diff --git a/thys/Goedel_HFSet_Semanticless/Coding.thy b/thys/Goedel_HFSet_Semanticless/Coding.thy
--- a/thys/Goedel_HFSet_Semanticless/Coding.thy
+++ b/thys/Goedel_HFSet_Semanticless/Coding.thy
@@ -1,681 +1,681 @@
-chapter{*De Bruijn Syntax, Quotations, Codes, V-Codes*}
+chapter\<open>De Bruijn Syntax, Quotations, Codes, V-Codes\<close>
 
 theory Coding
 imports SyntaxN
 begin
 
 declare fresh_Nil [iff]
 
-section {* de Bruijn Indices (locally-nameless version) *}
+section \<open>de Bruijn Indices (locally-nameless version)\<close>
 
 nominal_datatype dbtm = DBZero | DBVar name | DBInd nat | DBEats dbtm dbtm
 
 nominal_datatype dbfm =
     DBMem dbtm dbtm
   | DBEq dbtm dbtm
   | DBDisj dbfm dbfm
   | DBNeg dbfm
   | DBEx dbfm
 
 declare dbtm.supp [simp]
 declare dbfm.supp [simp]
 
 fun lookup :: "name list \<Rightarrow> nat \<Rightarrow> name \<Rightarrow> dbtm"
   where
     "lookup [] n x = DBVar x"
   | "lookup (y # ys) n x = (if x = y then DBInd n else (lookup ys (Suc n) x))"
 
 lemma fresh_imp_notin_env: "atom name \<sharp> e \<Longrightarrow> name \<notin> set e"
   by (metis List.finite_set fresh_finite_set_at_base fresh_set)
 
 lemma lookup_notin: "x \<notin> set e \<Longrightarrow> lookup e n x = DBVar x"
   by (induct e arbitrary: n) auto
 
 lemma lookup_in:
   "x \<in> set e \<Longrightarrow> \<exists>k. lookup e n x = DBInd k \<and> n \<le> k \<and> k < n + length e"
 apply (induct e arbitrary: n)
 apply (auto intro: Suc_leD)
 apply (metis Suc_leD add_Suc_right add_Suc_shift)
 done
 
 lemma lookup_fresh: "x \<sharp> lookup e n y \<longleftrightarrow> y \<in> set e \<or> x \<noteq> atom y"
   by (induct arbitrary: n rule: lookup.induct) (auto simp: pure_fresh fresh_at_base)
 
 lemma lookup_eqvt[eqvt]: "(p \<bullet> lookup xs n x) = lookup (p \<bullet> xs) (p \<bullet> n) (p \<bullet> x)"
   by (induct xs arbitrary: n) (simp_all add: permute_pure)
 
 lemma lookup_inject [iff]: "(lookup e n x = lookup e n y) \<longleftrightarrow> x = y"
 apply (induct e n x arbitrary: y rule: lookup.induct, force, simp)
 by (metis Suc_n_not_le_n dbtm.distinct(7) dbtm.eq_iff(3) lookup_in lookup_notin)
 
 nominal_function trans_tm :: "name list \<Rightarrow> tm \<Rightarrow> dbtm"
   where
    "trans_tm e Zero = DBZero"
  | "trans_tm e (Var k) = lookup e 0 k"
  | "trans_tm e (Eats t u) = DBEats (trans_tm e t) (trans_tm e u)"
 by (auto simp: eqvt_def trans_tm_graph_aux_def) (metis tm.strong_exhaust)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma fresh_trans_tm_iff [simp]: "i \<sharp> trans_tm e t \<longleftrightarrow> i \<sharp> t \<or> i \<in> atom ` set e"
   by (induct t rule: tm.induct, auto simp: lookup_fresh fresh_at_base)
 
 lemma trans_tm_forget: "atom i \<sharp> t \<Longrightarrow> trans_tm [i] t = trans_tm [] t"
   by (induct t rule: tm.induct, auto simp: fresh_Pair)
 
 nominal_function (invariant "\<lambda>(xs, _) y. atom ` set xs \<sharp>* y")
   trans_fm :: "name list \<Rightarrow> fm \<Rightarrow> dbfm"
   where
    "trans_fm e (Mem t u) = DBMem (trans_tm e t) (trans_tm e u)"
  | "trans_fm e (Eq t u)  = DBEq (trans_tm e t) (trans_tm e u)"
  | "trans_fm e (Disj A B) = DBDisj (trans_fm e A) (trans_fm e B)"
  | "trans_fm e (Neg A)   = DBNeg (trans_fm e A)"
  | "atom k \<sharp> e \<Longrightarrow> trans_fm e (Ex k A) = DBEx (trans_fm (k#e) A)"
 apply(simp add: eqvt_def trans_fm_graph_aux_def)
 apply(erule trans_fm_graph.induct)
 using [[simproc del: alpha_lst]]
 apply(auto simp: fresh_star_def)
 apply(rule_tac y=b and c=a in fm.strong_exhaust)
 apply(auto simp: fresh_star_def)
 apply(erule_tac c=ea in Abs_lst1_fcb2')
 apply (simp_all add: eqvt_at_def)
 apply (simp_all add: fresh_star_Pair perm_supp_eq)
 apply (simp add: fresh_star_def)
 done
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma fresh_trans_fm [simp]: "i \<sharp> trans_fm e A \<longleftrightarrow> i \<sharp> A \<or> i \<in> atom ` set e"
   by (nominal_induct A avoiding: e rule: fm.strong_induct, auto simp: fresh_at_base)
 
 abbreviation DBConj :: "dbfm \<Rightarrow> dbfm \<Rightarrow> dbfm"
   where "DBConj t u \<equiv> DBNeg (DBDisj (DBNeg t) (DBNeg u))"
 
 lemma trans_fm_Conj [simp]: "trans_fm e (Conj A B) = DBConj (trans_fm e A) (trans_fm e B)"
   by (simp add: Conj_def)
 
 lemma trans_tm_inject [iff]: "(trans_tm e t = trans_tm e u) \<longleftrightarrow> t = u"
 proof (induct t arbitrary: e u rule: tm.induct)
   case Zero show ?case
     apply (cases u rule: tm.exhaust, auto)
     apply (metis dbtm.distinct(1) dbtm.distinct(3) lookup_in lookup_notin)
     done
 next
   case (Var i) show ?case
     apply (cases u rule: tm.exhaust, auto)
     apply (metis dbtm.distinct(1) dbtm.distinct(3) lookup_in lookup_notin)
     apply (metis dbtm.distinct(10) dbtm.distinct(11) lookup_in lookup_notin)
     done
 next
   case (Eats tm1 tm2) thus ?case
     apply (cases u rule: tm.exhaust, auto)
     apply (metis dbtm.distinct(12) dbtm.distinct(9) lookup_in lookup_notin)
     done
 qed
 
 lemma trans_fm_inject [iff]: "(trans_fm e A = trans_fm e B) \<longleftrightarrow> A = B"
 proof (nominal_induct A avoiding: e B rule: fm.strong_induct)
   case (Mem tm1 tm2) thus ?case
     by (rule fm.strong_exhaust [where y=B and c=e]) (auto simp: fresh_star_def)
 next
   case (Eq tm1 tm2) thus ?case
     by (rule fm.strong_exhaust [where y=B and c=e]) (auto simp: fresh_star_def)
 next
   case (Disj fm1 fm2) show ?case
     by (rule fm.strong_exhaust [where y=B and c=e]) (auto simp: Disj fresh_star_def)
 next
   case (Neg fm) show ?case
     by (rule fm.strong_exhaust [where y=B and c=e]) (auto simp: Neg fresh_star_def)
 next
   case (Ex name fm)
   thus ?case  using [[simproc del: alpha_lst]]
   proof (cases rule: fm.strong_exhaust [where y=B and c="(e, name)"], simp_all add: fresh_star_def)
     fix name'::name and fm'::fm
     assume name': "atom name' \<sharp> (e, name)"
     assume "atom name \<sharp> fm' \<or> name = name'"
     thus "(trans_fm (name # e) fm = trans_fm (name' # e) fm') = ([[atom name]]lst. fm = [[atom name']]lst. fm')"
          (is "?lhs = ?rhs")
     proof (rule disjE)
       assume "name = name'"
       thus "?lhs = ?rhs"
         by (metis fresh_Pair fresh_at_base(2) name')
     next
       assume name: "atom name \<sharp> fm'"
       have eq1: "(name \<leftrightarrow> name') \<bullet> trans_fm (name' # e) fm' = trans_fm (name' # e) fm'"
         by (simp add: flip_fresh_fresh name)
       have eq2: "(name \<leftrightarrow> name') \<bullet> ([[atom name']]lst. fm') = [[atom name']]lst. fm'"
         by (rule flip_fresh_fresh) (auto simp: Abs_fresh_iff name)
       show "?lhs = ?rhs" using name' eq1 eq2 Ex(1) Ex(3) [of "name#e" "(name \<leftrightarrow> name') \<bullet> fm'"]
         by (simp add: flip_fresh_fresh) (metis Abs1_eq(3))
     qed
   qed
 qed
 
 lemma trans_fm_perm:
   assumes c: "atom c \<sharp> (i,j,A,B)"
   and     t: "trans_fm [i] A = trans_fm [j] B"
   shows "(i \<leftrightarrow> c) \<bullet> A = (j \<leftrightarrow> c) \<bullet> B"
 proof -
   have c_fresh1: "atom c \<sharp> trans_fm [i] A"
     using c by (auto simp: supp_Pair)
   moreover
   have i_fresh: "atom i \<sharp> trans_fm [i] A"
     by auto
   moreover
   have c_fresh2: "atom c \<sharp> trans_fm [j] B"
     using c by (auto simp: supp_Pair)
   moreover
   have j_fresh: "atom j \<sharp> trans_fm [j] B"
     by auto
   ultimately have "((i \<leftrightarrow> c) \<bullet> (trans_fm [i] A)) = ((j \<leftrightarrow> c) \<bullet> trans_fm [j] B)"
     by (simp only: flip_fresh_fresh t)
   then have "trans_fm [c] ((i \<leftrightarrow> c) \<bullet> A) = trans_fm [c] ((j \<leftrightarrow> c) \<bullet> B)"
     by simp
   then show "(i \<leftrightarrow> c) \<bullet> A = (j \<leftrightarrow> c) \<bullet> B" by simp
 qed
 
-section{*Characterising the Well-Formed de Bruijn Formulas*}
+section\<open>Characterising the Well-Formed de Bruijn Formulas\<close>
 
-subsection{*Well-Formed Terms*}
+subsection\<open>Well-Formed Terms\<close>
 
 inductive wf_dbtm :: "dbtm \<Rightarrow> bool"
   where
     Zero:  "wf_dbtm DBZero"
   | Var:   "wf_dbtm (DBVar name)"
   | Eats:  "wf_dbtm t1 \<Longrightarrow> wf_dbtm t2 \<Longrightarrow> wf_dbtm (DBEats t1 t2)"
 
 equivariance wf_dbtm
 
 inductive_cases Zero_wf_dbtm [elim!]: "wf_dbtm DBZero"
 inductive_cases Var_wf_dbtm [elim!]:  "wf_dbtm (DBVar name)"
 inductive_cases Ind_wf_dbtm [elim!]:  "wf_dbtm (DBInd i)"
 inductive_cases Eats_wf_dbtm [elim!]: "wf_dbtm (DBEats t1 t2)"
 
 declare wf_dbtm.intros [intro]
 
 lemma wf_dbtm_imp_is_tm:
   assumes "wf_dbtm x"
   shows "\<exists>t::tm. x = trans_tm [] t"
 using assms
 proof (induct rule: wf_dbtm.induct)
   case Zero thus ?case
     by (metis trans_tm.simps(1))
 next
   case (Var i) thus ?case
     by (metis lookup.simps(1) trans_tm.simps(2))
 next
   case (Eats dt1 dt2) thus ?case
     by (metis trans_tm.simps(3))
 qed
 
 lemma wf_dbtm_trans_tm: "wf_dbtm (trans_tm [] t)"
   by (induct t rule: tm.induct) auto
 
 theorem wf_dbtm_iff_is_tm: "wf_dbtm x \<longleftrightarrow> (\<exists>t::tm. x = trans_tm [] t)"
   by (metis wf_dbtm_imp_is_tm wf_dbtm_trans_tm)
 
 nominal_function abst_dbtm :: "name \<Rightarrow> nat \<Rightarrow> dbtm \<Rightarrow> dbtm"
   where
    "abst_dbtm name i DBZero = DBZero"
  | "abst_dbtm name i (DBVar name') = (if name = name' then DBInd i else DBVar name')"
  | "abst_dbtm name i (DBInd j) = DBInd j"
  | "abst_dbtm name i (DBEats t1 t2) = DBEats (abst_dbtm name i t1) (abst_dbtm name i t2)"
 apply (simp add: eqvt_def abst_dbtm_graph_aux_def, auto)
 apply (metis dbtm.exhaust)
 done
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 nominal_function subst_dbtm :: "dbtm \<Rightarrow> name \<Rightarrow> dbtm \<Rightarrow> dbtm"
   where
    "subst_dbtm u i DBZero = DBZero"
  | "subst_dbtm u i (DBVar name) = (if i = name then u else DBVar name)"
  | "subst_dbtm u i (DBInd j) = DBInd j"
  | "subst_dbtm u i (DBEats t1 t2) = DBEats (subst_dbtm u i t1) (subst_dbtm u i t2)"
 by (auto simp: eqvt_def subst_dbtm_graph_aux_def) (metis dbtm.exhaust)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma fresh_iff_non_subst_dbtm: "subst_dbtm DBZero i t = t \<longleftrightarrow> atom i \<sharp> t"
   by (induct t rule: dbtm.induct) (auto simp: pure_fresh fresh_at_base(2))
 
 lemma lookup_append: "lookup (e @ [i]) n j = abst_dbtm i (length e + n) (lookup e n j)"
   by (induct e arbitrary: n) (auto simp: fresh_Cons)
 
 lemma trans_tm_abs: "trans_tm (e@[name]) t = abst_dbtm name (length e) (trans_tm e t)"
   by (induct t rule: tm.induct) (auto simp: lookup_notin lookup_append)
 
-subsection{*Well-Formed Formulas*}
+subsection\<open>Well-Formed Formulas\<close>
 
 nominal_function abst_dbfm :: "name \<Rightarrow> nat \<Rightarrow> dbfm \<Rightarrow> dbfm"
   where
    "abst_dbfm name i (DBMem t1 t2) = DBMem (abst_dbtm name i t1) (abst_dbtm name i t2)"
  | "abst_dbfm name i (DBEq t1 t2) =  DBEq (abst_dbtm name i t1) (abst_dbtm name i t2)"
  | "abst_dbfm name i (DBDisj A1 A2) = DBDisj (abst_dbfm name i A1) (abst_dbfm name i A2)"
  | "abst_dbfm name i (DBNeg A) = DBNeg (abst_dbfm name i A)"
  | "abst_dbfm name i (DBEx A) = DBEx (abst_dbfm name (i+1) A)"
 apply (simp add: eqvt_def abst_dbfm_graph_aux_def, auto)
 apply (metis dbfm.exhaust)
 done
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 nominal_function subst_dbfm :: "dbtm \<Rightarrow> name \<Rightarrow> dbfm \<Rightarrow> dbfm"
   where
    "subst_dbfm u i (DBMem t1 t2) = DBMem (subst_dbtm u i t1) (subst_dbtm u i t2)"
  | "subst_dbfm u i (DBEq t1 t2) =  DBEq (subst_dbtm u i t1) (subst_dbtm u i t2)"
  | "subst_dbfm u i (DBDisj A1 A2) = DBDisj (subst_dbfm u i A1) (subst_dbfm u i A2)"
  | "subst_dbfm u i (DBNeg A) = DBNeg (subst_dbfm u i A)"
  | "subst_dbfm u i (DBEx A) = DBEx (subst_dbfm u i A)"
 by (auto simp: eqvt_def subst_dbfm_graph_aux_def) (metis dbfm.exhaust)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma fresh_iff_non_subst_dbfm: "subst_dbfm DBZero i t = t \<longleftrightarrow> atom i \<sharp> t"
   by (induct t rule: dbfm.induct) (auto simp: fresh_iff_non_subst_dbtm)
 
 
-section{*Well formed terms and formulas (de Bruijn representation)*}
+section\<open>Well formed terms and formulas (de Bruijn representation)\<close>
 
 inductive wf_dbfm :: "dbfm \<Rightarrow> bool"
   where
     Mem:   "wf_dbtm t1 \<Longrightarrow> wf_dbtm t2 \<Longrightarrow> wf_dbfm (DBMem t1 t2)"
   | Eq:    "wf_dbtm t1 \<Longrightarrow> wf_dbtm t2 \<Longrightarrow> wf_dbfm (DBEq t1 t2)"
   | Disj:  "wf_dbfm A1 \<Longrightarrow> wf_dbfm A2 \<Longrightarrow> wf_dbfm (DBDisj A1 A2)"
   | Neg:   "wf_dbfm A \<Longrightarrow> wf_dbfm (DBNeg A)"
   | Ex:    "wf_dbfm A \<Longrightarrow> wf_dbfm (DBEx (abst_dbfm name 0 A))"
 
 equivariance wf_dbfm
 
 lemma atom_fresh_abst_dbtm [simp]: "atom i \<sharp> abst_dbtm i n t"
   by (induct t rule: dbtm.induct) (auto simp: pure_fresh)
 
 lemma atom_fresh_abst_dbfm [simp]: "atom i \<sharp> abst_dbfm i n A"
   by (nominal_induct A arbitrary: n rule: dbfm.strong_induct) auto
 
-text{*Setting up strong induction: "avoiding" for name. Necessary to allow some proofs to go through*}
+text\<open>Setting up strong induction: "avoiding" for name. Necessary to allow some proofs to go through\<close>
 nominal_inductive wf_dbfm
   avoids Ex: name
   by (auto simp: fresh_star_def)
 
 inductive_cases Mem_wf_dbfm [elim!]:  "wf_dbfm (DBMem t1 t2)"
 inductive_cases Eq_wf_dbfm [elim!]:   "wf_dbfm (DBEq t1 t2)"
 inductive_cases Disj_wf_dbfm [elim!]: "wf_dbfm (DBDisj A1 A2)"
 inductive_cases Neg_wf_dbfm [elim!]:  "wf_dbfm (DBNeg A)"
 inductive_cases Ex_wf_dbfm [elim!]:   "wf_dbfm (DBEx z)"
 
 declare wf_dbfm.intros [intro]
 
 lemma trans_fm_abs: "trans_fm (e@[name]) A = abst_dbfm name (length e) (trans_fm e A)"
   apply (nominal_induct A avoiding: name e rule: fm.strong_induct)
   apply (auto simp: trans_tm_abs fresh_Cons fresh_append)
   apply (metis One_nat_def Suc_eq_plus1 append_Cons list.size(4))
   done
 
 lemma abst_trans_fm: "abst_dbfm name 0 (trans_fm [] A) = trans_fm [name] A"
   by (metis append_Nil list.size(3) trans_fm_abs)
 
 lemma abst_trans_fm2: "i \<noteq> j \<Longrightarrow> abst_dbfm i (Suc 0) (trans_fm [j] A) = trans_fm [j,i] A"
   using trans_fm_abs [where e="[j]" and name=i]
   by auto
 
 lemma wf_dbfm_imp_is_fm:
   assumes "wf_dbfm x" shows "\<exists>A::fm. x = trans_fm [] A"
 using assms
 proof (induct rule: wf_dbfm.induct)
   case (Mem t1 t2) thus ?case
     by (metis trans_fm.simps(1) wf_dbtm_imp_is_tm)
 next
   case (Eq t1 t2) thus ?case
     by (metis trans_fm.simps(2) wf_dbtm_imp_is_tm)
 next
   case (Disj fm1 fm2) thus ?case
     by (metis trans_fm.simps(3))
 next
   case (Neg fm) thus ?case
     by (metis trans_fm.simps(4))
 next
   case (Ex fm name) thus ?case
     apply auto
     apply (rule_tac x="Ex name A" in exI)
     apply (auto simp: abst_trans_fm)
     done
 qed
 
 lemma wf_dbfm_trans_fm: "wf_dbfm (trans_fm [] A)"
   apply (nominal_induct A rule: fm.strong_induct)
   apply (auto simp: wf_dbtm_trans_tm abst_trans_fm)
   apply (metis abst_trans_fm wf_dbfm.Ex)
   done
 
 lemma wf_dbfm_iff_is_fm: "wf_dbfm x \<longleftrightarrow> (\<exists>A::fm. x = trans_fm [] A)"
   by (metis wf_dbfm_imp_is_fm wf_dbfm_trans_fm)
 
 lemma dbtm_abst_ignore [simp]:
   "abst_dbtm name i (abst_dbtm name j t) = abst_dbtm name j t"
   by (induct t rule: dbtm.induct) auto
 
 lemma abst_dbtm_fresh_ignore [simp]: "atom name \<sharp> u \<Longrightarrow> abst_dbtm name j u = u"
   by (induct u rule: dbtm.induct) auto
 
 lemma dbtm_subst_ignore [simp]:
   "subst_dbtm u name (abst_dbtm name j t) = abst_dbtm name j t"
   by (induct t rule: dbtm.induct) auto
 
 lemma dbtm_abst_swap_subst:
   "name \<noteq> name' \<Longrightarrow> atom name' \<sharp> u \<Longrightarrow>
    subst_dbtm u name (abst_dbtm name' j t) = abst_dbtm name' j (subst_dbtm u name t)"
   by (induct t rule: dbtm.induct) auto
 
 lemma dbfm_abst_swap_subst:
   "name \<noteq> name' \<Longrightarrow> atom name' \<sharp> u \<Longrightarrow>
    subst_dbfm u name (abst_dbfm name' j A) = abst_dbfm name' j (subst_dbfm u name A)"
   by (induct A arbitrary: j rule: dbfm.induct) (auto simp: dbtm_abst_swap_subst)
 
 lemma subst_trans_commute [simp]:
   "atom i \<sharp> e \<Longrightarrow> subst_dbtm (trans_tm e u) i (trans_tm e t) = trans_tm e (subst i u t)"
   apply (induct t rule: tm.induct)
   apply (auto simp: lookup_notin fresh_imp_notin_env)
   apply (metis abst_dbtm_fresh_ignore dbtm_subst_ignore lookup_fresh lookup_notin subst_dbtm.simps(2))
   done
 
 lemma subst_fm_trans_commute [simp]:
   "subst_dbfm (trans_tm [] u) name (trans_fm [] A) = trans_fm [] (A (name::= u))"
   apply (nominal_induct A avoiding: name u rule: fm.strong_induct)
   apply (auto simp: lookup_notin abst_trans_fm [symmetric])
   apply (metis dbfm_abst_swap_subst fresh_at_base(2) fresh_trans_tm_iff)
   done
 
 lemma subst_fm_trans_commute_eq:
   "du = trans_tm [] u \<Longrightarrow> subst_dbfm du i (trans_fm [] A) = trans_fm [] (A(i::=u))"
   by (metis subst_fm_trans_commute)
 
