\<^sup>2 - 4 * a * c" apply(simp add: insertion_four insertion_mult insertion_sub insertion_pow b_def a_def c_def) using assms
       by (metis insertion_isovarspars_free list_update_id)
     have "\<And>xa. insertion (nth_default 0 (xs' @ xa # xs)) (2::real mpoly)  = (2::real)"
       by (metis MPoly_Type.insertion_one insertion_add one_add_one)  
     then have h8 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (2 * isolate_variable_sparse p var 2) = 2 * a"
       unfolding insertion_mult a_def apply auto using assms
-      by (metis (no_types, hide_lams) MPoly_Type.insertion_one add.inverse_inverse add_uminus_conv_diff arith_special(3) insertion_isovarspars_free insertion_neg insertion_sub list_update_id) 
+      by (metis (no_types, opaque_lifting) MPoly_Type.insertion_one add.inverse_inverse add_uminus_conv_diff arith_special(3) insertion_isovarspars_free insertion_neg insertion_sub list_update_id) 
     have h9 : "var\<notin>vars(- isolate_variable_sparse p var (Suc 0))"
       by (simp add: not_in_isovarspar not_in_neg)
     have h10 : "var\<notin>vars(- 1::real mpoly)"
       by (metis h9 not_in_neg not_in_pow power.simps(1))
     have h11 : "var\<notin>vars(2 * isolate_variable_sparse p var 2)"
       by (metis isovarspar_sum mult_2 not_in_isovarspar)
     have h :  "eval
       (substInfinitesimalQuadratic var (- isolate_variable_sparse p var (Suc 0)) (-1)
         ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
          4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)
         (2 * isolate_variable_sparse p var 2) At)
       (xs' @ x # xs) =  evalUni
       (substInfinitesimalQuadraticUni (- b) (-1) (b\<^sup>2 - 4 * a * c) (2 * a) At') x"
     proof (cases "convert_atom var At (xs' @ x # xs)")
       case None
       then show ?thesis using At' apply(cases At) by auto
     next
       case (Some aT)
       have h1 : "insertion (nth_default 0 (xs' @ x # xs)) (- isolate_variable_sparse p var (Suc 0)) = (-b)" unfolding b_def insertion_neg by auto
       have h2 : "insertion (nth_default 0 (xs' @ x # xs)) (-1) = -1" unfolding insertion_neg by auto
       have h3 : "insertion (nth_default 0 (xs' @ x # xs)) (((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
         4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)) = (b\<^sup>2 - 4 * a * c)"
         unfolding insertion_mult insertion_pow insertion_four insertion_neg insertion_sub a_def b_def c_def using assms
         by auto
       have h4 : "insertion (nth_default 0 (xs' @ x # xs)) (2 * isolate_variable_sparse p var 2) = 2 * a"
         unfolding insertion_mult a_def
         by (metis insertion_add insertion_mult mult_2)
       have h5 : "2 * a \<noteq> 0" using Cons by auto
       have h6 : "0 \<le> b\<^sup>2 - 4 * a * c" using Cons by auto
       have h7 : "var\<notin>vars(- isolate_variable_sparse p var (Suc 0))"
         by (simp add: not_in_isovarspar not_in_neg)
       have h8 : "var\<notin>vars(- 1::real mpoly)"
         by (simp add: h10 not_in_neg)
       have h9 : "var \<notin> vars ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
              4 * isolate_variable_sparse p var 2 *
              isolate_variable_sparse p var 0)"
         by (metis add_uminus_conv_diff not_in_add not_in_isovarspar not_in_mult not_in_neg not_in_pow num_double numeral_times_numeral one_add_one power_0)
       have h10 : "var\<notin>vars(2 * isolate_variable_sparse p var 2)"
         by (metis isovarspar_sum mult_2 not_in_isovarspar)
       have h : "evalUni (substInfinitesimalQuadraticUni (- b) (-1) (b\<^sup>2 - 4 * a * c) (2 * a) aT)
      x =
     evalUni (substInfinitesimalQuadraticUni (- b) (-1) (b\<^sup>2 - 4 * a * c) (2 * a) At')
      x"proof(cases At)
         case (Less p)
         then show ?thesis using At'[symmetric] Some[symmetric] apply(cases "MPoly_Type.degree p var < 3") by auto
       next
         case (Eq p)
         then show ?thesis using At'[symmetric] Some[symmetric] apply(cases "MPoly_Type.degree p var < 3") by auto
       next
         case (Leq x3)
         then show ?thesis using At'
           using Some option.inject by auto 
       next
         case (Neq x4)
         then show ?thesis using At'
           using Some by auto 
       qed
       show ?thesis unfolding convert_substInfinitesimalQuadratic[OF Some h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 assms(3)]
         using h .
     qed
 
 
     show ?case
       unfolding L2' apply(simp del : substInfinitesimalQuadratic.simps substInfinitesimalQuadraticUni.simps)
       unfolding 
         Cons(1)[OF Cons(2) Cons(3) L2's]
       unfolding h
       by auto
   qed
 
   show ?thesis using assms(1)[symmetric] unfolding Less apply(cases "MPoly_Type.degree p var < 3") apply simp_all
     apply(simp del : substInfinitesimalLinear.simps substInfinitesimalLinearUni.simps substInfinitesimalQuadratic.simps substInfinitesimalQuadraticUni.simps
         add: insertion_neg insertion_mult insertion_add insertion_pow insertion_sub insertion_four
         a_def[symmetric] b_def[symmetric] c_def[symmetric] a_def'[symmetric] b_def'[symmetric] c_def'[symmetric] eval_list_conj
         eval_list_conj_Uni
         ) using linear quadratic_1 quadratic_2 by smt
 next
   case (Eq p)
   define a where "a = insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 2)"
   have a_def' : "a = insertion (nth_default 0 (xs' @ 0 # xs)) (isolate_variable_sparse p var 2)" unfolding a_def
     using insertion_isovarspars_free[of "xs' @x#xs" var x p 2 0] using assms by auto
   define b where "b = insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var (Suc 0))"
   have b_def' : "b = insertion (nth_default 0 (xs' @ 0 # xs)) (isolate_variable_sparse p var (Suc 0))" unfolding b_def
     using insertion_isovarspars_free[of "xs' @x#xs" var x p "(Suc 0)" 0] using assms by auto
   define c where "c = insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 0)"
   have c_def' : "c = insertion (nth_default 0 (xs' @ 0 # xs)) (isolate_variable_sparse p var 0)" unfolding c_def
     using insertion_isovarspars_free[of "xs' @x#xs" var x p 0 0]using assms by auto
   have linear : "a=0 \<Longrightarrow> b\<noteq>0 \<Longrightarrow> (\<forall>f\<in>set L2.
          aEval
           (linear_substitution var 
             (-isolate_variable_sparse p var 0) (isolate_variable_sparse p var (Suc 0)) f)
           (xs' @ x # xs)) = (\<forall>l\<in>set L2'. evalUni (linearSubstitutionUni b c l) x)"
 
     using assms(2)
   proof(induction L2 arbitrary: L2')
     case Nil
     then show ?case by auto
   next
     case (Cons At L2)
     have "\<exists>At'. convert_atom var At (xs' @ x # xs) = Some At'" proof(cases At)
       case (Less p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Eq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Leq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     next
       case (Neq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     qed 
     then obtain At' where At' : "convert_atom var At (xs' @ x # xs) = Some At'" by auto
     have "\<exists>L2's. convert_atom_list var L2 (xs' @ x # xs) = Some L2's"
       using Cons(4) At'
       apply(cases "convert_atom_list var L2 (xs' @ x # xs)") by auto
     then obtain L2's where L2's : "convert_atom_list var L2 (xs' @ x # xs) = Some L2's" by auto
     have L2' : "L2' = At' # L2's"
       using Cons(4) At' apply(cases At) apply auto
       by (simp_all add: L2's)
     have h1 : "var \<notin> vars (isolate_variable_sparse p var (Suc 0))"
       by (simp add: not_in_isovarspar) 
     have h2 : "var \<notin> vars (isolate_variable_sparse p var 0)"by (simp add: not_in_isovarspar) 
     have h : "aEval
          (linear_substitution var
            (-isolate_variable_sparse p var 0) (isolate_variable_sparse p var (Suc 0)) At)
          (xs' @ x # xs) = evalUni (linearSubstitutionUni b c At') x"
     proof(cases "convert_atom var At (xs' @ x # xs)")
       case None
       then show ?thesis using At' apply(cases At) by auto
     next
       case (Some a)
       have h : "a=At'"
         using At' Some by auto
       show ?thesis unfolding convert_linearSubstitutionUni[OF Some b_def[symmetric] c_def[symmetric] Cons(3) h1 h2 assms(3)] 
         unfolding h by auto
     qed 
     have "(\<forall>f\<in>set (At # L2).
         aEval
          (linear_substitution var
            (-isolate_variable_sparse p var 0) (isolate_variable_sparse p var (Suc 0)) f)
          (xs' @ x # xs)) = (aEval
          (linear_substitution var 
            (-isolate_variable_sparse p var 0)(isolate_variable_sparse p var (Suc 0)) At)
          (xs' @ x # xs)\<and> (\<forall>f\<in>set (L2).
         aEval
          (linear_substitution var
            (-isolate_variable_sparse p var 0) (isolate_variable_sparse p var (Suc 0)) f)
          (xs' @ x # xs)))" by auto
     also have "... = (evalUni (linearSubstitutionUni b c At') x \<and>
      (\<forall>l\<in>set L2's. evalUni (linearSubstitutionUni b c l) x))"
       unfolding h Cons(1)[OF Cons(2) Cons(3) L2's]  by auto
     finally show ?case   unfolding L2' by auto
   qed
 
   have quadratic_1 : "(a \<noteq> 0) \<Longrightarrow>
      (4 * a * c \<le> b\<^sup>2) \<Longrightarrow>(\<forall>f\<in>set L2.
           eval
            (quadratic_sub var (- isolate_variable_sparse p var (Suc 0)) 1
              ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
               4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)
              (2 * isolate_variable_sparse p var 2) f)
            (xs' @ x # xs)) = (\<forall>l\<in>set L2'.
           evalUni (quadraticSubUni (- b) 1 (b\<^sup>2 - 4 * a * c) (2 * a) l) x)"
     using assms(2) proof(induction L2 arbitrary: L2')
     case Nil
     then show ?case by auto
   next
     case (Cons At L2)
     have "\<exists>At'. convert_atom var At (xs' @ x # xs) = Some At'" proof(cases At)
       case (Less p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Eq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Leq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     next
       case (Neq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     qed 
     then obtain At' where At' : "convert_atom var At (xs' @ x # xs) = Some At'" by auto
     have "\<exists>L2's. convert_atom_list var L2 (xs' @ x # xs) = Some L2's"
       using Cons(4) At'
       apply(cases "convert_atom_list var L2 (xs' @ x # xs)") by auto
     then obtain L2's where L2's : "convert_atom_list var L2 (xs' @ x # xs) = Some L2's" by auto
     have L2' : "L2' = At' # L2's"
       using Cons(4) At' apply(cases At) apply auto
       by (simp_all add: L2's)
     have h1 :  "var < length (xs' @ x # xs)" using assms by auto
     have h2 : "2*a \<noteq>0" using Cons by auto
     have h3 : "0\<le>b^2-4*a*c" using Cons(3) by auto
     have h4 : "var\<notin>vars ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
             4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)"
       by (metis add_uminus_conv_diff not_in_add not_in_isovarspar not_in_mult not_in_neg not_in_pow num_double numeral_times_numeral one_add_one power_0)
     have h5 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (- isolate_variable_sparse p var (Suc 0)) = -b"
       unfolding insertion_neg b_def
       by (metis insertion_isovarspars_free list_update_id) 
     have h6 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) 1 = 1" by auto
     have h7 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa]))
         ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
          4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0) =
        b\<^sup>2 - 4 * a * c" apply(simp add: insertion_four insertion_mult insertion_sub insertion_pow b_def a_def c_def)
       by (metis insertion_isovarspars_free list_update_id)
     have "\<And>xa. insertion (nth_default 0 (xs' @ xa # xs)) (2::real mpoly)  = (2::real)"
       by (metis MPoly_Type.insertion_one insertion_add one_add_one)  
     then have h8 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (2 * isolate_variable_sparse p var 2) = 2 * a"
       unfolding insertion_mult a_def apply auto
       by (metis assms(3) insertion_add insertion_isovarspars_free insertion_mult list_update_length mult_2)
     have h9 : "var\<notin>vars(- isolate_variable_sparse p var (Suc 0))"
       by (simp add: not_in_isovarspar not_in_neg)
     have h10 : "var\<notin>vars(1::real mpoly)"
       by (metis h9 not_in_pow power.simps(1))
     have h11 : "var\<notin>vars(2 * isolate_variable_sparse p var 2)"
       by (metis isovarspar_sum mult_2 not_in_isovarspar)
     have h :  "eval
      (quadratic_sub var (- isolate_variable_sparse p var (Suc 0)) 1
        ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
         4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)
        (2 * isolate_variable_sparse p var 2) At)
      (xs' @ x # xs) =  aEval At (xs' @ (((- b + 1 * sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)) # xs))"
       using quadratic_sub[OF h1 h2 h3 h4 h5 h6 h7 h8, symmetric, of At]
         free_in_quad[OF h9 h10 h4 h11]
       by (metis assms(3) list_update_length var_not_in_eval3) 
     have h2 : "aEval At (xs' @ (- b + 1 * sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) # xs) = evalUni (quadraticSubUni (- b) 1 (b\<^sup>2 - 4 * a * c) (2 * a) At') x"
     proof(cases At)
       case (Less p)
       then show ?thesis
       proof(cases "convert_atom var At (xs' @ x # xs)")
         case None
         then show ?thesis
           using At'[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Some aT)
         then have Some : "\<And>x. convert_atom var At (xs' @ x # xs) = Some aT"
           by (metis assms(3) convert_atom_change) 
         show ?thesis unfolding aEval_aEvalUni[OF Some assms(3)]
           using At'[symmetric] Some[symmetric]
           unfolding Less apply(cases "MPoly_Type.degree p var < 3")  by simp_all
       qed
     next
       case (Eq p)
       then show ?thesis proof(cases "convert_atom var At (xs' @ x # xs)")
         case None
         then show ?thesis
           using At'[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Some aT)
         then have Some : "\<And>x. convert_atom var At (xs' @ x # xs) = Some aT"
           by (metis assms(3) convert_atom_change) 
         show ?thesis unfolding aEval_aEvalUni[OF Some assms(3)]
           using At'[symmetric] Some[symmetric]
           unfolding Eq apply(cases "MPoly_Type.degree p var < 3")  by simp_all
       qed
     next
       case (Leq x3)
       then show ?thesis proof(cases "convert_atom var At (xs' @ x # xs)")
         case None
         then show ?thesis
           using At'[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Some aT)
         then have Some : "\<And>x. convert_atom var At (xs' @ x # xs) = Some aT"
           by (metis assms(3) convert_atom_change) 
         show ?thesis unfolding aEval_aEvalUni[OF Some assms(3)]
           using At'[symmetric] Some[symmetric]
           unfolding Leq apply(cases "MPoly_Type.degree p var < 3") by auto
       qed
     next
       case (Neq x4)
       then show ?thesis 
       proof(cases "convert_atom var At (xs' @ x # xs)")
         case None
         then show ?thesis
           using At'[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Some aT)
         then have Some : "\<And>x. convert_atom var At (xs' @ x # xs) = Some aT"
           by (metis assms(3) convert_atom_change) 
         show ?thesis unfolding aEval_aEvalUni[OF Some assms(3)]
           using At'[symmetric] Some[symmetric]
           unfolding Neq apply(cases "MPoly_Type.degree p var < 3") by auto
       qed
     qed
     show ?case
       unfolding L2' apply(simp del : quadratic_sub.simps quadraticSubUni.simps)
       unfolding 
         Cons(1)[OF Cons(2) Cons(3) L2's]
       unfolding h h2
       by auto
   qed
   have quadratic_2 : "(a \<noteq> 0) \<Longrightarrow>
      (4 * a * c \<le> b\<^sup>2) \<Longrightarrow> (\<forall>f\<in>set L2.
           eval
            (quadratic_sub var (- isolate_variable_sparse p var (Suc 0)) (- 1)
              ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
               4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)
              (2 * isolate_variable_sparse p var 2) f)
            (xs' @ x # xs)) = (\<forall>l\<in>set L2'.
           evalUni (quadraticSubUni (- b) (- 1) (b\<^sup>2 - 4 * a * c) (2 * a) l) x)"
     using assms(2) proof(induction L2 arbitrary: L2')
     case Nil
     then show ?case by auto
   next
     case (Cons At L2)
     have "\<exists>At'. convert_atom var At (xs' @ x # xs) = Some At'" proof(cases At)
       case (Less p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Eq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Leq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     next
       case (Neq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     qed 
     then obtain At' where At' : "convert_atom var At (xs' @ x # xs) = Some At'" using assms by auto
     have "\<exists>L2's. convert_atom_list var L2 (xs' @ x # xs) = Some L2's"
       using Cons(4) At'
       apply(cases "convert_atom_list var L2 (xs' @ x # xs)") by auto
     then obtain L2's where L2's : "convert_atom_list var L2 (xs' @ x # xs) = Some L2's" by auto
     have L2' : "L2' = At' # L2's"
       using Cons(4) At' apply(cases At) apply auto
       by (simp_all add: L2's)
     have h1 :  "var < length (xs' @ x # xs)" using assms by auto
     have h2 : "2*a \<noteq>0" using Cons by auto
     have h3 : "0\<le>b^2-4*a*c" using Cons(3) by auto
     have h4 : "var\<notin>vars ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
             4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)"
       by (metis add_uminus_conv_diff not_in_add not_in_isovarspar not_in_mult not_in_neg not_in_pow num_double numeral_times_numeral one_add_one power_0)
     have h5 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (- isolate_variable_sparse p var (Suc 0)) = -b"
       unfolding insertion_neg b_def
       by (metis insertion_isovarspars_free list_update_id) 
     have h6 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (-1) = -1"
       unfolding insertion_neg
       by auto
     have h7 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa]))
         ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
          4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0) =
        b\<^sup>2 - 4 * a * c" apply(simp add: insertion_four insertion_mult insertion_sub insertion_pow b_def a_def c_def)
       by (metis insertion_isovarspars_free list_update_id)
     have "\<And>xa. insertion (nth_default 0 (xs' @xa # xs)) (2::real mpoly)  = (2::real)"
       by (metis MPoly_Type.insertion_one insertion_add one_add_one)  
     then have h8 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (2 * isolate_variable_sparse p var 2) = 2 * a"
       unfolding insertion_mult a_def apply auto
       by (metis assms(3) insertion_lowerPoly1 list_update_length not_in_isovarspar)
     have h9 : "var\<notin>vars(- isolate_variable_sparse p var (Suc 0))"
       by (simp add: not_in_isovarspar not_in_neg)
     have h10 : "var\<notin>vars(-1::real mpoly)"
       by (metis h9 not_in_neg not_in_pow power.simps(1))
     have h11 : "var\<notin>vars(2 * isolate_variable_sparse p var 2)"
       by (metis isovarspar_sum mult_2 not_in_isovarspar)
     have h :  "eval
      (quadratic_sub var (- isolate_variable_sparse p var (Suc 0)) (-1)
        ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
         4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)
        (2 * isolate_variable_sparse p var 2) At)
      (xs' @ x # xs) =  aEval At (xs' @ (((- b - 1 * sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)) # xs))"
       using quadratic_sub[OF h1 h2 h3 h4 h5 h6 h7 h8, symmetric, of At]
         var_not_in_eval3 free_in_quad[OF h9 h10 h4 h11]
       using assms(3) by fastforce 
     have h2 : "aEval At (xs'  @ (- b - 1 * sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) # xs) = evalUni (quadraticSubUni (- b) (-1) (b\<^sup>2 - 4 * a * c) (2 * a) At') x"
     proof(cases At)
       case (Less p)
       then show ?thesis
       proof(cases "convert_atom var At (xs' @ x # xs)")
         case None
         then show ?thesis
           using At'[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Some aT)
         then have Some : "\<And>x. convert_atom var At (xs' @ x # xs) = Some aT"
           by (metis assms(3) convert_atom_change) 
         show ?thesis unfolding aEval_aEvalUni[OF Some assms(3)]
           using At'[symmetric] Some[symmetric]
           unfolding Less apply(cases "MPoly_Type.degree p var < 3")  by simp_all
       qed
     next
       case (Eq p)
       then show ?thesis proof(cases "convert_atom var At (xs' @ x # xs)")
         case None
         then show ?thesis
           using At'[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Some aT)
         then have Some : "\<And>x. convert_atom var At (xs' @ x # xs) = Some aT"
           by (metis assms(3) convert_atom_change) 
         show ?thesis unfolding aEval_aEvalUni[OF Some assms(3)]
           using At'[symmetric] Some[symmetric]
           unfolding Eq apply(cases "MPoly_Type.degree p var < 3") by simp_all
       qed
     next
       case (Leq x3)
       then show ?thesis proof(cases "convert_atom var At (xs' @ x # xs)")
         case None
         then show ?thesis
           using At'[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Some aT)
         then have Some : "\<And>x. convert_atom var At (xs' @ x # xs) = Some aT"
           by (metis assms(3) convert_atom_change) 
         show ?thesis unfolding aEval_aEvalUni[OF Some assms(3)]
           using At'[symmetric] Some[symmetric]
           unfolding Leq apply(cases "MPoly_Type.degree p var < 3") by auto
       qed
     next
       case (Neq x4)
       then show ?thesis proof(cases "convert_atom var At (xs' @ x # xs)")
         case None
         then show ?thesis
           using At'[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Some aT)
         then have Some : "\<And>x. convert_atom var At (xs' @ x # xs) = Some aT"
           by (metis assms(3) convert_atom_change) 
         show ?thesis unfolding aEval_aEvalUni[OF Some assms(3)]
           using At'[symmetric] Some[symmetric]
           unfolding Neq apply(cases "MPoly_Type.degree p var < 3") by auto
       qed
     qed
     show ?case
       unfolding L2' apply(simp del : quadratic_sub.simps quadraticSubUni.simps)
       unfolding 
         Cons(1)[OF Cons(2) Cons(3) L2's]
       unfolding h h2
       by auto
   qed
   show ?thesis using assms(1)[symmetric] unfolding Eq apply(cases "MPoly_Type.degree p var < 3") apply simp_all
     apply(simp del : linearSubstitutionUni.simps quadraticSubUni.simps
         add: insertion_neg insertion_mult insertion_add insertion_pow insertion_sub insertion_four
         a_def[symmetric] b_def[symmetric] c_def[symmetric] a_def'[symmetric] b_def'[symmetric] c_def'[symmetric] eval_list_conj
         eval_list_conj_Uni )using linear
     using quadratic_1 quadratic_2
     by smt
 next
   case (Leq p)
   define a where "a = insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 2)"
   have a_def' : "a = insertion (nth_default 0 (xs' @ 0 # xs)) (isolate_variable_sparse p var 2)" unfolding a_def
     using insertion_isovarspars_free[of "xs'@ x#xs" var x p 2 0] using assms by auto
   define b where "b = insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var (Suc 0))"
   have b_def' : "b = insertion (nth_default 0 (xs'@ 0 # xs)) (isolate_variable_sparse p var (Suc 0))" unfolding b_def
     using insertion_isovarspars_free[of "xs'@x#xs" var x p "(Suc 0)" 0] using assms by auto
   define c where "c = insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 0)"
   have c_def' : "c = insertion (nth_default 0 (xs'@ 0 # xs)) (isolate_variable_sparse p var 0)" unfolding c_def
     using insertion_isovarspars_free[of "xs'@ x#xs" var x p 0 0] using assms by auto
   have linear : "a=0 \<Longrightarrow> b\<noteq>0 \<Longrightarrow> (\<forall>f\<in>set L2.
          aEval
           (linear_substitution var 
             (-isolate_variable_sparse p var 0) (isolate_variable_sparse p var (Suc 0)) f)
           (xs' @ x # xs)) = (\<forall>l\<in>set L2'. evalUni (linearSubstitutionUni b c l) x)"
     using assms(2)
   proof(induction L2 arbitrary: L2')
     case Nil
     then show ?case by auto
   next
     case (Cons At L2)
     have "\<exists>At'. convert_atom var At (xs' @ x # xs) = Some At'" proof(cases At)
       case (Less p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Eq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Leq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     next
       case (Neq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     qed 
     then obtain At' where At' : "convert_atom var At (xs' @ x # xs) = Some At'" by auto
     have "\<exists>L2's. convert_atom_list var L2 (xs' @ x # xs) = Some L2's"
       using Cons(4) At'
       apply(cases "convert_atom_list var L2 (xs' @ x # xs)") by auto
     then obtain L2's where L2's : "convert_atom_list var L2 (xs' @ x # xs) = Some L2's" by auto
     have L2' : "L2' = At' # L2's"
       using Cons(4) At' apply(cases At) apply auto
       by (simp_all add: L2's)
     have h1 : "var \<notin> vars (isolate_variable_sparse p var (Suc 0))"
       by (simp add: not_in_isovarspar) 
     have h2 : "var \<notin> vars (isolate_variable_sparse p var 0)"by (simp add: not_in_isovarspar) 
     have h : "aEval
          (linear_substitution var 
            (-isolate_variable_sparse p var 0) (isolate_variable_sparse p var (Suc 0)) At)
          (xs' @ x # xs) = evalUni (linearSubstitutionUni b c At') x"
     proof(cases "convert_atom var At (xs' @ x # xs)")
       case None
       then show ?thesis using At' apply(cases At) by auto
     next
       case (Some a)
       have h : "a=At'"
         using At' Some by auto
       show ?thesis unfolding convert_linearSubstitutionUni[OF Some b_def[symmetric] c_def[symmetric] Cons(3) h1 h2 assms(3)] 
         unfolding h by auto
     qed 
     have "(\<forall>f\<in>set (At # L2).
         aEval
          (linear_substitution var 
            (-isolate_variable_sparse p var 0) (isolate_variable_sparse p var (Suc 0)) f)
          (xs' @ x # xs)) = (aEval
          (linear_substitution var 
            (-isolate_variable_sparse p var 0) (isolate_variable_sparse p var (Suc 0)) At)
          (xs' @ x # xs)\<and> (\<forall>f\<in>set (L2).
         aEval
          (linear_substitution var 
            (-isolate_variable_sparse p var 0) (isolate_variable_sparse p var (Suc 0)) f)
          (xs' @ x # xs)))" by auto
     also have "... = (evalUni (linearSubstitutionUni b c At') x \<and>
      (\<forall>l\<in>set L2's. evalUni (linearSubstitutionUni b c l) x))"
       unfolding h Cons(1)[OF Cons(2) Cons(3) L2's]  by auto
     finally show ?case   unfolding L2' by auto
   qed
 
   have quadratic_1 : "(a \<noteq> 0) \<Longrightarrow>
      (4 * a * c \<le> b\<^sup>2) \<Longrightarrow>(\<forall>f\<in>set L2.
           eval
            (quadratic_sub var (- isolate_variable_sparse p var (Suc 0)) 1
              ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
               4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)
              (2 * isolate_variable_sparse p var 2) f)
            (xs' @ x # xs)) = (\<forall>l\<in>set L2'.
           evalUni (quadraticSubUni (- b) 1 (b\<^sup>2 - 4 * a * c) (2 * a) l) x)"
     using assms(2) proof(induction L2 arbitrary: L2')
     case Nil
     then show ?case by auto
   next
     case (Cons At L2)
     have "\<exists>At'. convert_atom var At (xs' @ x # xs) = Some At'" proof(cases At)
       case (Less p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Eq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Leq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     next
       case (Neq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     qed 
     then obtain At' where At' : "convert_atom var At (xs' @ x # xs) = Some At'" by auto
     have "\<exists>L2's. convert_atom_list var L2 (xs' @ x # xs) = Some L2's"
       using Cons(4) At'
       apply(cases "convert_atom_list var L2 (xs' @ x # xs)") by auto
     then obtain L2's where L2's : "convert_atom_list var L2 (xs' @ x # xs) = Some L2's" by auto
     have L2' : "L2' = At' # L2's"
       using Cons(4) At' apply(cases At) apply auto
       by (simp_all add: L2's)
     have h1 :  "var < length (xs' @ x # xs)" using assms by auto
     have h2 : "2*a \<noteq>0" using Cons by auto
     have h3 : "0\<le>b^2-4*a*c" using Cons(3) by auto
     have h4 : "var\<notin>vars ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
             4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)"
       by (metis add_uminus_conv_diff not_in_add not_in_isovarspar not_in_mult not_in_neg not_in_pow num_double numeral_times_numeral one_add_one power_0)
     have h5 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (- isolate_variable_sparse p var (Suc 0)) = -b"
       unfolding insertion_neg b_def
       by (metis insertion_isovarspars_free list_update_id) 
     have h6 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) 1 = 1" by auto
     have h7 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa]))
         ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
          4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0) =
        b\<^sup>2 - 4 * a * c" apply(simp add: insertion_four insertion_mult insertion_sub insertion_pow b_def a_def c_def)
       by (metis insertion_isovarspars_free list_update_id)
     have "\<And>xa. insertion (nth_default 0 (xs' @xa # xs)) (2::real mpoly)  = (2::real)"
       by (metis MPoly_Type.insertion_one insertion_add one_add_one)  
     then have h8 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (2 * isolate_variable_sparse p var 2) = 2 * a"
       unfolding insertion_mult a_def apply auto
       by (metis assms(3) insertion_lowerPoly1 list_update_length not_in_isovarspar)
     have h9 : "var\<notin>vars(- isolate_variable_sparse p var (Suc 0))"
       by (simp add: not_in_isovarspar not_in_neg)
     have h10 : "var\<notin>vars(1::real mpoly)"
       by (metis h9 not_in_pow power.simps(1))
     have h11 : "var\<notin>vars(2 * isolate_variable_sparse p var 2)"
       by (metis isovarspar_sum mult_2 not_in_isovarspar)
     have h :  "eval
      (quadratic_sub var (- isolate_variable_sparse p var (Suc 0)) 1
        ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
         4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)
        (2 * isolate_variable_sparse p var 2) At)
      (xs' @ x # xs) =  aEval At (xs' @ (((- b + 1 * sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)) # xs))"
       using quadratic_sub[OF h1 h2 h3 h4 h5 h6 h7 h8, symmetric, of At]
         var_not_in_eval3 free_in_quad[OF h9 h10 h4 h11]
       by (metis assms(3) list_update_length) 
     have h2 : "aEval At (xs' @ (- b + 1 * sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) # xs) = evalUni (quadraticSubUni (- b) 1 (b\<^sup>2 - 4 * a * c) (2 * a) At') x"
     proof(cases At)
       case (Less p)
       then show ?thesis
       proof(cases "convert_atom var At (xs' @ x # xs)")
         case None
         then show ?thesis
           using At'[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Some aT)
         then have Some : "\<And>x. convert_atom var At (xs' @ x # xs) = Some aT"
           by (metis assms(3) convert_atom_change) 
         show ?thesis unfolding aEval_aEvalUni[OF Some assms(3)]
           using At'[symmetric] Some[symmetric]
           unfolding Less apply(cases "MPoly_Type.degree p var < 3") by simp_all
       qed
     next
       case (Eq p)
       then show ?thesis proof(cases "convert_atom var At (xs' @ x # xs)")
         case None
         then show ?thesis
           using At'[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Some aT)
         then have Some : "\<And>x. convert_atom var At (xs' @ x # xs) = Some aT"
           by (metis assms(3) convert_atom_change) 
         show ?thesis unfolding aEval_aEvalUni[OF Some assms(3)]
           using At'[symmetric] Some[symmetric]
           unfolding Eq apply(cases "MPoly_Type.degree p var < 3") by simp_all
       qed
     next
       case (Leq x3)
       then show ?thesis proof(cases "convert_atom var At (xs' @ x # xs)")
         case None
         then show ?thesis
           using At'[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Some aT)
         then have Some : "\<And>x. convert_atom var At (xs' @ x # xs) = Some aT"
           by (metis assms(3) convert_atom_change) 
         show ?thesis unfolding aEval_aEvalUni[OF Some assms(3)]
           using At'[symmetric] Some[symmetric]
           unfolding Leq apply(cases "MPoly_Type.degree p var < 3") by auto
       qed
     next
       case (Neq x4)
       then show ?thesis 
       proof(cases "convert_atom var At (xs' @ x # xs)")
         case None
         then show ?thesis
           using At'[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Some aT)
         then have Some : "\<And>x. convert_atom var At (xs' @ x # xs) = Some aT"
           by (metis assms(3) convert_atom_change) 
         show ?thesis unfolding aEval_aEvalUni[OF Some assms(3)]
           using At'[symmetric] Some[symmetric]
           unfolding Neq apply(cases "MPoly_Type.degree p var < 3") by auto
       qed
     qed
     show ?case
       unfolding L2' apply(simp del : quadratic_sub.simps quadraticSubUni.simps)
       unfolding 
         Cons(1)[OF Cons(2) Cons(3) L2's]
       unfolding h h2
       by auto
   qed
   have quadratic_2 : "(a \<noteq> 0) \<Longrightarrow>
      (4 * a * c \<le> b\<^sup>2) \<Longrightarrow> (\<forall>f\<in>set L2.
           eval
            (quadratic_sub var (- isolate_variable_sparse p var (Suc 0)) (- 1)
              ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
               4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)
              (2 * isolate_variable_sparse p var 2) f)
            (xs' @ x # xs)) = (\<forall>l\<in>set L2'.
           evalUni (quadraticSubUni (- b) (- 1) (b\<^sup>2 - 4 * a * c) (2 * a) l) x)"
     using assms(2) proof(induction L2 arbitrary: L2')
     case Nil
     then show ?case by auto
   next
     case (Cons At L2)
     have "\<exists>At'. convert_atom var At (xs' @ x # xs) = Some At'" proof(cases At)
       case (Less p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Eq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Leq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     next
       case (Neq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     qed 
     then obtain At' where At' : "convert_atom var At (xs' @ x # xs) = Some At'" by auto
     have "\<exists>L2's. convert_atom_list var L2 (xs' @ x # xs) = Some L2's"
       using Cons(4) At'
       apply(cases "convert_atom_list var L2 (xs' @ x # xs)") by auto
     then obtain L2's where L2's : "convert_atom_list var L2 (xs' @ x # xs) = Some L2's" by auto
     have L2' : "L2' = At' # L2's"
       using Cons(4) At' apply(cases At) apply auto
       by (simp_all add: L2's)
     have h1 :  "var < length (xs' @ x # xs)" using assms by auto
     have h2 : "2*a \<noteq>0" using Cons by auto
     have h3 : "0\<le>b^2-4*a*c" using Cons(3) by auto
     have h4 : "var\<notin>vars ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
             4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)"
       by (metis add_uminus_conv_diff not_in_add not_in_isovarspar not_in_mult not_in_neg not_in_pow num_double numeral_times_numeral one_add_one power_0)
     have h5 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (- isolate_variable_sparse p var (Suc 0)) = -b"
       unfolding insertion_neg b_def
       by (metis insertion_isovarspars_free list_update_id) 
     have h6 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (-1) = -1"
       unfolding insertion_neg
       by auto
     have h7 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa]))
         ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
          4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0) =
        b\<^sup>2 - 4 * a * c" apply(simp add: insertion_four insertion_mult insertion_sub insertion_pow b_def a_def c_def)
       by (metis insertion_isovarspars_free list_update_id)
     have "\<And>xa. insertion (nth_default 0 (xs' @ xa # xs)) (2::real mpoly)  = (2::real)"
       by (metis MPoly_Type.insertion_one insertion_add one_add_one)  
     then have h8 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (2 * isolate_variable_sparse p var 2) = 2 * a"
       unfolding insertion_mult a_def apply auto
       by (metis assms(3) insertion_lowerPoly1 list_update_length not_in_isovarspar)
     have h9 : "var\<notin>vars(- isolate_variable_sparse p var (Suc 0))"
       by (simp add: not_in_isovarspar not_in_neg)
     have h10 : "var\<notin>vars(-1::real mpoly)"
       by (metis h9 not_in_neg not_in_pow power.simps(1))
     have h11 : "var\<notin>vars(2 * isolate_variable_sparse p var 2)"
       by (metis isovarspar_sum mult_2 not_in_isovarspar)
     have h :  "eval
      (quadratic_sub var (- isolate_variable_sparse p var (Suc 0)) (-1)
        ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
         4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)
        (2 * isolate_variable_sparse p var 2) At)
      (xs' @ x # xs) =  aEval At (xs' @(((- b - 1 * sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)) # xs))"
       using quadratic_sub[OF h1 h2 h3 h4 h5 h6 h7 h8, symmetric, of At]
         var_not_in_eval3 free_in_quad[OF h9 h10 h4 h11]
       using assms(3) by fastforce 
     have h2 : "aEval At (xs'  @(- b - 1 * sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) # xs) = evalUni (quadraticSubUni (- b) (-1) (b\<^sup>2 - 4 * a * c) (2 * a) At') x"
     proof(cases At)
       case (Less p)
       then show ?thesis
       proof(cases "convert_atom var At (xs' @ x # xs)")
         case None
         then show ?thesis
           using At'[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Some aT)
         then have Some : "\<And>x. convert_atom var At (xs' @ x # xs) = Some aT"
           by (metis assms(3) convert_atom_change) 
         show ?thesis unfolding aEval_aEvalUni[OF Some assms(3)]
           using At'[symmetric] Some[symmetric]
           unfolding Less apply(cases "MPoly_Type.degree p var < 3")  by simp_all 
       qed
     next
       case (Eq p)
       then show ?thesis proof(cases "convert_atom var At (xs' @ x # xs)")
         case None
         then show ?thesis
           using At'[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Some aT)
         then have Some : "\<And>x. convert_atom var At (xs' @ x # xs) = Some aT"
           by (metis assms(3) convert_atom_change) 
         show ?thesis unfolding aEval_aEvalUni[OF Some assms(3)]
           using At'[symmetric] Some[symmetric]
           unfolding Eq apply(cases "MPoly_Type.degree p var < 3") by simp_all
       qed
     next
       case (Leq x3)
       then show ?thesis proof(cases "convert_atom var At (xs' @ x # xs)")
         case None
         then show ?thesis
           using At'[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Some aT)
         then have Some : "\<And>x. convert_atom var At (xs' @ x # xs) = Some aT"
           by (metis assms(3) convert_atom_change) 
         show ?thesis unfolding aEval_aEvalUni[OF Some assms(3)]
           using At'[symmetric] Some[symmetric]
           unfolding Leq apply(cases "MPoly_Type.degree p var < 3")
           by (auto)
       qed
     next
       case (Neq x4)
       then show ?thesis proof(cases "convert_atom var At (xs' @ x # xs)")
         case None
         then show ?thesis
           using At'[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Some aT)
         then have Some : "\<And>x. convert_atom var At (xs' @ x # xs) = Some aT"
           by (metis assms(3) convert_atom_change) 
         show ?thesis unfolding aEval_aEvalUni[OF Some assms(3)]
           using At'[symmetric] Some[symmetric]
           unfolding Neq apply(cases "MPoly_Type.degree p var < 3") by auto
       qed
     qed
     show ?case
       unfolding L2' apply(simp del : quadratic_sub.simps quadraticSubUni.simps)
       unfolding 
         Cons(1)[OF Cons(2) Cons(3) L2's]
       unfolding h h2
       by auto
   qed
   show ?thesis using assms(1)[symmetric] unfolding Leq apply(cases "MPoly_Type.degree p var < 3") apply simp_all
     apply(simp del : linearSubstitutionUni.simps quadraticSubUni.simps
         add: insertion_neg insertion_mult insertion_add insertion_pow insertion_sub insertion_four
         a_def[symmetric] b_def[symmetric] c_def[symmetric] a_def'[symmetric] b_def'[symmetric] c_def'[symmetric] eval_list_conj
         eval_list_conj_Uni ) using linear
     using quadratic_1 quadratic_2
     by smt
 next
   case (Neq p)
   define a where "a = insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 2)"
   have a_def' : "a = insertion (nth_default 0 (xs' @ 0 # xs)) (isolate_variable_sparse p var 2)" unfolding a_def
     using insertion_isovarspars_free[of "xs'  @x#xs" var x p 2 0] using assms by auto
   define b where "b = insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var (Suc 0))"
   have b_def' : "b = insertion (nth_default 0 (xs' @ 0 # xs)) (isolate_variable_sparse p var (Suc 0))" unfolding b_def
     using insertion_isovarspars_free[of "xs'@x#xs" var x p "(Suc 0)" 0] using assms by auto
   define c where "c = insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 0)"
   have c_def' : "c = insertion (nth_default 0 (xs'@0 # xs)) (isolate_variable_sparse p var 0)" unfolding c_def
     using insertion_isovarspars_free[of "xs'@x#xs" var x p 0 0] using assms by auto
   have linear : "b\<noteq>0 \<Longrightarrow> (\<forall>f\<in>set L2.
          eval
           (substInfinitesimalLinear var 
             (-isolate_variable_sparse p var 0) (isolate_variable_sparse p var (Suc 0)) f)
           (xs' @ x # xs)) = (\<forall>l\<in>set L2'. evalUni (substInfinitesimalLinearUni b c l) x)"
     using assms(2) proof(induction L2 arbitrary : L2')
     case Nil
     then show ?case by auto
   next
     case (Cons At L2)
     have "\<exists>At'. convert_atom var At (xs' @ x # xs) = Some At'" proof(cases At)
       case (Less p)
       then show ?thesis using Cons(3) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Eq p)
       then show ?thesis using Cons(3) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Leq p)
       then show ?thesis using Cons(3) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     next
       case (Neq p)
       then show ?thesis using Cons(3) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     qed 
     then obtain At' where At' : "convert_atom var At (xs' @ x # xs) = Some At'" by auto
     have "\<exists>L2's. convert_atom_list var L2 (xs' @ x # xs) = Some L2's"
       using Cons(3) At'
       apply(cases "convert_atom_list var L2 (xs' @ x # xs)") by auto
     then obtain L2's where L2's : "convert_atom_list var L2 (xs' @ x # xs) = Some L2's" by auto
     have L2' : "L2' = At' # L2's"
       using Cons(3) At'
       by (simp_all add: L2's)
     have h : "eval
          (substInfinitesimalLinear var 
            (-isolate_variable_sparse p var 0) (isolate_variable_sparse p var (Suc 0)) At)
          (xs' @ x # xs) = evalUni (substInfinitesimalLinearUni b c At') x"
     proof(cases "convert_atom var At (xs' @ x # xs)")
       case None
       then show ?thesis using At' apply(cases At) by simp_all
     next
       case (Some a)
       have h1 : "var \<notin> vars (isolate_variable_sparse p var (Suc 0))"
         by (simp add: not_in_isovarspar) 
       have h2 : "var \<notin> vars (isolate_variable_sparse p var 0)"by (simp add: not_in_isovarspar) 
       have h :  "evalUni (substInfinitesimalLinearUni b c a) x =
     evalUni (substInfinitesimalLinearUni b c At') x"
       proof(cases At)
         case (Less p)
         then show ?thesis using At'[symmetric] Some[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Eq p)
         then show ?thesis using At'[symmetric] Some[symmetric] apply(cases "MPoly_Type.degree p var < 3") by simp_all
       next
         case (Leq x3)
         then show ?thesis using At' Some by auto 
       next
         case (Neq x4)
         then show ?thesis using At' Some by auto 
       qed
       show ?thesis unfolding convert_substInfinitesimalLinear[OF Some b_def[symmetric] c_def[symmetric] Cons(2) h1 h2 assms(3)]
         using h .
     qed
     show ?case unfolding L2' using h Cons(1)[OF Cons(2) L2's] by auto
   qed
   have quadratic_1 : "(a \<noteq> 0) \<Longrightarrow>
      (4 * a * c \<le> b\<^sup>2) \<Longrightarrow> (\<forall>f\<in>set L2.
           eval
            (substInfinitesimalQuadratic var
              (- isolate_variable_sparse p var (Suc 0)) 1
              ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
               4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)
              (2 * isolate_variable_sparse p var 2) f)
            (xs' @ x # xs)) = (\<forall>l\<in>set L2'.
           evalUni
            (substInfinitesimalQuadraticUni (- b) 1 (b\<^sup>2 - 4 * a * c) (2 * a) l)
            x)"
     using assms(2) proof(induction L2 arbitrary: L2')
     case Nil
     then show ?case by auto
   next
     case (Cons At L2)
     have "\<exists>At'. convert_atom var At (xs' @ x # xs) = Some At'" proof(cases At)
       case (Less p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Eq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Leq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     next
       case (Neq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     qed 
     then obtain At' where At' : "convert_atom var At (xs' @ x # xs) = Some At'" by auto
     have "\<exists>L2's. convert_atom_list var L2 (xs' @ x # xs) = Some L2's"
       using Cons(4) At'
       apply(cases "convert_atom_list var L2 (xs' @ x # xs)") by auto
     then obtain L2's where L2's : "convert_atom_list var L2 (xs' @ x # xs) = Some L2's" by auto
     have L2' : "L2' = At' # L2's"
       using Cons(4) At' apply(cases At) apply auto
       by (simp_all add: L2's)
     have h1 :  "var < length (xs' @ x # xs)" using assms by auto
     have h2 : "2*a \<noteq>0" using Cons by auto
     have h3 : "0\<le>b^2-4*a*c" using Cons(3) by auto
     have h4 : "var\<notin>vars ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
             4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)"
       by (metis add_uminus_conv_diff not_in_add not_in_isovarspar not_in_mult not_in_neg not_in_pow num_double numeral_times_numeral one_add_one power_0)
     have h5 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (- isolate_variable_sparse p var (Suc 0)) = -b"
       unfolding insertion_neg b_def
       by (metis insertion_isovarspars_free list_update_id) 
     have h6 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) 1 = 1" by auto
     have h7 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa]))
         ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
          4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0) =
        b\<^sup>2 - 4 * a * c" apply(simp add: insertion_four insertion_mult insertion_sub insertion_pow b_def a_def c_def)
       by (metis insertion_isovarspars_free list_update_id)
     have "\<And>xa. insertion (nth_default 0 (xs' @xa # xs)) (2::real mpoly)  = (2::real)"
       by (metis MPoly_Type.insertion_one insertion_add one_add_one)  
     then have h8 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (2 * isolate_variable_sparse p var 2) = 2 * a"
       unfolding insertion_mult a_def apply auto
       by (metis assms(3) insertion_lowerPoly1 list_update_length not_in_isovarspar)
     have h9 : "var\<notin>vars(- isolate_variable_sparse p var (Suc 0))"
       by (simp add: not_in_isovarspar not_in_neg)
     have h10 : "var\<notin>vars(1::real mpoly)"
       by (metis h9 not_in_pow power.simps(1))
     have h11 : "var\<notin>vars(2 * isolate_variable_sparse p var 2)"
       by (metis isovarspar_sum mult_2 not_in_isovarspar)
     have h :  "eval
       (substInfinitesimalQuadratic var (- isolate_variable_sparse p var (Suc 0)) 1
         ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
          4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)
         (2 * isolate_variable_sparse p var 2) At)
       (xs' @ x # xs) =  evalUni
       (substInfinitesimalQuadraticUni (- b) 1 (b\<^sup>2 - 4 * a * c) (2 * a) At') x"
     proof (cases "convert_atom var At (xs' @ x # xs)")
       case None
       then show ?thesis using At' apply(cases At) by auto
     next
       case (Some aT)
       have h1 : "insertion (nth_default 0 (xs' @ x # xs)) (- isolate_variable_sparse p var (Suc 0)) = (-b)" unfolding b_def insertion_neg by auto
       have h2 : "insertion (nth_default 0 (xs' @ x # xs)) 1 = 1" by auto
       have h3 : "insertion (nth_default 0 (xs' @ x # xs)) (((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
         4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)) = (b\<^sup>2 - 4 * a * c)"
         unfolding insertion_mult insertion_pow insertion_four insertion_neg insertion_sub a_def b_def c_def
         by auto
       have h4 : "insertion (nth_default 0 (xs' @ x # xs)) (2 * isolate_variable_sparse p var 2) = 2 * a"
         unfolding insertion_mult a_def
         by (metis insertion_add insertion_mult mult_2)
       have h5 : "2 * a \<noteq> 0" using Cons by auto
       have h6 : "0 \<le> b\<^sup>2 - 4 * a * c" using Cons by auto
       have h7 : "var\<notin>vars(- isolate_variable_sparse p var (Suc 0))"
         by (simp add: not_in_isovarspar not_in_neg)
       have h8 : "var\<notin>vars(1::real mpoly)"
         by (metis h9 not_in_pow power.simps(1))
       have h9 : "var \<notin> vars ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
              4 * isolate_variable_sparse p var 2 *
              isolate_variable_sparse p var 0)"
         by (metis add_uminus_conv_diff not_in_add not_in_isovarspar not_in_mult not_in_neg not_in_pow num_double numeral_times_numeral one_add_one power_0)
       have h10 : "var\<notin>vars(2 * isolate_variable_sparse p var 2)"
         by (metis isovarspar_sum mult_2 not_in_isovarspar)
       have h : "evalUni (substInfinitesimalQuadraticUni (- b) 1 (b\<^sup>2 - 4 * a * c) (2 * a) aT)
      x =
     evalUni (substInfinitesimalQuadraticUni (- b) 1 (b\<^sup>2 - 4 * a * c) (2 * a) At')
      x"proof(cases At)
         case (Less p)
         then show ?thesis using At'[symmetric] Some[symmetric] apply(cases "MPoly_Type.degree p var < 3") by auto
       next
         case (Eq p)
         then show ?thesis using At'[symmetric] Some[symmetric] apply(cases "MPoly_Type.degree p var < 3") by auto
       next
         case (Leq x3)
         then show ?thesis using At' using Some by auto
       next
         case (Neq x4)
         then show ?thesis using At' using Some by auto 
       qed
       show ?thesis unfolding convert_substInfinitesimalQuadratic[OF Some h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 assms(3)]
         using h .
     qed
 
 
     show ?case
       unfolding L2' apply(simp del : substInfinitesimalQuadratic.simps substInfinitesimalQuadraticUni.simps)
       unfolding 
         Cons(1)[OF Cons(2) Cons(3) L2's]
       unfolding h
       by auto
   qed
   have quadratic_2 : "(a \<noteq> 0) \<Longrightarrow>
      (4 * a * c \<le> b\<^sup>2) \<Longrightarrow> (\<forall>f\<in>set L2.
           eval
            (substInfinitesimalQuadratic var
              (- isolate_variable_sparse p var (Suc 0)) (- 1)
              ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
               4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)
              (2 * isolate_variable_sparse p var 2) f)
            (xs' @ x # xs)) = (\<forall>l\<in>set L2'.
           evalUni
            (substInfinitesimalQuadraticUni (- b) (- 1) (b\<^sup>2 - 4 * a * c) (2 * a)
              l)
            x)" 
     using assms(2) proof(induction L2 arbitrary: L2')
     case Nil
     then show ?case by auto
   next
     case (Cons At L2)
     have "\<exists>At'. convert_atom var At (xs' @ x # xs) = Some At'" proof(cases At)
       case (Less p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Eq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by simp_all
     next
       case (Leq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     next
       case (Neq p)
       then show ?thesis using Cons(4) apply simp apply(cases "MPoly_Type.degree p var < 3") by auto
     qed 
     then obtain At' where At' : "convert_atom var At (xs' @ x # xs) = Some At'" by auto
     have "\<exists>L2's. convert_atom_list var L2 (xs' @ x # xs) = Some L2's"
       using Cons(4) At'
       apply(cases "convert_atom_list var L2 (xs' @ x # xs)") by auto
     then obtain L2's where L2's : "convert_atom_list var L2 (xs' @ x # xs) = Some L2's" by auto
     have L2' : "L2' = At' # L2's"
       using Cons(4) At' apply(cases At) apply auto
       by (simp_all add: L2's)
     have h1 :  "var < length ((xs' @ x # xs))" using assms by auto
     have h2 : "2*a \<noteq>0" using Cons by auto
     have h3 : "0\<le>b^2-4*a*c" using Cons(3) by auto
     have h4 : "var\<notin>vars ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
             4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)"
       by (metis add_uminus_conv_diff not_in_add not_in_isovarspar not_in_mult not_in_neg not_in_pow num_double numeral_times_numeral one_add_one power_0)
     have h5 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (- isolate_variable_sparse p var (Suc 0)) = -b"
       unfolding insertion_neg b_def
       by (metis insertion_isovarspars_free list_update_id) 
     have h6 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (-1) = (-1)" unfolding insertion_neg by auto
     have h7 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa]))
         ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
          4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0) =
        b\<^sup>2 - 4 * a * c" apply(simp add: insertion_four insertion_mult insertion_sub insertion_pow b_def a_def c_def)
       by (metis insertion_isovarspars_free list_update_id)
     have "\<And>xa. insertion (nth_default 0 (xs' @ xa # xs)) (2::real mpoly)  = (2::real)"
       by (metis MPoly_Type.insertion_one insertion_add one_add_one)  
     then have h8 : "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) (2 * isolate_variable_sparse p var 2) = 2 * a"
       unfolding insertion_mult a_def apply auto
       by (metis assms(3) insertion_lowerPoly1 list_update_length not_in_isovarspar)
     have h9 : "var\<notin>vars(- isolate_variable_sparse p var (Suc 0))"
       by (simp add: not_in_isovarspar not_in_neg)
     have h10 : "var\<notin>vars(- 1::real mpoly)"
       by (metis h9 not_in_neg not_in_pow power.simps(1))
     have h11 : "var\<notin>vars(2 * isolate_variable_sparse p var 2)"
       by (metis isovarspar_sum mult_2 not_in_isovarspar)
     have h :  "eval
       (substInfinitesimalQuadratic var (- isolate_variable_sparse p var (Suc 0)) (-1)
         ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
          4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)
         (2 * isolate_variable_sparse p var 2) At)
       (xs' @ x # xs) =  evalUni
       (substInfinitesimalQuadraticUni (- b) (-1) (b\<^sup>2 - 4 * a * c) (2 * a) At') x"
     proof (cases "convert_atom var At (xs' @ x # xs)")
       case None
       then show ?thesis using At' apply(cases At) by auto
     next
       case (Some aT)
       have h1 : "insertion (nth_default 0 (xs' @ x # xs)) (- isolate_variable_sparse p var (Suc 0)) = (-b)" unfolding b_def insertion_neg by auto
       have h2 : "insertion (nth_default 0 (xs' @ x # xs)) (-1) = -1" unfolding insertion_neg by auto
       have h3 : "insertion (nth_default 0 (xs' @ x # xs)) (((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
         4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)) = (b\<^sup>2 - 4 * a * c)"
         unfolding insertion_mult insertion_pow insertion_four insertion_neg insertion_sub a_def b_def c_def
         by auto
       have h4 : "insertion (nth_default 0 (xs' @ x # xs)) (2 * isolate_variable_sparse p var 2) = 2 * a"
         unfolding insertion_mult a_def
         by (metis insertion_add insertion_mult mult_2)
       have h5 : "2 * a \<noteq> 0" using Cons by auto
       have h6 : "0 \<le> b\<^sup>2 - 4 * a * c" using Cons by auto
       have h7 : "var\<notin>vars(- isolate_variable_sparse p var (Suc 0))"
         by (simp add: not_in_isovarspar not_in_neg)
       have h8 : "var\<notin>vars(- 1::real mpoly)"
         by (simp add: h10 not_in_neg)
       have h9 : "var \<notin> vars ((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
              4 * isolate_variable_sparse p var 2 *
              isolate_variable_sparse p var 0)"
         by (metis add_uminus_conv_diff not_in_add not_in_isovarspar not_in_mult not_in_neg not_in_pow num_double numeral_times_numeral one_add_one power_0)
       have h10 : "var\<notin>vars(2 * isolate_variable_sparse p var 2)"
         by (metis isovarspar_sum mult_2 not_in_isovarspar)
       have h : "evalUni (substInfinitesimalQuadraticUni (- b) (-1) (b\<^sup>2 - 4 * a * c) (2 * a) aT)
      x =
     evalUni (substInfinitesimalQuadraticUni (- b) (-1) (b\<^sup>2 - 4 * a * c) (2 * a) At')
      x"proof(cases At)
         case (Less p)
         then show ?thesis using At'[symmetric] Some[symmetric] apply(cases "MPoly_Type.degree p var < 3") by auto
       next
         case (Eq p)
         then show ?thesis using At'[symmetric] Some[symmetric] apply(cases "MPoly_Type.degree p var < 3") by auto
       next
         case (Leq x3)
         then show ?thesis using At'
           using Some option.inject by auto 
       next
         case (Neq x4)
         then show ?thesis using At'
           using Some by auto 
       qed
       show ?thesis unfolding convert_substInfinitesimalQuadratic[OF Some h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 assms(3)]
         using h .
     qed
 
 
     show ?case
       unfolding L2' apply(simp del : substInfinitesimalQuadratic.simps substInfinitesimalQuadraticUni.simps)
       unfolding 
         Cons(1)[OF Cons(2) Cons(3) L2's]
       unfolding h
       by auto
   qed
 
   show ?thesis using assms(1)[symmetric] unfolding Neq apply(cases "MPoly_Type.degree p var < 3") apply simp_all
     apply(simp del : substInfinitesimalLinear.simps substInfinitesimalLinearUni.simps substInfinitesimalQuadratic.simps substInfinitesimalQuadraticUni.simps
         add: insertion_neg insertion_mult insertion_add insertion_pow insertion_sub insertion_four
         a_def[symmetric] b_def[symmetric] c_def[symmetric] a_def'[symmetric] b_def'[symmetric] c_def'[symmetric] eval_list_conj
         eval_list_conj_Uni
         ) using linear quadratic_1 quadratic_2 by smt
 qed
 
 lemma convert_list : 
   assumes "convert_atom_list var L (xs' @ x # xs) = Some L'"
   assumes "l\<in>set(L)"
   shows "\<exists>l'\<in> set L'. convert_atom var l (xs' @ x # xs) = Some l'"
   using assms
 proof(induction L arbitrary : L')
   case Nil
   then show ?case by auto
 next
   case (Cons At L)
   then show ?case proof(cases At)
     case (Less p)
     then show ?thesis  using Cons(2)[symmetric] Cons(1) Cons(3) unfolding Less apply simp apply(cases "MPoly_Type.degree p var < 3") apply simp_all
       apply(cases "convert_atom_list var L (xs' @ x # xs)") apply simp_all apply(cases "l = Less p") by simp_all
   next
     case (Eq p)
     show ?thesis  using Cons(2)[symmetric] Cons(1) Cons(3) unfolding Eq apply simp apply(cases "MPoly_Type.degree p var < 3") apply simp_all
       apply(cases "convert_atom_list var L (xs' @ x # xs)") apply simp_all apply(cases "l = Eq p") by simp_all
   next
     case (Leq p)
     then show ?thesis using Cons(2)[symmetric] Cons(1) Cons(3) unfolding Leq apply simp apply(cases "MPoly_Type.degree p var < 3") apply simp_all
       apply(cases "convert_atom_list var L (xs' @ x # xs)") apply simp_all apply(cases "l = Leq p") by simp_all
   next
     case (Neq p)
     then show ?thesis using Cons(2)[symmetric] Cons(1) Cons(3) unfolding Neq apply simp apply(cases "MPoly_Type.degree p var < 3") apply simp_all
       apply(cases "convert_atom_list var L (xs' @ x # xs)") apply simp_all apply(cases "l = Neq p") by simp_all
   qed
 qed
 
 lemma convert_list2 : 
   assumes "convert_atom_list var L (xs' @ x # xs) = Some L'"
   assumes "l'\<in>set(L')"
   shows "\<exists>l\<in> set L. convert_atom var l (xs' @ x # xs) = Some l'"
   using assms
 proof(induction L arbitrary : L')
   case Nil
   then show ?case by auto
 next
   case (Cons At L)
   then show ?case proof(cases At)
     case (Less p)
     then show ?thesis  using Cons(2)[symmetric] Cons(1) Cons(3) unfolding Less apply simp apply(cases "MPoly_Type.degree p var < 3") apply simp_all
       apply(cases "convert_atom_list var L (xs' @ x # xs)") apply simp_all
       by blast
   next
     case (Eq p)
     show ?thesis  using Cons(2)[symmetric] Cons(1) Cons(3) unfolding Eq apply simp apply(cases "MPoly_Type.degree p var < 3") apply simp_all
       apply(cases "convert_atom_list var L (xs' @ x # xs)") apply simp_all by blast
   next
     case (Leq p)
     then show ?thesis  using Cons(2)[symmetric] Cons(1) Cons(3) unfolding Leq apply simp apply(cases "MPoly_Type.degree p var < 3") apply simp_all
       apply(cases "convert_atom_list var L (xs' @ x # xs)") apply simp_all by blast
   next
     case (Neq p)
     then show ?thesis  using Cons(2)[symmetric] Cons(1) Cons(3) unfolding Neq apply simp apply(cases "MPoly_Type.degree p var < 3") apply simp_all
       apply(cases "convert_atom_list var L (xs' @ x # xs)") apply simp_all by blast
   qed
 qed
 
 lemma elimVar_atom_convert : 
   assumes "convert_atom_list var L (xs' @ x # xs) = Some L'"
   assumes "convert_atom_list var L2 (xs' @ x # xs) = Some L2'"
   assumes "length xs' = var"
   shows "(\<exists>f\<in>set L. eval (elimVar var L2 [] f) (xs' @ x # xs))
          = (\<exists>f\<in>set L'. evalUni (elimVarUni_atom L2' f) x)"
 proof safe
   fix f
   assume h : "f \<in> set L"
     "eval (elimVar var L2 [] f) (xs' @ x # xs)"
   have "\<exists>f'\<in>set L'. convert_atom var f (xs' @ x # xs) = Some f'"
     using convert_list h assms by auto
   then obtain f' where f' : "f'\<in>set L'" "convert_atom var f (xs' @ x # xs) = Some f'" by metis
   show "\<exists>f\<in>set L'. evalUni (elimVarUni_atom L2' f) x"
     apply(rule bexI[where x=f']) using f' elimVar_atom_single[OF f'(2) assms(2) assms(3)] h by auto
 next
   fix f'
   assume h : "f' \<in> set L'"
     "evalUni (elimVarUni_atom L2' f') x"
   have "\<exists>f\<in>set L. convert_atom var f (xs' @ x # xs) = Some f'" using convert_list2 h assms by auto
   then obtain f where f : "f\<in>set L" "convert_atom var f (xs' @ x # xs) = Some f'" by metis
   show "\<exists>f\<in>set L. eval (elimVar var L2 [] f) (xs' @ x # xs)"
     apply(rule bexI[where x=f]) using f elimVar_atom_single[OF f(2) assms(2) assms(3)] h by auto
 qed
 
 
 lemma eval_convert : 
   assumes "convert_atom_list var L (xs' @ x # xs) = Some L'"
   assumes "length xs' = var"
   shows "(\<forall>f\<in>set L. aEval f (xs' @ x # xs)) = (\<forall>f\<in>set L'. aEvalUni f x)"
   using assms
 proof(induction L arbitrary : L')
   case Nil
   then show ?case by auto
 next
   case (Cons a L)
   then show ?case proof(cases a)
     case (Less p)
     then show ?thesis using Cons(2)[symmetric] Cons(1) Cons(3) unfolding Less apply(cases " MPoly_Type.degree p var < 3")
       apply simp_all apply(cases "convert_atom_list var L (xs' @ x # xs)") apply simp_all
       by (simp add: poly_to_univar) 
   next
     case (Eq p)
     then show ?thesis using Cons(2)[symmetric] Cons(1) Cons(3) unfolding Eq apply(cases " MPoly_Type.degree p var < 3")
       apply simp_all apply(cases "convert_atom_list var L (xs' @ x # xs)") apply simp_all
       by (simp add: poly_to_univar) 
   next
     case (Leq p)
     show ?thesis  using Cons(2)[symmetric] Cons(1) Cons(3) unfolding Leq apply(cases " MPoly_Type.degree p var < 3") 
       apply simp_all apply(cases "convert_atom_list var L (xs' @ x # xs)") apply simp_all
       by (simp add: poly_to_univar) 
   next
     case (Neq p)
     show ?thesis  using Cons(2)[symmetric] Cons(1) Cons(3) unfolding Neq apply(cases " MPoly_Type.degree p var < 3") 
       apply simp_all apply(cases "convert_atom_list var L (xs' @ x # xs)") apply simp_all
       by (simp add: poly_to_univar) 
   qed
 qed
 lemma all_degree_2_convert : 
   assumes "all_degree_2 var L"
   shows "\<exists>L'. convert_atom_list var L xs = Some L'"
   using assms
 proof(induction L)
   case Nil
   then show ?case by auto
 next
   case (Cons a L)
   then show ?case proof(cases a)
     case (Less p)
     show ?thesis using Cons unfolding Less all_degree_2.simps convert_atom_list.simps convert_atom.simps
       using degree_convert_eq[of var p xs] by auto
   next
     case (Eq p)
     then show ?thesis using Cons unfolding Eq all_degree_2.simps convert_atom_list.simps convert_atom.simps
       using degree_convert_eq[of var p xs] by auto
   next
     case (Leq x3)
     then show ?thesis using Cons by auto
   next
     case (Neq x4)
     then show ?thesis using Cons by auto
   qed
 qed
 lemma gen_qe_eval :
   assumes hlength : "length xs = var"
   shows "(\<exists>x. (eval (list_conj ((map Atom L) @ F)) (xs @ (x#\<Gamma>)))) = (\<exists>x.(eval (gen_qe var L F) (xs @ (x#\<Gamma>))))"
 proof(cases "luckyFind var L []")
   case None
   then have notLucky : "luckyFind var L [] = None" by auto 
   then show ?thesis proof(cases F)
     case Nil
     then show ?thesis proof(cases "all_degree_2 var L")
       case True
       then have "\<And>x.\<exists>L'. convert_atom_list var L (xs@x#\<Gamma>) = Some L'" using all_degree_2_convert[of var L "xs@_#\<Gamma>"] by auto
       then obtain L' where L' : "convert_atom_list var L (xs@x#\<Gamma>) = Some L'" by metis
       then have L' : "\<And>x. convert_atom_list var L (xs@x#\<Gamma>) = Some L'"
         by (metis convert_atom_list_change hlength)
       show ?thesis
         unfolding Nil apply (simp add:eval_list_conj eval_list_disj True del:luckyFind.simps) unfolding notLucky apply (simp add:eval_list_conj eval_list_disj)
         using negInf_convert[OF L' assms] elimVar_atom_convert[OF L' L' assms] eval_convert[OF L' assms]
         using eval_generalVS''[of L'] unfolding eval_list_conj_Uni generalVS_DNF.simps eval_list_conj_Uni eval_list_disj_Uni eval_append eval_map eval_map_all
           evalUni.simps 
 
         by auto
     next
       case False
       then show ?thesis using notLucky unfolding Nil  False apply simp
         by (metis append_Nil2 hlength notLucky option.simps(4) qe_eq_repeat.simps qe_eq_repeat_eval) 
     qed
   next
     case (Cons a list)
     show ?thesis
       apply(simp add:Cons del:qe_eq_repeat.simps)
       apply(rule qe_eq_repeat_eval[of xs var L "a # list" \<Gamma>])
       using assms .
   qed
 next
   case (Some a)
   then show ?thesis
     using luckyFind_eval[OF Some assms] apply(cases F) apply simp 
     apply(simp add:Cons del:qe_eq_repeat.simps)
     using qe_eq_repeat_eval[of xs var L _ \<Gamma>]
     using assms  by auto
 qed
 
 
 lemma freeIn_elimVar : "freeIn var (elimVar var L F A)"
 proof(cases A)
   case (Less p)
   have two: "2 = Suc(Suc 0)" by auto
   have notIn4: "var \<notin> vars (4::real mpoly)"
     by (metis isolate_var_one not_in_add not_in_isovarspar numeral_plus_numeral one_add_one semiring_norm(2) semiring_norm(6)) 
   show ?thesis using Less apply auto
     using not_in_isovarspar apply force+
     apply (rule freeIn_list_conj)
     apply auto
     defer defer
     using not_in_isovarspar apply force+
     using not_in_sub[OF not_in_mult[of var 4, OF _ not_in_mult[of var "isolate_variable_sparse p var 2" "isolate_variable_sparse p var 0"]], of "(isolate_variable_sparse p var (Suc 0))\<^sup>2"]
     apply (simp add:not_in_isovarspar two)
     using not_in_mult[of var "isolate_variable_sparse p var (Suc 0)" "isolate_variable_sparse p var (Suc 0)"]
     apply (simp add:not_in_isovarspar notIn4)
     apply (simp add: ideal.scale_scale)
     apply(rule freeIn_list_conj)
     apply auto
     defer defer
     apply(rule freeIn_list_conj)
     apply auto
     apply(rule freeIn_substInfinitesimalQuadratic) apply auto
     using not_in_isovarspar not_in_neg apply blast
     apply (metis not_in_isovarspar not_in_neg not_in_pow power_0)
     using notIn4 not_in_isovarspar not_in_mult not_in_pow not_in_sub apply auto[1]
     apply (metis isovarspar_sum mult_2 not_in_isovarspar)
     using freeIn_substInfinitesimalQuadratic_fm[of var "(- isolate_variable_sparse p var (Suc 0))" "-1" "((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
                       4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)" "(2 * isolate_variable_sparse p var 2)"] apply auto[1]
     apply (metis (no_types, lifting) mult_2 notIn4 not_in_add not_in_isovarspar not_in_mult not_in_neg not_in_pow not_in_sub power_0)
     apply(rule freeIn_substInfinitesimalLinear)
     apply (meson not_in_isovarspar not_in_neg)
     apply (simp add: not_in_isovarspar)
     using freeIn_substInfinitesimalLinear_fm
     using not_in_isovarspar not_in_neg apply force
     apply (metis (no_types, lifting) \<open>\<lbrakk>var \<notin> vars 4; var \<notin> vars (isolate_variable_sparse p var 2); var \<notin> vars (isolate_variable_sparse p var 0); var \<notin> vars ((isolate_variable_sparse p var (Suc 0))\<^sup>2)\<rbrakk> \<Longrightarrow> var \<notin> vars (4 * (isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0) - (isolate_variable_sparse p var (Suc 0))\<^sup>2)\<close> freeIn_substInfinitesimalQuadratic minus_diff_eq mult.assoc mult_2 notIn4 not_in_add not_in_isovarspar not_in_neg not_in_pow power_0)
     using freeIn_substInfinitesimalQuadratic_fm[of var "(- isolate_variable_sparse p var (Suc 0))" 1 "((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
                       4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)" "(2 * isolate_variable_sparse p var 2)"]
     apply auto
     by (metis (no_types, lifting) \<open>\<lbrakk>var \<notin> vars 4; var \<notin> vars (isolate_variable_sparse p var 2); var \<notin> vars (isolate_variable_sparse p var 0); var \<notin> vars ((isolate_variable_sparse p var (Suc 0))\<^sup>2)\<rbrakk> \<Longrightarrow> var \<notin> vars (4 * (isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0) - (isolate_variable_sparse p var (Suc 0))\<^sup>2)\<close> ideal.scale_scale minus_diff_eq mult_2 notIn4 not_in_add not_in_isovarspar not_in_neg not_in_pow power_0)
 next
   case (Eq p)
   then show ?thesis using freeIn_elimVar_eq by auto
 next
   case (Leq p)
   then show ?thesis using freeIn_elimVar_eq by auto
 next
   case (Neq p)
   have two: "2 = Suc(Suc 0)" by auto
   have notIn4: "var \<notin> vars (4::real mpoly)"
     by (metis isolate_var_one not_in_add not_in_isovarspar numeral_plus_numeral one_add_one semiring_norm(2) semiring_norm(6)) 
   show ?thesis using Neq apply auto
     using not_in_isovarspar apply force+
     apply (rule freeIn_list_conj)
     apply auto
     defer defer
     using not_in_isovarspar apply force+
     using not_in_sub[OF not_in_mult[of var 4, OF _ not_in_mult[of var "isolate_variable_sparse p var 2" "isolate_variable_sparse p var 0"]], of "(isolate_variable_sparse p var (Suc 0))\<^sup>2"]
     apply (simp add:not_in_isovarspar two)
     using not_in_mult[of var "isolate_variable_sparse p var (Suc 0)" "isolate_variable_sparse p var (Suc 0)"]
     apply (simp add:not_in_isovarspar notIn4)
     apply (simp add: ideal.scale_scale)
     apply(rule freeIn_list_conj)
     apply auto
     defer defer
     apply(rule freeIn_list_conj)
     apply auto
     apply(rule freeIn_substInfinitesimalQuadratic) apply auto
     using not_in_isovarspar not_in_neg apply blast
     apply (metis not_in_isovarspar not_in_neg not_in_pow power_0)
     using notIn4 not_in_isovarspar not_in_mult not_in_pow not_in_sub apply auto[1]
     apply (metis isovarspar_sum mult_2 not_in_isovarspar)
     using freeIn_substInfinitesimalQuadratic_fm[of var "(- isolate_variable_sparse p var (Suc 0))" "-1" "((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
                       4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)" "(2 * isolate_variable_sparse p var 2)"] apply auto[1]
     apply (metis (no_types, lifting) mult_2 notIn4 not_in_add not_in_isovarspar not_in_mult not_in_neg not_in_pow not_in_sub power_0)
     apply(rule freeIn_substInfinitesimalLinear)
     apply (meson not_in_isovarspar not_in_neg)
     apply (simp add: not_in_isovarspar)
     using freeIn_substInfinitesimalLinear_fm
     using not_in_isovarspar not_in_neg apply force
     apply (metis (no_types, lifting) \<open>\<lbrakk>var \<notin> vars 4; var \<notin> vars (isolate_variable_sparse p var 2); var \<notin> vars (isolate_variable_sparse p var 0); var \<notin> vars ((isolate_variable_sparse p var (Suc 0))\<^sup>2)\<rbrakk> \<Longrightarrow> var \<notin> vars (4 * (isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0) - (isolate_variable_sparse p var (Suc 0))\<^sup>2)\<close> freeIn_substInfinitesimalQuadratic minus_diff_eq mult.assoc mult_2 notIn4 not_in_add not_in_isovarspar not_in_neg not_in_pow power_0)
     using freeIn_substInfinitesimalQuadratic_fm[of var "(- isolate_variable_sparse p var (Suc 0))" 1 "((isolate_variable_sparse p var (Suc 0))\<^sup>2 -
                       4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0)" "(2 * isolate_variable_sparse p var 2)"]
     apply auto
     by (metis (no_types, lifting) \<open>\<lbrakk>var \<notin> vars 4; var \<notin> vars (isolate_variable_sparse p var 2); var \<notin> vars (isolate_variable_sparse p var 0); var \<notin> vars ((isolate_variable_sparse p var (Suc 0))\<^sup>2)\<rbrakk> \<Longrightarrow> var \<notin> vars (4 * (isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0) - (isolate_variable_sparse p var (Suc 0))\<^sup>2)\<close> ideal.scale_scale minus_diff_eq mult_2 notIn4 not_in_add not_in_isovarspar not_in_neg not_in_pow power_0)
 qed
 
 lemma freeInDisj: "freeIn var (list_disj (list_conj (map (substNegInfinity var) L) # map (elimVar var L []) L))"
   apply(rule freeIn_list_disj)
   apply(auto)
   apply(rule freeIn_list_conj)
   apply simp
 
   using freeIn_substNegInfinity[of var]
   apply simp
   using freeIn_elimVar
   by simp
 
 lemma gen_qe_eval' :
   assumes "all_degree_2 var L"
   assumes "length xs' = var"
   shows "(\<exists>x. (eval (list_conj (map Atom L)) (xs'@x#\<Gamma>))) = (\<forall>x.(eval (gen_qe var L []) (xs'@x#\<Gamma>)))"
     "freeIn var (gen_qe var L [])"
 proof-
   have h : "(\<exists>x. (eval (list_conj (map Atom L)) (xs'@x#\<Gamma>))) = (\<exists>x. eval (gen_qe var L []) (xs'@x # \<Gamma>))"
     using gen_qe_eval[OF assms(2), of L "[]" \<Gamma>] unfolding List.append.left_neutral  by auto
   show "(\<exists>x. (eval (list_conj (map Atom L)) (xs'@x#\<Gamma>))) = (\<forall>x.(eval (gen_qe var L []) (xs'@x#\<Gamma>)))"
     unfolding h
     apply (simp add:assms)
     apply(cases "find_lucky_eq var L")
     apply simp using freeInDisj[of var L]
     using var_not_in_eval3[OF _ assms(2)] apply blast
     subgoal for a
       using freeIn_elimVar_eq[of var L "[]" a]
       apply(simp del:elimVar.simps)
       using var_not_in_eval3[OF _ assms(2)] by blast
     done
 next
   show "freeIn var (gen_qe var L []) "
     apply(simp add:assms)
     apply(cases "find_lucky_eq var L") apply (simp add:freeInDisj)
     subgoal for a
       using freeIn_elimVar_eq[of var L "[]" a]
       by(simp del:elimVar.simps)
     done
 qed
 
 
 
 lemma gen_qe_eval'' :
   assumes "all_degree_2 var L"
   assumes "length xs' = var"
   shows "(\<exists>x. (eval (list_conj (map Atom L)) (xs'@x#\<Gamma>))) = (\<forall>x.(eval (list_disj
                           (list_conj (map (substNegInfinity var) L) # map (elimVar var L []) L)) (xs'@x#\<Gamma>)))"
 proof(cases "convert_atom_list var L (xs'@x#\<Gamma>)")
   case None
   then show ?thesis using all_degree_2_convert[OF assms(1), of "(xs' @ x # \<Gamma>)"] by auto
 next
   case (Some a)
   then have Some : "\<And>x. convert_atom_list var L (xs'@x#\<Gamma>) = Some a"  using convert_atom_list_change[OF assms(2), of L x \<Gamma>]
     by fastforce
 
   show ?thesis
     apply (simp add: eval_list_conj eval_list_disj)
     using negInf_convert[OF Some assms(2)] elimVar_atom_convert[OF Some Some assms(2)] eval_convert[OF Some assms(2)]
     using eval_generalVS''[of a] unfolding eval_list_conj_Uni generalVS_DNF.simps eval_list_conj_Uni eval_list_disj_Uni eval_append eval_map eval_map_all
       evalUni.simps 
     by auto
 qed
 
 end
\ No newline at end of file
diff --git a/thys/Virtual_Substitution/InfinitesimalsUni.thy b/thys/Virtual_Substitution/InfinitesimalsUni.thy
--- a/thys/Virtual_Substitution/InfinitesimalsUni.thy
+++ b/thys/Virtual_Substitution/InfinitesimalsUni.thy
@@ -1,1301 +1,1301 @@
 theory InfinitesimalsUni
   imports Infinitesimals UniAtoms NegInfinityUni QE
 
 begin
 
 
 
 fun convertDerivativeUni ::  "real * real * real \<Rightarrow> atomUni fmUni" where
   "convertDerivativeUni (a,b,c) = 
   OrUni(AtomUni(LessUni(a,b,c)))(AndUni(AtomUni(EqUni(a,b,c)))(
     OrUni(AtomUni(LessUni(0,2*a,b)))(AndUni(AtomUni(EqUni(0,2*a,b)))(
       (AtomUni(LessUni(0,0,2*a)))
     ))
   ))
 "
 
 
 
 lemma convert_convertDerivative : 
   assumes "convert_poly var p (xs'@x#xs) = Some(a,b,c)"
   assumes "length xs' = var"
   shows "eval (convertDerivative var p) (xs'@x#xs) = evalUni (convertDerivativeUni (a,b,c)) x"
 proof(cases "MPoly_Type.degree p var = 0")
   case True
   then show ?thesis using assms apply (simp add: isovar_greater_degree eval_or eval_and insertion_mult insertion_const)
     using sum_over_zero[of p var] by auto
 next
   case False
   then have nonzero: "MPoly_Type.degree p var \<noteq> 0" by auto
   then show ?thesis proof(cases "MPoly_Type.degree p var = 1")
     case True
     have h1 : "MPoly_Type.degree p var < 3" using True by auto
     have h2 : "get_coeffs var p = (isolate_variable_sparse p var 2, isolate_variable_sparse p var 1, isolate_variable_sparse p var 0)" by auto
     have h : "insertion (nth_default 0 (xs' @ x # xs)) p = b * x + c"
       using poly_to_univar[OF h1 h2 _ _ _ assms(2), of  a x xs b c x] using assms(1) apply(cases "MPoly_Type.degree p var < 3") apply simp_all
       using isovar_greater_degree[of p var] unfolding True by simp
     have h3: "MPoly_Type.degree (isolate_variable_sparse p var (Suc 0) * Const 1) var = 0"
       using degree_mult[of "isolate_variable_sparse p var (Suc 0)" "Const 1" var]
       using degree_isovarspar mult_one_right by presburger
     show ?thesis
       using assms True
       unfolding convertDerivative.simps[of _ p] convertDerivative.simps[of _ "(derivative var p)"]
       apply (simp add: derivative_def isovar_greater_degree eval_or eval_and insertion_add insertion_mult insertion_const HOL.arg_cong[OF sum_over_zero[of p var], of "insertion (nth_default var (xs'@x#xs))"] insertion_var_zero del:convertDerivative.simps)    
       unfolding h     h3 by(simp del:convertDerivative.simps)
   next
     case False
     then have deg2 :  "MPoly_Type.degree p var = 2"
       by (metis One_nat_def assms(1) convert_poly.simps le_SucE less_Suc0 less_Suc_eq_le nonzero numeral_2_eq_2 numeral_3_eq_3 option.distinct(1)) 
     then have first : "isolate_variable_sparse p var (Suc (Suc 0)) \<noteq> 0"
       by (metis MPoly_Type.degree_zero isolate_variable_sparse_degree_eq_zero_iff nat.distinct(1) numeral_2_eq_2)
     have second : "(isolate_variable_sparse p var (Suc (Suc 0)) * Var var)\<noteq>0"
       by (metis MPoly_Type.degree_zero One_nat_def ExecutiblePolyProps.degree_one Zero_not_Suc first mult_eq_0_iff)
     have other : "Const (2::real)\<noteq>0"
       by (simp add: nonzero_const_is_nonzero)
     have thing: "(Var var::real mpoly) \<noteq> 0"
       using second by auto 
     have degree: "MPoly_Type.degree
                   (isolate_variable_sparse p var (Suc 0) * Const 1 +
                    isolate_variable_sparse p var (Suc (Suc 0)) * Var var * Const 2)
                   var -
                  Suc 0 = 0"
       apply simp apply(rule Nat.eq_imp_le) apply(rule degree_less_sum'[of _ _ 0])
       apply (simp add: degree_isovarspar mult_one_right)  apply auto
       unfolding degree_mult[OF second other, of var] degree_const apply simp
       unfolding degree_mult[OF first thing] degree_one
       using degree_isovarspar by force
     have insertion: "insertion (nth_default 0 (xs'@x#xs)) (\<Sum>(i::nat)\<le>2. isolate_variable_sparse p var i * Var var ^ i) = a * x^2 + b * x + c"
     proof(cases "MPoly_Type.degree p var = 1")
       case True
       then show ?thesis
         using False by blast
     next
       case False
       then have deg2 :  "MPoly_Type.degree p var = 2"
         by (metis One_nat_def assms(1) convert_poly.simps le_SucE less_Suc0 less_Suc_eq_le nonzero numeral_2_eq_2 numeral_3_eq_3 option.distinct(1)) 
       then have degless3 : "MPoly_Type.degree p var < 3" by auto
       have thing : "var<length(xs'@x # xs)" using assms by auto
       have h : "(\<Sum>i\<le>2. isolate_variable_sparse p var i * Var var ^ i) = p"
         using deg2
         by (metis sum_over_zero)
       have three: "(3::nat) = Suc(Suc(Suc(0)))" by auto
       have "(\<Sum>i = 0..<3. insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var i) * x ^ i) =
           (insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var 0) * x ^ 0) + 
           (insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var (1::nat)) * x ^ (1::nat)) + 
           (insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var (2::nat)) * x ^ (2::nat))"
         unfolding Set_Interval.comm_monoid_add_class.sum.atLeast0_lessThan_Suc three
       proof -
         have "insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var (MPoly_Type.degree p var)) * x ^ MPoly_Type.degree p var + (x * insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var 1) + (insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var 0) + (\<Sum>n = 0..<0. insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var n) * x ^ n))) = insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var 0) + x * insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var 1) + insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var (MPoly_Type.degree p var)) * x ^ MPoly_Type.degree p var"
           by auto
         then show "(\<Sum>n = 0..<0. insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var n) * x ^ n) + insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var 0) * x ^ 0 + insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var (Suc 0)) * x ^ Suc 0 + insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var (Suc (Suc 0))) * x ^ Suc (Suc 0) = insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var 0) * x ^ 0 + insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var 1) * x ^ 1 + insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var 2) * x\<^sup>2"
           by (metis (no_types) One_nat_def Suc_1 add.commute deg2 mult.commute mult.left_neutral power_0 power_one_right)
       qed
       also have "... =  a * x\<^sup>2 + b * x + c"
         unfolding Power.power_class.power.power_0 Power.monoid_mult_class.power_one_right Groups.monoid_mult_class.mult_1_right
         using assms unfolding convert_poly.simps using degless3 by simp
       finally have h2  :"(\<Sum>i = 0..<3. insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var i) * x ^ i) =  a * x\<^sup>2 + b * x + c "
         .
       show ?thesis using sum_over_degree_insertion[OF thing deg2, of x] apply auto  using h h2 using assms(2) by auto
     qed
     have insertionb: "insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var (Suc 0)) = b"
       using assms apply(cases "MPoly_Type.degree p var < 3") by simp_all
     have insertiona : "insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var (Suc (Suc 0))) = a" 
       using assms apply(cases "MPoly_Type.degree p var < 3")  apply simp_all
       by (simp add: numeral_2_eq_2)
     have x :  "insertion (nth_default 0 (xs' @ x # xs)) (Var var) = x" using insertion_var[of var "(xs' @ x # xs)" x] using assms(2) by auto
     have zero1 : "insertion (nth_default 0 (xs' @ x # xs))
      (isolate_variable_sparse (isolate_variable_sparse p var (Suc 0)) var (Suc 0)) = 0"
       by (simp add: degree_isovarspar isovar_greater_degree)
     have zero2 : "insertion (nth_default 0 (xs' @ x # xs))
       (isolate_variable_sparse (isolate_variable_sparse p var (Suc (Suc 0))) var 0) = a"
       using degree0isovarspar degree_isovarspar insertiona by presburger
     have zero3 : "insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse (Var var) var (Suc 0)) = 1" using isolate_var_one
       using MPoly_Type.insertion_one by fastforce
     have zero4 : "insertion (nth_default 0 (xs' @ x # xs))
       (isolate_variable_sparse (isolate_variable_sparse p var (Suc (Suc 0))) var (Suc 0)) = 0"
       by (simp add: degree_isovarspar isovar_greater_degree)
     have insertion_deriv : "insertion (nth_default 0 (xs'@x#xs))
        (isolate_variable_sparse
          (isolate_variable_sparse p var (Suc 0) * Const 1 +
           isolate_variable_sparse p var (Suc (Suc 0)) * Var var * Const 2)
          var (Suc 0)) = 2*a" 
       unfolding isovarspar_sum isolate_variable_sparse_mult apply auto
       unfolding const_lookup_suc const_lookup_zero Rings.mult_zero_class.mult_zero_right
         Groups.group_add_class.add.group_left_neutral
       unfolding insertion_add insertion_mult insertion_const apply auto
       unfolding zero1 zero2 zero3 zero4 by simp
     have deg2 : "MPoly_Type.degree p var = 2" using degree_convert_eq[OF assms(1)] False nonzero by auto
     then show ?thesis
       using assms 
       unfolding convertDerivative.simps[of _ p] convertDerivative.simps[of _ "(derivative var p)"] convertDerivative.simps[of _ "(derivative var (derivative var p))"]
       apply (simp add: x insertionb insertiona insertion_deriv insertion degree derivative_def isovar_greater_degree eval_or eval_and insertion_add insertion_mult insertion_const HOL.arg_cong[OF sum_over_zero[of p var], of "insertion (nth_default 0 (xs'@x#xs))"] insertion_var_zero del:convertDerivative.simps)
       by (smt (z3) One_nat_def degree_isovarspar distrib_right insertion_deriv isolate_variable_sparse_ne_zeroD mult_one_right neq0_conv not_one_le_zero zero1)
   qed
 qed
 
 fun linearSubstitutionUni :: "real \<Rightarrow> real \<Rightarrow> atomUni \<Rightarrow> atomUni fmUni" where
   "linearSubstitutionUni b c a = (if aEvalUni a (-c/b) then TrueFUni else FalseFUni)"
 
 lemma convert_linearSubstitutionUni: 
   assumes "convert_atom var a (xs'@x#xs) = Some(a')"
   assumes "insertion (nth_default 0 (xs'@x#xs)) b = B"
   assumes "insertion (nth_default 0 (xs'@x#xs)) c = C"
   assumes "B \<noteq> 0"
   assumes "var\<notin>(vars b)"
   assumes "var\<notin>(vars c)"
   assumes "length xs' = var"
   shows "aEval (linear_substitution var (-c) b a) (xs'@x#xs) = evalUni (linearSubstitutionUni B C a') x"
   using assms
 proof -
   have "aEval a (xs'@(-C/B)#xs) = evalUni (linearSubstitutionUni B C a') x"
     using assms(1) proof(cases "a")
     case (Less p)
     then have "MPoly_Type.degree p var < 3" using assms(1)apply(cases "MPoly_Type.degree p var < 3") by auto
     then show ?thesis
       using Less assms apply simp
       using poly_to_univar by force
   next
     case (Eq p)
     then have "MPoly_Type.degree p var < 3" using assms(1)apply(cases "MPoly_Type.degree p var < 3") by auto
     then show ?thesis
       using Eq assms apply simp
       using poly_to_univar by force
   next
     case (Leq p)
     then have "MPoly_Type.degree p var < 3" using assms(1)apply(cases "MPoly_Type.degree p var < 3") by auto
     then show ?thesis
       using Leq assms apply simp
       using poly_to_univar by force
   next
     case (Neq p)
     then have "MPoly_Type.degree p var < 3" using assms(1)apply(cases "MPoly_Type.degree p var < 3") by auto
     then show ?thesis
       using Neq assms apply simp
       using poly_to_univar by force
   qed
   then have subst : "aEval a ((xs'@x#xs)[var := - C / B]) = evalUni (linearSubstitutionUni B C a') x" using assms by auto
   have hlength : "var< length (xs'@x#xs)" using assms by auto
   have inB : "insertion (nth_default 0 ((xs'@x#xs)[var := - C / B])) b = B" using assms apply auto apply(cases a) apply auto
     by (simp add: insertion_lowerPoly1)+ 
   have inC : "insertion (nth_default 0 ((xs'@x#xs)[var := - C / B])) (-c) = -C" using assms apply auto apply(cases a) apply auto
     by (simp add: insertion_lowerPoly1 insertion_neg)+
   have freenegc : "var\<notin>vars(-c)" using assms not_in_neg by auto
   show ?thesis using linear[OF hlength assms(4)  freenegc assms(5) inC inB, of a ] subst
     using  var_not_in_eval3[OF var_not_in_linear[OF freenegc assms(5)] assms(7)]
     by (metis assms(7) list_update_length)
 qed
   (*
   substInfinitesimalLinear var b c A
   substitutes -c/b+epsilon for variable var in atom A
   assumes b is nonzero
   defined in page 615
 *)
 fun substInfinitesimalLinearUni :: "real \<Rightarrow> real \<Rightarrow> atomUni \<Rightarrow> atomUni fmUni" where
   "substInfinitesimalLinearUni b c (EqUni p) = allZero' p"|
   "substInfinitesimalLinearUni b c (LessUni p) = 
   map_atomUni (linearSubstitutionUni b c) (convertDerivativeUni p)"|
   "substInfinitesimalLinearUni b c (LeqUni p) = 
 OrUni
   (allZero' p)
   (map_atomUni (linearSubstitutionUni b c) (convertDerivativeUni p))"|
   "substInfinitesimalLinearUni b c (NeqUni p) = negUni (allZero' p)"
 
 
 lemma convert_linear_subst_fm :
   assumes "convert_atom var a (xs'@x#xs) = Some a'"
   assumes "insertion (nth_default 0 (xs'@x#xs)) b = B"
   assumes "insertion (nth_default 0 (xs'@x#xs)) c = C"
   assumes "B \<noteq> 0"
   assumes "var\<notin>(vars b)"
   assumes "var\<notin>(vars c)"
   assumes "length xs' = var"
   shows "aEval (linear_substitution (var + 0) (liftPoly 0 0 (-c)) (liftPoly 0 0 b) a) (xs'@x#xs) =
      evalUni (linearSubstitutionUni B C a') x"
 proof-
   have lb : "insertion (nth_default 0 (xs'@x#xs)) (liftPoly 0 0 b) = B" unfolding lift00 using assms(2) by auto
   have lc : "insertion (nth_default 0 (xs'@x#xs)) (liftPoly 0 0 c) = C" unfolding lift00 using assms(3) insertion_neg by auto
   have nb : "var \<notin> vars (liftPoly 0 0 b)" using not_in_lift[OF assms(5), of 0] by auto
   have nc : "var \<notin> vars (liftPoly 0 0 c)" using not_in_lift[OF assms(6)] not_in_neg
     using assms(6) lift00 by auto
   then show ?thesis using assms using lb lc convert_linearSubstitutionUni[OF assms(1)  lb lc assms(4) nb nc]
     by (simp add: lift00)
 qed
 
 lemma evalUni_if : "evalUni (if cond then TrueFUni else FalseFUni) x = cond"
   by(cases cond)(auto)
 
 lemma degree_less_sum' : "MPoly_Type.degree (p::real mpoly) var = n \<Longrightarrow> MPoly_Type.degree (q::real mpoly) var = m \<Longrightarrow> n < m \<Longrightarrow> MPoly_Type.degree (p + q) var = m"
   using degree_less_sum[of q var m p n]
   by (simp add: add.commute) 
 
 lemma convert_substInfinitesimalLinear_less : 
   assumes "convert_poly var p (xs'@x#xs) = Some(p')"
   assumes "insertion (nth_default 0 (xs'@x#xs)) b = B"
   assumes "insertion (nth_default 0 (xs'@x#xs)) c = C"
   assumes "B \<noteq> 0"
   assumes "var\<notin>(vars b)"
   assumes "var\<notin>(vars c)"
   assumes "length xs' = var"
   shows "
 eval (liftmap
     (\<lambda>x. \<lambda>A. Atom(linear_substitution (var+x) (liftPoly 0 x (-c)) (liftPoly 0 x b) A)) 
     (convertDerivative var p)
     0) (xs'@x#xs) =
 evalUni (map_atomUni (linearSubstitutionUni B C) (convertDerivativeUni p')) x"
 proof(cases p')
   case (fields a' b' c')
   have h : "convert_poly var p (xs'@x#xs) = Some (a', b', c')"
     using assms fields by auto
   have h2 : "\<exists>F'. convert_fm var (convertDerivative var p) xs = Some F'"
     unfolding convertDerivative.simps[of _ p] convertDerivative.simps[of _ "derivative var p"] convertDerivative.simps[of _ "derivative var (derivative var p)"]
     apply( auto simp del: convertDerivative.simps)
     using degree_convert_eq h apply blast
     using assms(1) degree_convert_eq apply blast
     apply (metis Suc_eq_plus1 degree_derivative gr_implies_not0 less_trans_Suc nat_neq_iff)
     using assms(1) degree_convert_eq apply blast
     apply (meson assms(1) degree_convert_eq)
     apply (metis One_nat_def Suc_eq_plus1 degree_derivative less_2_cases less_Suc_eq nat_neq_iff numeral_3_eq_3 one_add_one)
     using assms(1) degree_convert_eq apply blast
     using degree_derivative apply force
     using assms(1) degree_convert_eq apply blast
     apply (meson assms(1) degree_convert_eq)
     apply (metis degree_derivative eq_numeral_Suc less_add_one less_trans_Suc not_less_eq numeral_2_eq_2 pred_numeral_simps(3))
     apply (meson assms(1) degree_convert_eq)
     using degree_derivative apply fastforce
     by (meson assms(1) degree_convert_eq)
   have c'_insertion : "insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var 0) = c'"
     using assms fields unfolding convert_poly.simps apply(cases "MPoly_Type.degree p var < 3") by auto
   have b'_insertion : "insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var (Suc 0)) = b'"
     using assms fields unfolding convert_poly.simps apply(cases "MPoly_Type.degree p var < 3") by auto
   then have b'_insertion2 : "insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var 1) = b'"
     by auto
   have a'_insertion : "insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var 2) = a'"
     using assms fields unfolding convert_poly.simps apply(cases "MPoly_Type.degree p var < 3") by auto
   have liftb : "liftPoly 0 0 b = b" using lift00 by auto
   have liftc : "liftPoly 0 0 c = c" using lift00 by auto
   have b'_insertion' : "insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse (isolate_variable_sparse p var (Suc 0)) var 0) = b'"
     using assms fields unfolding convert_poly.simps apply(cases "MPoly_Type.degree p var < 3") apply auto
     by (simp add: degree0isovarspar degree_isovarspar)
   have insertion_into_1 : "insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse (Const 1) var 0) = 1"
     by (simp add: const_lookup_zero insertion_const)
   have twominusone : "((2-1)::nat) = 1" by auto
   show ?thesis
   proof(cases "MPoly_Type.degree p var = 0")
     case True
     have simp: "(convertDerivative var p)=Atom(Less p)"
       using True
       by auto
     have azero : "a'=0"
       by (metis MPoly_Type.insertion_zero True a'_insertion isolate_variable_sparse_ne_zeroD nat.simps(3) not_less numeral_2_eq_2 zero_less_iff_neq_zero)
     have bzero : "b'=0"
       using True b'_insertion isovar_greater_degree by fastforce
     show ?thesis unfolding fields substInfinitesimalLinearUni.simps
         convertDerivativeUni.simps linearSubstitutionUni.simps map_atomUni.simps evalUni.simps evalUni_if aEvalUni.simps
         Rings.mult_zero_class.mult_zero_left Rings.mult_zero_class.mult_zero_right Groups.add_0 azero bzero
         substInfinitesimalLinear.simps convertDerivative.simps[of _ p] True simp liftmap.simps 
         linear_substitution.simps
       apply (auto simp add:True) 
       unfolding c'_insertion by auto
   next
     case False
     then have degnot0 : "MPoly_Type.degree p var \<noteq> 0" by auto
     then show ?thesis
     proof(cases "MPoly_Type.degree p var = 1")
       case True
       then have simp : "convertDerivative var p = Or (fm.Atom (Less p)) (And (fm.Atom (Eq p)) (fm.Atom (Less (derivative var p))))"
         by (metis One_nat_def Suc_eq_plus1 add_right_imp_eq convertDerivative.simps degnot0 degree_derivative zero_less_one)
       have azero : "a'=0"
         by (metis MPoly_Type.insertion_zero One_nat_def True a'_insertion isovar_greater_degree lessI numeral_2_eq_2)
       have degderiv : "MPoly_Type.degree (isolate_variable_sparse p var (Suc 0) * Const 1) var = 0"
         using degree_mult
         by (simp add: degree_isovarspar mult_one_right) 
       show ?thesis
         unfolding fields substInfinitesimalLinearUni.simps
           convertDerivativeUni.simps linearSubstitutionUni.simps map_atomUni.simps evalUni.simps evalUni_if aEvalUni.simps
           Rings.mult_zero_class.mult_zero_left Rings.mult_zero_class.mult_zero_right Groups.add_0 azero
           substInfinitesimalLinear.simps True simp liftmap.simps 
           linear_substitution.simps eval_Or eval_And liftb liftc 
         apply auto
         unfolding derivative_def True insertion_sub insertion_mult c'_insertion b'_insertion assms lift00 apply auto
         unfolding insertion_sub insertion_mult c'_insertion b'_insertion assms lift00
         apply (smt diff_divide_eq_iff divide_less_0_iff mult_less_0_iff)
         apply (smt mult_imp_less_div_pos neg_less_divide_eq zero_le_mult_iff)
         using assms(4) mult.commute nonzero_mult_div_cancel_left
         apply smt
         unfolding degderiv apply auto
         unfolding isolate_variable_sparse_mult apply auto
         unfolding insertion_mult defer
         apply (smt assms(4) diff_divide_eq_iff divide_less_0_iff mult_less_0_iff)
         defer
         using assms(4) apply blast
         unfolding b'_insertion' insertion_into_1 apply auto
         by (smt assms(4) less_divide_eq mult_pos_neg2 no_zero_divisors zero_less_mult_pos)
     next
       case False
       then have degreetwo : "MPoly_Type.degree p var = 2" using degnot0
         by (metis One_nat_def degree_convert_eq h less_2_cases less_Suc_eq numeral_2_eq_2 numeral_3_eq_3) 
       have two : "(2::nat) = Suc(Suc 0)" by auto
       have sum : "(\<Sum>i = 0..<2. isolate_variable_sparse p var i * (- c) ^ i * b ^ (2 - i)) =
                   isolate_variable_sparse p var 0 * (- c) ^ 0 * b ^ (2 - 0) + isolate_variable_sparse p var 1 * (- c) ^ 1 * b ^ (2 - 1) "
         unfolding Set_Interval.comm_monoid_add_class.sum.atLeast0_lessThan_Suc two by auto
       have a : "isolate_variable_sparse p var (Suc (Suc 0)) \<noteq> 0"
         by (metis degnot0 degree_isovarspar degreetwo isolate_variable_sparse_degree_eq_zero_iff numeral_2_eq_2) 
       have b : "((Var var * Const 2) :: real mpoly) \<noteq> (0::real mpoly)"
         by (metis MPoly_Type.degree_zero ExecutiblePolyProps.degree_one mult_eq_0_iff nonzero_const_is_nonzero zero_neq_numeral zero_neq_one)
       have degreedeg1 : "MPoly_Type.degree
                (isolate_variable_sparse p var (Suc 0) * Const 1 +
                 isolate_variable_sparse p var (Suc (Suc 0)) * Var var * Const 2)
                var  = 1"
         apply(rule degree_less_sum'[where n ="0"])
         apply (simp add: degree_isovarspar mult_one_right) defer
         apply simp
         using degree_mult[OF a b, of var]
-        by (metis (no_types, hide_lams) ExecutiblePolyProps.degree_one add.left_neutral b degree_const degree_isovarspar degree_mult mult.commute mult_zero_class.mult_zero_right)
+        by (metis (no_types, opaque_lifting) ExecutiblePolyProps.degree_one add.left_neutral b degree_const degree_isovarspar degree_mult mult.commute mult_zero_class.mult_zero_right)
       have simp : "(convertDerivative var p) = Or (fm.Atom (Less p))
      (And (fm.Atom (Eq p))
        (Or (fm.Atom (Less (derivative var p)))
          (And (fm.Atom (Eq (derivative var p))) (fm.Atom (Less (derivative var (derivative var p)))))))"
         using degreetwo
         by (metis One_nat_def Suc_1 Suc_eq_plus1 add_diff_cancel_right' convertDerivative.simps degree_derivative neq0_conv zero_less_Suc)
       have a : "insertion (nth_default 0 (xs'@x#xs))
      (isolate_variable_sparse
        (isolate_variable_sparse p var (Suc 0) * Const 1 +
         isolate_variable_sparse p var (Suc (Suc 0)) * Var var * Const 2)
        var 0) = b'" unfolding isovarspar_sum isolate_variable_sparse_mult apply auto
         unfolding const_lookup_suc const_lookup_zero Rings.mult_zero_class.mult_zero_right
           Groups.group_add_class.add.group_left_neutral
         by (simp add: b'_insertion' isolate_var_0 mult_one_right)
       have b : "insertion (nth_default 0 (xs'@x#xs))
      (isolate_variable_sparse
        (isolate_variable_sparse p var (Suc 0) * Const 1 +
         isolate_variable_sparse p var (Suc (Suc 0)) * Var var * Const 2)
        var (Suc 0)) = 2 * a'"
         unfolding isovarspar_sum isolate_variable_sparse_mult apply auto
         unfolding const_lookup_suc const_lookup_zero Rings.mult_zero_class.mult_zero_right
           Groups.group_add_class.add.group_left_neutral
         unfolding insertion_add insertion_mult insertion_const
         by (metis MPoly_Type.insertion_one MPoly_Type.insertion_zero One_nat_def a'_insertion add.commute add.right_neutral degree0isovarspar degree_isovarspar isolate_var_one isovar_greater_degree mult.commute mult.right_neutral mult_zero_class.mult_zero_right numeral_2_eq_2 zero_less_one)
       have simp_insertion_blob : "insertion (nth_default 0 (xs'@x#xs))
      (isolate_variable_sparse
         (isolate_variable_sparse p var (Suc 0) * Const 1 +
          isolate_variable_sparse p var (Suc (Suc 0)) * Var var * Const 2)
         var 0 *
        b -
        isolate_variable_sparse
         (isolate_variable_sparse p var (Suc 0) * Const 1 +
          isolate_variable_sparse p var (Suc (Suc 0)) * Var var * Const 2)
         var (Suc 0) *
        c) = b' * B - 2 * a' * C"
         unfolding insertion_sub insertion_mult assms a b by auto
       have a : "isolate_variable_sparse
        (isolate_variable_sparse p var (Suc 0) * Const 1 +
         isolate_variable_sparse p var (Suc (Suc 0)) * Var var * Const 2)
        var (Suc 0) \<noteq> 0"
         by (metis MPoly_Type.degree_zero One_nat_def degreedeg1 isolate_variable_sparse_degree_eq_zero_iff zero_neq_one) 
       have b' : "(Const 1::real mpoly) \<noteq> 0"
         by (simp add: nonzero_const_is_nonzero)
       have degreeblob : "MPoly_Type.degree
                (isolate_variable_sparse
                  (isolate_variable_sparse p var (Suc 0) * Const 1 +
                   isolate_variable_sparse p var (Suc (Suc 0)) * Var var * Const 2)
                  var (Suc 0) *
                 Const 1)
                var = 0" 
         unfolding degree_mult[OF a b', of var]
         by (simp add: degree_isovarspar degree_eq_iff monomials_Const)
       have otherblob : "insertion (nth_default 0 (xs'@x#xs))
       (isolate_variable_sparse
         (isolate_variable_sparse
           (isolate_variable_sparse p var (Suc 0) * Const 1 +
            isolate_variable_sparse p var (Suc (Suc 0)) * Var var * Const 2)
           var (Suc 0) *
          Const 1)
         var 0) = 2 * a'" using b
         by (simp add: degree0isovarspar degree_isovarspar mult_one_right)
 
       have "(c' * B\<^sup>2 - b' * C * B + a' * C\<^sup>2 < 0) = ((c' * B\<^sup>2 - b' * C * B + a' * C\<^sup>2)/(B\<^sup>2) < 0)"
         by (simp add: assms(4) divide_less_0_iff)
       also have "... = (((c' * B\<^sup>2)/(B\<^sup>2) - (b' * C * B)/(B\<^sup>2) + (a' * C\<^sup>2)/(B\<^sup>2)) < 0)"
         by (metis (no_types, lifting) add_divide_distrib diff_divide_distrib )
       also have "... = (a' * (C / B)\<^sup>2 - b' * C / B + c' < 0)"
       proof -
         { assume "c' + a' * (C / B)\<^sup>2 - b' * (C / B) < 0"
           then have ?thesis
             by (simp add: assms(4) power2_eq_square) }
         moreover
         { assume "\<not> c' + a' * (C / B)\<^sup>2 - b' * (C / B) < 0"
           then have ?thesis
             by (simp add: power2_eq_square) }
         ultimately show ?thesis
           by fastforce
       qed
       finally have h1: "(c' * B\<^sup>2 - b' * C * B + a' * C\<^sup>2 < 0) = (a' * (C / B)\<^sup>2 - b' * C / B + c' < 0)"
         .
       have "(c' * B\<^sup>2 - b' * C * B + a' * C\<^sup>2 = 0) = ((c' * B\<^sup>2 - b' * C * B + a' * C\<^sup>2)/(B\<^sup>2) = 0)"
         by (simp add: assms(4))
       also have "... = (((c' * B\<^sup>2)/(B\<^sup>2) - (b' * C * B)/(B\<^sup>2) + (a' * C\<^sup>2)/(B\<^sup>2)) = 0)"
         by (metis (no_types, lifting) add_divide_distrib diff_divide_distrib )
       also have "... = (a' * (C / B)\<^sup>2 - b' * C / B + c' = 0)"
       proof -
         { assume "c' + a' * (C * (C / (B * B))) - b' * (C / B) \<noteq> 0"
           then have ?thesis
             by (simp add: assms(4) power2_eq_square) }
         moreover
         { assume "c' + a' * (C * (C / (B * B))) - b' * (C / B) = 0"
           then have ?thesis
             by (simp add: power2_eq_square) }
         ultimately show ?thesis
           by fastforce
       qed
       finally have h2 : "(c' * B\<^sup>2 - b' * C * B + a' * C\<^sup>2 = 0) = (a' * (C / B)\<^sup>2 - b' * C / B + c' = 0)"
         .
       have h3 : "((b' * B - 2 * a' * C) * B < 0) = (b' < 2 * a' * C / B)"
         by (smt assms(4) less_divide_eq zero_le_mult_iff)
       have h4 : "(b' * B = 2 * a' * C) = (b' = 2 * a' * C / B)"
         by (simp add: assms(4) nonzero_eq_divide_eq)
       show ?thesis unfolding fields substInfinitesimalLinearUni.simps
           convertDerivativeUni.simps linearSubstitutionUni.simps map_atomUni.simps evalUni.simps evalUni_if aEvalUni.simps
           Rings.mult_zero_class.mult_zero_left Rings.mult_zero_class.mult_zero_right Groups.add_0
           substInfinitesimalLinear.simps degreetwo simp liftmap.simps 
           linear_substitution.simps eval_Or eval_And liftb liftc 
         apply simp 
         unfolding derivative_def degreetwo insertion_sub insertion_mult c'_insertion b'_insertion assms  apply simp
         unfolding sum insertion_add insertion_mult insertion_pow insertion_neg assms
         unfolding b'_insertion2 c'_insertion a'_insertion
         unfolding Power.power_class.power.power_0  Groups.monoid_mult_class.mult_1_right
           Groups.cancel_comm_monoid_add_class.diff_zero Power.monoid_mult_class.power_one_right
           twominusone degreedeg1 apply simp
         unfolding insertion_mult assms simp_insertion_blob degreeblob 
         unfolding insertion_mult insertion_sub assms otherblob apply simp
         unfolding otherblob h1 h2 h3 h4 unfolding lift00 insertion_neg assms insertion_isovarspars_free insertion_sum insertion_mult insertion_sub degree0isovarspar degree_isovarspar mult_one_right insertion_sum_var insertion_pow insertion_neg sum
         unfolding assms b'_insertion c'_insertion a'_insertion insertion_neg insertion_mult insertion_add insertion_pow apply simp
         by (smt assms(2) assms(3) b'_insertion h1 h2 h3 h4 insertion_mult insertion_sub mult_one_right simp_insertion_blob)
     qed
   qed
 qed
 lemma convert_substInfinitesimalLinear: 
   assumes "convert_atom var a (xs'@x#xs) = Some(a')"
   assumes "insertion (nth_default 0 (xs'@x#xs)) b = B"
   assumes "insertion (nth_default 0 (xs'@x#xs)) c = C"
   assumes "B \<noteq> 0"
   assumes "var\<notin>(vars b)"
   assumes "var\<notin>(vars c)"
   assumes "length xs' = var"
   shows "eval (substInfinitesimalLinear var (-c) b a) (xs'@x#xs) = evalUni (substInfinitesimalLinearUni B C a') x"
   using assms
 proof(cases a)
   case (Less p)
   have "\<exists>p'. convert_poly var p (xs'@x#xs) = Some p'"
     using Less assms(1) apply(cases "MPoly_Type.degree p var < 3") by auto
   then obtain p' where p'_def : "convert_poly var p (xs'@x#xs) = Some p'" by auto
   have A'_simp :  "a' = LessUni p'"
     using assms Less
     using p'_def by auto 
   have h1 : "eval (convertDerivative var p) (xs'@x#xs) = evalUni (convertDerivativeUni p') x" using convert_convertDerivative
     apply ( cases p')
     using A'_simp Less assms by auto 
   show ?thesis unfolding A'_simp using convert_substInfinitesimalLinear_less[OF p'_def assms(2-7)] unfolding Less by auto
 next
   case (Eq p)
   define p' where "p' = (case convert_poly var p (xs'@x#xs) of Some p' \<Rightarrow> p')"
   have A'_simp :  "a' = EqUni p'"
     using assms Eq
     using p'_def by auto 
   show ?thesis
     unfolding Eq A'_simp substInfinitesimalLinear.simps substInfinitesimalLinearUni.simps
     using convert_allZero A'_simp Eq assms by auto
 next
   case (Leq p)
   have "\<exists>p'. convert_poly var p (xs' @ x # xs) = Some p'"
     using assms(1) unfolding Leq apply auto apply(cases "MPoly_Type.degree p var < 3") by auto
   then obtain p' where p'_def : "convert_poly var p (xs' @ x # xs) = Some p'" by metis
   have A'_simp :  "a' = LeqUni p'"
     using assms Leq
     using p'_def by auto 
   have h1 : "eval (convertDerivative var p) (xs'@x#xs) = evalUni (convertDerivativeUni p') x" using convert_convertDerivative
     apply(cases p')
     using A'_simp Leq assms by auto
   show ?thesis unfolding A'_simp Leq substInfinitesimalLinear.simps eval_Or substInfinitesimalLinearUni.simps evalUni.simps
     using convert_substInfinitesimalLinear_less[OF p'_def assms(2-7)]
       convert_allZero[OF p'_def assms(7)] by simp
 next
   case (Neq p)
   have "\<exists>p'. convert_poly var p (xs' @ x # xs) = Some p'"
     using assms(1) unfolding Neq apply auto apply(cases "MPoly_Type.degree p var < 3") by auto
   then obtain p' where p'_def : "convert_poly var p (xs' @ x # xs) = Some p'" by metis
   have A'_simp :  "a' = NeqUni p'"
     using assms Neq
     using p'_def by auto 
   show ?thesis
     unfolding Neq A'_simp substInfinitesimalLinear.simps substInfinitesimalLinearUni.simps
     using convert_allZero[OF p'_def assms(7)]
     by (metis A'_simp Neq assms(1) assms(7) convert_substNegInfinity eval.simps(6) eval_neg substNegInfinityUni.simps(4) substNegInfinity.simps(4))  
 qed
 
 
 lemma either_or:
   fixes r :: "real"
   assumes a: "(\<exists>y'>r. \<forall>x\<in>{r<..y'}.  (aEvalUni (EqUni (a, b, c)) x) \<or> (aEvalUni (LessUni (a, b, c)) x))"
   shows "(\<exists>y'>r. \<forall>x\<in>{r<..y'}.  (aEvalUni (EqUni (a, b, c)) x)) \<or> 
   (\<exists>y'>r. \<forall>x\<in>{r<..y'}.  (aEvalUni (LessUni (a, b, c)) x))"
 proof  (cases "a = 0 \<and> b = 0 \<and> c= 0")
   case True
   then have "(\<exists>y'>r. \<forall>x\<in>{r<..y'}.  (aEvalUni (EqUni (a, b, c)) x))"
     using assms by auto
   then show ?thesis
     by blast 
 next
   case False 
   then have noz: "a\<noteq>0 \<or> b\<noteq>0 \<or> c\<noteq>0" by auto
   obtain y1' where y1prop: "y1' > r \<and> (\<forall>x\<in>{r<..y1'}. (aEvalUni (EqUni (a, b, c)) x) \<or> (aEvalUni (LessUni (a, b, c)) x))"
     using a by auto
   obtain y2' where y2prop: "y2' > r \<and> (\<forall>x\<in>{r<..y2'}. a * x\<^sup>2 + b * x + c \<noteq> 0)"
     using noz nonzcoeffs[of a b c] by auto
   let ?y = "min y1' y2'"
   have ygt: "?y > r" using y1prop y2prop by auto
   have "\<forall>x\<in>{r<..?y}. (aEvalUni (LessUni (a, b, c)) x)"
     using y1prop y2prop greaterThanAtMost_iff
     by force 
   then show ?thesis using ygt
     by blast 
 qed
 
 lemma infinitesimal_linear'_helper :
   assumes at_is: "At = LessUni p \<or> At = EqUni p"
   assumes "B \<noteq> 0"
   shows "((\<exists>y'::real>-C/B. \<forall>x::real \<in>{-C/B<..y'}. aEvalUni At x)
       = evalUni (substInfinitesimalLinearUni B C At) x)"
 proof (cases "At = LessUni p")
   case True
   then have LessUni: "At = LessUni p" by auto
   then show ?thesis proof(cases p)
     case (fields a b c) 
     then show ?thesis 
       unfolding LessUni fields 
       using one_root_a_lt0[where r="C/B", where a= "a", where b="b",where c= "c"] apply(auto) 
       using continuity_lem_lt0_expanded[where a= "a", where b = "2 * a * C / B ", where c = "c"]  apply (auto) 
       using continuity_lem_gt0_expanded[where a = "a", where b = "2 * a * C / B",where c = "c", where r = "- (C / B)"] apply (auto) 
       apply (meson less_eq_real_def linorder_not_less) 
       using one_root_a_gt0[where r = "C/B", where a = "a", where b="b", where c="c"] apply (auto) 
       using continuity_lem_lt0_expanded[where a= "a", where b = "2 * a * C / B", where c= "c"]
       apply (auto)
       using continuity_lem_gt0_expanded[where a = "a", where b = "2 * a * C / B",where c = "c", where r = "- (C / B)"]
       apply (auto) apply (meson less_eq_real_def linorder_not_less) 
       using case_d1 apply (auto)
       using continuity_lem_lt0_expanded[where a= "a", where b = "b", where c= "c"]
       apply (auto)
       using continuity_lem_gt0_expanded[where a = "a", where b = "b",where c = "c", where r = "- (C / B)"]
       apply (auto) apply (meson less_eq_real_def linorder_not_less) 
       using case_d4 apply (auto) 
       using continuity_lem_lt0_expanded[where a= "a", where b = "b", where c= "c"]
       apply (auto)
       using continuity_lem_gt0_expanded[where a = "a", where b = "b",where c = "c", where r = "- (C / B)"]
       apply (auto)
       by (meson less_eq_real_def linorder_not_le) 
   qed    
 next
   case False
   then have EqUni: "At = EqUni p" using at_is by auto
   then show ?thesis proof(cases p)
     case (fields a b c)
     show ?thesis
       apply(auto simp add:EqUni fields) 
       using continuity_lem_eq0[where r= "-(C/B)"] apply blast
       using continuity_lem_eq0[where r= "-(C/B)"] apply blast
       using continuity_lem_eq0[where r= "-(C/B)"] apply blast
       using linordered_field_no_ub by blast
   qed
 qed 
 
 (* I assume substInfinitesimalLinearUni' was supposed to be substInfinitesimalLinearUni?*)
 lemma infinitesimal_linear' :
   assumes "B \<noteq> 0"
   shows "(\<exists>y'::real>-C/B. \<forall>x::real \<in>{-C/B<..y'}. aEvalUni At x)
       = evalUni (substInfinitesimalLinearUni B C At) x"
 proof(cases At)
   case (LessUni p) 
   then show ?thesis using infinitesimal_linear'_helper[of At p B C] assms by auto
 next
   case (EqUni p)
   then show ?thesis  using infinitesimal_linear'_helper[of At p B C] assms by (auto) 
 next
   case (LeqUni p)
   then show ?thesis proof(cases p)
     case (fields a b c) 
     have same: "\<forall>x. aEvalUni (LeqUni p) x = (aEvalUni (EqUni p) x) \<or> (aEvalUni (LessUni p) x)" 
       apply (simp add: fields)
       by force 
     have "\<And>a b c.
        At = LeqUni p \<Longrightarrow>
        p = (a, b, c) \<Longrightarrow>
        (\<exists>y'>- C / B. \<forall>x\<in>{- C / B<..y'}. aEvalUni At x) =
        evalUni (substInfinitesimalLinearUni B C At) x "
     proof - 
       fix a b c
       assume atis: "At = LeqUni p"
       assume p_is: " p = (a, b, c)"
       have s1: "(\<exists>y'>- C / B. \<forall>x\<in>{- C / B<..y'}. aEvalUni At x) = (\<exists>y'>- C / B. \<forall>x\<in>{- C / B<..y'}.  (aEvalUni (EqUni p) x) \<or> (aEvalUni (LessUni p) x))"
         using atis same aEvalUni.simps(2) aEvalUni.simps(3) fields less_eq_real_def
         by blast
       have s2: "... = (\<exists>y'>- C / B. \<forall>x\<in>{- C / B<..y'}.  (aEvalUni (EqUni p) x)) \<or> (\<exists>y'>- C / B. \<forall>x\<in>{- C / B<..y'}. (aEvalUni (LessUni p) x))"
         using either_or[where r = "-C/B"] p_is 
         by blast 
       have eq1: "(\<exists>y'>- C / B. \<forall>x\<in>{- C / B<..y'}.  (aEvalUni (EqUni p) x)) =  (evalUni (substInfinitesimalLinearUni B C (EqUni p)) x)"
         using infinitesimal_linear'_helper[where At = "EqUni p", where p = "p", where B = "B", where C= "C"] 
           assms by auto
       have eq2: "(\<exists>y'>- C / B. \<forall>x\<in>{- C / B<..y'}. (aEvalUni (LessUni p) x)) = (evalUni (substInfinitesimalLinearUni B C (LessUni p)) x)"
         using infinitesimal_linear'_helper[where At = "LessUni p", where p = "p", where B = "B", where C= "C"] 
           assms by auto
       have z1: "(\<exists>y'>- C / B. \<forall>x\<in>{- C / B<..y'}. aEvalUni At x) = ((evalUni (substInfinitesimalLinearUni B C (EqUni p)) x) \<or> (evalUni (substInfinitesimalLinearUni B C (LessUni p)) x))"
         using s1 s2 eq1 eq2  by auto
       have z2: "(evalUni (substInfinitesimalLinearUni B C (EqUni p)) x) \<or> (evalUni (substInfinitesimalLinearUni B C (LessUni p)) x) = evalUni (substInfinitesimalLinearUni B C (LeqUni p)) x" 
         by auto
       have z3: "(evalUni (substInfinitesimalLinearUni B C At) x) = evalUni (substInfinitesimalLinearUni B C (LeqUni p)) x"
         using LeqUni by auto
       then have z4: "(evalUni (substInfinitesimalLinearUni B C (EqUni p)) x) \<or> (evalUni (substInfinitesimalLinearUni B C (LessUni p)) x) = (evalUni (substInfinitesimalLinearUni B C At) x) "
         using z2 z3 by auto
       let ?a = "(evalUni (substInfinitesimalLinearUni B C (EqUni p)) x) \<or> (evalUni (substInfinitesimalLinearUni B C (LessUni p)) x)"
       let ?b = "(\<exists>y'>- C / B. \<forall>x\<in>{- C / B<..y'}. aEvalUni At x)"
       let ?c = "(evalUni (substInfinitesimalLinearUni B C At) x)"
       have t1: "?b = ?a" using z1 by auto
       have t2: "?a = ?c" using z4
         by (simp add: atis)
       then have "?b = ?c" using t1 t2 by auto
       then show "(\<exists>y'>- C / B. \<forall>x\<in>{- C / B<..y'}. aEvalUni At x) = evalUni (substInfinitesimalLinearUni B C At) x"
         by auto
     qed
     then show ?thesis
       using LeqUni fields by blast 
   qed
 next
   case (NeqUni p)
   then show ?thesis proof(cases p)
     case (fields a b c)
     then show ?thesis unfolding NeqUni fields using nonzcoeffs  by (auto) 
   qed
 qed
 
 fun quadraticSubUni :: "real \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real \<Rightarrow> atomUni \<Rightarrow> atomUni fmUni" where
   "quadraticSubUni a b c d A = (if aEvalUni A ((a+b*sqrt(c))/d) then TrueFUni else FalseFUni)"
 
 fun substInfinitesimalQuadraticUni :: "real \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real \<Rightarrow> atomUni \<Rightarrow> atomUni fmUni" where
   "substInfinitesimalQuadraticUni a b c d (EqUni p) = allZero' p"|
   "substInfinitesimalQuadraticUni a b c d (LessUni p) = map_atomUni (quadraticSubUni a b c d) (convertDerivativeUni p)"|
   "substInfinitesimalQuadraticUni a b c d (LeqUni p) = OrUni(map_atomUni (quadraticSubUni a b c d) (convertDerivativeUni p)) (allZero' p)"|
   "substInfinitesimalQuadraticUni a b c d (NeqUni p) = negUni (allZero' p)"
 
 
 lemma weird :
   fixes D::"real"
   assumes dneq: "D \<noteq> (0::real)"
   shows 
     "((a'::real) * (((A::real) + (B::real) * sqrt (C::real)) / (D::real))\<^sup>2 + (b'::real) * (A + B * sqrt C) / D + c' < 0 \<or>
      a' * ((A + B * sqrt C) / D)\<^sup>2 + b' * (A + B * sqrt C) / D + (c'::real) = 0 \<and>
      (b' + a' * (A + B * sqrt C) * 2 / D < 0 \<or>
       b' + a' * (A + B * sqrt C) * 2 / D = 0 \<and> 2 * a' < 0)) =
     (a' * ((A + B * sqrt C) / D)\<^sup>2 + b' * (A + B * sqrt C) / D + c' < 0 \<or>
      a' * ((A + B * sqrt C) / D)\<^sup>2 + b' * (A + B * sqrt C) / D + c' = 0 \<and>
      (2 * a' * (A + B * sqrt C) / D + b' < 0 \<or>
       2 * a' * (A + B * sqrt C) / D + b' = 0 \<and> a' < 0))"
 proof (cases "(a' * ((A + B * sqrt C) / D)\<^sup>2 + b' * (A + B * sqrt C) / D + c' < 0)")
   case True
   then show ?thesis
     by auto
 next
   case False
   have "a' * (A + B * sqrt C) * 2 = 2 * a' * (A + B * sqrt C)" by auto
   then have "a' * (A + B * sqrt C) * 2 / D =2 * a' * (A + B * sqrt C) / D "
     using dneq by simp 
   then have "b' + a' * (A + B * sqrt C) * 2 / D = 2 * a' * (A + B * sqrt C) / D + b'"
     using add.commute by simp
   then have "(b' + a' * (A + B * sqrt C) * 2 / D < 0 \<or> b' + a' * (A + B * sqrt C) * 2 / D = 0 \<and> a' < 0)
    = (2 * a' * (A + B * sqrt C) / D + b' < 0 \<or> 2 * a' * (A + B * sqrt C) / D + b' = 0 \<and> a' < 0)"
     by (simp add: \<open>b' + a' * (A + B * sqrt C) * 2 / D = 2 * a' * (A + B * sqrt C) / D + b'\<close>)
   then have "(a' * ((A + B * sqrt C) / D)\<^sup>2 + b' * (A + B * sqrt C) / D + c' = 0 \<and>
      (b' + a' * (A + B * sqrt C) * 2 / D < 0 \<or> b' + a' * (A + B * sqrt C) * 2 / D = 0 \<and> a' < 0)) =
     (a' * ((A + B * sqrt C) / D)\<^sup>2 + b' * (A + B * sqrt C) / D + c' = 0 \<and>
      (2 * a' * (A + B * sqrt C) / D + b' < 0 \<or> 2 * a' * (A + B * sqrt C) / D + b' = 0 \<and> a' < 0))"
     by blast
   then show ?thesis using False by simp
 qed 
 
 lemma convert_substInfinitesimalQuadratic_less :
   assumes "convert_poly var p (xs'@x#xs) = Some p'"
   assumes "insertion (nth_default 0 (xs'@x#xs)) a = A"
   assumes "insertion (nth_default 0 (xs'@x#xs)) b = B"
   assumes "insertion (nth_default 0 (xs'@x#xs)) c = C"
   assumes "insertion (nth_default 0 (xs'@x#xs)) d = D"
   assumes "D \<noteq> 0"
   assumes "0 \<le> C"
   assumes "var\<notin>(vars a)"
   assumes "var\<notin>(vars b)"
   assumes "var\<notin>(vars c)"
   assumes "var\<notin>(vars d)"
   assumes "length xs' = var"
   shows "eval (quadratic_sub_fm var a b c d (convertDerivative var p)) (xs'@x#xs) = evalUni (map_atomUni (quadraticSubUni A B C D) (convertDerivativeUni p')) x"
 proof(cases p')
   case (fields a' b' c')
   have h : "convert_poly var p (xs'@x#xs) = Some (a', b', c')"
     using assms fields by auto
   have h2 : "\<exists>F'. convert_fm var (convertDerivative var p) (xs'@x#xs) = Some F'"
     unfolding convertDerivative.simps[of _ p] convertDerivative.simps[of _ "derivative var p"] convertDerivative.simps[of _ "derivative var (derivative var p)"]
     apply (auto simp del: convertDerivative.simps)
     using degree_convert_eq h apply blast
     using assms(1) degree_convert_eq apply blast
     using degree_derivative apply fastforce
     apply (metis degree_convert_eq h   numeral_3_eq_3 )
     apply (metis (no_types, lifting) One_nat_def add.right_neutral add_Suc_right degree_derivative less_Suc_eq_0_disj less_Suc_eq_le neq0_conv numeral_3_eq_3)
     apply (metis One_nat_def Suc_eq_plus1 degree_derivative less_2_cases less_Suc_eq nat_neq_iff numeral_3_eq_3 one_add_one)
     apply (meson assms(1) degree_convert_eq)
     using degree_derivative apply fastforce
     using assms(1) degree_convert_eq apply blast
     apply (meson assms(1) degree_convert_eq)
     apply (metis degree_derivative less_Suc_eq less_add_one not_less_eq numeral_3_eq_3)
     apply (meson assms(1) degree_convert_eq)
-    apply (metis (no_types, hide_lams) Suc_1 Suc_eq_plus1 degree_derivative less_2_cases less_Suc_eq numeral_3_eq_3)
+    apply (metis (no_types, opaque_lifting) Suc_1 Suc_eq_plus1 degree_derivative less_2_cases less_Suc_eq numeral_3_eq_3)
     using assms(1) degree_convert_eq by blast
   have c'_insertion : "insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var 0) = c'"
     using assms fields unfolding convert_poly.simps apply(cases "MPoly_Type.degree p var < 3") by auto
   then have c'_insertion'' : "\<And>x. insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 0) = c'"
     using assms(12) not_in_isovarspar[of p var 0 "isolate_variable_sparse p var 0", OF HOL.refl]
     by (metis list_update_length not_contains_insertion)
   have b'_insertion : "insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var (Suc 0)) = b'"
     using assms fields unfolding convert_poly.simps apply(cases "MPoly_Type.degree p var < 3") by auto
   then have b'_insertion'' : "\<And>x. insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var (Suc 0)) = b'"
     using assms(12) not_in_isovarspar[of p var "Suc 0" "isolate_variable_sparse p var (Suc 0)", OF HOL.refl]
     by (metis list_update_length not_contains_insertion)
   then have b'_insertion2 : "insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var 1) = b'"
     by auto
   have a'_insertion : "insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse p var 2) = a'"
     using assms fields unfolding convert_poly.simps apply(cases "MPoly_Type.degree p var < 3") by auto
   then have a'_insertion'' : "\<And>x. insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 2) = a'"
     using assms(12) not_in_isovarspar[of p var 2 "isolate_variable_sparse p var 2", OF HOL.refl]
     by (metis list_update_length not_contains_insertion)
   have liftb : "liftPoly 0 0 b = b" using lift00 by auto
   have liftc : "liftPoly 0 0 c = c" using lift00 by auto
   have b'_insertion' : "insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse (isolate_variable_sparse p var (Suc 0)) var 0) = b'"
     using assms fields unfolding convert_poly.simps apply(cases "MPoly_Type.degree p var < 3") apply auto
     using degree0isovarspar degree_isovarspar by auto
   have insertion_into_1 : "insertion (nth_default 0 (xs'@x#xs)) (isolate_variable_sparse (Const 1) var 0) = 1"
     by (simp add: const_lookup_zero insertion_const)
   have twominusone : "((2-1)::nat) = 1" by auto
   have length0 : "var < length (xs'@x#xs)" using assms by auto
   have altinserta : "\<forall>xa. insertion (nth_default 0 ((xs'@x#xs)[var := xa])) a = A"
     using assms by (metis list_update_length not_contains_insertion)
   have altinserta' : "\<And>xa. insertion (nth_default 0 ((xs'@x#xs)[var := xa])) a = A"
     using assms by (metis list_update_length not_contains_insertion)
   have altinsertb : "\<forall>xa. insertion (nth_default 0 ((xs'@x#xs)[var := xa])) b = B"
     using assms by (metis list_update_length not_contains_insertion)
   have altinsertb' : "\<And>xa. insertion (nth_default 0 ((xs'@x#xs)[var := xa])) b = B"
     using assms by (metis list_update_length not_contains_insertion)
   have altinsertc : "\<forall>xa. insertion (nth_default 0 ((xs'@x#xs)[var := xa])) c = C"
     using assms by (metis list_update_length not_contains_insertion)
   have altinsertc' : "\<And>xa. insertion (nth_default 0 ((xs'@x#xs)[var := xa])) c = C"
     using assms by (metis list_update_length not_contains_insertion)
   have altinsertd : "\<forall>xa. insertion (nth_default 0 ((xs'@x#xs)[var := xa])) d = D" 
     using assms by (metis list_update_length not_contains_insertion)
   have altinsertd' : "\<And>xa. insertion (nth_default 0 ((xs'@x#xs)[var := xa])) d = D" 
     using assms by (metis list_update_length not_contains_insertion)
   have freeInQuadraticSub : "\<forall>At. eval (quadratic_sub var a b c d At) ((xs'@x#xs)[var := sqrt C]) = eval (quadratic_sub var a b c d At) ((xs'@x#xs))"
     by (metis assms(10) assms(11) assms(8) assms(9) free_in_quad list_update_id var_not_in_eval)
   have quad : "\<And>At. (eval (quadratic_sub var a b c d At) (xs'@x#xs) =
   aEval At ((xs'@x#xs)[var := (A + B * sqrt C) / D]))"
     using quadratic_sub[OF length0 assms(6-7) assms(10) altinserta altinsertb altinsertc altinsertd, symmetric]
     using freeInQuadraticSub  by auto 
   show ?thesis
   proof(cases "MPoly_Type.degree p var = 0")
     case True
     then have simp: "(convertDerivative var p)=Atom(Less p)"
       by auto
     have azero : "a'=0"
       by (metis MPoly_Type.insertion_zero True a'_insertion isolate_variable_sparse_ne_zeroD nat.simps(3) not_less numeral_2_eq_2 zero_less_iff_neq_zero)
     have bzero : "b'=0"
       using True b'_insertion isovar_greater_degree by fastforce
     define p1 where "p1 = isolate_variable_sparse p var 0"
     have degree_p1: "MPoly_Type.degree p1 var = 0" unfolding p1_def
       by (simp add: degree_isovarspar)
     define p2 where "p2 = isolate_variable_sparse p1 var 0 * Const 0 * Var var + isolate_variable_sparse p1 var 0 * Const 1"
     define A where "A = isolate_variable_sparse p2 var 0"
     define B where "B = isolate_variable_sparse p2 var (Suc 0)"
     show ?thesis
       unfolding substInfinitesimalQuadratic.simps substInfinitesimalQuadraticUni.simps
         fields
         convertDerivativeUni.simps map_atomUni.simps quadraticSubUni.simps aEvalUni.simps evalUni.simps evalUni_if
         Rings.mult_zero_class.mult_zero_left Groups.add_0 Rings.mult_zero_class.mult_zero_right
         True simp azero bzero 
         quadratic_sub_fm.simps quadratic_sub_fm_helper.simps liftmap.simps lift00  
         quad aEval.simps
       apply (simp add:True c'_insertion p1_def[symmetric] degree_p1 p2_def[symmetric] A_def[symmetric] B_def[symmetric]) 
       unfolding A_def B_def p2_def p1_def  degree0isovarspar[OF True] isovarspar_sum mult_one_right mult_zero_right mult_zero_left const_lookup_zero const_lookup_suc
       apply simp
       unfolding insertion_add insertion_sub insertion_mult insertion_pow insertion_const c'_insertion apply simp
       using \<open>isolate_variable_sparse p var 0 = p\<close> b'_insertion2 bzero c'_insertion by force
   next
     case False
     then have degreenonzero : "MPoly_Type.degree p var \<noteq>0" by auto
     show ?thesis
     proof(cases "MPoly_Type.degree p var = 1")
       case True
       then have simp : "convertDerivative var p = Or (fm.Atom (Less p)) (And (fm.Atom (Eq p)) (fm.Atom (Less (derivative var p))))"
         by (metis One_nat_def Suc_eq_plus1 add_right_imp_eq convertDerivative.simps degree_derivative degreenonzero less_numeral_extra(1))
       have azero : "a'=0"
         by (metis MPoly_Type.insertion_zero One_nat_def True a'_insertion isovar_greater_degree lessI numeral_2_eq_2)
       have degderiv : "MPoly_Type.degree (isolate_variable_sparse p var (Suc 0) * Const 1) var = 0"
         using degree_mult
         by (simp add: degree_isovarspar mult_one_right)
       have thing : "var<length (xs'@((A + B * sqrt C) / D # xs))" using assms by auto
       have insertp : "insertion (nth_default 0 (xs'@((A + B * sqrt C) / D # xs))) p = b' * (A + B * sqrt C) / D + c'"
         using sum_over_degree_insertion[OF thing True, of "(A + B * sqrt C) / D", symmetric] unfolding list_update_length  assms(12)[symmetric] apply simp
         unfolding assms(12) unfolding c'_insertion'' b'_insertion''  by auto
       have insertb : "insertion (nth_default 0 (xs'@(A + B * sqrt C) / D # xs))
       (isolate_variable_sparse p var (Suc 0) * Const 1) = b'"
         unfolding insertion_mult insertion_const b'_insertion'' by simp
       show ?thesis
         unfolding substInfinitesimalQuadratic.simps substInfinitesimalQuadraticUni.simps
           fields
           convertDerivativeUni.simps map_atomUni.simps quadraticSubUni.simps aEvalUni.simps evalUni.simps evalUni_if
           Rings.mult_zero_class.mult_zero_left Groups.add_0 Rings.mult_zero_class.mult_zero_right
           True simp azero 
           quadratic_sub_fm.simps quadratic_sub_fm_helper.simps liftmap.simps lift00  
           quad aEval.simps eval.simps derivative_def Groups.monoid_add_class.add_0_right
         apply simp
         unfolding insertp insertb insertion_mult insertion_const
         using assms(12) b'_insertion'' insertp by force
     next
       case False
       then have degree2 : "MPoly_Type.degree p var = 2" using degreenonzero
         using h less_Suc_eq by fastforce 
       have simp : "(convertDerivative var p) = Or (fm.Atom (Less p))
      (And (fm.Atom (Eq p))
        (Or (fm.Atom (Less (derivative var p)))
          (And (fm.Atom (Eq (derivative var p))) (fm.Atom (Less (derivative var (derivative var p)))))))"
         by (metis One_nat_def Suc_eq_plus1 add_diff_cancel_right' convertDerivative.simps degree2 degree_derivative degreenonzero neq0_conv one_add_one)
       have insertionp : "var < length (xs'@(A + B * sqrt C) / D # xs)" using assms by auto
       have three : "3 = Suc(Suc(Suc(0)))" by auto
       have two : "2 = Suc(Suc(0))" by auto
       have insertionp : "insertion (nth_default 0 ((xs'@x # xs)[var := (A + B * sqrt C) / D])) p = a' * ((A + B * sqrt C) / D)\<^sup>2 + b' * (A + B * sqrt C) / D + c'"
         using sum_over_degree_insertion[OF insertionp degree2, of "(A + B * sqrt C) / D", symmetric] unfolding  
           a'_insertion[symmetric] b'_insertion[symmetric] c'_insertion[symmetric] 
           insertion_isovarspars_free[of _ _ "(A + B * sqrt C) / D" _ _ x]
         unfolding two apply simp
         using assms(12) by force
       have insertion_simp : "insertion (nth_default 0 ((xs' @ x # xs)[var := (A + B * sqrt C) / D])) = insertion (nth_default 0 ((xs' @ ((A + B * sqrt C) / D) # xs)))"
         using assms
         by (metis list_update_length) 
       have degreeone : "MPoly_Type.degree
                   (isolate_variable_sparse p var (Suc 0) * Const 1 +
                    isolate_variable_sparse p var (Suc (Suc 0)) * Var var * Const 2)
                   var = 1"
         apply(rule degree_less_sum'[where n=0])
         apply (simp add: degree_isovarspar mult_one_right)
         apply (smt One_nat_def ExecutiblePolyProps.degree_one degree2 degree_const degree_isovarspar degree_mult degreenonzero isolate_variable_sparse_degree_eq_zero_iff mult.commute nonzero_const_is_nonzero numeral_2_eq_2 plus_1_eq_Suc)
         by simp
       have zero1 : " insertion (nth_default 0 (xs' @ (A + B * sqrt C) / D # xs))
      (isolate_variable_sparse (isolate_variable_sparse p var (Suc 0)) var (Suc 0)) = 0"
         by (simp add: degree_isovarspar isovar_greater_degree) 
       have zero2 : "insertion (nth_default 0 (xs' @ (A + B * sqrt C) / D # xs))
       (isolate_variable_sparse (isolate_variable_sparse p var (Suc (Suc 0))) var 0) = a'"
         using a'_insertion'' degree0isovarspar degree_isovarspar numeral_2_eq_2 by force
       have zero3 : "insertion (nth_default 0 (xs' @ (A + B * sqrt C) / D # xs)) (isolate_variable_sparse (Var var) var (Suc 0)) = 1"
         using isolate_var_one by fastforce
       have zero4 : "insertion (nth_default 0 (xs' @ (A + B * sqrt C) / D # xs))
       (isolate_variable_sparse (isolate_variable_sparse p var (Suc (Suc 0))) var (Suc 0)) = 0"
         by (simp add: degree_isovarspar isovar_greater_degree)
       have insertiona' : " insertion (nth_default 0 (xs' @ (A + B * sqrt C) / D # xs))
        (isolate_variable_sparse
          (isolate_variable_sparse p var (Suc 0) * Const 1 +
           isolate_variable_sparse p var (Suc (Suc 0)) * Var var * Const 2)
          var (Suc 0) *
         Const 1) = 2 * a'"
         unfolding isovarspar_sum isolate_variable_sparse_mult apply auto
         unfolding const_lookup_suc const_lookup_zero Rings.mult_zero_class.mult_zero_right
           Groups.group_add_class.add.group_left_neutral
         unfolding insertion_add insertion_mult insertion_const b'_insertion' apply auto
         unfolding zero1 zero2 zero3 zero4 by auto
       have a' :  "insertion (nth_default 0 (xs' @ (A + B * sqrt C) / D # xs)) (isolate_variable_sparse p var (Suc (Suc 0))) = a'"
         unfolding two[symmetric] unfolding a'_insertion'' by auto
       have var: "insertion (nth_default 0 (xs' @ (A + B * sqrt C) / D # xs)) (Var var) = (A + B * sqrt C) / D" using assms
         by (metis insertion_simp insertion_var length0)
       show ?thesis
         unfolding substInfinitesimalQuadratic.simps substInfinitesimalQuadraticUni.simps
           fields
           convertDerivativeUni.simps map_atomUni.simps quadraticSubUni.simps aEvalUni.simps evalUni.simps evalUni_if
           Rings.mult_zero_class.mult_zero_left Groups.add_0 Rings.mult_zero_class.mult_zero_right
           degree2 simp
           quadratic_sub_fm.simps quadratic_sub_fm_helper.simps liftmap.simps lift00   Groups.monoid_add_class.add_0_right
           quad aEval.simps eval.simps derivative_def apply (simp add:insertion_sum insertion_add insertion_mult insertion_const insertion_var_zero)
         unfolding insertionp 
         unfolding insertion_simp
         unfolding b'_insertion'' a'_insertion'' 
         unfolding 
           degreeone apply simp
         unfolding a' var
         unfolding insertiona'
         using weird[OF assms(6)] by auto
     qed
   qed
 qed
 
 lemma convert_substInfinitesimalQuadratic: 
   assumes "convert_atom var At (xs'@ x#xs) = Some(At')"
   assumes "insertion (nth_default 0 (xs'@ x#xs)) a = A"
   assumes "insertion (nth_default 0 (xs'@ x#xs)) b = B"
   assumes "insertion (nth_default 0 (xs'@ x#xs)) c = C"
   assumes "insertion (nth_default 0 (xs'@ x#xs)) d = D"
   assumes "D \<noteq> 0"
   assumes "0 \<le> C"
   assumes "var\<notin>(vars a)"
   assumes "var\<notin>(vars b)"
   assumes "var\<notin>(vars c)"
   assumes "var\<notin>(vars d)"
   assumes "length xs' = var"
   shows "eval (substInfinitesimalQuadratic var a b c d At) (xs'@ x#xs) = evalUni (substInfinitesimalQuadraticUni A B C D At') x"
   using assms
 proof(cases At)
   case (Less p)
   define p' where "p' = (case convert_poly var p (xs'@ x#xs) of Some p' \<Rightarrow> p')"
   have At'_simp :  "At' = LessUni p'"
     using assms Less
     using p'_def by auto 
   show ?thesis 
     using convert_substInfinitesimalQuadratic_less[OF _ assms(2-12)]
     by (metis At'_simp Less None_eq_map_option_iff assms(1) convert_atom.simps(1) option.distinct(1) option.exhaust_sel option.the_def p'_def substInfinitesimalQuadraticUni.simps(2) substInfinitesimalQuadratic.simps(2))
 next
   case (Eq p)
   define p' where "p' = (case convert_poly var p (xs'@ x#xs) of Some p' \<Rightarrow> p')"
   have At'_simp :  "At' = EqUni p'"
     using assms Eq
     using p'_def by auto 
   show ?thesis 
     unfolding At'_simp Eq  substInfinitesimalQuadraticUni.simps substInfinitesimalQuadratic.simps
     using At'_simp Eq assms(1) convert_substNegInfinity assms(12) by fastforce
 next
   case (Leq p)
   define p' where "p' = (case convert_poly var p (xs'@ x#xs) of Some p' \<Rightarrow> p')"
   have At'_simp :  "At' = LeqUni p'"
     using assms Leq
     using p'_def by auto 
   have allzero : "eval (allZero p var) (xs'@ x#xs) = evalUni (allZero' p') x"
     using Leq assms(1) convert_allZero p'_def assms(12) by force
   have less : "eval (quadratic_sub_fm var a b c d (convertDerivative var p)) (xs'@ x#xs) = evalUni (map_atomUni (quadraticSubUni A B C D) (convertDerivativeUni p')) x"
     using convert_substInfinitesimalQuadratic_less[OF _ assms(2-12)]
     by (metis Leq assms(1) convert_atom.simps(3) option.distinct(1) option.exhaust_sel option.map(1) option.the_def p'_def)
   show ?thesis 
     unfolding At'_simp Leq substInfinitesimalQuadraticUni.simps substInfinitesimalQuadratic.simps
       eval.simps evalUni.simps
     using allzero less by auto
 next
   case (Neq p)
   define p' where "p' = (case convert_poly var p (xs'@ x#xs) of Some p' \<Rightarrow> p')"
   have At'_simp :  "At' = NeqUni p'"
     using assms Neq
     using p'_def by auto 
   show ?thesis 
     unfolding At'_simp Neq substInfinitesimalQuadraticUni.simps substInfinitesimalQuadratic.simps
     by (metis assms(12) At'_simp Neq assms(1) convert_substNegInfinity eval.simps(6) eval_neg substNegInfinityUni.simps(4) substNegInfinity.simps(4))
 qed
 