 
-section{*Quotations*}
+section\<open>Quotations\<close>
 
 fun HTuple :: "nat \<Rightarrow> tm"  where
    "HTuple 0 = HPair Zero Zero"
  | "HTuple (Suc k) = HPair Zero (HTuple k)"
 
 lemma fresh_HTuple [simp]: "x \<sharp> HTuple n"
   by (induct n) auto
 
 lemma HTuple_eqvt[eqvt]: "(p \<bullet> HTuple n) = HTuple (p \<bullet> n)"
   by (induct n, auto simp: HPair_eqvt permute_pure)
 
-subsection {*Quotations of de Bruijn terms *}
+subsection \<open>Quotations of de Bruijn terms\<close>
 
 definition nat_of_name :: "name \<Rightarrow> nat"
   where "nat_of_name x = nat_of (atom x)"
 
 lemma nat_of_name_inject [simp]: "nat_of_name n1 = nat_of_name n2 \<longleftrightarrow> n1 = n2"
   by (metis nat_of_name_def atom_components_eq_iff atom_eq_iff sort_of_atom_eq)
 
 definition name_of_nat :: "nat \<Rightarrow> name"
   where "name_of_nat n \<equiv> Abs_name (Atom (Sort ''SyntaxN.name'' []) n)"
 
 lemma nat_of_name_Abs_eq [simp]: "nat_of_name (Abs_name (Atom (Sort ''SyntaxN.name'' []) n)) = n"
   by (auto simp: nat_of_name_def atom_name_def Abs_name_inverse)
 
 lemma nat_of_name_name_eq [simp]: "nat_of_name (name_of_nat n) = n"
   by (simp add: name_of_nat_def)
 
 lemma name_of_nat_nat_of_name [simp]: "name_of_nat (nat_of_name i) = i"
   by (metis nat_of_name_inject nat_of_name_name_eq)
 
 lemma HPair_neq_ORD_OF [simp]: "HPair x y \<noteq> ORD_OF i"
   by (metis HPair_def ORD_OF.elims SUCC_def tm.distinct(3) tm.eq_iff(3))
 
-text{*Infinite support, so we cannot use nominal primrec.*}
+text\<open>Infinite support, so we cannot use nominal primrec.\<close>
 function quot_dbtm :: "dbtm \<Rightarrow> tm"
   where
    "quot_dbtm DBZero = Zero"
  | "quot_dbtm (DBVar name) = ORD_OF (Suc (nat_of_name name))"
  | "quot_dbtm (DBInd k) = HPair (HTuple 6) (ORD_OF k)"
  | "quot_dbtm (DBEats t u) = HPair (HTuple 1) (HPair (quot_dbtm t) (quot_dbtm u))"
 by (rule dbtm.exhaust) auto
 
 termination
   by lexicographic_order
 
-subsection {*Quotations of de Bruijn formulas *}
+subsection \<open>Quotations of de Bruijn formulas\<close>
 
-text{*Infinite support, so we cannot use nominal primrec.*}
+text\<open>Infinite support, so we cannot use nominal primrec.\<close>
 function quot_dbfm :: "dbfm \<Rightarrow> tm"
   where
    "quot_dbfm (DBMem t u) = HPair (HTuple 0) (HPair (quot_dbtm t) (quot_dbtm u))"
  | "quot_dbfm (DBEq t u) = HPair (HTuple 2) (HPair (quot_dbtm t) (quot_dbtm u))"
  | "quot_dbfm (DBDisj A B) = HPair (HTuple 3) (HPair (quot_dbfm A) (quot_dbfm B))"
  | "quot_dbfm (DBNeg A) = HPair (HTuple 4) (quot_dbfm A)"
  | "quot_dbfm (DBEx A) = HPair (HTuple 5) (quot_dbfm A)"
 by (rule_tac y=x in dbfm.exhaust, auto)
 
 termination
   by lexicographic_order
 
 lemma HTuple_minus_1: "n > 0 \<Longrightarrow> HTuple n = HPair Zero (HTuple (n - 1))"
   by (metis Suc_diff_1 HTuple.simps(2))
 
 lemmas HTS = HTuple_minus_1 HTuple.simps \<comment> \<open>for freeness reasoning on codes\<close>
 
 class quot =
   fixes quot :: "'a \<Rightarrow> tm"  ("\<lceil>_\<rceil>")
 
 instantiation tm :: quot
 begin
   definition quot_tm :: "tm \<Rightarrow> tm"
     where "quot_tm t = quot_dbtm (trans_tm [] t)"
 
   instance ..
 end
 
 lemma quot_dbtm_fresh [simp]: "s \<sharp> (quot_dbtm t)"
   by (induct t rule: dbtm.induct) auto
 
 lemma quot_tm_fresh [simp]: fixes t::tm shows "s \<sharp> \<lceil>t\<rceil>"
   by (simp add: quot_tm_def)
 
 lemma quot_Zero [simp]: "\<lceil>Zero\<rceil> = Zero"
   by (simp add: quot_tm_def)
 
 lemma quot_Var: "\<lceil>Var x\<rceil> = SUCC (ORD_OF (nat_of_name x))"
   by (simp add: quot_tm_def)
 
 lemma quot_Eats: "\<lceil>Eats x y\<rceil> = HPair (HTuple 1) (HPair \<lceil>x\<rceil> \<lceil>y\<rceil>)"
   by (simp add: quot_tm_def)
 
 
 instantiation fm :: quot
 begin
   definition quot_fm :: "fm \<Rightarrow> tm"
     where "quot_fm A = quot_dbfm (trans_fm [] A)"
 
   instance ..
 end
 
 lemma quot_dbfm_fresh [simp]: "s \<sharp> (quot_dbfm A)"
   by (induct A rule: dbfm.induct) auto
 
 lemma quot_fm_fresh [simp]: fixes A::fm shows "s \<sharp> \<lceil>A\<rceil>"
   by (simp add: quot_fm_def)
 
 lemma quot_fm_permute [simp]: fixes A:: fm shows "p \<bullet> \<lceil>A\<rceil> = \<lceil>A\<rceil>"
   by (metis fresh_star_def perm_supp_eq quot_fm_fresh)
 
 lemma quot_Mem: "\<lceil>x IN y\<rceil> = HPair (HTuple 0) (HPair (\<lceil>x\<rceil>) (\<lceil>y\<rceil>))"
   by (simp add: quot_fm_def quot_tm_def)
 
 lemma quot_Eq: "\<lceil>x EQ y\<rceil> = HPair (HTuple 2) (HPair (\<lceil>x\<rceil>) (\<lceil>y\<rceil>))"
   by (simp add: quot_fm_def quot_tm_def)
 
 lemma quot_Disj: "\<lceil>A OR B\<rceil> = HPair (HTuple 3) (HPair (\<lceil>A\<rceil>) (\<lceil>B\<rceil>))"
   by (simp add: quot_fm_def)
 
 lemma quot_Neg: "\<lceil>Neg A\<rceil> = HPair (HTuple 4) (\<lceil>A\<rceil>)"
   by (simp add: quot_fm_def)
 
 lemma quot_Ex: "\<lceil>Ex i A\<rceil> = HPair (HTuple 5) (quot_dbfm (trans_fm [i] A))"
   by (simp add: quot_fm_def)
 
 lemmas quot_simps = quot_Var quot_Eats quot_Eq quot_Mem quot_Disj quot_Neg quot_Ex
 
-section{*Definitions Involving Coding*}
+section\<open>Definitions Involving Coding\<close>
 
 abbreviation Q_Eats :: "tm \<Rightarrow> tm \<Rightarrow> tm"
   where "Q_Eats t u \<equiv> HPair (HTuple (Suc 0)) (HPair t u)"
 
 abbreviation Q_Succ :: "tm \<Rightarrow> tm"
   where "Q_Succ t \<equiv> Q_Eats t t"
 
 lemma quot_Succ: "\<lceil>SUCC x\<rceil> = Q_Succ \<lceil>x\<rceil>"
   by (auto simp: SUCC_def quot_Eats)
 
 abbreviation Q_HPair :: "tm \<Rightarrow> tm \<Rightarrow> tm"
   where "Q_HPair t u \<equiv>
            Q_Eats (Q_Eats Zero (Q_Eats (Q_Eats Zero u) t))
                   (Q_Eats (Q_Eats Zero t) t)"
 
 abbreviation Q_Mem :: "tm \<Rightarrow> tm \<Rightarrow> tm"
   where "Q_Mem t u \<equiv> HPair (HTuple 0) (HPair t u)"
 
 abbreviation Q_Eq :: "tm \<Rightarrow> tm \<Rightarrow> tm"
   where "Q_Eq t u \<equiv> HPair (HTuple 2) (HPair t u)"
 
 abbreviation Q_Disj :: "tm \<Rightarrow> tm \<Rightarrow> tm"
   where "Q_Disj t u \<equiv> HPair (HTuple 3) (HPair t u)"
 
 abbreviation Q_Neg :: "tm \<Rightarrow> tm"
   where "Q_Neg t \<equiv> HPair (HTuple 4) t"
 
 abbreviation Q_Conj :: "tm \<Rightarrow> tm \<Rightarrow> tm"
   where "Q_Conj t u \<equiv> Q_Neg (Q_Disj (Q_Neg t) (Q_Neg u))"
 
 abbreviation Q_Imp :: "tm \<Rightarrow> tm \<Rightarrow> tm"
   where "Q_Imp t u \<equiv> Q_Disj (Q_Neg t) u"
 
 abbreviation Q_Ex :: "tm \<Rightarrow> tm"
   where "Q_Ex t \<equiv> HPair (HTuple 5) t"
 
 abbreviation Q_All :: "tm \<Rightarrow> tm"
   where "Q_All t \<equiv> Q_Neg (Q_Ex (Q_Neg t))"
 
 lemma quot_subst_eq: "\<lceil>A(i::=t)\<rceil> = quot_dbfm (subst_dbfm (trans_tm [] t) i (trans_fm [] A))"
   by (metis quot_fm_def subst_fm_trans_commute)
 
 lemma Q_Succ_cong: "H \<turnstile> x EQ x' \<Longrightarrow> H \<turnstile> Q_Succ x EQ Q_Succ x'"
   by (metis HPair_cong Refl)
 
-subsection{*The set @{text \<Gamma>} of Definition 1.1, constant terms used for coding*}
+subsection\<open>The set \<open>\<Gamma>\<close> of Definition 1.1, constant terms used for coding\<close>
 
 inductive coding_tm :: "tm \<Rightarrow> bool"
   where
     Ord:    "\<exists>i. x = ORD_OF i \<Longrightarrow> coding_tm x"
   | HPair:  "coding_tm x \<Longrightarrow> coding_tm y \<Longrightarrow> coding_tm (HPair x y)"
 
 declare coding_tm.intros [intro]
 
 lemma coding_tm_Zero [intro]: "coding_tm Zero"
   by (metis ORD_OF.simps(1) Ord)
 
 lemma coding_tm_HTuple [intro]: "coding_tm (HTuple k)"
   by (induct k, auto)
 
 inductive_simps coding_tm_HPair [simp]: "coding_tm (HPair x y)"
 
 lemma quot_dbtm_coding [simp]: "coding_tm (quot_dbtm t)"
   apply (induct t rule: dbtm.induct, auto)
   apply (metis ORD_OF.simps(2) Ord)
   done
 
 lemma quot_dbfm_coding [simp]: "coding_tm (quot_dbfm fm)"
   by (induct fm rule: dbfm.induct, auto)
 
 lemma quot_fm_coding: fixes A::fm shows "coding_tm \<lceil>A\<rceil>"
   by (metis quot_dbfm_coding quot_fm_def)
 
 
-section {*V-Coding for terms and formulas, for the Second Theorem *}
+section \<open>V-Coding for terms and formulas, for the Second Theorem\<close>
 
-text{*Infinite support, so we cannot use nominal primrec.*}
+text\<open>Infinite support, so we cannot use nominal primrec.\<close>
 function vquot_dbtm :: "name set \<Rightarrow> dbtm \<Rightarrow> tm"
   where
    "vquot_dbtm V DBZero = Zero"
  | "vquot_dbtm V (DBVar name) = (if name \<in> V then Var name
                                  else ORD_OF (Suc (nat_of_name name)))"
  | "vquot_dbtm V (DBInd k) = HPair (HTuple 6) (ORD_OF k)"
  | "vquot_dbtm V (DBEats t u) = HPair (HTuple 1) (HPair (vquot_dbtm V t) (vquot_dbtm V u))"
 by (auto, rule_tac y=b in dbtm.exhaust, auto)
 
 termination
   by lexicographic_order
 
 lemma fresh_vquot_dbtm [simp]: "i \<sharp> vquot_dbtm V tm \<longleftrightarrow> i \<sharp> tm \<or> i \<notin> atom ` V"
   by (induct tm rule: dbtm.induct) (auto simp: fresh_at_base pure_fresh)
 
-text{*Infinite support, so we cannot use nominal primrec.*}
+text\<open>Infinite support, so we cannot use nominal primrec.\<close>
 function vquot_dbfm :: "name set \<Rightarrow> dbfm \<Rightarrow> tm"
   where
    "vquot_dbfm V (DBMem t u) = HPair (HTuple 0) (HPair (vquot_dbtm V t) (vquot_dbtm V u))"
  | "vquot_dbfm V (DBEq t u) = HPair (HTuple 2) (HPair (vquot_dbtm V t) (vquot_dbtm V u))"
  | "vquot_dbfm V (DBDisj A B) = HPair (HTuple 3) (HPair (vquot_dbfm V A) (vquot_dbfm V B))"
  | "vquot_dbfm V (DBNeg A) = HPair (HTuple 4) (vquot_dbfm V A)"
  | "vquot_dbfm V (DBEx A) = HPair (HTuple 5) (vquot_dbfm V A)"
 by (auto, rule_tac y=b in dbfm.exhaust, auto)
 
 termination
   by lexicographic_order
 
 lemma fresh_vquot_dbfm [simp]: "i \<sharp> vquot_dbfm V fm \<longleftrightarrow> i \<sharp> fm \<or> i \<notin> atom ` V"
   by (induct fm rule: dbfm.induct) (auto simp: HPair_def HTuple_minus_1)
 
 class vquot =
   fixes vquot :: "'a \<Rightarrow> name set \<Rightarrow> tm"  ("\<lfloor>_\<rfloor>_"  [0,1000]1000)
 
 instantiation tm :: vquot
 begin
   definition vquot_tm :: "tm \<Rightarrow> name set \<Rightarrow> tm"
     where "vquot_tm t V = vquot_dbtm V (trans_tm [] t)"
   instance ..
 end
 
 lemma vquot_dbtm_empty [simp]: "vquot_dbtm {} t = quot_dbtm t"
   by (induct t rule: dbtm.induct) auto
 
 lemma vquot_tm_empty [simp]: fixes t::tm shows "\<lfloor>t\<rfloor>{} = \<lceil>t\<rceil>"
   by (simp add: vquot_tm_def quot_tm_def)
 
 lemma vquot_dbtm_eq: "atom ` V \<inter> supp t = atom ` W \<inter> supp t \<Longrightarrow> vquot_dbtm V t = vquot_dbtm W t"
   by (induct t rule: dbtm.induct) (auto simp: image_iff, blast+)
 
 instantiation fm :: vquot
 begin
   definition vquot_fm :: "fm \<Rightarrow> name set \<Rightarrow> tm"
     where "vquot_fm A V = vquot_dbfm V (trans_fm [] A)"
   instance ..
 end
 
 lemma vquot_fm_fresh [simp]: fixes A::fm shows "i \<sharp> \<lfloor>A\<rfloor>V \<longleftrightarrow> i \<sharp> A \<or> i \<notin> atom ` V"
   by (simp add: vquot_fm_def)
 
 lemma vquot_dbfm_empty [simp]: "vquot_dbfm {} A = quot_dbfm A"
   by (induct A rule: dbfm.induct) auto
 
 lemma vquot_fm_empty [simp]: fixes A::fm shows "\<lfloor>A\<rfloor>{} = \<lceil>A\<rceil>"
   by (simp add: vquot_fm_def quot_fm_def)
 
 lemma vquot_dbfm_eq: "atom ` V \<inter> supp A = atom ` W \<inter> supp A \<Longrightarrow> vquot_dbfm V A = vquot_dbfm W A"
   by (induct A rule: dbfm.induct) (auto simp: intro!: vquot_dbtm_eq, blast+)
 
 lemma vquot_fm_insert:
   fixes A::fm shows "atom i \<notin> supp A \<Longrightarrow> \<lfloor>A\<rfloor>(insert i V) = \<lfloor>A\<rfloor>V"
   by (auto simp: vquot_fm_def supp_conv_fresh intro: vquot_dbfm_eq)
 
 declare HTuple.simps [simp del]
 
 end
diff --git a/thys/Goedel_HFSet_Semanticless/Coding_Predicates.thy b/thys/Goedel_HFSet_Semanticless/Coding_Predicates.thy
--- a/thys/Goedel_HFSet_Semanticless/Coding_Predicates.thy
+++ b/thys/Goedel_HFSet_Semanticless/Coding_Predicates.thy
@@ -1,852 +1,852 @@
-chapter {*Predicates for Terms, Formulas and Substitution*}
+chapter \<open>Predicates for Terms, Formulas and Substitution\<close>
 
 theory Coding_Predicates
 imports Coding Sigma
 begin
 
 declare succ_iff [simp del]
 
-text {*This material comes from Section 3, greatly modified for de Bruijn syntax.*}
+text \<open>This material comes from Section 3, greatly modified for de Bruijn syntax.\<close>
 
-section {*Predicates for atomic terms*}
+section \<open>Predicates for atomic terms\<close>
 
-subsection {*Free Variables*}
+subsection \<open>Free Variables\<close>
 
 definition VarP :: "tm \<Rightarrow> fm" where "VarP x \<equiv> OrdP x AND Zero IN x"
 
 lemma VarP_eqvt [eqvt]: "(p \<bullet> VarP x) = VarP (p \<bullet> x)"
   by (simp add: VarP_def)
 
 lemma VarP_fresh_iff [simp]: "a \<sharp> VarP x \<longleftrightarrow> a \<sharp> x"
   by (simp add: VarP_def)
 
 lemma VarP_sf [iff]: "Sigma_fm (VarP x)"
   by (auto simp: VarP_def)
 
 lemma VarP_subst [simp]: "(VarP x)(i::=t) = VarP (subst i t x) "
   by (simp add: VarP_def)
 
 lemma VarP_cong: "H \<turnstile> x EQ x' \<Longrightarrow> H \<turnstile> VarP x IFF VarP x'"
   by (rule P1_cong) auto
 
 lemma VarP_HPairE [intro!]: "insert (VarP (HPair x y)) H \<turnstile> A"
   by (auto simp: VarP_def)
 
-subsection {*De Bruijn Indexes*}
+subsection \<open>De Bruijn Indexes\<close>
 
 abbreviation Q_Ind :: "tm \<Rightarrow> tm"
   where "Q_Ind k \<equiv> HPair (HTuple 6) k"
 
 nominal_function IndP :: "tm \<Rightarrow> fm"
   where "atom m \<sharp> x \<Longrightarrow>
     IndP x = Ex m (OrdP (Var m) AND x EQ HPair (HTuple 6) (Var m))"
   by (auto simp: eqvt_def IndP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows IndP_fresh_iff [simp]: "a \<sharp> IndP x \<longleftrightarrow> a \<sharp> x"                (is ?thesis1)
     and IndP_sf [iff]:         "Sigma_fm (IndP x)"                   (is ?thsf)
     and OrdP_IndP_Q_Ind:       "{OrdP x} \<turnstile> IndP (Q_Ind x)"           (is ?thqind)
 proof -
   obtain m::name where "atom m \<sharp> x"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf ?thqind
     by (auto intro: Ex_I [where x=x])
 qed
 
 lemma IndP_Q_Ind: "H \<turnstile> OrdP x \<Longrightarrow> H \<turnstile> IndP (Q_Ind x)"
   by (rule cut1 [OF OrdP_IndP_Q_Ind])
 
 lemma subst_fm_IndP [simp]: "(IndP t)(i::=x) = IndP (subst i x t)"
 proof -
   obtain m::name where "atom m \<sharp> (i,t,x)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: IndP.simps [of m])
 qed
 
 lemma IndP_cong: "H \<turnstile> x EQ x' \<Longrightarrow> H \<turnstile> IndP x IFF IndP x'"
   by (rule P1_cong) auto
 
-subsection {*Various syntactic lemmas*}
+subsection \<open>Various syntactic lemmas\<close>
 
-section {*The predicate @{text SeqCTermP}, for Terms and Constants*}
+section \<open>The predicate \<open>SeqCTermP\<close>, for Terms and Constants\<close>
 