 lemma infinitesimal_quad_helper:
   fixes A B C D:: "real"
   assumes at_is: "At = LessUni p \<or> At = EqUni p"
   assumes "D\<noteq>0"
   assumes "C\<ge>0"
   shows "(\<exists>y'::real>((A+B * sqrt(C))/(D)). \<forall>x::real \<in>{((A+B * sqrt(C))/(D))<..y'}. aEvalUni At x)
       = (evalUni (substInfinitesimalQuadraticUni A B C D At) x)"
 proof(cases "At = LessUni p")
   case True
   then have LessUni: "At = LessUni p" by auto
   then show ?thesis proof(cases p)
     case (fields a b c)
     show ?thesis 
     proof(cases "2 * (a::real) * (A + B * sqrt C) / D + b = 0")
       case True
       then have True1 : "2 * a * (A + B * sqrt C) / D + b = 0" by auto
       show ?thesis proof(cases "a * ((A + B * sqrt C) / D)\<^sup>2 + b * (A + B * sqrt C) / D + c = 0")
         case True
         then have True2 : "a * ((A + B * sqrt C) / D)\<^sup>2 + b * (A + B * sqrt C) / D + c = 0" by auto
         then show ?thesis proof(cases "a<0")
           case True
           then show ?thesis unfolding LessUni fields apply (simp add:True1 True2 True)
             using True1 True2 True  
           proof - 
             assume beq: "2 * a * (A + B * sqrt C) / D + b = 0"
             assume root: "a * ((A + B * sqrt C) / D)\<^sup>2 + b * (A + B * sqrt C) / D + c = 0"
             assume alt: "a < 0 "
             let ?r = "-((A + B * sqrt C) / D)"
             have beq_var: "b = 2 * a * ?r" using beq
               by auto 
             have root_var: " a * ?r^2 - 2*a*?r*?r + c = 0" using root
               by (simp add: beq_var)
             have "\<exists>y'>- ?r. \<forall>x\<in>{- ?r<..y'}. a * x\<^sup>2 + 2 * a *?r * x + c < 0" 
               using beq_var root_var alt one_root_a_lt0[where a = "a", where b="b", where c="c", where r="?r"]
               by auto
             then show "\<exists>y'>(A + B * sqrt C) / D. \<forall>x\<in>{(A + B * sqrt C) / D<..y'}. a * x\<^sup>2 + b * x + c < 0"
               using beq_var by auto
           qed
         next
           case False
           then show ?thesis unfolding LessUni fields apply (simp add:True1 True2 False)
             using True1 True2 False 
           proof clarsimp 
             fix y'
             assume beq: " 2 * a * (A + B * sqrt C) / D + b = 0"
             assume root: " a * ((A + B * sqrt C) / D)\<^sup>2 + b * (A + B * sqrt C) / D + c = 0"
             assume agteq: "\<not> a < 0 "
             assume y_prop: "(A + B * sqrt C) / D < y'"
             have beq_var: "b = 2 * a * (- A - B * sqrt C) / D" using beq
-              by (metis (no_types, hide_lams) ab_group_add_class.ab_diff_conv_add_uminus add.left_neutral add_diff_cancel_left' divide_inverse mult.commute mult_minus_right)             
+              by (metis (no_types, opaque_lifting) ab_group_add_class.ab_diff_conv_add_uminus add.left_neutral add_diff_cancel_left' divide_inverse mult.commute mult_minus_right)             
             have root_var: " a * ((- A - B * sqrt C) / D)\<^sup>2 - 2 * a * (- A - B * sqrt C) * (- A - B * sqrt C) / (D * D) + c =  0"
               using root
             proof -
               have f1: "\<And>r ra. - ((r::real) + ra) = - r - ra"
                 by auto
               have f2: "\<And>r ra. r * (a * 2 * (- A - B * sqrt C)) / (ra * D) = r / (ra / b)"
                 by (simp add: beq_var)
               have f3: "c - 0 + a * ((A + B * sqrt C) / D)\<^sup>2 = - (b * (A + B * sqrt C) / D)"
                 using root by force
               have f4: "\<And>r ra rb. ((- (r::real) - ra) / rb)\<^sup>2 = ((r + ra) / rb)\<^sup>2"
                 using f1 by (metis (no_types) divide_minus_left power2_minus)
               have "\<And>r ra rb rc. - ((r::real) * (ra + rb) / rc) = r * (- ra - rb) / rc"
                 using f1 by (metis divide_divide_eq_right divide_minus_left mult.commute)
               then show ?thesis
                 using f4 f3 f2 by (simp add: mult.commute)
             qed 
             have y_prop_var: "- ((- A - B * sqrt C) / D) < y'" using y_prop
               by (metis add.commute diff_minus_eq_add divide_minus_left minus_diff_eq)
             have "\<exists>x\<in>{- (- (A + B * sqrt C) / D)<..y'}. \<not> a * x\<^sup>2 + 2 * a * (- (A + B * sqrt C) / D) * x + c < 0"
               using y_prop_var beq_var root_var agteq one_root_a_gt0[where a = "a", where b ="b", where c = "c", where r= "-(A + B * sqrt C) / D"]
               by auto
             then show " \<exists>x\<in>{(A + B * sqrt C) / D<..y'}. \<not> a * x\<^sup>2 + b * x + c < 0"
             proof -
               have f1: "2 * a * (A + B * sqrt C) * inverse D + b = 0"
                 by (metis True1 divide_inverse)
               obtain rr :: real where
                 f2: "rr \<in> {- (- (A + B * sqrt C) / D)<..y'} \<and> \<not> a * rr\<^sup>2 + 2 * a * (- (A + B * sqrt C) / D) * rr + c < 0"
                 using \<open>\<exists>x\<in>{- (- (A + B * sqrt C) / D)<..y'}. \<not> a * x\<^sup>2 + 2 * a * (- (A + B * sqrt C) / D) * x + c < 0\<close> by blast
               have f3: "a * ((A + B * sqrt C) * (inverse D * 2)) = - b"
                 using f1 by linarith
               have f4: "\<forall>r. - (- (r::real)) = r"
                 by simp
               have f5: "\<forall>r ra. (ra::real) * - r = r * - ra"
                 by simp
               have f6: "a * ((A + B * sqrt C) * (inverse D * - 2)) = b"
                 using f3 by simp
               have f7: "\<forall>r ra rb. (rb::real) * (ra * r) = r * (rb * ra)"
                 by auto
               have f8: "\<forall>r ra. - (ra::real) * r = ra * - r"
                 by linarith
               then have f9: "a * (inverse D * ((A + B * sqrt C) * - 2)) = b"
                 using f7 f6 f5 by presburger
               have f10: "rr \<in> {inverse D * (A + B * sqrt C)<..y'}"
                 using f4 f2 by (metis (no_types) divide_inverse mult.commute mult_minus_right)
               have "\<not> c + (rr * b + a * rr\<^sup>2) < 0"
                 using f9 f8 f7 f2 by (metis (no_types) add.commute divide_inverse mult.commute mult_minus_right)
               then show ?thesis
                 using f10 by (metis (no_types) add.commute divide_inverse mult.commute)
             qed
           qed
         qed
       next
         case False
         then have False1 : "a * ((A + B * sqrt C) / D)\<^sup>2 + b * (A + B * sqrt C) / D + c \<noteq> 0" by auto 
         show ?thesis proof(cases "a * ((A + B * sqrt C) / D)\<^sup>2 + b * (A + B * sqrt C) / D + c < 0")
           case True
           show ?thesis unfolding LessUni fields apply (simp add: True1 True)
             using True1 True   
           proof -
             let ?r = "(A + B * sqrt C) / D"
             assume " 2 * a * (A + B * sqrt C) / D + b = 0"
             assume "a * ((A + B * sqrt C) / D)\<^sup>2 + b * (A + B * sqrt C) / D + c < 0 "
             then have " \<exists>y'>(A + B * sqrt C) / D. \<forall>x\<in>{(A + B * sqrt C) / D<..y'}. poly [:c, b, a:] x < 0"  using continuity_lem_lt0[where r= "(A + B * sqrt C) / D", where c = "c", where b = "b", where a="a"]
                 quadratic_poly_eval[of c b a ?r]  by auto
             then show "\<exists>y'>(A + B * sqrt C) / D. \<forall>x\<in>{(A + B * sqrt C) / D<..y'}. a * x\<^sup>2 + b * x + c < 0"
               using quadratic_poly_eval[of c b a]
               by fastforce 
           qed
         next
           case False
           then have False' : "a * ((A + B * sqrt C) / D)\<^sup>2 + b * (A + B * sqrt C) / D + c > 0" using False1 by auto
           show ?thesis unfolding LessUni fields apply(simp add: True1 False False1) 
             using True1 False' continuity_lem_gt0_expanded[where a = "a", where b = "b",where c = "c", where r = "((A + B * sqrt C) / D)"]
             by (metis mult_less_0_iff not_square_less_zero times_divide_eq_right)
         qed
       qed
     next
       case False
       then have False1 : "2 * a * (A + B * sqrt C) / D + b \<noteq> 0" by auto
       have c1: "a * ((A + B * sqrt C) / D)\<^sup>2 + b * (A + B * sqrt C) / D + c = 0 \<Longrightarrow>
     2 * a * (A + B * sqrt C) / D + b < 0 \<Longrightarrow>
     \<exists>y'>(A + B * sqrt C) / D. \<forall>x\<in>{(A + B * sqrt C) / D<..y'}. a * x\<^sup>2 + b * x + c < 0"
       proof -
         assume root: "a * ((A + B * sqrt C) / D)\<^sup>2 + b * (A + B * sqrt C) / D + c = 0"
         assume blt: " 2 * a * (A + B * sqrt C) / D + b < 0"
         let ?r = "-(A + B * sqrt C) / D"
         have bltvar: "b < 2 * a * ?r" using blt divide_minus_left mult_2 mult_minus_right real_add_less_0_iff
           by (metis times_divide_eq_right)
         have rootvar: "a * ?r^2 - b * ?r + c = 0" using root
         proof -
           have f1: "\<forall>r ra. - (ra::real) * r = ra * - r"
             by simp
           have f2: "\<forall>r ra. ((ra::real) * - r)\<^sup>2 = (ra * r)\<^sup>2"
             by simp
           have f3: "a * (inverse D * (A - B * - sqrt C))\<^sup>2 - inverse D * (b * - (A - B * - sqrt C)) - - c = 0"
             by (metis (no_types) diff_minus_eq_add divide_inverse mult.commute mult_minus_left root)
           have f4: "\<forall>r ra rb. (rb::real) * (ra * r) = ra * (r * rb)"
             by simp
           have "\<forall>r ra. (ra::real) * - r = r * - ra"
             by simp
           then have "a * (inverse D * (A - B * - sqrt C))\<^sup>2 - b * (inverse D * - (A - B * - sqrt C)) - - c = 0"
             using f4 f3 f1 by (metis (no_types))
           then have "a * (inverse D * - (A - B * - sqrt C))\<^sup>2 - b * (inverse D * - (A - B * - sqrt C)) - - c = 0"
             using f2 by presburger
           then show ?thesis
             by (simp add: divide_inverse mult.commute)
         qed
         have "\<exists>y'> ((A + B * sqrt C) / D). \<forall>x\<in>{((A + B * sqrt C) / D)<..y'}. a * x\<^sup>2 + b * x + c < 0"
           using rootvar bltvar case_d1[where a= "a", where b = "b", where c = "c", where r = ?r]
           by (metis add.inverse_inverse divide_inverse mult_minus_left)
         then show ?thesis
           by blast 
       qed
       have c2: " \<And>y'. a * ((A + B * sqrt C) / D)\<^sup>2 + b * (A + B * sqrt C) / D + c = 0 \<Longrightarrow>
           \<not> 2 * a * (A + B * sqrt C) / D + b < 0 \<Longrightarrow>
           (A + B * sqrt C) / D < y' \<Longrightarrow>
           \<exists>x\<in>{(A + B * sqrt C) / D<..y'}. \<not> a * x\<^sup>2 + b * x + c < 0"
       proof - 
         let ?r = "(A + B * sqrt C) / D"
         fix y'
         assume h1: "a * ((A + B * sqrt C) / D)\<^sup>2 + b * (A + B * sqrt C) / D + c = 0"
         assume h2: "\<not> 2 * a * (A + B * sqrt C) / D + b < 0"
         assume h3: " (A + B * sqrt C) / D < y'"
         have eq: "2 * a * (A + B * sqrt C) / D + b = 0 \<Longrightarrow> \<exists>x\<in>{(A + B * sqrt C) / D..y'}. \<not> a * x\<^sup>2 + b * x + c < 0"
           using False1 by blast
         have "2 * a * (A + B * sqrt C) / D + b > 0 \<Longrightarrow> \<exists>x\<in>{?r<..y'}. \<not> a * x\<^sup>2 + b * x + c < 0" 
           using case_d4[where a = "a", where b = "b", where c= "c", where r = "-?r"] h1 h2 h3 by auto
         then show "\<exists>x\<in>{(A + B * sqrt C) / D<..y'}. \<not> a * x\<^sup>2 + b * x + c < 0" using h2 eq
           using False1 by linarith
       qed
       have c3: "((a::real) * ((A + B * sqrt C) / D)\<^sup>2 + b * (A + B * sqrt C) / D + c < 0) \<longrightarrow>
     (\<exists>y'>((A + B * sqrt C) / D). \<forall>x\<in>{(A + B * sqrt C) / D<..y'}. a * x\<^sup>2 + b * x + c < 0)"
       proof clarsimp 
         assume assump: "a * ((A + B * sqrt C) / D)\<^sup>2 + b * (A + B * sqrt C) / D + c < 0 "
         have "a * ((A + B * sqrt C) / D)\<^sup>2 + b * ((A + B * sqrt C) / D) + c < 0 \<Longrightarrow>
   \<exists>y'>(A + B * sqrt C) / D. \<forall>x\<in>{(A + B * sqrt C) / D<..y'}. a * x\<^sup>2 + b * x + c < 0" 
           using continuity_lem_lt0_expanded[where a= "a", where b = "b", where c = "c", where r = "((A + B * sqrt C) / D)::real"]
           by auto
         then have "\<exists>y'>(A + B * sqrt C) / D. \<forall>x\<in>{(A + B * sqrt C) / D<..y'}. a * x\<^sup>2 + b * x + c < 0" using assump by auto
         then obtain y where y_prop: "y >(A + B * sqrt C) / D \<and> (\<forall>x\<in>{(A + B * sqrt C) / D<..y}. a * x\<^sup>2 + b * x + c < 0)" by auto
         then have h: "\<exists> k. k >(A + B * sqrt C) / D \<and> k < y" using dense
           by blast 
         then obtain k where k_prop: "k >(A + B * sqrt C) / D  \<and> k < y" by auto
         then have "\<forall>x\<in>{(A + B * sqrt C) / D..k}. a * x\<^sup>2 + b * x + c < 0" using y_prop
           using assump by force 
         then show "\<exists>y'>((A + B * sqrt C) / D::real). \<forall>x\<in>{(A + B * sqrt C) / D<..y'}. a * x\<^sup>2 + b * x + c < 0"
           using k_prop by auto
       qed
       have c4: "\<And>y'. a * ((A + B * sqrt C) / D)\<^sup>2 + b * (A + B * sqrt C) / D + c \<noteq> 0 \<Longrightarrow>
           \<not> a * ((A + B * sqrt C) / D)\<^sup>2 + b * (A + B * sqrt C) / D + c < 0 \<Longrightarrow>
           (A + B * sqrt C) / D < y' \<Longrightarrow>
           \<exists>x\<in>{(A + B * sqrt C) / D<..y'}. \<not> a * x\<^sup>2 + b * x + c < 0"
         using continuity_lem_gt0_expanded[where a= "a", where b = "b", where c = "c", where r= "(A + B * sqrt C) / D"]
         by (metis Groups.mult_ac(1) divide_inverse less_eq_real_def linorder_not_le)
       show ?thesis unfolding LessUni fields apply(simp add: False1) 
         using c1 c2 c3 c4 by auto
     qed
   qed
 next
   case False
   then have EqUni: "At = EqUni p" using at_is by auto
   then show ?thesis proof(cases p) 
     case (fields a b c) 
     have " \<And>y'. (A + B * sqrt C) / D < y' \<Longrightarrow>
           \<forall>x\<in>{(A + B * sqrt C) / D<..y'}. a * x\<^sup>2 + b * x + c = 0 \<Longrightarrow> (a = 0 \<and> b = 0 \<and> c = 0)"
     proof -
       fix y'
       assume "(A + B * sqrt C) / D < y'"
       then show " \<forall>x\<in>{(A + B * sqrt C) / D<..y'}. a * x\<^sup>2 + b * x + c = 0 \<Longrightarrow> (a = 0 \<and> b = 0 \<and> c = 0)" using assms continuity_lem_eq0[where r = "(A + B * sqrt C) / D", where p = "y'", where a= "a", where b ="b", where c="c"]
         by auto
     qed
     then show ?thesis
       apply (auto simp add:EqUni fields )
       using linordered_field_no_ub by blast
   qed
 qed
 
 lemma infinitesimal_quad:
   fixes A B C D:: "real"
   assumes "D\<noteq>0"
   assumes "C\<ge>0"
   shows "(\<exists>y'::real>((A+B * sqrt(C))/(D)). \<forall>x::real \<in>{((A+B * sqrt(C))/(D))<..y'}. aEvalUni At x)
       = (evalUni (substInfinitesimalQuadraticUni A B C D At) x)"
 proof(cases At)
   case (LessUni p)
   then show ?thesis using infinitesimal_quad_helper assms
     by blast 
 next
   case (EqUni p)
   then show ?thesis
     using infinitesimal_quad_helper assms
     by blast 
 next
   case (LeqUni p)
   then show ?thesis 
   proof (cases p)
     case (fields a b c) 
     have same: "\<forall>x. aEvalUni (LeqUni p) x = (aEvalUni (EqUni p) x) \<or> (aEvalUni (LessUni p) x)" 
       apply (simp add: fields)
       by force 
     let ?r = "(A + B * sqrt C) / D"
     have "\<And>a b c.
        At = LeqUni p \<Longrightarrow>
        p = (a, b, c) \<Longrightarrow>
        (\<exists>y'>(A + B * sqrt C) / D. \<forall>x\<in>{(A + B * sqrt C) / D<..y'}. aEvalUni At x) =
        evalUni (substInfinitesimalQuadraticUni A B C D At) x"
     proof - 
       fix a b c
       assume atis: "At = LeqUni p"
       assume p_is: " p = (a, b, c)"
       have s1: "(\<exists>y'>?r. \<forall>x\<in>{?r<..y'}. aEvalUni At x) = (\<exists>y'>?r. \<forall>x\<in>{?r<..y'}.  (aEvalUni (EqUni p) x) \<or> (aEvalUni (LessUni p) x))"
         using atis same aEvalUni.simps(2) aEvalUni.simps(3) fields less_eq_real_def 
         by blast
       have s2: "... = (\<exists>y'>?r. \<forall>x\<in>{?r<..y'}.  (aEvalUni (EqUni p) x)) \<or> (\<exists>y'>?r. \<forall>x\<in>{?r<..y'}. (aEvalUni (LessUni p) x))"
         using either_or[where r = "?r"] p_is 
         by blast 
       have eq1: "(\<exists>y'>?r. \<forall>x\<in>{?r<..y'}.  (aEvalUni (EqUni p) x)) =  (evalUni (substInfinitesimalQuadraticUni A B C D (EqUni p)) x)"
         using infinitesimal_quad_helper[where At = "EqUni p", where p = "p", where B = "B", where C= "C", where A= "A", where D="D"] 
           assms  by auto
       have eq2: "(\<exists>y'>?r. \<forall>x\<in>{?r<..y'}.  (aEvalUni (LessUni p) x)) =  (evalUni (substInfinitesimalQuadraticUni A B C D (LessUni p)) x)"
         using infinitesimal_quad_helper[where At = "LessUni p", where p = "p", where B = "B", where C= "C", where A= "A", where D="D"] 
           assms by auto
       have z1: "(\<exists>y'>?r. \<forall>x\<in>{?r<..y'}. aEvalUni At x) = ((evalUni (substInfinitesimalQuadraticUni A B C D (EqUni p)) x) \<or> (evalUni (substInfinitesimalQuadraticUni A B C D (LessUni p)) x))"
         using s1 s2 eq1 eq2 by auto
       have z2: "(evalUni (substInfinitesimalQuadraticUni A B C D (EqUni p)) x) \<or> (evalUni (substInfinitesimalQuadraticUni A B C D (LessUni p)) x) = evalUni (substInfinitesimalQuadraticUni A B C D (LeqUni p)) x" 
         by auto
       have z3: "(evalUni (substInfinitesimalQuadraticUni A B C D At) x) = evalUni (substInfinitesimalQuadraticUni A B C D (LeqUni p)) x"
         using LeqUni by auto
       then have z4: "(evalUni (substInfinitesimalQuadraticUni A B C D (EqUni p)) x) \<or> (evalUni (substInfinitesimalQuadraticUni A B C D (LessUni p)) x) = (evalUni (substInfinitesimalQuadraticUni A B C D At) x) "
         using z2 z3 by auto
       let ?a = "(evalUni (substInfinitesimalQuadraticUni A B C D (EqUni p)) x) \<or> (evalUni (substInfinitesimalQuadraticUni A B C D (LessUni p)) x)"
       let ?b = "(\<exists>y'>?r. \<forall>x\<in>{?r<..y'}. aEvalUni At x)"
       let ?c = "(evalUni (substInfinitesimalQuadraticUni A B C D At) x)"
       have t1: "?b = ?a" using z1 by auto
       have t2: "?a = ?c" using z4
         using atis by auto
       then have "?b = ?c" using t1 t2 by auto
       then show "(\<exists>y'>?r. \<forall>x\<in>{?r<..y'}. aEvalUni At x) = evalUni (substInfinitesimalQuadraticUni A B C D At) x"
         by auto
     qed
     then show ?thesis
       using LeqUni fields by blast 
   qed
 next
   case (NeqUni p)
   then show ?thesis
   proof (cases p)
     case (fields a b c) 
     then show ?thesis unfolding NeqUni fields using nonzcoeffs by auto
   qed
 qed
 
 
 end
\ No newline at end of file
diff --git a/thys/Virtual_Substitution/MPolyExtension.thy b/thys/Virtual_Substitution/MPolyExtension.thy
--- a/thys/Virtual_Substitution/MPolyExtension.thy
+++ b/thys/Virtual_Substitution/MPolyExtension.thy
@@ -1,192 +1,192 @@
 section "Multivariate Polynomials Extension"
 theory MPolyExtension
   imports Polynomials.Polynomials (*MPoly_Type_Efficient_Code*) Polynomials.MPoly_Type_Class_FMap
 begin
 
 subsection "Definition Lifting"
 
 lift_definition coeff_code::"'a::zero mpoly \<Rightarrow> (nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow> 'a" is
   "lookup" .
 
 lemma coeff_code[code]: "coeff = coeff_code"
   unfolding coeff_def apply(transfer) by auto
 
 lemma coeff_transfer[transfer_rule]:\<comment> \<open>TODO: coeff should be defined via
 lifting, this gives us the illusion\<close>
   "rel_fun cr_mpoly (=) lookup coeff" using coeff_code.transfer[folded
       coeff_code] .
 
 lemmas coeff_add = coeff_add[symmetric]
 
 lemma plus_monom_zero[simp]: "p + MPoly_Type.monom m 0 = p"
   by transfer auto
 
 lift_definition monomials::"'a::zero mpoly \<Rightarrow> (nat \<Rightarrow>\<^sub>0 nat) set" is
   "Poly_Mapping.keys::((nat\<Rightarrow>\<^sub>0nat) \<Rightarrow>\<^sub>0 'a) \<Rightarrow> _ set" .
 
 lemma mpoly_induct [case_names monom sum]:\<comment> \<open>TODO: overwrites @{thm
 mpoly_induct}\<close>
   assumes monom:"\<And>m a. P (MPoly_Type.monom m a)"
     and sum:"(\<And>p1 p2 m a. P p1 \<Longrightarrow> P p2 \<Longrightarrow> p2 = (MPoly_Type.monom m a) \<Longrightarrow> m \<notin> monomials p1
 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> P (p1+p2))"
   shows "P p" using assms
 proof (induction p rule: mpoly_induct)
   case (sum p1 p2 m a)
   then show ?case
     by (cases "a = 0") (auto simp: monomials.rep_eq)
 qed simp
 
 value "monomials ((Var 0 + Const (3::int) * Var 1)^2)"
 
 lemma Sum_any_lookup_times_eq:
   "(\<Sum>k. ((lookup (x::'a\<Rightarrow>\<^sub>0('b::comm_semiring_1)) (k::'a)) * ((f:: 'a\<Rightarrow>('b::comm_semiring_1)) k))) = (\<Sum>k\<in>keys
 x. (lookup x (k::'a)) * (f k))"
   by (subst Sum_any.conditionalize) (auto simp: in_keys_iff intro!:
       Sum_any.cong)
 
 lemma Prod_any_power_lookup_eq:
   "(\<Prod>k::'a. f k ^ lookup (x::'a\<Rightarrow>\<^sub>0nat) k) = (\<Prod>k\<in>keys x. f k ^ lookup x k)"
   by (subst Prod_any.conditionalize) (auto simp: in_keys_iff intro!:
       Prod_any.cong)
 
 lemma insertion_monom: "insertion i (monom m a) = a * (\<Prod>k\<in>keys m. i k ^
 lookup m k)"
   by transfer (auto simp: insertion_aux_def insertion_fun_def
       Sum_any_lookup_times_eq Prod_any_power_lookup_eq)
 
 lemma monomials_monom[simp]: "monomials (monom m a) = (if a = 0 then {}
 else {m})"
   by transfer auto
 
 lemma finite_monomials[simp]: "finite (monomials m)"
   by transfer auto
 
 lemma monomials_add_disjoint:
   "monomials (a + b) = monomials a \<union> monomials b" if "monomials a \<inter>
 monomials b = {}"
   using that
   by transfer (auto simp add: keys_plus_eqI)
 
 lemma coeff_monom[simp]: "coeff (monom m a) m = a"
   by transfer simp
 
 lemma coeff_not_in_monomials[simp]: "coeff m x = 0" if "x \<notin> monomials m"
   using that
   by transfer (simp add: in_keys_iff)
 
 code_thms insertion
 
 lemma insertion_code[code]: "insertion i mp =
    (\<Sum>m\<in>monomials mp. (coeff mp m) * (\<Prod>k\<in>keys m. i k ^ lookup m k))"
 proof (induction mp rule: mpoly_induct)
   case (monom m a)
   show ?case
     by (simp add: insertion_monom)
 next
   case (sum p1 p2 m a)
   have monomials_add: "monomials (p1 + p2) = insert m (monomials p1)"
     using sum.hyps
     by (auto simp: monomials_add_disjoint)
   have *: "coeff (monom m a) x = 0" if "x \<in> monomials p1" for x
     using sum.hyps that
     by (subst coeff_not_in_monomials) auto
   show ?case
     unfolding insertion_add monomials_add sum.IH
     using sum.hyps
     by (auto simp: coeff_add * algebra_simps)
 qed
 
 
 (* insertion f p
   takes in a mapping from natural numbers to the type of the polynomial and substitutes in
   the constant (f var) for each var variable in polynomial p
 *)
 code_thms insertion
 
 value "insertion (nth [1, 2.3]) ((Var 0 + Const (3::int) * Var 1)^2)"
 
 
 (* isolate_variable_sparse p var degree
     returns the coefficient of the term a*var^degree in polynomial p
  *)
 lift_definition isolate_variable_sparse::"'a::comm_monoid_add mpoly \<Rightarrow>
 nat \<Rightarrow> nat \<Rightarrow> 'a mpoly" is
   "\<lambda>(mp::'a mpoly) x d. sum
      (\<lambda>m. monomial (coeff mp m) (Poly_Mapping.update x 0 m))
      {m \<in> monomials mp. lookup m x = d}" .
 
 lemma Poly_Mapping_update_code[code]: "Poly_Mapping.update a b (Pm_fmap
 fm) = Pm_fmap (fmupd a b fm)"
   by (auto intro!: poly_mapping_eqI simp: update.rep_eq
       fmlookup_default_def)
 
 
 lemma monom_zero [simp] : "monom m 0 = 0"
   by (simp add: coeff_all_0)
 
 
 lemma remove_key_code[code]: "remove_key v (Pm_fmap fm) = Pm_fmap
 (fmdrop v fm)"
   by (auto simp: remove_key_lookup fmlookup_default_def intro!:
       poly_mapping_eqI)
 lemma extract_var_code[code]:
   "extract_var p v =
      (\<Sum>m\<in>monomials p. monom (remove_key v m) (monom (Poly_Mapping.single
 v (lookup m v)) (coeff p m)))"
   apply (rule extract_var_finite_set[where S="monomials p"])
   using coeff_not_in_monomials by auto
 value "extract_var ((Var 0 + Const (3::real) * Var 1)^2) 0"
 
 
 
 (* degree p var
   takes in polynomial p and a variable var and finds the degree of that variable in the polynomial
   missing code theorems? still manages to evaluate
 *)
 code_thms degree
 value "degree ((Var 0 + Const (3::real) * Var 1)^2) 0"
 
 
 (* this function gives a set of all the variables in the polynomial
 *)
 lemma vars_code[code]: "vars p = \<Union> (keys ` monomials p)"
   unfolding monomials.rep_eq vars_def ..
 
 value "vars ((Var 0 + Const (3::real) * Var 1)^2)"
 
 
 (* return true if the polynomial contains no variables
 *)
 fun is_constant :: "'a::zero mpoly \<Rightarrow> bool" where
   "is_constant p = Set.is_empty (vars p)"
 
 value "is_constant (Const (0::int))"
 
 
 (*
   if the polynomial is constant, returns the real value associated with the polynomial,
   otherwise returns none
 *)
 fun get_if_const :: "real mpoly \<Rightarrow> real option" where
   "get_if_const p = (if is_constant p then Some (coeff p 0) else None)"
 
 term "coeff p 0"
 
 
-lemma insertionNegative : "insertion f p = - insertion f (-p)"
-  by (metis (no_types, hide_lams) add_eq_0_iff cancel_comm_monoid_add_class.diff_cancel insertion_add insertion_zero uminus_add_conv_diff)  
+lemma insertionNegative : "insertion f p = - insertion f (-p)" try
+  by (metis add.right_inverse eq_neg_iff_add_eq_0 insertion_add insertion_zero)
 
 
 definition derivative :: "nat \<Rightarrow> real mpoly \<Rightarrow> real mpoly" where
   "derivative x p = (\<Sum>i\<in>{0..degree p x-1}. isolate_variable_sparse p x (i+1) * (Var x)^i * (Const (i+1)))"
 
 text "get\\_coeffs $x$ $p$
   gets the tuple of coefficients $a$ $b$ $c$ of the term $a*x^2+b*x+c$ in polynomial $p$"
 fun get_coeffs :: "nat \<Rightarrow> real mpoly \<Rightarrow> real mpoly * real mpoly * real mpoly" where
   "get_coeffs var x = (
   isolate_variable_sparse x var 2,
   isolate_variable_sparse x var 1,
   isolate_variable_sparse x var 0)
 "
 
 end
diff --git a/thys/Virtual_Substitution/NegInfinityUni.thy b/thys/Virtual_Substitution/NegInfinityUni.thy
--- a/thys/Virtual_Substitution/NegInfinityUni.thy
+++ b/thys/Virtual_Substitution/NegInfinityUni.thy
@@ -1,632 +1,632 @@
 theory NegInfinityUni
   imports UniAtoms NegInfinity QE
 begin
 
 fun allZero' :: "real * real * real \<Rightarrow> atomUni fmUni" where
   "allZero' (a,b,c) = AndUni(AndUni(AtomUni(EqUni(0,0,a))) (AtomUni(EqUni(0,0,b)))) (AtomUni(EqUni(0,0,c)))"
 
 lemma convert_allZero : 
   assumes "convert_poly var p (xs'@x#xs) = Some p'"
   assumes "length xs' = var"
   shows "eval (allZero p var) (xs'@x#xs) = evalUni (allZero' p') x"
 proof(cases p')
   case (fields a b c)
   then show ?thesis proof(cases "MPoly_Type.degree p var = 0")
     case True
     then show ?thesis
       using assms fields
       by(simp add:eval_list_conj isovar_greater_degree)
   next
     case False
     then have nonzero : "MPoly_Type.degree p var \<noteq> 0" by auto
     then show ?thesis
     proof(cases "MPoly_Type.degree p var = 1")
       case True
       then show ?thesis
         using assms fields
         apply(simp add:eval_list_conj isovar_greater_degree)
         by (metis)
     next
       case False
       then have degree2 : "MPoly_Type.degree p var = 2" using degree_convert_eq[OF assms(1)] nonzero by auto
       then show ?thesis
         using assms
         apply(simp add:eval_list_conj isovar_greater_degree)
         using insertion_isovarspars_free list_update_code(2)
         apply auto
         by (metis One_nat_def Suc_1 less_2_cases less_Suc_eq numeral_3_eq_3)
     qed
   qed
 qed
 
 
 
 fun alternateNegInfinity' :: "real * real * real \<Rightarrow> atomUni fmUni" where
   "alternateNegInfinity' (a,b,c) = 
 OrUni(AtomUni(LessUni(0,0,a)))(
 AndUni(AtomUni(EqUni(0,0,a))) (
   OrUni(AtomUni(LessUni(0,0,-b)))(
   AndUni(AtomUni(EqUni(0,0,b)))(
     AtomUni(LessUni(0,0,c))
   ))
 ))
 "
 
 lemma convert_alternateNegInfinity : 
   assumes "convert_poly var p (xs'@x#xs) = Some X"
   assumes "length xs' = var"
   shows "eval (alternateNegInfinity p var) (xs'@x#xs) = evalUni (alternateNegInfinity' X) x"
 proof(cases X)
   case (fields a b c)
   then show ?thesis proof(cases "MPoly_Type.degree p var = 0")
     case True
     then show ?thesis
       using assms
       apply (simp add: isovar_greater_degree)
       apply auto
       apply (metis aEval.simps(2) eval.simps(1) eval_and eval_false eval_or  mult_one_left)
       by (metis aEval.simps(2) eval.simps(1) eval_or  mult_one_left)
   next
     case False
     then have nonzero : "MPoly_Type.degree p var \<noteq> 0" by auto
     then show ?thesis
     proof(cases "MPoly_Type.degree p var = 1")
       case True
       have letbind: "eval
      (let a_n = isolate_variable_sparse p var (Suc 0)
       in or (fm.Atom (Less (Const (- 1) * a_n)))
           (and (fm.Atom (Eq a_n))
             (let a_n = isolate_variable_sparse p var 0
              in or (fm.Atom (Less (Const 1 * a_n))) (and (fm.Atom (Eq a_n)) FalseF))))
      (xs'@x#xs) = 
     eval
      (or (fm.Atom (Less (Const (- 1) * (isolate_variable_sparse p var (Suc 0)))))
           (and (fm.Atom (Eq (isolate_variable_sparse p var (Suc 0))))
             (or (fm.Atom (Less (Const 1 * (isolate_variable_sparse p var 0)))) (and (fm.Atom (Eq (isolate_variable_sparse p var 0))) FalseF))))
      (xs'@x#xs)"
         by meson 
       show ?thesis
         using assms True unfolding fields
         by (simp add: isovar_greater_degree letbind eval_or eval_and insertion_mult insertion_const)
     next
       case False
       then have degree2 : "MPoly_Type.degree p var = 2" using degree_convert_eq[OF assms(1)] nonzero by auto
       have "[0..<3] = [0,1,2]"
         by (simp add: upt_rec)
       then have unfold : " (foldl
        (\<lambda>F i. let a_n = isolate_variable_sparse p var i
                in or (fm.Atom (Less ((if i mod 2 = 0 then Const 1 else Const (- 1)) * a_n)))
                    (and (fm.Atom (Eq a_n)) F))
        FalseF [0..<3]) =  
      (let a_n = isolate_variable_sparse p var 2
                in or (fm.Atom (Less ((Const 1) * a_n)))
                    (and (fm.Atom (Eq a_n))
        (let a_n = isolate_variable_sparse p var (Suc 0)
       in or (fm.Atom (Less (Const (- 1) * a_n)))
           (and (fm.Atom (Eq a_n))
             (let a_n = isolate_variable_sparse p var 0
              in or (fm.Atom (Less (Const 1 * a_n))) (and (fm.Atom (Eq a_n)) FalseF))))))" 
         by auto
       have letbind : "eval
      (foldl
        (\<lambda>F i. let a_n = isolate_variable_sparse p var i
                in or (fm.Atom (Less ((if i mod 2 = 0 then Const 1 else Const (- 1)) * a_n)))
                    (and (fm.Atom (Eq a_n)) F))
        FalseF [0..<3]) (xs'@x#xs) = 
       eval
      
 (or (fm.Atom (Less ( Const 1 * (isolate_variable_sparse p var 2))))
                    (and (fm.Atom (Eq (isolate_variable_sparse p var 2)))
 (or (fm.Atom (Less (Const (- 1) * (isolate_variable_sparse p var (Suc 0)))))
           (and (fm.Atom (Eq (isolate_variable_sparse p var (Suc 0))))
             (or (fm.Atom (Less (Const 1 * (isolate_variable_sparse p var 0)))) (and (fm.Atom (Eq (isolate_variable_sparse p var 0))) FalseF))))))
 (xs'@x#xs)" apply (simp add:unfold) by metis
       show ?thesis
         using degree2 assms unfolding fields by (simp add: isovar_greater_degree eval_or eval_and letbind insertion_mult insertion_const)
     qed
   qed
 qed
 
 
 
 fun substNegInfinityUni :: "atomUni \<Rightarrow> atomUni fmUni" where
   "substNegInfinityUni (EqUni p) = allZero' p " |
   "substNegInfinityUni (LessUni p) = alternateNegInfinity' p"|
   "substNegInfinityUni (LeqUni p) = OrUni (alternateNegInfinity' p) (allZero' p)"|
   "substNegInfinityUni (NeqUni p) = negUni (allZero' p)"
 
 
 lemma convert_substNegInfinity : 
   assumes "convert_atom var A (xs'@x#xs) = Some(A')"
   assumes "length xs' = var"
   shows "eval (substNegInfinity var A) (xs'@x#xs) = evalUni (substNegInfinityUni A') x"
   using assms
 proof(cases A)
   case (Less p)
   have "\<exists>X. convert_poly var p (xs' @ x # xs) = Some X" using assms Less apply(cases "MPoly_Type.degree p var < 3") by (simp_all)
   then obtain X where X_def: "convert_poly var p (xs' @ x # xs) = Some X" by auto
   then have A' : "A' = LessUni X" using assms Less apply(cases "MPoly_Type.degree p var < 3") by (simp_all)
   show ?thesis unfolding Less substNegInfinity.simps
     unfolding convert_alternateNegInfinity[OF X_def assms(2)] A'
     apply(cases X) by simp
 next
   case (Eq p)
   then show ?thesis using assms convert_allZero by auto
 next
   case (Leq p)
   define p' where "p' = (case convert_poly var p (xs'@x#xs) of Some p' \<Rightarrow> p')"
   have A'_simp :  "A' = LeqUni p'"
     using assms Leq
     using p'_def by auto 
   have allZ : "eval (allZero p var) (xs'@x#xs) = evalUni (allZero' p') x" using convert_allZero
     using Leq assms p'_def by auto 
   have altNeg : "eval (alternateNegInfinity p var) (xs'@x#xs) = evalUni (alternateNegInfinity' p') x" using convert_alternateNegInfinity
     using Leq assms p'_def by auto
   show ?thesis
     unfolding Leq substNegInfinity.simps eval_Or A'_simp substNegInfinityUni.simps evalUni.simps
     using allZ altNeg by auto
 next
   case (Neq p)
   then show ?thesis using assms convert_allZero convert_neg by auto
 qed
 
 lemma change_eval_eq:
   fixes x y:: "real"
   assumes "((aEvalUni (EqUni(a,b,c)) x \<noteq> aEvalUni (EqUni(a,b,c)) y) \<and> x < y)"
   shows "(\<exists>w. x \<le> w \<and> w \<le> y \<and> a*w^2 + b*w + c = 0)"
   using assms by auto
 lemma change_eval_lt:
   fixes x y:: "real"
   assumes "((aEvalUni (LessUni (a,b,c)) x \<noteq> aEvalUni (LessUni (a,b,c)) y) \<and> x < y)"
   shows "(\<exists>w. x \<le> w \<and> w \<le> y \<and> a*w^2 + b*w + c = 0)"
 proof - 
   let ?p = "[:c, b, a:]"
   have "sign ?p x \<noteq> sign ?p y"
     using assms unfolding sign_def 
     apply (simp add: distrib_left mult.commute mult.left_commute power2_eq_square)
     by linarith
   then have "(\<exists>w \<in> (root_list ?p). x \<le> w \<and> w \<le> y)" using changes_sign 
       assms by auto
   then obtain w where w_prop: "w \<in> (root_list ?p) \<and> x \<le> w \<and> w \<le> y" by auto
   then have "a*w^2 + b*w + c = 0" unfolding root_list_def 
     using add.commute distrib_right mult.assoc mult.commute power2_eq_square
     using quadratic_poly_eval by force
   then show ?thesis using w_prop by auto
 qed
 
 lemma no_change_eval_lt:
   fixes x y:: "real"
   assumes "x < y"
   assumes "\<not>(\<exists>w. x \<le> w \<and> w \<le> y \<and> a*w^2 + b*w + c = 0)"
   shows "((aEvalUni (LessUni (a,b,c)) x = aEvalUni (LessUni (a,b,c)) y))"
   using change_eval_lt
   using assms(1) assms(2) by blast 
 
 
 lemma change_eval_neq:
   fixes x y:: "real"
   assumes "((aEvalUni (NeqUni (a,b,c)) x \<noteq> aEvalUni (NeqUni (a,b,c)) y) \<and> x < y)"
   shows "(\<exists>w. x \<le> w \<and> w \<le> y \<and> a*w^2 + b*w + c = 0)"
   using assms by auto 
 
 lemma change_eval_leq:
   fixes x y:: "real"
   assumes "((aEvalUni (LeqUni (a,b,c)) x \<noteq> aEvalUni (LeqUni (a,b,c)) y) \<and> x < y)"
   shows "(\<exists>w. x \<le> w \<and> w \<le> y \<and> a*w^2 + b*w + c = 0)"
 proof - 
   let ?p = "[:c, b, a:]"
   have "sign ?p x \<noteq> sign ?p y"
     using assms unfolding sign_def
     apply (simp add: distrib_left mult.commute mult.left_commute power2_eq_square)
     by linarith
   then have "(\<exists>w \<in> (root_list ?p). x \<le> w \<and> w \<le> y)" using changes_sign 
       assms by auto
   then obtain w where w_prop: "w \<in> (root_list ?p) \<and> x \<le> w \<and> w \<le> y" by auto
   then have "a*w^2 + b*w + c = 0" unfolding root_list_def
     using add.commute distrib_right mult.assoc mult.commute power2_eq_square
     using quadratic_poly_eval by force  
   then show ?thesis using w_prop by auto
 qed
 
 lemma change_eval:
   fixes x y:: "real"
   fixes At:: "atomUni" 
   assumes xlt: "x < y"
   assumes noteq: "((aEvalUni At) x \<noteq> aEvalUni (At) y)"
   assumes "getPoly At = (a, b, c)"
   shows "(\<exists>w. x \<le> w \<and> w \<le> y \<and> a*w^2 + b*w + c = 0)"
 proof - 
   have four_types: "At = (EqUni (a,b,c)) \<or> At = (LessUni (a,b,c)) \<or> At = (LeqUni (a,b,c)) \<or> At = (NeqUni (a,b,c))"
     using getPoly.elims assms(3) by force 
   have eq_h: "At = (EqUni (a,b,c)) \<longrightarrow> (\<exists>w. x \<le> w \<and> w \<le> y \<and> a*w^2 + b*w + c = 0)"
     using assms(1) assms(2) change_eval_eq 
     by blast
   have less_h: "At = (LessUni(a,b,c)) \<longrightarrow> (\<exists>w. x \<le> w \<and> w \<le> y \<and> a*w^2 + b*w + c = 0)"
     using change_eval_lt assms
     by blast
   have leq_h: "At = (LeqUni(a,b,c)) \<longrightarrow> (\<exists>w. x \<le> w \<and> w \<le> y \<and> a*w^2 + b*w + c = 0)"
     using change_eval_leq assms
     by blast
   have neq_h: "At = (NeqUni(a,b,c)) \<longrightarrow> (\<exists>w. x \<le> w \<and> w \<le> y \<and> a*w^2 + b*w + c = 0)"
     using change_eval_neq assms
     by blast
   show ?thesis
     using four_types eq_h less_h leq_h neq_h
     by blast 
 qed 
 
 lemma no_change_eval:
   fixes x y:: "real"
   assumes "getPoly At = (a, b, c)"
   assumes "x < y"
   assumes "\<not>(\<exists>w. x \<le> w \<and> w \<le> y \<and> a*w^2 + b*w + c = 0)"
   shows  "((aEvalUni At) x = aEvalUni (At) y \<and> x < y)"
   using change_eval
   using assms(1) assms(2) assms(3) by blast 
 
 
 lemma same_eval'' :
   assumes "getPoly At = (a, b, c)"
   assumes nonz: "a \<noteq> 0 \<or> b \<noteq> 0 \<or> c \<noteq> 0"
   shows "\<exists>x. \<forall>y<x. (aEvalUni At y = aEvalUni At x)"
 proof - 
   let ?p = "[:c, b, a:]"
   have poly_eval: "\<forall>y. poly ?p y = a*y^2 + b*y + c" 
     by (simp add: distrib_left power2_eq_square) 
   have "?p \<noteq> 0" using nonz by auto
   then have "finite {y. poly ?p y = 0}"
     using poly_roots_finite
     by blast
   then have "finite {y. c + (a * y\<^sup>2 + b * y) = 0} \<Longrightarrow>
     \<forall>y. y * (b + y * a) = a * y\<^sup>2 + b * y \<Longrightarrow>
     finite {y. a * y\<^sup>2 + b * y + c = 0}"
   proof -
     assume a1: "finite {y. c + (a * y\<^sup>2 + b * y) = 0}"
     have "\<forall>x0. c + (a * x0\<^sup>2 + b * x0) = a * x0\<^sup>2 + b * x0 + c"
       by simp
     then show ?thesis
       using a1 by presburger
   qed 
   then have finset: "finite {y. a*y^2 + b*y + c = 0}" 
     using poly_eval
     by (metis \<open>finite {y. poly [:c, b, a:] y = 0}\<close> poly_roots_set_same) 
   then have "\<exists>x. (\<forall>y. a*y^2 + b*y + c = 0 \<longrightarrow> x < y)" 
   proof - 
     let ?l = "sorted_list_of_set {y. a*y^2 + b*y + c = 0}"
     have "\<exists>x. x < ?l ! 0" 
       using infzeros nonz by blast 
     then obtain x where x_prop: "x < ?l! 0" by auto
     then have "\<forall> y. List.member ?l y \<longrightarrow> x < y"
     proof clarsimp
       fix y
       assume "List.member ?l y"
       then have "\<exists>n. n < length ?l \<and> ?l ! n = y"
         by (meson in_set_conv_nth in_set_member)
       then obtain n where n_prop: "n < length ?l \<and> ?l ! n = y"
         by auto
       have h: "\<forall>n < length ?l. ?l ! n \<ge> ?l !0" using sorted_iff_nth_mono
         using sorted_sorted_list_of_set by blast
       then have h: "y \<ge> ?l ! 0" using n_prop by auto
       then show "x < y" using x_prop by auto
     qed
     then show ?thesis
       using finset set_sorted_list_of_set in_set_member
       by (metis (mono_tags, lifting) mem_Collect_eq)
   qed
   then obtain x where x_prop: "(\<forall>y. a*y^2 + b*y + c = 0 \<longrightarrow> x < y)" by auto
   then have same_as: "\<forall>y<x. (aEvalUni At y = aEvalUni At x)"
     using no_change_eval change_eval assms
   proof -
     have f1: "\<forall>x0. (x0 < x) = (\<not> 0 \<le> x0 + - 1 * x)"
       by auto
     have f2: "(0 \<le> - 1 * x + v0_0) = (x + - 1 * v0_0 \<le> 0)"
       by auto
     have f3: "v0_0 + - 1 * x = - 1 * x + v0_0"
       by auto
     have f4: "\<forall>x0 x1 x2 x3. (x3::real) * x0\<^sup>2 + x2 * x0 + x1 = x1 + x3 * x0\<^sup>2 + x2 * x0"
       by auto
     have f5: "\<forall>x3 x4 x5. (aEvalUni x3 x5 \<noteq> aEvalUni x3 x4) = ((\<not> aEvalUni x3 x5) = aEvalUni x3 x4)"
       by fastforce
     have f6: "\<forall>x0 x1 x2 x3 x4 x5. (x5 < x4 \<and> (\<not> aEvalUni x3 x5) = aEvalUni x3 x4 \<and> getPoly x3 = (x2, x1, x0) \<longrightarrow> (\<exists>v6\<ge>x5. v6 \<le> x4 \<and> x0 + x2 * v6\<^sup>2 + x1 * v6 = 0)) = ((\<not> x5 < x4 \<or> (\<not> aEvalUni x3 x5) \<noteq> aEvalUni x3 x4 \<or> getPoly x3 \<noteq> (x2, x1, x0)) \<or> (\<exists>v6\<ge>x5. v6 \<le> x4 \<and> x0 + x2 * v6\<^sup>2 + x1 * v6 = 0))"
       by fastforce
     have f7: "\<forall>x0 x5. ((x0::real) \<le> x5) = (x0 + - 1 * x5 \<le> 0)"
       by auto
     have f8: "\<forall>x0 x6. ((x6::real) \<le> x0) = (0 \<le> x0 + - 1 * x6)"
       by auto
     have "\<forall>x4 x5. ((x5::real) < x4) = (\<not> x4 + - 1 * x5 \<le> 0)"
       by auto
     then have "(\<forall>r ra a rb rc rd. r < ra \<and> aEvalUni a r \<noteq> aEvalUni a ra \<and> getPoly a = (rb, rc, rd) \<longrightarrow> (\<exists>re\<ge>r. re \<le> ra \<and> rb * re\<^sup>2 + rc * re + rd = 0)) = (\<forall>r ra a rb rc rd. (ra + - 1 * r \<le> 0 \<or> (\<not> aEvalUni a r) \<noteq> aEvalUni a ra \<or> getPoly a \<noteq> (rb, rc, rd)) \<or> (\<exists>re. 0 \<le> re + - 1 * r \<and> re + - 1 * ra \<le> 0 \<and> rd + rb * re\<^sup>2 + rc * re = 0))"
       using f8 f7 f6 f5 f4 by presburger
     then have f9: "\<forall>r ra a rb rc rd. (ra + - 1 * r \<le> 0 \<or> (\<not> aEvalUni a r) \<noteq> aEvalUni a ra \<or> getPoly a \<noteq> (rb, rc, rd)) \<or> (\<exists>re. 0 \<le> re + - 1 * r \<and> re + - 1 * ra \<le> 0 \<and> rd + rb * re\<^sup>2 + rc * re = 0)"
       by (meson change_eval)
     obtain rr :: "real \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real" where
       "\<forall>x0 x1 x2 x4 x5. (\<exists>v6. 0 \<le> v6 + - 1 * x5 \<and> v6 + - 1 * x4 \<le> 0 \<and> x0 + x2 * v6\<^sup>2 + x1 * v6 = 0) = (0 \<le> rr x0 x1 x2 x4 x5 + - 1 * x5 \<and> rr x0 x1 x2 x4 x5 + - 1 * x4 \<le> 0 \<and> x0 + x2 * (rr x0 x1 x2 x4 x5)\<^sup>2 + x1 * rr x0 x1 x2 x4 x5 = 0)"
       by moura
     then have f10: "\<forall>r ra a rb rc rd. ra + - 1 * r \<le> 0 \<or> aEvalUni a r = aEvalUni a ra \<or> getPoly a \<noteq> (rb, rc, rd) \<or> r + - 1 * rr rd rc rb ra r \<le> 0 \<and> 0 \<le> ra + - 1 * rr rd rc rb ra r \<and> rd + rb * (rr rd rc rb ra r)\<^sup>2 + rc * rr rd rc rb ra r = 0"
       using f9 by simp
     have f11: "(rr c b a x v0_0 + - 1 * x \<le> 0) = (0 \<le> x + - 1 * rr c b a x v0_0)"
       by force
     have "\<forall>x0. (x < x0) = (\<not> x0 + - 1 * x \<le> 0)"
       by auto
     then have f12: "\<forall>r. c + a * r\<^sup>2 + b * r \<noteq> 0 \<or> \<not> r + - 1 * x \<le> 0"
       using x_prop by auto
     obtain rra :: real where
       "(\<exists>v0. \<not> 0 \<le> v0 + - 1 * x \<and> aEvalUni At v0 \<noteq> aEvalUni At x) = (\<not> 0 \<le> rra + - 1 * x \<and> aEvalUni At rra \<noteq> aEvalUni At x)"
       by moura
     then show ?thesis
       using f12 f11 f10 f3 f2 f1
     proof -
       have f1: "\<forall>x0. (x0 < x) = (\<not> 0 \<le> x0 + - 1 * x)"
         by auto
       have f2: "(0 \<le> v0_0a + - 1 * x) = (x + - 1 * v0_0a \<le> 0)"
         by auto
       have f3: "(rr c b a x v0_0a + - 1 * x \<le> 0) = (0 \<le> x + - 1 * rr c b a x v0_0a)"
         by auto
       obtain rrb :: real where
         "(\<exists>v0. \<not> 0 \<le> v0 + - 1 * x \<and> aEvalUni At v0 \<noteq> aEvalUni At x) = (\<not> 0 \<le> rrb + - 1 * x \<and> aEvalUni At rrb \<noteq> aEvalUni At x)"
         by blast
       then show ?thesis
         using f3 f2 f1 assms(1) f10 f12
         by smt
     qed
   qed
   then show ?thesis by auto
 qed
 
 
 lemma inequality_case : "(\<exists>(x::real). \<forall>(y::real)<x. (a::real) * y\<^sup>2 + (b::real) * y + (c::real) < 0) =
     (a < 0 \<or> a = 0 \<and> (0 < b \<or> b = 0 \<and> c < 0))"
 proof-
   let ?At = "(LessUni (a, b, c))"
   have firsth : "\<And>x. (\<forall>y<x. a * y\<^sup>2 + b * y + c < 0 \<Longrightarrow> a\<le>0)"
   proof -
     fix x
     assume lt: "\<forall>y<x. a * y\<^sup>2 + b * y + c < 0"
     have "\<exists>w. \<forall>y < w. y^2 > (-b/a)*y - c/a"  using ysq_dom_y_plus_coeff[where b = "-b/a", where c = "-c/a"]
       by auto   
     then have gtdomhelp: "a > 0 \<Longrightarrow> \<exists>w. \<forall>y < w. a*y^2 > a*((-b/a)*y - c/a)"
       by auto
     have "\<forall>y. (a > 0 \<longrightarrow> a*((-b/a)*y - c/a) = -b*y - c)"
       by (simp add: right_diff_distrib') 
     then have gtdom: "a > 0 \<Longrightarrow> \<exists>w.\<forall>y < w. a*y^2 > -b*y - c"
       using gtdomhelp
       by simp 
     then have " a > 0 \<Longrightarrow> False"
     proof - 
       assume "a > 0"
       then have "\<exists>w.\<forall>y < w. a*y^2 > -b*y - c" using gtdom by auto
       then obtain w where w_prop: "\<forall>y < w. a*y^2 + b*y + c > 0"
         by fastforce 
       let ?mn = "min w x - 1"
       have gtz: "a*?mn^2 + b*?mn + c > 0" using w_prop
         by auto
       have ltz: "a*?mn^2 + b*?mn + c < 0" using lt by auto
       then show "False" using gtz ltz by auto
     qed
     then show "a \<le> 0"
       by fastforce 
   qed
   have bleq0 : "\<And>x. (\<forall>y<x. b * y + c < 0 \<Longrightarrow> b\<ge>0)"
   proof -
     fix x
     assume lt: "\<forall>y<x. b * y + c < 0"
     have gtdom: "b < 0 \<Longrightarrow> \<exists>w.\<forall>y < w. b*y > - c"
       by (metis mult.commute neg_less_divide_eq)
     then have "b < 0 \<Longrightarrow> False"
     proof - 
       assume "b < 0"
       then have "\<exists>w.\<forall>y < w. b*y > - c" using gtdom by auto
       then obtain w where w_prop: "\<forall>y < w .b*y + c > 0"
         by fastforce 
       let ?mn = "min w x - 1"
       have gtz: "b*?mn + c > 0" using w_prop
         by auto
       have ltz: "b*?mn + c < 0" using lt by auto
       then show "False" using gtz ltz by auto
     qed
     then show "b \<ge> 0"
       by fastforce 
   qed
   have secondh: "\<And>x. (\<forall>y<x. a * y\<^sup>2 + b * y + c < 0 \<Longrightarrow> \<not> a < 0 \<Longrightarrow> \<not> 0 < b \<Longrightarrow> b = 0)"
     using firsth bleq0
     by (metis add.commute add.right_neutral less_eq_real_def mult_zero_class.mult_zero_left) 
   have thirdh : "\<And>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0 \<Longrightarrow> \<not> a < 0 \<Longrightarrow> \<not> 0 < b \<Longrightarrow> c < 0"
     using firsth secondh add.commute add.right_neutral infzeros mult_zero_class.mult_zero_left not_numeral_le_zero order_refl
     by (metis less_eq_real_def)
   have fourthh : "a < 0 \<Longrightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0"
   proof - 
     assume aleq: "a < 0"
     have "\<exists>(w::real). \<forall>(y::real). (y < w \<longrightarrow> y^2 > (-b/a)*y + (-c/a))"
       using ysq_dom_y_plus_coeff[where b = "-b/a", where c = "-c/a"]
       by blast 
     then have hyp:"\<exists>(w::real). \<forall>(y::real). (y < w \<longrightarrow> a*y^2 \<le> a*(-b/a)*y + a*(-c/a))"
-      by (metis (no_types, hide_lams) \<open>a < 0\<close> distrib_left less_eq_real_def linorder_not_le mult.assoc mult_less_cancel_left)
+      by (metis (no_types, opaque_lifting) \<open>a < 0\<close> distrib_left less_eq_real_def linorder_not_le mult.assoc mult_less_cancel_left)
     have "\<forall>y. a*(-b/a)*y + a*(-c/a) = -b*y -c"
       using \<open>a < 0\<close> by auto
     then have "\<exists>(w::real). \<forall>(y::real). (y < w \<longrightarrow> a*y^2 \<le> -b*y - c)"
       using hyp by auto
     then have "\<exists>(w::real). \<forall>(y::real). (y < w \<longrightarrow> a*y^2 + b*y + c \<le> 0)"
       by (metis add.commute add_uminus_conv_diff le_diff_eq mult_minus_left real_add_le_0_iff)
     then obtain w where w_prop: "\<forall>(y::real). (y < w \<longrightarrow> a*y^2 + b*y + c \<le> 0)" by auto
     have "\<exists>x. \<forall>y < x. aEvalUni ?At x = aEvalUni ?At y" using same_eval''
     proof -
       have f1: "\<forall>x0 x1. ((x0::real) < x1) = (\<not> 0 \<le> x0 + - 1 * x1)"
         by linarith
       have "a \<noteq> 0"
         using \<open>a < 0\<close> by force
       then obtain rr :: "atomUni \<Rightarrow> real" where
         "\<forall>r. 0 \<le> r + - 1 * rr (LessUni (a, b, c)) \<or> aEvalUni (LessUni (a, b, c)) r = aEvalUni (LessUni (a, b, c)) (rr (LessUni (a, b, c)))"
         using f1 by (metis getPoly.simps(4) same_eval'')
       then show ?thesis
         using f1 by meson
     qed
     then obtain x where x_prop: "\<forall>y < x. aEvalUni ?At x = aEvalUni ?At y" by auto
     let ?mn = "min x w - 1"
     have "\<forall>y < ?mn.  aEvalUni ?At y = True \<or> aEvalUni ?At y = False"
       using x_prop by auto
     have "\<forall> y < ?mn. aEvalUni ?At y = False \<longrightarrow> a*y^2 + b*y + c \<ge> 0"
       by auto
     then have "\<And>y. \<forall>y<w. a * y\<^sup>2 + b * y + c \<le> 0 \<Longrightarrow>
          y < min x w - 1 \<Longrightarrow>
          \<not> a * y\<^sup>2 + b * y + c < 0 \<Longrightarrow>
          a * y\<^sup>2 + b * y + c = 0"
     proof -
       fix y :: real
       assume a1: "y < min x w - 1"
       assume a2: "\<not> a * y\<^sup>2 + b * y + c < 0"
       assume a3: "\<forall>y<w. a * y\<^sup>2 + b * y + c \<le> 0"
       have "y < w"
         using a1 by linarith
       then show "a * y\<^sup>2 + b * y + c = 0"
         using a3 a2 less_eq_real_def by blast
     qed 
     then have "\<forall> y < ?mn. aEvalUni ?At y = False \<longrightarrow> a*y^2 + b*y + c = 0"
       using w_prop by auto    
     then have "\<forall> y < ?mn. aEvalUni ?At y = False \<Longrightarrow> False" using infzeros aleq
       by (metis power_zero_numeral zero_less_power2)
     then have "\<forall> y < ?mn. aEvalUni ?At y = True"
     proof -
       { fix rr :: real
         have "\<forall>r ra. (ra::real) < r \<or> \<not> ra < r + - 1"
           by linarith
         then have "\<not> rr < min x w - 1 \<or> aEvalUni (LessUni (a, b, c)) rr"
           by (metis (no_types) \<open>\<forall>y<min x w - 1. aEvalUni (LessUni (a, b, c)) y = False \<Longrightarrow> False\<close> ab_group_add_class.ab_diff_conv_add_uminus less_eq_real_def min_less_iff_disj not_le x_prop) }
       then show ?thesis
         by blast
     qed 
     then show ?thesis by auto
   qed
   have fifthh : "b > 0 \<Longrightarrow> \<exists>x. \<forall>y<x. b * y + c < 0"
   proof-
     assume bh : "b > 0"
     show "\<exists>x. \<forall>y<x. b * y + c < 0"
       apply(rule exI[where x="-c/b"])
       apply auto
       using bh
       by (simp add: mult.commute pos_less_minus_divide_eq) 
   qed
   show ?thesis
     apply(auto)
     using firsth apply simp
     using secondh apply simp 
     using thirdh apply simp
     using fourthh apply simp
     using fifthh by simp
 qed
 
 lemma inequality_case_geq : "(\<exists>(x::real). \<forall>(y::real)<x. (a::real) * y\<^sup>2 + (b::real) * y + (c::real) > 0) =
     (a > 0 \<or> a = 0 \<and> (0 > b \<or> b = 0 \<and> c > 0))"
 proof - 
   have s1: "\<forall>y. - 1 * a * y\<^sup>2 + - 1 * b * y + - 1 * c < 0 \<longleftrightarrow>  a * y\<^sup>2 +  b * y +  c > 0"
     by auto
   have s2: "(- 1 * a < 0 \<or> - 1 * a = 0 \<and> (0 < - 1 * b \<or> - 1 * b = 0 \<and> - 1 * c < 0)) \<longleftrightarrow>
    (a > 0 \<or> a = 0 \<and> (0 > b \<or>  b = 0 \<and> c > 0))  "
     by auto
   have "(\<exists>x. \<forall>y<x. - 1 * a * y\<^sup>2 + - 1 * b * y + - 1 * c < 0) =
   (- 1 * a < 0 \<or> - 1 * a = 0 \<and> (0 < - 1 * b \<or> - 1 * b = 0 \<and> - 1 * c < 0))"
     using inequality_case[where a = "-1*a", where b = "-1*b", where c= "-1*c"]
     by auto
   then show ?thesis
     using s1 s2 by auto
 qed
 
 lemma infinity_evalUni_LessUni : "(\<exists>x. \<forall>y<x. aEvalUni (LessUni p) y) = (evalUni (substNegInfinityUni (LessUni p)) x)"
 proof(cases p)
   case (fields a b c)
   show ?thesis 
     unfolding fields  apply simp
     using inequality_case[of a b c] .
 qed
 
 lemma infinity_evalUni_EqUni : "(\<exists>x. \<forall>y<x. aEvalUni (EqUni p) y) = (evalUni (substNegInfinityUni (EqUni p)) x)"
 proof(cases p)
   case (fields a b c)
   show ?thesis
     using infzeros[of _ a b c] by(auto simp add: fields)
 qed
 
 lemma infinity_evalUni_NeqUni : "(\<exists>x. \<forall>y<x. aEvalUni (NeqUni p) y) = (evalUni (substNegInfinityUni (NeqUni p)) x)"
 proof(cases p)
   case (fields a b c)
   show ?thesis
     unfolding fields  apply simp 
     using inequality_case[of a b c] 
     using inequality_case_geq[of a b c]
     by (metis less_numeral_extra(3) linorder_neqE_linordered_idom mult_eq_0_iff)
 
 qed
 
 lemma infinity_evalUni_LeqUni : "(\<exists>x. \<forall>y<x. aEvalUni (LeqUni p) y) = (evalUni (substNegInfinityUni (LeqUni p)) x)"
 proof(cases p)
   case (fields a b c)
   show ?thesis
     unfolding fields  apply simp 
   proof -
     have h1: "((\<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or> (\<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c = 0)) \<longrightarrow> (\<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0)"
       using less_eq_real_def
       by auto
     have h2: "(\<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) \<Longrightarrow> ((\<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or> (\<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c = 0))"
     proof -
       assume a1: "(\<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0)"
       have "\<not>(\<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c = 0) \<Longrightarrow> (\<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) " 
       proof - 
         assume a2: "\<not>(\<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c = 0)"
         then have "a \<noteq> 0 \<or> b \<noteq> 0 \<or> c \<noteq> 0" by auto
         then have "(a < 0 \<or> a = 0 \<and> (0 < b \<or> b = 0 \<and> c < 0))"
         proof - 
           have x1: "a > 0 \<Longrightarrow> False"
           proof - 
             assume "a > 0"
             then have "(\<exists>(x::real). \<forall>(y::real)<x. (a::real) * y\<^sup>2 + (b::real) * y + (c::real) > 0)" using inequality_case_geq
               by auto
             then  show ?thesis
               using a1 
               by (meson a2 linorder_not_le min_less_iff_conj)  
           qed
           then have x2: "a = 0 \<and> 0 > b \<Longrightarrow> False"
           proof - 
             assume "a = 0 \<and> 0 > b"
             then have "(\<exists>(x::real). \<forall>(y::real)<x. (a::real) * y\<^sup>2 + (b::real) * y + (c::real) > 0)" using inequality_case_geq
               by blast
             then show ?thesis
               using a1
               by (meson a2 linorder_not_le min_less_iff_conj) 
           qed
           then have x3: "a = 0 \<and> b = 0 \<and> c > 0 \<Longrightarrow> False "
             using a1 a2 by auto  
           show ?thesis using x1 x2 x3
             by (meson \<open>a \<noteq> 0 \<or> b \<noteq> 0 \<or> c \<noteq> 0\<close> linorder_neqE_linordered_idom) 
         qed
         then show "(\<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0)" using inequality_case
           by auto
       qed
       then show ?thesis
         by auto
     qed
     have "(\<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) = (\<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or> (\<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c = 0)"
       using h1 h2 by auto
     then show "(\<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) =
     (a < 0 \<or> a = 0 \<and> (0 < b \<or> b = 0 \<and> c < 0) \<or> a = 0 \<and> b = 0 \<and> c = 0)"
       using inequality_case[of a b c] infzeros[of _ a b c] by auto
   qed
 qed
 
 text "This is the vertical translation for substNegInfinityUni where we represent the virtual
 substution of negative infinity in the univariate case"
 lemma infinity_evalUni :
   shows "(\<exists>x. \<forall>y<x. aEvalUni At y) = (evalUni (substNegInfinityUni At) x)"
 proof(cases At)
   case (LessUni p)
   then show ?thesis using infinity_evalUni_LessUni by auto
 next
   case (EqUni p)
   then show ?thesis using infinity_evalUni_EqUni by auto
 next
   case (LeqUni p)
   then show ?thesis using infinity_evalUni_LeqUni by auto
 next
   case (NeqUni p)
   then show ?thesis using infinity_evalUni_NeqUni by auto
 qed
 
 end
\ No newline at end of file
diff --git a/thys/Virtual_Substitution/QE.thy b/thys/Virtual_Substitution/QE.thy
--- a/thys/Virtual_Substitution/QE.thy
+++ b/thys/Virtual_Substitution/QE.thy
@@ -1,8838 +1,8837 @@
 section "QE lemmas"
 theory QE
   imports Polynomials.MPoly_Type_Univariate
     Polynomials.Polynomials  Polynomials.MPoly_Type_Class_FMap 
     "HOL-Library.Quadratic_Discriminant"
 begin
 
 (* This file may take some time to load *)
 
 subsection "Useful Definitions/Setting Up"
 
 definition sign:: "real Polynomial.poly \<Rightarrow> real \<Rightarrow> int"
   where "sign p x \<equiv> (if poly p x = 0 then 0 else (if poly p x > 0 then 1 else -1))"
 
 definition sign_num:: "real \<Rightarrow> int"
   where "sign_num x \<equiv> (if x = 0 then 0 else (if x > 0 then 1 else -1))"
 
 definition root_list:: "real Polynomial.poly \<Rightarrow> real set"
   where "root_list p \<equiv> ({(x::real). poly p x = 0}::real set)" 
 
 definition root_set:: "(real \<times> real \<times> real) set \<Rightarrow> real set"
   where "root_set les \<equiv> ({(x::real). (\<exists> (a, b, c) \<in> les. a*x^2 + b*x + c = 0)}::real set)" 
 
 definition sorted_root_list_set:: "(real \<times> real \<times> real) set \<Rightarrow> real list"
   where "sorted_root_list_set p \<equiv> sorted_list_of_set (root_set p)" 
 
 definition nonzero_root_set:: "(real \<times> real \<times> real) set \<Rightarrow> real set"
   where "nonzero_root_set les \<equiv> ({(x::real). (\<exists> (a, b, c) \<in> les. (a \<noteq> 0 \<or> b \<noteq> 0) \<and> a*x^2 + b*x + c = 0)}::real set)" 
 
 definition sorted_nonzero_root_list_set:: "(real \<times> real \<times> real) set \<Rightarrow> real list"
   where "sorted_nonzero_root_list_set p \<equiv> sorted_list_of_set (nonzero_root_set p)" 
 
 
 (* Important property of sorted lists *)
 lemma sorted_list_prop:
   fixes l::"real list"
   fixes x::"real"
   assumes sorted: "sorted l"
   assumes lengt: "length l > 0"
   assumes xgt: "x > l ! 0"
   assumes xlt: "x \<le> l ! (length l - 1)"
   shows "\<exists>n. (n+1) < (length l) \<and> x \<ge> l !n \<and> x \<le> l ! (n + 1)"
 proof - 
   have "\<not>(\<exists>n. (n+1) < (length l) \<and> x \<ge> l !n \<and> x \<le> l ! (n + 1)) \<Longrightarrow> False"
   proof clarsimp 
     fix n
     assume alln: "\<forall>n. l ! n \<le> x \<longrightarrow> Suc n < length l \<longrightarrow> \<not> x \<le> l ! Suc n"
     have "\<forall>k. (k < length l \<longrightarrow> x > l ! k)" 
     proof clarsimp 
       fix k
       show "k < length l \<Longrightarrow> l ! k < x"
       proof (induct k)
         case 0
         then show ?case using xgt by auto
       next
         case (Suc k)
         then show ?case using alln
           using less_eq_real_def by auto 
       qed
     qed
     then show "False"
       using xlt diff_Suc_less lengt not_less
       by (metis One_nat_def) 
   qed
   then show ?thesis by auto
 qed
 
 
 subsection "Quadratic polynomial properties"
 lemma quadratic_poly_eval: 
   fixes a b c::"real"
   fixes x::"real"
   shows "poly [:c, b, a:] x = a*x^2 + b*x + c"    
 proof - 
   have "x * (b + x * a) = a * x\<^sup>2 + b * x" by (metis add.commute distrib_right mult.assoc mult.commute power2_eq_square)
   then show ?thesis by auto
 qed
 
 lemma poly_roots_set_same:
   fixes a b c:: "real"
   shows "{(x::real). a * x\<^sup>2 + b * x + c = 0} = {x. poly [:c, b, a:] x = 0}"
 proof - 
   have "\<forall>x. a*x^2 + b*x + c = poly [:c, b, a:] x"
   proof clarsimp 
     fix x
     show "a * x\<^sup>2 + b * x = x * (b + x * a)"
       using quadratic_poly_eval[of c b a x] by auto
   qed
   then show ?thesis 
     by auto
 qed
 
 lemma root_set_finite:
   assumes fin: "finite les"
   assumes nex: "\<not>(\<exists> (a, b, c) \<in> les. a = 0 \<and> b = 0 \<and> c = 0 )"
   shows "finite (root_set les)"
 proof - 
   have "\<forall>(a, b, c) \<in> les. finite {(x::real). a*x^2 + b*x + c = 0}" 
   proof clarsimp 
     fix a b c
     assume "(a, b, c) \<in> les"
     then have "[:c, b, a:] \<noteq> 0" using nex by auto
     then have "finite {x. poly [:c, b, a:] x = 0}"
       using poly_roots_finite[where p = "[:c, b, a:]"] by auto
     then show "finite {x. a * x\<^sup>2 + b * x + c = 0}"
       using  poly_roots_set_same by auto
   qed
   then show ?thesis using fin
     unfolding root_set_def by auto
 qed
 
 lemma nonzero_root_set_finite:
   assumes fin: "finite les"
   shows "finite (nonzero_root_set les)"
 proof - 
   have "\<forall>(a, b, c) \<in> les. (a \<noteq> 0 \<or> b \<noteq> 0) \<longrightarrow> finite {(x::real). a*x^2 + b*x + c = 0}" 
   proof clarsimp 
     fix a b c
     assume ins: "(a, b, c) \<in> les"
     assume "a = 0 \<longrightarrow> b \<noteq> 0"
     then have "[:c, b, a:] \<noteq> 0" using ins by auto
     then have "finite {x. poly [:c, b, a:] x = 0}"
       using poly_roots_finite[where p = "[:c, b, a:]"] by auto
     then show "finite {x. a * x\<^sup>2 + b * x + c = 0}"
       using  poly_roots_set_same by auto
   qed
   then show ?thesis using fin
     unfolding nonzero_root_set_def by auto
 qed
 
 lemma discriminant_lemma:
   fixes a b c r::"real"
   assumes aneq: "a \<noteq> 0"
   assumes beq: "b = 2 * a * r"
   assumes root: " a * r^2 - 2 * a * r*r + c = 0"
   shows "\<forall>x. a * x\<^sup>2 + b * x + c = 0 \<longleftrightarrow> x = -r"
 proof - 
   have "c = a*r^2" using root
     by (simp add: power2_eq_square)
   then have same: "b^2 - 4*a*c = (2 * a * r)^2 - 4*a*(a*r^2)" using beq
     by blast 
   have "(2 * a * r)^2 = 4*a^2*r^2"
     by (simp add: mult.commute power2_eq_square)
   then have "(2 * a * r)^2 - 4*a*(a*(r)^2) = 0"
     using power2_eq_square by auto 
   then have "b^2 - 4*a*c = 0" using same
     by simp 
   then have "\<forall>x. a * x\<^sup>2 + b * x + c = 0 \<longleftrightarrow> x = -b / (2 * a)"
     using discriminant_zero aneq unfolding discrim_def by auto
   then show ?thesis using beq
     by (simp add: aneq) 
 qed
 
 (* Show a polynomial only changes sign when it passes through a root *)
 lemma changes_sign:
   fixes p:: "real Polynomial.poly"
   shows "\<forall>x::real. \<forall> y::real. ((sign p x \<noteq> sign p y  \<and> x < y) \<longrightarrow> (\<exists>c \<in> (root_list p). x \<le> c \<and> c \<le> y))"
 proof clarsimp
   fix x y
   assume "sign p x \<noteq> sign p y"
   assume "x < y"
   then show "\<exists>c\<in>root_list p. x \<le> c \<and> c \<le> y"
     using poly_IVT_pos[of x y p] poly_IVT_neg[of x y p] 
     by (metis (mono_tags) \<open>sign p x \<noteq> sign p y\<close> less_eq_real_def linorder_neqE_linordered_idom mem_Collect_eq root_list_def sign_def)
 qed
 
 (* Show a polynomial only changes sign if it passes through a root *)
 lemma changes_sign_var:
   fixes a b c x y:: "real"
   shows "((sign_num (a*x^2 + b*x + c) \<noteq> sign_num (a*y^2 + b*y + c)  \<and> x < y) \<Longrightarrow> (\<exists>q. (a*q^2 + b*q + c = 0 \<and> x \<le> q \<and> q \<le> y)))"
 proof  clarsimp
   assume sn: "sign_num (a * x\<^sup>2 + b * x + c) \<noteq> sign_num (a * y\<^sup>2 + b * y + c)"
   assume xy: " x < y"
   let ?p = "[:c, b, a:]"
   have cs: "((sign ?p x \<noteq> sign ?p y  \<and> x < y) \<longrightarrow> (\<exists>c \<in> (root_list ?p). x \<le> c \<and> c \<le> y))"
     using changes_sign[of ?p] by auto
   have "(sign ?p x \<noteq> sign ?p y  \<and> x < y)"
     using sn xy unfolding sign_def sign_num_def using quadratic_poly_eval
     by presburger 
   then have "(\<exists>c \<in> (root_list ?p). x \<le> c \<and> c \<le> y)" 
     using cs 
     by auto 
   then obtain q where "q \<in> root_list ?p \<and> x \<le> q \<and> q \<le> y" 
     by auto
   then have "a*q^2 + b*q + c = 0 \<and> x \<le> q \<and> q \<le> y"
     unfolding root_list_def using quadratic_poly_eval[of c b a q]
     by auto
   then show "\<exists>q. a * q\<^sup>2 + b * q + c = 0 \<and> x \<le> q \<and> q \<le> y"  
     by auto
 qed
 
 subsection "Continuity Properties"
 lemma continuity_lem_eq0:
   fixes p::"real"
   shows "r < p \<Longrightarrow> \<forall>x\<in>{r <..p}. a * x\<^sup>2 + b * x + c = 0 \<Longrightarrow> (a = 0 \<and> b = 0 \<and> c = 0)" 
 proof - 
   assume r_lt: "r < p"
   assume inf_zer: "\<forall>x\<in>{r <..p}. a * x\<^sup>2 + b * x + c = 0"
   have nf: "\<not>finite {r..<p}" using Set_Interval.dense_linorder_class.infinite_Ioo r_lt by auto
   have "\<not>(a = 0 \<and> b = 0 \<and> c = 0) \<Longrightarrow> False"
   proof - 
     assume "\<not>(a = 0 \<and> b = 0 \<and> c = 0)"
     then have "[:c, b, a:] \<noteq> 0" by auto
     then have fin: "finite {x. poly [:c, b, a:] x = 0}" using poly_roots_finite[where p = "[:c, b, a:]"] by auto
     have "{x. a*x^2 + b*x + c = 0} = {x. poly [:c, b, a:] x = 0}" using quadratic_poly_eval by auto
     then have finset: "finite {x. a*x^2 + b*x + c = 0}"  using fin by auto
     have "{r <..p} \<subseteq> {x. a*x^2 + b*x + c = 0}" using inf_zer by blast
     then show "False" using finset nf
       using finite_subset
       by (metis (no_types, lifting) infinite_Ioc_iff r_lt) 
   qed
   then show "(a = 0 \<and> b = 0 \<and> c = 0)" by auto
 qed
 
 lemma continuity_lem_lt0: 
   fixes r:: "real"
   fixes a b c:: "real"
   shows "poly [:c, b, a:] r < 0 \<Longrightarrow>
     \<exists>y'> r. \<forall>x\<in>{r<..y'}. poly [:c, b, a:] x < 0"
 proof - 
   let ?f = "poly [:c,b,a:]"
   assume r_ltz: "poly [:c, b, a:] r < 0"
   then have "[:c, b, a:] \<noteq> 0" by auto
   then have "finite {x. poly [:c, b, a:] x = 0}" using poly_roots_finite[where p = "[:c, b, a:]"]    
     by auto
   then have fin: "finite  {x. x > r \<and> poly [:c, b, a:] x = 0}"
     using finite_Collect_conjI by blast 
   let ?l = "sorted_list_of_set {x. x > r \<and> poly [:c, b, a:] x = 0}"
   show ?thesis proof (cases "length ?l = 0")
     case True
     then have no_zer: "\<not>(\<exists>x>r. poly [:c, b, a:] x = 0)" using sorted_list_of_set_eq_Nil_iff fin by auto
     then have "\<And>y. y > r \<and> y < (r + 1) \<Longrightarrow> poly [:c, b, a:] y < 0 " 
     proof - 
       fix y
       assume "y > r \<and> y < r + 1"
       then show "poly [:c, b, a:] y < 0"
         using r_ltz no_zer poly_IVT_pos[where a = "r", where p = "[:c, b, a:]", where b = "y"]
         by (meson linorder_neqE_linordered_idom)
     qed
     then show ?thesis
-      by (metis (no_types, hide_lams) \<open>\<not> (\<exists>x>r. poly [:c, b, a:] x = 0)\<close> \<open>poly [:c, b, a:] r < 0\<close> greaterThanAtMost_iff linorder_neqE_linordered_idom linordered_field_no_ub poly_IVT_pos) 
+      by (metis greaterThanAtMost_iff less_add_one less_eq_real_def linorder_not_le no_zer poly_IVT_pos r_ltz)
   next
     case False
     then have len_nonz: "length (sorted_list_of_set {x. r < x \<and> poly [:c, b, a:] x = 0}) \<noteq> 0"
       by blast 
     then have "\<forall>n \<in> {x. x > r \<and> poly [:c, b, a:] x = 0}. (nth_default 0 ?l 0) \<le> n"
       using fin set_sorted_list_of_set sorted_sorted_list_of_set
       using  in_set_conv_nth leI not_less0 sorted_nth_mono
       by (smt not_less_iff_gr_or_eq nth_default_def)
     then have no_zer: "\<not>(\<exists>x>r. (x < (nth_default 0 ?l 0) \<and> poly [:c, b, a:] x = 0))"
       using sorted_sorted_list_of_set by auto
     then have fa: "\<And>y. y > r \<and> y < (nth_default 0 ?l 0) \<Longrightarrow> poly [:c, b, a:] y < 0 " 
     proof - 
       fix y
       assume "y > r \<and> y < (nth_default 0 ?l 0)"
       then show "poly [:c, b, a:] y < 0"
         using r_ltz no_zer poly_IVT_pos[where a = "r", where p = "[:c, b, a:]", where b = "y"]
         by (meson less_imp_le less_le_trans linorder_neqE_linordered_idom)
     qed
     have "nth_default 0 ?l 0 > r" using fin set_sorted_list_of_set
       using len_nonz length_0_conv length_greater_0_conv mem_Collect_eq nth_mem
       by (metis (no_types, lifting) nth_default_def)
     then have "\<exists>(y'::real). r < y' \<and> y' < (nth_default 0 ?l 0)"
       using dense by blast
     then obtain y' where y_prop:"r < y' \<and>y' < (nth_default 0 ?l 0)" by auto
     then have "\<forall>x\<in>{r<..y'}. poly [:c, b, a:] x < 0"
       using fa by auto
     then show ?thesis using y_prop by blast 
   qed
 qed
 
 lemma continuity_lem_gt0: 
   fixes r:: "real"
   fixes a b c:: "real"
   shows "poly [:c, b, a:] r > 0 \<Longrightarrow>
     \<exists>y'> r. \<forall>x\<in>{r<..y'}. poly [:c, b, a:] x > 0"
 proof -
   assume r_gtz: "poly [:c, b, a:] r > 0 "
   let ?p = "[:-c, -b, -a:]"
-  have revpoly: "\<forall>x. -1*(poly [:c, b, a:] x) = poly [:-c, -b, -a:] x"
-    by (metis (no_types, hide_lams) Polynomial.poly_minus add.inverse_neutral minus_pCons mult_minus1)
+  have revpoly: "\<forall>x. -1*(poly [:c, b, a:] x) = poly [:-c, -b, -a:] x"    
+    by (metis (no_types, opaque_lifting) Polynomial.poly_minus minus_pCons mult_minus1 neg_equal_0_iff_equal)
   then have "poly ?p r < 0" using r_gtz
     by (metis mult_minus1 neg_less_0_iff_less)
   then have "\<exists>y'> r. \<forall>x\<in>{r<..y'}. poly ?p x < 0" using continuity_lem_lt0
     by blast
   then obtain y' where y_prop: "y' > r \<and> (\<forall>x\<in>{r<..y'}. poly ?p x < 0)" by auto
   then have "\<forall>x\<in>{r<..y'}. poly [:c, b, a:] x > 0 " using revpoly
     using neg_less_0_iff_less by fastforce
   then show ?thesis 
     using y_prop  by blast 
 qed
 
 lemma continuity_lem_lt0_expanded: 
   fixes r:: "real"
   fixes a b c:: "real"
   shows "a*r^2 + b*r + c < 0 \<Longrightarrow>
     \<exists>y'> r. \<forall>x\<in>{r<..y'}. a*x^2 + b*x + c < 0"
   using quadratic_poly_eval continuity_lem_lt0 
   by (simp add: add.commute) 
 
 lemma continuity_lem_gt0_expanded: 
   fixes r:: "real"
   fixes a b c:: "real"
   fixes k::"real"
   assumes kgt: "k > r"
   shows "a*r^2 + b*r + c > 0 \<Longrightarrow>
     \<exists>x\<in>{r<..k}. a*x^2 + b*x + c > 0"
 proof -
   assume "a*r^2 + b*r + c > 0"
   then have "\<exists>y'> r. \<forall>x\<in>{r<..y'}. poly [:c, b, a:] x > 0"
     using continuity_lem_gt0 quadratic_poly_eval[of c b a r] by auto 
   then obtain y' where y_prop: "y' > r \<and> (\<forall>x\<in>{r<..y'}. poly [:c, b, a:] x > 0)" by auto
   then have "\<exists>q. q > r \<and> q < min k y'" using kgt dense
     by (metis min_less_iff_conj)   
   then obtain q where q_prop: "q > r \<and>q < min k y'" by auto
   then have "a*q^2 + b*q + c > 0" using y_prop  quadratic_poly_eval[of c b a q]
     by (metis greaterThanAtMost_iff less_eq_real_def min_less_iff_conj)
   then show ?thesis
     using q_prop by auto
 qed
 
 subsection "Negative Infinity (Limit) Properties"
 
 lemma ysq_dom_y: 
   fixes b:: "real"
   fixes c:: "real"
   shows "\<exists>(w::real). \<forall>(y:: real). (y < w \<longrightarrow> y^2 > b*y)"
 proof - 
   have c1: "b \<ge> 0 ==> \<exists>(w::real). \<forall>(y:: real). (y < w \<longrightarrow> y^2 > b*y)"
   proof - 
     assume "b \<ge> 0"
     then have p1: "\<forall>(y:: real). (y < -1 \<longrightarrow> y*b \<le> 0)"
       by (simp add: mult_nonneg_nonpos2)
     have p2: "\<forall>(y:: real). (y < -1 \<longrightarrow> y^2 > 0)"
       by auto 
     then have h1: "\<forall>(y:: real). (y < -1 \<longrightarrow> y^2 > b*y)"  
       using p1 p2
       by (metis less_eq_real_def less_le_trans mult.commute) 
     then show ?thesis by auto
   qed
   have c2: "b < 0 \<and> b > -1 ==> \<exists>(w::real). \<forall>(y:: real). (y < w \<longrightarrow> y^2 > b*y)"
   proof - 
     assume "b < 0 \<and> b > -1 "
     then have h1: "\<forall>(y:: real). (y < -1 \<longrightarrow> y^2 > b*y)"
       by (simp add: power2_eq_square)  
     then show ?thesis by auto
   qed   
   have c3: "b \<le> -1 ==> \<exists>(w::real). \<forall>(y:: real). (y < w \<longrightarrow> y^2 > b*y)"
   proof - 
     assume "b \<le> -1 "
     then have h1: "\<forall>(y:: real). (y < b \<longrightarrow> y^2 > b*y)"
       by (metis le_minus_one_simps(3) less_irrefl less_le_trans mult.commute mult_less_cancel_left power2_eq_square)
     then show ?thesis by auto
   qed   
   then  show ?thesis using c1 c2 c3
     by (metis less_trans linorder_not_less) 
 qed
 
 lemma ysq_dom_y_plus_coeff: 
   fixes b:: "real"
   fixes c:: "real"
   shows "\<exists>(w::real). \<forall>(y::real). (y < w \<longrightarrow> y^2 > b*y + c)"
 proof - 
   have "\<exists>(w::real). \<forall>(y:: real). (y < w \<longrightarrow> y^2 > b*y)" using ysq_dom_y by auto
   then obtain w where w_prop: "\<forall>(y:: real). (y < w \<longrightarrow> y^2 > b*y)" by auto
   have "c \<le> 0 \<Longrightarrow>  \<forall>(y::real). (y < w \<longrightarrow> y^2 > b*y + c)"
     using w_prop by auto 
   then have c1: "c \<le> 0 \<Longrightarrow> \<exists>(w::real). \<forall>(y::real). (y < w \<longrightarrow> y^2 > b*y + c)" by auto
   have "\<exists>(w::real). \<forall>(y:: real). (y < w \<longrightarrow> y^2 > (b-c)*y)" using ysq_dom_y by auto
   then obtain k where k_prop: "\<forall>(y:: real). (y < k \<longrightarrow> y^2 > (b-c)*y)" by auto
   let ?mn = "min k (-1)"
   have "(c> 0 \<Longrightarrow> (\<forall> y < -1. -c*y > c))" 
   proof - 
     assume cgt: " c> 0"
     show "\<forall>(y::real) < -1. -c*y > c"
     proof clarsimp
       fix y::"real"
       assume "y < -1"
       then have "-y > 1"
         by auto
       then have "c < c*(-y)" using cgt 
         by (metis \<open>1 < - y\<close> mult.right_neutral mult_less_cancel_left_pos)
       then show " c < - (c * y) "
         by auto
     qed
   qed
   then have "(c> 0 \<longrightarrow> (\<forall> y < ?mn. (b-c)*y > b*y + c))" 
     by (simp add: left_diff_distrib) 
   then have c2:  "c > 0 \<Longrightarrow>  \<forall>(y::real). (y < ?mn \<longrightarrow> y^2 > b*y + c)"
     using k_prop
     by force
   then have c2:  "c > 0 \<Longrightarrow> \<exists>(w::real). \<forall>(y::real). (y < w \<longrightarrow> y^2 > b*y + c)"
     by blast
   show ?thesis using c1 c2
     by fastforce
 qed
 
 lemma ysq_dom_y_plus_coeff_2: 
   fixes b:: "real"
   fixes c:: "real"
   shows "\<exists>(w::real). \<forall>(y::real). (y > w \<longrightarrow> y^2 > b*y + c)"
 proof - 
   have "\<exists>(w::real). \<forall>(y::real). (y < w \<longrightarrow> y^2 > -b*y + c)"
     using ysq_dom_y_plus_coeff[where b = "-b", where c = "c"] by auto
   then obtain w where w_prop: "\<forall>(y::real). (y < w \<longrightarrow> y^2 > -b*y + c)" by auto
   let ?mn = "min w (-1)"
   have "\<forall>(y::real). (y < ?mn \<longrightarrow> y^2 > -b*y + c)" using w_prop by auto
   then have "\<forall>(y::real). (y > (-1*?mn) \<longrightarrow> y^2 > b*y + c)"
-    by (metis (no_types, hide_lams) add.inverse_inverse minus_less_iff mult_minus1 mult_minus_left mult_minus_right power2_eq_square) 
+    by (metis (no_types, opaque_lifting) minus_less_iff minus_mult_commute mult_1 power2_eq_iff)
   then show ?thesis by auto
 qed
 
 lemma neg_lc_dom_quad: 
   fixes a:: "real"
   fixes b:: "real"
   fixes c:: "real"
   assumes alt: "a < 0"
   shows "\<exists>(w::real). \<forall>(y::real). (y > w \<longrightarrow> a*y^2 + b*y + c < 0)"
 proof -
   have "\<exists>(w::real). \<forall>(y::real). (y > w \<longrightarrow> y^2 > (-b/a)*y + (-c/a))"
     using ysq_dom_y_plus_coeff_2[where b = "-b/a", where c = "-c/a"] by auto
   then have keyh: "\<exists>(w::real). \<forall>(y::real). (y > w \<longrightarrow> a*y^2 < a*((-b/a)*y + (-c/a)))"
     using alt by auto
   have simp1: "\<forall>y. a*((-b/a)*y + (-c/a)) = a*(-b/a)*y + a*(-c/a)"
     using diff_divide_eq_iff by fastforce
   have simp2: "\<forall>y. a*(-b/a)*y + a*(-c/a) = -b*y + a*(-c/a)"
     using assms by auto
   have simp3: "\<forall>y. -b*y + a*(-c/a) = -b*y - c"
     using assms by auto
   then have "\<forall>y. a*((-b/a)*y + (-c/a)) = -b*y - c" using simp1 simp2 simp3 by auto
   then have keyh2: "\<exists>(w::real). \<forall>(y::real). (y > w \<longrightarrow> a*y^2 < -b*y-c)"
     using keyh by auto
   have "\<forall>y. a*y^2 < -b*y-c \<longleftrightarrow> a*y^2 + b*y + c < 0" by auto
   then show ?thesis using keyh2 by auto
 qed
 
 lemma pos_lc_dom_quad: 
   fixes a:: "real"
   fixes b:: "real"
   fixes c:: "real"
   assumes alt: "a > 0"
   shows "\<exists>(w::real). \<forall>(y::real). (y > w \<longrightarrow> a*y^2 + b*y + c > 0)"
 proof -
   have "-a < 0" using alt
     by simp 
   then have "\<exists>(w::real). \<forall>(y::real). (y > w \<longrightarrow> -a*y^2 - b*y - c < 0)"
     using neg_lc_dom_quad[where a = "-a", where b = "-b", where c = "-c"] by auto
   then obtain w where w_prop: "\<forall>(y::real). (y > w \<longrightarrow> -a*y^2 - b*y - c < 0)" by auto
   then have "\<forall>(y::real). (y > w \<longrightarrow> a*y^2 + b*y + c > 0)"
     by auto
   then show ?thesis by auto
 qed
 
 (* lemma interval_infinite: 
   fixes r p::"real"
   assumes "r < p"
   shows "infinite {r<..<p}"
   using Set_Interval.dense_linorder_class.infinite_Ioo using assms by blast 
 *)
 
 subsection "Infinitesimal and Continuity Properties"
 lemma les_qe_inf_helper: 
   fixes q:: "real"
   shows"(\<forall>(d, e, f)\<in>set les. \<exists>y'> q. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f < 0) \<Longrightarrow> 
     (\<exists>y'>q. (\<forall>(d, e, f)\<in>set les. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f < 0))"
 proof (induct les)
   case Nil
   then show ?case using gt_ex by auto 
 next
   case (Cons z les)
   have "\<forall>a\<in>set les. case a of (d, e, f) \<Rightarrow> \<exists>y'>q. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f < 0"
     using  Cons.prems by auto
   then have " \<exists>y'>q. \<forall>a\<in>set les. case a of (d, e, f) \<Rightarrow> \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f < 0"
     using Cons.hyps by auto
   then obtain y1 where y1_prop : "y1>q \<and> (\<forall>a\<in>set les. case a of (d, e, f) \<Rightarrow> \<forall>x\<in>{q<..y1}. d * x\<^sup>2 + e * x + f < 0)"
     by auto
   have "case z of (d, e, f) \<Rightarrow> \<exists>y'>q. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f < 0"
     using Cons.prems by auto
   then obtain y2 where y2_prop: "y2>q \<and> (case z of (d, e, f) \<Rightarrow> (\<forall>x\<in>{q<..y2}. d * x\<^sup>2 + e * x + f < 0))"
     by auto
   let ?y = "min y1 y2"
   have "?y > q \<and> (\<forall>a\<in>set (z#les). case a of (d, e, f) \<Rightarrow> \<forall>x\<in>{q<..?y}. d * x\<^sup>2 + e * x + f < 0)"
     using y1_prop y2_prop
     by force
   then show ?case
     by blast
 qed 
 
 lemma have_inbetween_point_les:
   fixes r::"real"
   assumes "(\<forall>(d, e, f)\<in>set les. \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f < 0)"
   shows "(\<exists>x. (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
 proof -
   have "(\<forall>(d, e, f)\<in>set les. \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f < 0) \<Longrightarrow> 
     (\<exists>y'>r. (\<forall>(d, e, f)\<in>set les. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f < 0))"
     using les_qe_inf_helper assms by auto
   then have "(\<exists>y'>r. (\<forall>(d, e, f)\<in>set les. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f < 0))"
     using assms
     by blast 
   then obtain y where y_prop: "y > r \<and>  (\<forall>(d, e, f)\<in>set les. \<forall>x\<in>{r<..y}. d * x\<^sup>2 + e * x + f < 0)"
     by auto
   have "\<exists>q. q >  r \<and>q < y" using y_prop dense by auto
   then obtain q where q_prop: "q > r \<and> q < y" by auto
   then have "(\<forall>(d, e, f)\<in>set les. d*q^2 + e*q + f < 0)"
     using y_prop by auto
   then show ?thesis
     by auto
 qed
 
 lemma one_root_a_gt0: 
   fixes a b c r:: "real"
   shows "\<And>y'. b = 2 * a * r \<Longrightarrow>
           \<not> a < 0 \<Longrightarrow>
           a * r^2 - 2 * a *r*r + c = 0 \<Longrightarrow>
           - r < y' \<Longrightarrow>
           \<exists>x\<in>{-r<..y'}. \<not> a * x\<^sup>2 + 2 * a * r*x + c < 0"
 proof - 
   fix y'
   assume beq: "b = 2 * a * r"
   assume aprop: " \<not> a < 0"
   assume root: " a * r\<^sup>2 - 2 * a *r*r + c = 0"
   assume rootlt: "- r < y'"
   show " \<exists>x\<in>{- r<..y'}. \<not> a * x\<^sup>2 + 2 * a* r*x+ c < 0"
   proof - 
     have h: "a = 0 \<Longrightarrow> (b = 0 \<and> c = 0)" using beq root   
       by auto
     then have aeq: "a = 0 \<Longrightarrow> \<exists>x\<in>{- r<..y'}. \<not> a * x\<^sup>2 + 2 * a*r*x + c < 0"
       using rootlt
       by (metis add.left_neutral continuity_lem_eq0 less_numeral_extra(3) mult_zero_left mult_zero_right) 
     then have alt: "a > 0 \<Longrightarrow> \<exists>x\<in>{- r<..y'}. \<not> a * x\<^sup>2 + 2 * a *r*x + c < 0"
     proof - 
       assume agt: "a > 0"
       then have "\<exists>(w::real). \<forall>(y::real). (y > w \<longrightarrow> a*y^2 + b*y + c > 0)"
         using pos_lc_dom_quad by auto
       then obtain w where w_prop: "\<forall>y::real. (y > w \<longrightarrow> a*y^2 + b*y + c > 0)" by auto
       have isroot: "a*(-r)^2 + b*(-r) + c = 0" using root beq by auto
       then have wgteq: "w \<ge> -(r)"
       proof -
         have "w < -r \<Longrightarrow> False"
           using w_prop isroot by auto
         then show ?thesis
           using not_less by blast 
       qed
       then have w1: "w + 1 > -r"
         by auto 
       have w2: "a*(w + 1)^2 + b*(w+1) + c > 0" using w_prop by auto
       have rootiff: "\<forall>x. a * x\<^sup>2 + b * x + c = 0 \<longleftrightarrow> x = -r" using  discriminant_lemma[where a = "a", where b = "b", where c= "c", where r = "r"]
           isroot agt beq by auto
       have allgt: "\<forall>x > -r. a*x^2 + b*x + c > 0"
       proof clarsimp 
         fix x
         assume "x > -r"
         have xgtw:  "x > w + 1 \<Longrightarrow> a*x^2 + b*x + c > 0 "
           using w1 w2 rootiff  poly_IVT_neg[where a = "w+1", where b = "x", where p = "[:c,b,a:]"] 
             quadratic_poly_eval
           by (metis less_eq_real_def linorder_not_less) 
         have xltw: "x < w + 1 \<Longrightarrow> a*x^2 + b*x + c > 0"
           using w1 w2 rootiff poly_IVT_pos[where a= "x", where b = "w + 1", where p = "[:c,b,a:]"]
             quadratic_poly_eval less_eq_real_def linorder_not_less
           by (metis \<open>- r < x\<close>)
         then show "a*x^2 + b*x + c > 0"
           using w2 xgtw xltw by fastforce 
       qed
       have "\<exists>z. z > -r \<and> z < y'" using rootlt dense[where x = "-r", where y = "y'"] 
         by auto
       then obtain z where z_prop: " z > -r \<and> z  < y'" by auto
       then have "a*z^2 + b*z + c > 0" using allgt by auto
       then show ?thesis using z_prop
         using beq greaterThanAtMost_iff by force 
     qed
     then show ?thesis using aeq alt aprop
       by linarith
   qed
 qed
 
 lemma leq_qe_inf_helper: 
   fixes q:: "real"
   shows"(\<forall>(d, e, f)\<in>set leq. \<exists>y'> q. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<Longrightarrow> 
     (\<exists>y'>q. (\<forall>(d, e, f)\<in>set leq. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<le> 0))"
 proof (induct leq)
   case Nil
   then show ?case using gt_ex by auto 
 next
   case (Cons z leq)
   have "\<forall>a\<in>set leq. case a of (d, e, f) \<Rightarrow> \<exists>y'>q. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<le> 0"
     using  Cons.prems by auto
   then have " \<exists>y'>q. \<forall>a\<in>set leq. case a of (d, e, f) \<Rightarrow> \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<le> 0"
     using Cons.hyps by auto
   then obtain y1 where y1_prop : "y1>q \<and> (\<forall>a\<in>set leq. case a of (d, e, f) \<Rightarrow> \<forall>x\<in>{q<..y1}. d * x\<^sup>2 + e * x + f \<le> 0)"
     by auto
   have "case z of (d, e, f) \<Rightarrow> \<exists>y'>q. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<le> 0"
     using Cons.prems by auto
   then obtain y2 where y2_prop: "y2>q \<and> (case z of (d, e, f) \<Rightarrow> (\<forall>x\<in>{q<..y2}. d * x\<^sup>2 + e * x + f \<le> 0))"
     by auto
   let ?y = "min y1 y2"
   have "?y > q \<and> (\<forall>a\<in>set (z#leq). case a of (d, e, f) \<Rightarrow> \<forall>x\<in>{q<..?y}. d * x\<^sup>2 + e * x + f \<le> 0)"
     using y1_prop y2_prop
     by force
   then show ?case
     by blast
 qed 
 
 lemma neq_qe_inf_helper: 
   fixes q:: "real"
   shows"(\<forall>(d, e, f)\<in>set neq. \<exists>y'> q. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0) \<Longrightarrow> 
     (\<exists>y'>q. (\<forall>(d, e, f)\<in>set neq. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
 proof (induct neq)
   case Nil
   then show ?case using gt_ex by auto 
 next
   case (Cons z neq)
   have "\<forall>a\<in>set neq. case a of (d, e, f) \<Rightarrow> \<exists>y'>q. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0"
     using  Cons.prems by auto
   then have " \<exists>y'>q. \<forall>a\<in>set neq. case a of (d, e, f) \<Rightarrow> \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0"
     using Cons.hyps by auto
   then obtain y1 where y1_prop : "y1>q \<and> (\<forall>a\<in>set neq. case a of (d, e, f) \<Rightarrow> \<forall>x\<in>{q<..y1}. d * x\<^sup>2 + e * x + f \<noteq> 0)"
     by auto
   have "case z of (d, e, f) \<Rightarrow> \<exists>y'>q. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0"
     using Cons.prems by auto
   then obtain y2 where y2_prop: "y2>q \<and> (case z of (d, e, f) \<Rightarrow> (\<forall>x\<in>{q<..y2}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
     by auto
   let ?y = "min y1 y2"
   have "?y > q \<and> (\<forall>a\<in>set (z#neq). case a of (d, e, f) \<Rightarrow> \<forall>x\<in>{q<..?y}. d * x\<^sup>2 + e * x + f \<noteq> 0)"
     using y1_prop y2_prop
     by force
   then show ?case
     by blast
 qed 
 
 
 subsection "Some Casework"
 
 lemma quadratic_shape1a:
   fixes a b c x y::"real"
   assumes agt: "a > 0"
   assumes xyroots: "x < y \<and> a*x^2 + b*x + c = 0 \<and> a*y^2 + b*y + c = 0"
   shows "\<And>z. (z > x \<and> z < y \<Longrightarrow> a*z^2 + b*z + c < 0)"
 proof clarsimp 
   fix z
   assume zgt: "z > x"
   assume zlt: "z < y"
   have frac_gtz: "(1/(2*a)) > 0" using agt
     by simp 
   have xy_prop:"(x = (-b + sqrt(b^2 - 4*a*c))/(2*a) \<and> y = (-b - sqrt(b^2 - 4*a*c))/(2*a))
     \<or> (y = (-b + sqrt(b^2 - 4*a*c))/(2*a) \<and> x = (-b - sqrt(b^2 - 4*a*c))/(2*a))" 
     using xyroots agt discriminant_iff unfolding discrim_def by auto   
   have "b^2 - 4*a*c \<ge> 0" using xyroots discriminant_iff
     using assms(1) discrim_def by auto 
   then have pos_discrim: "b^2 - 4*a*c > 0" using xyroots discriminant_zero
     using \<open>0 \<le> b\<^sup>2 - 4 * a * c\<close> assms(1) discrim_def less_eq_real_def linorder_not_less
     by metis
   then have sqrt_gt: "sqrt(b^2 - 4*a*c) > 0"
     using real_sqrt_gt_0_iff by blast 
   then have "(- b - sqrt(b^2 - 4*a*c)) < (- b + sqrt(b^2 - 4*a*c))"
     by auto
   then have "(- b - sqrt(b^2 - 4*a*c))*(1/(2*a)) < (- b + sqrt(b^2 - 4*a*c))*(1/(2*a)) "
     using frac_gtz
     by (simp add: divide_strict_right_mono) 
   then have "(- b - sqrt(b^2 - 4*a*c))/(2*a) < (- b + sqrt(b^2 - 4*a*c))/(2*a)"
     by auto
   then have xandy: "x = (- b - sqrt(b^2 - 4*a*c))/(2*a) \<and> y = (- b + sqrt(b^2 - 4*a*c))/(2*a)"
     using xy_prop xyroots by auto
   let ?mdpt = "-b/(2*a)"
   have xlt: "x < ?mdpt"
     using xandy sqrt_gt using frac_gtz divide_minus_left divide_strict_right_mono sqrt_gt
     by (smt (verit) agt)
   have ylt: "?mdpt < y"
     using xandy sqrt_gt frac_gtz
     by (smt (verit, del_insts) divide_strict_right_mono zero_less_divide_1_iff) 
   have mdpt_val: "a*?mdpt^2 + b*?mdpt + c < 0"
   proof - 
     have firsteq: "a*?mdpt^2 + b*?mdpt + c = (a*b^2)/(4*a^2) - (b^2)/(2*a) + c"
       by (simp add: power2_eq_square) 
     have h1: "(a*b^2)/(4*a^2) = (b^2)/(4*a)"
       by (simp add: power2_eq_square)
     have h2: "(b^2)/(2*a) = (2*b^2)/(4*a)"
       by linarith
     have h3: "c = (4*a*c)/(4*a)"
       using agt by auto 
     have "a*?mdpt^2 + b*?mdpt + c = (b^2)/(4*a) - (2*b^2)/(4*a) + (4*a*c)/(4*a) "
       using firsteq h1 h2 h3
       by linarith 
     then have "a*?mdpt^2 + b*?mdpt + c = (b^2 - 2*b^2 + 4*a*c)/(4*a)"
       by (simp add: diff_divide_distrib)
     then have eq2: "a*?mdpt^2 + b*?mdpt + c = (4*a*c - b^2)/(4*a)"
       by simp
     have h: "4*a*c - b^2 < 0" using pos_discrim by auto
     have "1/(4*a) > 0" using agt by auto
     then have "(4*a*c - b^2)*(1/(4*a)) < 0"
       using h
       using mult_less_0_iff by blast 
     then show ?thesis using eq2 by auto
   qed
   have nex: "\<not> (\<exists>k> x. k < y \<and> poly [:c, b, a:] k = 0)"
     using discriminant_iff agt
-    by (metis (no_types, hide_lams) discrim_def order_less_irrefl quadratic_poly_eval xandy) 
+    by (metis discrim_def less_irrefl quadratic_poly_eval xandy)
   have nor2: "\<not> (\<exists>x>z. x < - b / (2 * a) \<and> poly [:c, b, a:] x = 0)"
     using nex xlt ylt zgt zlt by auto
   have nor: "\<not> (\<exists>x>- b / (2 * a). x < z \<and> poly [:c, b, a:] x = 0)"
     using nex xlt ylt zgt zlt discriminant_iff agt  by auto 
   then have mdpt_lt: "?mdpt < z \<Longrightarrow> a*z^2 + b*z + c < 0 "
     using mdpt_val zgt zlt xlt ylt poly_IVT_pos[where p = "[:c, b, a:]", where a= "?mdpt", where b = "z"] 
       quadratic_poly_eval[of c b a]
     by (metis \<open>\<not> (\<exists>k>x. k < y \<and> poly [:c, b, a:] k = 0)\<close> linorder_neqE_linordered_idom) 
   have mdpt_gt: "?mdpt > z \<Longrightarrow> a*z^2 + b*z + c < 0 "
     using zgt zlt mdpt_val xlt ylt nor2 poly_IVT_neg[where p = "[:c, b, a:]", where a = "z", where b = "?mdpt"] quadratic_poly_eval[of c b a]
     by (metis linorder_neqE_linordered_idom nex) 
   then show "a*z^2 + b*z + c < 0" 
     using mdpt_lt mdpt_gt mdpt_val by fastforce 
 qed
 
 lemma quadratic_shape1b:
   fixes a b c x y::"real"
   assumes agt: "a > 0"
   assumes xy_roots: "x < y \<and> a*x^2 + b*x + c = 0 \<and> a*y^2 + b*y + c = 0"
   shows "\<And>z. (z > y \<Longrightarrow> a*z^2 + b*z + c > 0)"
 proof - 
   fix z
   assume z_gt :"z > y"
   have nogt: "\<not>(\<exists>w. w > y \<and> a*w^2 + b*w + c = 0)" using xy_roots discriminant_iff
     by (metis agt less_eq_real_def linorder_not_less)
   have "\<exists>(w::real). \<forall>(y::real). (y > w \<longrightarrow> a*y^2 + b*y + c > 0)"
     using agt pos_lc_dom_quad by auto
   then have "\<exists>k > y.  a*k^2 + b*k + c > 0"
     by (metis add.commute agt less_add_same_cancel1 linorder_neqE_linordered_idom pos_add_strict) 
   then obtain k where k_prop: "k > y \<and> a*k^2 + b*k + c > 0" by auto
   have kgt: "k > z \<Longrightarrow> a*z^2 + b*z + c > 0" 
   proof - 
     assume kgt: "k > z"
     then have zneq: "a*z^2 + b*z + c = 0 \<Longrightarrow> False"
       using nogt  using z_gt by blast 
     have znlt: "a*z^2 + b*z + c < 0 \<Longrightarrow> False"
       using kgt k_prop quadratic_poly_eval[of c b a] z_gt  nogt poly_IVT_pos[where a= "z", where b = "k", where p = "[:c, b, a:]"]
       by (metis less_eq_real_def less_le_trans)
     then show "a*z^2 + b*z + c > 0" using zneq znlt
       using linorder_neqE_linordered_idom by blast 
   qed
   have klt: "k < z \<Longrightarrow> a*z^2 + b*z + c > 0" 
   proof - 
     assume klt: "k < z"
     then have zneq: "a*z^2 + b*z + c = 0 \<Longrightarrow> False"
       using nogt using z_gt by blast 
     have znlt: "a*z^2 + b*z + c < 0 \<Longrightarrow> False"
       using klt k_prop quadratic_poly_eval[of c b a] z_gt  nogt poly_IVT_neg[where a= "k", where b = "z", where p = "[:c, b, a:]"]
       by (metis add.commute add_less_cancel_left add_mono_thms_linordered_field(3) less_eq_real_def)
     then show "a*z^2 + b*z + c > 0" using zneq znlt
       using linorder_neqE_linordered_idom by blast 
   qed
   then show "a*z^2 + b*z + c > 0" using k_prop kgt klt
     by fastforce 
 qed
 
 lemma quadratic_shape2a:
   fixes a b c x y::"real"
   assumes "a < 0"
   assumes "x < y \<and> a*x^2 + b*x + c = 0 \<and> a*y^2 + b*y + c = 0"
   shows "\<And>z. (z > x \<and> z < y \<Longrightarrow> a*z^2 + b*z + c > 0)"
   using quadratic_shape1a[where a= "-a", where b = "-b", where c = "-c", where x = "x", where y = "y"]
   using assms(1) assms(2) by fastforce 
 
 lemma quadratic_shape2b:
   fixes a b c x y::"real"
   assumes "a < 0"
   assumes "x < y \<and> a*x^2 + b*x + c = 0 \<and> a*y^2 + b*y + c = 0"
   shows "\<And>z. (z > y \<Longrightarrow> a*z^2 + b*z + c < 0)"
   using quadratic_shape1b[where a= "-a", where b = "-b", where c = "-c", where x = "x", where y = "y"]
   using assms(1) assms(2) by force 
 
 lemma case_d1:
   fixes a b c r::"real"
   shows "b < 2 * a * r \<Longrightarrow>
     a * r^2 - b*r + c = 0 \<Longrightarrow>
     \<exists>y'>- r. \<forall>x\<in>{-r<..y'}. a * x\<^sup>2 + b * x + c < 0"
 proof - 
   assume b_lt: "b < 2*a*r"
   assume root: "a*r^2 - b*r + c = 0"
   then have "c = b*r - a*r^2" by auto
   have aeq: "a = 0 \<Longrightarrow> \<exists>y'>- r. \<forall>x\<in>{-r<..y'}. a * x\<^sup>2 + b * x + c < 0"
   proof - 
     assume azer: "a = 0"
     then have bltz: "b < 0" using b_lt by auto
     then have "c = b*r" using azer root by auto
     then have eval: "\<forall>x. a*x^2 + b*x + c = b*(x + r)" using azer
       by (simp add: distrib_left) 
     have "\<forall>x > -r. b*(x + r) < 0" 
     proof clarsimp 
       fix x
       assume xgt: "- r < x"
       then have "x + r > 0"
         by linarith 
       then show "b * (x + r) < 0"
         using bltz using mult_less_0_iff by blast 
     qed
     then show ?thesis using eval
       using less_add_same_cancel1 zero_less_one
       by (metis greaterThanAtMost_iff)
   qed
   have aneq: "a \<noteq> 0 \<Longrightarrow>\<exists>y'>- r. \<forall>x\<in>{-r<..y'}. a * x\<^sup>2 + b * x + c < 0"
   proof - 
     assume aneq: "(a::real) \<noteq> 0"
     have "b^2 - 4*a*c < 0 \<Longrightarrow> a * r\<^sup>2 + b * r + c \<noteq> 0" using root discriminant_negative[of a b c r] unfolding discrim_def 
       using aneq by auto
     then have " a * r\<^sup>2 + b * r + c \<noteq> 0 \<Longrightarrow>
     a * r\<^sup>2 - b * r + c = 0 \<Longrightarrow>
     b\<^sup>2 < 4 * a * c \<Longrightarrow> False"
     proof -
       assume a1: "a * r\<^sup>2 - b * r + c = 0"
       assume a2: "b\<^sup>2 < 4 * a * c"
       have f3: "(0 \<le> - 1 * (4 * a * c) + (- 1 * b)\<^sup>2) = (4 * a * c + - 1 * (- 1 * b)\<^sup>2 \<le> 0)"
         by simp
       have f4: "(- 1 * b)\<^sup>2 + - 1 * (4 * a * c) = - 1 * (4 * a * c) + (- 1 * b)\<^sup>2"
         by auto
       have f5: "c + a * r\<^sup>2 + - 1 * b * r = a * r\<^sup>2 + c + - 1 * b * r"
         by auto
       have f6: "\<forall>x0 x1 x2 x3. (x3::real) * x0\<^sup>2 + x2 * x0 + x1 = x1 + x3 * x0\<^sup>2 + x2 * x0"
         by simp
       have f7: "\<forall>x1 x2 x3. (discrim x3 x2 x1 < 0) = (\<not> 0 \<le> discrim x3 x2 x1)"
         by auto
       have f8: "\<forall>r ra rb. discrim r ra rb = ra\<^sup>2 + - 1 * (4 * r * rb)"
         using discrim_def by auto
       have "\<not> 4 * a * c + - 1 * (- 1 * b)\<^sup>2 \<le> 0"
         using a2 by simp
       then have "a * r\<^sup>2 + c + - 1 * b * r \<noteq> 0"
         using f8 f7 f6 f5 f4 f3 by (metis (no_types) aneq discriminant_negative)
       then show False
         using a1 by linarith
     qed 
     then have "\<not>(b^2 - 4*a*c < 0)" using root
       using \<open>b\<^sup>2 - 4 * a * c < 0 \<Longrightarrow> a * r\<^sup>2 + b * r + c \<noteq> 0\<close> by auto
     then have discrim: "b\<^sup>2 \<ge> 4 * a * c " by auto
     then have req: "r = (b + sqrt(b^2 - 4*a*c))/(2*a) \<or> r = (b - sqrt(b^2 - 4*a*c))/(2*a)"
       using aneq root discriminant_iff[where a="a", where b ="-b", where c="c", where x="r"] unfolding discrim_def
       by auto
     then have "r = (b - sqrt(b^2 - 4*a*c))/(2*a) \<Longrightarrow> b > 2*a*r"
     proof - 
       assume req: "r = (b - sqrt(b^2 - 4*a*c))/(2*a)"
       then have h1: "2*a*r = 2*a*((b - sqrt(b^2 - 4*a*c))/(2*a))" by auto
       then have h2: "2*a*((b - sqrt(b^2 - 4*a*c))/(2*a)) = b - sqrt(b^2 - 4*a*c)"
         using aneq by auto
       have h3: "sqrt(b^2 - 4*a*c) \<ge> 0" using discrim  by auto
       then have "b - sqrt(b^2 - 4*a*c) < b"
         using b_lt h1 h2 by linarith
       then show ?thesis using req h2 by auto
     qed
     then have req: "r = (b + sqrt(b^2 - 4*a*c))/(2*a)" using req b_lt by auto
     then have discrim2: "b^2 - 4 *a*c > 0" using aneq b_lt  by auto
     then have "\<exists>x y. x \<noteq> y \<and> a * x\<^sup>2 + b * x + c = 0 \<and> a * y\<^sup>2 + b * y + c = 0"
       using aneq discriminant_pos_ex[of a b c] unfolding discrim_def
       by auto
     then obtain x y where xy_prop: "x < y \<and> a * x\<^sup>2 + b * x + c = 0 \<and> a * y\<^sup>2 + b * y + c = 0"
       by (meson linorder_neqE_linordered_idom)
     then have "(x = (-b + sqrt(b^2 - 4*a*c))/(2*a) \<and> y = (-b - sqrt(b^2 - 4*a*c))/(2*a))
 \<or> (y = (-b + sqrt(b^2 - 4*a*c))/(2*a) \<and> x = (-b - sqrt(b^2 - 4*a*c))/(2*a))" 
       using aneq discriminant_iff unfolding discrim_def by auto   
     then have xy_prop2: "(x = (-b + sqrt(b^2 - 4*a*c))/(2*a) \<and> y = -r)
     \<or> (y = (-b + sqrt(b^2 - 4*a*c))/(2*a) \<and> x = -r)" using req
       by (simp add: \<open>x = (- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<and> y = (- b - sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<or> y = (- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<and> x = (- b - sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)\<close> minus_divide_left)
         (* When a < 0, -r is the bigger root *)
     have alt: "a < 0 \<Longrightarrow> \<forall>k > -r. a * k^2 + b * k + c < 0"
     proof clarsimp 
       fix k
       assume alt: " a < 0"
       assume "- r < k"
       have alt2: " (1/(2*a)::real) < 0" using alt
         by simp 
       have "(-b - sqrt(b^2 - 4*a*c)) < (-b + sqrt(b^2 - 4*a*c))"
         using discrim2 by auto
       then have "(-b - sqrt(b^2 - 4*a*c))* (1/(2*a)::real) > (-b + sqrt(b^2 - 4*a*c))* (1/(2*a)::real)"
         using alt2
         using mult_less_cancel_left_neg by fastforce 
       then have rgtroot: "-r >  (-b + sqrt(b^2 - 4*a*c))/(2*a)"
         using req  \<open>x = (- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<and> y = (- b - sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<or> y = (- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<and> x = (- b - sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)\<close> xy_prop2
         by auto 
       then have "(y = -r \<and> x = (-b + sqrt(b^2 - 4*a*c))/(2*a))"
         using xy_prop xy_prop2 by auto
       then show "a * k^2 + b * k + c < 0"
         using xy_prop \<open>- r < k\<close> alt quadratic_shape2b xy_prop 
         by blast 
     qed
       (* When a > 0, -r is the smaller root *)
     have agt: "a > 0 \<Longrightarrow> \<exists>y'>- r. \<forall>x\<in>{-r<..y'}. a * x\<^sup>2 + b * x + c < 0"
     proof - 
       assume agt: "a> 0"
       have alt2: " (1/(2*a)::real) > 0" using agt
         by simp 
       have "(-b - sqrt(b^2 - 4*a*c)) < (-b + sqrt(b^2 - 4*a*c))"
         using discrim2 by auto
       then have "(-b - sqrt(b^2 - 4*a*c))* (1/(2*a)::real) < (-b + sqrt(b^2 - 4*a*c))* (1/(2*a)::real)"
         using alt2
       proof -
         have f1: "- b - sqrt (b\<^sup>2 - c * (4 * a)) < - b + sqrt (b\<^sup>2 - c * (4 * a))"
           by (metis \<open>- b - sqrt (b\<^sup>2 - 4 * a * c) < - b + sqrt (b\<^sup>2 - 4 * a * c)\<close> mult.commute)
         have "0 < a * 2"
           using \<open>0 < 1 / (2 * a)\<close> by auto
         then show ?thesis
           using f1 by (simp add: divide_strict_right_mono mult.commute)
       qed
       then have rlltroot: "-r <  (-b + sqrt(b^2 - 4*a*c))/(2*a)"
         using req \<open>x = (- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<and> y = (- b - sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<or> y = (- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<and> x = (- b - sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)\<close> xy_prop2
         by auto
       then have "(x = -r \<and> y = (-b + sqrt(b^2 - 4*a*c))/(2*a))"
         using xy_prop xy_prop2 by auto
       have "\<exists>k. x < k \<and> k < y" using xy_prop dense by auto
       then obtain k where k_prop: "x < k \<and> k < y" by auto
       then have "\<forall>x\<in>{-r<..k}. a * x\<^sup>2 + b * x + c < 0"
         using agt quadratic_shape1a[where a= "a", where b = "b", where c= "c", where x = "x", where y = "y"]
         using \<open>x = - r \<and> y = (- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)\<close> greaterThanAtMost_iff xy_prop by auto 
       then show "\<exists>y'>- r. \<forall>x\<in>{-r<..y'}. a * x\<^sup>2 + b * x + c < 0" 
         using k_prop using \<open>x = - r \<and> y = (- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)\<close> by blast 
     qed
     show ?thesis
       using alt agt
       by (metis aneq greaterThanAtMost_iff less_add_same_cancel1 linorder_neqE_linordered_idom zero_less_one) 
   qed
   show "\<exists>y'>- r. \<forall>x\<in>{-r<..y'}. a * x\<^sup>2 + b * x + c < 0" using aeq aneq
     by blast 
 qed
 
 lemma case_d4:
   fixes a b c r::"real"
   shows "\<And>y'. b \<noteq> 2 * a * r \<Longrightarrow>
           \<not> b < 2 * a * r \<Longrightarrow>
           a *r^2 - b * r + c = 0 \<Longrightarrow>
           -r < y' \<Longrightarrow> \<exists>x\<in>{-r<..y'}. \<not> a * x\<^sup>2 + b * x + c < 0"
 proof - 
   fix y'
   assume bneq: "b \<noteq> 2 * a * r"
   assume bnotless: "\<not> b < 2 * a * r"
   assume root: "a *r^2 - b * r + c = 0"
   assume y_prop: "-r < y'"
   have b_gt: "b > 2*a*r" using bneq bnotless by auto
   have aeq: "a = 0 \<Longrightarrow> \<exists>y'>- r. \<forall>x\<in>{-r<..y'}. a * x\<^sup>2 + b * x + c > 0"
   proof - 
     assume azer: "a = 0"
     then have bgt: "b > 0" using b_gt by auto
     then have "c = b*r" using azer root by auto
     then have eval: "\<forall>x. a*x^2 + b*x + c = b*(x + r)" using azer
       by (simp add: distrib_left) 
     have "\<forall>x > -r. b*(x + r) > 0" 
     proof clarsimp 
       fix x
       assume xgt: "- r < x"
       then have "x + r > 0"
         by linarith 
       then show "b * (x + r) > 0"
         using bgt by auto 
     qed
     then show ?thesis using eval 
       using less_add_same_cancel1 zero_less_one
       by (metis greaterThanAtMost_iff) 
   qed
   have aneq: "a \<noteq> 0 \<Longrightarrow>\<exists>y'>- r. \<forall>x\<in>{-r<..y'}. a * x\<^sup>2 + b * x + c > 0"
   proof - 
     assume aneq: "a\<noteq>0"  
     {
       assume a1: "a * r\<^sup>2 - b * r + c = 0"
       assume a2: "b\<^sup>2 < 4 * a * c"
       have f3: "(0 \<le> - 1 * (4 * a * c) + (- 1 * b)\<^sup>2) = (4 * a * c + - 1 * (- 1 * b)\<^sup>2 \<le> 0)"
         by simp
       have f4: "(- 1 * b)\<^sup>2 + - 1 * (4 * a * c) = - 1 * (4 * a * c) + (- 1 * b)\<^sup>2"
         by auto
       have f5: "c + a * r\<^sup>2 + - 1 * b * r = a * r\<^sup>2 + c + - 1 * b * r"
         by auto
       have f6: "\<forall>x0 x1 x2 x3. (x3::real) * x0\<^sup>2 + x2 * x0 + x1 = x1 + x3 * x0\<^sup>2 + x2 * x0"
         by simp
       have f7: "\<forall>x1 x2 x3. (discrim x3 x2 x1 < 0) = (\<not> 0 \<le> discrim x3 x2 x1)"
         by auto
       have f8: "\<forall>r ra rb. discrim r ra rb = ra\<^sup>2 + - 1 * (4 * r * rb)"
         using discrim_def by auto
       have "\<not> 4 * a * c + - 1 * (- 1 * b)\<^sup>2 \<le> 0"
         using a2 by simp
       then have "a * r\<^sup>2 + c + - 1 * b * r \<noteq> 0"
         using f8 f7 f6 f5 f4 f3 by (metis (no_types) aneq discriminant_negative)
       then have False
         using a1 by linarith
     } note * = this
     have "b^2 - 4*a*c < 0 \<Longrightarrow> a * r\<^sup>2 + b * r + c \<noteq> 0" using root discriminant_negative[of a b c r] unfolding discrim_def 
       using aneq by auto
     then have "\<not>(b^2 - 4*a*c < 0)" using root * by auto
     then have discrim: "b\<^sup>2 \<ge> 4 * a * c " by auto
     then have req: "r = (b + sqrt(b^2 - 4*a*c))/(2*a) \<or> r = (b - sqrt(b^2 - 4*a*c))/(2*a)"
       using aneq root discriminant_iff[where a="a", where b ="-b", where c="c", where x="r"] unfolding discrim_def
       by auto
     then have "r = (b + sqrt(b^2 - 4*a*c))/(2*a) \<Longrightarrow> b < 2*a*r"
     proof - 
       assume req: "r = (b + sqrt(b^2 - 4*a*c))/(2*a)"
       then have h1: "2*a*r = 2*a*((b + sqrt(b^2 - 4*a*c))/(2*a))" by auto
       then have h2: "2*a*((b + sqrt(b^2 - 4*a*c))/(2*a)) = b + sqrt(b^2 - 4*a*c)"
         using aneq by auto
       have h3: "sqrt(b^2 - 4*a*c) \<ge> 0" using discrim  by auto
       then have "b + sqrt(b^2 - 4*a*c) > b"
         using b_gt h1 h2 by linarith
       then show ?thesis using req h2 by auto
     qed
     then have req: "r = (b - sqrt(b^2 - 4*a*c))/(2*a)" using req b_gt
       using aneq discrim by auto
     then have discrim2: "b^2 - 4 *a*c > 0" using aneq b_gt  by auto
     then have "\<exists>x y. x \<noteq> y \<and> a * x\<^sup>2 + b * x + c = 0 \<and> a * y\<^sup>2 + b * y + c = 0"
       using aneq discriminant_pos_ex[of a b c] unfolding discrim_def
       by auto
     then obtain x y where xy_prop: "x < y \<and> a * x\<^sup>2 + b * x + c = 0 \<and> a * y\<^sup>2 + b * y + c = 0"
       by (meson linorder_neqE_linordered_idom)
     then have "(x = (-b + sqrt(b^2 - 4*a*c))/(2*a) \<and> y = (-b - sqrt(b^2 - 4*a*c))/(2*a))
 \<or> (y = (-b + sqrt(b^2 - 4*a*c))/(2*a) \<and> x = (-b - sqrt(b^2 - 4*a*c))/(2*a))" 
       using aneq discriminant_iff unfolding discrim_def by auto   
     then have xy_prop2: "(x = (-b - sqrt(b^2 - 4*a*c))/(2*a) \<and> y = -r)
     \<or> (y = (-b - sqrt(b^2 - 4*a*c))/(2*a) \<and> x = -r)" using req divide_inverse minus_diff_eq mult.commute mult_minus_right
       by (smt (verit, ccfv_SIG) uminus_add_conv_diff)
         (* When a > 0, -r is the greater root *)
     have agt: "a > 0 \<Longrightarrow> \<forall>k > -r. a * k^2 + b * k + c > 0"
     proof clarsimp 
       fix k
       assume agt: " a > 0"
       assume "- r < k"
       have agt2: " (1/(2*a)::real) > 0" using agt
         by simp 
       have "(-b - sqrt(b^2 - 4*a*c)) < (-b + sqrt(b^2 - 4*a*c))"
         using discrim2 by auto
       then have "(-b - sqrt(b^2 - 4*a*c))* (1/(2*a)::real) < (-b + sqrt(b^2 - 4*a*c))* (1/(2*a)::real)"
         using agt2 by (simp add: divide_strict_right_mono) 
       then have rgtroot: "-r >  (-b - sqrt(b^2 - 4*a*c))/(2*a)"
         using  req \<open>x = (- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<and> y = (- b - sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<or> y = (- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<and> x = (- b - sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)\<close> xy_prop2
         by auto 
       then have "(x = (-b - sqrt(b^2 - 4*a*c))/(2*a)) \<and> y = -r"
         using xy_prop xy_prop2 
         by auto
       then show "a * k^2 + b * k + c > 0"
         using \<open>- r < k\<close> xy_prop agt quadratic_shape1b[where a= "a", where b ="b", where c="c", where x = "x", where y = "-r", where z = "k"]    
         by blast 
     qed
       (* When a < 0, -r is the smaller root *)
     have agt2: "a < 0 \<Longrightarrow> \<exists>y'>- r. \<forall>x\<in>{-r<..y'}. a * x\<^sup>2 + b * x + c > 0"
     proof - 
       assume alt: "a<0"
       have alt2: " (1/(2*a)::real) < 0" using alt
         by simp 
       have "(-b - sqrt(b^2 - 4*a*c)) < (-b + sqrt(b^2 - 4*a*c))"
         using discrim2 by auto
       then have "(-b - sqrt(b^2 - 4*a*c))* (1/(2*a)::real) > (-b + sqrt(b^2 - 4*a*c))* (1/(2*a)::real)"
         using alt2  using mult_less_cancel_left_neg by fastforce 
       then have rlltroot: "-r < (-b - sqrt(b^2 - 4*a*c))/(2*a)"
         using req 
         using \<open>x = (- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<and> y = (- b - sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<or> y = (- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a) \<and> x = (- b - sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)\<close> 
           xy_prop2 
         by auto
       then have h: "(x = -r \<and> y = (-b - sqrt(b^2 - 4*a*c))/(2*a))"
         using xy_prop xy_prop2 
         by auto
       have "\<exists>k. x < k \<and> k < y" using xy_prop dense by auto
       then obtain k where k_prop: "x < k \<and> k < y" by auto
       then have "\<forall>x\<in>{-r<..k}. a * x\<^sup>2 + b * x + c > 0"
         using alt quadratic_shape2a[where a= "a", where b = "b", where c= "c", where x = "x", where y = "y"]
           xy_prop  h greaterThanAtMost_iff  by auto 
       then show "\<exists>y'>- r. \<forall>x\<in>{-r<..y'}. a * x\<^sup>2 + b * x + c > 0" 
         using k_prop using h by blast 
     qed
     show ?thesis
       using aneq agt agt2
       by (meson greaterThanAtMost_iff linorder_neqE_linordered_idom y_prop) 
   qed
   show "\<exists>x\<in>{-r<..y'}. \<not> a * x\<^sup>2 + b * x + c < 0" using aneq aeq
     by (metis greaterThanAtMost_iff less_eq_real_def linorder_not_less y_prop)
 qed
 
 lemma one_root_a_lt0:
   fixes a b c r y'::"real"
   assumes alt: "a < 0"
   assumes beq: "b = 2 * a * r"
   assumes root: " a * r^2 - 2*a*r*r + c = 0"
   shows "\<exists>y'>- r. \<forall>x\<in>{- r<..y'}. a * x\<^sup>2 + 2*a*r*x + c < 0"
 proof - 
   have root_iff: "\<forall>x. a * x\<^sup>2 + b * x + c = 0 \<longleftrightarrow> x = -r" using alt root discriminant_lemma[where r = "r"] beq 
     by auto
   have "a < 0 \<longrightarrow> (\<exists>y. \<forall>x > y. a*x^2 + b*x + c < 0)" using neg_lc_dom_quad 
     by auto
   then obtain k where k_prop: "\<forall>x > k. a*x^2 + b*x + c < 0" using alt by auto
   let ?mx = "max (k+1) (-r + 1)"
   have "a*?mx^2 + b*?mx + c < 0" using k_prop by auto
   then have "\<exists>y > -r. a*y^2 + b*y + c < 0"
     by force
   then obtain z where z_prop: "z > -r \<and> a*z^2 + b*z + c < 0" by auto
   have poly_eval_prop: "\<forall>(x::real). poly [:c, b, a :] x = a*x^2 + b*x + c" 
     using quadratic_poly_eval by auto
   then have nozer: "\<not>(\<exists>x. (x > -r \<and> poly [:c, b, a :] x = 0))" using root_iff 
     by (simp add: add.commute) 
   have poly_z: "poly [:c, b, a:] z < 0" using z_prop poly_eval_prop by auto
   have "\<forall>y > -r. a*y^2 + b*y + c < 0" 
   proof clarsimp 
     fix y
     assume ygt: "- r < y"
     have h1: "y = z \<Longrightarrow> a * y\<^sup>2 + b * y + c < 0" using z_prop by auto
     have h2: "y < z \<Longrightarrow> a * y\<^sup>2 + b * y + c < 0" proof -
       assume ylt: "y < z"
       have notz: "a*y^2 + b*y + c \<noteq> 0" using ygt nozer poly_eval_prop by auto
       have h1: "a *y^2 + b*y + c > 0 \<Longrightarrow> poly [:c, b, a:] y > 0" using poly_eval_prop by auto
       have ivtprop: "poly [:c, b, a:] y > 0 \<Longrightarrow> (\<exists>x. y < x \<and> x < z \<and> poly [:c, b, a:] x = 0)" 
         using ylt poly_z poly_IVT_neg[where a = "y", where b = "z", where p = "[:c, b, a:]"]
         by auto
       then have "a*y^2 + b*y + c > 0 \<Longrightarrow> False" using h1 ivtprop ygt nozer by auto
       then show "a*y^2 + b*y + c < 0" using notz
         using linorder_neqE_linordered_idom by blast 
     qed
     have h3: "y > z \<Longrightarrow> a * y\<^sup>2 + b * y + c < 0" 
     proof -
       assume ygtz: "y > z"
       have notz: "a*y^2 + b*y + c \<noteq> 0" using ygt nozer poly_eval_prop by auto
       have h1: "a *y^2 + b*y + c > 0 \<Longrightarrow> poly [:c, b, a:] y > 0" using poly_eval_prop by auto
       have ivtprop: "poly [:c, b, a:] y > 0 \<Longrightarrow> (\<exists>x. z < x \<and> x < y \<and> poly [:c, b, a:] x = 0)" 
         using ygtz poly_z using poly_IVT_pos by blast 
       then have "a*y^2 + b*y + c > 0 \<Longrightarrow> False" using h1 ivtprop z_prop nozer by auto
       then show "a*y^2 + b*y + c < 0" using notz
         using linorder_neqE_linordered_idom by blast 
     qed
     show "a * y\<^sup>2 + b * y + c < 0" using h1 h2 h3
       using linorder_neqE_linordered_idom by blast
   qed
   then show ?thesis
     using \<open>\<exists>y>- r. a * y\<^sup>2 + b * y + c < 0\<close> beq by auto 
 qed
 
 
 lemma one_root_a_lt0_var:
   fixes a b c r y'::"real"
   assumes alt: "a < 0"
   assumes beq: "b = 2 * a * r"
   assumes root: " a * r^2 - 2*a*r*r + c = 0"
   shows "\<exists>y'>- r. \<forall>x\<in>{- r<..y'}. a * x\<^sup>2 + 2*a*r*x + c \<le> 0"
 proof - 
   have h1: "\<exists>y'>- r. \<forall>x\<in>{- r<..y'}. a * x\<^sup>2 + 2 * a * r * x + c < 0 \<Longrightarrow>
      \<exists>y'>-r. \<forall>x\<in>{- r<..y'}. a * x\<^sup>2 + 2 * a *r * x + c \<le> 0"
     using less_eq_real_def by blast
   then show ?thesis
     using one_root_a_lt0[of a b r] assms by auto
 qed
 
 subsection "More Continuity Properties"
 lemma continuity_lem_gt0_expanded_var: 
   fixes r:: "real"
   fixes a b c:: "real"
   fixes k::"real"
   assumes kgt: "k > r"
   shows "a*r^2 + b*r + c > 0 \<Longrightarrow>
     \<exists>x\<in>{r<..k}. a*x^2 + b*x + c \<ge> 0"
 proof -
   assume a: "a*r^2 + b*r + c > 0  "
   have h: "\<exists>x\<in>{r<..k}. a*x^2 + b*x + c > 0  \<Longrightarrow> \<exists>x\<in>{r<..k}. a*x^2 + b*x + c \<ge> 0"
     using less_eq_real_def by blast 
   have "\<exists>x\<in>{r<..k}. a*x^2 + b*x + c > 0" using a continuity_lem_gt0_expanded[of r k a b c] assms by auto
   then show "\<exists>x\<in>{r<..k}. a*x^2 + b*x + c \<ge> 0"
     using h by auto
 qed
 
 lemma continuity_lem_lt0_expanded_var: 
   fixes r:: "real"
   fixes a b c:: "real"
   shows "a*r^2 + b*r + c < 0 \<Longrightarrow>
     \<exists>y'> r. \<forall>x\<in>{r<..y'}. a*x^2 + b*x + c \<le> 0"
 proof - 
   assume "a*r^2 + b*r + c < 0 "
   then have " \<exists>y'> r. \<forall>x\<in>{r<..y'}. a*x^2 + b*x + c < 0"
     using continuity_lem_lt0_expanded by auto
   then show " \<exists>y'> r. \<forall>x\<in>{r<..y'}. a*x^2 + b*x + c \<le> 0"
     using less_eq_real_def by auto
 qed
 
 lemma nonzcoeffs:
   fixes a b c r::"real"
   shows "a\<noteq>0 \<or> b\<noteq>0 \<or> c\<noteq>0 \<Longrightarrow> \<exists>y'>r. \<forall>x\<in>{r<..y'}. a * x\<^sup>2 + b * x + c \<noteq> 0 "
 proof - 
   assume "a\<noteq>0 \<or> b\<noteq>0 \<or> c\<noteq>0"
   then have fin: "finite {x. a*x^2 + b*x + c = 0}"
     by (metis pCons_eq_0_iff poly_roots_finite poly_roots_set_same) 
       (* then have fin2: "finite {x. a*x^2 + b*x + c = 0 \<and> x > r}"
     using finite_Collect_conjI by blast *)
   let ?s = "{x. a*x^2 + b*x + c = 0}"
   have imp: "(\<exists>q \<in> ?s. q > r) \<Longrightarrow> (\<exists>q \<in> ?s. (q > r \<and> (\<forall>x \<in> ?s. x > r \<longrightarrow> x \<ge> q)))"
   proof - 
     assume asm: "(\<exists>q \<in> ?s. q > r)"
     then have none: "{q. q \<in> ?s \<and> q > r} \<noteq> {}"
       by blast 
     have fin2: "finite {q. q \<in> ?s \<and> q > r}" using fin
       by simp
     have "\<exists>k. k = Min  {q. q \<in> ?s \<and> q > r}" using fin2 none
       by blast
     then obtain k where k_prop: "k =  Min  {q. q \<in> ?s \<and> q > r}" by auto
     then have kp1: "k \<in> ?s \<and> k > r" 
       using Min_in fin2 none
       by blast 
     then  have kp2: "\<forall>x \<in> ?s. x > r \<longrightarrow> x \<ge> k" 
       using Min_le fin2  using k_prop by blast 
     show "(\<exists>q \<in> ?s. (q > r \<and> (\<forall>x \<in> ?s. x > r \<longrightarrow> x \<ge> q)))" 
       using kp1 kp2 by blast
   qed
   have h2: "(\<exists>q \<in> ?s. q > r) \<Longrightarrow> \<exists>y'>r. \<forall>x\<in>{r<..y'}. a * x\<^sup>2 + b * x + c \<noteq> 0" 
   proof - 
     assume "(\<exists>q \<in> ?s. q > r)"
     then obtain q where q_prop: "q \<in> ?s \<and> (q > r \<and> (\<forall>x \<in> ?s. x > r \<longrightarrow> x \<ge> q))"
       using imp by blast 
     then have "\<exists>w. w > r \<and> w < q" using dense
       by blast
     then obtain w where w_prop: "w > r \<and> w < q" by auto
     then have "\<not>(\<exists>x\<in>{r<..w}. x \<in> ?s)"
       using w_prop q_prop by auto
     then have "\<forall>x\<in>{r<..w}. a * x\<^sup>2 + b * x + c \<noteq> 0"
       by blast
     then show "\<exists>y'>r. \<forall>x\<in>{r<..y'}. a * x\<^sup>2 + b * x + c \<noteq> 0"
       using w_prop by blast
   qed
   have h1: "\<not>(\<exists>q \<in> ?s. q > r) \<Longrightarrow> \<exists>y'>r. \<forall>x\<in>{r<..y'}. a * x\<^sup>2 + b * x + c \<noteq> 0"
   proof - 
     assume "\<not>(\<exists>q \<in> ?s. q > r)"
     then have "\<forall>x\<in>{r<..(r+1)}. a * x\<^sup>2 + b * x + c \<noteq> 0"
       using greaterThanAtMost_iff by blast
     then show ?thesis
       using less_add_same_cancel1 less_numeral_extra(1) by blast 
   qed
   then show "\<exists>y'>r. \<forall>x\<in>{r<..y'}. a * x\<^sup>2 + b * x + c \<noteq> 0"
     using h2 by blast
 qed
 
 
 (* Show if there are infinitely many values of x where a*x^2 + b*x + c is 0,
 then the a*x^2 + b*x + c is the zero polynomial *)
 lemma infzeros : 
   fixes y:: "real"
   assumes "\<forall>x::real < (y::real). a * x\<^sup>2 + b * x + c = 0"
   shows "a = 0 \<and> b=0 \<and> c=0"
 proof - 
   let ?A = "{(x::real). x < y}"
   have "\<exists> (n::nat) f. ?A = f ` {i. i < n} \<and> inj_on f {i. i < n} \<Longrightarrow> False"
   proof clarsimp 
     fix n:: "nat" 
     fix f
     assume xlt: "{x. x < y} = f ` {i. i < n}"
     assume injh: "inj_on f {i. i < n}"
     have "?A \<noteq> {}"
       by (simp add: linordered_field_no_lb)
     then have ngtz: "n > 0" 
       using xlt injh using gr_implies_not_zero by auto 
     have cardisn: "card ?A = n" using xlt injh
       by (simp add: card_image) 
     have "\<forall>k::nat. ((y - (k::nat) - 1) \<in> ?A)" 
       by auto
     then have subs: "{k. \<exists>(x::nat). k = y - x - 1 \<and> 0 \<le> x \<and> x \<le> n} \<subseteq> ?A"
       by auto
     have seteq: "(\<lambda>x. y - real x - 1) ` {0..n} ={k. \<exists>(x::nat). k = y - x - 1 \<and> 0 \<le> x \<and> x \<le> n}"
       by auto
     have injf: "inj_on (\<lambda>x. y - real x - 1) {0..n}"
       unfolding inj_on_def by auto
     have "card {k. \<exists>(x::nat). k = y - x - 1 \<and> 0 \<le> x \<and> x \<le> n} = n + 1"
       using  injf seteq card_atMost inj_on_iff_eq_card[where A = "{0..n}", where f = "\<lambda>x. y - x - 1"] 
       by auto
     then have if_fin: "finite ?A \<Longrightarrow> card ?A \<ge> n + 1" 
       using subs card_mono
       by (metis (lifting) card_mono)
     then have if_inf: "infinite ?A \<Longrightarrow> card ?A = 0"
       by (meson card.infinite)
     then show "False" using if_fin if_inf cardisn ngtz by auto
   qed
   then have nfin: "\<not> finite {(x::real). x < y}"
     using finite_imp_nat_seg_image_inj_on by blast
   have "{(x::real). x < y} \<subseteq> {x. a*x^2 + b*x + c = 0}"
     using assms by auto
   then have nfin2: "\<not> finite {x. a*x^2 + b*x + c = 0}"
     using nfin finite_subset by blast 
   {
     fix x
     assume "a * x\<^sup>2 + b * x + c = 0"
     then have f1: "a * (x * x) + x * b + c = 0"
       by (simp add: Groups.mult_ac(2) power2_eq_square)
     have f2: "\<forall>r. c + (r + (c + - c)) = r + c"
       by simp
     have f3: "\<forall>r ra rb. (rb::real) * ra + ra * r = (rb + r) * ra"
       by (metis Groups.mult_ac(2) Rings.ring_distribs(2))
     have "\<forall>r. r + (c + - c) = r"
       by simp
     then have "c + x * (b + x * a) = 0"
       using f3 f2 f1 by (metis Groups.add_ac(3) Groups.mult_ac(1) Groups.mult_ac(2))
   }
   hence "{x. a*x^2 + b*x + c = 0} \<subseteq> {x. poly [:c, b, a:] x = 0}"
     by auto
   then have " \<not> finite {x. poly [:c, b, a:] x = 0}" 
     using nfin2 using finite_subset by blast 
   then have "[:c, b, a:] = 0" 
     using poly_roots_finite[where p = "[:c, b, a:]"]  by auto
   then show ?thesis
     by auto
 qed
 
 
 
 lemma have_inbetween_point_leq:
   fixes r::"real"
   assumes "(\<forall>((d::real), (e::real), (f::real))\<in>set leq. \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0)"
   shows "(\<exists>x. (\<forall>(a, b, c)\<in>set leq. a * x\<^sup>2 + b * x + c \<le> 0))"
 proof -
   have "(\<forall>(d, e, f)\<in>set leq. \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<Longrightarrow> 
     (\<exists>y'>r. (\<forall>(d, e, f)\<in>set leq. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0))"
     using leq_qe_inf_helper assms by auto
   then have "(\<exists>y'>r. (\<forall>(d, e, f)\<in>set leq. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0))"
     using assms
     by blast 
   then obtain y where y_prop: "y > r \<and>  (\<forall>(d, e, f)\<in>set leq. \<forall>x\<in>{r<..y}. d * x\<^sup>2 + e * x + f \<le> 0)"
     by auto
   have "\<exists>q. q >  r \<and>q < y" using y_prop dense by auto
   then obtain q where q_prop: "q > r \<and> q < y" by auto
   then have "(\<forall>(d, e, f)\<in>set leq. d*q^2 + e*q + f \<le> 0)"
     using y_prop by auto
   then show ?thesis
     by auto
 qed
 
 
 lemma have_inbetween_point_neq:
   fixes r::"real"
   assumes "(\<forall>((d::real), (e::real), (f::real))\<in>set neq. \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)"
   shows "(\<exists>x. (\<forall>(a, b, c)\<in>set neq. a * x\<^sup>2 + b * x + c \<noteq> 0))"
 proof -
   have "(\<forall>(d, e, f)\<in>set neq. \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0) \<Longrightarrow> 
     (\<exists>y'>r. (\<forall>(d, e, f)\<in>set neq. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
     using neq_qe_inf_helper assms by auto
   then have "(\<exists>y'>r. (\<forall>(d, e, f)\<in>set neq. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
     using assms
     by blast 
   then obtain y where y_prop: "y > r \<and>  (\<forall>(d, e, f)\<in>set neq. \<forall>x\<in>{r<..y}. d * x\<^sup>2 + e * x + f \<noteq> 0)"
     by auto
   have "\<exists>q. q >  r \<and>q < y" using y_prop dense by auto
   then obtain q where q_prop: "q > r \<and> q < y" by auto
   then have "(\<forall>(d, e, f)\<in>set neq. d*q^2 + e*q + f \<noteq> 0)"
     using y_prop by auto
   then show ?thesis
     by auto
 qed
 
 subsection "Setting up and Helper Lemmas"
 subsubsection "The les\\_qe lemma"
 lemma les_qe_forward : 
   shows "(((\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or>
      (\<exists>(a', b', c')\<in>set les.
          a' = 0 \<and>
          b' \<noteq> 0 \<and>
          (\<forall>(d, e, f)\<in>set les.
              \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<or>
          a' \<noteq> 0 \<and>
          4 * a' * c' \<le> b'\<^sup>2 \<and>
          ((\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a').
                  \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<or>
           (\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0))))) \<Longrightarrow> ((\<exists>x. (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)))"
 proof -
   assume big_asm: "(((\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or>
      (\<exists>(a', b', c')\<in>set les.
          a' = 0 \<and>
          b' \<noteq> 0 \<and>
          (\<forall>(d, e, f)\<in>set les.
              \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<or>
          a' \<noteq> 0 \<and>
          4 * a' * c' \<le> b'\<^sup>2 \<and>
          ((\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a').
                  \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<or>
           (\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0)))))"
   then have big_or: "(\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or> 
                (\<exists>(a', b', c')\<in>set les.
                a' = 0 \<and>
                b' \<noteq> 0 \<and> (\<forall>(d, e, f)\<in>set les. \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0)) 
         \<or>
         (\<exists>(a', b', c')\<in>set les.
                a' \<noteq> 0 \<and>
                4 * a' * c' \<le> b'\<^sup>2 \<and>
                (\<forall>(d, e, f)\<in>set les.
                    \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a').
                       \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0)) 
       \<or>
       (\<exists>(a', b', c')\<in>set les.  a' \<noteq> 0 \<and>
                4 * a' * c' \<le> b'\<^sup>2 \<and>
                 (\<forall>(d, e, f)\<in>set les.
                     \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                        \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                           d * x\<^sup>2 + e * x + f < 0)) " 
     by auto
   have h1_helper: "(\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<Longrightarrow> (\<exists>y.\<forall>x<y. (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
   proof - 
     show "(\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<Longrightarrow> (\<exists>y.\<forall>x<y. (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))" 
     proof (induct les)
       case Nil
       then show ?case
         by auto
     next
       case (Cons q les) 
       have ind: " \<forall>a\<in>set (q # les). case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0"
         using Cons.prems
         by auto
       then have "case q of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0 "
         by simp      
       then obtain y2 where y2_prop: "case q of (a, ba, c) \<Rightarrow>  (\<forall>y<y2. a * y\<^sup>2 + ba * y + c < 0)"
         by auto
       have "\<forall>a\<in>set les. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0"
         using ind by simp
       then have " \<exists>y. \<forall>x<y. \<forall>a\<in>set les. case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c < 0"
         using Cons.hyps by blast 
       then obtain y1 where y1_prop: "\<forall>x<y1. \<forall>a\<in>set les. case a of (a, ba, c) \<Rightarrow> a * x^2 + ba * x + c < 0"
         by blast
       let ?y = "min y1 y2"
       have "\<forall>x < ?y.  (\<forall>a\<in>set (q #les). case a of (a, ba, c) \<Rightarrow> a * x^2 + ba * x + c < 0)"
         using y1_prop y2_prop 
         by fastforce 
       then show ?case
         by blast 
     qed
   qed
   then have h1: "(\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<Longrightarrow>(\<exists>x. (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
     by (smt (z3) infzeros less_eq_real_def not_numeral_le_zero)
       (* apply (auto)
     by (metis (lifting) infzeros zero_neq_numeral) *)
   have h2: " (\<exists>(a', b', c')\<in>set les.
                a' = 0 \<and>
                b' \<noteq> 0 \<and> (\<forall>(d, e, f)\<in>set les. \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0))
     \<Longrightarrow> (\<exists>x. (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
   proof -
     assume asm: "(\<exists>(a', b', c')\<in>set les. a' = 0 \<and> b' \<noteq> 0 \<and>
          (\<forall>(d, e, f)\<in>set les. \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0))"
     then obtain a' b' c' where abc_prop: "(a', b', c') \<in>set les \<and> a' = 0 \<and> b' \<noteq> 0 \<and>
          (\<forall>(d, e, f)\<in>set les. \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0)"
       by auto
     then show "(\<exists>x. (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
       using have_inbetween_point_les by auto
   qed
   have h3: " (\<exists>(a', b', c')\<in>set les.
                a' \<noteq> 0 \<and>
                4 * a' * c' \<le> b'\<^sup>2 \<and>
                (\<forall>(d, e, f)\<in>set les.
                    \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a').
                       \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0)) \<Longrightarrow> ((\<exists>x. (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)))"
   proof - 
     assume asm: "\<exists>(a', b', c')\<in>set les.  a' \<noteq> 0 \<and>
            4 * a' * c' \<le> b'\<^sup>2 \<and>
            (\<forall>(d, e, f)\<in>set les.
                 \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a').
                    \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}.
                       d * x\<^sup>2 + e * x + f < 0)"
     then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set les \<and> a' \<noteq> 0 \<and> 4 * a' * c' \<le> b'\<^sup>2 \<and>
        (\<forall>(d, e, f)\<in>set les.
            \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a').
               \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0)"
       by auto
     then show "(\<exists>x. (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
       using have_inbetween_point_les by auto
   qed
   have h4: "(\<exists>(a', b', c')\<in>set les.  a' \<noteq> 0 \<and>
                4 * a' * c' \<le> b'\<^sup>2 \<and>
                 (\<forall>(d, e, f)\<in>set les.
                     \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                        \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                           d * x\<^sup>2 + e * x + f < 0)) \<Longrightarrow> (\<exists>x. (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
   proof - 
     assume asm: "(\<exists>(a', b', c')\<in>set les.  a' \<noteq> 0 \<and>
          4 * a' * c' \<le> b'\<^sup>2 \<and>
           (\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0)) "
     then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set les \<and> a' \<noteq> 0 \<and> 4 * a' * c' \<le> b'\<^sup>2 \<and>
      (\<forall>(d, e, f)\<in>set les.
          \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
             \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0)"
       by auto
     then have "(\<exists>x. (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
       using have_inbetween_point_les by auto
     then show ?thesis using asm by auto
   qed 
   show ?thesis using big_or h1 h2 h3 h4
     by blast 
 qed
 