 (*SeqCTerm(s,k,t) \<equiv> LstSeq(s,k,t) \<and> (\<forall>l\<in>k)[s l=0 \<or> Var(s l)\<or>(\<exists>m,n\<in>l)[s l = \<langle>Eats, s m, s n\<rangle>]]*)
 nominal_function SeqCTermP :: "bool \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom l \<sharp> (s,k,sl,m,n,sm,sn);  atom sl \<sharp> (s,m,n,sm,sn);
           atom m \<sharp> (s,n,sm,sn);  atom n \<sharp> (s,sm,sn);
           atom sm \<sharp> (s,sn);  atom sn \<sharp> (s)\<rbrakk> \<Longrightarrow>
     SeqCTermP vf s k t =
       LstSeqP s k t AND
       All2 l (SUCC k) (Ex sl (HPair (Var l) (Var sl) IN s AND
                (Var sl EQ Zero OR (if vf then VarP (Var sl) else Fls) OR
                 Ex m (Ex n (Ex sm (Ex sn (Var m IN Var l AND Var n IN Var l AND
                        HPair (Var m) (Var sm) IN s AND HPair (Var n) (Var sn) IN s AND
                        Var sl EQ Q_Eats (Var sm) (Var sn))))))))"
   by (auto simp: eqvt_def SeqCTermP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows SeqCTermP_fresh_iff [simp]:
        "a \<sharp> SeqCTermP vf s k t \<longleftrightarrow> a \<sharp> s \<and> a \<sharp> k \<and> a \<sharp> t"  (is ?thesis1)
     and SeqCTermP_sf [iff]:
        "Sigma_fm (SeqCTermP vf s k t)"   (is ?thsf)
     and SeqCTermP_imp_LstSeqP:
       "{ SeqCTermP vf s k t } \<turnstile> LstSeqP s k t"  (is ?thlstseq)
     and SeqCTermP_imp_OrdP [simp]:
       "{ SeqCTermP vf s k t } \<turnstile> OrdP k"  (is ?thord)
 proof -
   obtain l::name and sl::name and m::name and n::name and sm::name and sn::name
     where atoms: "atom l \<sharp> (s,k,sl,m,n,sm,sn)"   "atom sl \<sharp> (s,m,n,sm,sn)"
         "atom m \<sharp> (s,n,sm,sn)"   "atom n \<sharp> (s,sm,sn)"
         "atom sm \<sharp> (s,sn)"   "atom sn \<sharp> (s)"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf ?thlstseq ?thord
     by (auto simp: LstSeqP.simps)
 qed
 
 lemma SeqCTermP_subst [simp]:
       "(SeqCTermP vf s k t)(j::=w) = SeqCTermP vf (subst j w s) (subst j w k) (subst j w t)"
 proof -
   obtain l::name and sl::name and m::name and n::name and sm::name and sn::name
     where "atom l \<sharp> (j,w,s,k,sl,m,n,sm,sn)"   "atom sl \<sharp> (j,w,s,m,n,sm,sn)"
           "atom m \<sharp> (j,w,s,n,sm,sn)"   "atom n \<sharp> (j,w,s,sm,sn)"
           "atom sm \<sharp> (j,w,s,sn)"   "atom sn \<sharp> (j,w,s)"
     by (metis obtain_fresh)
   thus ?thesis
     by (force simp add: SeqCTermP.simps [of l _ _ sl m n sm sn])
 qed
 
 declare SeqCTermP.simps [simp del]
 
 abbreviation SeqTermP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "SeqTermP \<equiv> SeqCTermP True"
 
 abbreviation SeqConstP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "SeqConstP \<equiv> SeqCTermP False"
 
 lemma SeqConstP_imp_SeqTermP: "{SeqConstP s k t} \<turnstile> SeqTermP s k t"
 proof -
   obtain l::name and sl::name and m::name and n::name and sm::name and sn::name
     where "atom l \<sharp> (s,k,t,sl,m,n,sm,sn)"   "atom sl \<sharp> (s,k,t,m,n,sm,sn)"
           "atom m \<sharp> (s,k,t,n,sm,sn)"   "atom n \<sharp> (s,k,t,sm,sn)"
           "atom sm \<sharp> (s,k,t,sn)"   "atom sn \<sharp> (s,k,t)"
     by (metis obtain_fresh)
   thus ?thesis
     apply (auto simp: SeqCTermP.simps [of l s k sl m n sm sn])
     apply (rule Ex_I [where x="Var l"], auto)
     apply (rule Ex_I [where x = "Var sl"], force intro: Disj_I1)
     apply (rule Ex_I [where x = "Var sl"], simp)
     apply (rule Conj_I, blast)
     apply (rule Disj_I2)+
     apply (rule Ex_I [where x = "Var m"], simp)
     apply (rule Ex_I [where x = "Var n"], simp)
     apply (rule Ex_I [where x = "Var sm"], simp)
     apply (rule Ex_I [where x = "Var sn"], auto)
     done
 qed
 
 
-section {*The predicates @{text TermP} and @{text ConstP}*}
+section \<open>The predicates \<open>TermP\<close> and \<open>ConstP\<close>\<close>
 
-subsection {*Definition*}
+subsection \<open>Definition\<close>
 
 nominal_function CTermP :: "bool \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom k \<sharp> (s,t); atom s \<sharp> t\<rbrakk> \<Longrightarrow>
     CTermP vf t = Ex s (Ex k (SeqCTermP vf (Var s) (Var k) t))"
   by (auto simp: eqvt_def CTermP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows CTermP_fresh_iff [simp]: "a \<sharp> CTermP vf t \<longleftrightarrow> a \<sharp> t"            (is ?thesis1)
     and CTermP_sf [iff]: "Sigma_fm (CTermP vf t)"                      (is ?thsf)
 proof -
   obtain k::name and s::name  where "atom k \<sharp> (s,t)" "atom s \<sharp> t"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf
     by auto
 qed
 
 lemma CTermP_subst [simp]: "(CTermP vf i)(j::=w) = CTermP vf (subst j w i)"
 proof -
   obtain k::name and s::name  where "atom k \<sharp> (s,i,j,w)" "atom s \<sharp> (i,j,w)"
     by (metis obtain_fresh)
   thus ?thesis
     by (simp add: CTermP.simps [of k s])
 qed
 
 abbreviation TermP :: "tm \<Rightarrow> fm"
   where "TermP \<equiv> CTermP True"
 
 abbreviation ConstP :: "tm \<Rightarrow> fm"
   where "ConstP \<equiv> CTermP False"
 
-subsection {*Correctness properties for constants*}
+subsection \<open>Correctness properties for constants\<close>
 
 lemma ConstP_imp_TermP: "{ConstP t} \<turnstile> TermP t"
 proof -
   obtain k::name and s::name  where "atom k \<sharp> (s,t)" "atom s \<sharp> t"
     by (metis obtain_fresh)
   thus ?thesis
     apply auto
     apply (rule Ex_I [where x = "Var s"], simp)
     apply (rule Ex_I [where x = "Var k"], auto intro: SeqConstP_imp_SeqTermP [THEN cut1])
     done
 qed
 
 
-section {*Abstraction over terms*}
+section \<open>Abstraction over terms\<close>
 
 nominal_function SeqStTermP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom l \<sharp> (s,k,v,i,sl,sl',m,n,sm,sm',sn,sn');
           atom sl \<sharp> (s,v,i,sl',m,n,sm,sm',sn,sn'); atom sl' \<sharp> (s,v,i,m,n,sm,sm',sn,sn');
           atom m \<sharp> (s,n,sm,sm',sn,sn'); atom n \<sharp> (s,sm,sm',sn,sn');
           atom sm \<sharp> (s,sm',sn,sn'); atom sm' \<sharp> (s,sn,sn');
           atom sn \<sharp> (s,sn'); atom sn' \<sharp> s\<rbrakk> \<Longrightarrow>
     SeqStTermP v i t u s k =
       VarP v AND LstSeqP s k (HPair t u) AND
       All2 l (SUCC k) (Ex sl (Ex sl' (HPair (Var l) (HPair (Var sl) (Var sl')) IN s AND
                 (((Var sl EQ v AND Var sl' EQ i) OR
                   ((IndP (Var sl) OR Var sl NEQ v) AND Var sl' EQ Var sl)) OR
                 Ex m (Ex n (Ex sm (Ex sm' (Ex sn (Ex sn' (Var m IN Var l AND Var n IN Var l AND
                        HPair (Var m) (HPair (Var sm) (Var sm')) IN s AND
                        HPair (Var n) (HPair (Var sn) (Var sn')) IN s AND
                        Var sl EQ Q_Eats (Var sm) (Var sn) AND
                        Var sl' EQ Q_Eats (Var sm') (Var sn')))))))))))"
   apply (simp_all add: eqvt_def SeqStTermP_graph_aux_def flip_fresh_fresh)
   by auto (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows SeqStTermP_fresh_iff [simp]:
       "a \<sharp> SeqStTermP v i t u s k \<longleftrightarrow> a \<sharp> v \<and> a \<sharp> i \<and> a \<sharp> t \<and> a \<sharp> u \<and> a \<sharp> s \<and> a \<sharp> k"  (is ?thesis1)
     and SeqStTermP_sf [iff]:
       "Sigma_fm (SeqStTermP v i t u s k)"  (is ?thsf)
     and SeqStTermP_imp_OrdP:
       "{ SeqStTermP v i t u s k } \<turnstile> OrdP k"  (is ?thord)
     and SeqStTermP_imp_VarP:
       "{ SeqStTermP v i t u s k } \<turnstile> VarP v"  (is ?thvar)
     and SeqStTermP_imp_LstSeqP:
       "{ SeqStTermP v i t u s k } \<turnstile> LstSeqP s k (HPair t u)"  (is ?thlstseq)
 proof -
   obtain l::name and sl::name and sl'::name and m::name and n::name and
          sm::name and sm'::name and sn::name and sn'::name
     where atoms:
        "atom l \<sharp> (s,k,v,i,sl,sl',m,n,sm,sm',sn,sn')"
        "atom sl \<sharp> (s,v,i,sl',m,n,sm,sm',sn,sn')" "atom sl' \<sharp> (s,v,i,m,n,sm,sm',sn,sn')"
        "atom m \<sharp> (s,n,sm,sm',sn,sn')" "atom n \<sharp> (s,sm,sm',sn,sn')"
        "atom sm \<sharp> (s,sm',sn,sn')" "atom sm' \<sharp> (s,sn,sn')"
        "atom sn \<sharp> (s,sn')" "atom sn' \<sharp> (s)"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf ?thord ?thvar ?thlstseq
     by (auto intro: LstSeqP_OrdP)
 qed
 
 lemma SeqStTermP_subst [simp]:
       "(SeqStTermP v i t u s k)(j::=w) =
        SeqStTermP (subst j w v) (subst j w i) (subst j w t) (subst j w u) (subst j w s) (subst j w k)"
 proof -
   obtain l::name and sl::name and sl'::name and m::name and n::name and
          sm::name and sm'::name and sn::name and sn'::name
     where "atom l \<sharp> (s,k,v,i,w,j,sl,sl',m,n,sm,sm',sn,sn')"
          "atom sl \<sharp> (s,v,i,w,j,sl',m,n,sm,sm',sn,sn')"
          "atom sl' \<sharp> (s,v,i,w,j,m,n,sm,sm',sn,sn')"
          "atom m \<sharp> (s,w,j,n,sm,sm',sn,sn')" "atom n \<sharp> (s,w,j,sm,sm',sn,sn')"
          "atom sm \<sharp> (s,w,j,sm',sn,sn')" "atom sm' \<sharp> (s,w,j,sn,sn')"
          "atom sn \<sharp> (s,w,j,sn')" "atom sn' \<sharp> (s,w,j)"
     by (metis obtain_fresh)
   thus ?thesis
     by (force simp add: SeqStTermP.simps [of l _ _ _ _ sl sl' m n sm sm' sn sn'])
 qed
 
 lemma SeqStTermP_cong:
   "\<lbrakk>H \<turnstile> t EQ t'; H \<turnstile> u EQ u'; H \<turnstile> s EQ s'; H \<turnstile> k EQ k'\<rbrakk>
    \<Longrightarrow> H \<turnstile> SeqStTermP v i t u s k IFF SeqStTermP v i t' u' s' k'"
    by (rule P4_cong [where tms="[v,i]"]) (auto simp: fresh_Cons)
 
 declare SeqStTermP.simps [simp del]
 
-subsection {*Defining the syntax: main predicate*}
+subsection \<open>Defining the syntax: main predicate\<close>
 
 nominal_function AbstTermP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom s \<sharp> (v,i,t,u,k); atom k \<sharp> (v,i,t,u)\<rbrakk> \<Longrightarrow>
     AbstTermP v i t u =
      OrdP i AND Ex s (Ex k (SeqStTermP v (Q_Ind i) t u (Var s) (Var k)))"
   by (auto simp: eqvt_def AbstTermP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows AbstTermP_fresh_iff [simp]:
       "a \<sharp> AbstTermP v i t u \<longleftrightarrow> a \<sharp> v \<and> a \<sharp> i \<and> a \<sharp> t \<and> a \<sharp> u"  (is ?thesis1)
     and AbstTermP_sf [iff]:
       "Sigma_fm (AbstTermP v i t u)"  (is ?thsf)
     and AbstTermP_imp_VarP:
        "{ AbstTermP v i t u } \<turnstile> VarP v"   (is ?thvar)
     and AbstTermP_imp_OrdP:
        "{ AbstTermP v i t u } \<turnstile> OrdP i"   (is ?thord)
 proof -
   obtain s::name and k::name where "atom s \<sharp> (v,i,t,u,k)"  "atom k \<sharp> (v,i,t,u)"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf ?thvar ?thord
     by (auto intro: SeqStTermP_imp_VarP thin2)
 qed
 
 lemma AbstTermP_subst [simp]:
       "(AbstTermP v i t u)(j::=w) = AbstTermP (subst j w v) (subst j w i) (subst j w t) (subst j w u)"
 proof -
   obtain s::name and k::name  where "atom s \<sharp> (v,i,t,u,w,j,k)"  "atom k \<sharp> (v,i,t,u,w,j)"
     by (metis obtain_fresh)
   thus ?thesis
     by (simp add: AbstTermP.simps [of s _ _ _ _ k])
 qed
 
 declare AbstTermP.simps [simp del]
 
-section {*Substitution over terms*}
+section \<open>Substitution over terms\<close>
 
-subsection {*Defining the syntax*}
+subsection \<open>Defining the syntax\<close>
 
 nominal_function SubstTermP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom s \<sharp> (v,i,t,u,k); atom k \<sharp> (v,i,t,u)\<rbrakk> \<Longrightarrow>
     SubstTermP v i t u = TermP i AND Ex s (Ex k (SeqStTermP v i t u (Var s) (Var k)))"
   by (auto simp: eqvt_def SubstTermP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows SubstTermP_fresh_iff [simp]:
        "a \<sharp> SubstTermP v i t u \<longleftrightarrow> a \<sharp> v \<and> a \<sharp> i \<and> a \<sharp> t \<and> a \<sharp> u"  (is ?thesis1)
     and SubstTermP_sf [iff]:
        "Sigma_fm (SubstTermP v i t u)"     (is ?thsf)
     and SubstTermP_imp_TermP:
        "{ SubstTermP v i t u } \<turnstile> TermP i"  (is ?thterm)
     and SubstTermP_imp_VarP:
        "{ SubstTermP v i t u } \<turnstile> VarP v"   (is ?thvar)
 proof -
   obtain s::name and k::name  where "atom s \<sharp> (v,i,t,u,k)" "atom k \<sharp> (v,i,t,u)"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf ?thterm ?thvar
     by (auto intro: SeqStTermP_imp_VarP thin2)
 qed
 
 lemma SubstTermP_subst [simp]:
       "(SubstTermP v i t u)(j::=w) = SubstTermP (subst j w v) (subst j w i) (subst j w t) (subst j w u)"
 proof -
   obtain s::name and k::name
     where "atom s \<sharp> (v,i,t,u,w,j,k)"  "atom k \<sharp> (v,i,t,u,w,j)"
     by (metis obtain_fresh)
   thus ?thesis
     by (simp add: SubstTermP.simps [of s _ _ _ _ k])
 qed
 
 lemma SubstTermP_cong:
   "\<lbrakk>H \<turnstile> v EQ v'; H \<turnstile> i EQ i'; H \<turnstile> t EQ t'; H \<turnstile> u EQ u'\<rbrakk>
    \<Longrightarrow> H \<turnstile> SubstTermP v i t u IFF SubstTermP v' i' t' u'"
   by (rule P4_cong) auto
 
 declare SubstTermP.simps [simp del]
 
-section {*Abstraction over formulas*}
+section \<open>Abstraction over formulas\<close>
 
-subsection {*The predicate @{text AbstAtomicP}*}
+subsection \<open>The predicate \<open>AbstAtomicP\<close>\<close>
 
 nominal_function AbstAtomicP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom t \<sharp> (v,i,y,y',t',u,u'); atom t' \<sharp> (v,i,y,y',u,u');
           atom u \<sharp> (v,i,y,y',u'); atom u' \<sharp> (v,i,y,y')\<rbrakk> \<Longrightarrow>
     AbstAtomicP v i y y' =
          Ex t (Ex u (Ex t' (Ex u'
            (AbstTermP v i (Var t) (Var t') AND AbstTermP v i (Var u) (Var u') AND
                       ((y EQ Q_Eq (Var t) (Var u) AND y' EQ Q_Eq (Var t') (Var u')) OR
                        (y EQ Q_Mem (Var t) (Var u) AND y' EQ Q_Mem (Var t') (Var u')))))))"
   by (auto simp: eqvt_def AbstAtomicP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows AbstAtomicP_fresh_iff [simp]:
        "a \<sharp> AbstAtomicP v i y y' \<longleftrightarrow> a \<sharp> v \<and> a \<sharp> i \<and> a \<sharp> y \<and> a \<sharp> y'"         (is ?thesis1)
     and AbstAtomicP_sf [iff]: "Sigma_fm (AbstAtomicP v i y y')"              (is ?thsf)
 proof -
   obtain t::name and u::name and t'::name  and u'::name
     where "atom t \<sharp> (v,i,y,y',t',u,u')" "atom t' \<sharp> (v,i,y,y',u,u')"
           "atom u \<sharp> (v,i,y,y',u')" "atom u' \<sharp> (v,i,y,y')"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf
     by auto
 qed
 
 lemma AbstAtomicP_subst [simp]:
       "(AbstAtomicP v tm y y')(i::=w) = AbstAtomicP (subst i w v) (subst i w tm) (subst i w y) (subst i w y')"
 proof -
   obtain t::name and u::name and t'::name  and u'::name
     where "atom t \<sharp> (v,tm,y,y',w,i,t',u,u')"  "atom t' \<sharp> (v,tm,y,y',w,i,u,u')"
           "atom u \<sharp> (v,tm,y,y',w,i,u')"       "atom u' \<sharp> (v,tm,y,y',w,i)"
     by (metis obtain_fresh)
   thus ?thesis
     by (simp add: AbstAtomicP.simps [of t _ _ _ _ t' u u'])
 qed
 
 declare AbstAtomicP.simps [simp del]
 
-subsection {*The predicate @{text AbsMakeForm}*}
+subsection \<open>The predicate \<open>AbsMakeForm\<close>\<close>
 
 nominal_function SeqAbstFormP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom l \<sharp> (s,k,v,sli,sl,sl',m,n,smi,sm,sm',sni,sn,sn');
           atom sli \<sharp> (s,v,sl,sl',m,n,smi,sm,sm',sni,sn,sn');
           atom sl \<sharp> (s,v,sl',m,n,smi,sm,sm',sni,sn,sn');
           atom sl' \<sharp> (s,v,m,n,smi,sm,sm',sni,sn,sn');
           atom m \<sharp> (s,n,smi,sm,sm',sni,sn,sn');
           atom n \<sharp> (s,smi,sm,sm',sni,sn,sn'); atom smi \<sharp> (s,sm,sm',sni,sn,sn');
           atom sm \<sharp> (s,sm',sni,sn,sn'); atom sm' \<sharp> (s,sni,sn,sn');
           atom sni \<sharp> (s,sn,sn'); atom sn \<sharp> (s,sn'); atom sn' \<sharp> (s)\<rbrakk> \<Longrightarrow>
     SeqAbstFormP v i x x' s k =
       LstSeqP s k (HPair i (HPair x x')) AND
       All2 l (SUCC k) (Ex sli (Ex sl (Ex sl' (HPair (Var l) (HPair (Var sli) (HPair (Var sl) (Var sl'))) IN s AND
                 (AbstAtomicP v (Var sli) (Var sl) (Var sl') OR
                 OrdP (Var sli) AND
                 Ex m (Ex n (Ex smi (Ex sm (Ex sm' (Ex sni (Ex sn (Ex sn'
                       (Var m IN Var l AND Var n IN Var l AND
                        HPair (Var m) (HPair (Var smi) (HPair (Var sm) (Var sm'))) IN s AND
                        HPair (Var n) (HPair (Var sni) (HPair (Var sn) (Var sn'))) IN s AND
                        ((Var sli EQ Var smi AND Var sli EQ Var sni AND
                          Var sl EQ Q_Disj (Var sm) (Var sn) AND
                          Var sl' EQ Q_Disj (Var sm') (Var sn')) OR
                         (Var sli EQ Var smi AND
                          Var sl EQ Q_Neg (Var sm) AND Var sl' EQ Q_Neg (Var sm')) OR
                         (SUCC (Var sli) EQ Var smi AND
                          Var sl EQ Q_Ex (Var sm) AND Var sl' EQ Q_Ex (Var sm'))))))))))))))))"
   by (auto simp: eqvt_def SeqAbstFormP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows SeqAbstFormP_fresh_iff [simp]:
        "a \<sharp> SeqAbstFormP v i x x' s k \<longleftrightarrow> a \<sharp> v \<and> a \<sharp> i \<and> a \<sharp> x \<and> a \<sharp> x' \<and> a \<sharp> s \<and> a \<sharp> k"  (is ?thesis1)
     and SeqAbstFormP_sf [iff]:
        "Sigma_fm (SeqAbstFormP v i x x' s k)"  (is ?thsf)
     and SeqAbstFormP_imp_OrdP:
        "{ SeqAbstFormP v u x x' s k } \<turnstile> OrdP k"  (is ?thOrd)
     and SeqAbstFormP_imp_LstSeqP:
        "{ SeqAbstFormP v u x x' s k } \<turnstile> LstSeqP s k (HPair u (HPair x x'))"  (is ?thLstSeq)
 proof -
   obtain l::name and sli::name and sl::name and sl'::name and m::name and n::name and
          smi::name and sm::name and sm'::name and sni::name and sn::name and sn'::name
     where atoms:
          "atom l \<sharp> (s,k,v,sli,sl,sl',m,n,smi,sm,sm',sni,sn,sn')"
          "atom sli \<sharp> (s,v,sl,sl',m,n,smi,sm,sm',sni,sn,sn')"
          "atom sl \<sharp> (s,v,sl',m,n,smi,sm,sm',sni,sn,sn')"
          "atom sl' \<sharp> (s,v,m,n,smi,sm,sm',sni,sn,sn')"
          "atom m \<sharp> (s,n,smi,sm,sm',sni,sn,sn')" "atom n \<sharp> (s,smi,sm,sm',sni,sn,sn')"
          "atom smi \<sharp> (s,sm,sm',sni,sn,sn')"
          "atom sm \<sharp> (s,sm',sni,sn,sn')"
          "atom sm' \<sharp> (s,sni,sn,sn')"
          "atom sni \<sharp> (s,sn,sn')" "atom sn \<sharp> (s,sn')" "atom sn' \<sharp> s"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf ?thOrd ?thLstSeq
     by (auto intro: LstSeqP_OrdP)
 qed
 
 lemma SeqAbstFormP_subst [simp]:
       "(SeqAbstFormP v u x x' s k)(i::=t) =
        SeqAbstFormP (subst i t v) (subst i t u) (subst i t x) (subst i t x') (subst i t s) (subst i t k)"
 proof -
   obtain l::name and sli::name and sl::name and sl'::name and m::name and n::name and
          smi::name and sm::name and sm'::name and sni::name and sn::name and sn'::name
    where "atom l \<sharp> (i,t,s,k,v,sli,sl,sl',m,n,smi,sm,sm',sni,sn,sn')"
          "atom sli \<sharp> (i,t,s,v,sl,sl',m,n,smi,sm,sm',sni,sn,sn')"
          "atom sl \<sharp> (i,t,s,v,sl',m,n,smi,sm,sm',sni,sn,sn')"
          "atom sl' \<sharp> (i,t,s,v,m,n,smi,sm,sm',sni,sn,sn')"
          "atom m \<sharp> (i,t,s,n,smi,sm,sm',sni,sn,sn')"
          "atom n \<sharp> (i,t,s,smi,sm,sm',sni,sn,sn')"
          "atom smi \<sharp> (i,t,s,sm,sm',sni,sn,sn')"
          "atom sm \<sharp> (i,t,s,sm',sni,sn,sn')" "atom sm' \<sharp> (i,t,s,sni,sn,sn')"
          "atom sni \<sharp> (i,t,s,sn,sn')" "atom sn \<sharp> (i,t,s,sn')" "atom sn' \<sharp> (i,t,s)"
     by (metis obtain_fresh)
   thus ?thesis
     by (force simp add: SeqAbstFormP.simps [of l _ _ _ sli sl sl' m n smi sm sm' sni sn sn'])
 qed
 
 declare SeqAbstFormP.simps [simp del]
 
-subsection {*Defining the syntax: the main AbstForm predicate*}
+subsection \<open>Defining the syntax: the main AbstForm predicate\<close>
 
 nominal_function AbstFormP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom s \<sharp> (v,i,x,x',k);
           atom k \<sharp> (v,i,x,x')\<rbrakk> \<Longrightarrow>
     AbstFormP v i x x' = VarP v AND OrdP i AND Ex s (Ex k (SeqAbstFormP v i x x' (Var s) (Var k)))"
   by (auto simp: eqvt_def AbstFormP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows AbstFormP_fresh_iff [simp]:
        "a \<sharp> AbstFormP v i x x' \<longleftrightarrow> a \<sharp> v \<and> a \<sharp> i \<and> a \<sharp> x \<and> a \<sharp> x'" (is ?thesis1)
     and AbstFormP_sf [iff]:
        "Sigma_fm (AbstFormP v i x x')"    (is ?thsf)
 proof -
   obtain s::name and k::name  where "atom s \<sharp> (v,i,x,x',k)" "atom k \<sharp> (v,i,x,x')"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf
     by auto
 qed
 
 lemma AbstFormP_subst [simp]:
      "(AbstFormP v i x x')(j::=t) = AbstFormP (subst j t v) (subst j t i) (subst j t x) (subst j t x')"
 proof -
   obtain s::name and k::name  where "atom s \<sharp> (v,i,x,x',t,j,k)" "atom k \<sharp> (v,i,x,x',t,j)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: AbstFormP.simps [of s _ _ _ _ k])
 qed
 
 declare AbstFormP.simps [simp del]
 
-section {*Substitution over formulas*}
+section \<open>Substitution over formulas\<close>
 
-subsection {*The predicate @{text SubstAtomicP}*}
+subsection \<open>The predicate \<open>SubstAtomicP\<close>\<close>
 
 nominal_function SubstAtomicP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom t \<sharp> (v,tm,y,y',t',u,u');
           atom t' \<sharp> (v,tm,y,y',u,u');
           atom u \<sharp> (v,tm,y,y',u');
           atom u' \<sharp> (v,tm,y,y')\<rbrakk> \<Longrightarrow>
     SubstAtomicP v tm y y' =
          Ex t (Ex u (Ex t' (Ex u'
            (SubstTermP v tm (Var t) (Var t') AND SubstTermP v tm (Var u) (Var u') AND
                       ((y EQ Q_Eq (Var t) (Var u) AND y' EQ Q_Eq (Var t') (Var u')) OR
                        (y EQ Q_Mem (Var t) (Var u) AND y' EQ Q_Mem (Var t') (Var u')))))))"
 by (auto simp: eqvt_def SubstAtomicP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows SubstAtomicP_fresh_iff [simp]:
        "a \<sharp> SubstAtomicP v tm y y' \<longleftrightarrow> a \<sharp> v \<and> a \<sharp> tm \<and> a \<sharp> y \<and> a \<sharp> y'"           (is ?thesis1)
     and SubstAtomicP_sf [iff]: "Sigma_fm (SubstAtomicP v tm y y')"               (is ?thsf)
 proof -
   obtain t::name and u::name and t'::name  and u'::name
     where "atom t \<sharp> (v,tm,y,y',t',u,u')" "atom t' \<sharp> (v,tm,y,y',u,u')"
           "atom u \<sharp> (v,tm,y,y',u')" "atom u' \<sharp> (v,tm,y,y')"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf
     by auto
 qed
 
 lemma SubstAtomicP_subst [simp]:
   "(SubstAtomicP v tm y y')(i::=w) = SubstAtomicP (subst i w v) (subst i w tm) (subst i w y) (subst i w y')"
 proof -
   obtain t::name and u::name and t'::name  and u'::name
     where "atom t \<sharp> (v,tm,y,y',w,i,t',u,u')" "atom t' \<sharp> (v,tm,y,y',w,i,u,u')"
           "atom u \<sharp> (v,tm,y,y',w,i,u')" "atom u' \<sharp> (v,tm,y,y',w,i)"
     by (metis obtain_fresh)
   thus ?thesis
     by (simp add: SubstAtomicP.simps [of t _ _ _ _ t' u u'])
 qed
 
 lemma SubstAtomicP_cong:
   "\<lbrakk>H \<turnstile> v EQ v'; H \<turnstile> tm EQ tm'; H \<turnstile> x EQ x'; H \<turnstile> y EQ y'\<rbrakk>
    \<Longrightarrow> H \<turnstile> SubstAtomicP v tm x y IFF SubstAtomicP v' tm' x' y'"
   by (rule P4_cong) auto
 
 
-subsection {*The predicate @{text SubstMakeForm}*}
+subsection \<open>The predicate \<open>SubstMakeForm\<close>\<close>
 
 nominal_function SeqSubstFormP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom l \<sharp> (s,k,v,u,sl,sl',m,n,sm,sm',sn,sn');
           atom sl \<sharp> (s,v,u,sl',m,n,sm,sm',sn,sn');
           atom sl' \<sharp> (s,v,u,m,n,sm,sm',sn,sn');
           atom m \<sharp> (s,n,sm,sm',sn,sn'); atom n \<sharp> (s,sm,sm',sn,sn');
           atom sm \<sharp> (s,sm',sn,sn'); atom sm' \<sharp> (s,sn,sn');
           atom sn \<sharp> (s,sn'); atom sn' \<sharp> s\<rbrakk> \<Longrightarrow>
     SeqSubstFormP v u x x' s k =
       LstSeqP s k (HPair x x') AND
       All2 l (SUCC k) (Ex sl (Ex sl' (HPair (Var l) (HPair (Var sl) (Var sl')) IN s AND
                 (SubstAtomicP v u (Var sl) (Var sl') OR
                 Ex m (Ex n (Ex sm (Ex sm' (Ex sn (Ex sn' (Var m IN Var l AND Var n IN Var l AND
                        HPair (Var m) (HPair (Var sm) (Var sm')) IN s AND
                        HPair (Var n) (HPair (Var sn) (Var sn')) IN s AND
                        ((Var sl EQ Q_Disj (Var sm) (Var sn) AND
                         Var sl' EQ Q_Disj (Var sm') (Var sn')) OR
                         (Var sl EQ Q_Neg (Var sm) AND Var sl' EQ Q_Neg (Var sm')) OR
                         (Var sl EQ Q_Ex (Var sm) AND Var sl' EQ Q_Ex (Var sm')))))))))))))"
   apply (simp_all add: eqvt_def SeqSubstFormP_graph_aux_def flip_fresh_fresh)
   by auto (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows SeqSubstFormP_fresh_iff [simp]:
        "a \<sharp> SeqSubstFormP v u x x' s k \<longleftrightarrow> a \<sharp> v \<and> a \<sharp> u \<and> a \<sharp> x \<and> a \<sharp> x' \<and> a \<sharp> s \<and> a \<sharp> k"  (is ?thesis1)
     and SeqSubstFormP_sf [iff]:
        "Sigma_fm (SeqSubstFormP v u x x' s k)"  (is ?thsf)
     and SeqSubstFormP_imp_OrdP:
        "{ SeqSubstFormP v u x x' s k } \<turnstile> OrdP k"  (is ?thOrd)
     and SeqSubstFormP_imp_LstSeqP:
        "{ SeqSubstFormP v u x x' s k } \<turnstile> LstSeqP s k (HPair x x')"  (is ?thLstSeq)
 proof -
   obtain l::name and sl::name and sl'::name and m::name and n::name and
          sm::name and sm'::name and sn::name and sn'::name
     where atoms:
          "atom l \<sharp> (s,k,v,u,sl,sl',m,n,sm,sm',sn,sn')"
          "atom sl \<sharp> (s,v,u,sl',m,n,sm,sm',sn,sn')"
          "atom sl' \<sharp> (s,v,u,m,n,sm,sm',sn,sn')"
          "atom m \<sharp> (s,n,sm,sm',sn,sn')" "atom n \<sharp> (s,sm,sm',sn,sn')"
          "atom sm \<sharp> (s,sm',sn,sn')" "atom sm' \<sharp> (s,sn,sn')"
          "atom sn \<sharp> (s,sn')" "atom sn' \<sharp> (s)"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf ?thOrd ?thLstSeq
     by (auto intro: LstSeqP_OrdP)
 qed
 
 lemma SeqSubstFormP_subst [simp]:
       "(SeqSubstFormP v u x x' s k)(i::=t) =
        SeqSubstFormP (subst i t v) (subst i t u) (subst i t x) (subst i t x') (subst i t s) (subst i t k)"
 proof -
   obtain l::name and sl::name and sl'::name and m::name and n::name and
          sm::name and sm'::name and sn::name and sn'::name
    where "atom l \<sharp> (s,k,v,u,t,i,sl,sl',m,n,sm,sm',sn,sn')"
          "atom sl \<sharp> (s,v,u,t,i,sl',m,n,sm,sm',sn,sn')"
          "atom sl' \<sharp> (s,v,u,t,i,m,n,sm,sm',sn,sn')"
          "atom m \<sharp> (s,t,i,n,sm,sm',sn,sn')" "atom n \<sharp> (s,t,i,sm,sm',sn,sn')"
          "atom sm \<sharp> (s,t,i,sm',sn,sn')" "atom sm' \<sharp> (s,t,i,sn,sn')"
          "atom sn \<sharp> (s,t,i,sn')" "atom sn' \<sharp> (s,t,i)"
     by (metis obtain_fresh)
   thus ?thesis
     by (force simp add: SeqSubstFormP.simps [of l _ _ _ _ sl sl' m n sm sm' sn sn'])
 qed
 
 lemma SeqSubstFormP_cong:
   "\<lbrakk>H \<turnstile> t EQ t'; H \<turnstile> u EQ u'; H \<turnstile> s EQ s'; H \<turnstile> k EQ k'\<rbrakk>
    \<Longrightarrow> H \<turnstile> SeqSubstFormP v i t u s k IFF SeqSubstFormP v i t' u' s' k'"
    by (rule P4_cong [where tms="[v,i]"]) (auto simp: fresh_Cons)
 
 declare SeqSubstFormP.simps [simp del]
 
-subsection {*Defining the syntax: the main SubstForm predicate*}
+subsection \<open>Defining the syntax: the main SubstForm predicate\<close>
 
 nominal_function SubstFormP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom s \<sharp> (v,i,x,x',k); atom k \<sharp> (v,i,x,x')\<rbrakk> \<Longrightarrow>
     SubstFormP v i x x' =
       VarP v AND TermP i AND Ex s (Ex k (SeqSubstFormP v i x x' (Var s) (Var k)))"
   by (auto simp: eqvt_def SubstFormP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows SubstFormP_fresh_iff [simp]:
        "a \<sharp> SubstFormP v i x x' \<longleftrightarrow> a \<sharp> v \<and> a \<sharp> i \<and> a \<sharp> x \<and> a \<sharp> x'"  (is ?thesis1)
     and SubstFormP_sf [iff]:
        "Sigma_fm (SubstFormP v i x x')"  (is ?thsf)
 proof -
   obtain s::name and k::name
     where "atom s \<sharp> (v,i,x,x',k)"  "atom k \<sharp> (v,i,x,x')"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf
     by auto
 qed
 
 lemma SubstFormP_subst [simp]:
      "(SubstFormP v i x x')(j::=t) = SubstFormP (subst j t v) (subst j t i) (subst j t x) (subst j t x')"
 proof -
   obtain s::name and k::name  where "atom s \<sharp> (v,i,x,x',t,j,k)" "atom k \<sharp> (v,i,x,x',t,j)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: SubstFormP.simps [of s _ _ _ _ k])
 qed
 
 lemma SubstFormP_cong:
   "\<lbrakk>H \<turnstile> v EQ v'; H \<turnstile> i EQ i'; H \<turnstile> t EQ t'; H \<turnstile> u EQ u'\<rbrakk>
    \<Longrightarrow> H \<turnstile> SubstFormP v i t u IFF SubstFormP v' i' t' u'"
   by (rule P4_cong) auto
 
 lemma ground_SubstFormP [simp]: "ground_fm (SubstFormP v y x x') \<longleftrightarrow> ground v \<and> ground y \<and> ground x \<and> ground x'"
   by (auto simp: ground_aux_def ground_fm_aux_def supp_conv_fresh)
 
 declare SubstFormP.simps [simp del]
 
-section {*The predicate @{text AtomicP}*}
+section \<open>The predicate \<open>AtomicP\<close>\<close>
 
 nominal_function AtomicP :: "tm \<Rightarrow> fm"
   where "\<lbrakk>atom t \<sharp> (u,y); atom u \<sharp> y\<rbrakk> \<Longrightarrow>
     AtomicP y = Ex t (Ex u (TermP (Var t) AND TermP (Var u) AND
                       (y EQ Q_Eq (Var t) (Var u) OR
                        y EQ Q_Mem (Var t) (Var u))))"
   by (auto simp: eqvt_def AtomicP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows AtomicP_fresh_iff [simp]: "a \<sharp> AtomicP y \<longleftrightarrow> a \<sharp> y"    (is ?thesis1)
     and AtomicP_sf [iff]: "Sigma_fm (AtomicP y)"  (is ?thsf)
 proof -
   obtain t::name and u::name  where "atom t \<sharp> (u,y)"  "atom u \<sharp> y"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf
     by auto
 qed
 
 lemma AtompicP_subst [simp]: "(AtomicP t)(j::=w) = AtomicP (subst j w t)"
 proof -
   obtain x y :: name where "atom x \<sharp> (j,w,t,y)"   "atom y \<sharp> (j,w,t)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: AtomicP.simps [of x y])
 qed
 
 
-section {*The predicate @{text MakeForm} *}
+section \<open>The predicate \<open>MakeForm\<close>\<close>
 
 nominal_function MakeFormP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom v \<sharp> (y,u,w,au); atom au \<sharp> (y,u,w)\<rbrakk> \<Longrightarrow>
     MakeFormP y u w =
       y EQ Q_Disj u w OR y EQ Q_Neg u OR
       Ex v (Ex au (AbstFormP (Var v) Zero u (Var au) AND y EQ Q_Ex (Var au)))"
   by (auto simp: eqvt_def MakeFormP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows MakeFormP_fresh_iff [simp]:
        "a \<sharp> MakeFormP y u w \<longleftrightarrow> a \<sharp> y \<and> a \<sharp> u \<and> a \<sharp> w"  (is ?thesis1)
     and MakeFormP_sf [iff]:
        "Sigma_fm (MakeFormP y u w)"  (is ?thsf)
 proof -
   obtain v::name and au::name  where "atom v \<sharp> (y,u,w,au)"  "atom au \<sharp> (y,u,w)"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf
     by auto
 qed
 
 declare MakeFormP.simps [simp del]
 
 lemma MakeFormP_subst [simp]: "(MakeFormP y u t)(j::=w) = MakeFormP (subst j w y) (subst j w u) (subst j w t)"
 proof -
   obtain a b :: name where "atom a \<sharp> (j,w,y,u,t,b)"   "atom b \<sharp> (j,w,y,u,t)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: MakeFormP.simps [of a _ _ _ b])
 qed
 
 
-section {*The predicate @{text SeqFormP} *}
+section \<open>The predicate \<open>SeqFormP\<close>\<close>
 
 (*SeqForm(s,k,t) \<equiv> LstSeq(s,k,t) \<and> (\<forall>n\<in>k)[Atomic (s n) \<or> (\<exists>m,l\<in>n)[MakeForm (s m) (s l) (s n)]]*)
 
 nominal_function SeqFormP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom l \<sharp> (s,k,t,sl,m,n,sm,sn); atom sl \<sharp> (s,k,t,m,n,sm,sn);
           atom m \<sharp> (s,k,t,n,sm,sn); atom n \<sharp> (s,k,t,sm,sn);
           atom sm \<sharp> (s,k,t,sn); atom sn \<sharp> (s,k,t)\<rbrakk> \<Longrightarrow>
     SeqFormP s k t =
       LstSeqP s k t AND
       All2 n (SUCC k) (Ex sn (HPair (Var n) (Var sn) IN s AND (AtomicP (Var sn) OR
                 Ex m (Ex l (Ex sm (Ex sl (Var m IN Var n AND Var l IN Var n AND
                        HPair (Var m) (Var sm) IN s AND HPair (Var l) (Var sl) IN s AND
                        MakeFormP (Var sn) (Var sm) (Var sl))))))))"
   by (auto simp: eqvt_def SeqFormP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows SeqFormP_fresh_iff [simp]:
        "a \<sharp> SeqFormP s k t \<longleftrightarrow> a \<sharp> s \<and> a \<sharp> k \<and> a \<sharp> t" (is ?thesis1)
     and SeqFormP_sf [iff]: "Sigma_fm (SeqFormP s k t)"          (is ?thsf)
     and SeqFormP_imp_OrdP:
        "{ SeqFormP s k t } \<turnstile> OrdP k"  (is ?thOrd)
     and SeqFormP_imp_LstSeqP:
        "{ SeqFormP s k t } \<turnstile> LstSeqP s k t"  (is ?thLstSeq)
 proof -
   obtain l::name and sl::name and m::name and n::name and sm::name and sn::name
     where atoms: "atom l \<sharp> (s,k,t,sl,m,n,sm,sn)"   "atom sl \<sharp> (s,k,t,m,n,sm,sn)"
         "atom m \<sharp> (s,k,t,n,sm,sn)"   "atom n \<sharp> (s,k,t,sm,sn)"
         "atom sm \<sharp> (s,k,t,sn)"       "atom sn \<sharp> (s,k,t)"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf ?thOrd ?thLstSeq
     by (auto intro: LstSeqP_OrdP)
 qed
 
 lemma SeqFormP_subst [simp]:
       "(SeqFormP s k t)(j::=w) = SeqFormP (subst j w s) (subst j w k) (subst j w t)"
 proof -
   obtain l::name and sl::name and m::name and n::name and sm::name and sn::name
     where "atom l \<sharp> (j,w,s,t,k,sl,m,n,sm,sn)"   "atom sl \<sharp> (j,w,s,k,t,m,n,sm,sn)"
         "atom m \<sharp> (j,w,s,k,t,n,sm,sn)"   "atom n \<sharp> (j,w,s,k,t,sm,sn)"
         "atom sm \<sharp> (j,w,s,k,t,sn)"   "atom sn \<sharp> (j,w,s,k,t)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: SeqFormP.simps [of l _ _ _ sl m n sm sn])
 qed
 
-section {*The predicate @{text FormP}*}
+section \<open>The predicate \<open>FormP\<close>\<close>
 
-subsection {*Definition*}
+subsection \<open>Definition\<close>
 
 nominal_function FormP :: "tm \<Rightarrow> fm"
   where "\<lbrakk>atom k \<sharp> (s,y); atom s \<sharp> y\<rbrakk> \<Longrightarrow>
     FormP y = Ex k (Ex s (SeqFormP (Var s) (Var k) y))"
   by (auto simp: eqvt_def FormP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows FormP_fresh_iff [simp]: "a \<sharp> FormP y \<longleftrightarrow> a \<sharp> y"              (is ?thesis1)
     and FormP_sf [iff]:         "Sigma_fm (FormP y)"                 (is ?thsf)
 proof -
   obtain k::name and s::name  where k: "atom k \<sharp> (s,y)" "atom s \<sharp> y"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf
     by auto
 qed
 
 lemma FormP_subst [simp]: "(FormP y)(j::=w) = FormP (subst j w y)"
 proof -
   obtain k::name and s::name where "atom k \<sharp> (s,j,w,y)"  "atom s \<sharp> (j,w,y)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: FormP.simps [of k s])
 qed
 
 
-subsection {*The predicate @{text VarNonOccFormP} (Derived from @{text SubstFormP})*}
+subsection \<open>The predicate \<open>VarNonOccFormP\<close> (Derived from \<open>SubstFormP\<close>)\<close>
 
 nominal_function VarNonOccFormP :: "tm \<Rightarrow> tm \<Rightarrow> fm"
   where "VarNonOccFormP v x = FormP x AND SubstFormP v Zero x x"
   by (auto simp: eqvt_def VarNonOccFormP_graph_aux_def)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows VarNonOccFormP_fresh_iff [simp]: "a \<sharp> VarNonOccFormP v y \<longleftrightarrow> a \<sharp> v \<and> a \<sharp> y" (is ?thesis1)
     and VarNonOccFormP_sf [iff]: "Sigma_fm (VarNonOccFormP v y)" (is ?thsf)
 proof -
   show ?thesis1 ?thsf
     by auto
 qed
 
 declare VarNonOccFormP.simps [simp del]
 
 end
 
diff --git a/thys/Goedel_HFSet_Semanticless/Functions.thy b/thys/Goedel_HFSet_Semanticless/Functions.thy
--- a/thys/Goedel_HFSet_Semanticless/Functions.thy
+++ b/thys/Goedel_HFSet_Semanticless/Functions.thy
@@ -1,497 +1,497 @@
-chapter{*Uniqueness Results: Syntactic Relations are Functions*}
+chapter\<open>Uniqueness Results: Syntactic Relations are Functions\<close>
 
 theory Functions
 imports Coding_Predicates
 begin
 
-subsection {*SeqStTermP*}
+subsection \<open>SeqStTermP\<close>
 
 lemma not_IndP_VarP: "{IndP x, VarP x} \<turnstile> A"
 proof -
   obtain m::name  where "atom m \<sharp> (x,A)"
     by (metis obtain_fresh)
   thus ?thesis
     by  (auto simp: fresh_Pair) (blast intro: ExFalso cut_same [OF VarP_cong [THEN Iff_MP_same]])
 qed
 