 (*sample points, some starter proofs below in comments *)
 lemma les_qe_backward : 
   shows "(\<exists>x. (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)) \<Longrightarrow>
     ((\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or>
      (\<exists>(a', b', c')\<in>set les.
          a' = 0 \<and>
          b' \<noteq> 0 \<and>
          (\<forall>(d, e, f)\<in>set les.
              \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<or>
          a' \<noteq> 0 \<and>
          4 * a' * c' \<le> b'\<^sup>2 \<and>
          ((\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a').
                  \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<or>
           (\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0))))"
 
 proof - 
   assume havex: "(\<exists>x. (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
   then obtain x where x_prop: "\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0" by auto
   have h: "(\<not> (\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<and> \<not> (\<exists>(a', b', c')\<in>set les.
            a' = 0 \<and>
            b' \<noteq> 0 \<and>
            (\<forall>(d, e, f)\<in>set les. \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0)) \<and>
         \<not> (\<exists>(a', b', c')\<in>set les.
            a' \<noteq> 0 \<and>
            4 * a' * c' \<le> b'\<^sup>2 \<and>
            (\<forall>(d, e, f)\<in>set les.
                \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a').
                   \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0)) \<and>
         \<not> (\<exists>(a', b', c')\<in>set les.
            a' \<noteq> 0 \<and>
            4 * a' * c' \<le> b'\<^sup>2 \<and>
            (\<forall>(d, e, f)\<in>set les.
                \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0))) \<Longrightarrow> False"
   proof -
     assume big: "(\<not> (\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<and> \<not> (\<exists>(a', b', c')\<in>set les.
            a' = 0 \<and>
            b' \<noteq> 0 \<and>
            (\<forall>(d, e, f)\<in>set les. \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0)) \<and>
         \<not> (\<exists>(a', b', c')\<in>set les.
            a' \<noteq> 0 \<and>
            4 * a' * c' \<le> b'\<^sup>2 \<and>
            (\<forall>(d, e, f)\<in>set les.
                \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a').
                   \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0)) \<and>
         \<not> (\<exists>(a', b', c')\<in>set les.
            a' \<noteq> 0 \<and>
            4 * a' * c' \<le> b'\<^sup>2 \<and>
            (\<forall>(d, e, f)\<in>set les.
                \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0)))"
     have notneginf: "\<not> (\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0)" using big by auto
     have notlinroot: "\<not> (\<exists>(a', b', c')\<in>set les.
            a' = 0 \<and>
            b' \<noteq> 0 \<and>
            (\<forall>(d, e, f)\<in>set les. \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0))" 
       using big by auto
     have notquadroot1: " \<not> (\<exists>(a', b', c')\<in>set les.
            a' \<noteq> 0 \<and>
            4 * a' * c' \<le> b'\<^sup>2 \<and>
            (\<forall>(d, e, f)\<in>set les.
                \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a').
                   \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0))"
       using big by auto
     have notquadroot2:" \<not> (\<exists>(a', b', c')\<in>set les.
            a' \<noteq> 0 \<and>
            4 * a' * c' \<le> b'\<^sup>2 \<and>
            (\<forall>(d, e, f)\<in>set les.
                \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0))"
       using big by auto
     have nok: "\<not> (\<exists>k. \<exists>(a, b, c)\<in>set les. a*k^2 + b*k + c = 0 \<and>
                 (\<forall>(d, e, f)\<in>set les. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f < 0))"
     proof - 
       have "(\<exists>k. \<exists>(a, b, c)\<in>set les. a*k^2 + b*k + c = 0 \<and>
                 (\<forall>(d, e, f)\<in>set les. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f < 0)) \<Longrightarrow> False"
       proof - 
         assume "(\<exists>k. \<exists>(a, b, c)\<in>set les. a*k^2 + b*k + c = 0 \<and>
                 (\<forall>(d, e, f)\<in>set les. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f < 0))"
         then obtain k a b c where k_prop: "(a, b, c) \<in> set les \<and>  a*k^2 + b*k + c = 0 \<and>
                 (\<forall>(d, e, f)\<in>set les. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f < 0)"
           by auto
         have azer:  "a = 0 \<Longrightarrow> False"
         proof - 
           assume azer: "a = 0"
           then have "b = 0 \<Longrightarrow> c = 0" using k_prop by auto
           then have bnonz: "b\<noteq> 0"
             using azer x_prop k_prop 
             by auto 
           then have "k = -c/b" using k_prop azer
-            by (metis (no_types, hide_lams) add.commute add.left_neutral add_uminus_conv_diff diff_le_0_iff_le divide_non_zero less_eq_real_def mult_zero_left neg_less_iff_less order_less_irrefl real_add_less_0_iff)
+            by (smt (verit, best) mult_eq_0_iff nonzero_mult_div_cancel_left)
           then have " (\<exists>(a', b', c')\<in>set les.
            a' = 0 \<and> b' \<noteq> 0 \<and> (\<forall>(d, e, f)\<in>set les. \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0))"
             using k_prop azer bnonz by auto
           then show "False" using notlinroot
             by auto
         qed
         have anonz: "a \<noteq> 0 \<Longrightarrow> False"
         proof - 
           assume anonz: "a \<noteq> 0 "
           let ?r1 = "(- b - sqrt (b^2 - 4 * a * c)) / (2 * a)"
           let ?r2 = "(- b + sqrt (b^2 - 4 * a * c)) / (2 * a)"
           have discr: "4 * a * c \<le> b^2" using anonz k_prop discriminant_negative[of a b c] 
             unfolding discrim_def 
             by fastforce 
           then have "k = ?r1 \<or> k = ?r2" using k_prop discriminant_nonneg[of a b c] unfolding discrim_def
             using anonz
             by auto 
           then have "(\<exists>(a', b', c')\<in>set les.
            a' \<noteq> 0 \<and>
            4 * a' * c' \<le> b'\<^sup>2 \<and>
            (\<forall>(d, e, f)\<in>set les.
                \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a').
                   \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0)) \<or>
            (\<exists>(a', b', c')\<in>set les.
            a' \<noteq> 0 \<and>
            4 * a' * c' \<le> b'\<^sup>2 \<and>
            (\<forall>(d, e, f)\<in>set les.
                \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0))"
             using discr anonz notquadroot1 notquadroot2 k_prop 
             by auto
           then show "False" using  notquadroot1 notquadroot2
             by auto
         qed
         show "False"
           using azer anonz  by auto
       qed
       then show ?thesis by auto
     qed
     have finset: "finite (set les)"
       by blast
     have h1: "(\<exists>(a, b, c)\<in>set les. a = 0 \<and> b = 0 \<and> c = 0)  \<Longrightarrow> False"
       using x_prop by fastforce
     then have h2: "\<not>(\<exists>(a, b, c)\<in>set les. a = 0 \<and> b = 0 \<and> c = 0) \<Longrightarrow> False"
     proof - 
       assume nozer: "\<not>(\<exists>(a, b, c)\<in>set les. a = 0 \<and> b = 0 \<and> c = 0)"
       then have same_set: "root_set (set les) = set (sorted_root_list_set (set les))"
         using root_set_finite finset set_sorted_list_of_set
         by (simp add: nozer root_set_finite sorted_root_list_set_def)
       have xnotin: "x \<notin> root_set (set les)"
         unfolding root_set_def using x_prop by auto
       let ?srl = "sorted_root_list_set (set les)"
       have notinlist: "\<not> List.member ?srl x"
         using xnotin same_set
         by (simp add: in_set_member)
       then have notmem: "\<forall>n < (length ?srl). x \<noteq> nth_default 0 ?srl n"
         using nth_mem same_set xnotin nth_default_def
         by metis  
       show ?thesis 
       proof (induct ?srl)
         case Nil
         then have "(\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0)"
         proof clarsimp 
           fix a b c
           assume noroots: "[] = sorted_root_list_set (set les)"
           assume inset: "(a, b, c) \<in> set les"
           have "{} = root_set (set les)" 
             using noroots same_set 
             by auto 
           then have nozero: "\<not>(\<exists>x. a*x^2  + b*x + c = 0)"
             using inset unfolding root_set_def by auto
           have "\<forall>y<x. a * y\<^sup>2 + b * y + c < 0" 
           proof clarsimp 
             fix y
             assume "y < x"
             then have "sign_num (a*x^2 + b*x + c) = sign_num (a*y^2 + b*y + c)"
               using nozero  by (metis changes_sign_var) 
             then show "a * y\<^sup>2 + b * y + c < 0"
               unfolding sign_num_def using x_prop inset 
               by (smt split_conv) 
           qed
           then show "\<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0"
             by auto
         qed
         then show ?case using notneginf by auto
       next
         case (Cons w xa) 
           (* Need to argue that x isn't greater than the largest element of ?srl *)
           (* that if srl has length \<ge> 2, x isn't in between any of the roots of ?srl*)
           (* and that x isn't less than the lowest root in ?srl *)
         then have lengthsrl: "length ?srl > 0" by auto
         have neginf: "x < nth_default 0 ?srl 0 \<Longrightarrow> False"
         proof -
           assume xlt: "x < nth_default 0 ?srl 0"
           have all: "(\<forall>(a, b, c)\<in>set les. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0)"
           proof clarsimp 
             fix a b c y
             assume inset: "(a, b, c) \<in> set les"
             assume "y < x"
             have xl: "a*x^2 + b*x + c < 0" using x_prop inset by auto
             have "\<not>(\<exists>q. q < nth_default 0 ?srl 0  \<and> a*q^2 + b*q + c = 0)"
             proof - 
               have "(\<exists>q. q < nth_default 0 ?srl 0  \<and> a*q^2 + b*q + c = 0) \<Longrightarrow> False"
               proof - assume "\<exists>q. q < nth_default 0 ?srl 0  \<and> a*q^2 + b*q + c = 0"
                 then obtain q where q_prop: "q < nth_default 0 ?srl 0 \<and>a*q^2 + b*q + c = 0" by auto
                 then have " q \<in> root_set (set les)" unfolding root_set_def using inset by auto
                 then have "List.member ?srl q" using same_set
                   by (simp add: in_set_member)
                 then have "q \<ge> nth_default 0 ?srl 0"
                   using sorted_sorted_list_of_set[where A = "root_set (set les)"]
                   unfolding sorted_root_list_set_def
                   by (metis \<open>q \<in> root_set (set les)\<close> in_set_conv_nth le_less_linear lengthsrl not_less0 nth_default_nth same_set sorted_nth_mono sorted_root_list_set_def)
                 then show "False" using q_prop by auto
               qed
               then show ?thesis by auto
             qed
             then have "\<not>(\<exists>q. q < x \<and> a*q^2 + b*q + c = 0)" using xlt by auto
             then show " a * y\<^sup>2 + b * y + c < 0" 
               using xl changes_sign_var[where a = "a", where b = "b", where c = "c", where x = "y", where y = "x"]
               unfolding sign_num_def using \<open>y < x\<close> less_eq_real_def zero_neq_numeral 
               by fastforce
           qed
           have "(\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0)"
           proof clarsimp 
             fix a b c
             assume "(a, b, c)\<in>set les"
             then show "\<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0" 
               using all by blast 
           qed
           then show "False" using notneginf by auto
         qed
         have "x > nth_default 0 ?srl (length ?srl - 1) \<Longrightarrow> (\<exists>k. \<exists>(a, b, c)\<in>set les. a*k^2 + b*k + c = 0 \<and>
                 (\<forall>(d, e, f)\<in>set les. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f < 0))"
         proof - 
           assume xgt: "x > nth_default 0 ?srl (length ?srl - 1)"
           let ?lg = "nth_default 0 ?srl (length ?srl - 1)"
           have "List.member ?srl ?lg"
             by (metis diff_less in_set_member lengthsrl nth_default_def nth_mem zero_less_one)
           then have "?lg \<in> root_set (set les) "
             using same_set in_set_member[of ?lg ?srl]  by auto
           then have exabc: "\<exists>(a, b, c)\<in>set les. a*?lg^2 + b*?lg + c = 0"
             unfolding root_set_def by auto
           have "(\<forall>(d, e, f)\<in>set les. \<forall>q\<in>{?lg<..x}. d * q^2 + e * q + f < 0)"
           proof clarsimp 
             fix d e f q 
             assume inset: "(d, e, f) \<in> set les"
             assume qgt: "(nth_default 0) (sorted_root_list_set (set les)) (length (sorted_root_list_set (set les)) - Suc 0) < q"
             assume qlt: "q \<le> x"
             have nor: "\<not>(\<exists>r. d * r^2 + e * r + f = 0 \<and> r > ?lg)"
             proof - 
               have "(\<exists>r. d * r^2 + e * r + f = 0 \<and> r > ?lg) \<Longrightarrow> False "
               proof - 
                 assume "\<exists>r. d * r^2 + e * r + f = 0 \<and> r > ?lg"
                 then obtain r where r_prop: "d*r^2 + e*r + f = 0 \<and> r > ?lg" by auto
                 then have "r \<in> root_set (set les)" using inset unfolding root_set_def by auto
                 then have "List.member ?srl r"
                   using same_set in_set_member
                   by (simp add: in_set_member) 
                 then have " r \<le> ?lg" using sorted_sorted_list_of_set nth_default_def
                   by (metis One_nat_def Suc_pred \<open>r \<in> root_set (set les)\<close> in_set_conv_nth lengthsrl lessI less_Suc_eq_le same_set sorted_nth_mono sorted_root_list_set_def)
                 then show "False" using r_prop by auto
               qed
               then show ?thesis by auto
             qed
             then have xltz_helper: "\<not>(\<exists>r. r \<ge> q \<and> d * r^2 + e * r + f = 0)"
               using qgt by auto
             then have xltz: "d*x^2 + e*x + f < 0" using inset x_prop by auto
             show "d * q\<^sup>2 + e * q + f < 0"
               using qlt qgt nor changes_sign_var[of d _ e f _] xltz xltz_helper unfolding sign_num_def
               apply (auto) 
               by smt
           qed
           then have " (\<forall>(d, e, f)\<in>set les. \<exists>y'>?lg. \<forall>x\<in>{?lg<..y'}. d * x\<^sup>2 + e * x + f < 0)"
             using xgt by auto
           then have "(\<exists>(a, b, c)\<in>set les. a*?lg^2 + b*?lg + c = 0 \<and>
                 (\<forall>(d, e, f)\<in>set les. \<exists>y'>?lg. \<forall>x\<in>{?lg<..y'}. d * x\<^sup>2 + e * x + f < 0))"
             using exabc by auto
           then show "(\<exists>k. \<exists>(a, b, c)\<in>set les. a*k^2 + b*k + c = 0 \<and>
                 (\<forall>(d, e, f)\<in>set les. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f < 0))"
             by auto
         qed
         then have posinf: "x > nth_default 0 ?srl (length ?srl - 1) \<Longrightarrow> False" 
           using nok by auto
         have "(\<exists>n. (n+1) < (length ?srl) \<and> x > (nth_default 0 ?srl) n \<and> x < (nth_default 0 ?srl (n + 1))) \<Longrightarrow> (\<exists>k. \<exists>(a, b, c)\<in>set les. a*k^2 + b*k + c = 0 \<and>
                 (\<forall>(d, e, f)\<in>set les. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f < 0))"
         proof - 
           assume "\<exists>n. (n+1) < (length ?srl) \<and> x > nth_default 0 ?srl n \<and> x < nth_default 0 ?srl (n + 1)"
           then obtain n where n_prop: "(n+1) < (length ?srl) \<and> x > nth_default 0 ?srl n \<and> x < nth_default 0 ?srl (n + 1)" by auto
           let ?elt = "nth_default 0 ?srl n"
           let ?elt2 = "nth_default 0 ?srl (n + 1)"
           have "List.member ?srl ?elt"
             using n_prop nth_default_def
             by (metis add_lessD1 in_set_member nth_mem) 
           then have "?elt \<in> root_set (set les) "
             using same_set in_set_member[of ?elt ?srl]  by auto
           then have exabc: "\<exists>(a, b, c)\<in>set les. a*?elt^2 + b*?elt + c = 0"
             unfolding root_set_def by auto
           then obtain a b c where "(a, b, c)\<in>set les \<and> a*?elt^2 + b*?elt + c = 0"
             by auto
           have xltel2: "x < ?elt2" using n_prop by auto
           have xgtel: "x > ?elt " using n_prop by auto
           have "(\<forall>(d, e, f)\<in>set les. \<forall>q\<in>{?elt<..x}. d * q^2 + e * q + f < 0)"
           proof clarsimp 
             fix d e f q 
             assume inset: "(d, e, f) \<in> set les"
             assume qgt: "nth_default 0 (sorted_root_list_set (set les)) n < q"
             assume qlt: "q \<le> x"
 
             have nor: "\<not>(\<exists>r. d * r^2 + e * r + f = 0 \<and> r > ?elt \<and>r < ?elt2)"
             proof - 
               have "(\<exists>r. d * r^2 + e * r + f = 0 \<and> r > ?elt \<and> r < ?elt2) \<Longrightarrow> False "
               proof - 
                 assume "\<exists>r. d * r^2 + e * r + f = 0 \<and> r > ?elt  \<and> r < ?elt2"
                 then obtain r where r_prop: "d*r^2 + e*r + f = 0 \<and> r > ?elt  \<and> r < ?elt2" by auto
                 then have "r \<in> root_set (set les)" using inset unfolding root_set_def by auto
                 then have "List.member ?srl r"
                   using same_set in_set_member
                   by (simp add: in_set_member) 
                 then have "\<exists>i < (length ?srl). r = nth_default 0 ?srl i"
                   by (metis \<open>r \<in> root_set (set les)\<close> in_set_conv_nth same_set nth_default_def)
                 then obtain i where i_prop: "i < (length ?srl) \<and> r = nth_default 0 ?srl i"
                   by auto
                 have "r > ?elt" using r_prop by auto
                 then  have igt: " i > n" using i_prop sorted_sorted_list_of_set
                   by (smt add_lessD1 leI n_prop nth_default_def sorted_nth_mono sorted_root_list_set_def)
                 have "r < ?elt2" using r_prop by auto
                 then have ilt: " i < n + 1" using i_prop sorted_sorted_list_of_set
                   by (smt leI n_prop nth_default_def sorted_nth_mono sorted_root_list_set_def) 
                 then show "False"  using igt ilt
                   by auto
               qed
               then show ?thesis by auto
             qed
             then have nor: "\<not>(\<exists>r. d * r^2 + e * r + f = 0 \<and> r > ?elt \<and>r \<le> x)"
               using xltel2 xgtel by auto
             then have xltz: "d*x^2 + e*x + f < 0" using inset x_prop by auto
             show "d * q\<^sup>2 + e * q + f < 0"
               using qlt qgt nor changes_sign_var[of d _ e f _] xltz unfolding sign_num_def
               by smt 
           qed
           then have " (\<forall>(d, e, f)\<in>set les. \<exists>y'>?elt. \<forall>x\<in>{?elt<..y'}. d * x\<^sup>2 + e * x + f < 0)"
             using xgtel xltel2 by auto
           then have "(\<exists>(a, b, c)\<in>set les. a*?elt^2 + b*?elt + c = 0 \<and>
                 (\<forall>(d, e, f)\<in>set les. \<exists>y'>?elt. \<forall>x\<in>{?elt<..y'}. d * x\<^sup>2 + e * x + f < 0))"
             using exabc by auto
           then show "(\<exists>k. \<exists>(a, b, c)\<in>set les. a*k^2 + b*k + c = 0 \<and>
                 (\<forall>(d, e, f)\<in>set les. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f < 0))"
             by auto
         qed
         then have inbetw: "(\<exists>n. (n+1) < (length ?srl) \<and> x > nth_default 0 ?srl n \<and> x < nth_default 0 ?srl (n + 1)) \<Longrightarrow> False"
           using nok by auto
         have lenzer: "length xa = 0 \<Longrightarrow> False" 
         proof - 
           assume "length xa = 0"
           have xis: "x > w \<or> x < w"
             using notmem Cons.hyps
             by (smt list.set_intros(1) same_set xnotin) 
           have xgt: "x > w \<Longrightarrow> False"
           proof - 
             assume xgt: "x > w"
             show "False" using posinf Cons.hyps
               by (metis One_nat_def Suc_eq_plus1 \<open>length xa = 0\<close> cancel_comm_monoid_add_class.diff_cancel list.size(4) nth_default_Cons_0 xgt)
           qed
           have xlt: "x < w \<Longrightarrow> False"
           proof - 
             assume xlt: "x < w"
             show "False" using neginf Cons.hyps
               by (metis nth_default_Cons_0 xlt) 
           qed
           show "False" using xis xgt xlt by auto
         qed
         have lengt: "length xa > 0 \<Longrightarrow> False"
         proof - 
           assume "length xa > 0"
           have "x \<ge> nth_default 0 ?srl 0" using neginf
             by fastforce
           then have xgtf: "x > nth_default 0 ?srl 0" using notmem
             using Cons.hyps(2) by fastforce
           have "x \<le> nth_default 0 ?srl (length ?srl - 1)" using posinf by fastforce
           then have "(\<exists>n. (n+1) < (length ?srl) \<and> x \<ge> nth_default 0 ?srl n \<and> x \<le> nth_default 0 ?srl (n + 1))"
             using lengthsrl xgtf notmem sorted_list_prop[where l = ?srl, where x = "x"]
             by (metis add_lessD1 diff_less nth_default_nth sorted_root_list_set_def sorted_sorted_list_of_set zero_less_one)
           then obtain n where n_prop: "(n+1) < (length ?srl) \<and> x \<ge> nth_default 0 ?srl n \<and> x \<le> nth_default 0 ?srl (n + 1)" by auto
           then have "x > nth_default 0 ?srl n \<and> x < nth_default 0 ?srl (n+1)"
             using notmem
             by (metis Suc_eq_plus1 Suc_lessD less_eq_real_def)
           then have "(\<exists>n. (n+1) < (length ?srl) \<and> x > nth_default 0 ?srl n \<and> x < nth_default 0 ?srl (n + 1))"
             using n_prop
             by blast  
           then show "False" using inbetw by auto
         qed
         then show ?case using lenzer lengt by auto
       qed 
     qed
     show "False"
       using h1 h2 by auto
   qed
   then have equiv_false: "\<not>((\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or> 
                (\<exists>(a', b', c')\<in>set les.
                a' = 0 \<and>
                b' \<noteq> 0 \<and> (\<forall>(d, e, f)\<in>set les. \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0)) 
         \<or>
         (\<exists>(a', b', c')\<in>set les.
                a' \<noteq> 0 \<and>
                4 * a' * c' \<le> b'\<^sup>2 \<and>
                (\<forall>(d, e, f)\<in>set les.
                    \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a').
                       \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0)) 
       \<or>
       (\<exists>(a', b', c')\<in>set les.  a' \<noteq> 0 \<and>
                4 * a' * c' \<le> b'\<^sup>2 \<and>
                 (\<forall>(d, e, f)\<in>set les.
                     \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                        \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                           d * x\<^sup>2 + e * x + f < 0))) \<Longrightarrow> False"
     by linarith
   have "\<not>((\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or>
      (\<exists>(a', b', c')\<in>set les.
          a' = 0 \<and>
          b' \<noteq> 0 \<and>
          (\<forall>(d, e, f)\<in>set les.
              \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<or>
          a' \<noteq> 0 \<and>
          4 * a' * c' \<le> b'\<^sup>2 \<and>
          ((\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a').
                  \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<or>
           (\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0)))) \<Longrightarrow> False"
   proof - 
     assume "\<not>((\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or>
      (\<exists>(a', b', c')\<in>set les.
          a' = 0 \<and>
          b' \<noteq> 0 \<and>
          (\<forall>(d, e, f)\<in>set les.
              \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<or>
          a' \<noteq> 0 \<and>
          4 * a' * c' \<le> b'\<^sup>2 \<and>
          ((\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a').
                  \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<or>
           (\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0))))"
     then have "\<not>((\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or> 
                (\<exists>(a', b', c')\<in>set les.
                a' = 0 \<and>
                b' \<noteq> 0 \<and> (\<forall>(d, e, f)\<in>set les. \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0)) 
         \<or>
         (\<exists>(a', b', c')\<in>set les.
                a' \<noteq> 0 \<and>
                4 * a' * c' \<le> b'\<^sup>2 \<and>
                (\<forall>(d, e, f)\<in>set les.
                    \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a').
                       \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0)) 
       \<or>
       (\<exists>(a', b', c')\<in>set les.  a' \<noteq> 0 \<and>
                4 * a' * c' \<le> b'\<^sup>2 \<and>
                 (\<forall>(d, e, f)\<in>set les.
                     \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                        \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                           d * x\<^sup>2 + e * x + f < 0)))"
       by auto
     then show ?thesis
       using equiv_false by auto
   qed
   then show ?thesis
     by blast 
 qed
 
 lemma les_qe : 
   shows "(\<exists>x. (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)) =
     ((\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or>
      (\<exists>(a', b', c')\<in>set les.
          a' = 0 \<and>
          b' \<noteq> 0 \<and>
          (\<forall>(d, e, f)\<in>set les.
              \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<or>
          a' \<noteq> 0 \<and>
          4 * a' * c' \<le> b'\<^sup>2 \<and>
          ((\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a').
                  \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<or>
           (\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0))))"
 proof -
   have first: "(\<exists>x. (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)) \<longrightarrow>
     ((\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or>
      (\<exists>(a', b', c')\<in>set les.
          a' = 0 \<and>
          b' \<noteq> 0 \<and>
          (\<forall>(d, e, f)\<in>set les.
              \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<or>
          a' \<noteq> 0 \<and>
          4 * a' * c' \<le> b'\<^sup>2 \<and>
          ((\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a').
                  \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<or>
           (\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0))))"
     using les_qe_backward by auto
   have second: "((\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or>
      (\<exists>(a', b', c')\<in>set les.
          a' = 0 \<and>
          b' \<noteq> 0 \<and>
          (\<forall>(d, e, f)\<in>set les.
              \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<or>
          a' \<noteq> 0 \<and>
          4 * a' * c' \<le> b'\<^sup>2 \<and>
          ((\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a').
                  \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<or>
           (\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0)))) \<longrightarrow> (\<exists>x. (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)) "
     using les_qe_forward by auto
   have "(\<exists>x. (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)) \<longleftrightarrow>
   ((\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or>
      (\<exists>(a', b', c')\<in>set les.
          a' = 0 \<and>
          b' \<noteq> 0 \<and>
          (\<forall>(d, e, f)\<in>set les.
              \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<or>
          a' \<noteq> 0 \<and>
          4 * a' * c' \<le> b'\<^sup>2 \<and>
          ((\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a').
                  \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<or>
           (\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0))))"
     using first second
     by meson 
   then show ?thesis
     by blast
 qed
 
 
 subsubsection "equiv\\_lemma"
 lemma equiv_lemma: 
   assumes big_asm: "(\<exists>(a', b', c')\<in>set eq.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0)) \<or>
   (\<exists>(a', b', c')\<in>set eq.
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0))) \<or>
           
       (\<exists>(a', b', c')\<in>set eq. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0)) \<or>
      ((\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and>
      (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
   shows "((\<exists>(a', b', c')\<in>set eq.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<or>
           (\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0))) \<or>
      (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and>
      (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))" 
 proof - 
   let ?t = " ((\<exists>(a', b', c')\<in>set eq.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<or>
           (\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0))) \<or>
      (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and>
      (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
   have h1: "(\<exists>(a', b', c')\<in>set eq.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0)) \<Longrightarrow> ?t"
     by auto
   have h2: "(\<exists>(a', b', c')\<in>set eq.
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0))) \<Longrightarrow> ?t" 
     by auto
   have h3: "(\<exists>(a', b', c')\<in>set eq. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0)) \<Longrightarrow> ?t" 
     by auto
   show ?thesis
     using big_asm h1 h2 h3
     by presburger 
 qed
 
 subsubsection "The eq\\_qe lemma"
 lemma eq_qe_forwards: 
   shows "(\<exists>x. (\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))  \<Longrightarrow>
     ((\<exists>(a', b', c')\<in>set eq.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<or>
           (\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0))) \<or>
      (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and>
      (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
 proof - 
   let ?big_or = "(\<exists>(a', b', c')\<in>set eq.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0)) \<or>
   (\<exists>(a', b', c')\<in>set eq.
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0))) \<or>
           
       (\<exists>(a', b', c')\<in>set eq. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0)) \<or>
      ((\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and>
      (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
   assume asm: "(\<exists>x. (\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)) " 
   then obtain x where x_prop: "(\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)" by auto
   have "\<not> (\<exists>(a', b', c')\<in>set eq.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0)) \<and>
   \<not> (\<exists>(a', b', c')\<in>set eq.
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0))) \<and>
           \<not> (\<exists>(a', b', c')\<in>set eq. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0)) \<and>
      \<not> ((\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and>
      (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)) \<Longrightarrow> False" 
   proof - 
     assume big_conj: "\<not> (\<exists>(a', b', c')\<in>set eq.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0)) \<and>
   \<not> (\<exists>(a', b', c')\<in>set eq.
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0))) \<and>
           \<not> (\<exists>(a', b', c')\<in>set eq. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0)) \<and>
      \<not> ((\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and>
      (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
     have not_lin: "\<not>(\<exists>(a', b', c')\<in>set eq.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0))"
       using big_conj by auto
     have not_quad1: "\<not>(\<exists>(a', b', c')\<in>set eq.
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0)))"
       using big_conj by auto
     have not_quad2: "\<not>(\<exists>(a', b', c')\<in>set eq. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0))"
       using big_conj by auto
     have not_zer: "\<not>((\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and>
      (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
       using big_conj by auto
     then have not_zer1: "\<not>(\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<or>
      \<not> (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)" by auto
     have "(\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)" using asm 
       by auto 
     then have "\<not>(\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0)" using not_zer1 by auto
     then have "\<exists> (d, e, f)\<in>set eq. d \<noteq> 0 \<or> e \<noteq> 0 \<or> f \<noteq> 0 "
       by auto
     then obtain d e f where def_prop: "(d, e, f) \<in> set eq \<and> (d \<noteq> 0 \<or> e \<noteq> 0 \<or> f \<noteq> 0)" by auto
     then have eval_at_x: "d*x^2 + e*x + f = 0" using x_prop by auto
     have dnonz: "d \<noteq> 0 \<Longrightarrow> False"
     proof - 
       assume dneq: "d \<noteq> 0"
       then have discr: "-(e^2) + 4 *d *f \<le> 0" using  discriminant_negative[of d e f x] eval_at_x unfolding discrim_def
         by linarith
       let ?r1 = "(- e + - 1 * sqrt (e^2 - 4 *d *f)) / (2 * d)"
       let ?r2 = "(- e + 1 * sqrt (e^2 - 4 *d *f)) / (2 * d)"
       have xis: "x = ?r1 \<or> x = ?r2"
         using dneq discr discriminant_nonneg[of d e f x] eval_at_x unfolding discrim_def
         by auto
       have xr1: "x = ?r1 \<Longrightarrow> False"
         using not_quad2 x_prop discr def_prop dneq by auto
       have xr2: "x = ?r2 \<Longrightarrow> False"
         using not_quad1 x_prop discr def_prop dneq by auto
       show "False" using xr1 xr2 xis by auto
     qed
     then have dz: "d = 0" by auto
     have enonz: "e \<noteq> 0 \<Longrightarrow> False" 
     proof - 
       assume enonz: "e\<noteq> 0"
       then have "x = -f/e" using dz eval_at_x
         by (metis add.commute minus_add_cancel mult.commute mult_zero_class.mult_zero_left nonzero_eq_divide_eq) 
       then show "False"
         using not_lin x_prop enonz dz def_prop by auto
     qed
     then have ez: "e = 0" by auto
     have fnonz: "f \<noteq> 0 \<Longrightarrow> False" using ez dz eval_at_x by auto
     show "False"
       using def_prop dnonz enonz fnonz by auto
   qed
   then have h: "\<not>(?big_or) \<Longrightarrow> False"
     by auto
   then show ?thesis using equiv_lemma
     by presburger
 qed
 
 lemma eq_qe_backwards: "((\<exists>(a', b', c')\<in>set eq.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<or>
           (\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0))) \<or>
      (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and>
      (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)) \<Longrightarrow> 
     (\<exists>x. ((\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)))
     "
 proof - 
   assume "((\<exists>(a', b', c')\<in>set eq.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<or>
           (\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0))) \<or>
      (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and>
      (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
   then have bigor: "(\<exists>(a', b', c')\<in>set eq.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0)) \<or>
   (\<exists>(a', b', c')\<in>set eq.
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0))) \<or>
           
       (\<exists>(a', b', c')\<in>set eq. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0)) \<or>
      ((\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and>
      (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
     by auto
   have h1: "(\<exists>(a', b', c')\<in>set eq.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0)) \<Longrightarrow>
     (\<exists>(x::real). (\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))" 
   proof - 
     assume "(\<exists>(a', b', c')\<in>set eq.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0))"
     then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set eq \<and>
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0)" by auto
     let ?x = "(-c' /b')::real"
     have "(\<forall>(d, e, f)\<in>set eq. d * ?x\<^sup>2 + e * ?x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * ?x^2 + e * ?x + f < 0)" using abc_prop by auto
     then  show ?thesis using abc_prop by blast 
   qed
   have h2: " (\<exists>(a', b', c')\<in>set eq.
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0))) \<Longrightarrow> (\<exists>x. (\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
   proof - 
     assume "(\<exists>(a', b', c')\<in>set eq.
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0)))"
     then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set eq \<and> a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0))" by auto
     let ?x = "((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')::real)"
     have anonz: "a' \<noteq> 0" using abc_prop by auto
     then have "\<exists>(q::real). q = ?x" by auto
     then obtain q where q_prop: "q = ?x" by auto
     have "(\<forall>(d, e, f)\<in>set eq. d * (?x)\<^sup>2 + e * (?x) + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set les. d * (?x)\<^sup>2 + e * (?x) + f < 0)"
       using abc_prop by auto
     then have "(\<forall>(d, e, f)\<in>set eq. d * q\<^sup>2 + e * q + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set les. d * q\<^sup>2 + e * q + f < 0)" using q_prop by auto
     then show ?thesis by auto
   qed
   have h3: "(\<exists>(a', b', c')\<in>set eq. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0)) \<Longrightarrow> (\<exists>x. (\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))" 
   proof - 
     assume "(\<exists>(a', b', c')\<in>set eq. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0))"
     then obtain a' b' c' where abc_prop: "a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (a', b', c')\<in>set eq \<and> (\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0)" by auto
     let ?x = "(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')"
     have anonz: "a' \<noteq> 0" using abc_prop by auto
     then have "\<exists>(q::real). q = ?x" by auto
     then obtain q where q_prop: "q = ?x" by auto
     have "(\<forall>(d, e, f)\<in>set eq. d * (?x)\<^sup>2 + e * (?x) + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set les. d * (?x)\<^sup>2 + e * (?x) + f < 0)"
       using abc_prop by auto
     then have "(\<forall>(d, e, f)\<in>set eq. d * q\<^sup>2 + e * q + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set les. d * q\<^sup>2 + e * q + f < 0)" using q_prop by auto
     then show ?thesis by auto
   qed
   have h4: "((\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and>
      (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)) \<Longrightarrow> (\<exists>x. (\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
   proof - 
     assume asm: "((\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and>
      (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
     then have allzer: "(\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0)" by auto
     have "(\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)" using asm by auto
     then obtain x where x_prop: " \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0" by auto
     then have "\<forall>(d, e, f)\<in>set eq. d*x^2 + e*x + f = 0"
       using allzer by auto
     then show ?thesis using x_prop by auto
   qed
   show ?thesis
     using bigor h1 h2 h3 h4
     by blast 
 qed
 
 
 lemma eq_qe : "(\<exists>x. ((\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))) =
     ((\<exists>(a', b', c')\<in>set eq.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<or>
           (\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0))) \<or>
      (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and>
      (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
 proof - 
   have h1: "(\<exists>x. (\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)) \<longrightarrow>
     ((\<exists>(a', b', c')\<in>set eq.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<or>
           (\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0))) \<or>
      (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and>
      (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
     using eq_qe_forwards by auto
   have h2: "((\<exists>(a', b', c')\<in>set eq.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<or>
           (\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0))) \<or>
      (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and>
      (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)) \<longrightarrow> (\<exists>x. (\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
     using eq_qe_backwards by auto
   have h3: "(\<exists>x. (\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)) \<longleftrightarrow>
     ((\<exists>(a', b', c')\<in>set eq.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<or>
           (\<forall>(d, e, f)\<in>set eq.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0))) \<or>
      (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and>
      (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
     using h1 h2
     by smt 
   then  show ?thesis
     by (auto) 
 qed
 
 subsubsection "The qe\\_forwards lemma"
 lemma qe_forwards_helper_gen:
   fixes r:: "real"
   assumes f8: "\<not>(\<exists>((a'::real), (b'::real), (c'::real))\<in>set c. 
          ((a'\<noteq> 0 \<or> b'\<noteq> 0) \<and> a'*r^2 + b'*r + c' = 0) \<and>
          ((\<forall>(d, e, f)\<in>set a. d * r\<^sup>2 + e * r + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * r^2 + e * r + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * r^2 + e * r + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * r^2 + e * r + f \<noteq> 0)))"
   assumes alleqset: "\<forall>x. (\<forall>(d, e, f)\<in>set a. d * x^2 + e * x + f = 0)"
   assumes f5: "\<not>(\<exists>(a', b', c')\<in>set b.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
   assumes f6: "\<not> (\<exists>(a', b', c')\<in>set b. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))"
   assumes f7: "\<not> (\<exists>(a', b', c')\<in>set b. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))"
   assumes f10: "\<not>(\<exists>(a', b', c')\<in>set d.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
   assumes f11: "\<not>(\<exists>(a', b', c')\<in>set d.
           a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))"
   assumes f12: "\<not>(\<exists>(a', b', c')\<in>set d.
           a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))"
   shows "\<not>(\<exists>(a', b', c')\<in>set c.
          ((a'\<noteq> 0 \<or> b'\<noteq> 0) \<and> a'*r^2 + b'*r + c' = 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))" 
 proof - 
   have "(\<exists>(a', b', c')\<in>set c.
          ((a'\<noteq> 0 \<or> b'\<noteq> 0) \<and> a'*r^2 + b'*r + c' = 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)) \<Longrightarrow> False"
   proof - 
     assume "(\<exists>(a', b', c')\<in>set c.
          ((a'\<noteq> 0 \<or> b'\<noteq> 0) \<and> a'*r^2 + b'*r + c' = 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
     then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set c \<and>
          ((a'\<noteq> 0 \<or> b'\<noteq> 0) \<and> a'*r^2 + b'*r + c' = 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)"
       by auto
     have h1: "(\<forall>(d, e, f)\<in>set a. d * r^2 + e * r + f = 0)" 
       using alleqset
       by blast
     have c_prop: "(\<forall>(d, e, f)\<in>set c.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0)" 
       using abc_prop by auto 
     have h2: "(\<forall>(d, e, f)\<in>set c. d *r^2 + e * r + f \<le> 0)" 
     proof - 
       have c1: "\<exists> (d, e, f) \<in> set c.  d * (r)\<^sup>2 + e * (r) + f > 0 \<Longrightarrow> False"
       proof - 
         assume "\<exists> (d, e, f) \<in> set c.  d * (r)\<^sup>2 + e * (r) + f > 0"
         then obtain d e f where def_prop: "(d, e, f) \<in> set c \<and> d * (r)\<^sup>2 + e * r + f > 0"
           by auto
         have "\<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0"
           using def_prop c_prop   by auto
         then obtain y' where y_prop: " y' >r \<and> (\<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0)" by auto
         have "\<exists>x\<in>{r<..y'}. d*x^2 + e*x + f > 0"
           using def_prop continuity_lem_gt0_expanded[of "r" y' d e f]
           using y_prop by linarith 
         then show "False" using  y_prop
           by auto  
       qed
       then show ?thesis
         by fastforce 
     qed
     have b_prop: "(\<forall>(d, e, f)\<in>set b.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f < 0)" 
       using abc_prop by auto
     have h3: "(\<forall>(d, e, f)\<in>set b. d * r\<^sup>2 + e * r + f < 0)" 
     proof - 
       have c1: "\<exists> (d, e, f) \<in> set b.  d * r\<^sup>2 + e * r + f > 0 \<Longrightarrow> False"
       proof - 
         assume "\<exists> (d, e, f) \<in> set b.  d * r\<^sup>2 + e * r + f > 0"
         then obtain d e f where def_prop: "(d, e, f) \<in> set b \<and> d * r\<^sup>2 + e * r + f > 0"
           by auto
         then have "\<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f < 0"
           using b_prop by auto
         then obtain y' where y_prop: " y' >r \<and> (\<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f < 0)" by auto
         then have "\<exists>k. k > r \<and> k < y' \<and> d * k^2 + e * k + f < 0" using dense
           by (meson dense greaterThanAtMost_iff less_eq_real_def) 
         then obtain k where k_prop: "k > r \<and> k < y' \<and> d * k^2 + e * k + f < 0" 
           by auto
         then have "\<not>(\<exists>x>r. x < y' \<and> d * x\<^sup>2 + e * x + f = 0)"
           using y_prop by force 
         then show "False" using k_prop def_prop y_prop poly_IVT_neg[of "r" "k" "[:f, e, d:]"] poly_IVT_pos[of "-c'/b'" "k" "[:f, e, d:]"]
           by (smt quadratic_poly_eval)
       qed
       have c2: "\<exists> (d, e, f) \<in> set b.  d * r\<^sup>2 + e * r + f = 0 \<Longrightarrow> False" 
       proof - 
         assume "\<exists> (d, e, f) \<in> set b.  d * r\<^sup>2 + e * r + f = 0"
         then obtain d' e f where def_prop: "(d', e, f) \<in> set b \<and> d' * r\<^sup>2 + e * r + f = 0"
           by auto
         then have same: "(d' = 0 \<and> e \<noteq> 0) \<Longrightarrow> (-f/e = r)" 
         proof - 
           assume asm: "(d' = 0 \<and> e \<noteq> 0)" 
           then have " e * r + f = 0" using def_prop 
             by auto
           then show "-f/e = r" using asm
             by (metis (no_types) add.commute diff_0 divide_minus_left minus_add_cancel nonzero_mult_div_cancel_left uminus_add_conv_diff) 
         qed
         let ?r = "-f/e"
         have "(d' = 0 \<and> e \<noteq> 0) \<Longrightarrow> ((d', e, f) \<in> set b \<and> ((\<forall>(d, e, f)\<in>set a. \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
     (\<forall>(d, e, f)\<in>set b. \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
     (\<forall>(d, e, f)\<in>set c. \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
     (\<forall>(d, e, f)\<in>set d. \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
           using same def_prop abc_prop by auto
         then have "(d' = 0 \<and> e \<noteq> 0) \<Longrightarrow> (\<exists>(a', b', c')\<in>set b.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))" 
           by auto
         then have f1: "(d' = 0 \<and> e \<noteq> 0) \<Longrightarrow> False" using f5 
           by auto 
         have f2: "(d' = 0 \<and> e = 0 \<and> f = 0) \<Longrightarrow> False" proof - 
           assume "(d' = 0 \<and> e = 0 \<and> f = 0)"
           then have allzer: "\<forall>x. d'*x^2 + e*x + f = 0" by auto
           have "\<exists>y'>r. \<forall>x\<in>{r<..y'}. d' * x\<^sup>2 + e * x + f < 0"
             using b_prop def_prop  by auto
           then obtain y' where y_prop: " y' >r \<and> (\<forall>x\<in>{r<..y'}. d' * x\<^sup>2 + e * x + f < 0)" by auto
           then have "\<exists>k. k > r \<and> k < y' \<and> d' * k^2 + e * k + f < 0" using dense
             by (meson dense greaterThanAtMost_iff less_eq_real_def) 
           then show "False" using allzer
             by auto
         qed
         have f3: "d' \<noteq> 0 \<Longrightarrow> False" 
         proof - 
           assume dnonz: "d' \<noteq> 0"
           have discr: " - e\<^sup>2 + 4 * d' * f \<le> 0"
             using def_prop discriminant_negative[of d' e f] unfolding discrim_def
             using def_prop by fastforce
           then have two_cases: "r = (- e + - 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')
           \<or> r = (- e + 1 * sqrt (e\<^sup>2 - 4 * d' * f)) / (2 * d')"
             using def_prop dnonz discriminant_nonneg[of d' e f] unfolding discrim_def
             by fastforce
           have some_props: "((d', e, f) \<in> set b \<and> d' \<noteq> 0 \<and> - e\<^sup>2 + 4 * d' * f \<le> 0)"
             using dnonz def_prop discr by auto
           let ?r1 = "(- e + - 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')"
           let ?r2 = "(- e + 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')"
           have cf1: "r = (- e + - 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d') \<Longrightarrow> False" 
           proof - 
             assume "r = (- e + - 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')"
             then have "(d', e, f) \<in> set b \<and> d' \<noteq> 0 \<and> - e\<^sup>2 + 4 * d' * f \<le> 0 \<and> 
     ((\<forall>(d, e, f)\<in>set a. \<exists>y'>?r1. \<forall>x\<in>{?r1<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
     (\<forall>(d, e, f)\<in>set b. \<exists>y'>?r1. \<forall>x\<in>{?r1<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
     (\<forall>(d, e, f)\<in>set c. \<exists>y'>?r1. \<forall>x\<in>{?r1<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
     (\<forall>(d, e, f)\<in>set d. \<exists>y'>?r1. \<forall>x\<in>{?r1<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
               using abc_prop some_props by auto
             then show "False" using f7 by auto
           qed
           have cf2: "r = (- e +  1 * sqrt (e^2 - 4 * d' * f)) / (2 * d') \<Longrightarrow> False" 
           proof - 
             assume "r = (- e + 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')"
             then have "(d', e, f) \<in> set b \<and> d' \<noteq> 0 \<and> - e\<^sup>2 + 4 * d' * f \<le> 0 \<and> 
     ((\<forall>(d, e, f)\<in>set a. \<exists>y'>?r2. \<forall>x\<in>{?r2<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
     (\<forall>(d, e, f)\<in>set b. \<exists>y'>?r2. \<forall>x\<in>{?r2<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
     (\<forall>(d, e, f)\<in>set c. \<exists>y'>?r2. \<forall>x\<in>{?r2<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
     (\<forall>(d, e, f)\<in>set d. \<exists>y'>?r2. \<forall>x\<in>{?r2<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
               using abc_prop some_props by auto
             then show "False" using f6 by auto
           qed
           then show "False" using two_cases cf1 cf2 by auto
         qed
           (* discriminant_nonnegative *)
         have eo: "(d' \<noteq> 0) \<or> (d' = 0 \<and> e \<noteq> 0) \<or> (d' = 0 \<and> e = 0 \<and> f = 0)" 
           using def_prop by auto
         then show "False" using f1 f2 f3 by auto
       qed
       show ?thesis using c1 c2
         by fastforce
     qed
     have d_prop: "(\<forall>(d, e, f)\<in>set d.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)" 
       using abc_prop by auto
     have h4: "(\<forall>(d, e, f)\<in>set d. d * r\<^sup>2 + e * r + f \<noteq> 0)"
     proof - 
       have "(\<exists>(d, e, f)\<in>set d. d * r\<^sup>2 + e * r + f = 0) \<Longrightarrow> False"
       proof - 
         assume "\<exists> (d, e, f) \<in> set d.  d * r\<^sup>2 + e * r + f = 0"
         then obtain d' e f where def_prop: "(d', e, f) \<in> set d \<and> d' * r\<^sup>2 + e * r + f = 0"
           by auto
         then have same: "(d' = 0 \<and> e \<noteq> 0) \<Longrightarrow> (-f/e = r)" 
         proof - 
           assume asm: "(d' = 0 \<and> e \<noteq> 0)" 
           then have " e * r + f = 0" using def_prop 
             by auto
           then show "-f/e = r" using asm
             by (metis (no_types) add.commute diff_0 divide_minus_left minus_add_cancel nonzero_mult_div_cancel_left uminus_add_conv_diff) 
         qed
         let ?r = "-f/e"
         have "(d' = 0 \<and> e \<noteq> 0) \<Longrightarrow> ((d', e, f) \<in> set d \<and> ((\<forall>(d, e, f)\<in>set a. \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
     (\<forall>(d, e, f)\<in>set b. \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
     (\<forall>(d, e, f)\<in>set c. \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
     (\<forall>(d, e, f)\<in>set d. \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
           using same def_prop abc_prop by auto
         then have "(d' = 0 \<and> e \<noteq> 0) \<Longrightarrow> (\<exists>(a', b', c')\<in>set d.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'> -c'/b'. \<forall>x\<in>{ -c'/b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'> -c'/b'. \<forall>x\<in>{ -c'/b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'> -c'/b'. \<forall>x\<in>{ -c'/b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'> -c'/b'. \<forall>x\<in>{ -c'/b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))" 
           by auto
         then have f1: "(d' = 0 \<and> e \<noteq> 0) \<Longrightarrow> False" using f10 
           by auto 
         have f2: "(d' = 0 \<and> e = 0 \<and> f = 0) \<Longrightarrow> False" proof - 
           assume "(d' = 0 \<and> e = 0 \<and> f = 0)"
           then have allzer: "\<forall>x. d'*x^2 + e*x + f = 0" by auto
           have "\<exists>y'> r. \<forall>x\<in>{ r<..y'}. d' * x\<^sup>2 + e * x + f \<noteq> 0"
             using d_prop def_prop 
             by auto
           then obtain y' where y_prop: " y' >r \<and> (\<forall>x\<in>{r<..y'}. d' * x\<^sup>2 + e * x + f \<noteq> 0)" by auto
           then have "\<exists>k. k > r \<and> k < y' \<and> d' * k^2 + e * k + f \<noteq> 0" using dense
             by (meson dense greaterThanAtMost_iff less_eq_real_def) 
           then show "False" using allzer
             by auto
         qed
         have f3: "d' \<noteq> 0 \<Longrightarrow> False" 
         proof - 
           assume dnonz: "d' \<noteq> 0"
           have discr: " - e\<^sup>2 + 4 * d' * f \<le> 0"
             using def_prop discriminant_negative[of d' e f] unfolding discrim_def
             by fastforce
           then have two_cases: "r = (- e + - 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')
           \<or> r = (- e + 1 * sqrt (e\<^sup>2 - 4 * d' * f)) / (2 * d')"
             using def_prop dnonz discriminant_nonneg[of d' e f] unfolding discrim_def
             by fastforce
           have some_props: "((d', e, f) \<in> set d \<and> d' \<noteq> 0 \<and> - e\<^sup>2 + 4 * d' * f \<le> 0)"
             using dnonz def_prop discr by auto
           let ?r1 = "(- e + - 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')"
           let ?r2 = "(- e + 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')"
           have cf1: "r = (- e + - 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d') \<Longrightarrow> False" 
           proof - 
             assume "r = (- e + - 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')"
             then have "(d', e, f) \<in> set d \<and> d' \<noteq> 0 \<and> - e\<^sup>2 + 4 * d' * f \<le> 0 \<and> 
     ((\<forall>(d, e, f)\<in>set a. \<exists>y'>?r1. \<forall>x\<in>{?r1<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
     (\<forall>(d, e, f)\<in>set b. \<exists>y'>?r1. \<forall>x\<in>{?r1<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
     (\<forall>(d, e, f)\<in>set c. \<exists>y'>?r1. \<forall>x\<in>{?r1<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
     (\<forall>(d, e, f)\<in>set d. \<exists>y'>?r1. \<forall>x\<in>{?r1<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
               using abc_prop some_props by auto
             then show "False" using f12 by auto
           qed
           have cf2: "r = (- e +  1 * sqrt (e^2 - 4 * d' * f)) / (2 * d') \<Longrightarrow> False" 
           proof - 
             assume "r = (- e + 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')"
             then have "(d', e, f) \<in> set d \<and> d' \<noteq> 0 \<and> - e\<^sup>2 + 4 * d' * f \<le> 0 \<and> 
     ((\<forall>(d, e, f)\<in>set a. \<exists>y'>?r2. \<forall>x\<in>{?r2<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
     (\<forall>(d, e, f)\<in>set b. \<exists>y'>?r2. \<forall>x\<in>{?r2<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
     (\<forall>(d, e, f)\<in>set c. \<exists>y'>?r2. \<forall>x\<in>{?r2<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
     (\<forall>(d, e, f)\<in>set d. \<exists>y'>?r2. \<forall>x\<in>{?r2<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
               using abc_prop some_props by auto
             then show "False" using f11 by auto
           qed
           then show "False" using two_cases cf1 cf2 by auto
         qed
           (* discriminant_nonnegative *)
         have eo: "(d' \<noteq> 0) \<or> (d' = 0 \<and> e \<noteq> 0) \<or> (d' = 0 \<and> e = 0 \<and> f = 0)" 
           using def_prop by auto
         then show "False" using f1 f2 f3 by auto
       qed
       then show ?thesis by auto
     qed
     have "(\<exists>(a', b', c')\<in>set c. ((a'\<noteq> 0 \<or> b'\<noteq> 0) \<and> a'*r^2 + b'*r + c' = 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * r\<^sup>2 + e * r + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * r\<^sup>2 + e * r + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * r\<^sup>2 + e * r + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * r\<^sup>2 + e * r + f \<noteq> 0))"
       using h1 h2 h3 h4 abc_prop by auto
     then show "False" using f8 by auto
   qed
   then show ?thesis by auto
 qed
 
 
 
 lemma qe_forwards_helper_lin:
   assumes alleqset: "\<forall>x. (\<forall>(d, e, f)\<in>set a. d * x^2 + e * x + f = 0)"
   assumes f5: "\<not>(\<exists>(a', b', c')\<in>set b.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
   assumes f6: "\<not> (\<exists>(a', b', c')\<in>set b. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))"
   assumes f7: "\<not> (\<exists>(a', b', c')\<in>set b. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))"
   assumes f8: "\<not>(\<exists>(a', b', c')\<in>set c. (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0))"
   assumes f10: "\<not>(\<exists>(a', b', c')\<in>set d.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
   assumes f11: "\<not>(\<exists>(a', b', c')\<in>set d.
           a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))"
   assumes f12: "\<not>(\<exists>(a', b', c')\<in>set d.
           a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))"
   shows "\<not>(\<exists>(a', b', c')\<in>set c.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))" 
 proof - 
   have "(\<exists>(a', b', c')\<in>set c.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)) \<Longrightarrow> False"
   proof - 
     assume "(\<exists>(a', b', c')\<in>set c.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
     then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set c \<and>
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)"
       by auto
     then have bnonz: "b'\<noteq>0" by auto
     have h1: "(\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0)" 
       using bnonz alleqset
       by blast
     have c_prop: "(\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0)" 
       using abc_prop by auto 
     have h2: "(\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0)" 
     proof - 
       have c1: "\<exists> (d, e, f) \<in> set c.  d * (- c' / b')\<^sup>2 + e * (- c' / b') + f > 0 \<Longrightarrow> False"
       proof - 
         assume "\<exists> (d, e, f) \<in> set c.  d * (- c' / b')\<^sup>2 + e * (- c' / b') + f > 0"
         then obtain d e f where def_prop: "(d, e, f) \<in> set c \<and> d * (- c' / b')\<^sup>2 + e * (- c' / b') + f > 0"
           by auto
         have "\<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0"
           using def_prop c_prop   by auto
         then obtain y' where y_prop: " y' >- c' / b' \<and> (\<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0)" by auto
         have "\<exists>x\<in>{(-c'/b')<..y'}. d*x^2 + e*x + f > 0"
           using def_prop continuity_lem_gt0_expanded[of "(-c'/b')" y' d e f]
           using y_prop by linarith 
         then show "False" using  y_prop
           by auto  
       qed
       then show ?thesis
         by fastforce 
     qed
     have b_prop: "(\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0)" 
       using abc_prop by auto
     have h3: "(\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0)" 
     proof - 
       have c1: "\<exists> (d, e, f) \<in> set b.  d * (- c' / b')\<^sup>2 + e * (- c' / b') + f > 0 \<Longrightarrow> False"
       proof - 
         assume "\<exists> (d, e, f) \<in> set b.  d * (- c' / b')\<^sup>2 + e * (- c' / b') + f > 0"
         then obtain d e f where def_prop: "(d, e, f) \<in> set b \<and> d * (- c' / b')\<^sup>2 + e * (- c' / b') + f > 0"
           by auto
         then have "\<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0"
           using b_prop by auto
         then obtain y' where y_prop: " y' >- c' / b' \<and> (\<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0)" by auto
         then have "\<exists>k. k > -c'/b' \<and> k < y' \<and> d * k^2 + e * k + f < 0" using dense
           by (meson dense greaterThanAtMost_iff less_eq_real_def) 
         then obtain k where k_prop: "k > -c'/b' \<and> k < y' \<and> d * k^2 + e * k + f < 0" 
           by auto
         then have "\<not>(\<exists>x>(-c'/b'). x < y' \<and> d * x\<^sup>2 + e * x + f = 0)"
           using y_prop by force 
         then show "False" using k_prop def_prop y_prop poly_IVT_neg[of "-c'/b'" "k" "[:f, e, d:]"] poly_IVT_pos[of "-c'/b'" "k" "[:f, e, d:]"]
           by (smt quadratic_poly_eval)
       qed
       have c2: "\<exists> (d, e, f) \<in> set b.  d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0 \<Longrightarrow> False" 
       proof - 
         assume "\<exists> (d, e, f) \<in> set b.  d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0"
         then obtain d' e f where def_prop: "(d', e, f) \<in> set b \<and> d' * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0"
           by auto
         then have same: "(d' = 0 \<and> e \<noteq> 0) \<Longrightarrow> (-f/e = -c'/b')" 
         proof - 
           assume asm: "(d' = 0 \<and> e \<noteq> 0)" 
           then have " e * (- c' / b') + f = 0" using def_prop 
             by auto
           then show "-f/e = -c'/b'" using asm 
             by auto
         qed
         let ?r = "-f/e"
         have "(d' = 0 \<and> e \<noteq> 0) \<Longrightarrow> ((d', e, f) \<in> set b \<and> ((\<forall>(d, e, f)\<in>set a. \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
     (\<forall>(d, e, f)\<in>set b. \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
     (\<forall>(d, e, f)\<in>set c. \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
     (\<forall>(d, e, f)\<in>set d. \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
           using same def_prop abc_prop by auto
         then have "(d' = 0 \<and> e \<noteq> 0) \<Longrightarrow> (\<exists>(a', b', c')\<in>set b.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))" 
           by auto
         then have f1: "(d' = 0 \<and> e \<noteq> 0) \<Longrightarrow> False" using f5 
           by auto 
         have f2: "(d' = 0 \<and> e = 0 \<and> f = 0) \<Longrightarrow> False" proof - 
           assume "(d' = 0 \<and> e = 0 \<and> f = 0)"
           then have allzer: "\<forall>x. d'*x^2 + e*x + f = 0" by auto
           have "\<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d' * x\<^sup>2 + e * x + f < 0"
             using b_prop def_prop  by auto
           then obtain y' where y_prop: " y' >- c' / b' \<and> (\<forall>x\<in>{- c' / b'<..y'}. d' * x\<^sup>2 + e * x + f < 0)" by auto
           then have "\<exists>k. k > -c'/b' \<and> k < y' \<and> d' * k^2 + e * k + f < 0" using dense
             by (meson dense greaterThanAtMost_iff less_eq_real_def) 
           then show "False" using allzer
             by auto
         qed
         have f3: "d' \<noteq> 0 \<Longrightarrow> False" 
         proof - 
           assume dnonz: "d' \<noteq> 0"
           have discr: " - e\<^sup>2 + 4 * d' * f \<le> 0"
             using def_prop discriminant_negative[of d' e f] unfolding discrim_def
             by fastforce
           then have two_cases: "(- c' / b') = (- e + - 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')
           \<or> (- c' / b') = (- e + 1 * sqrt (e\<^sup>2 - 4 * d' * f)) / (2 * d')"
             using def_prop dnonz discriminant_nonneg[of d' e f] unfolding discrim_def
             by fastforce
           have some_props: "((d', e, f) \<in> set b \<and> d' \<noteq> 0 \<and> - e\<^sup>2 + 4 * d' * f \<le> 0)"
             using dnonz def_prop discr by auto
           let ?r1 = "(- e + - 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')"
           let ?r2 = "(- e + 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')"
           have cf1: "(- c' / b') = (- e + - 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d') \<Longrightarrow> False" 
           proof - 
             assume "(- c' / b') = (- e + - 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')"
             then have "(d', e, f) \<in> set b \<and> d' \<noteq> 0 \<and> - e\<^sup>2 + 4 * d' * f \<le> 0 \<and> 
     ((\<forall>(d, e, f)\<in>set a. \<exists>y'>?r1. \<forall>x\<in>{?r1<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
     (\<forall>(d, e, f)\<in>set b. \<exists>y'>?r1. \<forall>x\<in>{?r1<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
     (\<forall>(d, e, f)\<in>set c. \<exists>y'>?r1. \<forall>x\<in>{?r1<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
     (\<forall>(d, e, f)\<in>set d. \<exists>y'>?r1. \<forall>x\<in>{?r1<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
               using abc_prop some_props by auto
             then show "False" using f7 by auto
           qed
           have cf2: "(- c' / b') = (- e +  1 * sqrt (e^2 - 4 * d' * f)) / (2 * d') \<Longrightarrow> False" 
           proof - 
             assume "(- c' / b') = (- e + 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')"
             then have "(d', e, f) \<in> set b \<and> d' \<noteq> 0 \<and> - e\<^sup>2 + 4 * d' * f \<le> 0 \<and> 
     ((\<forall>(d, e, f)\<in>set a. \<exists>y'>?r2. \<forall>x\<in>{?r2<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
     (\<forall>(d, e, f)\<in>set b. \<exists>y'>?r2. \<forall>x\<in>{?r2<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
     (\<forall>(d, e, f)\<in>set c. \<exists>y'>?r2. \<forall>x\<in>{?r2<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
     (\<forall>(d, e, f)\<in>set d. \<exists>y'>?r2. \<forall>x\<in>{?r2<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
               using abc_prop some_props by auto
             then show "False" using f6 by auto
           qed
           then show "False" using two_cases cf1 cf2 by auto
         qed
           (* discriminant_nonnegative *)
         have eo: "(d' \<noteq> 0) \<or> (d' = 0 \<and> e \<noteq> 0) \<or> (d' = 0 \<and> e = 0 \<and> f = 0)" 
           using def_prop by auto
         then show "False" using f1 f2 f3 by auto
       qed
       show ?thesis using c1 c2
         by fastforce
     qed
     have d_prop: "(\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)" 
       using abc_prop by auto
     have h4: "(\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0)"
     proof - 
       have "(\<exists>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<Longrightarrow> False"
         (* begin *)
       proof - 
         assume "\<exists> (d, e, f) \<in> set d.  d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0"
         then obtain d' e f where def_prop: "(d', e, f) \<in> set d \<and> d' * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0"
           by auto
         then have same: "(d' = 0 \<and> e \<noteq> 0) \<Longrightarrow> (-f/e = -c'/b')" 
         proof - 
           assume asm: "(d' = 0 \<and> e \<noteq> 0)" 
           then have " e * (- c' / b') + f = 0" using def_prop 
             by auto
           then show "-f/e = -c'/b'" using asm 
             by auto
         qed
         let ?r = "-f/e"
         have "(d' = 0 \<and> e \<noteq> 0) \<Longrightarrow> ((d', e, f) \<in> set d \<and> ((\<forall>(d, e, f)\<in>set a. \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
     (\<forall>(d, e, f)\<in>set b. \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
     (\<forall>(d, e, f)\<in>set c. \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
     (\<forall>(d, e, f)\<in>set d. \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
           using same def_prop abc_prop by auto
         then have "(d' = 0 \<and> e \<noteq> 0) \<Longrightarrow> (\<exists>(a', b', c')\<in>set d.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))" 
           by auto
         then have f1: "(d' = 0 \<and> e \<noteq> 0) \<Longrightarrow> False" using f10 
           by auto 
         have f2: "(d' = 0 \<and> e = 0 \<and> f = 0) \<Longrightarrow> False" proof - 
           assume "(d' = 0 \<and> e = 0 \<and> f = 0)"
           then have allzer: "\<forall>x. d'*x^2 + e*x + f = 0" by auto
           have "\<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d' * x\<^sup>2 + e * x + f \<noteq> 0"
             using d_prop def_prop  by auto
           then obtain y' where y_prop: " y' >- c' / b' \<and> (\<forall>x\<in>{- c' / b'<..y'}. d' * x\<^sup>2 + e * x + f \<noteq> 0)" by auto
           then have "\<exists>k. k > -c'/b' \<and> k < y' \<and> d' * k^2 + e * k + f \<noteq> 0" using dense
             by (meson dense greaterThanAtMost_iff less_eq_real_def) 
           then show "False" using allzer
             by auto
         qed
         have f3: "d' \<noteq> 0 \<Longrightarrow> False" 
         proof - 
           assume dnonz: "d' \<noteq> 0"
           have discr: " - e\<^sup>2 + 4 * d' * f \<le> 0"
             using def_prop discriminant_negative[of d' e f] unfolding discrim_def
             by fastforce
           then have two_cases: "(- c' / b') = (- e + - 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')
           \<or> (- c' / b') = (- e + 1 * sqrt (e\<^sup>2 - 4 * d' * f)) / (2 * d')"
             using def_prop dnonz discriminant_nonneg[of d' e f] unfolding discrim_def
             by fastforce
           have some_props: "((d', e, f) \<in> set d \<and> d' \<noteq> 0 \<and> - e\<^sup>2 + 4 * d' * f \<le> 0)"
             using dnonz def_prop discr by auto
           let ?r1 = "(- e + - 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')"
           let ?r2 = "(- e + 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')"
           have cf1: "(- c' / b') = (- e + - 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d') \<Longrightarrow> False" 
           proof - 
             assume "(- c' / b') = (- e + - 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')"
             then have "(d', e, f) \<in> set d \<and> d' \<noteq> 0 \<and> - e\<^sup>2 + 4 * d' * f \<le> 0 \<and> 
     ((\<forall>(d, e, f)\<in>set a. \<exists>y'>?r1. \<forall>x\<in>{?r1<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
     (\<forall>(d, e, f)\<in>set b. \<exists>y'>?r1. \<forall>x\<in>{?r1<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
     (\<forall>(d, e, f)\<in>set c. \<exists>y'>?r1. \<forall>x\<in>{?r1<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
     (\<forall>(d, e, f)\<in>set d. \<exists>y'>?r1. \<forall>x\<in>{?r1<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
               using abc_prop some_props by auto
             then show "False" using f12 by auto
           qed
           have cf2: "(- c' / b') = (- e +  1 * sqrt (e^2 - 4 * d' * f)) / (2 * d') \<Longrightarrow> False" 
           proof - 
             assume "(- c' / b') = (- e + 1 * sqrt (e^2 - 4 * d' * f)) / (2 * d')"
             then have "(d', e, f) \<in> set d \<and> d' \<noteq> 0 \<and> - e\<^sup>2 + 4 * d' * f \<le> 0 \<and> 
     ((\<forall>(d, e, f)\<in>set a. \<exists>y'>?r2. \<forall>x\<in>{?r2<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
     (\<forall>(d, e, f)\<in>set b. \<exists>y'>?r2. \<forall>x\<in>{?r2<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
     (\<forall>(d, e, f)\<in>set c. \<exists>y'>?r2. \<forall>x\<in>{?r2<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
     (\<forall>(d, e, f)\<in>set d. \<exists>y'>?r2. \<forall>x\<in>{?r2<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
               using abc_prop some_props by auto
             then show "False" using f11 by auto
           qed
           then show "False" using two_cases cf1 cf2 by auto
         qed
           (* discriminant_nonnegative *)
         have eo: "(d' \<noteq> 0) \<or> (d' = 0 \<and> e \<noteq> 0) \<or> (d' = 0 \<and> e = 0 \<and> f = 0)" 
           using def_prop by auto
         then show "False" using f1 f2 f3 by auto
       qed
       then show ?thesis by auto
     qed
     have "(\<exists>(a', b', c')\<in>set c. (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0))"
       using h1 h2 h3 h4 bnonz abc_prop by auto
     then show "False" using f8 by auto
   qed
   then show ?thesis by auto
 qed
 