-text{*It IS a pair, but not just any pair.*}
+text\<open>It IS a pair, but not just any pair.\<close>
 lemma IndP_HPairE: "insert (IndP (HPair (HPair Zero (HPair Zero Zero)) x)) H \<turnstile> A"
 proof -
   obtain m::name  where "atom m \<sharp> (x,A)"
     by (metis obtain_fresh)
   hence "{ IndP (HPair (HPair Zero (HPair Zero Zero)) x) } \<turnstile> A"
     by (auto simp: IndP.simps [of m] HTuple_minus_1 intro: thin1)
   thus ?thesis
     by (metis Assume cut1)
 qed
 
 lemma atom_HPairE:
   assumes "H \<turnstile> x EQ HPair (HPair Zero (HPair Zero Zero)) y"
     shows "insert (IndP x OR x NEQ v) H \<turnstile> A"
 proof -
   have "{ IndP x OR x NEQ v, x EQ HPair (HPair Zero (HPair Zero Zero)) y } \<turnstile> A"
     by (auto intro!: OrdNotEqP_OrdP_E IndP_HPairE
              intro: cut_same [OF IndP_cong [THEN Iff_MP_same]]
                     cut_same [OF OrdP_cong [THEN Iff_MP_same]])
   thus ?thesis
     by (metis Assume assms rcut2)
 qed
 
 lemma SeqStTermP_lemma:
   assumes "atom m \<sharp> (v,i,t,u,s,k,n,sm,sm',sn,sn')" "atom n \<sharp> (v,i,t,u,s,k,sm,sm',sn,sn')"
           "atom sm \<sharp> (v,i,t,u,s,k,sm',sn,sn')" "atom sm' \<sharp> (v,i,t,u,s,k,sn,sn')"
           "atom sn \<sharp> (v,i,t,u,s,k,sn')" "atom sn' \<sharp> (v,i,t,u,s,k)"
     shows "{ SeqStTermP v i t u s k }
            \<turnstile> ((t EQ v AND u EQ i) OR
               ((IndP t OR t NEQ v) AND u EQ t)) OR
               Ex m (Ex n (Ex sm (Ex sm' (Ex sn (Ex sn' (Var m IN k AND Var n IN k AND
                        SeqStTermP v i (Var sm) (Var sm') s (Var m) AND
                        SeqStTermP v i (Var sn) (Var sn') s (Var n) AND
                        t EQ Q_Eats (Var sm) (Var sn) AND
                        u EQ Q_Eats (Var sm') (Var sn')))))))"
 proof -
   obtain l::name and sl::name and sl'::name
     where "atom l \<sharp> (v,i,t,u,s,k,sl,sl',m,n,sm,sm',sn,sn')"
           "atom sl \<sharp> (v,i,t,u,s,k,sl',m,n,sm,sm',sn,sn')"
           "atom sl' \<sharp> (v,i,t,u,s,k,m,n,sm,sm',sn,sn')"
     by (metis obtain_fresh)
   thus ?thesis using assms
     apply (simp add: SeqStTermP.simps [of l s k v i sl sl' m n sm sm' sn sn'])
     apply (rule Conj_EH Ex_EH All2_SUCC_E [THEN rotate2] | simp)+
     apply (rule cut_same [where A = "HPair t u EQ HPair (Var sl) (Var sl')"])
     apply (metis Assume AssumeH(4) LstSeqP_EQ)
     apply clarify
     apply (rule Disj_EH)
     apply (rule Disj_I1)
     apply (rule anti_deduction)
     apply (rule Var_Eq_subst_Iff [THEN Sym_L, THEN Iff_MP_same])
     apply (rule Sym_L [THEN rotate2])
     apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same], force)
     \<comment> \<open>now the quantified case\<close>
     \<comment> \<open>auto could be used but is VERY SLOW\<close>
     apply (rule Ex_EH Conj_EH)+
     apply simp_all
     apply (rule Disj_I2)
     apply (rule Ex_I [where x = "Var m"], simp)
     apply (rule Ex_I [where x = "Var n"], simp)
     apply (rule Ex_I [where x = "Var sm"], simp)
     apply (rule Ex_I [where x = "Var sm'"], simp)
     apply (rule Ex_I [where x = "Var sn"], simp)
     apply (rule Ex_I [where x = "Var sn'"], simp)
     apply (simp_all add: SeqStTermP.simps [of l s _ v i sl sl' m n sm sm' sn sn'])
     apply ((rule Conj_I)+, blast intro: LstSeqP_Mem)+
     \<comment> \<open>first SeqStTermP subgoal\<close>
     apply (rule All2_Subset [OF Hyp], blast)
     apply (blast intro!: SUCC_Subset_Ord LstSeqP_OrdP, blast, simp)
     \<comment> \<open>next SeqStTermP subgoal\<close>
     apply ((rule Conj_I)+, blast intro: LstSeqP_Mem)+
     apply (rule All2_Subset [OF Hyp], blast)
     apply (blast intro!: SUCC_Subset_Ord LstSeqP_OrdP, blast, simp)
     \<comment> \<open>finally, the equality pair\<close>
     apply (blast intro: Trans)
     done
 qed
 
 
 lemma SeqStTermP_unique: "{SeqStTermP v a t u s kk, SeqStTermP v a t u' s' kk'} \<turnstile> u' EQ u"
 proof -
   obtain i::name and j::name and j'::name and k::name and k'::name and l::name
      and m::name and n::name and sm::name and sn::name and sm'::name and sn'::name
      and m2::name and n2::name and sm2::name and sn2::name and sm2'::name and sn2'::name
     where atoms:  "atom i \<sharp> (s,s',v,a,t,u,u')"   "atom j \<sharp> (s,s',v,a,t,i,t,u,u')"
                   "atom j' \<sharp> (s,s',v,a,t,i,j,t,u,u')"
                   "atom k \<sharp> (s,s',v,a,t,u,u',kk',i,j,j')" "atom k' \<sharp> (s,s',v,a,t,u,u',k,i,j,j')"
                   "atom l \<sharp> (s,s',v,a,t,i,j,j',k,k')"
                   "atom m \<sharp> (s,s',v,a,i,j,j',k,k',l)"   "atom n \<sharp> (s,s',v,a,i,j,j',k,k',l,m)"
                   "atom sm \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n)"  "atom sn \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n,sm)"
                   "atom sm' \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n,sm,sn)"   "atom sn' \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n,sm,sn,sm')"
                   "atom m2 \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n,sm,sn,sm',sn')"   "atom n2 \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n,sm,sn,sm',sn',m2)"
                   "atom sm2 \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n,sm,sn,sm',sn',m2,n2)"  "atom sn2 \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n,sm,sn,sm',sn',m2,n2,sm2)"
                   "atom sm2' \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n,sm,sn,sm',sn',m2,n2,sm2,sn2)"   "atom sn2' \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n,sm,sn,sm',sn',m2,n2,sm2,sn2,sm2')"
     by (metis obtain_fresh)
   have "{ OrdP (Var k), VarP v }
        \<turnstile> All i (All j (All j' (All k' (SeqStTermP v a (Var i) (Var j) s (Var k)
                                   IMP (SeqStTermP v a (Var i) (Var j') s' (Var k') IMP Var j' EQ Var j)))))"
     apply (rule OrdIndH [where j=l])
     using atoms apply auto
     apply (rule Swap)
     apply (rule cut_same)
     apply (rule cut1 [OF SeqStTermP_lemma [of m v a "Var i" "Var j" s "Var k" n sm sm' sn sn']], simp_all, blast)
     apply (rule cut_same)
     apply (rule cut1 [OF SeqStTermP_lemma [of m2 v a "Var i" "Var j'" s' "Var k'" n2 sm2 sm2' sn2 sn2']], simp_all, blast)
     apply (rule Disj_EH Conj_EH)+
     \<comment> \<open>case 1, both sides equal "v"\<close>
     apply (blast intro: Trans Sym)
     \<comment> \<open>case 2, @{term "Var i EQ v"} and also @{term "IndP (Var i) OR Var i NEQ v"}\<close>
     apply (rule Conj_EH Disj_EH)+
     apply (blast intro: IndP_cong [THEN Iff_MP_same] not_IndP_VarP [THEN cut2])
     apply (metis Assume OrdNotEqP_E)
     \<comment> \<open>case 3, both a variable and a pair\<close>
     apply (rule Ex_EH Conj_EH)+
     apply simp_all
     apply (rule cut_same [where A = "VarP (Q_Eats (Var sm) (Var sn))"])
     apply (blast intro: Trans Sym VarP_cong [where x=v, THEN Iff_MP_same] Hyp, blast)
     \<comment> \<open>towards remaining cases\<close>
     apply (rule Disj_EH Ex_EH)+
     \<comment> \<open>case 4, @{term "Var i EQ v"} and also @{term "IndP (Var i) OR Var i NEQ v"}\<close>
     apply (blast intro: IndP_cong [THEN Iff_MP_same] not_IndP_VarP [THEN cut2] OrdNotEqP_E)
     \<comment> \<open>case 5, @{term "Var i EQ v"} for both\<close>
     apply (blast intro: Trans Sym)
     \<comment> \<open>case 6, both an atom and a pair\<close>
     apply (rule Ex_EH Conj_EH)+
     apply simp_all
     apply (rule atom_HPairE)
     apply (simp add: HTuple.simps)
     apply (blast intro: Trans)
     \<comment> \<open>towards remaining cases\<close>
     apply (rule Conj_EH Disj_EH Ex_EH)+
     apply simp_all
     \<comment> \<open>case 7, both an atom and a pair\<close>
     apply (rule cut_same [where A = "VarP (Q_Eats (Var sm2) (Var sn2))"])
     apply (blast intro: Trans Sym VarP_cong [where x=v, THEN Iff_MP_same] Hyp, blast)
     \<comment> \<open>case 8, both an atom and a pair\<close>
     apply (rule Ex_EH Conj_EH)+
     apply simp_all
     apply (rule atom_HPairE)
     apply (simp add: HTuple.simps)
     apply (blast intro: Trans)
     \<comment> \<open>case 9, two Eats terms\<close>
     apply (rule Ex_EH Disj_EH Conj_EH)+
     apply simp_all
     apply (rule All_E' [OF Hyp, where x="Var m"], blast)
     apply (rule All_E' [OF Hyp, where x="Var n"], blast, simp)
     apply (rule Disj_EH, blast intro: thin1 ContraProve)+
     apply (rule All_E [where x="Var sm"], simp)
     apply (rule All_E [where x="Var sm'"], simp)
     apply (rule All_E [where x="Var sm2'"], simp)
     apply (rule All_E [where x="Var m2"], simp)
     apply (rule All_E [where x="Var sn", THEN rotate2], simp)
     apply (rule All_E [where x="Var sn'"], simp)
     apply (rule All_E [where x="Var sn2'"], simp)
     apply (rule All_E [where x="Var n2"], simp)
     apply (rule cut_same [where A = "Q_Eats (Var sm) (Var sn) EQ Q_Eats (Var sm2) (Var sn2)"])
     apply (blast intro: Sym Trans, clarify)
     apply (rule cut_same [where A = "SeqStTermP v a (Var sn) (Var sn2') s' (Var n2)"])
     apply (blast intro: Hyp SeqStTermP_cong [OF Hyp Refl Refl, THEN Iff_MP2_same])
     apply (rule cut_same [where A = "SeqStTermP v a (Var sm) (Var sm2') s' (Var m2)"])
     apply (blast intro: Hyp SeqStTermP_cong [OF Hyp Refl Refl, THEN Iff_MP2_same])
     apply (rule Disj_EH, blast intro: thin1 ContraProve)+
     apply (blast intro: HPair_cong Trans [OF Hyp Sym])
     done
   hence p1: "{OrdP (Var k), VarP v}
              \<turnstile> (All j (All j' (All k' (SeqStTermP v a (Var i) (Var j) s (Var k)
                  IMP (SeqStTermP v a (Var i) (Var j') s' (Var k') IMP Var j' EQ Var j)))))(i::=t)"
     by (metis All_D)
   have p2: "{OrdP (Var k), VarP v}
          \<turnstile> (All j' (All k' (SeqStTermP v a t (Var j) s (Var k)
                  IMP (SeqStTermP v a t (Var j') s' (Var k') IMP Var j' EQ Var j))))(j::=u)"
     apply (rule All_D)
     using atoms p1 by simp
   have p3: "{OrdP (Var k), VarP v}
              \<turnstile> (All k' (SeqStTermP v a t u s (Var k) IMP (SeqStTermP v a t (Var j') s' (Var k') IMP Var j' EQ u)))(j'::=u')"
     apply (rule All_D)
     using atoms p2 by simp
   have p4: "{OrdP (Var k), VarP v}
           \<turnstile> (SeqStTermP v a t u s (Var k) IMP (SeqStTermP v a t u' s' (Var k') IMP u' EQ u))(k'::=kk')"
     apply (rule All_D)
     using atoms p3 by simp
   hence "{SeqStTermP v a t u s (Var k), VarP v} \<turnstile> SeqStTermP v a t u s (Var k) IMP (SeqStTermP v a t u' s' kk' IMP u' EQ u)"
     using atoms apply simp
     by (metis SeqStTermP_imp_OrdP rcut1)
   hence "{VarP v} \<turnstile> ((SeqStTermP v a t u s (Var k) IMP (SeqStTermP v a t u' s' kk' IMP u' EQ u)))"
     by (metis Assume MP_same Imp_I)
   hence "{VarP v} \<turnstile> ((SeqStTermP v a t u s (Var k) IMP (SeqStTermP v a t u' s' kk' IMP u' EQ u)))(k::=kk)"
    using atoms by (force intro!: Subst)
   hence "{VarP v} \<turnstile> SeqStTermP v a t u s kk IMP (SeqStTermP v a t u' s' kk' IMP u' EQ u)"
     using atoms by simp
   hence "{SeqStTermP v a t u s kk} \<turnstile> SeqStTermP v a t u s kk IMP (SeqStTermP v a t u' s' kk' IMP u' EQ u)"
     by (metis SeqStTermP_imp_VarP rcut1)
   thus ?thesis
     by (metis Assume AssumeH(2) MP_same rcut1)
 qed
 
 
 theorem SubstTermP_unique: "{SubstTermP v tm t u, SubstTermP v tm t u'} \<turnstile> u' EQ u"
 proof -
   obtain s::name and s'::name and k::name and k'::name
     where "atom s \<sharp> (v,tm,t,u,u',k,k')" "atom s' \<sharp> (v,tm,t,u,u',k,k',s)"
           "atom k \<sharp> (v,tm,t,u,u')" "atom k' \<sharp> (v,tm,t,u,u',k)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: SubstTermP.simps [of s v tm t u k]  SubstTermP.simps [of s' v tm t u' k'])
        (metis SeqStTermP_unique rotate3 thin1)
 qed
 
-subsection{*@{term SubstAtomicP}*}
+subsection\<open>@{term SubstAtomicP}\<close>
 
 lemma SubstTermP_eq:
   "\<lbrakk>H \<turnstile> SubstTermP v tm x z; insert (SubstTermP v tm y z) H \<turnstile> A\<rbrakk> \<Longrightarrow> insert (x EQ y) H \<turnstile> A"
   by (metis Assume rotate2 Iff_E1 cut_same thin1 SubstTermP_cong [OF Refl Refl _ Refl])
 
 lemma SubstAtomicP_unique: "{SubstAtomicP v tm x y, SubstAtomicP v tm x y'} \<turnstile> y' EQ y"
 proof -
   obtain t::name and ts::name and u::name and us::name
      and t'::name and ts'::name and u'::name and us'::name
     where "atom t \<sharp> (v,tm,x,y,y',ts,u,us)"  "atom ts \<sharp> (v,tm,x,y,y',u,us)"
           "atom u \<sharp> (v,tm,x,y,y',us)"       "atom us \<sharp> (v,tm,x,y,y')"
           "atom t' \<sharp> (v,tm,x,y,y',t,ts,u,us,ts',u',us')" "atom ts' \<sharp> (v,tm,x,y,y',t,ts,u,us,u',us')"
           "atom u' \<sharp> (v,tm,x,y,y',t,ts,u,us,us')"        "atom us' \<sharp> (v,tm,x,y,y',t,ts,u,us)"
     by (metis obtain_fresh)
   thus ?thesis
     apply (simp add: SubstAtomicP.simps [of t v tm x y ts u us]
                      SubstAtomicP.simps [of t' v tm x y' ts' u' us'])
     apply (rule Ex_EH Disj_EH Conj_EH)+
     apply simp_all
     apply (rule Eq_Trans_E [OF Hyp], auto simp: HTS)
     apply (rule SubstTermP_eq [THEN thin1], blast)
     apply (rule SubstTermP_eq [THEN rotate2], blast)
     apply (rule Trans [OF Hyp Sym], blast)
     apply (rule Trans [OF Hyp], blast)
     apply (metis Assume AssumeH(8) HPair_cong Refl cut2 [OF SubstTermP_unique] thin1)
     apply (rule Eq_Trans_E [OF Hyp], blast, force simp add: HTS)
     apply (rule Eq_Trans_E [OF Hyp], blast, force simp add: HTS)
     apply (rule Eq_Trans_E [OF Hyp], auto simp: HTS)
     apply (rule SubstTermP_eq [THEN thin1], blast)
     apply (rule SubstTermP_eq [THEN rotate2], blast)
     apply (rule Trans [OF Hyp Sym], blast)
     apply (rule Trans [OF Hyp], blast)
     apply (metis Assume AssumeH(8) HPair_cong Refl cut2 [OF SubstTermP_unique] thin1)
     done
 qed
 
-subsection{*@{term SeqSubstFormP}*}
+subsection\<open>@{term SeqSubstFormP}\<close>
 
 lemma SeqSubstFormP_lemma:
   assumes "atom m \<sharp> (v,u,x,y,s,k,n,sm,sm',sn,sn')" "atom n \<sharp> (v,u,x,y,s,k,sm,sm',sn,sn')"
           "atom sm \<sharp> (v,u,x,y,s,k,sm',sn,sn')"  "atom sm' \<sharp> (v,u,x,y,s,k,sn,sn')"
           "atom sn \<sharp> (v,u,x,y,s,k,sn')"         "atom sn' \<sharp> (v,u,x,y,s,k)"
   shows "{ SeqSubstFormP v u x y s k }
          \<turnstile> SubstAtomicP v u x y  OR
            Ex m (Ex n (Ex sm (Ex sm' (Ex sn (Ex sn' (Var m IN k AND Var n IN k AND
                        SeqSubstFormP v u (Var sm) (Var sm') s (Var m) AND
                        SeqSubstFormP v u (Var sn) (Var sn') s (Var n) AND
                        (((x EQ Q_Disj (Var sm) (Var sn) AND y EQ Q_Disj (Var sm') (Var sn')) OR
                         (x EQ Q_Neg (Var sm) AND y EQ Q_Neg (Var sm')) OR
                         (x EQ Q_Ex (Var sm) AND y EQ Q_Ex (Var sm'))))))))))"
 proof -
   obtain l::name and sl::name and sl'::name
     where "atom l \<sharp> (v,u,x,y,s,k,sl,sl',m,n,sm,sm',sn,sn')"
           "atom sl \<sharp> (v,u,x,y,s,k,sl',m,n,sm,sm',sn,sn')"
           "atom sl' \<sharp> (v,u,x,y,s,k,m,n,sm,sm',sn,sn')"
     by (metis obtain_fresh)
   thus ?thesis using assms
     apply (simp add: SeqSubstFormP.simps [of l s k v u sl sl' m n sm sm' sn sn'])
     apply (rule Conj_EH Ex_EH All2_SUCC_E [THEN rotate2] | simp)+
     apply (rule cut_same [where A = "HPair x y EQ HPair (Var sl) (Var sl')"])
     apply (metis Assume AssumeH(4) LstSeqP_EQ)
     apply clarify
     apply (rule Disj_EH)
     apply (blast intro: Disj_I1 SubstAtomicP_cong [THEN Iff_MP2_same])
     \<comment> \<open>now the quantified cases\<close>
     apply (rule Ex_EH Conj_EH)+
     apply simp_all
     apply (rule Disj_I2)
     apply (rule Ex_I [where x = "Var m"], simp)
     apply (rule Ex_I [where x = "Var n"], simp)
     apply (rule Ex_I [where x = "Var sm"], simp)
     apply (rule Ex_I [where x = "Var sm'"], simp)
     apply (rule Ex_I [where x = "Var sn"], simp)
     apply (rule Ex_I [where x = "Var sn'"], simp)
     apply (simp_all add: SeqSubstFormP.simps [of l s _ v u sl sl' m n sm sm' sn sn'])
     apply ((rule Conj_I)+, blast intro: LstSeqP_Mem)+
     \<comment> \<open>first SeqSubstFormP subgoal\<close>
     apply (rule All2_Subset [OF Hyp], blast)
     apply (blast intro!: SUCC_Subset_Ord LstSeqP_OrdP, blast, simp)
     \<comment> \<open>next SeqSubstFormP subgoal\<close>
     apply ((rule Conj_I)+, blast intro: LstSeqP_Mem)+
     apply (rule All2_Subset [OF Hyp], blast)
     apply (blast intro!: SUCC_Subset_Ord LstSeqP_OrdP, blast, simp)
     \<comment> \<open>finally, the equality pairs\<close>
     apply (rule anti_deduction [THEN thin1])
     apply (rule Sym_L [THEN rotate4])
     apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same])
     apply (rule Sym_L [THEN rotate5])
     apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same], force)
     done
 qed
 