 
 
 lemma qe_forwards_helper:
   assumes alleqset: "\<forall>x. (\<forall>(d, e, f)\<in>set a. d * x^2 + e * x + f = 0)"
   assumes f1: "\<not>((\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0) \<and>
      (\<forall>(a, b, c)\<in>set b. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<and>
      (\<forall>(a, b, c)\<in>set c. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) \<and>
      (\<forall>(a, b, c)\<in>set d. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0))"
   assumes f5: "\<not>(\<exists>(a', b', c')\<in>set b.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
   assumes f6: "\<not> (\<exists>(a', b', c')\<in>set b. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))"
   assumes f7: "\<not> (\<exists>(a', b', c')\<in>set b. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))"
   assumes f8: "\<not>(\<exists>(a', b', c')\<in>set c. (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0))"
   assumes f13: "\<not>(\<exists>(a', b', c')\<in>set c.
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0)))"
   assumes f9: "\<not>(\<exists>(a', b', c')\<in>set c.  a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
         (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0))"
   assumes f10: "\<not>(\<exists>(a', b', c')\<in>set d.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
   assumes f11: "\<not>(\<exists>(a', b', c')\<in>set d.
           a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))"
   assumes f12: "\<not>(\<exists>(a', b', c')\<in>set d.
           a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))"
   shows "\<not>(\<exists>x. (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
          (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
          (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))" 
 proof - 
   have nor: "\<forall>r. \<not>(\<exists>(a', b', c')\<in>set c. 
          ((a'\<noteq> 0 \<or> b'\<noteq> 0) \<and> a'*r^2 + b'*r + c' = 0) \<and>
          ((\<forall>(d, e, f)\<in>set a. d * r\<^sup>2 + e * r + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * r^2 + e * r + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * r^2 + e * r + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * r^2 + e * r + f \<noteq> 0)))" 
   proof clarsimp
     fix r t u v
     assume inset: "(t, u, v) \<in> set c"
     assume eo: "t = 0 \<longrightarrow> u \<noteq> 0 "
     assume zero_eq: "t*r^2 + u*r + v = 0"
     assume ah: "\<forall>x\<in>set a. case x of (d, e, f) \<Rightarrow> d * r\<^sup>2 + e * r + f = 0"
     assume bh: "\<forall>x\<in>set b. case x of (d, e, f) \<Rightarrow> d * r\<^sup>2 + e * r + f < 0" 
     assume ch: "\<forall>x\<in>set c. case x of (d, e, f) \<Rightarrow> d * r\<^sup>2 + e * r + f \<le> 0"
     assume dh: "\<forall>x\<in>set d. case x of (d, e, f) \<Rightarrow> d * r\<^sup>2 + e * r + f \<noteq> 0"
     have two_cases: "t \<noteq> 0 \<or> (t = 0 \<and> u \<noteq> 0)" using eo by auto
     have c1: "t \<noteq> 0 \<Longrightarrow> False" 
     proof - 
       assume tnonz:  "t \<noteq> 0"
       then have discr_prop: "- u\<^sup>2 + 4 * t * v \<le> 0 "
         using discriminant_negative[of t u v] zero_eq unfolding discrim_def
         by force 
       then have ris: "r =  ((-u + - 1 * sqrt (u^2 - 4 * t * v)) / (2 * t)) \<or>
        r = ((-u + 1 * sqrt (u^2 - 4 * t * v)) / (2 * t)) "
         using tnonz discriminant_nonneg[of t u v] zero_eq unfolding discrim_def by auto 
       let ?r1 = "((-u + - 1 * sqrt (u^2 - 4 * t * v)) / (2 * t))"
       let ?r2 = "((-u + 1 * sqrt (u^2 - 4 * t * v)) / (2 * t))" 
       have ris1: "r = ?r1 \<Longrightarrow> False"
       proof - 
         assume "r = ?r1"
         then have "(\<exists>(a', b', c')\<in>set c.  a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
         (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0))" 
           using inset ah bh ch dh discr_prop tnonz by auto
         then show ?thesis 
           using f9 by auto
       qed
       have ris2: "r = ?r2 \<Longrightarrow> False" 
       proof - 
         assume "r = ?r2"
         then have "(\<exists>(a', b', c')\<in>set c.
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0)))" 
           using inset ah bh ch dh discr_prop tnonz by auto
         then show ?thesis 
           using f13 by auto
       qed
       show "False" using ris ris1 ris2 by auto 
     qed
     have c2: "(t = 0 \<and> u \<noteq> 0) \<Longrightarrow> False" 
     proof - 
       assume asm: "t = 0 \<and> u \<noteq> 0"
       then have "r = -v/u" using zero_eq add.right_neutral  nonzero_mult_div_cancel_left     
         by (metis add.commute divide_divide_eq_right divide_eq_0_iff neg_eq_iff_add_eq_0)
       then have "(\<exists>(a', b', c')\<in>set c. (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0))"
         using asm inset ah bh ch dh by auto
       then show "False" using f8
         by auto
     qed
     then show "False" using two_cases c1 c2 by auto
   qed
   have keyh: "\<And>r. \<not>(\<exists>(a', b', c')\<in>set c.
          ((a'\<noteq> 0 \<or> b'\<noteq> 0) \<and> a'*r^2 + b'*r + c' = 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
   proof - 
     fix r
     have h8: "\<not>(\<exists>(a', b', c')\<in>set c. 
          ((a'\<noteq> 0 \<or> b'\<noteq> 0) \<and> a'*r^2 + b'*r + c' = 0) \<and>
          ((\<forall>(d, e, f)\<in>set a. d * r\<^sup>2 + e * r + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * r^2 + e * r + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * r^2 + e * r + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * r^2 + e * r + f \<noteq> 0)))" 
       using nor by auto
     show "\<not>(\<exists>(a', b', c')\<in>set c.
          ((a'\<noteq> 0 \<or> b'\<noteq> 0) \<and> a'*r^2 + b'*r + c' = 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
       using qe_forwards_helper_gen[of c r a b d]
         alleqset  f5 f6 f7 h8 f10 f11 f12 
       by auto
   qed
   have f8a: "\<not>(\<exists>(a', b', c')\<in>set c.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
     using qe_forwards_helper_lin[of a b c d] alleqset f5 f6 f7 f8 f10 f11 f12
     by blast 
   have f13a: "\<not> (\<exists>(a', b', c')\<in>set c. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))"
   proof - 
     have "(\<exists>(a', b', c')\<in>set c. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))) \<Longrightarrow> False" 
     proof - 
       assume "(\<exists>(a', b', c')\<in>set c. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))"
       then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set c \<and> a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))" by auto
       let ?r = "(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')"
       have somek: "\<exists>k. k = ?r" by auto
       then obtain k where k_prop: "k = ?r" by auto
       have "(a'\<noteq> 0 \<or> b'\<noteq> 0) \<and> (a'*?r^2 + b'*?r + c' = 0)" 
         using abc_prop discriminant_nonneg[of a' b' c'] 
         unfolding discrim_def apply (auto)
         by (metis (mono_tags, lifting) times_divide_eq_right) 
       then have "(\<exists>(a', b', c')\<in>set c.
          ((a'\<noteq> 0 \<or> b'\<noteq> 0) \<and> a'*?r^2 + b'*?r + c' = 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
         using abc_prop by auto
       then have "(\<exists>(a', b', c')\<in>set c.
          ((a'\<noteq> 0 \<or> b'\<noteq> 0) \<and> a'*k^2 + b'*k + c' = 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
         using k_prop by auto
       then have "\<exists>k. (\<exists>(a', b', c')\<in>set c.
          ((a'\<noteq> 0 \<or> b'\<noteq> 0) \<and> a'*k^2 + b'*k + c' = 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
         by auto 
       then show "False" using keyh by auto
     qed
     then
     show ?thesis
       by auto
   qed
   have f9a: "\<not> (\<exists>(a', b', c')\<in>set c. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))" 
 
   proof - 
     have "(\<exists>(a', b', c')\<in>set c. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)) \<Longrightarrow> False" 
     proof - 
       assume "(\<exists>(a', b', c')\<in>set c. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))"
       then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set c \<and> a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)" by auto
       let ?r = "(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')"
       have somek: "\<exists>k. k = ?r" by auto
       then obtain k where k_prop: "k = ?r" by auto
       have "(a'\<noteq> 0 \<or> b'\<noteq> 0) \<and> (a'*?r^2 + b'*?r + c' = 0)" 
         using abc_prop discriminant_nonneg[of a' b' c'] 
         unfolding discrim_def apply (auto)
         by (metis (mono_tags, lifting) times_divide_eq_right) 
       then have "(\<exists>(a', b', c')\<in>set c.
          ((a'\<noteq> 0 \<or> b'\<noteq> 0) \<and> a'*?r^2 + b'*?r + c' = 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>?r. \<forall>x\<in>{?r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
         using abc_prop by auto
       then have "(\<exists>(a', b', c')\<in>set c.
          ((a'\<noteq> 0 \<or> b'\<noteq> 0) \<and> a'*k^2 + b'*k + c' = 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
         using k_prop by auto
       then have "\<exists>k. (\<exists>(a', b', c')\<in>set c.
          ((a'\<noteq> 0 \<or> b'\<noteq> 0) \<and> a'*k^2 + b'*k + c' = 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
         by auto 
       then show "False" using keyh by auto
     qed
     then
     show ?thesis
       by auto
   qed
     (* We need to show that the point is in one of these ranges *)
   have "(\<exists>x. (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
          (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
          (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0)) \<Longrightarrow> False"
   proof - 
     assume "(\<exists>x. (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
          (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
          (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))"
     then obtain x where x_prop: "(\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
          (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
          (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0)" by auto
     (* Need this sorted_nonzero_root_list_set in case some of the tuples from set c are (0, 0, 0) *)
     let ?srl = "sorted_nonzero_root_list_set (((set b) \<union> (set c))\<union> (set d))"
     have alleqsetvar: "\<forall>(t, u, v) \<in> set a. t = 0 \<and> u = 0 \<and> v = 0"
     proof clarsimp
       fix t u v
       assume "(t, u, v) \<in> set a"
       then have "\<forall>x. t*x^2 + u*x + v = 0" 
         using alleqset by auto 
       then have "\<forall>x\<in>{0<..1}. t * x\<^sup>2 + u * x + v = 0" 
         by auto
       then show "t = 0 \<and> u = 0 \<and> v = 0" 
         using continuity_lem_eq0[of 0 1 t u v] 
         by auto  
     qed
       (* Should violate f1 *)
     have lenzero: "length ?srl = 0 \<Longrightarrow> False"
     proof - 
       assume lenzero: "length ?srl = 0"
       have ina: "(\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0)"
         using alleqsetvar by auto 
       have inb: "(\<forall>(a, b, c)\<in>set b. \<forall>y. a * y\<^sup>2 + b * y + c < 0)"
       proof clarsimp 
         fix t u v y
         assume insetb: "(t, u, v) \<in> set b"
         then have "t * x\<^sup>2 + u * x + v < 0" using x_prop by auto
         then have tuv_prop: "t \<noteq> 0 \<or> u \<noteq> 0 \<or> v \<noteq> 0"
           by auto 
         then have tuzer: "(t = 0 \<and> u = 0) \<Longrightarrow> \<not>(\<exists>q. t * q\<^sup>2 + u * q + v = 0)"
           by simp
         then have tunonz: "(t \<noteq> 0 \<or> u \<noteq> 0) \<Longrightarrow> \<not>(\<exists>q. t * q\<^sup>2 + u * q + v = 0)"
         proof - 
           assume tuv_asm: "t \<noteq> 0 \<or> u \<noteq> 0"
           have "\<exists>q. t * q\<^sup>2 + u * q + v = 0 \<Longrightarrow> False"
           proof - 
             assume "\<exists> q. t * q\<^sup>2 + u * q + v = 0"
             then obtain q where "t * q\<^sup>2 + u * q + v = 0" by auto
             then have qin: "q \<in>  {x. \<exists>(a, b, c)\<in>set b \<union> set c \<union> set d. (a \<noteq> 0 \<or> b \<noteq> 0) \<and> a * x\<^sup>2 + b * x + c = 0}"
               using insetb tuv_asm tuv_prop by auto
             have "set ?srl = nonzero_root_set (set b \<union> set c \<union> set d)"
               unfolding sorted_nonzero_root_list_set_def
               using set_sorted_list_of_set[of "nonzero_root_set (set b \<union> set c \<union> set d)"]
                 nonzero_root_set_finite[of "(set b \<union> set c \<union> set d)"]
               by auto
             then have "q \<in> set ?srl" using qin unfolding nonzero_root_set_def
               by auto 
             then have "List.member ?srl q"     
               using in_set_member[of q ?srl]
               by auto
             then show "False"
               using lenzero
               by (simp add: member_rec(2)) 
           qed
           then show ?thesis by auto
         qed
         have nozer: "\<not>(\<exists>q. t * q\<^sup>2 + u * q + v = 0)" 
           using  tuzer tunonz
           by blast 
         have samesn: "sign_num (t*x^2 + u*x + v) = sign_num (t*y^2 + u*y + v)"
         proof - 
           have "x < y \<Longrightarrow> sign_num (t*x^2 + u*x + v) = sign_num (t*y^2 + u*y + v)"
             using changes_sign_var[of t x u v y] nozer by auto
           have "y < x \<Longrightarrow> sign_num (t*x^2 + u*x + v) = sign_num (t*y^2 + u*y + v)"
             using changes_sign_var[of t y u v x] nozer
           proof -
             assume "y < x"
             then show ?thesis
               using \<open>\<nexists>q. t * q\<^sup>2 + u * q + v = 0\<close> \<open>sign_num (t * y\<^sup>2 + u * y + v) \<noteq> sign_num (t * x\<^sup>2 + u * x + v) \<and> y < x \<Longrightarrow> \<exists>q. t * q\<^sup>2 + u * q + v = 0 \<and> y \<le> q \<and> q \<le> x\<close> by presburger
           qed
           show ?thesis
             using changes_sign_var using \<open>x < y \<Longrightarrow> sign_num (t * x\<^sup>2 + u * x + v) = sign_num (t * y\<^sup>2 + u * y + v)\<close> \<open>y < x \<Longrightarrow> sign_num (t * x\<^sup>2 + u * x + v) = sign_num (t * y\<^sup>2 + u * y + v)\<close> 
             by fastforce 
         qed
           (* changes_sign_var *)
         have "sign_num (t*x^2 + u*x + v) = -1" using insetb unfolding sign_num_def using x_prop 
           by auto 
         then have "sign_num (t*y^2 + u*y + v) = -1" using samesn by auto
         then show "t * y\<^sup>2 + u * y + v < 0" unfolding sign_num_def
           by smt
       qed
       have inc: "(\<forall>(a, b, c)\<in>set c. \<forall>y. a * y\<^sup>2 + b * y + c \<le> 0)"
       proof clarsimp 
         fix t u v y
         assume insetc: "(t, u, v) \<in> set c"
         then have "t * x\<^sup>2 + u * x + v \<le> 0" using x_prop by auto
         then have tuzer: "t = 0 \<and> u = 0 \<Longrightarrow> t*y^2 + u*y + v \<le> 0 " 
         proof - 
           assume tandu: "t = 0 \<and> u = 0"
           then have "v \<le> 0" using insetc x_prop 
             by auto
           then show "t*y^2 + u*y + v \<le> 0" using tandu 
             by auto
         qed
         have tunonz: "t \<noteq> 0 \<or> u \<noteq> 0 \<Longrightarrow> t*y^2 + u*y + v \<le> 0"
         proof - 
           assume tuv_asm: "t \<noteq> 0 \<or> u \<noteq> 0"
           have insetcvar: "t*x^2 + u*x + v < 0" 
           proof - 
             have "t*x^2 + u*x + v = 0 \<Longrightarrow> False" 
             proof -
               assume zer: "t*x^2 + u*x + v = 0"
               then have xin: "x \<in>  {x. \<exists>(a, b, c)\<in>set b \<union> set c \<union> set d. (a \<noteq> 0 \<or> b \<noteq> 0) \<and> a * x\<^sup>2 + b * x + c = 0}"
                 using insetc tuv_asm by auto
               have "set ?srl = nonzero_root_set (set b \<union> set c \<union> set d)"
                 unfolding sorted_nonzero_root_list_set_def
                 using set_sorted_list_of_set[of "nonzero_root_set (set b \<union> set c \<union> set d)"]
                   nonzero_root_set_finite[of "(set b \<union> set c \<union> set d)"]
                 by auto
               then have "x \<in> set ?srl" using xin unfolding nonzero_root_set_def
                 by auto 
               then have "List.member ?srl x"     
                 using in_set_member[of x ?srl]
                 by auto
               then show "False" using lenzero
                 by (simp add: member_rec(2)) 
             qed
             then show ?thesis
               using \<open>t * x\<^sup>2 + u * x + v \<le> 0\<close> by fastforce 
           qed
           then have tunonz: "\<not>(\<exists>q. t * q\<^sup>2 + u * q + v = 0)"
           proof - 
             have "\<exists>q. t * q\<^sup>2 + u * q + v = 0 \<Longrightarrow> False"
             proof - 
               assume "\<exists> q. t * q\<^sup>2 + u * q + v = 0"
               then obtain q where "t * q\<^sup>2 + u * q + v = 0" by auto
               then have qin: "q \<in>  {x. \<exists>(a, b, c)\<in>set b \<union> set c \<union> set d. (a \<noteq> 0 \<or> b \<noteq> 0) \<and> a * x\<^sup>2 + b * x + c = 0}"
                 using insetc tuv_asm by auto
               have "set ?srl = nonzero_root_set (set b \<union> set c \<union> set d)"
                 unfolding sorted_nonzero_root_list_set_def
                 using set_sorted_list_of_set[of "nonzero_root_set (set b \<union> set c \<union> set d)"]
                   nonzero_root_set_finite[of "(set b \<union> set c \<union> set d)"]
                 by auto
               then have "q \<in> set ?srl" using qin unfolding nonzero_root_set_def
                 by auto 
               then have "List.member ?srl q"     
                 using in_set_member[of q ?srl]
                 by auto
               then show "False"
                 using lenzero
                 by (simp add: member_rec(2)) 
             qed
             then show ?thesis by auto
           qed
           have nozer: "\<not>(\<exists>q. t * q\<^sup>2 + u * q + v = 0)" 
             using  tuzer tunonz
             by blast 
           have samesn: "sign_num (t*x^2 + u*x + v) = sign_num (t*y^2 + u*y + v)"
           proof - 
             have "x < y \<Longrightarrow> sign_num (t*x^2 + u*x + v) = sign_num (t*y^2 + u*y + v)"
               using changes_sign_var[of t x u v y] nozer by auto
             have "y < x \<Longrightarrow> sign_num (t*x^2 + u*x + v) = sign_num (t*y^2 + u*y + v)"
               using changes_sign_var[of t y u v x] nozer
             proof -
               assume "y < x"
               then show ?thesis
                 using \<open>\<nexists>q. t * q\<^sup>2 + u * q + v = 0\<close> \<open>sign_num (t * y\<^sup>2 + u * y + v) \<noteq> sign_num (t * x\<^sup>2 + u * x + v) \<and> y < x \<Longrightarrow> \<exists>q. t * q\<^sup>2 + u * q + v = 0 \<and> y \<le> q \<and> q \<le> x\<close> by presburger
             qed
             show ?thesis
               using changes_sign_var using \<open>x < y \<Longrightarrow> sign_num (t * x\<^sup>2 + u * x + v) = sign_num (t * y\<^sup>2 + u * y + v)\<close> \<open>y < x \<Longrightarrow> sign_num (t * x\<^sup>2 + u * x + v) = sign_num (t * y\<^sup>2 + u * y + v)\<close> 
               by fastforce 
           qed
             (* changes_sign_var *)
           have "sign_num (t*x^2 + u*x + v) = -1" using insetcvar unfolding sign_num_def using x_prop 
             by auto 
           then have "sign_num (t*y^2 + u*y + v) = -1" using samesn by auto
           then show "t * y\<^sup>2 + u * y + v \<le> 0" unfolding sign_num_def
             by smt
         qed
         then show "t * y\<^sup>2 + u * y + v \<le> 0" 
           using tuzer tunonz
           by blast 
       qed
       have ind: "(\<forall>(a, b, c)\<in>set d. \<forall>y. a * y\<^sup>2 + b * y + c \<noteq> 0)"
       proof clarsimp 
         fix t u v y
         assume insetd: "(t, u, v) \<in> set d"
         assume falseasm: "t * y\<^sup>2 + u * y + v = 0"
         then have snz: "sign_num (t*y^2 + u*y + v) = 0"
           unfolding sign_num_def by auto
         have "t * x\<^sup>2 + u * x + v \<noteq> 0" using insetd x_prop by auto
         then have tuv_prop: "t \<noteq> 0 \<or> u \<noteq> 0 \<or> v \<noteq> 0"
           by auto 
         then have tuzer: "(t = 0 \<and> u = 0) \<Longrightarrow> \<not>(\<exists>q. t * q\<^sup>2 + u * q + v = 0)"
           by simp
         then have tunonz: "(t \<noteq> 0 \<or> u \<noteq> 0) \<Longrightarrow> \<not>(\<exists>q. t * q\<^sup>2 + u * q + v = 0)"
         proof - 
           assume tuv_asm: "t \<noteq> 0 \<or> u \<noteq> 0"
           have "\<exists>q. t * q\<^sup>2 + u * q + v = 0 \<Longrightarrow> False"
           proof - 
             assume "\<exists> q. t * q\<^sup>2 + u * q + v = 0"
             then obtain q where "t * q\<^sup>2 + u * q + v = 0" by auto
             then have qin: "q \<in>  {x. \<exists>(a, b, c)\<in>set b \<union> set c \<union> set d. (a \<noteq> 0 \<or> b \<noteq> 0) \<and> a * x\<^sup>2 + b * x + c = 0}"
               using insetd tuv_asm tuv_prop by auto
             have "set ?srl = nonzero_root_set (set b \<union> set c \<union> set d)"
               unfolding sorted_nonzero_root_list_set_def
               using set_sorted_list_of_set[of "nonzero_root_set (set b \<union> set c \<union> set d)"]
                 nonzero_root_set_finite[of "(set b \<union> set c \<union> set d)"]
               by auto
             then have "q \<in> set ?srl" using qin unfolding nonzero_root_set_def
               by auto 
             then have "List.member ?srl q"     
               using in_set_member[of q ?srl]
               by auto
             then show "False"
               using lenzero
               by (simp add: member_rec(2)) 
           qed
           then show ?thesis by auto
         qed
         have nozer: "\<not>(\<exists>q. t * q\<^sup>2 + u * q + v = 0)" 
           using  tuzer tunonz
           by blast 
         have samesn: "sign_num (t*x^2 + u*x + v) = sign_num (t*y^2 + u*y + v)"
         proof - 
           have "x < y \<Longrightarrow> sign_num (t*x^2 + u*x + v) = sign_num (t*y^2 + u*y + v)"
             using changes_sign_var[of t x u v y] nozer by auto
           have "y < x \<Longrightarrow> sign_num (t*x^2 + u*x + v) = sign_num (t*y^2 + u*y + v)"
             using changes_sign_var[of t y u v x] nozer
           proof -
             assume "y < x"
             then show ?thesis
               using \<open>\<nexists>q. t * q\<^sup>2 + u * q + v = 0\<close> \<open>sign_num (t * y\<^sup>2 + u * y + v) \<noteq> sign_num (t * x\<^sup>2 + u * x + v) \<and> y < x \<Longrightarrow> \<exists>q. t * q\<^sup>2 + u * q + v = 0 \<and> y \<le> q \<and> q \<le> x\<close> by presburger
           qed
           show ?thesis
             using changes_sign_var using \<open>x < y \<Longrightarrow> sign_num (t * x\<^sup>2 + u * x + v) = sign_num (t * y\<^sup>2 + u * y + v)\<close> \<open>y < x \<Longrightarrow> sign_num (t * x\<^sup>2 + u * x + v) = sign_num (t * y\<^sup>2 + u * y + v)\<close> 
             by fastforce 
         qed
           (* changes_sign_var *)
         have "sign_num (t*x^2 + u*x + v) = -1 \<or> sign_num (t*x^2 + u*x + v) = 1 " 
           using insetd unfolding sign_num_def using x_prop 
           by auto 
         then have "sign_num (t*y^2 + u*y + v) = -1 \<or> sign_num (t*y^2 + u*y + v) = 1" using samesn by auto
         then show "False"  using snz by auto 
       qed
         (* Show all the polynomials never change sign *)
       have "((\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0) \<and>
      (\<forall>(a, b, c)\<in>set b. \<forall>y. a * y\<^sup>2 + b * y + c < 0) \<and>
      (\<forall>(a, b, c)\<in>set c. \<forall>y. a * y\<^sup>2 + b * y + c \<le> 0) \<and>
      (\<forall>(a, b, c)\<in>set d. \<forall>y. a * y\<^sup>2 + b * y + c \<noteq> 0))"
         using ina inb inc ind by auto
       then  show "False" 
         using f1
         by auto 
     qed
     have cases_mem: "(List.member ?srl x) \<Longrightarrow> False"
     proof - 
       assume "(List.member ?srl x)"
       then have "x \<in> {x. \<exists>(a, b, c)\<in>set b \<union> set c \<union> set d. (a \<noteq> 0 \<or> b \<noteq> 0) \<and> a * x\<^sup>2 + b * x + c = 0}"
         using set_sorted_list_of_set nonzero_root_set_finite in_set_member
         by (metis List.finite_set finite_Un nonzero_root_set_def sorted_nonzero_root_list_set_def)
       then have "\<exists> (a, b, c) \<in> (((set b) \<union> (set c))\<union> (set d)) . (a \<noteq> 0 \<or> b \<noteq> 0) \<and> a*x^2 + b*x + c = 0"
         by blast
       then obtain t u v where def_prop: "(t, u, v) \<in> (((set b) \<union> (set c))\<union> (set d)) \<and> (t \<noteq> 0 \<or> u \<noteq> 0) \<and> t*x^2 + u*x + v = 0"
         by auto
       have notinb: "(t, u, v) \<notin> (set b)"
       proof - 
         have "(t, u, v) \<in> (set b ) \<Longrightarrow> False"
         proof - 
           assume "(t, u, v) \<in> (set b)"
           then have "t*x^2 + u*x + v < 0" using x_prop
             by blast
           then show "False" using def_prop
             by simp 
         qed
         then  show ?thesis by auto
       qed
       have notind: "(t, u, v) \<notin> (set d)"
       proof - 
         have "(t, u, v) \<in> (set d) \<Longrightarrow> False"
         proof - 
           assume "(t, u, v) \<in> (set d)"
           then have "t*x^2 + u*x + v \<noteq> 0" using x_prop 
             by blast
           then show "False" using def_prop 
             by simp
         qed
         then show ?thesis by auto
       qed
       then have inset: "(t, u, v) \<in> (set c)"
         using def_prop notinb notind
         by blast 
       have case1: "t \<noteq> 0 \<Longrightarrow> False"
       proof - 
         assume tnonz: "t \<noteq> 0" 
         then have r1or2:"x = (- u + - 1 * sqrt (u\<^sup>2 - 4 * t * v)) / (2 * t) \<or>
             x = (- u + 1 * sqrt (u^2 - 4 * t * v)) / (2 * t) "
           using def_prop discriminant_negative[of t u v] discriminant_nonneg[of t u v]
           apply (auto) 
           using notinb apply (force)
            apply (simp add: discrim_def discriminant_iff)
           using notind by force 
         have discrh: "-1*u^2 + 4 * t * v \<le> 0" 
           using tnonz discriminant_negative[of t u v] unfolding discrim_def
           using def_prop by force 
         have r1: "x = (- u + - 1 * sqrt (u\<^sup>2 - 4 * t * v)) / (2 * t) \<Longrightarrow> False"
         proof - 
           assume xis: "x = (- u + - 1 * sqrt (u\<^sup>2 - 4 * t * v)) / (2 * t)"
           have " t \<noteq> 0 \<and>
          - 1*u^2 + 4 * t * v \<le> 0 \<and>
         (\<forall>(d, e, f)\<in>set a.
               d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
                d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * x\<^sup>2 + e * x + f \<noteq> 0)"
             using tnonz alleqset discrh x_prop
             by auto
           then have "(\<exists>(a', b', c')\<in>set c.  a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
         (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0))" 
             using xis inset 
             by auto
           then show "False"
             using f9 by auto
         qed
         have r2: "x = (- u + 1 * sqrt (u\<^sup>2 - 4 * t * v)) / (2 * t) \<Longrightarrow> False" 
         proof - 
           assume xis: "x = (- u + 1 * sqrt (u\<^sup>2 - 4 * t * v)) / (2 * t)"
           have " t \<noteq> 0 \<and>
          - 1*u^2 + 4 * t * v \<le> 0 \<and>
         (\<forall>(d, e, f)\<in>set a.
               d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
                d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * x\<^sup>2 + e * x + f \<noteq> 0)"
             using tnonz alleqset discrh x_prop
             by auto
           then have "(\<exists>(a', b', c')\<in>set c.  a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
         (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0))" 
             using xis inset 
             by auto
           then show "False"
             using f13 by auto
         qed
         then show "False"
           using r1or2 r1 r2 by auto
       qed
       have case2: "(t = 0 \<and> u \<noteq> 0) \<Longrightarrow> False" 
       proof -
         assume asm: "t = 0 \<and> u \<noteq> 0"
         then have xis: "x = - v / u" using def_prop notinb add.commute diff_0 divide_non_zero minus_add_cancel uminus_add_conv_diff
           by (metis mult_zero_left)
         have "((t = 0 \<and> u \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * x^2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * x^2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * x^2 + e * x + f \<noteq> 0))"
           using asm x_prop alleqset by auto
         then have "(\<exists>(a', b', c')\<in>set c. (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0))"
           using xis inset
           by auto
         then show "False"
           using f8 by auto
       qed
       show "False"
         using def_prop case1 case2 by auto 
     qed
     have lengt0: "length ?srl \<ge> 1 \<Longrightarrow> False" 
     proof- 
       assume asm: "length ?srl \<ge> 1"
         (* should violate f1 *)
       have cases_lt: "x < ?srl ! 0 \<Longrightarrow> False" 
       proof - 
         assume xlt: "x < ?srl ! 0"
         have samesign: "\<forall> (a, b, c) \<in> (set b \<union> set c \<union> set d).
               (\<forall>y < x. sign_num (a * y\<^sup>2 + b * y + c) =  sign_num (a*x^2 + b*x + c))"
         proof  clarsimp  
           fix t u v y
           assume insetunion: "(t, u, v) \<in> set b \<or> (t, u, v)  \<in> set c \<or> (t, u, v) \<in> set d"
           assume ylt: "y < x" 
           have tuzer: "t = 0 \<and> u = 0 \<Longrightarrow> sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
             unfolding sign_num_def 
             by auto
           have tunonzer: "t \<noteq> 0 \<or> u \<noteq> 0  \<Longrightarrow> sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
           proof - 
             assume tuv_asm: "t\<noteq> 0 \<or> u \<noteq> 0"
             have "\<not>(\<exists>q. q < ?srl ! 0 \<and> t * q\<^sup>2 + u * q + v = 0)"
             proof clarsimp 
               fix q
               assume qlt: "q < sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! 0"
               assume "t * q\<^sup>2 + u * q + v = 0"
               then have qin: "q \<in>  {x. \<exists>(a, b, c)\<in>set b \<union> set c \<union> set d. (a \<noteq> 0 \<or> b \<noteq> 0) \<and> a * x\<^sup>2 + b * x + c = 0}"
                 using insetunion tuv_asm by auto
               have "set ?srl = nonzero_root_set (set b \<union> set c \<union> set d)"
                 unfolding sorted_nonzero_root_list_set_def
                 using set_sorted_list_of_set[of "nonzero_root_set (set b \<union> set c \<union> set d)"]
                   nonzero_root_set_finite[of "(set b \<union> set c \<union> set d)"]
                 by auto
               then have "q \<in> set ?srl" using qin unfolding nonzero_root_set_def
                 by auto 
               then have lm: "List.member ?srl q"     
                 using in_set_member[of q ?srl]
                 by auto
               then have " List.member
                  (sorted_list_of_set (nonzero_root_set (set b \<union> set c \<union> set d)))
                  q \<Longrightarrow>
                 q < sorted_list_of_set (nonzero_root_set (set b \<union> set c \<union> set d)) !
                     0 \<Longrightarrow>
                 (\<And>x xs. (x \<in> set xs) = (\<exists>i<length xs. xs ! i = x)) \<Longrightarrow>
                 (\<And>x xs. (x \<in> set xs) = List.member xs x) \<Longrightarrow>
                 (\<And>y x. \<not> y \<le> x \<Longrightarrow> x < y) \<Longrightarrow>
                 (\<And>xs. sorted xs =
                        (\<forall>i j. i \<le> j \<longrightarrow> j < length xs \<longrightarrow> xs ! i \<le> xs ! j)) \<Longrightarrow>
                 (\<And>p. sorted_nonzero_root_list_set p \<equiv>
                       sorted_list_of_set (nonzero_root_set p)) \<Longrightarrow>
                 False"
               proof -
                 assume a1: "List.member (sorted_list_of_set (nonzero_root_set (set b \<union> set c \<union> set d))) q"
                 assume a2: "q < sorted_list_of_set (nonzero_root_set (set b \<union> set c \<union> set d)) ! 0"
                 have f3: "List.member (sorted_list_of_set {r. \<exists>p. p \<in> set b \<union> set c \<union> set d \<and> (case p of (ra, rb, rc) \<Rightarrow> (ra \<noteq> 0 \<or> rb \<noteq> 0) \<and> ra * r\<^sup>2 + rb * r + rc = 0)}) q"
                   using a1 by (metis nonzero_root_set_def)
                 have f4: "q < sorted_list_of_set {r. \<exists>p. p \<in> set b \<union> set c \<union> set d \<and> (case p of (ra, rb, rc) \<Rightarrow> (ra \<noteq> 0 \<or> rb \<noteq> 0) \<and> ra * r\<^sup>2 + rb * r + rc = 0)} ! 0"
                   using a2 by (metis nonzero_root_set_def)
                 have f5: "q \<in> set (sorted_list_of_set {r. \<exists>p. p \<in> set b \<union> set c \<union> set d \<and> (case p of (ra, rb, rc) \<Rightarrow> (ra \<noteq> 0 \<or> rb \<noteq> 0) \<and> ra * r\<^sup>2 + rb * r + rc = 0)})"
                   using f3 by (meson in_set_member)
                 have "\<forall>rs r. \<exists>n. ((r::real) \<notin> set rs \<or> n < length rs) \<and> (r \<notin> set rs \<or> rs ! n = r)"
                   by (metis in_set_conv_nth)
                 then obtain nn :: "real list \<Rightarrow> real \<Rightarrow> nat" where
                   f6: "\<And>r rs. (r \<notin> set rs \<or> nn rs r < length rs) \<and> (r \<notin> set rs \<or> rs ! nn rs r = r)"
                   by moura
                 then have "sorted_list_of_set {r. \<exists>p. p \<in> set b \<union> set c \<union> set d \<and> (case p of (ra, rb, rc) \<Rightarrow> (ra \<noteq> 0 \<or> rb \<noteq> 0) \<and> ra * r\<^sup>2 + rb * r + rc = 0)} ! nn (sorted_list_of_set {r. \<exists>p. p \<in> set b \<union> set c \<union> set d \<and> (case p of (ra, rb, rc) \<Rightarrow> (ra \<noteq> 0 \<or> rb \<noteq> 0) \<and> ra * r\<^sup>2 + rb * r + rc = 0)}) q = q"
                   using f5 by blast
                 then have "\<And>n. \<not> sorted (sorted_list_of_set {r. \<exists>p. p \<in> set b \<union> set c \<union> set d \<and> (case p of (ra, rb, rc) \<Rightarrow> (ra \<noteq> 0 \<or> rb \<noteq> 0) \<and> ra * r\<^sup>2 + rb * r + rc = 0)}) \<or> \<not> n \<le> nn (sorted_list_of_set {r. \<exists>p. p \<in> set b \<union> set c \<union> set d \<and> (case p of (ra, rb, rc) \<Rightarrow> (ra \<noteq> 0 \<or> rb \<noteq> 0) \<and> ra * r\<^sup>2 + rb * r + rc = 0)}) q \<or> \<not> nn (sorted_list_of_set {r. \<exists>p. p \<in> set b \<union> set c \<union> set d \<and> (case p of (ra, rb, rc) \<Rightarrow> (ra \<noteq> 0 \<or> rb \<noteq> 0) \<and> ra * r\<^sup>2 + rb * r + rc = 0)}) q < length (sorted_list_of_set {r. \<exists>p. p \<in> set b \<union> set c \<union> set d \<and> (case p of (ra, rb, rc) \<Rightarrow> (ra \<noteq> 0 \<or> rb \<noteq> 0) \<and> ra * r\<^sup>2 + rb * r + rc = 0)}) \<or> sorted_list_of_set {r. \<exists>p. p \<in> set b \<union> set c \<union> set d \<and> (case p of (ra, rb, rc) \<Rightarrow> (ra \<noteq> 0 \<or> rb \<noteq> 0) \<and> ra * r\<^sup>2 + rb * r + rc = 0)} ! n \<le> q"
                   using not_less not_less0 sorted_iff_nth_mono
                   by (metis (no_types, lifting)) 
                 then show ?thesis
                   using f6 f5 f4 by (meson le0 not_less sorted_sorted_list_of_set)
               qed 
               then show "False" using lm qlt in_set_conv_nth in_set_member not_le_imp_less not_less0 sorted_iff_nth_mono sorted_nonzero_root_list_set_def sorted_sorted_list_of_set
                 by auto
             qed   
             then have "\<not>(\<exists>q. q \<le> x \<and> t * q\<^sup>2 + u * q + v = 0)"
               using xlt
               by auto 
             then show " sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
               using ylt changes_sign_var[of t y u v x]
               by blast
           qed
           then show " sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
             using tuzer 
             by blast
         qed
         have bseth: "(\<forall>(a, b, c)\<in>set b. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0)" 
         proof clarsimp 
           fix t u v y
           assume insetb: "(t, u, v) \<in> set b"
           assume yltx: "y < x" 
           have "(t, u, v) \<in> (set b \<union> set c \<union> set d)" using insetb 
             by auto
           then have samesn: "sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
             using samesign insetb yltx
             by blast 
           have "sign_num (t*x^2 + u*x + v) = -1" 
             using x_prop insetb unfolding sign_num_def
             by auto
           then show  "t * y\<^sup>2 + u * y + v < 0"
             using samesn unfolding sign_num_def 
             by (metis add.right_inverse add.right_neutral linorder_neqE_linordered_idom one_add_one zero_neq_numeral) 
         qed
         have bset: " (\<forall>(a, b, c)\<in>set b. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0)"
         proof clarsimp 
           fix t u v
           assume inset: "(t, u, v) \<in> set b"
           then have " \<forall>y<x. t * y\<^sup>2 + u * y + v < 0 " using bseth by auto
           then show "\<exists>x. \<forall>y<x. t * y\<^sup>2 + u * y + v < 0"
             by auto
         qed
         have cseth: "(\<forall>(a, b, c)\<in>set c. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0)" 
         proof clarsimp 
           fix t u v y
           assume insetc: "(t, u, v) \<in> set c"
           assume yltx: "y < x" 
           have "(t, u, v) \<in> (set b \<union> set c \<union> set d)" using insetc
             by auto
           then have samesn: "sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
             using samesign insetc yltx
             by blast 
           have "sign_num (t*x^2 + u*x + v) = -1 \<or> sign_num (t*x^2 + u*x + v) = 0" 
             using x_prop insetc unfolding sign_num_def
             by auto
           then show  "t * y\<^sup>2 + u * y + v \<le> 0"
             using samesn unfolding sign_num_def
             using zero_neq_one by fastforce
         qed
         have cset: " (\<forall>(a, b, c)\<in>set c. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0)"
         proof clarsimp 
           fix t u v
           assume inset: "(t, u, v) \<in> set c"
           then have " \<forall>y<x. t * y\<^sup>2 + u * y + v \<le> 0 " using cseth by auto
           then show "\<exists>x. \<forall>y<x. t * y\<^sup>2 + u * y + v \<le>0"
             by auto
         qed
         have dseth: "(\<forall>(a, b, c)\<in>set d. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0)" 
         proof clarsimp 
           fix t u v y
           assume insetd: "(t, u, v) \<in> set d"
           assume yltx: "y < x" 
           assume contrad: "t * y\<^sup>2 + u * y + v = 0"
           have "(t, u, v) \<in> (set b \<union> set c \<union> set d)" using insetd
             by auto
           then have samesn: "sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
             using samesign insetd yltx
             by blast 
           have "sign_num (t*x^2 + u*x + v) = -1 \<or> sign_num (t*x^2 + u*x + v) = 1" 
             using x_prop insetd unfolding sign_num_def
             by auto
           then have  "t * y\<^sup>2 + u * y + v \<noteq> 0"
             using samesn unfolding sign_num_def 
             by auto
           then show "False" using contrad by auto
         qed
         have dset: " (\<forall>(a, b, c)\<in>set d. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0)"
         proof clarsimp 
           fix t u v
           assume inset: "(t, u, v) \<in> set d"
           then have " \<forall>y<x. t * y\<^sup>2 + u * y + v \<noteq> 0 " using dseth by auto
           then show "\<exists>x. \<forall>y<x. t * y\<^sup>2 + u * y + v \<noteq> 0"
             by auto
         qed
         have "(\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0)" 
           using alleqsetvar by auto 
         then have "((\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0) \<and>
      (\<forall>(a, b, c)\<in>set b. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<and>
      (\<forall>(a, b, c)\<in>set c. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) \<and>
      (\<forall>(a, b, c)\<in>set d. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0))" 
           using  bset cset dset by auto
         then show "False" using f1 by auto
       qed
         (* should violate one of the infinitesmials *)
       have cases_gt: " x > ?srl ! (length ?srl - 1) \<Longrightarrow> False" 
       proof - 
         assume xgt: "x > ?srl ! (length ?srl - 1)"
         let ?bgrt = "?srl ! (length ?srl - 1)"
         have samesign: "\<forall> (a, b, c) \<in> (set b \<union> set c \<union> set d).
               (\<forall>y > ?bgrt. sign_num (a * y\<^sup>2 + b * y + c) =  sign_num (a*x^2 + b*x + c))"
         proof  clarsimp  
           fix t u v y
           assume insetunion: "(t, u, v) \<in> set b \<or> (t, u, v)  \<in> set c \<or> (t, u, v) \<in> set d"
           assume ygt: "sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) !
               (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - Suc 0)  < y" 
           have tuzer: "t = 0 \<and> u = 0 \<Longrightarrow> sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
             unfolding sign_num_def 
             by auto
           have tunonzer: "t \<noteq> 0 \<or> u \<noteq> 0  \<Longrightarrow> sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
           proof - 
             assume tuv_asm: "t\<noteq> 0 \<or> u \<noteq> 0"
             have "\<not>(\<exists>q. q > ?srl ! (length ?srl - 1) \<and> t * q\<^sup>2 + u * q + v = 0)"
             proof clarsimp 
               fix q
               assume qgt: "sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) !
                 (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - Suc 0) < q"
               assume "t * q\<^sup>2 + u * q + v = 0"
               then have qin: "q \<in>  {x. \<exists>(a, b, c)\<in>set b \<union> set c \<union> set d. (a \<noteq> 0 \<or> b \<noteq> 0) \<and> a * x\<^sup>2 + b * x + c = 0}"
                 using insetunion tuv_asm by auto
               have "set ?srl = nonzero_root_set (set b \<union> set c \<union> set d)"
                 unfolding sorted_nonzero_root_list_set_def
                 using set_sorted_list_of_set[of "nonzero_root_set (set b \<union> set c \<union> set d)"]
                   nonzero_root_set_finite[of "(set b \<union> set c \<union> set d)"]
                 by auto
               then have "q \<in> set ?srl" using qin unfolding nonzero_root_set_def
                 by auto 
               then have "List.member ?srl q"     
                 using in_set_member[of q ?srl]
                 by auto
               then show "False" using qgt in_set_conv_nth in_set_member not_le_imp_less not_less0 sorted_iff_nth_mono sorted_nonzero_root_list_set_def sorted_sorted_list_of_set
                 by (smt (z3) Suc_diff_Suc Suc_n_not_le_n \<open>q \<in> set (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d))\<close> in_set_conv_nth length_0_conv length_greater_0_conv length_sorted_list_of_set lenzero less_Suc_eq_le minus_nat.diff_0 not_le sorted_nth_mono sorted_sorted_list_of_set) 
             qed   
             then have nor: "\<not>(\<exists>q. q > ?bgrt \<and> t * q\<^sup>2 + u * q + v = 0)"
               using xgt
               by auto 
             have c1: " x > y \<Longrightarrow>  sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
               using nor changes_sign_var[of t y u v x] xgt ygt
               by fastforce 
             then have c2: " y > x \<Longrightarrow>  sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
               using nor changes_sign_var[of t x u v y] xgt ygt
               by force
             then have c3: " x = y \<Longrightarrow>  sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
               unfolding sign_num_def 
               by auto
             then show "sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
               using c1 c2 c3
               by linarith
           qed
           then show " sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
             using tuzer 
             by blast
         qed
 
         have "(\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0)" 
           using alleqsetvar by auto 
         have " ?bgrt \<in> set ?srl" 
           using set_sorted_list_of_set nonzero_root_set_finite in_set_member
           using asm by auto
         then have "?bgrt \<in> nonzero_root_set (set b \<union> set c \<union> set d )"
           unfolding sorted_nonzero_root_list_set_def
           using  set_sorted_list_of_set nonzero_root_set_finite 
           by auto
         then have "\<exists>t u v. (t, u, v) \<in> set b \<union> set c \<union> set d \<and>(t \<noteq> 0 \<or> u \<noteq> 0) \<and> (t * ?bgrt\<^sup>2 + u * ?bgrt + v = 0)"
           unfolding nonzero_root_set_def by auto
         then obtain t u v where tuvprop1: "(t, u, v) \<in> set b \<union> set c \<union> set d \<and>(t \<noteq> 0 \<or> u \<noteq> 0) \<and> (t * ?bgrt\<^sup>2 + u * ?bgrt + v = 0)"
           by auto
         then have tuvprop: "((t, u, v) \<in> set b \<and> (t \<noteq> 0 \<or> u \<noteq> 0) \<and> (t * ?bgrt\<^sup>2 + u * ?bgrt + v = 0))
           \<or> ((t, u, v) \<in> set c \<and> (t \<noteq> 0 \<or> u \<noteq> 0) \<and> (t * ?bgrt\<^sup>2 + u * ?bgrt + v = 0)) \<or>
             ((t, u, v) \<in> set d \<and> (t \<noteq> 0 \<or> u \<noteq> 0) \<and> (t * ?bgrt\<^sup>2 + u * ?bgrt + v = 0))  "
           by auto
         have tnonz: "t\<noteq> 0 \<Longrightarrow> (-1*u^2 + 4 * t * v \<le> 0  \<and> (?bgrt = (- u + 1 * sqrt (u^2 - 4 * t * v)) / (2 * t) \<or> ?bgrt = (- u + -1 * sqrt (u^2 - 4 * t * v)) / (2 * t)))"
         proof - 
           assume "t\<noteq> 0"
           have "-1*u^2 + 4 * t * v \<le> 0 "  using tuvprop1 discriminant_negative[of t u v]
             unfolding discrim_def
             using \<open>t \<noteq> 0\<close> by force              
           then show ?thesis
             using tuvprop discriminant_nonneg[of t u v]
             unfolding discrim_def
             using \<open>t \<noteq> 0\<close> by auto 
         qed
         have unonz: "(t = 0 \<and> u \<noteq> 0) \<Longrightarrow> ?bgrt = - v / u"
         proof - 
           assume "(t = 0 \<and> u \<noteq> 0)"
           then have "u*?bgrt + v = 0" using tuvprop1
             by simp 
           then show "?bgrt = - v / u"
             by (simp add: \<open>t = 0 \<and> u \<noteq> 0\<close> eq_minus_divide_eq mult.commute) 
         qed
 
         have allpropb: "(\<forall>(d, e, f)\<in>set b.
               \<forall>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0)" 
         proof clarsimp 
           fix t1 u1 v1 y1 x1
           assume ins: "(t1, u1, v1) \<in> set b"
           assume x1gt: " sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) !
        (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - Suc 0) < x1"
           assume "x1 \<le> y1"
           have xsn: "sign_num (t1 * x^2 + u1 * x + v1 ) = -1" using ins x_prop unfolding sign_num_def
             by auto
           have "sign_num (t1 * x1\<^sup>2 + u1 * x1 + v1 ) = sign_num (t1 * x^2 + u1 * x + v1 ) " 
             using ins x1gt samesign
             apply (auto) 
             by blast 
           then show "t1 * x1\<^sup>2 + u1 * x1 + v1 < 0" using xsn unfolding sign_num_def
             by (metis add.right_inverse add.right_neutral linorder_neqE_linordered_idom one_add_one zero_neq_numeral) 
         qed
         have allpropbvar: "(\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0)" 
         proof clarsimp 
           fix t1 u1 v1
           assume "(t1, u1, v1) \<in> set b"
           then have "\<forall>x\<in>{?bgrt<..(?bgrt + 1)}. t1 * x\<^sup>2 + u1 * x + v1 < 0"
             using allpropb
             by force 
           then show "\<exists>y'>sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) !
            (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - Suc 0).
           \<forall>x\<in>{sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) !
                (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - Suc 0)<..y'}.
              t1 * x\<^sup>2 + u1 * x + v1 < 0"
             using less_add_one
             by (metis One_nat_def)        
         qed
         have allpropc: "(\<forall>(d, e, f)\<in>set c.
               \<forall>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0)" 
         proof clarsimp 
           fix t1 u1 v1 y1 x1
           assume ins: "(t1, u1, v1) \<in> set c"
           assume x1gt: " sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) !
        (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - Suc 0) < x1"
           assume "x1 \<le> y1"
           have xsn: "sign_num (t1 * x^2 + u1 * x + v1 ) = -1 \<or> sign_num (t1 * x^2 + u1 * x + v1 ) = 0" using ins x_prop unfolding sign_num_def
             by auto
           have "sign_num (t1 * x1\<^sup>2 + u1 * x1 + v1 ) = sign_num (t1 * x^2 + u1 * x + v1 ) " 
             using ins x1gt samesign  One_nat_def
           proof -
             have "case (t1, u1, v1) of (r, ra, rb) \<Rightarrow> \<forall>raa>sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - 1). sign_num (r * raa\<^sup>2 + ra * raa + rb) = sign_num (r * x\<^sup>2 + ra * x + rb)"
               by (smt (z3) Un_iff ins samesign)
             then show ?thesis
               by (simp add: x1gt)
           qed
           then show "t1 * x1\<^sup>2 + u1 * x1 + v1 \<le> 0" using xsn unfolding sign_num_def
             by (metis equal_neg_zero less_numeral_extra(3) linorder_not_less zero_neq_one)
         qed
         have allpropcvar: "(\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0)" 
         proof clarsimp 
           fix t1 u1 v1
           assume "(t1, u1, v1) \<in> set c"
           then have "\<forall>x\<in>{?bgrt<..(?bgrt + 1)}. t1 * x\<^sup>2 + u1 * x + v1 \<le> 0"
             using allpropc
             by force 
           then show "\<exists>y'>sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) !
            (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - Suc 0).
           \<forall>x\<in>{sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) !
                (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - Suc 0)<..y'}.
              t1 * x\<^sup>2 + u1 * x + v1 \<le> 0"
             using less_add_one One_nat_def
-            by (metis (no_types, hide_lams))        
+            by metis
         qed
         have allpropd: "(\<forall>(d, e, f)\<in>set d.
               \<forall>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)" 
         proof clarsimp 
           fix t1 u1 v1 y1 x1
           assume ins: "(t1, u1, v1) \<in> set d"
           assume contrad:"t1 * x1\<^sup>2 + u1 * x1 + v1 = 0"
           assume x1gt: " sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) !
        (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - Suc 0) < x1"
           assume "x1 \<le> y1"
           have xsn: "sign_num (t1 * x^2 + u1 * x + v1 ) = -1 \<or> sign_num (t1 * x^2 + u1 * x + v1 ) = 1" using ins x_prop unfolding sign_num_def
             by auto
           have "sign_num (t1 * x1\<^sup>2 + u1 * x1 + v1 ) = sign_num (t1 * x^2 + u1 * x + v1 ) " 
             using ins x1gt samesign apply (auto) 
             by blast 
           then have "t1 * x1\<^sup>2 + u1 * x1 + v1 \<noteq> 0" using xsn unfolding sign_num_def 
             by auto
           then show "False" using contrad by auto
         qed
         have allpropdvar: "(\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)" 
         proof clarsimp 
           fix t1 u1 v1
           assume "(t1, u1, v1) \<in> set d"
           then have "\<forall>x\<in>{?bgrt<..(?bgrt + 1)}. t1 * x\<^sup>2 + u1 * x + v1 \<noteq> 0"
             using allpropd
             by force 
           then show "\<exists>y'>sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) !
            (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - Suc 0).
           \<forall>x\<in>{sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) !
                (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - Suc 0)<..y'}.
              t1 * x\<^sup>2 + u1 * x + v1 \<noteq> 0"
-            using less_add_one
-            by (metis (no_types, hide_lams) One_nat_def)     
+            using less_add_one by force
         qed
         have "\<forall>x. (\<forall>(d, e, f)\<in>set a.
              d * x\<^sup>2 + e * x + f = 0)" using alleqsetvar
           by auto
         then have ast: "(\<forall>(d, e, f)\<in>set a.
              \<forall>x\<in>{?bgrt<..(?bgrt + 1)}. d * x\<^sup>2 + e * x + f = 0)"
           by auto
         have allpropavar: "(\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0)"
         proof clarsimp 
           fix t1 u1 v1 
           assume "(t1, u1, v1) \<in> set a"
           then have "\<forall>x\<in>{?bgrt<..(?bgrt + 1)}. t1 * x\<^sup>2 + u1 * x + v1 = 0 "
             using ast by auto 
           then show "\<exists>y'>sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) !
            (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - Suc 0).
           \<forall>x\<in>{sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) !
                (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - Suc 0)<..y'}.
              t1 * x\<^sup>2 + u1 * x + v1 = 0"
             using less_add_one One_nat_def
             by metis
         qed
         have quadsetb: "((t, u, v) \<in> set b \<and> t\<noteq> 0) \<Longrightarrow> False"
         proof - 
           assume asm: "(t, u, v) \<in> set b \<and> t\<noteq> 0"
           have bgrt1: "(?bgrt = (- u + 1 * sqrt (u^2 - 4 * t * v)) / (2 * t)) \<Longrightarrow> False "
           proof - 
             assume bgrtis: "?bgrt = (- u + 1 * sqrt (u^2 - 4 * t * v)) / (2 * t)"
             have discrim_prop: "-1*u^2 + 4 * t * v \<le> 0" using asm tnonz
               using \<open>sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - 1) = (- u + 1 * sqrt (u\<^sup>2 - 4 * t * v)) / (2 * t)\<close> 
               by auto 
             have "((t, u, v)\<in>set b \<and> t \<noteq> 0 \<and> - 1*u^2 + 4 * t * v \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
               using asm discrim_prop allpropavar allpropbvar allpropcvar allpropdvar
               by linarith
             then show "False" using f6 bgrtis 
               by auto
           qed
           have bgrt2: "(?bgrt = (- u + -1 * sqrt (u^2 - 4 * t * v)) / (2 * t)) \<Longrightarrow> False "
           proof -
             assume bgrtis: "?bgrt = (- u + -1 * sqrt (u^2 - 4 * t * v)) / (2 * t)"
             have discrim_prop: "-1*u^2 + 4 * t * v \<le> 0" using asm tnonz
               using \<open>sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - 1) = (- u + -1 * sqrt (u\<^sup>2 - 4 * t * v)) / (2 * t)\<close> 
               by auto
             have "((t, u, v)\<in>set b \<and> t \<noteq> 0 \<and> - 1*u^2 + 4 * t * v \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
               using asm discrim_prop allpropavar allpropbvar allpropcvar allpropdvar
               by linarith
             then show "False" using f7 bgrtis 
               by auto
           qed 
           show "False" using tnonz bgrt1 bgrt2 asm 
             by auto
         qed
         have linsetb: "((t, u, v) \<in> set b \<and> (t = 0 \<and> u \<noteq> 0)) \<Longrightarrow> False"
         proof - 
           assume asm: "(t, u, v) \<in> set b \<and> (t = 0 \<and> u \<noteq> 0)"
           then have bgrtis: "?bgrt = (- v / u)"
             using unonz
             by blast 
           have "((t, u, v)\<in>set b \<and> (t = 0 \<and> u \<noteq> 0) \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
             using asm allpropavar allpropbvar allpropcvar allpropdvar
             by linarith
           then show "False" using bgrtis f5  
             by auto
         qed
         have insetb: "((t, u, v) \<in> set b \<and> (t \<noteq> 0 \<or> u \<noteq> 0) \<and> (t * ?bgrt\<^sup>2 + u * ?bgrt + v = 0)) \<Longrightarrow> False"
           using quadsetb linsetb by auto
         have quadsetc: "(t, u, v) \<in> set c \<and> t\<noteq> 0 \<Longrightarrow> False"
         proof - 
           assume asm: "(t, u, v) \<in> set c \<and> t\<noteq> 0"
           have bgrt1: "(?bgrt = (- u + 1 * sqrt (u^2 - 4 * t * v)) / (2 * t)) \<Longrightarrow> False "
           proof - 
             assume bgrtis: "?bgrt = (- u + 1 * sqrt (u^2 - 4 * t * v)) / (2 * t)"
             have discrim_prop: "-1*u^2 + 4 * t * v \<le> 0" using asm tnonz
               using \<open>sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - 1) = (- u + 1 * sqrt (u\<^sup>2 - 4 * t * v)) / (2 * t)\<close> 
               by auto 
             have "((t, u, v)\<in>set c \<and> t \<noteq> 0 \<and> - 1*u^2 + 4 * t * v \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
               using asm discrim_prop allpropavar allpropbvar allpropcvar allpropdvar
               by linarith
             then show "False" using f13a bgrtis 
               by auto
           qed
           have bgrt2: "(?bgrt = (- u + -1 * sqrt (u^2 - 4 * t * v)) / (2 * t)) \<Longrightarrow> False "
           proof -
             assume bgrtis: "?bgrt = (- u + -1 * sqrt (u^2 - 4 * t * v)) / (2 * t)"
             have discrim_prop: "-1*u^2 + 4 * t * v \<le> 0" using asm tnonz
               using \<open>sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - 1) = (- u + -1 * sqrt (u\<^sup>2 - 4 * t * v)) / (2 * t)\<close> 
               by auto
             have "((t, u, v)\<in>set c \<and> t \<noteq> 0 \<and> - 1*u^2 + 4 * t * v \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
               using asm discrim_prop allpropavar allpropbvar allpropcvar allpropdvar
               by linarith
             then show "False" using f9a bgrtis 
               by auto
           qed 
           show "False" using tnonz bgrt1 bgrt2 asm 
             by auto
         qed
         have linsetc: "(t, u, v) \<in> set c \<and> (t = 0 \<and> u \<noteq> 0) \<Longrightarrow> False"
         proof - 
           assume asm: "(t, u, v) \<in> set c \<and> (t = 0 \<and> u \<noteq> 0)"
           then have bgrtis: "?bgrt = (- v / u)"
             using unonz
             by blast 
           have "((t, u, v)\<in>set c \<and> (t = 0 \<and> u \<noteq> 0) \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
             using asm allpropavar allpropbvar allpropcvar allpropdvar
             by linarith
           then show "False" using bgrtis f8a  
             by auto
         qed
         have insetc: "((t, u, v) \<in> set c \<and> (t \<noteq> 0 \<or> u \<noteq> 0) \<and> (t * ?bgrt\<^sup>2 + u * ?bgrt + v = 0)) \<Longrightarrow> False"
           using quadsetc linsetc by auto
         have quadsetd: "(t, u, v) \<in> set d \<and> t\<noteq> 0 \<Longrightarrow> False"
         proof - 
           assume asm: "(t, u, v) \<in> set d \<and> t\<noteq> 0"
           have bgrt1: "(?bgrt = (- u + 1 * sqrt (u^2 - 4 * t * v)) / (2 * t)) \<Longrightarrow> False "
           proof - 
             assume bgrtis: "?bgrt = (- u + 1 * sqrt (u^2 - 4 * t * v)) / (2 * t)"
             have discrim_prop: "-1*u^2 + 4 * t * v \<le> 0" using asm tnonz
               using \<open>sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - 1) = (- u + 1 * sqrt (u\<^sup>2 - 4 * t * v)) / (2 * t)\<close> 
               by auto 
             have "((t, u, v)\<in>set d \<and> t \<noteq> 0 \<and> - 1*u^2 + 4 * t * v \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
               using asm discrim_prop allpropavar allpropbvar allpropcvar allpropdvar
               by linarith
             then show "False" using f11 bgrtis 
               by auto
           qed
           have bgrt2: "(?bgrt = (- u + -1 * sqrt (u^2 - 4 * t * v)) / (2 * t)) \<Longrightarrow> False "
           proof -
             assume bgrtis: "?bgrt = (- u + -1 * sqrt (u^2 - 4 * t * v)) / (2 * t)"
             have discrim_prop: "-1*u^2 + 4 * t * v \<le> 0" using asm tnonz
               using \<open>sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! (length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - 1) = (- u + -1 * sqrt (u\<^sup>2 - 4 * t * v)) / (2 * t)\<close> 
               by auto
             have "((t, u, v)\<in>set d \<and> t \<noteq> 0 \<and> - 1*u^2 + 4 * t * v \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
               using asm discrim_prop allpropavar allpropbvar allpropcvar allpropdvar
               by linarith
             then show "False" using f12 bgrtis 
               by auto
           qed 
           show "False" using tnonz bgrt1 bgrt2 asm 
             by auto
         qed
         have linsetd: "(t, u, v) \<in> set d \<and> (t = 0 \<and> u \<noteq> 0) \<Longrightarrow> False"
         proof - 
           assume asm: "(t, u, v) \<in> set d \<and> (t = 0 \<and> u \<noteq> 0)"
           then have bgrtis: "?bgrt = (- v / u)"
             using unonz
             by blast 
           have "((t, u, v)\<in>set d \<and> (t = 0 \<and> u \<noteq> 0) \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
             using asm allpropavar allpropbvar allpropcvar allpropdvar
             by linarith
           then show "False" using bgrtis f10
             by auto
         qed
         have insetd: "((t, u, v) \<in> set d \<and> (t \<noteq> 0 \<or> u \<noteq> 0) \<and> (t * ?bgrt\<^sup>2 + u * ?bgrt + v = 0)) \<Longrightarrow> False"
           using quadsetd linsetd by auto
         then show "False" using insetb insetc insetd tuvprop 
           by auto
       qed
       have len1: "length ?srl = 1 \<Longrightarrow> False" 
       proof - 
         assume len1: "length ?srl = 1"
         have cases: "(List.member ?srl x) \<or> x < ?srl ! 0 \<or> x > ?srl ! 0"
           using in_set_member lenzero nth_mem by fastforce
         then show "False"
           using len1 cases_mem cases_lt cases_gt by auto
       qed
       have lengtone: "length ?srl > 1 \<Longrightarrow> False" 
       proof - 
         assume lengt1: "length ?srl > 1" 
         have cases: "(List.member ?srl x) \<or> x < ?srl ! 0 \<or> x > ?srl ! (length ?srl -1)
                   \<or> (\<exists>k \<le> (length ?srl - 2). (?srl ! k < x \<and> x <?srl ! (k + 1)))"
         proof -
           have eo: "x < ?srl ! 0 \<or> x > ?srl ! (length ?srl -1) \<or> (x \<ge> ?srl ! 0 \<and> x \<le> ?srl ! (length ?srl -1))"
             by auto
           have ifo: "(x \<ge> ?srl ! 0 \<and> x \<le> ?srl ! (length ?srl -1)) \<Longrightarrow> ((List.member ?srl x) \<or> (\<exists>k \<le> (length ?srl - 2). ?srl ! k < x \<and> x <?srl ! (k + 1)))"
           proof - 
             assume xinbtw: "x \<ge> ?srl ! 0 \<and> x \<le> ?srl ! (length ?srl -1)"
             then have "\<not>(List.member ?srl x) \<Longrightarrow>  (\<exists>k \<le> (length ?srl - 2). ?srl ! k < x \<and> x <?srl ! (k + 1))"
             proof - 
               assume nonmem: "\<not>(List.member ?srl x)"
               have "\<not>(\<exists>k \<le> (length ?srl - 2). ?srl ! k < x \<and> x <?srl ! (k + 1)) \<Longrightarrow> False"
               proof clarsimp
                 assume "\<forall>k. sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! k < x \<longrightarrow>
         k \<le> length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - 2 \<longrightarrow>
         \<not> x < sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! Suc k"
                 then have allk: "(\<forall>k \<le> length ?srl - 2. ?srl ! k < x \<longrightarrow>
         \<not> x < ?srl ! Suc k)" by auto 
                 have basec: "x \<ge> ?srl ! 0" using xinbtw by auto
                 have "\<forall>k \<le> length ?srl - 2. ?srl ! k < x" 
                 proof clarsimp 
                   fix k
                   assume klteq: "k \<le> length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - 2"
                   show "sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! k < x"
                     using nonmem klteq basec 
                   proof (induct k)
                     case 0
                     then show ?case
                       using in_set_member lenzero nth_mem by fastforce 
                   next
                     case (Suc k)
                     then show ?case
                       by (smt Suc_leD Suc_le_lessD \<open>\<forall>k. sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! k < x \<longrightarrow> k \<le> length (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d)) - 2 \<longrightarrow> \<not> x < sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! Suc k\<close> diff_less in_set_member length_0_conv length_greater_0_conv lenzero less_trans_Suc nth_mem pos2) 
                   qed
                 qed
                 then have "x \<ge> ?srl ! (length ?srl -1)" 
                   using allk
                   by (metis One_nat_def Suc_diff_Suc lengt1 less_eq_real_def less_or_eq_imp_le one_add_one plus_1_eq_Suc xinbtw) 
                 then have "x > ?srl ! (length ?srl - 1)" using nonmem
                   by (metis One_nat_def Suc_le_D asm diff_Suc_Suc diff_zero in_set_member lessI less_eq_real_def nth_mem) 
                 then show "False" using xinbtw by auto
               qed
               then show "(\<exists>k \<le> (length ?srl - 2). ?srl ! k < x \<and> x <?srl ! (k + 1))"
                 by blast
             qed
             then show "((List.member ?srl x) \<or> (\<exists>k \<le> (length ?srl - 2). ?srl ! k < x \<and> x <?srl ! (k + 1)))" 
               using sorted_nth_mono
               by auto 
           qed
           then show ?thesis using eo ifo by auto
         qed
           (* should violate one of the infinitesmials *)
         have cases_btw: "(\<exists>k \<le> (length ?srl - 2). ?srl ! k < x \<and> x <?srl ! (k + 1)) \<Longrightarrow> False"
         proof - 
           assume "(\<exists>k \<le> (length ?srl - 2). ?srl ! k < x \<and> x <?srl ! (k + 1))"
           then obtain k where k_prop: "k \<le> (length ?srl - 2) \<and> ?srl ! k < x \<and> x <?srl ! (k + 1)"
             by auto
           have samesign: "\<forall> (a, b, c) \<in> (set b \<union> set c \<union> set d).
               (\<forall>y. (?srl ! k < y \<and> y <?srl ! (k + 1)) \<longrightarrow> sign_num (a * y\<^sup>2 + b * y + c) =  sign_num (a*x^2 + b*x + c))"
           proof  clarsimp  
             fix t u v y
             assume insetunion: "(t, u, v) \<in> set b \<or> (t, u, v)  \<in> set c \<or> (t, u, v) \<in> set d"
             assume ygt: " sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! k < y" 
             assume ylt: "y < sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! Suc k"
             have tuzer: "t = 0 \<and> u = 0 \<Longrightarrow> sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
               unfolding sign_num_def 
               by auto
             have tunonzer: "t \<noteq> 0 \<or> u \<noteq> 0  \<Longrightarrow> sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
             proof - 
               assume tuv_asm: "t\<noteq> 0 \<or> u \<noteq> 0"
               have nor: "\<not>(\<exists>q. q > ?srl ! k \<and> q < ?srl ! (k + 1) \<and> t * q\<^sup>2 + u * q + v = 0)"
               proof clarsimp 
                 fix q
                 assume qlt: "q < sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! Suc k"
                 assume qgt: "sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! k < q"
                 assume "t * q\<^sup>2 + u * q + v = 0"
                 then have qin: "q \<in>  {x. \<exists>(a, b, c)\<in>set b \<union> set c \<union> set d. (a \<noteq> 0 \<or> b \<noteq> 0) \<and> a * x\<^sup>2 + b * x + c = 0}"
                   using insetunion tuv_asm by auto
                 have "set ?srl = nonzero_root_set (set b \<union> set c \<union> set d)"
                   unfolding sorted_nonzero_root_list_set_def
                   using set_sorted_list_of_set[of "nonzero_root_set (set b \<union> set c \<union> set d)"]
                     nonzero_root_set_finite[of "(set b \<union> set c \<union> set d)"]
                   by auto
                 then have "q \<in> set ?srl" using qin unfolding nonzero_root_set_def
                   by auto 
                 then have "List.member ?srl q"     
                   using in_set_member[of q ?srl]
                   by auto
                 then have "\<exists>n < length ?srl. q = ?srl ! n"
                   by (metis \<open>q \<in> set (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d))\<close> in_set_conv_nth) 
                 then obtain n where nprop: "n < length ?srl \<and> q = ?srl ! n" by auto
                 then have ngtk: "n > k"
                 proof - 
                   have sortedh: "sorted ?srl"
                     by (simp add: sorted_nonzero_root_list_set_def) 
                   then have nlteq: "n \<le> k \<Longrightarrow> ?srl ! n \<le> ?srl ! k" using nprop k_prop sorted_iff_nth_mono
                     using sorted_nth_mono
-                    by (metis (no_types, hide_lams) Suc_1 \<open>q \<in> set (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d))\<close> diff_Suc_less length_pos_if_in_set sup.absorb_iff2 sup.strict_boundedE) 
+                    by (metis Suc_1 \<open>q \<in> set (sorted_nonzero_root_list_set (set b \<union> set c \<union> set d))\<close> diff_less length_pos_if_in_set sup.absorb_iff2 sup.strict_boundedE zero_less_Suc)
                   have "?srl ! n > ?srl ! k" using nprop qgt by auto
                   then show ?thesis
                     using nlteq
                     by linarith 
                 qed
                 then have nltkp1: "n < k+1"
                 proof - 
                   have sortedh: "sorted ?srl"
                     by (simp add: sorted_nonzero_root_list_set_def) 
                   then have ngteq: "k+1 \<le> n \<Longrightarrow> ?srl ! (k+1) \<le> ?srl ! n" using nprop k_prop sorted_iff_nth_mono
                     by auto
                   have "?srl ! n < ?srl ! (k + 1)" using nprop qlt by auto
                   then show ?thesis
                     using ngteq by linarith
                 qed
                 then show "False" using ngtk nltkp1 by auto
               qed
               have c1: " x > y \<Longrightarrow>  sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
                 using nor changes_sign_var[of t y u v x] k_prop ygt ylt
                 by fastforce 
               then have c2: " y > x \<Longrightarrow>  sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
                 using nor changes_sign_var[of t x u v y] k_prop ygt ylt
                 by force
               then have c3: " x = y \<Longrightarrow>  sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
                 unfolding sign_num_def 
                 by auto
               then show "sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
                 using c1 c2 c3
                 by linarith
             qed
             then show " sign_num (t * y\<^sup>2 + u * y + v) = sign_num (t * x\<^sup>2 + u * x + v)"
               using tuzer 
               by blast
           qed
 
           let ?bgrt = "?srl ! k"
 
           have "(\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0)" 
             using alleqsetvar by auto 
           have " ?bgrt \<in> set ?srl" 
             using set_sorted_list_of_set nonzero_root_set_finite in_set_member k_prop asm
             by (smt diff_Suc_less le_eq_less_or_eq less_le_trans nth_mem one_add_one plus_1_eq_Suc zero_less_one)
           then have "?bgrt \<in> nonzero_root_set (set b \<union> set c \<union> set d )"
             unfolding sorted_nonzero_root_list_set_def
             using  set_sorted_list_of_set nonzero_root_set_finite 
             by auto
           then have "\<exists>t u v. (t, u, v) \<in> set b \<union> set c \<union> set d \<and>(t \<noteq> 0 \<or> u \<noteq> 0) \<and> (t * ?bgrt\<^sup>2 + u * ?bgrt + v = 0)"
             unfolding nonzero_root_set_def by auto
           then obtain t u v where tuvprop1: "(t, u, v) \<in> set b \<union> set c \<union> set d \<and>(t \<noteq> 0 \<or> u \<noteq> 0) \<and> (t * ?bgrt\<^sup>2 + u * ?bgrt + v = 0)"
             by auto
           then have tuvprop: "((t, u, v) \<in> set b \<and> (t \<noteq> 0 \<or> u \<noteq> 0) \<and> (t * ?bgrt\<^sup>2 + u * ?bgrt + v = 0))
           \<or> ((t, u, v) \<in> set c \<and> (t \<noteq> 0 \<or> u \<noteq> 0) \<and> (t * ?bgrt\<^sup>2 + u * ?bgrt + v = 0)) \<or>
             ((t, u, v) \<in> set d \<and> (t \<noteq> 0 \<or> u \<noteq> 0) \<and> (t * ?bgrt\<^sup>2 + u * ?bgrt + v = 0))  "
             by auto
           have tnonz: "t\<noteq> 0 \<Longrightarrow> (-1*u^2 + 4 * t * v \<le> 0  \<and> (?bgrt = (- u + 1 * sqrt (u^2 - 4 * t * v)) / (2 * t) \<or> ?bgrt = (- u + -1 * sqrt (u^2 - 4 * t * v)) / (2 * t)))"
           proof - 
             assume "t\<noteq> 0"
             have "-1*u^2 + 4 * t * v \<le> 0 "  using tuvprop1 discriminant_negative[of t u v]
               unfolding discrim_def
               using \<open>t \<noteq> 0\<close> by force              
             then show ?thesis
               using tuvprop discriminant_nonneg[of t u v]
               unfolding discrim_def
               using \<open>t \<noteq> 0\<close> by auto 
           qed
           have unonz: "(t = 0 \<and> u \<noteq> 0) \<Longrightarrow> ?bgrt = - v / u"
           proof - 
             assume "(t = 0 \<and> u \<noteq> 0)"
             then have "u*?bgrt + v = 0" using tuvprop1
               by simp 
             then show "?bgrt = - v / u"
               by (simp add: \<open>t = 0 \<and> u \<noteq> 0\<close> eq_minus_divide_eq mult.commute) 
           qed
 
           have "\<exists>y'. y' > x \<and> y' < ?srl ! (k+1)" using k_prop dense
             by blast 
           then obtain y1 where y1_prop: "y1 > x \<and> y1 < ?srl ! (k+1)" by auto
           then have y1inbtw: "y1 > ?srl ! k \<and> y1 < ?srl ! (k+1)" using k_prop
             by auto
 
           have allpropb: "(\<forall>(d, e, f)\<in>set b.
              \<forall>x\<in>{?bgrt<..y1}. d * x\<^sup>2 + e * x + f < 0)" 
           proof clarsimp 
             fix t1 u1 v1 x1
             assume ins: "(t1, u1, v1) \<in> set b"
             assume x1gt: "sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! k < x1"
             assume x1lt: "x1 \<le> y1"
             have x1inbtw: "x1 > ?srl ! k \<and> x1 < ?srl ! (k+1)"
               using x1gt x1lt y1inbtw
               by (smt One_nat_def cases_gt k_prop) 
             have xsn: "sign_num (t1 * x^2 + u1 * x + v1 ) = -1" using ins x_prop unfolding sign_num_def
               by auto
             have "sign_num (t1 * x1\<^sup>2 + u1 * x1 + v1 ) = sign_num (t1 * x^2 + u1 * x + v1 ) " 
               using ins x1inbtw samesign
               by blast 
             then show "t1 * x1\<^sup>2 + u1 * x1 + v1 < 0" using xsn unfolding sign_num_def 
               by (metis add.right_inverse add.right_neutral linorder_neqE_linordered_idom one_add_one zero_neq_numeral) 
           qed
           have allpropbvar: "(\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0)" 
           proof clarsimp 
             fix t1 u1 v1
             assume "(t1, u1, v1) \<in> set b"
             then have "\<forall>x\<in>{?bgrt<..y1}. t1 * x\<^sup>2 + u1 * x + v1 < 0"
               using allpropb
               by force 
             then show " \<exists>y'>sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! k.
           \<forall>x\<in>{sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! k<..y'}.
              t1 * x\<^sup>2 + u1 * x + v1 < 0" 
               using y1inbtw by blast 
           qed
           have allpropc: "(\<forall>(d, e, f)\<in>set c.
                \<forall>x\<in>{?bgrt<..y1}. d * x\<^sup>2 + e * x + f \<le> 0)" 
           proof clarsimp 
             fix t1 u1 v1 x1
             assume ins: "(t1, u1, v1) \<in> set c"
             assume x1gt: " sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! k < x1"
             assume x1lt: "x1 \<le> y1"
             have x1inbtw: "x1 > ?srl ! k \<and> x1 < ?srl ! (k+1)"
               using x1gt x1lt y1inbtw
               by (smt One_nat_def cases_gt k_prop) 
             have xsn: "sign_num (t1 * x^2 + u1 * x + v1 ) = -1 \<or> sign_num (t1 * x^2 + u1 * x + v1 ) = 0" using ins x_prop unfolding sign_num_def
               by auto
             have "sign_num (t1 * x1\<^sup>2 + u1 * x1 + v1 ) = sign_num (t1 * x^2 + u1 * x + v1 ) " 
               using ins x1inbtw samesign
               by blast 
             then show "t1 * x1\<^sup>2 + u1 * x1 + v1 \<le> 0" using xsn unfolding sign_num_def
-              by (metis (no_types, hide_lams) equal_neg_zero less_eq_real_def linorder_not_less zero_neq_one) 
+              by (smt (verit, del_insts))
           qed
           have allpropcvar: "(\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0)" 
           proof clarsimp 
             fix t1 u1 v1
             assume "(t1, u1, v1) \<in> set c"
             then have "\<forall>x\<in>{?bgrt<..y1}. t1 * x\<^sup>2 + u1 * x + v1 \<le> 0"
               using allpropc 
               by force 
             then show " \<exists>y'>sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! k.
           \<forall>x\<in>{sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! k<..y'}.
              t1 * x\<^sup>2 + u1 * x + v1 \<le> 0" 
               using y1inbtw by blast 
           qed
           have allpropd: "(\<forall>(d, e, f)\<in>set d.
               \<forall>x\<in>{?bgrt<..y1}. d * x\<^sup>2 + e * x + f \<noteq> 0)" 
           proof clarsimp 
             fix t1 u1 v1 x1
             assume ins: "(t1, u1, v1) \<in> set d"
             assume contrad:"t1 * x1\<^sup>2 + u1 * x1 + v1 = 0"
             assume x1gt: " sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! k < x1"
             assume x1lt: "x1 \<le> y1"
             have x1inbtw: "x1 > ?srl ! k \<and> x1 < ?srl ! (k+1)"
               using x1gt x1lt y1inbtw
               by (smt One_nat_def cases_gt k_prop) 
             have xsn: "sign_num (t1 * x^2 + u1 * x + v1 ) = -1 \<or> sign_num (t1 * x^2 + u1 * x + v1 ) = 1" using ins x_prop unfolding sign_num_def
               by auto
             have "sign_num (t1 * x1\<^sup>2 + u1 * x1 + v1 ) = sign_num (t1 * x^2 + u1 * x + v1 ) " 
               using ins x1inbtw samesign
               by blast
             then have "t1 * x1\<^sup>2 + u1 * x1 + v1 \<noteq> 0" using xsn unfolding sign_num_def 
               by auto
             then show "False" using contrad by auto
           qed
           have allpropdvar: "(\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)" 
           proof clarsimp 
             fix t1 u1 v1
             assume "(t1, u1, v1) \<in> set d"
             then have "\<forall>x\<in>{?bgrt<..y1}. t1 * x\<^sup>2 + u1 * x + v1 \<noteq> 0"
               using allpropd
               by force 
             then show " \<exists>y'>sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! k.
           \<forall>x\<in>{sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! k<..y'}.
              t1 * x\<^sup>2 + u1 * x + v1 \<noteq> 0" 
               using y1inbtw by blast 
           qed
           have "\<forall>x. (\<forall>(d, e, f)\<in>set a.
              d * x\<^sup>2 + e * x + f = 0)" using alleqsetvar
             by auto
           then have ast: "(\<forall>(d, e, f)\<in>set a.
              \<forall>x\<in>{?bgrt<..(?bgrt + 1)}. d * x\<^sup>2 + e * x + f = 0)"
             by auto
           have allpropavar: "(\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0)"
           proof clarsimp 
             fix t1 u1 v1 
             assume "(t1, u1, v1) \<in> set a"
             then have "\<forall>x\<in>{?bgrt<..(?bgrt + 1)}. t1 * x\<^sup>2 + u1 * x + v1 = 0 "
               using ast by auto 
             then show "\<exists>y'>sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! k.
           \<forall>x\<in>{sorted_nonzero_root_list_set (set b \<union> set c \<union> set d) ! k<..y'}.
              t1 * x\<^sup>2 + u1 * x + v1 = 0" 
               using less_add_one by blast 
           qed
 
           have quadsetb: "((t, u, v) \<in> set b \<and> t\<noteq> 0) \<Longrightarrow> False"
           proof - 
             assume asm: "(t, u, v) \<in> set b \<and> t\<noteq> 0"
             have bgrt1: "(?bgrt = (- u + 1 * sqrt (u^2 - 4 * t * v)) / (2 * t)) \<Longrightarrow> False "
             proof - 
               assume bgrtis: "?bgrt = (- u + 1 * sqrt (u^2 - 4 * t * v)) / (2 * t)"
               have discrim_prop: "-1*u^2 + 4 * t * v \<le> 0" using asm tnonz
                 by blast           
               have "((t, u, v)\<in>set b \<and> t \<noteq> 0 \<and> - 1*u^2 + 4 * t * v \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
                 using asm discrim_prop allpropavar allpropbvar allpropcvar allpropdvar
                 by linarith
               then show "False" using f6 bgrtis 
                 by auto
             qed
             have bgrt2: "(?bgrt = (- u + -1 * sqrt (u^2 - 4 * t * v)) / (2 * t)) \<Longrightarrow> False "
             proof -
               assume bgrtis: "?bgrt = (- u + -1 * sqrt (u^2 - 4 * t * v)) / (2 * t)"
               have discrim_prop: "-1*u^2 + 4 * t * v \<le> 0" using asm tnonz
                 by blast
               have "((t, u, v)\<in>set b \<and> t \<noteq> 0 \<and> - 1*u^2 + 4 * t * v \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
                 using asm discrim_prop allpropavar allpropbvar allpropcvar allpropdvar
                 by linarith
               then show "False" using f7 bgrtis 
                 by auto
             qed 
             show "False" using tnonz bgrt1 bgrt2 asm 
               by auto
           qed
           have linsetb: "((t, u, v) \<in> set b \<and> (t = 0 \<and> u \<noteq> 0)) \<Longrightarrow> False"
           proof - 
             assume asm: "(t, u, v) \<in> set b \<and> (t = 0 \<and> u \<noteq> 0)"
             then have bgrtis: "?bgrt = (- v / u)"
               using unonz
               by blast 
             have "((t, u, v)\<in>set b \<and> (t = 0 \<and> u \<noteq> 0) \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
               using asm allpropavar allpropbvar allpropcvar allpropdvar
               by linarith
             then show "False" using bgrtis f5  
               by auto
           qed
           have insetb: "((t, u, v) \<in> set b \<and> (t \<noteq> 0 \<or> u \<noteq> 0) \<and> (t * ?bgrt\<^sup>2 + u * ?bgrt + v = 0)) \<Longrightarrow> False"
             using quadsetb linsetb by auto
           have quadsetc: "(t, u, v) \<in> set c \<and> t\<noteq> 0 \<Longrightarrow> False"
           proof - 
             assume asm: "(t, u, v) \<in> set c \<and> t\<noteq> 0"
             have bgrt1: "(?bgrt = (- u + 1 * sqrt (u^2 - 4 * t * v)) / (2 * t)) \<Longrightarrow> False "
             proof - 
               assume bgrtis: "?bgrt = (- u + 1 * sqrt (u^2 - 4 * t * v)) / (2 * t)"
               have discrim_prop: "-1*u^2 + 4 * t * v \<le> 0" using asm tnonz
                 by blast 
               have "((t, u, v)\<in>set c \<and> t \<noteq> 0 \<and> - 1*u^2 + 4 * t * v \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
                 using asm discrim_prop allpropavar allpropbvar allpropcvar allpropdvar
                 by linarith
               then show "False" using f13a bgrtis 
                 by auto
             qed
             have bgrt2: "(?bgrt = (- u + -1 * sqrt (u^2 - 4 * t * v)) / (2 * t)) \<Longrightarrow> False "
             proof -
               assume bgrtis: "?bgrt = (- u + -1 * sqrt (u^2 - 4 * t * v)) / (2 * t)"
               have discrim_prop: "-1*u^2 + 4 * t * v \<le> 0" using asm tnonz
                 by blast
               have "((t, u, v)\<in>set c \<and> t \<noteq> 0 \<and> - 1*u^2 + 4 * t * v \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
                 using asm discrim_prop allpropavar allpropbvar allpropcvar allpropdvar
                 by linarith
               then show "False" using f9a bgrtis 
                 by auto
             qed 
             show "False" using tnonz bgrt1 bgrt2 asm 
               by auto
           qed
           have linsetc: "(t, u, v) \<in> set c \<and> (t = 0 \<and> u \<noteq> 0) \<Longrightarrow> False"
           proof - 
             assume asm: "(t, u, v) \<in> set c \<and> (t = 0 \<and> u \<noteq> 0)"
             then have bgrtis: "?bgrt = (- v / u)"
               using unonz
               by blast 
             have "((t, u, v)\<in>set c \<and> (t = 0 \<and> u \<noteq> 0) \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
               using asm allpropavar allpropbvar allpropcvar allpropdvar
               by linarith
             then show "False" using bgrtis f8a  
               by auto
           qed
           have insetc: "((t, u, v) \<in> set c \<and> (t \<noteq> 0 \<or> u \<noteq> 0) \<and> (t * ?bgrt\<^sup>2 + u * ?bgrt + v = 0)) \<Longrightarrow> False"
             using quadsetc linsetc by auto
           have quadsetd: "(t, u, v) \<in> set d \<and> t\<noteq> 0 \<Longrightarrow> False"
           proof - 
             assume asm: "(t, u, v) \<in> set d \<and> t\<noteq> 0"
             have bgrt1: "(?bgrt = (- u + 1 * sqrt (u^2 - 4 * t * v)) / (2 * t)) \<Longrightarrow> False "
             proof - 
               assume bgrtis: "?bgrt = (- u + 1 * sqrt (u^2 - 4 * t * v)) / (2 * t)"
               have discrim_prop: "-1*u^2 + 4 * t * v \<le> 0" using asm tnonz
                 by blast
               have "((t, u, v)\<in>set d \<and> t \<noteq> 0 \<and> - 1*u^2 + 4 * t * v \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
                 using asm discrim_prop allpropavar allpropbvar allpropcvar allpropdvar
                 by linarith
               then show "False" using f11 bgrtis 
                 by auto
             qed
             have bgrt2: "(?bgrt = (- u + -1 * sqrt (u^2 - 4 * t * v)) / (2 * t)) \<Longrightarrow> False "
             proof -
               assume bgrtis: "?bgrt = (- u + -1 * sqrt (u^2 - 4 * t * v)) / (2 * t)"
               have discrim_prop: "-1*u^2 + 4 * t * v \<le> 0" using asm tnonz
                 by blast
               have "((t, u, v)\<in>set d \<and> t \<noteq> 0 \<and> - 1*u^2 + 4 * t * v \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
                 using asm discrim_prop allpropavar allpropbvar allpropcvar allpropdvar
                 by linarith
               then show "False" using f12 bgrtis 
                 by auto
             qed 
             show "False" using tnonz bgrt1 bgrt2 asm 
               by auto
           qed
           have linsetd: "(t, u, v) \<in> set d \<and> (t = 0 \<and> u \<noteq> 0) \<Longrightarrow> False"
           proof - 
             assume asm: "(t, u, v) \<in> set d \<and> (t = 0 \<and> u \<noteq> 0)"
             then have bgrtis: "?bgrt = (- v / u)"
               using unonz
               by blast 
             have "((t, u, v)\<in>set d \<and> (t = 0 \<and> u \<noteq> 0) \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>?bgrt. \<forall>x\<in>{?bgrt<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)))"
               using asm allpropavar allpropbvar allpropcvar allpropdvar
               by linarith
             then show "False" using bgrtis f10
               by auto
           qed
           have insetd: "((t, u, v) \<in> set d \<and> (t \<noteq> 0 \<or> u \<noteq> 0) \<and> (t * ?bgrt\<^sup>2 + u * ?bgrt + v = 0)) \<Longrightarrow> False"
             using quadsetd linsetd by auto
           then show "False" using insetb insetc insetd tuvprop 
             by auto
         qed
         show "False" using cases cases_btw cases_mem cases_lt cases_gt 
           by auto
       qed
       show "False" using asm len1 lengtone
         by linarith 
     qed
     show "False" using lenzero lengt0
       by linarith 
   qed
   then show ?thesis
     by blast  
 qed
 
 
 lemma qe_forwards: 
   assumes "(\<exists>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
          (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
          (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))"
   shows "((\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0) \<and>
      (\<forall>(a, b, c)\<in>set b. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<and>
      (\<forall>(a, b, c)\<in>set c. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) \<and>
      (\<forall>(a, b, c)\<in>set d. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0) \<or>
      (\<exists>(a', b', c')\<in>set a.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq>
               0) \<or>
           (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq>
               0))) \<or>
      (\<exists>(a', b', c')\<in>set b.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0) \<or>
           (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))) \<or>
      (\<exists>(a', b', c')\<in>set c.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq>
               0) \<or>
           (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq>
               0))) \<or>
      (\<exists>(a', b', c')\<in>set d.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0) \<or>
           (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))))" 
     (* using eq_qe_1 les_qe_1 *)
 proof -
   let ?e2 = "(((\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0) \<and>
      (\<forall>(a, b, c)\<in>set b. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<and>
      (\<forall>(a, b, c)\<in>set c. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) \<and>
      (\<forall>(a, b, c)\<in>set d. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0) 
      \<or>
      (\<exists>(a', b', c')\<in>set a.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0) 
         \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0)         
           \<or>
           (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0))) 
     \<or>
      (\<exists>(a', b', c')\<in>set b.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0) 
         \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0) 
       \<or>
           (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))) 
     \<or>
      (\<exists>(a', b', c')\<in>set c.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0) 
       \<or>
           (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0))) 
       \<or>
      (\<exists>(a', b', c')\<in>set d.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0) 
         \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0) 
       \<or>
           (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))))"
   let ?f10orf11orf12 = "(\<exists>(a', b', c')\<in>set d.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0) 
         \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0) 
       \<or>
           (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))"
   let ?f8orf9 = "(\<exists>(a', b', c')\<in>set c.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0) 
       \<or>
           (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0)))"
   let ?f5orf6orf7 = "(\<exists>(a', b', c')\<in>set b.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0) 
         \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0) 
       \<or>
           (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))"
   let ?f2orf3orf4 = "(\<exists>(a', b', c')\<in>set a.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0) 
         \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0)         
           \<or>
           (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0)))"
   let ?e1 = "(\<exists>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
          (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
          (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))"
   let ?f1 = "((\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0) \<and>
      (\<forall>(a, b, c)\<in>set b. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<and>
      (\<forall>(a, b, c)\<in>set c. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) \<and>
      (\<forall>(a, b, c)\<in>set d. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0))"
   let ?f2 = "(\<exists>(a', b', c')\<in>set a.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0))"
   let ?f3 = "(\<exists>(a', b', c')\<in>set a. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0))"
   let ?f4 = "(\<exists>(a', b', c')\<in>set a. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
  (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0)) "
   let ?f5 = "(\<exists>(a', b', c')\<in>set b.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
   let ?f6 = "(\<exists>(a', b', c')\<in>set b. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))"
   let ?f7 = "(\<exists>(a', b', c')\<in>set b. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))"
   let ?f8 = "(\<exists>(a', b', c')\<in>set c. (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0))"
   let ?f13 = "(\<exists>(a', b', c')\<in>set c.
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0)))"
   let ?f9 = "(\<exists>(a', b', c')\<in>set c.  a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
         (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0))"
   let ?f10 = "(\<exists>(a', b', c')\<in>set d.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
   let ?f11 = "(\<exists>(a', b', c')\<in>set d.
           a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))"
   let ?f12 = "(\<exists>(a', b', c')\<in>set d.
           a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))"
   have h1a: "(?f1 \<or> ?f2orf3orf4 \<or> ?f5orf6orf7 \<or> ?f8orf9 \<or> ?f10orf11orf12) \<longrightarrow> ?e2"
     by auto
   have h2: "(?f2 \<or> ?f3 \<or> ?f4) \<longrightarrow> ?f2orf3orf4" by auto
   then have h1b: "(?f1 \<or> ?f2 \<or> ?f3 \<or> ?f4 \<or> ?f5orf6orf7 \<or> ?f8orf9 \<or> ?f10orf11orf12) \<longrightarrow> ?e2"
     using h1a by auto
   have h3: "(?f5 \<or> ?f6 \<or> ?f7) \<longrightarrow> ?f5orf6orf7" by auto
   then have h1c: "(?f1 \<or> ?f2 \<or> ?f3 \<or> ?f4 \<or> ?f5 \<or> ?f6 \<or> ?f7 \<or> ?f8orf9 \<or> ?f10orf11orf12) \<longrightarrow> ?e2"
     using h1b by smt 
   have h4: "(?f8 \<or> ?f9 \<or> ?f13) \<longrightarrow> ?f8orf9" by auto
   then have h1d: "(?f1 \<or> ?f2 \<or> ?f3 \<or> ?f4 \<or> ?f5 \<or> ?f6 \<or> ?f7 \<or> ?f8 \<or> ?f9 \<or> ?f13 \<or> ?f10orf11orf12) \<longrightarrow> ?e2"
     using h1c
     by smt 
   have h5: "(?f10 \<or> ?f11 \<or> ?f12) \<longrightarrow> ?f10orf11orf12" 
     by auto
   then have bigor: "(?f1 \<or> ?f2 \<or> ?f3 \<or> ?f4 \<or> ?f5 \<or> ?f6 \<or> ?f7 \<or> ?f8 \<or> ?f13 \<or> ?f9 \<or> ?f10 \<or> ?f11 \<or> ?f12)
     \<longrightarrow> ?e2 "
     using h1d  by smt 
   then have bigor_var: "\<not>?e2 \<longrightarrow> \<not>(?f1 \<or> ?f2 \<or> ?f3 \<or> ?f4 \<or> ?f5 \<or> ?f6 \<or> ?f7 \<or> ?f8 \<or> ?f13 \<or> ?f9 \<or> ?f10 \<or> ?f11 \<or> ?f12)
    " using contrapos_nn
     by smt 
   have not_eq: "\<not>(?f1 \<or> ?f2 \<or> ?f3 \<or> ?f4 \<or> ?f5 \<or> ?f6 \<or> ?f7 \<or> ?f8 \<or> ?f13 \<or> ?f9 \<or> ?f10 \<or> ?f11 \<or> ?f12) 
 =(\<not>?f1 \<and> \<not>?f2  \<and> \<not>?f3  \<and> \<not>?f4  \<and> \<not>?f5 \<and> \<not>?f6 \<and> \<not>?f7 \<and> \<not>?f8 \<and> \<not>?f13 \<and> \<not>?f9 \<and> \<not>?f10 \<and> \<not>?f11 \<and> \<not>?f12) "
     by linarith
   obtain x where x_prop: "(\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
          (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
          (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0)" using assms by auto
   have "(\<not>?f1 \<and> \<not>?f2  \<and> \<not>?f3  \<and> \<not>?f4  \<and> \<not>?f5 \<and> \<not>?f6 \<and> \<not>?f7 \<and> \<not>?f8 \<and> \<not>?f13 \<and> \<not>?f9 \<and> \<not>?f10 \<and> \<not>?f11 \<and> \<not>?f12) \<Longrightarrow> False"
   proof - 
     assume big_not: " \<not> ((\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0) \<and>
         (\<forall>(a, b, c)\<in>set b. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<and>
         (\<forall>(a, b, c)\<in>set c. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) \<and>
         (\<forall>(a, b, c)\<in>set d. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0)) \<and>
     \<not> (\<exists>(a', b', c')\<in>set a.
            (a' = 0 \<and> b' \<noteq> 0) \<and>
            (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
            (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
            (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
            (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0)) \<and>
     \<not> (\<exists>(a', b', c')\<in>set a.
            a' \<noteq> 0 \<and>
            - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
            (\<forall>(d, e, f)\<in>set a.
                d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f = 0) \<and>
            (\<forall>(d, e, f)\<in>set b.
                d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f < 0) \<and>
            (\<forall>(d, e, f)\<in>set c.
                d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f \<le> 0) \<and>
            (\<forall>(d, e, f)\<in>set d.
                d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f \<noteq> 0)) \<and>
     \<not> (\<exists>(a', b', c')\<in>set a.
            a' \<noteq> 0 \<and>
            - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
            (\<forall>(d, e, f)\<in>set a.
                d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f = 0) \<and>
            (\<forall>(d, e, f)\<in>set b.
                d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f < 0) \<and>
            (\<forall>(d, e, f)\<in>set c.
                d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f \<le> 0) \<and>
            (\<forall>(d, e, f)\<in>set d.
                d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f \<noteq> 0)) \<and>
     \<not> (\<exists>(a', b', c')\<in>set b.
            (a' = 0 \<and> b' \<noteq> 0) \<and>
            (\<forall>(d, e, f)\<in>set a. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
            (\<forall>(d, e, f)\<in>set b. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
            (\<forall>(d, e, f)\<in>set c. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
            (\<forall>(d, e, f)\<in>set d. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)) \<and>
     \<not> (\<exists>(a', b', c')\<in>set b.
            a' \<noteq> 0 \<and>
            - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
            (\<forall>(d, e, f)\<in>set a.
                \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                      d * x\<^sup>2 + e * x + f = 0) \<and>
            (\<forall>(d, e, f)\<in>set b.
                \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                      d * x\<^sup>2 + e * x + f < 0) \<and>
            (\<forall>(d, e, f)\<in>set c.
                \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                      d * x\<^sup>2 + e * x + f \<le> 0) \<and>
            (\<forall>(d, e, f)\<in>set d.
                \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                      d * x\<^sup>2 + e * x + f \<noteq> 0)) \<and>
     \<not> (\<exists>(a', b', c')\<in>set b.
            a' \<noteq> 0 \<and>
            - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
            (\<forall>(d, e, f)\<in>set a.
                \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                      d * x\<^sup>2 + e * x + f = 0) \<and>
            (\<forall>(d, e, f)\<in>set b.
                \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                      d * x\<^sup>2 + e * x + f < 0) \<and>
            (\<forall>(d, e, f)\<in>set c.
                \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                      d * x\<^sup>2 + e * x + f \<le> 0) \<and>
            (\<forall>(d, e, f)\<in>set d.
                \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                      d * x\<^sup>2 + e * x + f \<noteq> 0)) \<and>
     \<not> (\<exists>(a', b', c')\<in>set c.
            (a' = 0 \<and> b' \<noteq> 0) \<and>
            (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
            (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
            (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
            (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0)) \<and>
     \<not> (\<exists>(a', b', c')\<in>set c.
            a' \<noteq> 0 \<and>
            - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
            (\<forall>(d, e, f)\<in>set a.
                d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f =
                0) \<and>
            (\<forall>(d, e, f)\<in>set b.
                d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f
                < 0) \<and>
            (\<forall>(d, e, f)\<in>set c.
                d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f
                \<le> 0) \<and>
            (\<forall>(d, e, f)\<in>set d.
                d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f \<noteq>
                0)) \<and>
     \<not> (\<exists>(a', b', c')\<in>set c.
            a' \<noteq> 0 \<and>
            - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
            (\<forall>(d, e, f)\<in>set a.
                d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f =
                0) \<and>
            (\<forall>(d, e, f)\<in>set b.
                d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f
                < 0) \<and>
            (\<forall>(d, e, f)\<in>set c.
                d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f
                \<le> 0) \<and>
            (\<forall>(d, e, f)\<in>set d.
                d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f \<noteq>
                0)) \<and>
     \<not> (\<exists>(a', b', c')\<in>set d.
            (a' = 0 \<and> b' \<noteq> 0) \<and>
            (\<forall>(d, e, f)\<in>set a. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
            (\<forall>(d, e, f)\<in>set b. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
            (\<forall>(d, e, f)\<in>set c. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
            (\<forall>(d, e, f)\<in>set d. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)) \<and>
     \<not> (\<exists>(a', b', c')\<in>set d.
            a' \<noteq> 0 \<and>
            - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
            (\<forall>(d, e, f)\<in>set a.
                \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                      d * x\<^sup>2 + e * x + f = 0) \<and>
            (\<forall>(d, e, f)\<in>set b.
                \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                      d * x\<^sup>2 + e * x + f < 0) \<and>
            (\<forall>(d, e, f)\<in>set c.
                \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                      d * x\<^sup>2 + e * x + f \<le> 0) \<and>
            (\<forall>(d, e, f)\<in>set d.
                \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                      d * x\<^sup>2 + e * x + f \<noteq> 0)) \<and>
     \<not> (\<exists>(a', b', c')\<in>set d.
            a' \<noteq> 0 \<and>
            - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
            (\<forall>(d, e, f)\<in>set a.
                \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                      d * x\<^sup>2 + e * x + f = 0) \<and>
            (\<forall>(d, e, f)\<in>set b.
                \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                      d * x\<^sup>2 + e * x + f < 0) \<and>
            (\<forall>(d, e, f)\<in>set c.
                \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                      d * x\<^sup>2 + e * x + f \<le> 0) \<and>
            (\<forall>(d, e, f)\<in>set d.
                \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                   \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                      d * x\<^sup>2 + e * x + f \<noteq> 0))"
     have c1: "(\<exists> (d, e, f) \<in> set a. d \<noteq> 0 \<and> - e\<^sup>2 + 4 * d * f \<le> 0) \<Longrightarrow> False"
     proof - 
       assume "(\<exists> (d, e, f) \<in> set a. d \<noteq> 0 \<and> - e\<^sup>2 + 4 * d * f \<le> 0)"
       then obtain a' b' c' where abc_prop:  "(a', b', c') \<in> set a \<and> a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0"
         by auto
       then have "a'*x^2 + b'*x + c' = 0" using x_prop by auto
       then have xis: "x = (- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a') \<or> x = (- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a') " 
         using abc_prop discriminant_nonneg[of a' b' c'] unfolding discrim_def
         by auto 
       then have "((\<forall>(d, e, f)\<in>set a.
                d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f = 0) \<and>
            (\<forall>(d, e, f)\<in>set b.
                d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f < 0) \<and>
            (\<forall>(d, e, f)\<in>set c.
                d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f \<le> 0) \<and>
            (\<forall>(d, e, f)\<in>set d.
                d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f \<noteq> 0)) \<or>
         ((\<forall>(d, e, f)\<in>set a.
                d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f = 0) \<and>
            (\<forall>(d, e, f)\<in>set b.
                d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f < 0) \<and>
            (\<forall>(d, e, f)\<in>set c.
                d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f \<le> 0) \<and>
            (\<forall>(d, e, f)\<in>set d.
                d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f \<noteq> 0))"
         using x_prop by auto
       then have "(\<exists>(a', b', c')\<in>set a.
            a' \<noteq> 0 \<and>
            - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
            (\<forall>(d, e, f)\<in>set a.
                d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f = 0) \<and>
            (\<forall>(d, e, f)\<in>set b.
                d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f < 0) \<and>
            (\<forall>(d, e, f)\<in>set c.
                d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f \<le> 0) \<and>
            (\<forall>(d, e, f)\<in>set d.
                d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f \<noteq> 0)) \<or>
         (\<exists>(a', b', c')\<in>set a.
            a' \<noteq> 0 \<and>
            - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
            (\<forall>(d, e, f)\<in>set a.
                d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f = 0) \<and>
            (\<forall>(d, e, f)\<in>set b.
                d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f < 0) \<and>
            (\<forall>(d, e, f)\<in>set c.
                d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f \<le> 0) \<and>
            (\<forall>(d, e, f)\<in>set d.
                d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                f \<noteq> 0))" using abc_prop xis by auto
       then show "False"
         using big_not by auto
     qed
     have c2: "(\<exists> (d, e, f) \<in> set a. d = 0 \<and> e \<noteq> 0) \<Longrightarrow> False"
     proof - 
       assume "(\<exists> (d, e, f) \<in> set a. d = 0 \<and> e \<noteq> 0)"
       then obtain a' b' c' where abc_prop: "(a', b', c') \<in> set a \<and> a' = 0 \<and> b' \<noteq> 0" by auto 
       then have "a'*x^2 + b'*x + c' = 0" using x_prop by auto
       then have "b'*x + c' = 0" using abc_prop by auto
       then have xis: "x = - c' / b'" using abc_prop
         by (smt divide_non_zero)
       then have "(\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
            (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
            (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
            (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0)" 
         using x_prop by auto 
       then have "(\<exists>(a', b', c')\<in>set a.
            (a' = 0 \<and> b' \<noteq> 0) \<and>
            (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
            (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
            (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
            (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0))"
         using abc_prop xis by auto
       then show "False"
         using big_not by auto 
     qed
     have c3: "(\<forall> (d, e, f) \<in> set a. d = 0 \<and> e = 0 \<and> f = 0) \<Longrightarrow> False"
     proof - 
       assume "(\<forall> (d, e, f) \<in> set a. d = 0 \<and> e = 0 \<and> f = 0)"
       then have equalset: "\<forall>x. (\<forall>(d, e, f)\<in>set a. d * x^2 + e * x + f = 0)"
         using case_prodE by auto
       have "\<not>?f5 \<and> \<not>?f6 \<and> \<not>?f7 \<and> \<not>?f8 \<and> \<not>?f13 \<and> \<not>?f9 \<and> \<not>?f10 \<and> \<not>?f11 \<and> \<not>?f12"
         using big_not by auto
       then have "\<not>(\<exists>x. (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
          (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
          (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))"
         using equalset big_not qe_forwards_helper[of a b c d] by auto
       then show "False"
         using x_prop by auto
     qed
     have eo: "(\<exists> (d, e, f) \<in> set a. d \<noteq> 0 \<and> - e\<^sup>2 + 4 * d * f \<le> 0) \<or> (\<exists> (d, e, f) \<in> set a. d = 0 \<and> e \<noteq> 0) \<or> (\<forall> (d, e, f) \<in> set a. d = 0 \<and> e = 0 \<and> f = 0)"
     proof - 
       have  "(\<forall> (d, e, f) \<in> set a. (d \<noteq> 0 \<longrightarrow>  - e\<^sup>2 + 4 * d * f \<le> 0))"
       proof clarsimp 
         fix d e f
         assume in_set: " (d, e, f) \<in> set a"
         assume dnonz: "d \<noteq> 0"
         have "d*x^2 + e*x + f = 0" using in_set x_prop by auto 
         then show " 4 * d * f \<le> e\<^sup>2"
           using dnonz discriminant_negative[of d e f] unfolding discrim_def
           by fastforce 
       qed
       then have discrim_prop: "\<not>(\<exists> (d, e, f) \<in> set a. d \<noteq> 0 \<and> - e\<^sup>2 + 4 * d * f \<le> 0) \<Longrightarrow> \<not>(\<exists> (d, e, f) \<in> set a. d \<noteq> 0)"
         by auto
       have "\<not>(\<exists> (d, e, f) \<in> set a. d \<noteq> 0) \<and> \<not>(\<exists> (d, e, f) \<in> set a. d = 0 \<and> e \<noteq> 0) \<Longrightarrow> (\<forall> (d, e, f) \<in> set a. d = 0 \<and> e = 0 \<and> f = 0)"
       proof - 
         assume ne: "\<not>(\<exists> (d, e, f) \<in> set a. d \<noteq> 0) \<and> \<not>(\<exists> (d, e, f) \<in> set a. d = 0 \<and> e \<noteq> 0)"
         show "(\<forall> (d, e, f) \<in> set a. d = 0 \<and> e = 0 \<and> f = 0)"
         proof clarsimp 
           fix d e f 
           assume in_set: "(d, e, f) \<in>set a"
           then have xzer: "d*x^2 + e*x + f = 0" using x_prop by auto
           have dzer: "d = 0" using ne in_set by auto
           have ezer: "e = 0" using ne in_set by auto
           show "d = 0 \<and> e = 0 \<and> f = 0" using xzer dzer ezer by auto 
         qed
       qed
       then show ?thesis using discrim_prop by auto
     qed
     show "False" using c1 c2 c3 eo by auto 
   qed
   then have " \<not>?e2 \<Longrightarrow> False" using bigor_var not_eq
     by presburger (* Takes a second *) 
   then have " \<not>?e2 \<longrightarrow> False" using impI[of "\<not>?e2" "False"]
     by blast 
   then show ?thesis 
     by auto
 qed
 
 subsubsection "Some Cases and Misc"
 lemma quadratic_linear :
   assumes "b\<noteq>0"
   assumes "a \<noteq> 0"
   assumes "4 * a * ba \<le> aa\<^sup>2"
   assumes "b * (sqrt (aa\<^sup>2 - 4 * a * ba) - aa) / (2 * a) + c = 0"
   assumes "\<forall>x\<in>set eq.
           case x of
           (d, e, f) \<Rightarrow>
             d * ((sqrt (aa\<^sup>2 - 4 * a * ba) - aa) / (2 * a))\<^sup>2 +
             e * (sqrt (aa\<^sup>2 - 4 * a * ba) - aa) / (2 * a) +
             f =
             0"
   assumes "(aaa, aaaa, baa) \<in> set eq"
   shows "aaa * (c / b)\<^sup>2 - aaaa * c / b + baa = 0"
 proof-
   have h:  "-(c/b) = (sqrt (aa\<^sup>2 - 4 * a * ba) - aa) / (2 * a)"
     using assms
     by (smt divide_minus_left nonzero_mult_div_cancel_left times_divide_eq_right)
   have h1 : "\<forall>x\<in>set eq. case x of (d, e, f) \<Rightarrow> d * (c / b)\<^sup>2 + e * - (c / b) + f = 0"
     using assms(5) unfolding h[symmetric] Fields.division_ring_class.times_divide_eq_right[symmetric]
       Power.ring_1_class.power2_minus .
   show ?thesis  
     using bspec[OF h1 assms(6)] by simp
 qed
 
 lemma quadratic_linear1:  
   assumes "b\<noteq>0"
   assumes "a \<noteq> 0"
   assumes "4 * a * ba \<le> aa\<^sup>2"
   assumes "(b::real) * (sqrt ((aa::real)\<^sup>2 - 4 * (a::real) * (ba::real)) - (aa::real)) / (2 * a) + (c::real) = 0"
   assumes "
        (\<forall>x\<in>set (les::(real*real*real)list).
           case x of
           (d, e, f) \<Rightarrow>
             d * ((sqrt (aa\<^sup>2 - 4 * a * ba) - aa) / (2 * a))\<^sup>2 +
             e * (sqrt (aa\<^sup>2 - 4 * a * ba) - aa) / (2 * a) +
             f
             < 0)"
   assumes "(aaa, aaaa, baa) \<in> set les"
   shows "aaa * (c / b)\<^sup>2 - aaaa * c / b + baa < 0"
 proof-
   have h:  "-(c/b) = (sqrt (aa\<^sup>2 - 4 * a * ba) - aa) / (2 * a)"
     using assms
     by (smt divide_minus_left nonzero_mult_div_cancel_left times_divide_eq_right)
   have h1 : "\<forall>x\<in>set les. case x of (d, e, f) \<Rightarrow> d * (c / b)\<^sup>2 + e * - (c / b) + f < 0"
     using assms(5) unfolding h[symmetric] Fields.division_ring_class.times_divide_eq_right[symmetric]
       Power.ring_1_class.power2_minus .
   show ?thesis  
     using bspec[OF h1 assms(6)] by simp
 qed
 
 lemma quadratic_linear2 :
   assumes "b\<noteq>0"
   assumes "a \<noteq> 0"
   assumes "4 * a * ba \<le> aa\<^sup>2"
   assumes "b * (- aa -sqrt (aa\<^sup>2 - 4 * a * ba)) / (2 * a) + c = 0"
   assumes "\<forall>x\<in>set eq.
           case x of
           (d, e, f) \<Rightarrow>
             d * ((- aa -sqrt (aa\<^sup>2 - 4 * a * ba)) / (2 * a))\<^sup>2 +
             e * (- aa -sqrt (aa\<^sup>2 - 4 * a * ba)) / (2 * a) +
             f =
             0"
   assumes "(aaa, aaaa, baa) \<in> set eq"
   shows "aaa * (c / b)\<^sup>2 - aaaa * c / b + baa = 0"
 proof-
   have h:  "-((c::real)/(b::real)) = (- (aa::real) -sqrt (aa\<^sup>2 - 4 * (a::real) * (ba::real))) / (2 * a)"
     using assms
     by (smt divide_minus_left nonzero_mult_div_cancel_left times_divide_eq_right)
   have h1 : "\<forall>x\<in>set eq. case x of (d, e, f) \<Rightarrow> d * (c / b)\<^sup>2 + e * - (c / b) + f = 0"
     using assms(5) unfolding h[symmetric] Fields.division_ring_class.times_divide_eq_right[symmetric]
       Power.ring_1_class.power2_minus .
   show ?thesis  
     using bspec[OF h1 assms(6)] by simp
 qed
 
 lemma quadratic_linear3:  
   assumes "b\<noteq>0"
   assumes "a \<noteq> 0"
   assumes "4 * a * ba \<le> aa\<^sup>2"
   assumes "(b::real) * (- (aa::real)- sqrt ((aa::real)\<^sup>2 - 4 * (a::real) * (ba::real)) ) / (2 * a) + (c::real) = 0"
   assumes "(\<forall>x\<in>set (les::(real*real*real)list).
           case x of
           (d, e, f) \<Rightarrow>
             d * ((- aa - sqrt (aa\<^sup>2 - 4 * a * ba)) / (2 * a))\<^sup>2 +
             e * (- aa - sqrt (aa\<^sup>2 - 4 * a * ba)) / (2 * a) +
             f
             < 0)"
   assumes "(aaa, aaaa, baa) \<in> set les"
   shows "aaa * (c / b)\<^sup>2 - aaaa * c / b + baa < 0"
 proof-
   have h:  "-((c::real)/(b::real)) = (- (aa::real) -sqrt (aa\<^sup>2 - 4 * (a::real) * (ba::real))) / (2 * a)"
     using assms
     by (smt divide_minus_left nonzero_mult_div_cancel_left times_divide_eq_right)
   have h1 : "\<forall>x\<in>set les. case x of (d, e, f) \<Rightarrow> d * (c / b)\<^sup>2 + e * - (c / b) + f < 0"
     using assms(5) unfolding h[symmetric] Fields.division_ring_class.times_divide_eq_right[symmetric]
       Power.ring_1_class.power2_minus .
   show ?thesis  
     using bspec[OF h1 assms(6)] by simp
 qed
 
 
 lemma h1b_helper_les: 
   "(\<forall>((a::real), (b::real), (c::real))\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<Longrightarrow> (\<exists>y.\<forall>x<y. (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))"
 proof - 
   show "(\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<Longrightarrow> (\<exists>y.\<forall>x<y. (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0))" 
   proof (induct les)
     case Nil
     then show ?case
       by auto
   next
     case (Cons q les) 
     have ind: " \<forall>a\<in>set (q # les). case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0"
       using Cons.prems
       by auto
     then have "case q of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0 "
       by simp      
     then obtain y2 where y2_prop: "case q of (a, ba, c) \<Rightarrow>  (\<forall>y<y2. a * y\<^sup>2 + ba * y + c < 0)"
       by auto
     have "\<forall>a\<in>set les. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0"
       using ind by simp
     then have " \<exists>y. \<forall>x<y. \<forall>a\<in>set les. case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c < 0"
       using Cons.hyps by blast 
     then obtain y1 where y1_prop: "\<forall>x<y1. \<forall>a\<in>set les. case a of (a, ba, c) \<Rightarrow> a * x^2 + ba * x + c < 0"
       by blast
     let ?y = "min y1 y2"
     have "\<forall>x < ?y.  (\<forall>a\<in>set (q #les). case a of (a, ba, c) \<Rightarrow> a * x^2 + ba * x + c < 0)"
       using y1_prop y2_prop 
       by fastforce 
     then show ?case
       by blast 
   qed
 qed
 
 lemma h1b_helper_leq: 
   "(\<forall>((a::real), (b::real), (c::real))\<in>set leq. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) \<Longrightarrow> (\<exists>y.\<forall>x<y. (\<forall>(a, b, c)\<in>set leq. a * x\<^sup>2 + b * x + c \<le> 0))"
 proof - 
   show "(\<forall>(a, b, c)\<in>set leq. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) \<Longrightarrow> (\<exists>y.\<forall>x<y. (\<forall>(a, b, c)\<in>set leq. a * x\<^sup>2 + b * x + c \<le> 0))" 
   proof (induct leq)
     case Nil
     then show ?case
       by auto
   next
     case (Cons q leq) 
     have ind: " \<forall>a\<in>set (q # leq). case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<le> 0"
       using Cons.prems
       by auto
     then have "case q of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<le> 0 "
       by simp      
     then obtain y2 where y2_prop: "case q of (a, ba, c) \<Rightarrow>  (\<forall>y<y2. a * y\<^sup>2 + ba * y + c \<le> 0)"
       by auto
     have "\<forall>a\<in>set leq. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<le> 0"
       using ind by simp
     then have " \<exists>y. \<forall>x<y. \<forall>a\<in>set leq. case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c \<le> 0"
       using Cons.hyps by blast 
     then obtain y1 where y1_prop: "\<forall>x<y1. \<forall>a\<in>set leq. case a of (a, ba, c) \<Rightarrow> a * x^2 + ba * x + c \<le> 0"
       by blast
     let ?y = "min y1 y2"
     have "\<forall>x < ?y.  (\<forall>a\<in>set (q #leq). case a of (a, ba, c) \<Rightarrow> a * x^2 + ba * x + c \<le> 0)"
       using y1_prop y2_prop 
       by fastforce 
     then show ?case
       by blast 
   qed
 qed
 
 lemma h1b_helper_neq: 
   "(\<forall>((a::real), (b::real), (c::real))\<in>set neq. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0) \<Longrightarrow> (\<exists>y.\<forall>x<y. (\<forall>(a, b, c)\<in>set neq. a * x\<^sup>2 + b * x + c \<noteq> 0))"
 proof - 
   show "(\<forall>(a, b, c)\<in>set neq. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0) \<Longrightarrow> (\<exists>y.\<forall>x<y. (\<forall>(a, b, c)\<in>set neq. a * x\<^sup>2 + b * x + c \<noteq> 0))" 
   proof (induct neq)
     case Nil
     then show ?case
       by auto
   next
     case (Cons q neq) 
     have ind: " \<forall>a\<in>set (q # neq). case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<noteq> 0"
       using Cons.prems
       by auto
     then have "case q of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<noteq> 0 "
       by simp      
     then obtain y2 where y2_prop: "case q of (a, ba, c) \<Rightarrow>  (\<forall>y<y2. a * y\<^sup>2 + ba * y + c \<noteq> 0)"
       by auto
     have "\<forall>a\<in>set neq. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<noteq> 0"
       using ind by simp
     then have " \<exists>y. \<forall>x<y. \<forall>a\<in>set neq. case a of (a, ba, c) \<Rightarrow> a * x\<^sup>2 + ba * x + c \<noteq> 0"
       using Cons.hyps by blast 
     then obtain y1 where y1_prop: "\<forall>x<y1. \<forall>a\<in>set neq. case a of (a, ba, c) \<Rightarrow> a * x^2 + ba * x + c \<noteq> 0"
       by blast
     let ?y = "min y1 y2"
     have "\<forall>x < ?y.  (\<forall>a\<in>set (q #neq). case a of (a, ba, c) \<Rightarrow> a * x^2 + ba * x + c \<noteq> 0)"
       using y1_prop y2_prop 
       by fastforce 
     then show ?case
       by blast 
   qed
 qed
 
 
 lemma min_lem: 
   fixes r::"real"
   assumes a1: "(\<exists>y'>r. (\<forall>((d::real), (e::real), (f::real))\<in>set b. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f < 0))" 
   assumes a2: "(\<exists>y'>r. (\<forall>((d::real), (e::real), (f::real))\<in>set c. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0))"
   assumes a3: "(\<exists>y'>r. (\<forall>((d::real), (e::real), (f::real))\<in>set d. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))" 
   shows "(\<exists>x. (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
          (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
          (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))"
 proof - 
   obtain y1 where y1_prop: "y1 > r \<and> (\<forall>(d, e, f)\<in>set b.  \<forall>x\<in>{r<..y1}. d * x\<^sup>2 + e * x + f < 0)"
     using a1 by auto
   obtain y2 where y2_prop: "y2 > r \<and> (\<forall>(d, e, f)\<in>set c.  \<forall>x\<in>{r<..y2}. d * x\<^sup>2 + e * x + f \<le> 0)"
     using a2 by auto
   obtain y3 where y3_prop: "y3 > r \<and> (\<forall>(d, e, f)\<in>set d.  \<forall>x\<in>{r<..y3}. d * x\<^sup>2 + e * x + f \<noteq> 0)"
     using a3 by auto
   let ?y = "(min (min y1 y2) y3)"
   have "?y > r" using y1_prop y2_prop y3_prop by auto
   then have "\<exists>x. x > r \<and> x < ?y" using dense[of r ?y] 
     by auto
   then obtain x where x_prop: "x > r \<and> x < ?y" by auto
   have bp: "(\<forall>(a, b, c)\<in>set b. a *x\<^sup>2 + b * x + c < 0)"  
     using x_prop y1_prop by auto 
   have cp: "(\<forall>(a, b, c)\<in>set c. a * x^2 + b * x + c \<le> 0)"  
     using x_prop y2_prop by auto 
   have dp: "(\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0)"  
     using x_prop y3_prop by auto 
   then have  "(\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
         (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
         (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0)"
     using  bp cp dp by auto
   then show ?thesis by auto
 qed 
 
 lemma qe_infinitesimals_helper:
   fixes k::"real"
   assumes asm: "(\<forall>(d, e, f)\<in>set a. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d. \<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)"
   shows "(\<exists>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
          (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
          (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))"
 proof -
   have "\<forall>(d, e, f)\<in>set a. d = 0 \<and> e = 0 \<and> f = 0" 
   proof clarsimp 
     fix d e f
     assume "(d, e, f) \<in> set a"
     then have "\<exists>y'>k. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f = 0"
       using asm by auto
     then obtain y' where y_prop: "y'>k \<and> (\<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f = 0)"
       by auto
     then show "d = 0 \<and> e = 0 \<and> f = 0" using continuity_lem_eq0[of "k" "y'" d e f]
       by auto
   qed
   then have eqprop: "\<forall>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) "
     by auto
   have lesprop: "(\<exists>y'>k. (\<forall>(d, e, f)\<in>set b. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f < 0))" 
     using les_qe_inf_helper[of b "k"] asm 
     by blast 
   have leqprop: "(\<exists>y'>k. (\<forall>(d, e, f)\<in>set c. \<forall>x\<in>{(k)<..y'}. d * x\<^sup>2 + e * x + f \<le> 0))" 
     using leq_qe_inf_helper[of c "k"] asm 
     by blast 
   have neqprop: "(\<exists>y'>(k). (\<forall>(d, e, f)\<in>set d. \<forall>x\<in>{(k)<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))" 
     using neq_qe_inf_helper[of d "k"] asm 
     by blast  
   then have "(\<exists>x. (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
          (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
          (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0)) " 
     using lesprop leqprop neqprop min_lem[of "k" b c d]
     by auto
   then show ?thesis
     using eqprop by auto 
 qed
 
 subsubsection "The qe\\_backwards lemma"
 lemma qe_backwards: 
   assumes "(((\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0) \<and>
      (\<forall>(a, b, c)\<in>set b. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<and>
      (\<forall>(a, b, c)\<in>set c. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) \<and>
      (\<forall>(a, b, c)\<in>set d. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0) 
      \<or>
      (\<exists>(a', b', c')\<in>set a.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0) 
         \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0)        
           \<or>
           (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0))) 
     \<or>
      (\<exists>(a', b', c')\<in>set b.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0) 
         \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0) 
       \<or>
           (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))) 
     \<or>
      (\<exists>(a', b', c')\<in>set c.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0) 
       \<or>
           (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0))) 
       \<or>
      (\<exists>(a', b', c')\<in>set d.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0) 
         \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0) 
       \<or>
           (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))))"
   shows " (\<exists>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
          (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
          (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))" 
 proof - 
   let ?e2 = "(((\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0) \<and>
      (\<forall>(a, b, c)\<in>set b. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<and>
      (\<forall>(a, b, c)\<in>set c. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) \<and>
      (\<forall>(a, b, c)\<in>set d. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0) 
      \<or>
      (\<exists>(a', b', c')\<in>set a.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0) 
         \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0)         
           \<or>
           (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0))) 
     \<or>
      (\<exists>(a', b', c')\<in>set b.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0) 
         \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0) 
       \<or>
           (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))) 
     \<or>
      (\<exists>(a', b', c')\<in>set c.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0) 
       \<or>
           (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0))) 
       \<or>
      (\<exists>(a', b', c')\<in>set d.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0) 
         \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0) 
       \<or>
           (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))))"
   let ?f10orf11orf12 = "(\<exists>(a', b', c')\<in>set d.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0) 
         \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0) 
       \<or>
           (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))"
   let ?f8orf9 = "(\<exists>(a', b', c')\<in>set c.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0) 
       \<or>
           (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0)))"
   let ?f5orf6orf7 = "(\<exists>(a', b', c')\<in>set b.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0) 
         \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0) 
       \<or>
           (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))"
   let ?f2orf3orf4 = "(\<exists>(a', b', c')\<in>set a.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0) 
         \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0)         
           \<or>
           (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0)))"
   let ?e1 = "(\<exists>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
          (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
          (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))"
   let ?f1 = "((\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0) \<and>
      (\<forall>(a, b, c)\<in>set b. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<and>
      (\<forall>(a, b, c)\<in>set c. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) \<and>
      (\<forall>(a, b, c)\<in>set d. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0))"
   let ?f2 = "(\<exists>(a', b', c')\<in>set a.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0))"
   let ?f3 = "(\<exists>(a', b', c')\<in>set a. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0))"
   let ?f4 = "(\<exists>(a', b', c')\<in>set a. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
  (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0)) "
   let ?f5 = "(\<exists>(a', b', c')\<in>set b.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
   let ?f6 = "(\<exists>(a', b', c')\<in>set b. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))"
   let ?f7 = "(\<exists>(a', b', c')\<in>set b. a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))"
   let ?f8 = "(\<exists>(a', b', c')\<in>set c. (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0))"
   let ?f13 = "(\<exists>(a', b', c')\<in>set c.
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0)))"
   let ?f9 = "(\<exists>(a', b', c')\<in>set c.  a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
         (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq> 0))"
   let ?f10 = "(\<exists>(a', b', c')\<in>set d.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
   let ?f11 = "(\<exists>(a', b', c')\<in>set d.
           a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))"
   let ?f12 = "(\<exists>(a', b', c')\<in>set d.
           a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))"
   have h1a: "?e2 \<longrightarrow> (?f1 \<or> ?f2orf3orf4 \<or> ?f5orf6orf7 \<or> ?f8orf9 \<or> ?f10orf11orf12)"
     by auto
   have h2: "?f2orf3orf4 \<longrightarrow> (?f2 \<or> ?f3 \<or> ?f4)" by auto
   then have h1b: "?e2 \<longrightarrow> (?f1 \<or> ?f2 \<or> ?f3 \<or> ?f4 \<or> ?f5orf6orf7 \<or> ?f8orf9 \<or> ?f10orf11orf12) "
     using h1a by auto
   have h3: "?f5orf6orf7 \<longrightarrow> (?f5 \<or> ?f6 \<or> ?f7)" by auto
   then have h1c: "?e2 \<longrightarrow> (?f1 \<or> ?f2 \<or> ?f3 \<or> ?f4 \<or> ?f5 \<or> ?f6 \<or> ?f7 \<or> ?f8orf9 \<or> ?f10orf11orf12) "
     using h1b by smt 
   have h4: "?f8orf9 \<longrightarrow> (?f8 \<or> ?f9 \<or> ?f13)" by auto
   then have h1d: "?e2 \<longrightarrow> (?f1 \<or> ?f2 \<or> ?f3 \<or> ?f4 \<or> ?f5 \<or> ?f6 \<or> ?f7 \<or> ?f8 \<or> ?f9 \<or> ?f13 \<or> ?f10orf11orf12) "
     using h1c
     by smt 
   have h5: "?f10orf11orf12 \<longrightarrow> (?f10 \<or> ?f11 \<or> ?f12)" 
     by auto
   then have bigor: "?e2 \<longrightarrow> (?f1 \<or> ?f2 \<or> ?f3 \<or> ?f4 \<or> ?f5 \<or> ?f6 \<or> ?f7 \<or> ?f8 \<or> ?f13 \<or> ?f9 \<or> ?f10 \<or> ?f11 \<or> ?f12) "
     using h1d  by smt 
   have "?f1 \<Longrightarrow> ?e1"
   proof - 
     assume asm: "(\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0) \<and>
     (\<forall>(a, b, c)\<in>set b. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<and>
     (\<forall>(a, b, c)\<in>set c. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) \<and>
     (\<forall>(a, b, c)\<in>set d. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0)"
     then have eqprop: "\<forall>x. \<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0" by auto
     have "\<exists>y. \<forall>x<y. \<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0"
       using asm h1b_helper_les by auto
     then obtain y1 where y1_prop: "\<forall>x<y1. \<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0" by auto
     have "\<exists>y. \<forall>x<y. \<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0"
       using asm h1b_helper_leq by auto
     then obtain y2 where y2_prop: "\<forall>x<y2. \<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0" by auto
     have "\<exists>y. \<forall>x<y. \<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0"
       using asm h1b_helper_neq by auto
     then obtain y3 where y3_prop: "\<forall>x<y3. \<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0" by auto
     let ?y = "(min (min y1 y2) y3) - 1"
     have y_prop: "?y < y1 \<and> ?y < y2 \<and> ?y < y3"
       by auto
     have ap: "(\<forall>(a, b, c)\<in>set a. a * ?y\<^sup>2 + b * ?y + c = 0)" 
       using eqprop by auto
     have bp: "(\<forall>(a, b, c)\<in>set b. a * ?y\<^sup>2 + b * ?y + c < 0)"  
       using y_prop y1_prop by auto 
     have cp: "(\<forall>(a, b, c)\<in>set c. a * ?y\<^sup>2 + b * ?y + c \<le> 0)"  
       using y_prop y2_prop by auto 
     have dp: "(\<forall>(a, b, c)\<in>set d. a * ?y\<^sup>2 + b * ?y + c \<noteq> 0)"  
       using y_prop y3_prop by auto 
     then have  "(\<forall>(a, b, c)\<in>set a. a * ?y\<^sup>2 + b * ?y + c = 0) \<and>
         (\<forall>(a, b, c)\<in>set b. a * ?y\<^sup>2 + b * ?y + c < 0) \<and>
         (\<forall>(a, b, c)\<in>set c. a * ?y\<^sup>2 + b * ?y + c \<le> 0) \<and>
         (\<forall>(a, b, c)\<in>set d. a * ?y\<^sup>2 + b * ?y + c \<noteq> 0)"
       using ap bp cp dp by auto
     then show ?thesis by auto
   qed
   then have h1: "?f1 \<longrightarrow> ?e1" 
     by auto
   have "?f2 \<Longrightarrow> ?e1" 
   proof - 
     assume " \<exists>(a', b', c')\<in>set a.
        (a' = 0 \<and> b' \<noteq> 0) \<and>
        (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0)"
     then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set a \<and>
        (a' = 0 \<and> b' \<noteq> 0) \<and>
        (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0)"
       by auto
     then have "\<exists>(x::real). x = -c'/b'" by auto
     then obtain x where x_prop: "x = - c' / b'" by auto
     then have "(\<forall>xa\<in>set a. case xa of (a, b, c) \<Rightarrow> a * x\<^sup>2 + b * x + c = 0) \<and>
         (\<forall>xa\<in>set b. case xa of (a, b, c) \<Rightarrow> a * x\<^sup>2 + b * x + c < 0) \<and>
         (\<forall>xa\<in>set c. case xa of (a, b, c) \<Rightarrow> a * x\<^sup>2 + b * x + c \<le> 0) \<and>
         (\<forall>xa\<in>set d. case xa of (a, b, c) \<Rightarrow> a * x\<^sup>2 + b * x + c \<noteq> 0)"
       using abc_prop by auto
     then show ?thesis by auto
   qed
   then have h2: "?f2 \<longrightarrow> ?e1" 
     by auto
   have "?f3 \<Longrightarrow> ?e1" 
   proof -
     assume "\<exists>(a', b', c')\<in>set a.
        a' \<noteq> 0 \<and>
        - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
        (\<forall>(d, e, f)\<in>set a.
            d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b.
            d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c.
            d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f  \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d.
            d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f \<noteq> 0)"
     then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set a \<and>
        a' \<noteq> 0 \<and>
        - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
        (\<forall>(d, e, f)\<in>set a.
            d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b.
            d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c.
            d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f  \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d.
            d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f \<noteq> 0)" by auto
     then have "\<exists>(x::real). x = (- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')" by auto
     then obtain x where x_prop: " x = (- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')" by auto
     then have "(\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and>
         (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
         (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
         (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0)" using abc_prop by auto
     then show ?thesis by auto
   qed
   then have h3: "?f3  \<longrightarrow> ?e1" 
     by auto
   have "?f4 \<Longrightarrow> ?e1" 
   proof -
     assume " \<exists>(a', b', c')\<in>set a.
        a' \<noteq> 0 \<and>
        - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
        (\<forall>(d, e, f)\<in>set a.
            d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b.
            d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c.
            d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d.
            d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f \<noteq> 0)"
     then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set a \<and>
        a' \<noteq> 0 \<and>
        - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
        (\<forall>(d, e, f)\<in>set a.
            d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b.
            d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c.
            d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d.
            d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f \<noteq> 0)" by auto
     then have "\<exists>(x::real). x = (- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')" by auto
     then obtain x where x_prop: " x = (- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')" by auto
     then have "(\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and>
         (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
         (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
         (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0)" using abc_prop by auto
     then show ?thesis by auto
   qed
   then have "?f4 \<longrightarrow> ?e1" by auto
   have "?f5 \<Longrightarrow> ?e1" 
   proof - 
     assume asm: "\<exists>(a', b', c')\<in>set b.
        (a' = 0 \<and> b' \<noteq> 0) \<and>
        (\<forall>(d, e, f)\<in>set a. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)"
     then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set b \<and>
        (a' = 0 \<and> b' \<noteq> 0) \<and>
        (\<forall>(d, e, f)\<in>set a. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)"
       by auto
     then show ?thesis using qe_infinitesimals_helper[of a "- c' / b'" b c d]
       by auto
   qed
   then have h5: "?f5 \<longrightarrow> ?e1" 
     by auto
   have "?f6 \<Longrightarrow> ?e1" 
   proof - 
     assume "\<exists>(a', b', c')\<in>set b.
        a' \<noteq> 0 \<and>
        - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
        (\<forall>(d, e, f)\<in>set a.
            \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b.
            \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c.
            \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d.
            \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f \<noteq> 0)"
     then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set b \<and>
        a' \<noteq> 0 \<and>
        - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
        (\<forall>(d, e, f)\<in>set a.
            \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b.
            \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c.
            \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d.
            \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f \<noteq> 0)" by auto
     then show ?thesis using qe_infinitesimals_helper[of a "(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')" b c d]
       by auto
   qed
   then have h6: "?f6  \<longrightarrow> ?e1" 
     by auto
   have "?f7 \<Longrightarrow> ?e1" 
   proof - 
     assume "\<exists>(a', b', c')\<in>set b.
        a' \<noteq> 0 \<and>
        - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
        (\<forall>(d, e, f)\<in>set a.
            \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b.
            \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c.
            \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d.
            \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f \<noteq> 0)"
     then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set b \<and>
        a' \<noteq> 0 \<and>
        - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
        (\<forall>(d, e, f)\<in>set a.
            \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b.
            \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c.
            \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d.
            \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f \<noteq> 0)"
       by auto
     then show ?thesis using qe_infinitesimals_helper[of a "(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')" b c d]
       by auto
   qed
   then have h7: "?f7 \<longrightarrow> ?e1"
     by auto
   have "?f8 \<Longrightarrow> ?e1"   
   proof -
     assume "\<exists>(a', b', c')\<in>set c.
        (a' = 0 \<and> b' \<noteq> 0) \<and>
        (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0)"
     then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set c \<and>
        (a' = 0 \<and> b' \<noteq> 0) \<and>
        (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0)" by auto
     then have "\<exists>(x::real). x = (- c' / b')" by auto
     then obtain x where x_prop: " x = - c' / b'" by auto
     then have "(\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and>
         (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
         (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
         (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0)" using abc_prop by auto
     then show ?thesis by auto
   qed
   then have h8: "?f8 \<longrightarrow> ?e1" by auto
   have "?f9 \<Longrightarrow> ?e1" 
   proof -
     assume "\<exists>(a', b', c')\<in>set c.
        a' \<noteq> 0 \<and>
        - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
        (\<forall>(d, e, f)\<in>set a.
            d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f =  0) \<and>
        (\<forall>(d, e, f)\<in>set b.
            d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c.
            d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d.
            d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f \<noteq> 0)"
     then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set c \<and>
        a' \<noteq> 0 \<and>
        - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
        (\<forall>(d, e, f)\<in>set a.
            d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f =  0) \<and>
        (\<forall>(d, e, f)\<in>set b.
            d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f  < 0) \<and>
        (\<forall>(d, e, f)\<in>set c.
            d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d.
            d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f \<noteq>  0)" by auto
     then have "\<exists>(x::real). x = (- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')" by auto
     then obtain x where x_prop: " x = (- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')" by auto
     then have "(\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and>
         (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
         (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
         (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0)" using abc_prop by auto
     then show ?thesis by auto
   qed
   then have h9:  "?f9 \<longrightarrow> ?e1" by auto
   have "?f10 \<Longrightarrow> ?e1"
   proof - 
     assume asm: "\<exists>(a', b', c')\<in>set d.
        (a' = 0 \<and> b' \<noteq> 0) \<and>
        (\<forall>(d, e, f)\<in>set a. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)"
     then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set d \<and>
        (a' = 0 \<and> b' \<noteq> 0) \<and>
        (\<forall>(d, e, f)\<in>set a. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)"
       by auto
     then show ?thesis using qe_infinitesimals_helper[of a "- c' / b'" b c d]
       by auto
   qed
   then have h10: "?f10 \<longrightarrow> ?e1" by auto
   have "?f11 \<Longrightarrow> ?e1"   
   proof - 
     assume "\<exists>(a', b', c')\<in>set d.
        a' \<noteq> 0 \<and>
        - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
        (\<forall>(d, e, f)\<in>set a.
            \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b.
            \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c.
            \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d.
            \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f \<noteq> 0)"
     then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set d \<and>
        a' \<noteq> 0 \<and>
        - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
        (\<forall>(d, e, f)\<in>set a.
            \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b.
            \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c.
            \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d.
            \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f \<noteq> 0)" by auto
     then show ?thesis using qe_infinitesimals_helper[of a "(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')" b c d]
       by auto
   qed
   then have h11: "?f11 \<longrightarrow> ?e1" by auto
   have "?f12 \<Longrightarrow> ?e1"  proof - 
     assume "\<exists>(a', b', c')\<in>set d.
        a' \<noteq> 0 \<and>
        - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
        (\<forall>(d, e, f)\<in>set a.
            \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b.
            \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c.
            \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d.
            \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f \<noteq> 0)"
     then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set d \<and>
        a' \<noteq> 0 \<and>
        - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
        (\<forall>(d, e, f)\<in>set a.
            \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b.
            \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c.
            \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d.
            \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
               \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                  d * x\<^sup>2 + e * x + f \<noteq> 0)"
       by auto
     then show ?thesis using qe_infinitesimals_helper[of a "(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')" b c d]
       by auto
   qed
   then have h12: "?f12 \<longrightarrow> ?e1" by auto
   have "?f13 \<Longrightarrow> ?e1" proof -
     assume " \<exists>(a', b', c')\<in>set c.
        a' \<noteq> 0 \<and>
        - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
        (\<forall>(d, e, f)\<in>set a.
            d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b.
            d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c.
            d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d.
            d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f \<noteq> 0)"
     then obtain a' b' c' where abc_prop: "(a', b', c')\<in>set c \<and>
        a' \<noteq> 0 \<and>
        - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
        (\<forall>(d, e, f)\<in>set a.
            d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f = 0) \<and>
        (\<forall>(d, e, f)\<in>set b.
            d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f < 0) \<and>
        (\<forall>(d, e, f)\<in>set c.
            d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f \<le> 0) \<and>
        (\<forall>(d, e, f)\<in>set d.
            d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
            e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
            f \<noteq> 0)" by auto
     then have "\<exists>(x::real). x = (- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')" by auto
     then obtain x where x_prop: " x = (- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')" by auto
     then have "(\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and>
         (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
         (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
         (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0)" using abc_prop by auto
     then show ?thesis by auto
   qed
   then have h13: "?f13 \<longrightarrow> ?e1" by auto
   show ?thesis using bigor h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 h11 h12 h13
     using assms
     by (smt \<open>\<exists>(a', b', c')\<in>set a. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f \<noteq> 0) \<Longrightarrow> \<exists>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and> (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0)\<close>) 
       (* by force *)
 qed
 
 subsection "General QE lemmas"
 
 lemma qe: "(\<exists>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
          (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
          (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0)) =
     ((\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0) \<and>
      (\<forall>(a, b, c)\<in>set b. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<and>
      (\<forall>(a, b, c)\<in>set c. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) \<and>
      (\<forall>(a, b, c)\<in>set d. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0) \<or>
      (\<exists>(a', b', c')\<in>set a.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq>
               0) \<or>
           (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq>
               0))) \<or>
      (\<exists>(a', b', c')\<in>set b.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0) \<or>
           (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))) \<or>
      (\<exists>(a', b', c')\<in>set c.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq>
               0) \<or>
           (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq>
               0))) \<or>
      (\<exists>(a', b', c')\<in>set d.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0) \<or>
           (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))))" 
 proof - 
   let ?e1 = "((\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0) \<and>
      (\<forall>(a, b, c)\<in>set b. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<and>
      (\<forall>(a, b, c)\<in>set c. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) \<and>
      (\<forall>(a, b, c)\<in>set d. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0) \<or>
      (\<exists>(a', b', c')\<in>set a.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq>
               0) \<or>
           (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq>
               0))) \<or>
      (\<exists>(a', b', c')\<in>set b.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0) \<or>
           (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))) \<or>
      (\<exists>(a', b', c')\<in>set c.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq>
               0) \<or>
           (\<forall>(d, e, f)\<in>set a.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f =
               0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f
               \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
               e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
               f \<noteq>
               0))) \<or>
      (\<exists>(a', b', c')\<in>set d.
          (a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set a.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set b.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set c.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set d.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0) \<or>
           (\<forall>(d, e, f)\<in>set a.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set b.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set c.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set d.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0))))" 
   let ?e2 = "(\<exists>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and>
          (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and>
          (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and>
          (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))"
   have h1: "?e1 \<longrightarrow> ?e2" using qe_backwards 
     by auto
   have h2: "?e2 \<longrightarrow> ?e1" using qe_forwards
     by auto
   have "(?e2 \<longrightarrow> ?e1) \<and> (?e1 \<longrightarrow> ?e2) " using h1 h2
     by force 
   then have  "?e2 \<longleftrightarrow> ?e1"
     using iff_conv_conj_imp[of ?e1 ?e2]
     by presburger
   then show ?thesis
     by auto
 qed
 
 
 fun eq_fun :: "real \<Rightarrow> real \<Rightarrow> real \<Rightarrow> (real*real*real) list \<Rightarrow> (real*real*real) list \<Rightarrow> (real*real*real) list \<Rightarrow> (real*real*real) list \<Rightarrow> bool" where
   "eq_fun a' b' c' eq les leq neq = ((a' = 0 \<and> b' \<noteq> 0) \<and>
           (\<forall>a\<in>set eq.
               case a of (d, e, f) \<Rightarrow> d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and>
           (\<forall>a\<in>set les.
               case a of (d, e, f) \<Rightarrow> d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<and>
           (\<forall>a\<in>set leq.
               case a of (d, e, f) \<Rightarrow> d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<le> 0) \<and>
           (\<forall>a\<in>set neq.
               case a of (d, e, f) \<Rightarrow> d * (- c' / b')\<^sup>2 + e * (- c' / b') + f \<noteq> 0) \<or>
           a' \<noteq> 0 \<and>
           - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
           ((\<forall>a\<in>set eq.
                case a of
                (d, e, f) \<Rightarrow>
                  d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                  e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                  f =
                  0) \<and>
            (\<forall>a\<in>set les.
                case a of
                (d, e, f) \<Rightarrow>
                  d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                  e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                  f
                  < 0) \<and>
            (\<forall>a\<in>set leq.
                case a of
                (d, e, f) \<Rightarrow>
                  d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                  e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                  f
                  \<le> 0) \<and>
            (\<forall>a\<in>set neq.
                case a of
                (d, e, f) \<Rightarrow>
                  d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                  e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                  f \<noteq>
                  0) \<or>
            (\<forall>a\<in>set eq.
                case a of
                (d, e, f) \<Rightarrow>
                  d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                  e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                  f =
                  0) \<and>
            (\<forall>a\<in>set les.
                case a of
                (d, e, f) \<Rightarrow>
                  d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                  e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                  f
                  < 0) \<and>
            (\<forall>a\<in>set leq.
                case a of
                (d, e, f) \<Rightarrow>
                  d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                  e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                  f
                  \<le> 0) \<and>
            (\<forall>a\<in>set neq.
                case a of
                (d, e, f) \<Rightarrow>
                  d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 +
                  e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) +
                  f \<noteq>
                  0)))"
 
 fun les_fun :: "real \<Rightarrow> real \<Rightarrow> real \<Rightarrow> (real*real*real) list \<Rightarrow> (real*real*real) list \<Rightarrow> (real*real*real) list \<Rightarrow> (real*real*real) list \<Rightarrow> bool" where 
   "les_fun a' b' c' eq les leq neq = ((a' = 0 \<and> b' \<noteq> 0) \<and>
          (\<forall>(d, e, f)\<in>set eq.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and>
          (\<forall>(d, e, f)\<in>set les.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and>
          (\<forall>(d, e, f)\<in>set leq.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and>
          (\<forall>(d, e, f)\<in>set neq.
              \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0) \<or>
          a' \<noteq> 0 \<and>
          - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and>
          ((\<forall>(d, e, f)\<in>set eq.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set leq.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set neq.
               \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0) \<or>
           (\<forall>(d, e, f)\<in>set eq.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f = 0) \<and>
           (\<forall>(d, e, f)\<in>set les.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f < 0) \<and>
           (\<forall>(d, e, f)\<in>set leq.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<le> 0) \<and>
           (\<forall>(d, e, f)\<in>set neq.
               \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a').
                  \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}.
                     d * x\<^sup>2 + e * x + f \<noteq> 0)))"
 
 lemma general_qe' :
   (* Direct substitution F(x) *)
   assumes "F = (\<lambda>x. 
                (\<forall>(a,b,c)\<in>set eq . a*x\<^sup>2+b*x+c=0)\<and>
                (\<forall>(a,b,c)\<in>set les. a*x\<^sup>2+b*x+c<0)\<and>
                (\<forall>(a,b,c)\<in>set leq. a*x\<^sup>2+b*x+c\<le>0)\<and>
                (\<forall>(a,b,c)\<in>set neq. a*x\<^sup>2+b*x+c\<noteq>0))"
     (* Substitution of r+\<epsilon> into F *)
   assumes "F\<epsilon> = (\<lambda>r. 
          (\<forall>(a,b,c)\<in>set eq.  \<exists>y>r.\<forall>x\<in>{r<..y}. a*x\<^sup>2+b*x+c=0) \<and>
          (\<forall>(a,b,c)\<in>set les. \<exists>y>r.\<forall>x\<in>{r<..y}. a*x\<^sup>2+b*x+c<0) \<and>
          (\<forall>(a,b,c)\<in>set leq. \<exists>y>r.\<forall>x\<in>{r<..y}. a*x\<^sup>2+b*x+c\<le>0) \<and>
          (\<forall>(a,b,c)\<in>set neq. \<exists>y>r.\<forall>x\<in>{r<..y}. a*x\<^sup>2+b*x+c\<noteq>0)
         )"
     (* Substitution of -\<infinity> into F *)
   assumes "F\<^sub>i\<^sub>n\<^sub>f = (
      (\<forall>(a,b,c)\<in>set eq.  \<exists>x. \<forall>y<x. a*y\<^sup>2+b*y+c=0) \<and>
      (\<forall>(a,b,c)\<in>set les. \<exists>x. \<forall>y<x. a*y\<^sup>2+b*y+c<0) \<and>
      (\<forall>(a,b,c)\<in>set leq. \<exists>x. \<forall>y<x. a*y\<^sup>2+b*y+c\<le>0) \<and>
      (\<forall>(a,b,c)\<in>set neq. \<exists>x. \<forall>y<x. a*y\<^sup>2+b*y+c\<noteq>0)
     )"
     (* finds linear or quadratic roots of a polynomial *)
   assumes "roots = (\<lambda>(a,b,c).
      if a=0 \<and> b\<noteq>0 then {-c/b}::real set else
      if a\<noteq>0 \<and> b\<^sup>2-4*a*c\<ge>0 then {(-b+sqrt(b\<^sup>2-4*a*c))/(2*a)}\<union>{(-b-sqrt(b\<^sup>2-4*a*c))/(2*a)} else {})"
     (* all the root of each atom *)
   assumes "A = \<Union>(roots ` (set eq))"
   assumes "B = \<Union>(roots ` (set les))"
   assumes "C = \<Union>(roots ` (set leq))"
   assumes "D = \<Union>(roots ` (set neq))"
     (* Quantifier Elimination *)
   shows "(\<exists>x. F(x)) = (F\<^sub>i\<^sub>n\<^sub>f\<or>(\<exists>r\<in>A. F r)\<or>(\<exists>r\<in>B. F\<epsilon> r)\<or>(\<exists>r\<in>C. F r)\<or>(\<exists>r\<in>D. F\<epsilon> r))"
 proof-
   { fix X
     have "(\<exists>(a, b, c)\<in>set X. eq_fun a b c eq les leq neq) = (\<exists>x\<in>F ` \<Union>(roots ` (set X)). x)"
     proof(induction X)
       case Nil
       then show ?case by auto
     next
       case (Cons p X)
       have h1: "(\<exists>x\<in>F ` \<Union> (roots ` set (p # X)). x) = ((\<exists>x\<in>F ` roots p. x) \<or> (\<exists>x\<in>F ` \<Union> (roots ` set X). x))"
         by auto
       have h2 :"(case p of (a,b,c) \<Rightarrow> eq_fun a b c eq les leq neq) = (\<exists>x\<in>F ` roots p. x)"
         apply(cases p) unfolding assms apply simp by linarith
       show ?case  unfolding h1 Cons[symmetric] using h2 by auto
     qed
   }
   then have eq : "\<And>X. (\<exists>(a, b, c)\<in>set X. eq_fun a b c eq les leq neq) = (\<exists>x\<in>F ` \<Union> (roots ` set X). x)" by auto
   { fix X
     have "(\<exists>(a, b, c)\<in>set X. les_fun a b c eq les leq neq) = (\<exists>x\<in>F\<epsilon> ` \<Union>(roots ` (set X)). x)"
     proof(induction X)
       case Nil
       then show ?case by auto
     next
       case (Cons p X)
       have h1: "(\<exists>x\<in>F\<epsilon> ` \<Union> (roots ` set (p # X)). x) = ((\<exists>x\<in>F\<epsilon> ` roots p. x) \<or> (\<exists>x\<in>F\<epsilon> ` \<Union> (roots ` set X). x))"
         by auto
       have h2 :"(case p of (a,b,c) \<Rightarrow> les_fun a b c eq les leq neq) = (\<exists>x\<in>F\<epsilon> ` roots p. x)"
         apply(cases p) unfolding assms apply simp by linarith
       show ?case  unfolding h1 Cons[symmetric] using h2 by auto
     qed
   }
   then have les : "\<And>X. (\<exists>(a, b, c)\<in>set X. les_fun a b c eq les leq neq) = (\<exists>x\<in>F\<epsilon> ` \<Union> (roots ` set X). x)" by auto
   have inf : "(\<forall>(a, b, c)\<in>set eq. a = 0 \<and> b = 0 \<and> c = 0) = (\<forall>x\<in>set eq. case x of (a, b, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c = 0)"
   proof(induction eq)
     case Nil
     then show ?case by auto
   next
     case (Cons p eq)
     then show ?case proof(cases p)
       case (fields a b c)
       show ?thesis unfolding fields using infzeros[of _ a b c] Cons by auto
     qed
   qed
   show ?thesis
     using qe[of "eq" "les" "leq" "neq"]
     using eq[of eq] eq[of leq] les[of les] les[of neq] unfolding inf assms
     by auto
 qed
 
 lemma general_qe'' :
   (* Direct substitution F(x) *)
   assumes "S = {(=), (<), (\<le>), (\<noteq>)}"
   assumes "finite (X (=))"
   assumes "finite (X (<))"
   assumes "finite (X (\<le>))"
   assumes "finite (X (\<noteq>))"
   assumes "F = (\<lambda>x. \<forall>rel\<in>S. \<forall>(a,b,c)\<in>(X rel). rel (a*x\<^sup>2+b*x+c) 0)"
     (* Substitution of r+\<epsilon> into F *)
   assumes "F\<epsilon> = (\<lambda>r. \<forall>rel\<in>S. \<forall>(a,b,c)\<in>(X rel). \<exists>y>r.\<forall>x\<in>{r<..y}. rel (a*x\<^sup>2+b*x+c) 0)"
     (* Substitution of -\<infinity> into F *)
   assumes "F\<^sub>i\<^sub>n\<^sub>f = (\<forall>rel\<in>S. \<forall>(a,b,c)\<in>(X rel). \<exists>x. \<forall>y<x. rel (a*y\<^sup>2+b*y+c) 0)"
     (* finds linear or quadratic roots of a polynomial *)
   assumes "roots = (\<lambda>(a,b,c).
      if a=0 \<and> b\<noteq>0 then {-c/b}::real set else
      if a\<noteq>0 \<and> b\<^sup>2-4*a*c\<ge>0 then {(-b+sqrt(b\<^sup>2-4*a*c))/(2*a)}\<union>{(-b-sqrt(b\<^sup>2-4*a*c))/(2*a)} else {})"
     (* all the root of each atom *)
   assumes "A = \<Union>(roots ` ((X (=))))"
   assumes "B = \<Union>(roots ` ((X (<))))"
   assumes "C = \<Union>(roots ` ((X (\<le>))))"
   assumes "D = \<Union>(roots ` ((X (\<noteq>))))"
     (* Quantifier Elimination *)
   shows "(\<exists>x. F(x)) = (F\<^sub>i\<^sub>n\<^sub>f\<or>(\<exists>r\<in>A. F r)\<or>(\<exists>r\<in>B. F\<epsilon> r)\<or>(\<exists>r\<in>C. F r)\<or>(\<exists>r\<in>D. F\<epsilon> r))"
 proof-
   define less where "less = (\<lambda>(a::real,b::real,c::real).\<lambda>(a',b',c'). a<a'\<or> (a=a'\<and> (b<b'\<or>(b=b'\<and>c<c'))))"
   define leq where "leq = (\<lambda>x.\<lambda>y. x=y \<or> less x y)"
   have linorder: "class.linorder leq less"
     unfolding class.linorder_def class.order_def class.preorder_def class.order_axioms_def class.linorder_axioms_def
       less_def leq_def by auto
   show ?thesis
     using assms(6-8) unfolding assms(1) apply simp
     using general_qe'[OF _ _ _ assms(9), of F "List.linorder.sorted_list_of_set leq (X (=))" "List.linorder.sorted_list_of_set leq (X (<))" "List.linorder.sorted_list_of_set leq (X (\<le>))" "List.linorder.sorted_list_of_set leq (X (\<noteq>))" F\<epsilon> F\<^sub>i\<^sub>n\<^sub>f A B C D]
     unfolding List.linorder.set_sorted_list_of_set[OF linorder assms(2)] List.linorder.set_sorted_list_of_set[OF linorder assms(3)] List.linorder.set_sorted_list_of_set[OF linorder assms(4)] List.linorder.set_sorted_list_of_set[OF linorder assms(5)]
     using assms(10-13)by auto
 qed
 
 
 theorem general_qe :
   (* finite sets of atoms involving = < \<le> and \<noteq>*)
   assumes "R = {(=), (<), (\<le>), (\<noteq>)}"
   assumes "\<forall>rel\<in>R. finite (Atoms rel)"
     (* Direct substitution F(x) *)
   assumes "F = (\<lambda>x. \<forall>rel\<in>R. \<forall>(a,b,c)\<in>(Atoms rel). rel (a*x\<^sup>2+b*x+c) 0)"
     (* Substitution of r+\<epsilon> into F *)
   assumes "F\<epsilon> = (\<lambda>r. \<forall>rel\<in>R. \<forall>(a,b,c)\<in>(Atoms rel). \<exists>y>r.\<forall>x\<in>{r<..y}. rel (a*x\<^sup>2+b*x+c) 0)"
     (* Substitution of -\<infinity> into F *)
   assumes "F\<^sub>i\<^sub>n\<^sub>f = (\<forall>rel\<in>R. \<forall>(a,b,c)\<in>(Atoms rel). \<exists>x. \<forall>y<x. rel (a*y\<^sup>2+b*y+c) 0)"
     (* finds linear or quadratic roots of a polynomial *)
   assumes "roots = (\<lambda>(a,b,c).
      if a=0 \<and> b\<noteq>0 then {-c/b} else
      if a\<noteq>0 \<and> b\<^sup>2-4*a*c\<ge>0 then {(-b+sqrt(b\<^sup>2-4*a*c))/(2*a)}\<union>{(-b-sqrt(b\<^sup>2-4*a*c))/(2*a)} else {})"
     (* Quantifier Elimination *)
   shows "(\<exists>x. F(x)) = 
             (F\<^sub>i\<^sub>n\<^sub>f \<or>
             (\<exists>r\<in>\<Union>(roots ` (Atoms (=) \<union> Atoms (\<le>))). F r) \<or>
             (\<exists>r\<in>\<Union>(roots ` (Atoms (<) \<union> Atoms (\<noteq>))). F\<epsilon> r))"
   using general_qe''[OF assms(1) _ _ _ _ assms(3-6) refl refl refl refl]
   using assms(2) unfolding assms(1) 
   by auto
 
 end
\ No newline at end of file
diff --git a/thys/Virtual_Substitution/QuadraticCase.thy b/thys/Virtual_Substitution/QuadraticCase.thy
--- a/thys/Virtual_Substitution/QuadraticCase.thy
+++ b/thys/Virtual_Substitution/QuadraticCase.thy
@@ -1,969 +1,969 @@
 subsection "Quadratic Case"
 theory QuadraticCase
   imports VSAlgos
 begin
 
 (*-------------------------------------------------------------------------------------------------------------*)
 lemma quad_part_1_eq :
   assumes lLength : "length L > var"
   assumes hdeg : "MPoly_Type.degree (p::real mpoly) var = (deg::nat)"
   assumes nonzero : "D \<noteq> 0"
   assumes ha : "\<forall>x. insertion (nth_default 0 (list_update L var x)) a = (A::real)"
   assumes hb : "\<forall>x. insertion (nth_default 0 (list_update L var x)) b = (B::real)"
   assumes hd : "\<forall>x. insertion (nth_default 0 (list_update L var x)) d = (D::real)"
   shows "aEval (Eq p) (list_update L var ((A+B*C)/D)) = aEval (Eq(quadratic_part_1 var a b d (Eq p))) (list_update L var C)"
 proof - 
   define f where "f i = insertion (nth_default 0 (list_update L var C)) (isolate_variable_sparse p var i) * ((A + B * C) ^ i)" for i
   have h1 : "\<forall>i. (insertion (nth_default 0 (list_update L var C)) (isolate_variable_sparse p var i)) = (insertion (nth_default 0 (list_update L var ((A+B*C)/D))) (isolate_variable_sparse p var i))"
     by(simp add: insertion_isovarspars_free)
   have h2 : "((\<Sum>i = 0..<deg+1. f i / D ^ i) = 0) =((\<Sum>i = 0..<deg+1. (f i) * D ^ (deg - i)) = 0)"
     using normalize_summation nonzero by(auto)
   have "aEval (Eq(quadratic_part_1 var a b d (Eq p))) (list_update L var C) = 
       ((\<Sum>i = 0..<deg+1. (insertion (nth_default 0 (list_update L var C)) (isolate_variable_sparse p var i) * ((A + B * C) ^ i)) * D ^ (deg - i)) = 0)"
     by(simp add: hdeg insertion_sum insertion_add insertion_mult insertion_var insertion_pow ha hb hd lLength)
   also have "... =((\<Sum>i = 0..<deg+1. (insertion (nth_default 0 (list_update L var C)) (isolate_variable_sparse p var i) * ((A + B * C) ^ i)) / D ^ i) = 0)"
     using f_def h2 by auto
   also have "... =((\<Sum>i = 0..<deg+1. (insertion (nth_default 0 (list_update L var C)) (isolate_variable_sparse p var i) * ((A + B * C)^i / (D ^ i)))) = 0)"
     by auto
   also have "... =((\<Sum>i = 0..<deg+1. (insertion (nth_default 0 (list_update L var C)) (isolate_variable_sparse p var i) * ((A + B * C)/D) ^ i)) = 0)"
     by (metis (no_types, lifting) power_divide sum.cong)
   also have "... =((\<Sum>i = 0..<deg+1. (insertion (nth_default 0 (list_update L var ((A+B*C)/D))) (isolate_variable_sparse p var i) * ((A + B * C)/D) ^ i))=0)"
     using h1 by auto 
   also have "... = (insertion (nth_default 0 (list_update L var ((A+B*C)/D))) p =0)"
     using sum_over_degree_insertion hdeg lLength by auto
   also have "... = aEval (Eq p) (list_update L var ((A+B*C)/D))"
     using aEval.simps(1) by blast
   finally show ?thesis using assms by auto
 qed
 