 lemma
   shows Neg_SubstAtomicP_Fls:  "{y EQ Q_Neg z,  SubstAtomicP v tm y y'} \<turnstile> Fls"    (is ?thesis1)
     and Disj_SubstAtomicP_Fls: "{y EQ Q_Disj z w, SubstAtomicP v tm y y'} \<turnstile> Fls"  (is ?thesis2)
     and Ex_SubstAtomicP_Fls:   "{y EQ Q_Ex z,   SubstAtomicP v tm y y'} \<turnstile> Fls"    (is ?thesis3)
 proof -
   obtain t::name and u::name and t'::name  and u'::name
     where "atom t \<sharp> (z,w,v,tm,y,y',t',u,u')" "atom t' \<sharp> (z,w,v,tm,y,y',u,u')"
           "atom u \<sharp> (z,w,v,tm,y,y',u')" "atom u' \<sharp> (z,w,v,tm,y,y')"
     by (metis obtain_fresh)
   thus ?thesis1 ?thesis2 ?thesis3
     by (auto simp: SubstAtomicP.simps [of t v tm y y' t' u u'] HTS  intro: Eq_Trans_E [OF Hyp])
 qed
 
 lemma SeqSubstFormP_eq:
   "\<lbrakk>H \<turnstile> SeqSubstFormP v tm x z s k; insert (SeqSubstFormP v tm y z s k) H \<turnstile> A\<rbrakk>
    \<Longrightarrow> insert (x EQ y) H \<turnstile> A"
   apply (rule cut_same [OF SeqSubstFormP_cong [OF Assume Refl Refl Refl, THEN Iff_MP_same]])
   apply (auto simp: insert_commute intro: thin1)
   done
 
 lemma SeqSubstFormP_unique: "{SeqSubstFormP v a x y s kk, SeqSubstFormP v a x y' s' kk'} \<turnstile> y' EQ y"
 proof -
   obtain i::name and j::name and j'::name and k::name and k'::name and l::name
      and m::name and n::name and sm::name and sn::name and sm'::name and sn'::name
      and m2::name and n2::name and sm2::name and sn2::name and sm2'::name and sn2'::name
     where atoms:  "atom i \<sharp> (s,s',v,a,x,y,y')"   "atom j \<sharp> (s,s',v,a,x,i,x,y,y')"
                   "atom j' \<sharp> (s,s',v,a,x,i,j,x,y,y')"
                   "atom k \<sharp> (s,s',v,a,x,y,y',kk',i,j,j')" "atom k' \<sharp> (s,s',v,a,x,y,y',k,i,j,j')"
                   "atom l \<sharp> (s,s',v,a,x,i,j,j',k,k')"
                   "atom m \<sharp> (s,s',v,a,i,j,j',k,k',l)"   "atom n \<sharp> (s,s',v,a,i,j,j',k,k',l,m)"
                   "atom sm \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n)"  "atom sn \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n,sm)"
                   "atom sm' \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n,sm,sn)"   "atom sn' \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n,sm,sn,sm')"
                   "atom m2 \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n,sm,sn,sm',sn')"   "atom n2 \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n,sm,sn,sm',sn',m2)"
                   "atom sm2 \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n,sm,sn,sm',sn',m2,n2)"  "atom sn2 \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n,sm,sn,sm',sn',m2,n2,sm2)"
                   "atom sm2' \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n,sm,sn,sm',sn',m2,n2,sm2,sn2)"   "atom sn2' \<sharp> (s,s',v,a,i,j,j',k,k',l,m,n,sm,sn,sm',sn',m2,n2,sm2,sn2,sm2')"
     by (metis obtain_fresh)
   have "{ OrdP (Var k) }
        \<turnstile> All i (All j (All j' (All k' (SeqSubstFormP v a (Var i) (Var j) s (Var k)
                                   IMP (SeqSubstFormP v a (Var i) (Var j') s' (Var k') IMP Var j' EQ Var j)))))"
     apply (rule OrdIndH [where j=l])
     using atoms apply auto
     apply (rule Swap)
     apply (rule cut_same)
     apply (rule cut1 [OF SeqSubstFormP_lemma [of m v a "Var i" "Var j" s "Var k" n sm sm' sn sn']], simp_all, blast)
     apply (rule cut_same)
     apply (rule cut1 [OF SeqSubstFormP_lemma [of m2 v a "Var i" "Var j'" s' "Var k'" n2 sm2 sm2' sn2 sn2']], simp_all, blast)
     apply (rule Disj_EH Conj_EH)+
     \<comment> \<open>case 1, both sides are atomic\<close>
     apply (blast intro: cut2 [OF SubstAtomicP_unique])
     \<comment> \<open>case 2, atomic and also not\<close>
     apply (rule Ex_EH Conj_EH Disj_EH)+
     apply simp_all
     apply (metis Assume AssumeH(7) Disj_I1 Neg_I anti_deduction cut2 [OF Disj_SubstAtomicP_Fls])
     apply (rule Conj_EH Disj_EH)+
     apply (metis Assume AssumeH(7) Disj_I1 Neg_I anti_deduction cut2 [OF Neg_SubstAtomicP_Fls])
     apply (rule Conj_EH)+
     apply (metis Assume AssumeH(7) Disj_I1 Neg_I anti_deduction cut2 [OF Ex_SubstAtomicP_Fls])
     \<comment> \<open>towards remaining cases\<close>
     apply (rule Conj_EH Disj_EH Ex_EH)+
     apply simp_all
     apply (metis Assume AssumeH(7) Disj_I1 Neg_I anti_deduction cut2 [OF Disj_SubstAtomicP_Fls])
     apply (rule Conj_EH Disj_EH)+
     apply (metis Assume AssumeH(7) Disj_I1 Neg_I anti_deduction cut2 [OF Neg_SubstAtomicP_Fls])
     apply (rule Conj_EH)+
     apply (metis Assume AssumeH(7) Disj_I1 Neg_I anti_deduction cut2 [OF Ex_SubstAtomicP_Fls])
     \<comment> \<open>towards remaining cases\<close>
     apply (rule Conj_EH Disj_EH Ex_EH)+
     apply simp_all
     \<comment> \<open>case two Disj terms\<close>
     apply (rule All_E' [OF Hyp, where x="Var m"], blast)
     apply (rule All_E' [OF Hyp, where x="Var n"], blast, simp)
     apply (rule Disj_EH, blast intro: thin1 ContraProve)+
     apply (rule All_E [where x="Var sm"], simp)
     apply (rule All_E [where x="Var sm'"], simp)
     apply (rule All_E [where x="Var sm2'"], simp)
     apply (rule All_E [where x="Var m2"], simp)
     apply (rule All_E [where x="Var sn", THEN rotate2], simp)
     apply (rule All_E [where x="Var sn'"], simp)
     apply (rule All_E [where x="Var sn2'"], simp)
     apply (rule All_E [where x="Var n2"], simp)
     apply (rule rotate3)
     apply (rule Eq_Trans_E [OF Hyp], blast)
     apply (clarsimp simp add: HTS)
     apply (rule thin1)
     apply (rule Disj_EH [OF ContraProve], blast intro: thin1 SeqSubstFormP_eq)+
     apply (blast intro: HPair_cong Trans [OF Hyp Sym])
     \<comment> \<open>towards remaining cases\<close>
     apply (rule Conj_EH Disj_EH)+
     \<comment> \<open>Negation = Disjunction?\<close>
     apply (rule Eq_Trans_E [OF Hyp], blast, force simp add: HTS)
    \<comment> \<open>Existential = Disjunction?\<close>
     apply (rule Conj_EH)
     apply (rule Eq_Trans_E [OF Hyp], blast, force simp add: HTS)
      \<comment> \<open>towards remaining cases\<close>
     apply (rule Conj_EH Disj_EH Ex_EH)+
     apply simp_all
     \<comment> \<open>Disjunction = Negation?\<close>
     apply (rule Eq_Trans_E [OF Hyp], blast, force simp add: HTS)
     apply (rule Conj_EH Disj_EH)+
     \<comment> \<open>case two Neg terms\<close>
     apply (rule Eq_Trans_E [OF Hyp], blast, clarify)
     apply (rule thin1)
     apply (rule All_E' [OF Hyp, where x="Var m"], blast, simp)
     apply (rule Disj_EH, blast intro: thin1 ContraProve)+
     apply (rule All_E [where x="Var sm"], simp)
     apply (rule All_E [where x="Var sm'"], simp)
     apply (rule All_E [where x="Var sm2'"], simp)
     apply (rule All_E [where x="Var m2"], simp)
     apply (rule Disj_EH [OF ContraProve], blast intro: SeqSubstFormP_eq Sym_L)+
     apply (blast intro: HPair_cong Sym Trans [OF Hyp])
     \<comment> \<open>Existential = Negation?\<close>
     apply (rule Conj_EH)+
     apply (rule Eq_Trans_E [OF Hyp], blast, force simp add: HTS)
      \<comment> \<open>towards remaining cases\<close>
     apply (rule Conj_EH Disj_EH Ex_EH)+
     apply simp_all
     \<comment> \<open>Disjunction = Existential\<close>
     apply (rule Eq_Trans_E [OF Hyp], blast, force simp add: HTS)
     apply (rule Conj_EH Disj_EH Ex_EH)+
     \<comment> \<open>Negation = Existential\<close>
     apply (rule Eq_Trans_E [OF Hyp], blast, force simp add: HTS)
    \<comment> \<open>case two Ex terms\<close>
     apply (rule Conj_EH)+
     apply (rule Eq_Trans_E [OF Hyp], blast, clarify)
     apply (rule thin1)
     apply (rule All_E' [OF Hyp, where x="Var m"], blast, simp)
     apply (rule Disj_EH, blast intro: thin1 ContraProve)+
     apply (rule All_E [where x="Var sm"], simp)
     apply (rule All_E [where x="Var sm'"], simp)
     apply (rule All_E [where x="Var sm2'"], simp)
     apply (rule All_E [where x="Var m2"], simp)
     apply (rule Disj_EH [OF ContraProve], blast intro: SeqSubstFormP_eq Sym_L)+
     apply (blast intro: HPair_cong Sym Trans [OF Hyp])
     done
   hence p1: "{OrdP (Var k)}
              \<turnstile> (All j (All j' (All k' (SeqSubstFormP v a (Var i) (Var j) s (Var k)
                  IMP (SeqSubstFormP v a (Var i) (Var j') s' (Var k') IMP Var j' EQ Var j)))))(i::=x)"
     by (metis All_D)
   have p2: "{OrdP (Var k)}
             \<turnstile> (All j' (All k' (SeqSubstFormP v a x (Var j) s (Var k)
                  IMP (SeqSubstFormP v a x (Var j') s' (Var k') IMP Var j' EQ Var j))))(j::=y)"
     apply (rule All_D)
     using atoms p1 by simp
   have p3: "{OrdP (Var k)}
             \<turnstile> (All k' (SeqSubstFormP v a x y s (Var k)
                  IMP (SeqSubstFormP v a x (Var j') s' (Var k') IMP Var j' EQ y)))(j'::=y')"
     apply (rule All_D)
     using atoms p2 by simp
   have p4: "{OrdP (Var k)}
           \<turnstile> (SeqSubstFormP v a x y s (Var k) IMP (SeqSubstFormP v a x y' s' (Var k') IMP y' EQ y))(k'::=kk')"
     apply (rule All_D)
     using atoms p3 by simp
   hence "{OrdP (Var k)} \<turnstile> SeqSubstFormP v a x y s (Var k) IMP (SeqSubstFormP v a x y' s' kk' IMP y' EQ y)"
     using atoms by simp
   hence "{SeqSubstFormP v a x y s (Var k)}
          \<turnstile> SeqSubstFormP v a x y s (Var k) IMP (SeqSubstFormP v a x y' s' kk' IMP y' EQ y)"
     by (metis SeqSubstFormP_imp_OrdP rcut1)
   hence "{} \<turnstile> SeqSubstFormP v a x y s (Var k) IMP (SeqSubstFormP v a x y' s' kk' IMP y' EQ y)"
     by (metis Assume Disj_Neg_2 Disj_commute anti_deduction Imp_I)
   hence "{} \<turnstile> ((SeqSubstFormP v a x y s (Var k) IMP (SeqSubstFormP v a x y' s' kk' IMP y' EQ y)))(k::=kk)"
     using atoms by (force intro!: Subst)
   thus ?thesis
     using atoms by simp (metis DisjAssoc2 Disj_commute anti_deduction)
 qed
 
-subsection{*@{term SubstFormP}*}
+subsection\<open>@{term SubstFormP}\<close>
 
 theorem SubstFormP_unique: "{SubstFormP v tm x y, SubstFormP v tm x y'} \<turnstile> y' EQ y"
 proof -
   obtain s::name and s'::name and k::name and k'::name
     where "atom s \<sharp> (v,tm,x,y,y',k,k')" "atom s' \<sharp> (v,tm,x,y,y',k,k',s)"
           "atom k \<sharp> (v,tm,x,y,y')" "atom k' \<sharp> (v,tm,x,y,y',k)"
     by (metis obtain_fresh)
   thus ?thesis
     by (force simp: SubstFormP.simps [of s v tm x y k]  SubstFormP.simps [of s' v tm x y' k']
                     SeqSubstFormP_unique rotate3 thin1)
 qed
 
 end
 
diff --git a/thys/Goedel_HFSet_Semanticless/Goedel_I.thy b/thys/Goedel_HFSet_Semanticless/Goedel_I.thy
--- a/thys/Goedel_HFSet_Semanticless/Goedel_I.thy
+++ b/thys/Goedel_HFSet_Semanticless/Goedel_I.thy
@@ -1,790 +1,790 @@
-chapter {* Section 6 Material and Gödel's First Incompleteness Theorem*}
+chapter \<open>Section 6 Material and Gödel's First Incompleteness Theorem\<close>
 
 theory Goedel_I
 imports Pf_Predicates Functions II_Prelims
 begin
 
-section{*The Function W and Lemma 6.1*}
+section\<open>The Function W and Lemma 6.1\<close>
 
-subsection{*Predicate form, defined on sequences*}
+subsection\<open>Predicate form, defined on sequences\<close>
 
 nominal_function SeqWRP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom l \<sharp> (s,k,sl); atom sl \<sharp> (s)\<rbrakk> \<Longrightarrow>
     SeqWRP s k y = LstSeqP s k y AND
           HPair Zero Zero IN s AND
           All2 l k (Ex sl (HPair (Var l) (Var sl) IN s AND
                            HPair (SUCC (Var l)) (Q_Succ (Var sl)) IN s))"
   by (auto simp: eqvt_def SeqWRP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows SeqWRP_fresh_iff [simp]: "a \<sharp> SeqWRP s k y \<longleftrightarrow> a \<sharp> s \<and> a \<sharp> k \<and> a \<sharp> y" (is ?thesis1)
     and SeqWRP_sf [iff]:         "Sigma_fm (SeqWRP s k y)"  (is ?thsf)
     and SeqWRP_imp_OrdP:         "{SeqWRP s k t} \<turnstile> OrdP k" (is ?thOrd)
     and SeqWRP_LstSeqP:          "{SeqWRP s k t} \<turnstile> LstSeqP s k t" (is ?thlstseq)
 proof -
   obtain l::name and sl::name where "atom l \<sharp> (s,k,sl)" "atom sl \<sharp> (s)"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf ?thOrd ?thlstseq
       by (auto intro: LstSeqP_OrdP[THEN cut1])
 qed
 
 lemma SeqWRP_subst [simp]:
       "(SeqWRP s k y)(i::=t) = SeqWRP (subst i t s) (subst i t k) (subst i t y)"
 proof -
   obtain l::name and sl::name
     where "atom l \<sharp> (s,k,sl,t,i)" "atom sl \<sharp> (s,k,t,i)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: SeqWRP.simps [where l=l and sl=sl])
 qed
 
 lemma SeqWRP_cong:
   assumes "H \<turnstile> s EQ s'" and  "H \<turnstile> k EQ k'" and  "H \<turnstile> y EQ y'"
   shows "H \<turnstile> SeqWRP s k y IFF SeqWRP s' k' y'"
   by (rule P3_cong [OF _ assms], auto)
 
 declare SeqWRP.simps [simp del]
 
-subsection{*Predicate form of W*}
+subsection\<open>Predicate form of W\<close>
 
 nominal_function WRP :: "tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom s \<sharp> (x,y)\<rbrakk> \<Longrightarrow>
     WRP x y = Ex s (SeqWRP (Var s) x y)"
   by (auto simp: eqvt_def WRP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows WRP_fresh_iff [simp]: "a \<sharp> WRP x y \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> y" (is ?thesis1)
     and sigma_fm_WRP [simp]:  "Sigma_fm (WRP x y)"  (is ?thsf)
 proof -
   obtain s::name where "atom s \<sharp> (x,y)"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf
     by auto
 qed
 
 lemma WRP_subst [simp]: "(WRP x y)(i::=t) = WRP (subst i t x) (subst i t y)"
 proof -
   obtain s::name where "atom s \<sharp> (x,y,t,i)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: WRP.simps [of s])
 qed
 
 lemma WRP_cong: "H \<turnstile> t EQ t' \<Longrightarrow> H \<turnstile> u EQ u' \<Longrightarrow> H \<turnstile> WRP t u IFF WRP t' u'"
   by (rule P2_cong) auto
 
 declare WRP.simps [simp del]
 
 lemma ground_WRP [simp]: "ground_fm (WRP x y) \<longleftrightarrow> ground x \<and> ground y"
   by (auto simp: ground_aux_def ground_fm_aux_def supp_conv_fresh)
 
 lemma SeqWRP_Zero: "{} \<turnstile> SyntaxN.Ex s (SeqWRP (Var s) Zero Zero)"
 proof -
   obtain l sl :: name where "atom l \<sharp> (s, sl)" "atom sl \<sharp> s" by (metis obtain_fresh)
   then show ?thesis
     apply (subst SeqWRP.simps[of l _ _ sl]; simp)
     apply (rule Ex_I[where x="(Eats Zero (HPair Zero Zero))"], simp)
     apply (auto intro!: Mem_Eats_I2)
     done
 qed
 
 lemma WRP_Zero: "{} \<turnstile> WRP Zero Zero"
   by (subst WRP.simps[of undefined]) (auto simp: SeqWRP_Zero)
 
 lemma SeqWRP_HPair_Zero_Zero: "{SeqWRP s k y} \<turnstile> HPair Zero Zero IN s"
 proof -
   let ?vs = "(s,k,y)"
   obtain l::name and sl::name
     where "atom l \<sharp> (?vs,sl)" "atom sl \<sharp> (?vs)" by (metis obtain_fresh)
   then show ?thesis
     by (subst SeqWRP.simps[of l _ _ sl]) auto
 qed
 
 lemma SeqWRP_Succ:
   assumes "atom s \<sharp> (s1,k1,y)"
   shows "{SeqWRP s1 k1 y} \<turnstile> SyntaxN.Ex s (SeqWRP (Var s) (SUCC k1) (Q_Succ y))"
 proof -
   let ?vs = "(s,s1,k1,y)"
   obtain l::name and sl::name and l1::name and sl1::name
     where atoms:
       "atom l \<sharp> (?vs,sl1,l1,sl)"
       "atom sl \<sharp> (?vs,sl1,l1)"
       "atom l1 \<sharp> (?vs,sl1)"
       "atom sl1 \<sharp> (?vs)"
     by (metis obtain_fresh)
   let ?hyp = "{RestrictedP s1 (SUCC k1) (Var s), OrdP k1, SeqWRP s1 k1 y}"
   show ?thesis
     using assms atoms
     apply (auto simp: SeqWRP.simps [of l "Var s" _ sl])
     apply (rule cut_same [where A="OrdP k1"])
     apply (rule SeqWRP_imp_OrdP)
     apply (rule cut_same [OF exists_RestrictedP [of s s1 "SUCC k1"]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
       apply (rule Ex_I [where x="Eats (Var s) (HPair (SUCC k1)  (Q_Succ y))"])
       apply (simp_all (no_asm_simp))
     apply (rule Conj_I)
     apply (blast intro: RestrictedP_LstSeqP_Eats[THEN cut2] SeqWRP_LstSeqP[THEN cut1])
     apply (rule Conj_I)
      apply (rule Mem_Eats_I1)
      apply (blast intro: RestrictedP_Mem[THEN cut3] SeqWRP_HPair_Zero_Zero[THEN cut1] Zero_In_SUCC[THEN cut1])
   proof (rule All2_SUCC_I, simp_all)
     show "?hyp \<turnstile>  SyntaxN.Ex sl
      (HPair k1 (Var sl) IN Eats (Var s) (HPair (SUCC k1) (Q_Succ y)) AND
       HPair (SUCC k1) (Q_Succ (Var sl)) IN
       Eats (Var s) (HPair (SUCC k1) (Q_Succ y)))"
       \<comment> \<open>verifying the final values\<close>
       apply (rule Ex_I [where x="y"])
       using assms atoms apply simp
       apply (rule Conj_I[rotated])
        apply (rule Mem_Eats_I2, rule Refl)
      apply (rule Mem_Eats_I1)
       apply (rule RestrictedP_Mem[THEN cut3])
         apply (rule AssumeH)
        apply (simp add: LstSeqP_imp_Mem SeqWRP_LstSeqP thin1)
       apply (rule Mem_SUCC_Refl)
       done
   next
     show "?hyp \<turnstile> All2 l k1
      (SyntaxN.Ex sl
        (HPair (Var l) (Var sl) IN
         Eats (Var s) (HPair (SUCC k1) (Q_Succ y)) AND
         HPair (SUCC (Var l)) (Q_Succ (Var sl)) IN
         Eats (Var s) (HPair (SUCC k1) (Q_Succ y))))"
       \<comment> \<open>verifying the sequence buildup\<close>
       apply (rule All_I Imp_I)+
       using assms atoms apply simp_all
         \<comment> \<open>... the sequence buildup via s1\<close>
     apply (simp add: SeqWRP.simps [of l s1 _ sl])
       apply (rule AssumeH Ex_EH Conj_EH)+
       apply (rule All2_E [THEN rotate2], auto del: Disj_EH)
       apply (rule Ex_I [where x="Var sl"], simp)
       apply (rule Conj_I)
        apply (blast intro: Mem_Eats_I1 [OF RestrictedP_Mem [THEN cut3]] Mem_SUCC_I1)
       apply (blast intro: Mem_Eats_I1 [OF RestrictedP_Mem [THEN cut3]] OrdP_IN_SUCC)
       done
   qed
 qed (*>*)
 
 lemma WRP_Succ: "{OrdP i, WRP i y} \<turnstile> WRP (SUCC i) (Q_Succ y)"
 proof -
   obtain s t :: name where "atom s \<sharp> (i, y)" "atom t \<sharp> (s,i, y)" by (metis obtain_fresh)
   then show ?thesis
     by (subst WRP.simps[of s], simp, subst WRP.simps[of t], simp) (force intro: SeqWRP_Succ[THEN cut1])
 qed
 
 lemma WRP: "{} \<turnstile> WRP (ORD_OF i) \<lceil>ORD_OF i\<rceil>"
   by (induct i)
     (auto simp: WRP_Zero quot_Succ intro!: WRP_Succ[THEN cut2])
 
 lemma prove_WRP:  "{} \<turnstile> WRP \<lceil>Var x\<rceil> \<lceil>\<lceil>Var x\<rceil>\<rceil>"
   unfolding quot_Var quot_Succ
   by (rule WRP_Succ[THEN cut2]) (auto simp: WRP)
 