 
 (*------------------------------------------------------------------------------------------------*)
 lemma quad_part_1_less :
   assumes lLength : "length L > var"
   assumes hdeg : "MPoly_Type.degree (p::real mpoly) var = (deg::nat)"
   assumes nonzero : "D \<noteq> 0"
   assumes ha : "\<forall>x. insertion (nth_default 0 (list_update L var x)) a = (A::real)"
   assumes hb : "\<forall>x. insertion (nth_default 0 (list_update L var x)) b = (B::real)"
   assumes hd : "\<forall>x. insertion (nth_default 0 (list_update L var x)) d = (D::real)"
   shows "aEval (Less p) (list_update L var ((A+B*C)/D)) = aEval (Less(quadratic_part_1 var a b d (Less p))) (list_update L var C)"
 proof - 
   define f where "f i = insertion (nth_default 0 (list_update L var C)) (isolate_variable_sparse p var i) * ((A + B * C) ^ i)" for i
   have h1a : "((\<Sum>i = 0..<deg+1. f i / D ^ i) < 0) =((\<Sum>i = 0..<deg+1. (f i) * D ^ (deg - i))  * D ^ (deg mod 2) < 0)"
     using normalize_summation_less nonzero by(auto)
   have h4a : "\<forall>i. (insertion (nth_default 0 (list_update L var C)) (isolate_variable_sparse p var i)) = (insertion (nth_default 0 (list_update L var ((A+B*C)/D))) (isolate_variable_sparse p var i))"
     by(simp add: insertion_isovarspars_free)
   have "((\<Sum>i = 0..<deg+1. (insertion (nth_default 0 (list_update L var C)) (isolate_variable_sparse p var i) * ((A + B * C) ^ i)) * D ^ (deg - i)) * D ^ (deg mod 2) < 0)
         =((\<Sum>i = 0..<deg+1. (insertion (nth_default 0 (list_update L var C)) (isolate_variable_sparse p var i) * ((A + B * C) ^ i)) / D ^ i) < 0)"
     using h1a f_def by auto
   also have "...=((\<Sum>i = 0..<deg+1. (insertion (nth_default 0 (list_update L var C)) (isolate_variable_sparse p var i) * ((A + B * C)^i / (D ^ i)))) < 0)"
     by auto
   also have "...=((\<Sum>i = 0..<deg+1. (insertion (nth_default 0 (list_update L var C)) (isolate_variable_sparse p var i) * ((A + B * C)/D) ^ i)) < 0)"
     by (metis (no_types, lifting) power_divide sum.cong)
   also have "... =((\<Sum>i = 0..<deg+1. (insertion (nth_default 0 (list_update L var ((A+B*C)/D)))  (isolate_variable_sparse p var i) * ((A + B * C)/D) ^ i))<0)"
     using h4a
     by auto 
   also have "... = (insertion (nth_default 0 (list_update L var ((A+B*C)/D))) p <0)"
     using sum_over_degree_insertion hdeg lLength by auto
   finally show ?thesis
     by(simp add: hdeg lLength insertion_add insertion_mult ha hb hd insertion_sum insertion_pow insertion_var)
 qed
 
 (*------------------------------------------------------------------------------------------------*)
 lemma quad_part_1_leq :
   assumes lLength : "length L > var"
   assumes hdeg : "MPoly_Type.degree (p::real mpoly) var = (deg::nat)"
   assumes nonzero : "D \<noteq> 0"
   assumes ha : "\<forall>x. insertion (nth_default 0 (list_update L var x)) a = (A::real)"
   assumes hb : "\<forall>x. insertion (nth_default 0 (list_update L var x)) b = (B::real)"
   assumes hd : "\<forall>x. insertion (nth_default 0 (list_update L var x)) d = (D::real)"
   shows "aEval (Leq p) (list_update L var ((A+B*C)/D)) = aEval (Leq(quadratic_part_1 var a b d (Leq p))) (list_update L var C)"
 proof - 
   define f where "f i = insertion (nth_default 0 (list_update L var C)) (isolate_variable_sparse p var i) * ((A + B * C) ^ i)" for i
   have h1a : "((\<Sum>i = 0..<deg+1. f i / D ^ i) < 0) =((\<Sum>i = 0..<deg+1. (f i) * D ^ (deg - i))  * D ^ (deg mod 2) < 0)"
     using normalize_summation_less nonzero by(auto)
   have h1b : "((\<Sum>i = 0..<deg+1. f i / D ^ i) = 0) =((\<Sum>i = 0..<deg+1. (f i) * D ^ (deg - i)) = 0)"
     using normalize_summation nonzero by(auto)
   have h1c : "((\<Sum>i = 0..<deg+1. f i / D ^ i) \<le> 0) =((\<Sum>i = 0..<deg+1. (f i) * D ^ (deg - i))  * D ^ (deg mod 2) \<le> 0)"
     using h1a h1b nonzero by auto 
   have h4a : "\<forall>i. (insertion (nth_default 0 (list_update L var C)) (isolate_variable_sparse p var i)) = (insertion (nth_default 0 (list_update L var ((A+B*C)/D))) (isolate_variable_sparse p var i))"
     by(simp add: insertion_isovarspars_free)
   have "((\<Sum>i = 0..<deg+1. (insertion (nth_default 0 (list_update L var C)) (isolate_variable_sparse p var i) * ((A + B * C) ^ i)) * D ^ (deg - i)) * D ^ (deg mod 2) \<le> 0)=
     ((\<Sum>i = 0..<deg+1. (insertion (nth_default 0 (list_update L var C)) (isolate_variable_sparse p var i) * ((A + B * C) ^ i)) / D ^ i) \<le> 0)"
     using h1c f_def by auto
   also have "...=((\<Sum>i = 0..<deg+1. (insertion (nth_default 0 (list_update L var C)) (isolate_variable_sparse p var i) * ((A + B * C)^i / (D ^ i)))) \<le> 0)"
     by auto
   also have "...=((\<Sum>i = 0..<deg+1. (insertion (nth_default 0 (list_update L var C)) (isolate_variable_sparse p var i) * ((A + B * C)/D) ^ i)) \<le> 0)"
     by (metis (no_types, lifting) power_divide sum.cong)
   also have "...=((\<Sum>i = 0..<deg+1. (insertion (nth_default 0 (list_update L var ((A+B*C)/D))) (isolate_variable_sparse p var i) * ((A + B * C)/D) ^ i))\<le>0)"
     using h4a by auto 
   also have "... = (insertion (nth_default 0 (list_update L var ((A+B*C)/D))) p\<le>0)"
     using sum_over_degree_insertion hdeg lLength by auto
   finally show ?thesis
     by(simp add: hdeg lLength insertion_add insertion_mult ha hb hd insertion_sum insertion_pow insertion_var)
 qed
 
 (*------------------------------------------------------------------------------------------------*)
 lemma quad_part_1_neq :
   assumes lLength : "length L > var"
   assumes hdeg : "MPoly_Type.degree (p::real mpoly) var = (deg::nat)"
   assumes nonzero : "D \<noteq> 0"
   assumes ha : "\<forall>x. insertion (nth_default 0 (list_update L var x)) a = (A::real)"
   assumes hb : "\<forall>x. insertion (nth_default 0 (list_update L var x)) b = (B::real)"
   assumes hd : "\<forall>x. insertion (nth_default 0 (list_update L var x)) d = (D::real)"
   shows "aEval (Neq p) (list_update L var ((A+B*C)/D)) = aEval (Neq(quadratic_part_1 var a b d (Neq p))) (list_update L var C)"
 proof -
   have "aEval (Eq(quadratic_part_1 var a b d (Eq p))) (list_update L var C) = aEval (Eq p) (list_update L var ((A+B*C)/D))"
     using quad_part_1_eq assms by blast
   then show ?thesis by auto
 qed
 
 (*------------------------------------------------------------------------------------------------*)
 
 
 lemma sqrt_case : 
   assumes detGreater0 : "SQ \<ge> 0"
   shows "((SQ^(i div 2)) * real (i mod 2) * sqrt SQ + SQ ^ (i div 2) * (1 - real (i mod 2))) = (sqrt SQ) ^ i"
 proof -
   have h1 : "i mod 2 = 0 \<or> (odd i \<and> (i mod 2 = 1))"
     by auto
   have h2 : "i mod 2 = 0 \<Longrightarrow> ((SQ^(i div 2)) * real (i mod 2) * sqrt SQ + SQ ^ (i div 2) * (1 - real (i mod 2))) = (sqrt SQ) ^ i"
     using detGreater0 apply auto
     by (simp add: real_sqrt_power_even)
   have h3 : "(odd i \<and> (i mod 2 = 1)) \<Longrightarrow> ((SQ^(i div 2)) * real (i mod 2) * sqrt SQ + SQ ^ (i div 2) * (1 - real (i mod 2))) = (sqrt SQ) ^ i"
     using detGreater0 apply auto
     by (smt One_nat_def add_Suc_right mult.commute nat_arith.rule0 odd_two_times_div_two_succ power.simps(2) power_mult real_sqrt_pow2)
   show ?thesis
     using h1 h2 h3
     by linarith 
 qed
 
 lemma sum_over_sqrt :
   assumes detGreater0 : "SQ \<ge> 0"
   shows "(\<Sum>i\<in>{0..<n+1}. ((f i::real) * (SQ^(i div 2)) * real (i mod 2) * sqrt SQ +f i * SQ ^ (i div 2) * (1 - real (i mod 2))))
         =(\<Sum>i\<in>{0..<n+1}. ((f i::real) * ((sqrt SQ)^i)))"
-  using sqrt_case detGreater0
-  by (metis (no_types, hide_lams) distrib_left mult.assoc) 
+  apply (intro sum.cong refl)
+  using sqrt_case[OF detGreater0] distrib_left mult.assoc by metis
 
 lemma quad_part_2_eq :
   assumes lLength : "length L > var"
   assumes detGreater0 : "SQ\<ge>0"
   assumes hdeg : "MPoly_Type.degree (p::real mpoly) var = (deg ::nat)"
   assumes hsq : "\<forall>x. insertion (nth_default 0 (list_update L var x)) sq = (SQ::real)"
   shows "aEval (Eq p) (list_update L var (sqrt SQ)) = aEval (Eq(quadratic_part_2 var sq p)) (list_update L var (sqrt SQ))"
 proof -
   define f where "f i = insertion (nth_default 0 (list_update L var (sqrt SQ))) (isolate_variable_sparse p var i)" for i
   have h1a : "(\<Sum>i\<in>{0..<deg+1}. (f i * (SQ^(i div 2)) * real (i mod 2) * sqrt SQ +f i * SQ ^ (i div 2) * (1 - real (i mod 2))))
              =(\<Sum>i\<in>{0..<deg+1}. (f i * ((sqrt SQ)^i)))"
     using sum_over_sqrt detGreater0 by auto
   have "(\<Sum>i\<in>{0..<deg+1}. (insertion (nth_default 0 (list_update L var (sqrt SQ))) (isolate_variable_sparse p var i) * (SQ^(i div 2)) * real (i mod 2) * sqrt SQ + (insertion (nth_default 0 (list_update L var (sqrt SQ))) (isolate_variable_sparse p var i)) * SQ ^ (i div 2) * (1 - real (i mod 2))))
              =(\<Sum>i\<in>{0..<deg+1}. (insertion (nth_default 0 (list_update L var (sqrt SQ))) (isolate_variable_sparse p var i) * ((sqrt SQ)^i)))"
     using h1a f_def by auto
   also have "... = insertion (nth_default 0 (list_update L var (sqrt SQ))) p"
     using sum_over_degree_insertion hdeg lLength by auto
   finally show ?thesis
     by(simp add:hdeg hsq insertion_add insertion_sum insertion_mult insertion_pow insertion_var insertion_const lLength)
 qed
 
 lemma quad_part_2_less :
   assumes lLength : "length L > var"
   assumes detGreater0 : "SQ\<ge>0"
   assumes hdeg : "MPoly_Type.degree (p::real mpoly) var = (deg ::nat)"
   assumes hsq : "\<forall>x. insertion (nth_default 0 (list_update L var x)) sq = (SQ::real)"
   shows "aEval (Less p) (list_update L var (sqrt SQ)) = aEval (Less(quadratic_part_2 var sq p)) (list_update L var (sqrt SQ))"
 proof -
   define f where "f i = insertion (nth_default 0 (list_update L var (sqrt SQ))) (isolate_variable_sparse p var i)" for i
   have h1a : "(\<Sum>i\<in>{0..<deg+1}. (f i * (SQ^(i div 2)) * real (i mod 2) * sqrt SQ +f i * SQ ^ (i div 2) * (1 - real (i mod 2))))
              =(\<Sum>i\<in>{0..<deg+1}. (f i * ((sqrt SQ)^i)))"
     using sum_over_sqrt detGreater0 by auto
   have "(\<Sum>i\<in>{0..<deg+1}. (insertion (nth_default 0 (list_update L var (sqrt SQ))) (isolate_variable_sparse p var i) * (SQ^(i div 2)) * real (i mod 2) * sqrt SQ + (insertion (nth_default 0 (list_update L var (sqrt SQ))) (isolate_variable_sparse p var i)) * SQ ^ (i div 2) * (1 - real (i mod 2))))
              =(\<Sum>i\<in>{0..<deg+1}. (insertion (nth_default 0 (list_update L var (sqrt SQ))) (isolate_variable_sparse p var i) * ((sqrt SQ)^i)))"
     using h1a f_def by auto
   also have "... = insertion (nth_default 0 (list_update L var (sqrt SQ))) p"
     using sum_over_degree_insertion hdeg lLength by auto
   finally show ?thesis
     by(simp add:hdeg hsq insertion_add insertion_sum insertion_mult insertion_pow insertion_var insertion_const lLength)
 qed
 
 lemma quad_part_2_neq :
   assumes lLength : "length L > var"
   assumes detGreater0 : "SQ\<ge>0"
   assumes hdeg : "MPoly_Type.degree (p::real mpoly) var = (deg ::nat)"
   assumes hsq : "\<forall>x. insertion (nth_default 0 (list_update L var x)) sq = (SQ::real)"
   shows "aEval (Neq p) (list_update L var (sqrt SQ)) = aEval (Neq(quadratic_part_2 var sq p)) (list_update L var (sqrt SQ))"
 proof -
   define f where "f i = insertion (nth_default 0 (list_update L var (sqrt SQ))) (isolate_variable_sparse p var i)" for i
   have h1a : "(\<Sum>i\<in>{0..<deg+1}. (f i * (SQ^(i div 2)) * real (i mod 2) * sqrt SQ +f i * SQ ^ (i div 2) * (1 - real (i mod 2))))
              =(\<Sum>i\<in>{0..<deg+1}. (f i * ((sqrt SQ)^i)))"
     using sum_over_sqrt detGreater0 by auto
   have "(\<Sum>i\<in>{0..<deg+1}. (insertion (nth_default 0 (list_update L var (sqrt SQ))) (isolate_variable_sparse p var i) * (SQ^(i div 2)) * real (i mod 2) * sqrt SQ + (insertion (nth_default 0 (list_update L var (sqrt SQ))) (isolate_variable_sparse p var i)) * SQ ^ (i div 2) * (1 - real (i mod 2))))
              =(\<Sum>i\<in>{0..<deg+1}. (insertion (nth_default 0 (list_update L var (sqrt SQ))) (isolate_variable_sparse p var i) * ((sqrt SQ)^i)))"
     using h1a f_def by auto
   also have "... = insertion (nth_default 0 (list_update L var (sqrt SQ))) p"
     using sum_over_degree_insertion hdeg lLength by auto
   finally show ?thesis
     by(simp add:hdeg hsq insertion_add insertion_sum insertion_mult insertion_pow insertion_var insertion_const lLength)
 qed
 
 lemma quad_part_2_leq :
   assumes lLength : "length L > var"
   assumes detGreater0 : "SQ\<ge>0"
   assumes hdeg : "MPoly_Type.degree (p::real mpoly) var = (deg ::nat)"
   assumes hsq : "\<forall>x. insertion (nth_default 0 (list_update L var x)) sq = (SQ::real)"
   shows "aEval (Leq p) (list_update L var (sqrt SQ)) = aEval (Leq(quadratic_part_2 var sq p)) (list_update L var (sqrt SQ))"
 proof -
   define f where "f i = insertion (nth_default 0 (list_update L var (sqrt SQ))) (isolate_variable_sparse p var i)" for i
   have h1a : "(\<Sum>i\<in>{0..<deg+1}. (f i * (SQ^(i div 2)) * real (i mod 2) * sqrt SQ +f i * SQ ^ (i div 2) * (1 - real (i mod 2))))
              =(\<Sum>i\<in>{0..<deg+1}. (f i * ((sqrt SQ)^i)))"
     using sum_over_sqrt detGreater0 by auto
   have "(\<Sum>i\<in>{0..<deg+1}. (insertion (nth_default 0 (list_update L var (sqrt SQ))) (isolate_variable_sparse p var i) * (SQ^(i div 2)) * real (i mod 2) * sqrt SQ + (insertion (nth_default 0 (list_update L var (sqrt SQ))) (isolate_variable_sparse p var i)) * SQ ^ (i div 2) * (1 - real (i mod 2))))
              =(\<Sum>i\<in>{0..<deg+1}. (insertion (nth_default 0 (list_update L var (sqrt SQ))) (isolate_variable_sparse p var i) * ((sqrt SQ)^i)))"
     using h1a f_def by auto
   also have "... = insertion (nth_default 0 (list_update L var (sqrt SQ))) p"
     using sum_over_degree_insertion hdeg lLength by auto
   finally show ?thesis
     by(simp add:hdeg hsq insertion_add insertion_sum insertion_mult insertion_pow insertion_var insertion_const lLength)
 qed
 
 lemma quad_part_2_deg :
   assumes sqfree : "(var::nat)\<notin>vars(sq::real mpoly)"
   shows "MPoly_Type.degree (quadratic_part_2 var sq p) var \<le> 1"
 proof -
   define deg where "deg = MPoly_Type.degree (p::real mpoly) var"
   define f where "f i = isolate_variable_sparse p var i * sq ^ (i div 2) * Const (real (i mod 2)) * Var var" for i
   define g where "g i = isolate_variable_sparse p var i * sq ^ (i div 2) * Const (1 - real (i mod 2))" for i
   have h1a : "\<forall>i. MPoly_Type.degree (isolate_variable_sparse p var i) var = 0"
     by (simp add: varNotIn_degree not_in_isovarspar) 
   have h1b : "\<forall>i. MPoly_Type.degree (sq ^ (i div 2)) var = 0"
     by (simp add: sqfree varNotIn_degree not_in_pow)
   have h1c : "\<forall>i. MPoly_Type.degree (Const (real (i mod 2))) var = 0"
     using degree_const by blast
   have h1d : "MPoly_Type.degree (Var var :: real mpoly) var = 1"
     using degree_one by auto
   have h1 : "\<forall>i<deg+1. MPoly_Type.degree (f i) var \<le> 1"
     using f_def degree_mult h1a h1b h1c h1d
     by (smt ExecutiblePolyProps.degree_one add.right_neutral mult.commute mult_eq_0_iff nat_le_linear not_one_le_zero)
   have h2a : "\<forall>i. MPoly_Type.degree (Const (1 - real (i mod 2))) var = 0"
     using degree_const by blast
   have h2 : "\<forall>i<deg+1. MPoly_Type.degree (g i) var = 0"
     using g_def degree_mult h1a h1b h2a
     by (metis (no_types, lifting) add.right_neutral mult_eq_0_iff)
   have h3 : "\<forall>i<deg+1. MPoly_Type.degree (f i + g i) var \<le> 1"
     using h1 h2 by (simp add: degree_add_leq)
   show ?thesis using atLeastLessThanSuc_atLeastAtMost degree_sum f_def g_def h3 deg_def by auto 
 qed
 
 
 
 (*------------------------------------------------------------------------------------------------*)
 
 lemma quad_equality_helper :
   assumes lLength : "length L > var"
   assumes detGreat0 : "Cv\<ge>0"
   assumes hC : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (C::real mpoly) = (Cv::real)"
   assumes hA : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (A::real mpoly) = (Av::real)"
   assumes hB : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (B::real mpoly) = (Bv::real)"
   shows "aEval (Eq (A + B * Var var)) (list_update L var (sqrt Cv)) = eval (And (Atom(Leq (A*B))) (Atom (Eq (A^2-B^2*C)))) (list_update L var (sqrt Cv))"
 proof-
   have h1 : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (A^2-(B^2)*C) = Av^2-(Bv^2)*Cv"
     by(simp add: hA hB hC insertion_add insertion_mult insertion_sub insertion_pow)
   have h2a : "(Av + Bv * sqrt Cv = 0) = (Av = - Bv * sqrt Cv)"
     by auto
   have h2b : "(Av = - Bv * sqrt Cv) \<Longrightarrow> (Av^2 = (- Bv * sqrt Cv)^2)"
     by simp
   have h2c : "(Av^2 = (- Bv * sqrt Cv)^2) = (Av^2 = Bv^2 * (sqrt Cv)^2)"
     by (simp add: power_mult_distrib)
   have h2d : "(Av^2 = Bv^2 * (sqrt Cv)^2) = (Av^2 = Bv^2 * Cv)"
     by (simp add: detGreat0)
   have h2 : "(Av + Bv * sqrt Cv = 0) \<Longrightarrow> (Av^2 = Bv^2 * Cv)"
     using h2a h2b h2c h2d by blast
   have h3a : "(Av*Bv > 0) \<Longrightarrow> (Av + Bv * sqrt Cv \<noteq> 0)"
     by (smt detGreat0 mult_nonneg_nonneg real_sqrt_ge_zero zero_less_mult_iff)
   have h3 : "(Av + Bv * sqrt Cv = 0) \<Longrightarrow> (Av*Bv\<le> 0)"
     using h3a by linarith
   have h4 : "(Av * Bv \<le> 0 \<and> Av\<^sup>2 = Bv\<^sup>2 * Cv) \<Longrightarrow> (Av + Bv * sqrt Cv = 0)"
     apply(cases "Av>0")
      apply (metis detGreat0 h2a h2c h2d mult_minus_left not_le power2_eq_iff real_sqrt_lt_0_iff zero_less_mult_iff)
     by (smt h2a real_sqrt_abs real_sqrt_mult zero_less_mult_iff)
   show ?thesis
     apply(simp add: hA hB h1 insertion_add insertion_mult insertion_var lLength)
     using h2 h3 h4 by blast
 qed
 
 lemma quadratic_sub_eq :
   assumes lLength : "length L > var"
   assumes nonzero : "Dv \<noteq> 0"
   assumes detGreater0 : "Cv \<ge> 0"
   assumes freeC : "var \<notin> vars c"
   assumes ha : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (a::real mpoly) = (Av :: real)"
   assumes hb : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (b::real mpoly) = (Bv :: real)"
   assumes hc : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (c::real mpoly) = (Cv :: real)"
   assumes hd : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (d::real mpoly) = (Dv :: real)"
   shows "aEval (Eq p) (list_update L var ((Av+Bv*sqrt(Cv))/Dv)) = eval (quadratic_sub var a b c d (Eq p)) (list_update L var (sqrt Cv))"
 proof -
   define p1 where "(p1::real mpoly) = quadratic_part_1 var a b d (Eq p)"
   define p2 where "(p2::real mpoly) = quadratic_part_2 var c p1"
   define A where "A = isolate_variable_sparse p2 var 0"
   define B where "B = isolate_variable_sparse p2 var 1"
   have h3c : "MPoly_Type.degree p2 var = 0 \<or> MPoly_Type.degree p2 var = 1"
     using freeC quad_part_2_deg p2_def by (meson le_neq_implies_less less_one)
   have h3d : "MPoly_Type.degree p2 var = 0 \<Longrightarrow> B = 0"
     by(simp add: B_def isovar_greater_degree)
   then have h3f : "MPoly_Type.degree p2 var = 0 \<Longrightarrow> p2 = A + B * Var var"
     by(simp add: h3d A_def degree0isovarspar)
   have h3g1 : "MPoly_Type.degree p2 var = 1 \<Longrightarrow> p2 = (\<Sum>i\<le>1. isolate_variable_sparse p2 var i * Var var ^ i)"
     using sum_over_zero by metis 
   have h3g2a : "\<forall>f. (\<Sum>i::nat\<le>1. f i) = f 0 + f 1" by simp
   have h3g2 : "(\<Sum>i::nat\<le>1. isolate_variable_sparse p2 var i * Var var ^ i) = 
                 isolate_variable_sparse p2 var 0 * Var var ^ 0 + isolate_variable_sparse p2 var 1 * Var var ^ 1"
     using h3g2a by blast 
   have h3g : "MPoly_Type.degree p2 var = 1 \<Longrightarrow> p2 = A + B * Var var"
     apply(simp add: sum_over_zero A_def B_def)
     using h3g1 h3g2
     by (metis (no_types, lifting) One_nat_def mult_cancel_left2 power_0 power_one_right)
   have h3h : "p2 = A + B * Var var"
     using h3c h3f h3g by auto
 
   have h4a : "\<exists>x::real. \<forall>y::real. insertion (nth_default 0 (list_update L var y)) A = x"
     using not_contains_insertion not_in_isovarspar A_def by blast
   have h4b : "\<exists>x::real. \<forall>y::real. insertion (nth_default 0 (list_update L var y)) B = x"
     using not_contains_insertion not_in_isovarspar B_def by blast
 
 
   have "aEval (Eq p) (list_update L var ((Av+Bv*sqrt(Cv))/Dv)) =  aEval (Eq p1) (list_update L var (sqrt Cv))"
     using p1_def quad_part_1_eq nonzero ha hb hd lLength by blast
   also have h2 : "... = aEval (Eq p2) (list_update L var (sqrt Cv))"
     using p2_def quad_part_2_eq lLength detGreater0 hc by metis
   also have "... = aEval (Eq (A + B * Var var)) (list_update L var (sqrt Cv))"
     using h3h by auto
   also have "... = eval (And (Atom(Leq (A*B))) (Atom (Eq (A^2-B^2*c)))) (list_update L var (sqrt Cv))"
     using quad_equality_helper hc detGreater0 h4a h4b lLength by blast
   also have "... = eval (quadratic_sub var a b c d (Eq p)) (list_update L var (sqrt Cv))"
     using p2_def A_def B_def p1_def quadratic_sub.simps(1) by metis
   finally show ?thesis by blast
 qed
   (*------------------------------------------------------------------------------------------------*)
 lemma quadratic_sub_less_helper :
   assumes lLength : "length L > var"
   assumes detGreat0 : "Cv\<ge>0"
   assumes hC : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (C::real mpoly) = (Cv::real)"
   assumes hA : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (A::real mpoly) = (Av::real)"
   assumes hB : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (B::real mpoly) = (Bv::real)"
   shows "aEval (Less (A + B * Var var)) (list_update L var (sqrt Cv)) = eval
      (Or (And (fm.Atom (Less A)) (fm.Atom (Less (B\<^sup>2 * C - A\<^sup>2))))
        (And (fm.Atom (Leq B)) (Or (fm.Atom (Less A)) (fm.Atom (Less (A\<^sup>2 - B\<^sup>2 * C))))))
      (list_update L var (sqrt Cv)) "
 proof-
   have h1 : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (A^2-(B^2)*C) = Av^2-(Bv^2)*Cv"
     by(simp add: hA hB hC insertion_add insertion_mult insertion_sub insertion_pow)
   have h2 : "\<forall>x. insertion (nth_default 0 (list_update L var x)) ((B^2)*C-A^2) = (Bv^2)*Cv-Av^2"
     by(simp add: hA hB hC insertion_add insertion_mult insertion_sub insertion_pow)
   have h3 : "Av=0 \<Longrightarrow> Bv=0 \<Longrightarrow> (Av + Bv * sqrt Cv < 0) =
     (Av < 0 \<and> Bv\<^sup>2 * Cv < Av\<^sup>2 \<or> Bv \<le> 0 \<and> (Av < 0 \<or> Av\<^sup>2 < Bv\<^sup>2 * Cv))"
     by simp
   have h4 : "Av<0 \<Longrightarrow> Bv\<le>0 \<Longrightarrow> (Av + Bv * sqrt Cv < 0) =
     (Av < 0 \<and> Bv\<^sup>2 * Cv < Av\<^sup>2 \<or> Bv \<le> 0 \<and> (Av < 0 \<or> Av\<^sup>2 < Bv\<^sup>2 * Cv))"
     by (metis add.right_neutral add_mono_thms_linordered_field(5) detGreat0 less_eq_real_def mult_less_0_iff mult_zero_class.mult_zero_left mult_zero_class.mult_zero_right real_sqrt_eq_zero_cancel_iff real_sqrt_gt_0_iff)
   have h5a : "Av\<ge>0 \<Longrightarrow> Bv\<le>0 \<Longrightarrow> (Av < -Bv * sqrt Cv) \<Longrightarrow> (Av\<^sup>2 < Bv\<^sup>2 * Cv)"
   proof -
     assume a1: "0 \<le> Av"
     assume a2: "Av < - Bv * sqrt Cv"
     assume "Bv \<le> 0"
     then have "Av < sqrt (Cv * (Bv * Bv))"
       using a2 by (simp add: mult.commute real_sqrt_mult)
     then show ?thesis
       using a1 by (metis (no_types) mult.commute power2_eq_square real_sqrt_less_iff real_sqrt_mult real_sqrt_pow2_iff)
   qed
   have h5b : "Av\<ge>0 \<Longrightarrow> Bv\<le>0 \<Longrightarrow> (Av\<^sup>2 < Bv\<^sup>2 * Cv) \<Longrightarrow> (Av < -Bv * sqrt Cv)"
     using real_less_rsqrt real_sqrt_mult by fastforce
   have h5 : "Av\<ge>0 \<Longrightarrow> Bv\<le>0 \<Longrightarrow> (Av + Bv * sqrt Cv < 0) =
     (Av < 0 \<and> Bv\<^sup>2 * Cv < Av\<^sup>2 \<or> Bv \<le> 0 \<and> (Av < 0 \<or> Av\<^sup>2 < Bv\<^sup>2 * Cv))"
     using h5a h5b by linarith
   have h6 : "Av\<ge>0 \<Longrightarrow> Bv>0 \<Longrightarrow> (Av + Bv * sqrt Cv < 0) =
     (Av < 0 \<and> Bv\<^sup>2 * Cv < Av\<^sup>2 \<or> Bv \<le> 0 \<and> (Av < 0 \<or> Av\<^sup>2 < Bv\<^sup>2 * Cv))"
     by (smt detGreat0 mult_nonneg_nonneg real_sqrt_ge_zero)
   have h7a : "Av<0 \<Longrightarrow> Bv>0 \<Longrightarrow> (Av < -Bv * sqrt Cv) \<Longrightarrow> (Bv\<^sup>2 * Cv < Av\<^sup>2)"
     by (smt mult_minus_left real_sqrt_abs real_sqrt_le_mono real_sqrt_mult)
   have h7b : "Av<0 \<Longrightarrow> Bv>0 \<Longrightarrow> (Bv\<^sup>2 * Cv < Av\<^sup>2) \<Longrightarrow> (Av < -Bv * sqrt Cv)"
     by (metis abs_of_nonneg abs_real_def add.commute less_eq_real_def mult.assoc mult_minus_left power2_eq_square real_add_less_0_iff real_sqrt_less_iff real_sqrt_mult real_sqrt_mult_self)
   have h7 : "Av<0 \<Longrightarrow> Bv>0 \<Longrightarrow> (Av + Bv * sqrt Cv < 0) =
     (Av < 0 \<and> Bv\<^sup>2 * Cv < Av\<^sup>2 \<or> Bv \<le> 0 \<and> (Av < 0 \<or> Av\<^sup>2 < Bv\<^sup>2 * Cv))"
     using h7a h7b by linarith
   show ?thesis
     apply(simp add: hA hB h1 h2 insertion_add insertion_mult insertion_var lLength)
     using h3 h4 h5 h6 h7 by smt 
 qed
 
 lemma quadratic_sub_less :
   assumes lLength : "length L > var"
   assumes nonzero : "Dv \<noteq> 0"
   assumes detGreater0 : "Cv \<ge> 0"
   assumes freeC : "var \<notin> vars c"
   assumes ha : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (a::real mpoly) = (Av :: real)"
   assumes hb : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (b::real mpoly) = (Bv :: real)"
   assumes hc : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (c::real mpoly) = (Cv :: real)"
   assumes hd : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (d::real mpoly) = (Dv :: real)"
   shows "aEval (Less p) (list_update L var ((Av+Bv*sqrt(Cv))/Dv)) = eval (quadratic_sub var a b c d (Less p)) (list_update L var (sqrt Cv))"
 proof -
   define p1 where "(p1::real mpoly) = quadratic_part_1 var a b d (Less p)"
   define p2 where "(p2::real mpoly) = quadratic_part_2 var c p1"
   define A where "A = isolate_variable_sparse p2 var 0"
   define B where "B = isolate_variable_sparse p2 var 1"
 
   have h3b : "MPoly_Type.degree p2 var \<le> 1"
     using freeC quad_part_2_deg p2_def by blast
   then have h3c : "MPoly_Type.degree p2 var = 0 \<or> MPoly_Type.degree p2 var = 1"
     by auto
   have h3d : "MPoly_Type.degree p2 var = 0 \<Longrightarrow> B = 0"
     by(simp add: B_def isovar_greater_degree)
   then have h3f : "MPoly_Type.degree p2 var = 0 \<Longrightarrow> p2 = A + B * Var var"
     by(simp add: h3d A_def degree0isovarspar)
   have h3g1 : "MPoly_Type.degree p2 var = 1 \<Longrightarrow> p2 = (\<Sum>i\<le>1. isolate_variable_sparse p2 var i * Var var ^ i)"
     using sum_over_zero by metis 
   have h3g2a : "\<forall>f. (\<Sum>i::nat\<le>1. f i) = f 0 + f 1" by simp
   have h3g2 : "(\<Sum>i::nat\<le>1. isolate_variable_sparse p2 var i * Var var ^ i) = 
                 isolate_variable_sparse p2 var 0 * Var var ^ 0 + isolate_variable_sparse p2 var 1 * Var var ^ 1"
     using h3g2a by blast 
   have h3g : "MPoly_Type.degree p2 var = 1 \<Longrightarrow> p2 = A + B * Var var"
     apply(simp add: sum_over_zero A_def B_def)
     using h3g1 h3g2
     by (metis (no_types, lifting) One_nat_def mult_cancel_left2 power_0 power_one_right)
   have h3h : "p2 = A + B * Var var"
     using h3c h3f h3g by auto
 
   have h4a : "\<exists>x::real. \<forall>y::real. insertion (nth_default 0(list_update L var y)) A = x"
     using not_contains_insertion not_in_isovarspar A_def by blast
   have h4b : "\<exists>x::real. \<forall>y::real. insertion (nth_default 0(list_update L var y)) B = x"
     using not_contains_insertion not_in_isovarspar B_def by blast
 
   have h1 : "aEval (Less p) (list_update L var ((Av+Bv*sqrt(Cv))/Dv)) = aEval (Less (quadratic_part_1 var a b d (Less p))) (list_update L var (sqrt Cv))"
     using quad_part_1_less assms by blast
   also have "... = aEval (Less p1) (list_update L var (sqrt Cv))"
     using p1_def by auto
   also have "... = aEval (Less (quadratic_part_2 var c p1)) (list_update L var (sqrt Cv))"
     using quad_part_2_less assms by blast
   also have "... = aEval (Less p2) (list_update L var (sqrt Cv))"
     using p2_def by auto
   also have "... = aEval (Less (A + B * Var var)) (list_update L var (sqrt Cv))"
     using h3h by auto
   also have "... = eval
      (Or (And (fm.Atom (Less A)) (fm.Atom (Less (B\<^sup>2 * c - A\<^sup>2))))
        (And (fm.Atom (Leq B)) (Or (fm.Atom (Less A)) (fm.Atom (Less (A\<^sup>2 - B\<^sup>2 * c))))))
      (list_update L var (sqrt Cv))"
     using quadratic_sub_less_helper hc detGreater0 h4a h4b lLength by blast
   also have  "... = eval (quadratic_sub var a b c d (Less p)) (list_update L var (sqrt Cv))"
     using p2_def A_def B_def p1_def quadratic_sub.simps(2) by metis
   finally show ?thesis by blast
 qed
 
 (*------------------------------------------------------------------------------------------------*) 
 lemma quadratic_sub_leq_helper :
   assumes lLength : "length L > var"
   assumes detGreat0 : "Cv\<ge>0"
   assumes hC : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (C::real mpoly) = (Cv::real)"
   assumes hA : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (A::real mpoly) = (Av::real)"
   assumes hB : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (B::real mpoly) = (Bv::real)"
   shows "aEval (Leq (A + B * Var var)) (list_update L var (sqrt Cv)) = 
   eval (Or(And(Atom(Leq(A)))(Atom (Leq(B^2*C-A^2))))(And (Atom(Leq B)) (Atom(Leq (A^2-B^2*C))))) (list_update L var (sqrt Cv))"
 proof-
   have h1 : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (A^2-(B^2)*C) = Av^2-(Bv^2)*Cv"
     by(simp add: hA hB hC insertion_add insertion_mult insertion_sub insertion_pow)
   have h2 : "\<forall>x. insertion (nth_default 0 (list_update L var x)) ((B^2)*C-A^2) = (Bv^2)*Cv-Av^2"
     by(simp add: hA hB hC insertion_add insertion_mult insertion_sub insertion_pow)
   have h3 : "Av=0 \<Longrightarrow> Bv=0 \<Longrightarrow> (Av + Bv * sqrt Cv \<le> 0) = (Av \<le> 0 \<and> Bv\<^sup>2 * Cv \<le> Av\<^sup>2 \<or> Bv \<le> 0 \<and> Av\<^sup>2 \<le> Bv\<^sup>2 * Cv)"
     by simp
   have h4 : "Av<0 \<Longrightarrow> Bv\<le>0 \<Longrightarrow> (Av + Bv * sqrt Cv \<le> 0) = (Av \<le> 0 \<and> Bv\<^sup>2 * Cv \<le> Av\<^sup>2 \<or> Bv \<le> 0 \<and> Av\<^sup>2 \<le> Bv\<^sup>2 * Cv)"
     by (smt detGreat0 real_sqrt_ge_zero zero_less_mult_iff)
   have h5 : "Av=0 \<Longrightarrow> Bv\<le>0 \<Longrightarrow> (Av + Bv * sqrt Cv \<le> 0) = (Av \<le> 0 \<and> Bv\<^sup>2 * Cv \<le> Av\<^sup>2 \<or> Bv \<le> 0 \<and> Av\<^sup>2 \<le> Bv\<^sup>2 * Cv)"
     by (smt detGreat0 real_sqrt_ge_zero zero_less_mult_iff)
   have h6 : "Av\<ge>0 \<Longrightarrow> Bv>0 \<Longrightarrow> (Av + Bv * sqrt Cv \<le> 0) = (Av \<le> 0 \<and> Bv\<^sup>2 * Cv \<le> Av\<^sup>2 \<or> Bv \<le> 0 \<and> Av\<^sup>2 \<le> Bv\<^sup>2 * Cv)"
     by (smt detGreat0 mult_nonneg_nonneg mult_pos_pos real_sqrt_gt_0_iff real_sqrt_zero zero_le_power2 zero_less_mult_pos zero_less_power2)
   have h7a : "Av<0 \<Longrightarrow> Bv>0 \<Longrightarrow> (Av + Bv * sqrt Cv \<le> 0) \<Longrightarrow> Bv\<^sup>2 * Cv \<le> Av\<^sup>2"
     by (smt real_sqrt_abs real_sqrt_less_mono real_sqrt_mult)
   have h7b : "Av<0 \<Longrightarrow> Bv>0 \<Longrightarrow>  Bv\<^sup>2 * Cv \<le> Av\<^sup>2 \<Longrightarrow> (Av + Bv * sqrt Cv \<le> 0) "
     by (smt real_sqrt_abs real_sqrt_less_mono real_sqrt_mult)
   have h7 : "Av<0 \<Longrightarrow> Bv>0 \<Longrightarrow> (Av + Bv * sqrt Cv \<le> 0) = (Av \<le> 0 \<and> Bv\<^sup>2 * Cv \<le> Av\<^sup>2 \<or> Bv \<le> 0 \<and> Av\<^sup>2 \<le> Bv\<^sup>2 * Cv)"
     using h7a h7b by linarith
   have h8c : "(-Bv * sqrt Cv)^2 = Bv\<^sup>2 * Cv"
     by (simp add: detGreat0 power_mult_distrib)
   have h8a : "Av>0 \<Longrightarrow> Bv\<le>0  \<Longrightarrow> (Av \<le> -Bv * sqrt Cv) \<Longrightarrow>  Av\<^sup>2 \<le> Bv\<^sup>2 * Cv"
     using detGreat0 h8c power_both_sides by smt 
   have h8b : "Av>0 \<Longrightarrow> Bv\<le>0  \<Longrightarrow>   Av\<^sup>2 \<le> Bv\<^sup>2 * Cv \<Longrightarrow> (Av + Bv * sqrt Cv \<le> 0) "
     using detGreat0 h8c power_both_sides
     by (smt mult_minus_left real_sqrt_ge_zero zero_less_mult_iff) 
   have h8 : "Av>0 \<Longrightarrow> Bv\<le>0 \<Longrightarrow> (Av + Bv * sqrt Cv \<le> 0) = (Av \<le> 0 \<and> Bv\<^sup>2 * Cv \<le> Av\<^sup>2 \<or> Bv \<le> 0 \<and> Av\<^sup>2 \<le> Bv\<^sup>2 * Cv)"
     using h8a h8b by linarith
   show ?thesis
     apply(simp add: hA hB h1 h2 insertion_add insertion_mult insertion_var lLength)
     using h3 h4 h5 h6 h7 h8 by smt
 qed
 
 lemma quadratic_sub_leq :
   assumes lLength : "length L > var"
   assumes nonzero : "Dv \<noteq> 0"
   assumes detGreater0 : "Cv \<ge> 0"
   assumes freeC : "var \<notin> vars c"
   assumes ha : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (a::real mpoly) = (Av :: real)"
   assumes hb : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (b::real mpoly) = (Bv :: real)"
   assumes hc : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (c::real mpoly) = (Cv :: real)"
   assumes hd : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (d::real mpoly) = (Dv :: real)"
   shows "aEval (Leq p) (list_update L var ((Av+Bv*sqrt(Cv))/Dv)) = eval (quadratic_sub var a b c d (Leq p)) (list_update L var (sqrt Cv))"
 proof -
   define p1 where "(p1::real mpoly) = quadratic_part_1 var a b d (Leq p)"
   define p2 where "(p2::real mpoly) = quadratic_part_2 var c p1"
   define A where "A = isolate_variable_sparse p2 var 0"
   define B where "B = isolate_variable_sparse p2 var 1"
 
   have h3b : "MPoly_Type.degree p2 var \<le> 1"
     using freeC quad_part_2_deg p2_def lLength by metis
   then have h3c : "MPoly_Type.degree p2 var = 0 \<or> MPoly_Type.degree p2 var = 1"
     by auto
   have h3d : "MPoly_Type.degree p2 var = 0 \<Longrightarrow> B = 0"
     by(simp add: B_def isovar_greater_degree)
   then have h3f : "MPoly_Type.degree p2 var = 0 \<Longrightarrow> p2 = A + B * Var var"
     by(simp add: h3d A_def degree0isovarspar)
   have h3g1 : "MPoly_Type.degree p2 var = 1 \<Longrightarrow> p2 = (\<Sum>i\<le>1. isolate_variable_sparse p2 var i * Var var ^ i)"
     using sum_over_zero by metis 
   have h3g2a : "\<forall>f. (\<Sum>i::nat\<le>1. f i) = f 0 + f 1" by simp
   have h3g2 : "(\<Sum>i::nat\<le>1. isolate_variable_sparse p2 var i * Var var ^ i) = 
                 isolate_variable_sparse p2 var 0 * Var var ^ 0 + isolate_variable_sparse p2 var 1 * Var var ^ 1"
     using h3g2a by blast 
   have h3g : "MPoly_Type.degree p2 var = 1 \<Longrightarrow> p2 = A + B * Var var"
     apply(simp add: sum_over_zero A_def B_def)
     using h3g1 h3g2
     by (metis (no_types, lifting) One_nat_def mult_cancel_left2 power_0 power_one_right)
   have h3h : "p2 = A + B * Var var"
     using h3c h3f h3g by auto
 
   have h4a : "\<exists>x::real. \<forall>y::real. insertion (nth_default 0 (list_update L var y)) A = x"
     using not_contains_insertion not_in_isovarspar A_def by blast
   have h4b : "\<exists>x::real. \<forall>y::real. insertion (nth_default 0 (list_update L var y)) B = x"
     using not_contains_insertion not_in_isovarspar B_def by blast
 
   have "aEval (Leq p) (list_update L var ((Av+Bv*sqrt(Cv))/Dv)) = aEval (Leq p1) (list_update L var (sqrt Cv))"
     using quad_part_1_leq nonzero ha hb hd p1_def lLength by metis
   also have "... = aEval (Leq p2) (list_update L var (sqrt Cv))"
     using p2_def quad_part_2_leq hc detGreater0 lLength by metis
   also have "... = aEval (Leq (A + B * Var var)) (list_update L var (sqrt Cv))"
     using h3h by auto
   also have h4 : "... = eval
      (Or
       (And
         (Atom(Leq(A)))
         (Atom (Leq(B^2*c-A^2))))
       (And
         (Atom(Leq B))
         (Atom(Leq (A^2-B^2*c)))))
      (list_update L var (sqrt Cv))"
     using quadratic_sub_leq_helper hc detGreater0 h4a h4b lLength by blast
   also have "... = eval (quadratic_sub var a b c d (Leq p)) (list_update L var (sqrt Cv))"
     using p1_def quadratic_sub.simps(3) p2_def A_def B_def by metis
   finally show ?thesis by blast
 qed
   (*------------------------------------------------------------------------------------------------*)
 lemma quadratic_sub_neq :
   assumes lLength : "length L > var"
   assumes nonzero : "Dv \<noteq> 0"
   assumes detGreater0 : "Cv \<ge> 0"
   assumes freeC : "var \<notin> vars c"
   assumes ha : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (a::real mpoly) = (Av :: real)"
   assumes hb : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (b::real mpoly) = (Bv :: real)"
   assumes hc : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (c::real mpoly) = (Cv :: real)"
   assumes hd : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (d::real mpoly) = (Dv :: real)"
   shows "aEval (Neq p) (list_update L var ((Av+Bv*sqrt(Cv))/Dv)) = eval (quadratic_sub var a b c d (Neq p)) (list_update L var (sqrt Cv))"
 proof -
   define p1 where "(p1::real mpoly) = quadratic_part_1 var a b d (Neq p)"
   define p2 where "(p2::real mpoly) = quadratic_part_2 var c p1"
   define A where "A = isolate_variable_sparse p2 var 0"
   define B where "B = isolate_variable_sparse p2 var 1"
 
   have h3b : "MPoly_Type.degree p2 var \<le> 1"
     using freeC quad_part_2_deg p2_def lLength by metis
   then have h3c : "MPoly_Type.degree p2 var = 0 \<or> MPoly_Type.degree p2 var = 1"
     by auto
   have h3d : "MPoly_Type.degree p2 var = 0 \<Longrightarrow> B = 0"
     by(simp add: B_def isovar_greater_degree)
   then have h3f : "MPoly_Type.degree p2 var = 0 \<Longrightarrow> p2 = A + B * Var var"
     by(simp add: h3d A_def degree0isovarspar)
   have h3g1 : "MPoly_Type.degree p2 var = 1 \<Longrightarrow> p2 = (\<Sum>i\<le>1. isolate_variable_sparse p2 var i * Var var ^ i)"
     using sum_over_zero by metis 
   have h3g2a : "\<forall>f. (\<Sum>i::nat\<le>1. f i) = f 0 + f 1" by simp
   have h3g2 : "(\<Sum>i::nat\<le>1. isolate_variable_sparse p2 var i * Var var ^ i) = 
                 isolate_variable_sparse p2 var 0 * Var var ^ 0 + isolate_variable_sparse p2 var 1 * Var var ^ 1"
     using h3g2a by blast 
   have h3g : "MPoly_Type.degree p2 var = 1 \<Longrightarrow> p2 = A + B * Var var"
     apply(simp add: sum_over_zero A_def B_def)
     using h3g1 h3g2
     by (metis (no_types, lifting) One_nat_def mult_cancel_left2 power_0 power_one_right)
   have h3h : "p2 = A + B * Var var"
     using h3c h3f h3g by auto
 
   have h4a : "\<exists>x::real. \<forall>y::real. insertion (nth_default 0 (list_update L var y)) A = x"
     using not_contains_insertion not_in_isovarspar A_def by blast
   have h4b : "\<exists>x::real. \<forall>y::real. insertion (nth_default 0 (list_update L var y)) B = x"
     using not_contains_insertion not_in_isovarspar B_def by blast
   have h4c : "aEval (Eq (A + B * Var var)) (list_update L var (sqrt Cv))
            = eval (And (Atom(Leq (A*B))) (Atom (Eq (A^2-B^2*c)))) (list_update L var (sqrt Cv))"
     using quad_equality_helper hc detGreater0 h4a h4b lLength by blast
   have h4d : "aEval (Neq (A + B * Var var)) (list_update L var (sqrt Cv))
            = (\<not> (eval (And (Atom(Leq (A*B))) (Atom (Eq (A^2-B^2*c)))) (list_update L var (sqrt Cv))))"
     using aEval.simps(1) aEval.simps(4) h4c by blast
   have h4e : "(\<not> (eval (And (Atom(Leq (A*B))) (Atom (Eq (A^2-B^2*c)))) (list_update L var (sqrt Cv))))
                 = eval (Or (Atom(Less(-A*B))) (Atom (Neq(A^2-B^2*c)))) (list_update L var (sqrt Cv))"
     by (metis aNeg.simps(2) aNeg.simps(3) aNeg_aEval eval.simps(1) eval.simps(4) eval.simps(5) mult_minus_left)
 
   have "aEval (Neq p) (list_update L var ((Av+Bv*sqrt(Cv))/Dv)) = aEval (Neq p1) (list_update L var (sqrt Cv))"
     using quad_part_1_neq nonzero ha hb hd p1_def lLength by blast
   also have "... = aEval (Neq p2) (list_update L var (sqrt Cv))"
     using p2_def quad_part_2_neq hc detGreater0 lLength by metis
   also have "... = aEval (Neq (A + B * Var var)) (list_update L var (sqrt Cv))"
     using h3h by auto
   also have "... = eval (Or
       (Atom(Less(-A*B)))
       (Atom (Neq(A^2-B^2*c)))) (list_update L var (sqrt Cv))"
     using h4c h4d h4e by auto
   also have "... = eval (quadratic_sub var a b c d (Neq p)) (list_update L var (sqrt Cv))"
     using p2_def A_def B_def p1_def quadratic_sub.simps(4) quadratic_part_1.simps(1) quadratic_part_1.simps(4)
     by (metis (no_types, lifting))
   finally show ?thesis by blast
 qed
   (*-----------------------------------------------------------------------------------------------*)
 theorem free_in_quad :
   assumes freeA : "var\<notin> vars a"
   assumes freeB : "var\<notin> vars b"
   assumes freeC : "var\<notin> vars c"
   assumes freeD : "var\<notin> vars d"
   shows "freeIn var (quadratic_sub var a b c d A)"
 proof(cases A)
   case (Less p)
   define p1 where "(p1::real mpoly) = quadratic_part_1 var a b d (Less p)"
   define p2 where "(p2::real mpoly) = quadratic_part_2 var c p1"
   define A where "A = isolate_variable_sparse p2 var 0"
   define B where "B = isolate_variable_sparse p2 var 1"
   have h1 : "freeIn var (quadratic_sub var a b c d (Less p)) = freeIn var (Or (And (fm.Atom (Less A)) (fm.Atom (Less (B\<^sup>2 * c - A\<^sup>2))))
        (And (fm.Atom (Leq B)) (Or (fm.Atom (Less A)) (fm.Atom (Less (A\<^sup>2 - B\<^sup>2 * c))))))"
     using p2_def A_def B_def p1_def quadratic_sub.simps(2) by metis
   have h2d : "var\<notin>vars(4::real mpoly)"
     by (metis freeB not_in_add not_in_pow numeral_Bit0 one_add_one power_0)
   have h2 : "freeIn var ((Or (And (fm.Atom (Less A)) (fm.Atom (Less (B\<^sup>2 * c - A\<^sup>2))))
        (And (fm.Atom (Leq B)) (Or (fm.Atom (Less A)) (fm.Atom (Less (A\<^sup>2 - B\<^sup>2 * c)))))))"
     using vars_mult not_in_isovarspar A_def B_def not_in_sub not_in_mult not_in_neg not_in_pow not_in_isovarspar h2d freeC
     by (simp)
   show ?thesis using h1 h2 Less by blast
 next
   case (Eq p)
   define p1 where "(p1::real mpoly) = quadratic_part_1 var a b d (Eq p)"
   define p2 where "(p2::real mpoly) = quadratic_part_2 var c p1"
   define A where "A = isolate_variable_sparse p2 var 0"
   define B where "B = isolate_variable_sparse p2 var 1"
   have h1 : "freeIn var (quadratic_sub var a b c d (Eq p)) = freeIn var (And (Atom(Leq (A*B))) (Atom (Eq (A\<^sup>2 - B\<^sup>2 * c))))"
     using p2_def A_def B_def p1_def quadratic_sub.simps(1) by metis
   have h2d : "var\<notin>vars(4::real mpoly)"
     by (metis freeB not_in_add not_in_pow numeral_Bit0 one_add_one power_0)
   have h2 : "freeIn var (And (Atom(Leq (A*B))) (Atom (Eq (A\<^sup>2 - B\<^sup>2 * c))))"
     using vars_mult not_in_isovarspar A_def B_def not_in_sub not_in_mult not_in_neg not_in_pow not_in_isovarspar h2d freeC
     by (simp)
   show ?thesis using h1 h2 Eq by blast
 next
   case (Leq p)
   define p1 where "(p1::real mpoly) = quadratic_part_1 var a b d (Leq p)"
   define p2 where "(p2::real mpoly) = quadratic_part_2 var c p1"
   define A where "A = isolate_variable_sparse p2 var 0"
   define B where "B = isolate_variable_sparse p2 var 1"
   have h1 : "freeIn var (quadratic_sub var a b c d (Leq p)) = freeIn var (Or(And(Atom(Leq(A)))(Atom (Leq(B^2*c-A^2))))(And(Atom(Leq B))(Atom(Leq (A^2-B^2*c)))))"
     using p2_def A_def B_def p1_def quadratic_sub.simps(3) by metis
   have h2d : "var\<notin>vars(4::real mpoly)"
     by (metis freeB not_in_add not_in_pow numeral_Bit0 one_add_one power_0)
   have h2 : "freeIn var (Or(And(Atom(Leq(A)))(Atom (Leq(B^2*c-A^2))))(And(Atom(Leq B))(Atom(Leq (A^2-B^2*c)))))"
     using vars_mult not_in_isovarspar A_def B_def not_in_sub not_in_mult not_in_neg not_in_pow not_in_isovarspar h2d freeC
     by (simp)
   show ?thesis using h1 h2 Leq by blast
 next
   case (Neq p)
   define p1 where "(p1::real mpoly) = quadratic_part_1 var a b d (Neq p)"
   define p2 where "(p2::real mpoly) = quadratic_part_2 var c p1"
   define A where "A = isolate_variable_sparse p2 var 0"
   define B where "B = isolate_variable_sparse p2 var 1"
   have h1 : "freeIn var (quadratic_sub var a b c d (Neq p)) = freeIn var (Or (Atom(Less(-A*B))) (Atom (Neq(A^2-B^2*c))))"
     using p2_def A_def B_def p1_def quadratic_sub.simps(4) by metis
   have h2d : "var\<notin>vars(4::real mpoly)"
     by (metis freeB not_in_add not_in_pow numeral_Bit0 one_add_one power_0)
   have h2 : "freeIn var (Or (Atom(Less(-A*B))) (Atom (Neq(A^2-B^2*c))))"
     using vars_mult not_in_isovarspar A_def B_def not_in_sub not_in_mult not_in_neg not_in_pow not_in_isovarspar h2d freeC
     by (simp)
   show ?thesis using h1 h2 Neq by blast
 qed
 
 theorem quadratic_sub :
   assumes lLength : "length L > var"
   assumes nonzero : "Dv \<noteq> 0"
   assumes detGreater0 : "Cv \<ge> 0"
   assumes freeC : "var \<notin> vars c"
   assumes ha : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (a::real mpoly) = (Av :: real)"
   assumes hb : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (b::real mpoly) = (Bv :: real)"
   assumes hc : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (c::real mpoly) = (Cv :: real)"
   assumes hd : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (d::real mpoly) = (Dv :: real)"
   shows "aEval A (list_update L var ((Av+Bv*sqrt(Cv))/Dv)) = eval (quadratic_sub var a b c d A) (list_update L var (sqrt Cv))"
 proof(cases A)
   case (Less x1)
   then show ?thesis using quadratic_sub_less assms by blast
 next
   case (Eq x2)
   then show ?thesis using quadratic_sub_eq assms by blast
 next
   case (Leq x3)
   then show ?thesis using quadratic_sub_leq assms by blast
 next
   case (Neq x4)
   then show ?thesis using quadratic_sub_neq assms by blast
 qed
 
 
 
 
 lemma free_in_quad_fm_helper :
   assumes freeA : "var\<notin> vars a"
   assumes freeB : "var\<notin> vars b"
   assumes freeC : "var\<notin> vars c"
   assumes freeD : "var\<notin> vars d"
   shows "freeIn (var+z) (quadratic_sub_fm_helper var a b c d F z)"
 proof(induction F arbitrary: z)
   case TrueF
   then show ?case by auto
 next
   case FalseF
   then show ?case by auto
 next
   case (Atom x)
   then show ?case using free_in_quad[OF not_in_lift[OF assms(1)] not_in_lift[OF assms(2)] not_in_lift[OF assms(3)] not_in_lift[OF assms(4)], of z] by auto
 next
   case (And F1 F2)
   then show ?case by auto
 next
   case (Or F1 F2)
   then show ?case by auto
 next
   case (Neg F)
   then show ?case by auto
 next
   case (ExQ F)
   show ?case using ExQ[of "z+1"] by simp
 next
   case (AllQ F)
   show ?case using AllQ[of "z+1"] by simp
 next
   case (ExN x1 F)
   then show ?case
     by (metis (no_types, lifting) add.assoc freeIn.simps(13) liftmap.simps(10) quadratic_sub_fm_helper.simps)
 next
   case (AllN x1 F)
   then show ?case
     by (metis (no_types, lifting) freeIn.simps(12) group_cancel.add1 liftmap.simps(9) quadratic_sub_fm_helper.simps)
 qed
 
 theorem free_in_quad_fm :
   assumes freeA : "var\<notin> vars a"
   assumes freeB : "var\<notin> vars b"
   assumes freeC : "var\<notin> vars c"
   assumes freeD : "var\<notin> vars d"
   shows "freeIn var (quadratic_sub_fm var a b c d A)"
   using free_in_quad_fm_helper[OF assms, of 0] by auto
 
 
 
 lemma quadratic_sub_fm_helper :
   assumes nonzero : "Dv \<noteq> 0"
   assumes detGreater0 : "Cv \<ge> 0"
   assumes freeC : "var \<notin> vars c"
   assumes lLength : "length L > var+z"
   assumes ha : "\<forall>x. insertion (nth_default 0 (list_update (drop z L) var x)) (a::real mpoly) = (Av :: real)"
   assumes hb : "\<forall>x. insertion (nth_default 0 (list_update (drop z L) var x)) (b::real mpoly) = (Bv :: real)"
   assumes hc : "\<forall>x. insertion (nth_default 0 (list_update (drop z L) var x)) (c::real mpoly) = (Cv :: real)"
   assumes hd : "\<forall>x. insertion (nth_default 0 (list_update (drop z L) var x)) (d::real mpoly) = (Dv :: real)"
   shows "eval F (list_update L (var+z) ((Av+Bv*sqrt(Cv))/Dv)) = eval (quadratic_sub_fm_helper var a b c d F z) (list_update L (var+z) (sqrt Cv))"
   using assms proof(induction F arbitrary: z L)
   case TrueF
   then show ?case by auto
 next
   case FalseF
   then show ?case by auto
 next
   case (Atom x)
   define L1 where "L1 = drop z L"
   define L2 where "L2 = take z L"
   have L_def : "L = L2 @ L1" using L1_def L2_def by auto
   have lengthl2 : "length L2 = z" using L2_def
     using Atom.prems(4) by auto
   have "eval (Atom(Eq (a-Const Av))) ([] @ L1) = eval (liftFm 0 z (Atom(Eq (a- Const Av)))) ([] @ L2 @ L1)"
     by (metis eval_liftFm_helper lengthl2 list.size(3))
   then have "(insertion (nth_default 0 (L2 @ L1)) (liftPoly 0 z (a - Const Av)) = 0)"
     apply(simp add: insertion_sub insertion_const)
     using Atom(5) unfolding L1_def
     by (metis list_update_id) 
   then have "insertion (nth_default 0 (L2 @ L1)) (liftPoly 0 z a) = Av"
     using lift_minus by blast
   then have a1 : "\<forall>x. insertion (nth_default 0 (L[var + z := x])) (liftPoly 0 z a) = Av"
-    unfolding L_def 
+    unfolding L_def
     by (metis (no_types, lifting) Atom.prems(5) L1_def add.right_neutral add_diff_cancel_right' append_eq_append_conv append_eq_append_conv2 length_append lengthl2 lift_insertion list.size(3) list_update_append not_add_less2) 
   have "eval (Atom(Eq (b-Const Bv))) ([] @ L1) = eval (liftFm 0 z (Atom(Eq (b- Const Bv)))) ([] @ L2 @ L1)"
     by (metis eval_liftFm_helper lengthl2 list.size(3))
   then have "(insertion (nth_default 0 (L2 @ L1)) (liftPoly 0 z (b - Const Bv)) = 0)"
     apply(simp add: insertion_sub insertion_const)
     using Atom(6) unfolding L1_def
     by (metis list_update_id) 
   then have "insertion (nth_default 0 (L2 @ L1)) (liftPoly 0 z b) = Bv"
     using lift_minus by blast
   then have a2 : "\<forall>x. insertion (nth_default 0 (L[var + z := x])) (liftPoly 0 z b) = Bv"
     unfolding L_def using Atom(6) L1_def
     by (metis L_def add_diff_cancel_right' append.simps(1) lengthl2 lift_insertion list.size(3) list_update_append not_add_less2)    
   have "eval (Atom(Eq (c-Const Cv))) ([] @ L1) = eval (liftFm 0 z (Atom(Eq (c- Const Cv)))) ([] @ L2 @ L1)"
     by (metis eval_liftFm_helper lengthl2 list.size(3))
   then have "(insertion (nth_default 0 (L2 @ L1)) (liftPoly 0 z (c - Const Cv)) = 0)"
     apply(simp add: insertion_sub insertion_const)
     using Atom(7) unfolding L1_def
     by (metis list_update_id) 
   then have "insertion (nth_default 0 (L2 @ L1)) (liftPoly 0 z c) = Cv"
     using lift_minus by blast
   then have a3 : "\<forall>x. insertion (nth_default 0 (L[var + z := x])) (liftPoly 0 z c) = Cv"
     unfolding L_def
   proof -
     obtain nn :: "(nat \<Rightarrow> real) \<Rightarrow> (nat \<Rightarrow> real) \<Rightarrow> real mpoly \<Rightarrow> nat" where
       "\<forall>x0 x1 x2. (\<exists>v3. v3 \<in> vars x2 \<and> x1 v3 \<noteq> x0 v3) = (nn x0 x1 x2 \<in> vars x2 \<and> x1 (nn x0 x1 x2) \<noteq> x0 (nn x0 x1 x2))"
       by moura
     then have f1: "\<forall>m f fa. nn fa f m \<in> vars m \<and> f (nn fa f m) \<noteq> fa (nn fa f m) \<or> insertion f m = insertion fa m"
       by (meson insertion_irrelevant_vars)
     obtain rr :: real where
       "(\<exists>v0. insertion (nth_default 0 ((L2 @ L1)[var + z := v0])) (liftPoly 0 z c) \<noteq> Cv) = (insertion (nth_default 0 ((L2 @ L1)[var + z := rr])) (liftPoly 0 z c) \<noteq> Cv)"
       by blast
     moreover
     { assume "var + z \<noteq> nn (nth_default 0 ((L2 @ L1)[var + z := rr])) (nth_default 0 (L2 @ L1)) (liftPoly 0 z c)"
       moreover
       { assume "(nth_default 0 (L2 @ L1) (nn (nth_default 0 ((L2 @ L1)[var + z := rr])) (nth_default 0 (L2 @ L1)) (liftPoly 0 z c)) = nth_default 0 ((L2 @ L1)[var + z := rr]) (nn (nth_default 0 ((L2 @ L1)[var + z := rr])) (nth_default 0 (L2 @ L1)) (liftPoly 0 z c))) \<noteq> ((L2 @ L1) ! nn (nth_default 0 ((L2 @ L1)[var + z := rr])) (nth_default 0 (L2 @ L1)) (liftPoly 0 z c) = (L2 @ L1)[var + z := rr] ! nn (nth_default 0 ((L2 @ L1)[var + z := rr])) (nth_default 0 (L2 @ L1)) (liftPoly 0 z c))"
         then have "nth_default 0 ((L2 @ L1)[var + z := rr]) (nn (nth_default 0 ((L2 @ L1)[var + z := rr])) (nth_default 0 (L2 @ L1)) (liftPoly 0 z c)) \<noteq> (L2 @ L1)[var + z := rr] ! nn (nth_default 0 ((L2 @ L1)[var + z := rr])) (nth_default 0 (L2 @ L1)) (liftPoly 0 z c) \<or> nth_default 0 (L2 @ L1) (nn (nth_default 0 ((L2 @ L1)[var + z := rr])) (nth_default 0 (L2 @ L1)) (liftPoly 0 z c)) \<noteq> (L2 @ L1) ! nn (nth_default 0 ((L2 @ L1)[var + z := rr])) (nth_default 0 (L2 @ L1)) (liftPoly 0 z c)"
           by linarith
         then have "nn (nth_default 0 ((L2 @ L1)[var + z := rr])) (nth_default 0 (L2 @ L1)) (liftPoly 0 z c) \<notin> vars (liftPoly 0 z c) \<or> nth_default 0 (L2 @ L1) (nn (nth_default 0 ((L2 @ L1)[var + z := rr])) (nth_default 0 (L2 @ L1)) (liftPoly 0 z c)) = nth_default 0 ((L2 @ L1)[var + z := rr]) (nn (nth_default 0 ((L2 @ L1)[var + z := rr])) (nth_default 0 (L2 @ L1)) (liftPoly 0 z c))"
           by (metis (no_types) append_Nil2 length_list_update nth_default_append) }
       ultimately have "nn (nth_default 0 ((L2 @ L1)[var + z := rr])) (nth_default 0 (L2 @ L1)) (liftPoly 0 z c) \<notin> vars (liftPoly 0 z c) \<or> nth_default 0 (L2 @ L1) (nn (nth_default 0 ((L2 @ L1)[var + z := rr])) (nth_default 0 (L2 @ L1)) (liftPoly 0 z c)) = nth_default 0 ((L2 @ L1)[var + z := rr]) (nn (nth_default 0 ((L2 @ L1)[var + z := rr])) (nth_default 0 (L2 @ L1)) (liftPoly 0 z c))"
         by force }
     ultimately show "\<forall>r. insertion (nth_default 0 ((L2 @ L1)[var + z := r])) (liftPoly 0 z c) = Cv"
       using f1 by (metis (full_types) Atom.prems(3) \<open>insertion (nth_default 0 (L2 @ L1)) (liftPoly 0 z c) = Cv\<close> not_in_lift)
   qed
   have "eval (Atom(Eq (d-Const Dv))) ([] @ L1) = eval (liftFm 0 z (Atom(Eq (d- Const Dv)))) ([] @ L2 @ L1)"
     by (metis eval_liftFm_helper lengthl2 list.size(3))
   then have "(insertion (nth_default 0 (L2 @ L1)) (liftPoly 0 z (d - Const Dv)) = 0)"
     apply(simp add: insertion_sub insertion_const)
     using Atom(8) unfolding L1_def
     by (metis list_update_id) 
   then have "insertion (nth_default 0 (L2 @ L1)) (liftPoly 0 z d) = Dv"
     using lift_minus by blast
   then have a4 : "\<forall>x. insertion (nth_default 0 (L[var + z := x])) (liftPoly 0 z d) = Dv"
     unfolding L_def
     by (metis Atom(8) L1_def L_def add_diff_cancel_right' append.simps(1) lengthl2 lift_insertion list.size(3) list_update_append not_add_less2)
   then show ?case  apply(simp)
     using quadratic_sub[OF Atom(4) Atom(1) Atom(2) not_in_lift[OF Atom(3)], of "(liftPoly 0 z a)" Av "(liftPoly 0 z b)" Bv "(liftPoly 0 z d)" x
         , OF a1 a2 a3 a4] .
 next
   case (And F1 F2)
   then show ?case by auto
 next
   case (Or F1 F2)
   then show ?case by auto
 next
   case (Neg F)
   then show ?case by auto
 next
   case (ExQ F)
   have lengthG : "var + (z + 1) < length (x#L)" for x using ExQ(5) by auto
   have forall : "\<forall>x. insertion (nth_default 0 ((drop z L)[var := x])) a = Av \<Longrightarrow> 
       \<forall>x. insertion (nth_default 0 ((drop (z + 1) (x1 # L))[var := x])) a = Av" for x1 a Av
     by auto
   have l : "x # L[var + z := v] = ((x#L)[var+(z+1):=v])" for x v
     by auto
   have "eval (ExQ F) (L[var + z := (Av + Bv * sqrt Cv) / Dv]) =
     (\<exists>x. eval F (x # L[var + z := (Av + Bv * sqrt Cv) / Dv]))"
     by(simp)
   also have "... = (\<exists>x. eval
           (liftmap
             (\<lambda>x. quadratic_sub (var + x) (liftPoly 0 x a) (liftPoly 0 x b) (liftPoly 0 x c) (liftPoly 0 x d))
             F (z + 1))
           (x # L[var + z := sqrt Cv]))"
     apply(rule ex_cong1)
     unfolding l
     using ExQ(1)[OF ExQ(2) ExQ(3) ExQ(4) lengthG forall[OF ExQ(6)] forall[OF ExQ(7)] forall[OF ExQ(8)] forall[OF ExQ(9)]]
     unfolding quadratic_sub_fm_helper.simps liftmap.simps
     by simp
   also have "... = eval (quadratic_sub_fm_helper var a b c d (ExQ F) z) (L[var + z := sqrt Cv])"
     unfolding quadratic_sub_fm_helper.simps liftmap.simps eval.simps by auto
   finally show ?case by simp
 next
   case (AllQ F)
   have lengthG : "var + (z + 1) < length (x#L)" for x using AllQ(5) by auto
   have forall : "\<forall>x. insertion (nth_default 0 ((drop z L)[var := x])) a = Av \<Longrightarrow> 
       \<forall>x. insertion (nth_default 0 ((drop (z + 1) (x1 # L))[var := x])) a = Av" for x1 a Av
     by auto
   have l : "x # L[var + z := v] = ((x#L)[var+(z+1):=v])" for x v
     by auto
   have "eval (AllQ F) (L[var + z := (Av + Bv * sqrt Cv) / Dv]) =
     (\<forall>x. eval F (x # L[var + z := (Av + Bv * sqrt Cv) / Dv]))"
     by(simp)
   also have "... = (\<forall>x. eval
           (liftmap
             (\<lambda>x. quadratic_sub (var + x) (liftPoly 0 x a) (liftPoly 0 x b) (liftPoly 0 x c) (liftPoly 0 x d))
             F (z + 1))
           (x # L[var + z := sqrt Cv]))"
     apply(rule all_cong1)
     unfolding l
     using AllQ(1)[OF AllQ(2) AllQ(3) AllQ(4) lengthG forall[OF AllQ(6)] forall[OF AllQ(7)] forall[OF AllQ(8)] forall[OF AllQ(9)]]
     unfolding quadratic_sub_fm_helper.simps liftmap.simps
     by simp
   also have "... = eval (quadratic_sub_fm_helper var a b c d (AllQ F) z) (L[var + z := sqrt Cv])"
     unfolding quadratic_sub_fm_helper.simps liftmap.simps eval.simps by auto
   finally show ?case by simp
 next
   case (ExN x1 F)
   have list : "\<And>l. length l=x1 \<Longrightarrow> ((drop (z + x1) l @ drop (z + x1 - length l) L)) = ((drop z L))"
     by auto
   have map : "\<And> z L. eval (liftmap (\<lambda>x A. (quadratic_sub (var + x) (liftPoly 0 x a) (liftPoly 0 x b) (liftPoly 0 x c) (liftPoly 0 x d) A)) F (z + x1))
       L = eval (liftmap (\<lambda>x A. (quadratic_sub (var + x1 + x) (liftPoly 0 (x+x1) a) (liftPoly 0 (x+x1) b) (liftPoly 0 (x+x1) c) (liftPoly 0 (x+x1) d) A)) F z)
       L"
     apply(induction F) apply(simp_all add:add.commute add.left_commute)
     apply force
     apply force
     by (metis (mono_tags, lifting) ab_semigroup_add_class.add_ac(1))+
   show ?case apply simp apply(rule ex_cong1)
     subgoal for l
       using map[of z] list[of l] ExN(1)[OF ExN(2-4), of "z+x1" "l@L"] ExN(5-9) list_update_append
       apply auto
       by (simp add: list_update_append) +
     done
 next
   case (AllN x1 F)
   have list : "\<And>l. length l=x1 \<Longrightarrow> ((drop (z + x1) l @ drop (z + x1 - length l) L)) = ((drop z L))"
     by auto
   have map : "\<And> z L. eval (liftmap (\<lambda>x A. (quadratic_sub (var + x) (liftPoly 0 x a) (liftPoly 0 x b) (liftPoly 0 x c) (liftPoly 0 x d) A)) F (z + x1))
       L = eval (liftmap (\<lambda>x A. (quadratic_sub (var + x1 + x) (liftPoly 0 (x+x1) a) (liftPoly 0 (x+x1) b) (liftPoly 0 (x+x1) c) (liftPoly 0 (x+x1) d) A)) F z)
       L"
     apply(induction F) apply(simp_all add:add.commute add.left_commute)
     apply force
     apply force
     by (metis (mono_tags, lifting) ab_semigroup_add_class.add_ac(1))+
   show ?case apply simp apply(rule all_cong1)
     subgoal for l
       using map[of z] list[of l] AllN(1)[OF AllN(2-4), of "z+x1" "l@L"] AllN(5-9)
       apply auto
       by (simp add: list_update_append) +
     done
 qed
 
 theorem quadratic_sub_fm :
   assumes lLength : "length L > var"
   assumes nonzero : "Dv \<noteq> 0"
   assumes detGreater0 : "Cv \<ge> 0"
   assumes freeC : "var \<notin> vars c"
   assumes ha : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (a::real mpoly) = (Av :: real)"
   assumes hb : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (b::real mpoly) = (Bv :: real)"
   assumes hc : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (c::real mpoly) = (Cv :: real)"
   assumes hd : "\<forall>x. insertion (nth_default 0 (list_update L var x)) (d::real mpoly) = (Dv :: real)"
   shows "eval F (list_update L var ((Av+Bv*sqrt(Cv))/Dv)) = eval (quadratic_sub_fm var a b c d F) (list_update L var (sqrt Cv))"
   unfolding quadratic_sub_fm.simps using quadratic_sub_fm_helper[OF assms(2) assms(3) assms(4), of 0 L a Av b Bv d F] assms(1) assms(5) assms(6) assms(7) assms(8)
   by (simp add: lLength)
 end
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