-subsection{*Proving that these relations are functions*}
+subsection\<open>Proving that these relations are functions\<close>
 
 lemma SeqWRP_Zero_E:
   assumes "insert (y EQ Zero) H \<turnstile> A"  "H \<turnstile> k EQ Zero"
   shows "insert (SeqWRP s k y) H \<turnstile> A"
 proof -
   obtain l::name and sl::name
     where "atom l \<sharp> (s,k,sl)" "atom sl \<sharp> (s)"
     by (metis obtain_fresh)
   thus ?thesis
     apply (auto simp: SeqWRP.simps [where s=s and l=l and sl=sl])
     apply (rule cut_same [where A = "LstSeqP s Zero y"])
     apply (blast intro: thin1 assms  LstSeqP_cong [OF Refl _ Refl, THEN Iff_MP_same])
     apply (rule cut_same [where A = "y EQ Zero"])
     apply (blast intro: LstSeqP_EQ)
     apply (metis rotate2 assms(1) thin1)
     done
 qed
 
 lemma SeqWRP_SUCC_lemma:
   assumes y': "atom y' \<sharp> (s,k,y)"
   shows "{SeqWRP s (SUCC k) y} \<turnstile> Ex y' (SeqWRP s k (Var y') AND y EQ Q_Succ (Var y'))"
 proof -
   obtain l::name and sl::name
     where atoms: "atom l \<sharp> (s,k,y,y',sl)" "atom sl \<sharp> (s,k,y,y')"
     by (metis obtain_fresh)
   thus ?thesis using y'
     apply (auto simp: SeqWRP.simps [where s=s and l=l and sl=sl])
     apply (rule All2_SUCC_E' [where t=k, THEN rotate2], auto)
     apply (rule Ex_I [where x = "Var sl"], auto)
     apply (blast intro: LstSeqP_SUCC) \<comment> \<open>showing @{term"SeqWRP s k (Var sl)"}\<close>
     apply (blast intro: ContraProve LstSeqP_EQ)
     done
 qed
 
 lemma SeqWRP_SUCC_E:
   assumes y': "atom y' \<sharp> (s,k,y)" and k': "H \<turnstile> k' EQ (SUCC k)"
   shows "insert (SeqWRP s k' y) H  \<turnstile> Ex y' (SeqWRP s k (Var y') AND y EQ Q_Succ (Var y'))"
   using SeqWRP_cong [OF Refl k' Refl] cut1 [OF SeqWRP_SUCC_lemma [of y' s k y]]
   by (metis Assume Iff_MP_left Iff_sym y')
 
 lemma SeqWRP_unique: "{OrdP x, SeqWRP s x y, SeqWRP s' x y'} \<turnstile> y' EQ y"
 proof -
   obtain i::name and j::name and j'::name and k::name and sl::name and sl'::name and l::name and pi::name
     where  i: "atom i \<sharp> (s,s',y,y')" and j: "atom j \<sharp> (s,s',i,x,y,y')" and j': "atom j' \<sharp> (s,s',i,j,x,y,y')"
       and atoms: "atom k \<sharp> (s,s',i,j,j')" "atom sl \<sharp> (s,s',i,j,j',k)" "atom sl' \<sharp> (s,s',i,j,j',k,sl)"
                  "atom pi \<sharp> (s,s',i,j,j',k,sl,sl')"
     by (metis obtain_fresh)
   have "{OrdP (Var i)} \<turnstile> All j (All j' (SeqWRP s (Var i) (Var j) IMP (SeqWRP s' (Var i) (Var j') IMP Var j' EQ Var j)))"
     apply (rule OrdIndH [where j=k])
     using i j j' atoms apply auto
     apply (rule rotate4)
     apply (rule OrdP_cases_E [where k=pi], simp_all)
     \<comment> \<open>Zero case\<close>
     apply (rule SeqWRP_Zero_E [THEN rotate3])
     prefer 2 apply blast
     apply (rule SeqWRP_Zero_E [THEN rotate4])
     prefer 2 apply blast
     apply (blast intro: ContraProve [THEN rotate4] Sym Trans)
     \<comment> \<open>SUCC case\<close>
     apply (rule Ex_I [where x = "Var pi"], auto)
     apply (metis ContraProve EQ_imp_SUBS2 Mem_SUCC_I2 Refl Subset_D)
     apply (rule cut_same)
     apply (rule SeqWRP_SUCC_E [of sl' s' "Var pi", THEN rotate4], auto)
     apply (rule cut_same)
     apply (rule SeqWRP_SUCC_E [of sl s "Var pi", THEN rotate7], auto)
     apply (rule All_E [where x = "Var sl", THEN rotate5], simp)
     apply (rule All_E [where x = "Var sl'"], simp)
     apply (rule Imp_E, blast)+
     apply (rule cut_same [OF Q_Succ_cong [OF Assume]])
     apply (blast intro: Trans [OF Hyp Sym] HPair_cong)
     done
   hence "{OrdP (Var i)} \<turnstile> (All j' (SeqWRP s (Var i) (Var j) IMP (SeqWRP s' (Var i) (Var j') IMP Var j' EQ Var j)))(j::=y)"
     by (metis All_D)
   hence "{OrdP (Var i)} \<turnstile> (SeqWRP s (Var i) y IMP (SeqWRP s' (Var i) (Var j') IMP Var j' EQ y))(j'::=y')"
     using j j'
     by simp (drule All_D [where x=y'], simp)
   hence "{} \<turnstile> OrdP (Var i) IMP (SeqWRP s (Var i) y IMP (SeqWRP s' (Var i) y' IMP y' EQ y))"
     using j j'
     by simp (metis Imp_I)
   hence "{} \<turnstile> (OrdP (Var i) IMP (SeqWRP s (Var i) y IMP (SeqWRP s' (Var i) y' IMP y' EQ y)))(i::=x)"
     by (metis Subst emptyE)
   thus ?thesis using i
     by simp (metis anti_deduction insert_commute)
 qed
 
 theorem WRP_unique: "{OrdP x, WRP x y, WRP x y'} \<turnstile> y' EQ y"
 proof -
   obtain s::name and s'::name
     where "atom s \<sharp> (x,y,y')"  "atom s' \<sharp> (x,y,y',s)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: SeqWRP_unique [THEN rotate3] WRP.simps [of s _ y]  WRP.simps [of s' _ y'])
 qed
 
-section{*The Function HF and Lemma 6.2*}
+section\<open>The Function HF and Lemma 6.2\<close>
 
-subsection {*Defining the syntax: quantified body*}
+subsection \<open>Defining the syntax: quantified body\<close>
 
 nominal_function SeqHRP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom l \<sharp> (s,k,sl,sl',m,n,sm,sm',sn,sn');
           atom sl \<sharp> (s,sl',m,n,sm,sm',sn,sn');
           atom sl' \<sharp> (s,m,n,sm,sm',sn,sn');
           atom m \<sharp> (s,n,sm,sm',sn,sn');
           atom n \<sharp> (s,sm,sm',sn,sn');
           atom sm \<sharp> (s,sm',sn,sn');
           atom sm' \<sharp> (s,sn,sn');
           atom sn \<sharp> (s,sn');
           atom sn' \<sharp> (s)\<rbrakk> \<Longrightarrow>
     SeqHRP x x' s k =
       LstSeqP s k (HPair x x') AND
       All2 l (SUCC k) (Ex sl (Ex sl' (HPair (Var l) (HPair (Var sl) (Var sl')) IN s AND
                 ((OrdP (Var sl) AND WRP (Var sl) (Var sl')) OR
                  Ex m (Ex n (Ex sm (Ex sm' (Ex sn (Ex sn' (Var m IN Var l AND Var n IN Var l AND
                        HPair (Var m) (HPair (Var sm) (Var sm')) IN s AND
                        HPair (Var n) (HPair (Var sn) (Var sn')) IN s AND
                        Var sl EQ HPair (Var sm) (Var sn) AND
                        Var sl' EQ Q_HPair (Var sm') (Var sn')))))))))))"
 by (auto simp: eqvt_def SeqHRP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
  shows SeqHRP_fresh_iff [simp]:
       "a \<sharp> SeqHRP x x' s k \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> x' \<and> a \<sharp> s \<and> a \<sharp> k"  (is ?thesis1)
   and SeqHRP_sf [iff]:  "Sigma_fm (SeqHRP x x' s k)"  (is ?thsf)
   and SeqHRP_imp_OrdP: "{ SeqHRP x y s k } \<turnstile> OrdP k"  (is ?thord)
   and SeqHRP_imp_LstSeqP: "{ SeqHRP x y s k } \<turnstile> LstSeqP s k (HPair x y)"  (is ?thlstseq)
 proof -
   obtain l::name and sl::name and sl'::name and m::name and n::name and
          sm::name and sm'::name and sn::name and sn'::name
     where atoms:
          "atom l \<sharp> (s,k,sl,sl',m,n,sm,sm',sn,sn')"
          "atom sl \<sharp> (s,sl',m,n,sm,sm',sn,sn')" "atom sl' \<sharp> (s,m,n,sm,sm',sn,sn')"
          "atom m \<sharp> (s,n,sm,sm',sn,sn')" "atom n \<sharp> (s,sm,sm',sn,sn')"
          "atom sm \<sharp> (s,sm',sn,sn')" "atom sm' \<sharp> (s,sn,sn')"
          "atom sn \<sharp> (s,sn')" "atom sn' \<sharp> (s)"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf ?thord ?thlstseq
     by (auto intro: LstSeqP_OrdP)
 qed
 
 lemma SeqHRP_subst [simp]:
       "(SeqHRP x x' s k)(i::=t) = SeqHRP (subst i t x) (subst i t x') (subst i t s) (subst i t k)"
 proof -
   obtain l::name and sl::name and sl'::name and m::name and n::name and
          sm::name and sm'::name and sn::name and sn'::name
     where "atom l \<sharp> (s,k,t,i,sl,sl',m,n,sm,sm',sn,sn')"
           "atom sl \<sharp> (s,t,i,sl',m,n,sm,sm',sn,sn')"
           "atom sl' \<sharp> (s,t,i,m,n,sm,sm',sn,sn')"
           "atom m \<sharp> (s,t,i,n,sm,sm',sn,sn')" "atom n \<sharp> (s,t,i,sm,sm',sn,sn')"
           "atom sm \<sharp> (s,t,i,sm',sn,sn')" "atom sm' \<sharp> (s,t,i,sn,sn')"
           "atom sn \<sharp> (s,t,i,sn')" "atom sn' \<sharp> (s,t,i)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: SeqHRP.simps [of l _ _ sl sl' m n sm sm' sn sn'])
 qed
 
 lemma SeqHRP_cong:
   assumes "H \<turnstile> x EQ x'" and "H \<turnstile> y EQ y'" "H \<turnstile> s EQ s'" and  "H \<turnstile> k EQ k'"
   shows "H \<turnstile> SeqHRP x y s k IFF SeqHRP x' y' s' k'"
   by (rule P4_cong [OF _ assms], auto)
 
-subsection {*Defining the syntax: main predicate*}
+subsection \<open>Defining the syntax: main predicate\<close>
 
 nominal_function HRP :: "tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom s \<sharp> (x,x',k); atom k \<sharp> (x,x')\<rbrakk> \<Longrightarrow>
          HRP x x' = Ex s (Ex k (SeqHRP x x' (Var s) (Var k)))"
   by (auto simp: eqvt_def HRP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
  shows HRP_fresh_iff [simp]: "a \<sharp> HRP x x' \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> x'"  (is ?thesis1)
    and HRP_sf [iff]:         "Sigma_fm (HRP x x')"  (is ?thsf)
 proof -
   obtain s::name and k::name  where "atom s \<sharp> (x,x',k)"  "atom k \<sharp> (x,x')"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf
     by auto
 qed
 
 lemma HRP_subst [simp]: "(HRP x x')(i::=t) = HRP (subst i t x) (subst i t x')"
 proof -
   obtain s::name and k::name where "atom s \<sharp> (x,x',t,i,k)"  "atom k \<sharp> (x,x',t,i)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: HRP.simps [of s _ _ k])
 qed
 
-subsection{*Proving that these relations are functions*}
+subsection\<open>Proving that these relations are functions\<close>
 
 lemma SeqHRP_lemma:
   assumes "atom m \<sharp> (x,x',s,k,n,sm,sm',sn,sn')" "atom n \<sharp> (x,x',s,k,sm,sm',sn,sn')"
           "atom sm \<sharp> (x,x',s,k,sm',sn,sn')" "atom sm' \<sharp> (x,x',s,k,sn,sn')"
           "atom sn \<sharp> (x,x',s,k,sn')" "atom sn' \<sharp> (x,x',s,k)"
     shows "{ SeqHRP x x' s k }
          \<turnstile> (OrdP x AND WRP x x') OR
              Ex m (Ex n (Ex sm (Ex sm' (Ex sn (Ex sn' (Var m IN k AND Var n IN k AND
                        SeqHRP (Var sm) (Var sm') s (Var m) AND
                        SeqHRP (Var sn) (Var sn') s (Var n) AND
                        x EQ HPair (Var sm) (Var sn) AND
                        x' EQ Q_HPair (Var sm') (Var sn')))))))"
 proof -
   obtain l::name and sl::name and sl'::name
     where atoms:
           "atom l \<sharp> (x,x',s,k,sl,sl',m,n,sm,sm',sn,sn')"
           "atom sl \<sharp> (x,x',s,k,sl',m,n,sm,sm',sn,sn')"
           "atom sl' \<sharp> (x,x',s,k,m,n,sm,sm',sn,sn')"
     by (metis obtain_fresh)
   thus ?thesis using atoms assms
     apply (simp add: SeqHRP.simps [of l s k sl sl' m n sm sm' sn sn'])
     apply (rule Conj_E)
     apply (rule All2_SUCC_E' [where t=k, THEN rotate2], simp_all)
     apply (rule rotate2)
     apply (rule Ex_E Conj_E)+
     apply (rule cut_same [where A = "HPair x x' EQ HPair (Var sl) (Var sl')"])
     apply (metis Assume LstSeqP_EQ rotate4, simp_all, clarify)
     apply (rule Disj_E [THEN rotate4])
     apply (rule Disj_I1)
     apply (metis Assume AssumeH(3) Sym thin1  Iff_MP_same [OF Conj_cong [OF OrdP_cong WRP_cong] Assume])
     \<comment> \<open>auto could be used but is VERY SLOW\<close>
     apply (rule Disj_I2)
     apply (rule Ex_E Conj_EH)+
     apply simp_all
     apply (rule Ex_I [where x = "Var m"], simp)
     apply (rule Ex_I [where x = "Var n"], simp)
     apply (rule Ex_I [where x = "Var sm"], simp)
     apply (rule Ex_I [where x = "Var sm'"], simp)
     apply (rule Ex_I [where x = "Var sn"], simp)
     apply (rule Ex_I [where x = "Var sn'"], simp)
     apply (simp add: SeqHRP.simps [of l _ _ sl sl' m n sm sm' sn sn'])
     apply (rule Conj_I, blast)+
     \<comment> \<open>first SeqHRP subgoal\<close>
     apply (rule Conj_I)+
     apply (blast intro: LstSeqP_Mem)
     apply (rule All2_Subset [OF Hyp], blast)
     apply (blast intro!: SUCC_Subset_Ord LstSeqP_OrdP, blast, simp)
     \<comment> \<open>next SeqHRP subgoal\<close>
     apply (rule Conj_I)+
     apply (blast intro: LstSeqP_Mem)
     apply (rule All2_Subset [OF Hyp], blast)
     apply (auto intro!: SUCC_Subset_Ord LstSeqP_OrdP)
     \<comment> \<open>finally, the equality pair\<close>
     apply (blast intro: Trans)+
     done
 qed
 
 lemma SeqHRP_unique: "{SeqHRP x y s u, SeqHRP x y' s' u'} \<turnstile> y' EQ y"
 proof -
   obtain i::name and j::name and j'::name and k::name and k'::name and l::name
      and m::name and n::name and sm::name and sn::name and sm'::name and sn'::name
      and m2::name and n2::name and sm2::name and sn2::name and sm2'::name and sn2'::name
     where atoms:  "atom i \<sharp> (s,s',y,y')"   "atom j \<sharp> (s,s',i,x,y,y')"  "atom j' \<sharp> (s,s',i,j,x,y,y')"
                   "atom k \<sharp> (s,s',x,y,y',u',i,j,j')" "atom k' \<sharp> (s,s',x,y,y',k,i,j,j')" "atom l \<sharp> (s,s',i,j,j',k,k')"
                   "atom m \<sharp> (s,s',i,j,j',k,k',l)"   "atom n \<sharp> (s,s',i,j,j',k,k',l,m)"
                   "atom sm \<sharp> (s,s',i,j,j',k,k',l,m,n)"  "atom sn \<sharp> (s,s',i,j,j',k,k',l,m,n,sm)"
                   "atom sm' \<sharp> (s,s',i,j,j',k,k',l,m,n,sm,sn)"   "atom sn' \<sharp> (s,s',i,j,j',k,k',l,m,n,sm,sn,sm')"
                   "atom m2 \<sharp> (s,s',i,j,j',k,k',l,m,n,sm,sn,sm',sn')"   "atom n2 \<sharp> (s,s',i,j,j',k,k',l,m,n,sm,sn,sm',sn',m2)"
                   "atom sm2 \<sharp> (s,s',i,j,j',k,k',l,m,n,sm,sn,sm',sn',m2,n2)"  "atom sn2 \<sharp> (s,s',i,j,j',k,k',l,m,n,sm,sn,sm',sn',m2,n2,sm2)"
                   "atom sm2' \<sharp> (s,s',i,j,j',k,k',l,m,n,sm,sn,sm',sn',m2,n2,sm2,sn2)"   "atom sn2' \<sharp> (s,s',i,j,j',k,k',l,m,n,sm,sn,sm',sn',m2,n2,sm2,sn2,sm2')"
     by (metis obtain_fresh)
   have "{OrdP (Var k)}
        \<turnstile> All i (All j (All j' (All k' (SeqHRP (Var i) (Var j) s (Var k) IMP (SeqHRP (Var i) (Var j') s' (Var k') IMP Var j' EQ Var j)))))"
     apply (rule OrdIndH [where j=l])
     using atoms apply auto
     apply (rule Swap)
     apply (rule cut_same)
     apply (rule cut1 [OF SeqHRP_lemma [of m "Var i" "Var j" s "Var k" n sm sm' sn sn']], simp_all, blast)
     apply (rule cut_same)
     apply (rule cut1 [OF SeqHRP_lemma [of m2 "Var i" "Var j'" s' "Var k'" n2 sm2 sm2' sn2 sn2']], simp_all, blast)
     apply (rule Disj_EH Conj_EH)+
     \<comment> \<open>case 1, both are ordinals\<close>
     apply (blast intro: cut3 [OF WRP_unique])
     \<comment> \<open>case 2, @{term "OrdP (Var i)"} but also a pair\<close>
     apply (rule Conj_EH Ex_EH)+
     apply simp_all
     apply (rule cut_same [where A = "OrdP (HPair (Var sm) (Var sn))"])
     apply (blast intro: OrdP_cong [OF Hyp, THEN Iff_MP_same], blast)
     \<comment> \<open>towards second two cases\<close>
     apply (rule Ex_E Disj_EH Conj_EH)+
     \<comment> \<open>case 3, @{term "OrdP (Var i)"} but also a pair\<close>
     apply (rule cut_same [where A = "OrdP (HPair (Var sm2) (Var sn2))"])
     apply (blast intro: OrdP_cong [OF Hyp, THEN Iff_MP_same], blast)
     \<comment> \<open>case 4, two pairs\<close>
     apply (rule Ex_E Disj_EH Conj_EH)+
     apply (rule All_E' [OF Hyp, where x="Var m"], blast)
     apply (rule All_E' [OF Hyp, where x="Var n"], blast, simp_all)
     apply (rule Disj_EH, blast intro: thin1 ContraProve)+
     apply (rule All_E [where x="Var sm"], simp)
     apply (rule All_E [where x="Var sm'"], simp)
     apply (rule All_E [where x="Var sm2'"], simp)
     apply (rule All_E [where x="Var m2"], simp)
     apply (rule All_E [where x="Var sn", THEN rotate2], simp)
     apply (rule All_E [where x="Var sn'"], simp)
     apply (rule All_E [where x="Var sn2'"], simp)
     apply (rule All_E [where x="Var n2"], simp)
     apply (rule cut_same [where A = "HPair (Var sm) (Var sn) EQ HPair (Var sm2) (Var sn2)"])
     apply (blast intro: Sym Trans)
     apply (rule cut_same [where A = "SeqHRP (Var sn) (Var sn2') s' (Var n2)"])
     apply (blast intro: SeqHRP_cong [OF Hyp Refl Refl, THEN Iff_MP2_same])
     apply (rule cut_same [where A = "SeqHRP (Var sm) (Var sm2') s' (Var m2)"])
     apply (blast intro: SeqHRP_cong [OF Hyp Refl Refl, THEN Iff_MP2_same])
     apply (rule Disj_EH, blast intro: thin1 ContraProve)+
     apply (blast intro: Trans [OF Hyp Sym] intro!: HPair_cong)
     done
   hence "{OrdP (Var k)}
          \<turnstile> All j (All j' (All k' (SeqHRP x (Var j) s (Var k)
                IMP (SeqHRP x (Var j') s' (Var k') IMP Var j' EQ Var j))))"
     apply (rule All_D [where x = x, THEN cut_same])
     using atoms by auto
   hence "{OrdP (Var k)}
          \<turnstile> All j' (All k' (SeqHRP x y s (Var k) IMP (SeqHRP x (Var j') s' (Var k') IMP Var j' EQ y)))"
     apply (rule All_D [where x = y, THEN cut_same])
     using atoms by auto
   hence "{OrdP (Var k)}
           \<turnstile> All k' (SeqHRP x y s (Var k) IMP (SeqHRP x y' s' (Var k') IMP y' EQ y))"
     apply (rule All_D [where x = y', THEN cut_same])
     using atoms by auto
   hence "{OrdP (Var k)} \<turnstile> SeqHRP x y s (Var k) IMP (SeqHRP x y' s' u' IMP y' EQ y)"
     apply (rule All_D [where x = u', THEN cut_same])
     using atoms by auto
   hence "{SeqHRP x y s (Var k)} \<turnstile> SeqHRP x y s (Var k) IMP (SeqHRP x y' s' u' IMP y' EQ y)"
     by (metis SeqHRP_imp_OrdP cut1)
   hence "{} \<turnstile> ((SeqHRP x y s (Var k) IMP (SeqHRP x y' s' u' IMP y' EQ y)))(k::=u)"
     by (metis Subst emptyE Assume MP_same Imp_I)
   hence "{} \<turnstile> SeqHRP x y s u IMP (SeqHRP x y' s' u' IMP y' EQ y)"
     using atoms by simp
   thus ?thesis
     by (metis anti_deduction insert_commute)
 qed
 
 theorem HRP_unique: "{HRP x y, HRP x y'} \<turnstile> y' EQ y"
 proof -
   obtain s::name and s'::name and k::name and k'::name
     where "atom s \<sharp> (x,y,y')" "atom s' \<sharp> (x,y,y',s)"
           "atom k \<sharp> (x,y,y',s,s')" "atom k' \<sharp> (x,y,y',s,s',k)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: SeqHRP_unique HRP.simps [of s x y k]  HRP.simps [of s' x y' k'])
 qed
 
 lemma HRP_ORD_OF: "{} \<turnstile> HRP (ORD_OF i) \<lceil>ORD_OF i\<rceil>"
 proof -
   let ?vs = "(i)"
   obtain s k l::name and sl::name and sl'::name and m::name and n::name and
     sm::name and sm'::name and sn::name and sn'::name
     where atoms:
       "atom s \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn',l,k)"
       "atom k \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn',l)"
       "atom l \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn')"
       "atom sl \<sharp> (?vs,sl',m,n,sm,sm',sn,sn')" "atom sl' \<sharp> (?vs,m,n,sm,sm',sn,sn')"
       "atom m \<sharp> (?vs,n,sm,sm',sn,sn')" "atom n \<sharp> (?vs,sm,sm',sn,sn')"
       "atom sm \<sharp> (?vs,sm',sn,sn')" "atom sm' \<sharp> (?vs,sn,sn')"
       "atom sn \<sharp> (?vs,sn')" "atom sn' \<sharp> ?vs"
     by (metis obtain_fresh)
   then show ?thesis
   apply (subst HRP.simps[of s _ _ k]; simp)
     apply (subst SeqHRP.simps[of l _ _ sl sl' m n sm sm' sn sn']; simp?)
     apply (rule Ex_I[where x="Eats Zero (HPair Zero (HPair (ORD_OF i) \<lceil>ORD_OF i\<rceil>))"]; simp)
     apply (rule Ex_I[where x="Zero"]; simp)
     apply (rule Conj_I[OF LstSeqP_single])
     apply (rule All2_SUCC_I, simp)
      apply auto [2]
     apply (rule Ex_I[where x="ORD_OF i"], simp)
     apply (rule Ex_I[where x="\<lceil>ORD_OF i\<rceil>"], simp)
     apply (auto intro!: Disj_I1 WRP Mem_Eats_I2)
     done
 qed
 
 lemma SeqHRP_HPair:
   assumes "atom s \<sharp> (k,s1,s2,k1,k2,x,y,x',y')" "atom k \<sharp> (s1,s2,k1,k2,x,y,x',y')"
   shows "{SeqHRP x x' s1 k1,
             SeqHRP y y' s2 k2}
            \<turnstile> Ex s (Ex k (SeqHRP (HPair x y) (Q_HPair x' y') (Var s) (Var k)))" (*<*)
 proof -
   let ?vs = "(s1,s2,s,k1,k2,k,x,y,x',y')"
   obtain km::name and kn::name and j::name and k'::name
     and l::name and sl::name and sl'::name and m::name and n::name
     and sm::name and sm'::name and sn::name and sn'::name
     where atoms2: "atom km \<sharp> (kn,j,k',l,s1,s2,s,k1,k2,k,x,y,x',y',sl,sl',m,n,sm,sm',sn,sn')"
       "atom kn \<sharp> (j,k',l,s1,s2,s,k1,k2,k,x,y,x',y',sl,sl',m,n,sm,sm',sn,sn')"
       "atom j \<sharp> (k',l,s1,s2,s,k1,k2,k,x,y,x',y',sl,sl',m,n,sm,sm',sn,sn')"
       and atoms: "atom k' \<sharp> (l,s1,s2,s,k1,k2,k,x,y,x',y',sl,sl',m,n,sm,sm',sn,sn')"
       "atom l \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',sl,sl',m,n,sm,sm',sn,sn')"
       "atom sl \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',sl',m,n,sm,sm',sn,sn')"
       "atom sl' \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',m,n,sm,sm',sn,sn')"
       "atom m \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',n,sm,sm',sn,sn')"
       "atom n \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',sm,sm',sn,sn')"
       "atom sm \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',sm',sn,sn')"
       "atom sm' \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',sn,sn')"
       "atom sn \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',sn')"
       "atom sn' \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y')"
     by (metis obtain_fresh)
   let ?hyp = "{HaddP k1 k2 (Var k'), OrdP k1, OrdP k2, SeqAppendP s1 (SUCC k1) s2 (SUCC k2) (Var s),
                SeqHRP x x' s1 k1, SeqHRP y y' s2 k2}"
   show ?thesis
     using assms atoms
     apply (auto simp: SeqHRP.simps [of l "Var s" _ sl sl' m n sm sm' sn sn'])
     apply (rule cut_same [where A="OrdP k1 AND OrdP k2"])
      apply (metis Conj_I SeqHRP_imp_OrdP thin1 thin2)
     apply (rule cut_same [OF exists_SeqAppendP [of s s1 "SUCC k1" s2 "SUCC k2"]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
       apply (rule cut_same [OF exists_HaddP [where j=k' and x=k1 and y=k2]])
         apply (rule AssumeH Ex_EH Conj_EH | simp)+
         apply (rule Ex_I [where x="Eats (Var s) (HPair (SUCC(SUCC(Var k'))) (HPair(HPair x y)(Q_HPair x' y')))"])
         apply (simp_all (no_asm_simp))
     apply (rule Ex_I [where x="SUCC (SUCC (Var k'))"], simp)
     apply (rule Conj_I)
      apply (blast intro: LstSeqP_SeqAppendP_Eats SeqHRP_imp_LstSeqP [THEN cut1])
   proof (rule All2_SUCC_I, simp_all)
     show "?hyp \<turnstile>     SyntaxN.Ex sl
      (SyntaxN.Ex sl'
        (HPair (SUCC (SUCC (Var k'))) (HPair (Var sl) (Var sl')) IN
         Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (HPair x y) (Q_HPair x' y'))) AND
         (OrdP (Var sl) AND WRP (Var sl) (Var sl') OR
          SyntaxN.Ex m
           (SyntaxN.Ex n
             (SyntaxN.Ex sm
               (SyntaxN.Ex sm'
                 (SyntaxN.Ex sn
                   (SyntaxN.Ex sn'
                     (Var m IN SUCC (SUCC (Var k')) AND
                      Var n IN SUCC (SUCC (Var k')) AND
                      HPair (Var m) (HPair (Var sm) (Var sm')) IN
                      Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (HPair x y) (Q_HPair x' y'))) AND
                      HPair (Var n) (HPair (Var sn) (Var sn')) IN
                      Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (HPair x y) (Q_HPair x' y'))) AND
                      Var sl EQ HPair (Var sm) (Var sn) AND Var sl' EQ Q_HPair (Var sm') (Var sn'))))))))))"
       \<comment> \<open>verifying the final values\<close>
       apply (rule Ex_I [where x="HPair x y"])
       using assms atoms apply simp
       apply (rule Ex_I [where x="Q_HPair x' y'"], simp)
       apply (rule Conj_I, metis Mem_Eats_I2 Refl)
       apply (rule Disj_I2)
       apply (rule Ex_I [where x=k1], simp)
       apply (rule Ex_I [where x="SUCC (Var k')"], simp)
       apply (rule Ex_I [where x=x], simp)
       apply (rule_tac x=x' in Ex_I, simp)
       apply (rule Ex_I [where x=y], simp)
       apply (rule_tac x=y' in Ex_I, simp)
       apply (rule Conj_I)
        apply (blast intro: HaddP_Mem_I LstSeqP_OrdP Mem_SUCC_I1)
       apply (rule Conj_I [OF Mem_SUCC_Refl])
       apply (blast intro: Disj_I1 Mem_Eats_I1 Mem_SUCC_Refl SeqHRP_imp_LstSeqP [THEN cut1]
           LstSeqP_imp_Mem SeqAppendP_Mem1 [THEN cut3] SeqAppendP_Mem2 [THEN cut4] HaddP_SUCC1 [THEN cut1])
       done
   next
     show "?hyp \<turnstile>  All2 l (SUCC (SUCC (Var k')))
      (SyntaxN.Ex sl
        (SyntaxN.Ex sl'
          (HPair (Var l) (HPair (Var sl) (Var sl')) IN
           Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (HPair x y) (Q_HPair x' y'))) AND
           (OrdP (Var sl) AND WRP (Var sl) (Var sl') OR
            SyntaxN.Ex m
             (SyntaxN.Ex n
               (SyntaxN.Ex sm
                 (SyntaxN.Ex sm'
                   (SyntaxN.Ex sn
                     (SyntaxN.Ex sn'
                       (Var m IN Var l AND
                        Var n IN Var l AND
                        HPair (Var m) (HPair (Var sm) (Var sm')) IN
                        Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (HPair x y) (Q_HPair x' y'))) AND
                        HPair (Var n) (HPair (Var sn) (Var sn')) IN
                        Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (HPair x y) (Q_HPair x' y'))) AND
                        Var sl EQ HPair (Var sm) (Var sn) AND Var sl' EQ Q_HPair (Var sm') (Var sn')))))))))))"
       \<comment> \<open>verifying the sequence buildup\<close>
       apply (rule cut_same [where A="HaddP (SUCC k1) (SUCC k2) (SUCC (SUCC (Var k')))"])
        apply (blast intro: HaddP_SUCC1 [THEN cut1] HaddP_SUCC2 [THEN cut1])
       apply (rule All_I Imp_I)+
        apply (rule HaddP_Mem_cases [where i=j])
       using assms atoms atoms2 apply simp_all
           apply (rule AssumeH)
          apply (blast intro: OrdP_SUCC_I LstSeqP_OrdP)
         \<comment> \<open>... the sequence buildup via s1\<close>
         apply (simp add: SeqHRP.simps [of l s1 _ sl sl' m n sm sm' sn sn'])
         apply (rule AssumeH Ex_EH Conj_EH)+
         apply (rule All2_E [THEN rotate2])
           apply (simp | rule AssumeH Ex_EH Conj_EH)+
             apply (rule Ex_I [where x="Var sl"], simp)
             apply (rule Ex_I [where x="Var sl'"], simp)
             apply (rule Conj_I [OF Mem_Eats_I1])
              apply (metis SeqAppendP_Mem1 rotate3 thin2 thin4)
             apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
                         apply (rule Ex_I [where x="Var m"], simp)
                         apply (rule Ex_I [where x="Var n"], simp)
                         apply (rule Ex_I [where x="Var sm"], simp)
                         apply (rule Ex_I [where x="Var sm'"], simp)
                         apply (rule Ex_I [where x="Var sn"], simp)
                         apply (rule Ex_I [where x="Var sn'"], simp_all (no_asm_simp))
        apply (rule Conj_I, rule AssumeH)+
        apply (rule Conj_I)
         apply (blast intro: OrdP_Trans [OF OrdP_SUCC_I] Mem_Eats_I1 [OF SeqAppendP_Mem1 [THEN cut3]] Hyp)
        apply (blast intro: Disj_I1 Disj_I2 OrdP_Trans [OF OrdP_SUCC_I] Mem_Eats_I1 [OF SeqAppendP_Mem1 [THEN cut3]] Hyp)
         \<comment> \<open>... the sequence buildup via s2\<close>
       apply (simp add: SeqHRP.simps [of l s2 _ sl sl' m n sm sm' sn sn'])
       apply (rule AssumeH Ex_EH Conj_EH)+
       apply (rule All2_E [THEN rotate2])
         apply (simp | rule AssumeH Ex_EH Conj_EH)+
           apply (rule Ex_I [where x="Var sl"], simp)
           apply (rule Ex_I [where x="Var sl'"], simp)
           apply (rule cut_same [where A="OrdP (Var j)"])
            apply (metis HaddP_imp_OrdP rotate2 thin2)
           apply (rule Conj_I)
            apply (blast intro: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] del: Disj_EH)
           apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
                       apply (rule cut_same [OF exists_HaddP [where j=km and x="SUCC k1" and y="Var m"]])
                         apply (blast intro: Ord_IN_Ord, simp)
                       apply (rule cut_same [OF exists_HaddP [where j=kn and x="SUCC k1" and y="Var n"]])
                         apply (metis AssumeH(6) Ord_IN_Ord0 rotate8, simp)
                       apply (rule AssumeH Ex_EH Conj_EH | simp)+
                           apply (rule Ex_I [where x="Var km"], simp)
                           apply (rule Ex_I [where x="Var kn"], simp)
                           apply (rule Ex_I [where x="Var sm"], simp)
                           apply (rule Ex_I [where x="Var sm'"], simp)
                           apply (rule Ex_I [where x="Var sn"], simp)
                           apply (rule Ex_I [where x="Var sn'"], simp_all (no_asm_simp))
       apply (rule Conj_I [OF _ Conj_I])
         apply (blast intro!: HaddP_Mem_cancel_left [THEN Iff_MP2_same] OrdP_SUCC_I intro: LstSeqP_OrdP Hyp)+
       apply (blast del: Disj_EH  intro: OrdP_Trans Hyp
           intro!: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] HaddP_imp_OrdP [THEN cut1])
       done
   qed
 qed (*>*)
 
 lemma HRP_HPair: "{HRP x x', HRP y y'} \<turnstile> HRP (HPair x y) (Q_HPair x' y')"
 proof -
   obtain k1::name and s1::name and k2::name and s2::name and k::name and s::name
     where "atom s1 \<sharp> (x,y,x',y')"        "atom k1 \<sharp> (x,y,x',y',s1)"
       "atom s2 \<sharp> (x,y,x',y',k1,s1)"  "atom k2 \<sharp> (x,y,x',y',s2,k1,s1)"
       "atom s  \<sharp> (x,y,x',y',k2,s2,k1,s1)" "atom k  \<sharp> (x,y,x',y',s,k2,s2,k1,s1)"
     by (metis obtain_fresh)
   thus ?thesis
     by (force simp: HRP.simps [of s "HPair x y" _ k]
         HRP.simps [of s1 x _ k1]
         HRP.simps [of s2 y _ k2]
         intro: SeqHRP_HPair [THEN cut2])
 qed
 
 lemma HRP_HPair_quot: "{HRP x \<lceil>x\<rceil>, HRP y \<lceil>y\<rceil>} \<turnstile> HRP (HPair x y) \<lceil>HPair x y\<rceil>"
   using HRP_HPair[of x "\<lceil>x\<rceil>" y "\<lceil>y\<rceil>"]
   unfolding HPair_def quot_simps by auto
 
 lemma prove_HRP_coding_tm: fixes t::tm shows "coding_tm t \<Longrightarrow> {} \<turnstile> HRP t \<lceil>t\<rceil>"
   by (induct t rule: coding_tm.induct)
     (auto simp: quot_simps HRP_ORD_OF HRP_HPair_quot[THEN cut2])
 
 lemmas prove_HRP = prove_HRP_coding_tm[OF quot_fm_coding]
 
-section{*The Function K and Lemma 6.3*}
+section\<open>The Function K and Lemma 6.3\<close>
 
 nominal_function KRP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "atom y \<sharp> (v,x,x') \<Longrightarrow>
          KRP v x x' = Ex y (HRP x (Var y) AND SubstFormP v (Var y) x x')"
   by (auto simp: eqvt_def KRP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma KRP_fresh_iff [simp]: "a \<sharp> KRP v x x' \<longleftrightarrow> a \<sharp> v \<and> a \<sharp> x \<and> a \<sharp> x'"
 proof -
   obtain y::name where "atom y \<sharp> (v,x,x')"
     by (metis obtain_fresh)
   thus ?thesis
     by auto
 qed
 
 lemma KRP_subst [simp]: "(KRP v x x')(i::=t) = KRP (subst i t v) (subst i t x) (subst i t x')"
 proof -
   obtain y::name where "atom y \<sharp> (v,x,x',t,i)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: KRP.simps [of y])
 qed
 
 declare KRP.simps [simp del]
 
 lemma prove_SubstFormP: "{} \<turnstile> SubstFormP \<lceil>Var i\<rceil> \<lceil>\<lceil>A\<rceil>\<rceil> \<lceil>A\<rceil> \<lceil>A(i::=\<lceil>A\<rceil>)\<rceil>"
   using SubstFormP by blast
 
 lemma prove_KRP: "{} \<turnstile> KRP \<lceil>Var i\<rceil> \<lceil>A\<rceil> \<lceil>A(i::=\<lceil>A\<rceil>)\<rceil>"
   by (auto simp: KRP.simps [of y]
            intro!: Ex_I [where x="\<lceil>\<lceil>A\<rceil>\<rceil>"] prove_HRP prove_SubstFormP)
 
 lemma KRP_unique: "{KRP v x y, KRP v x y'} \<turnstile> y' EQ y"
 proof -
   obtain u::name and u'::name where "atom u \<sharp> (v,x,y,y')" "atom u' \<sharp> (v,x,y,y',u)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: KRP.simps [of u v x y] KRP.simps [of u' v x y']
              intro: SubstFormP_cong [THEN Iff_MP2_same]
                     SubstFormP_unique [THEN cut2] HRP_unique [THEN cut2])
 qed
 
 lemma KRP_subst_fm: "{KRP \<lceil>Var i\<rceil> \<lceil>\<beta>\<rceil> (Var j)} \<turnstile> Var j EQ \<lceil>\<beta>(i::=\<lceil>\<beta>\<rceil>)\<rceil>"
   by (metis KRP_unique cut0 prove_KRP)
 
 end
 
diff --git a/thys/Goedel_HFSet_Semanticless/II_Prelims.thy b/thys/Goedel_HFSet_Semanticless/II_Prelims.thy
--- a/thys/Goedel_HFSet_Semanticless/II_Prelims.thy
+++ b/thys/Goedel_HFSet_Semanticless/II_Prelims.thy
@@ -1,4582 +1,4582 @@
-chapter{*Syntactic Preliminaries for the Second Incompleteness Theorem*}
+chapter\<open>Syntactic Preliminaries for the Second Incompleteness Theorem\<close>
 
 theory II_Prelims
 imports Pf_Predicates
 begin
 
 declare IndP.simps [simp del]
 
 lemma OrdP_ORD_OF [intro]: "H \<turnstile> OrdP (ORD_OF n)"
 proof -
   have "{} \<turnstile> OrdP (ORD_OF n)"
     by (induct n) (auto simp: OrdP_SUCC_I)
   thus ?thesis
     by (rule thin0)
 qed
 
 lemma VarP_Var [intro]: "H \<turnstile> VarP \<lceil>Var i\<rceil>"
   unfolding VarP_def
   by (auto simp: quot_Var OrdP_ORD_OF intro!: OrdP_SUCC_I cut1[OF Zero_In_SUCC])
 
 lemma VarP_neq_IndP: "{t EQ v, VarP v, IndP t} \<turnstile> Fls"
 proof -
   obtain m::name where "atom m \<sharp> (t,v)"
     by (metis obtain_fresh)
   thus ?thesis
     apply (auto simp: VarP_def IndP.simps [of m])
     apply (rule cut_same [of _ "OrdP (Q_Ind (Var m))"])
     apply (blast intro: Sym Trans OrdP_cong [THEN Iff_MP_same])
     by (metis OrdP_HPairE)
 qed
 
 lemma Mem_HFun_Sigma_OrdP: "{HPair t u IN f, HFun_Sigma f} \<turnstile> OrdP t"
 proof -
   obtain x::name and y::name and z::name and x'::name and y'::name and z'::name
     where "atom z \<sharp> (f,t,u,z',x,y,x',y')"  "atom z' \<sharp> (f,t,u,x,y,x',y')"
        "atom x \<sharp> (f,t,u,y,x',y')"  "atom y \<sharp> (f,t,u,x',y')"
        "atom x' \<sharp> (f,t,u,y')"  "atom y' \<sharp> (f,t,u)"
     by (metis obtain_fresh)
   thus ?thesis
   apply (simp add: HFun_Sigma.simps [of z f z' x y x' y'])
   apply (rule All2_E [where x="HPair t u", THEN rotate2], auto)
   apply (rule All2_E [where x="HPair t u"], auto intro: OrdP_cong [THEN Iff_MP2_same])
   done
 qed
 
-section {*NotInDom*}
+section \<open>NotInDom\<close>
 
 nominal_function NotInDom :: "tm \<Rightarrow> tm \<Rightarrow> fm"
   where "atom z \<sharp> (t, r) \<Longrightarrow> NotInDom t r = All z (Neg (HPair t (Var z) IN r))"
 by (auto simp: eqvt_def NotInDom_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma NotInDom_fresh_iff [simp]: "a \<sharp> NotInDom t r \<longleftrightarrow> a \<sharp> (t, r)"
 proof -
   obtain j::name where "atom j \<sharp> (t,r)"
     by (rule obtain_fresh)
   thus ?thesis
     by auto
 qed
 
 lemma subst_fm_NotInDom [simp]: "(NotInDom t r)(i::=x) = NotInDom (subst i x t) (subst i x r)"
 proof -
   obtain j::name where "atom j \<sharp> (i,x,t,r)"
     by (rule obtain_fresh)
   thus ?thesis
     by (auto simp: NotInDom.simps [of j])
 qed
 
 lemma NotInDom_cong: "H \<turnstile> t EQ t' \<Longrightarrow> H \<turnstile> r EQ r' \<Longrightarrow> H \<turnstile> NotInDom t r IFF NotInDom t' r'"
   by (rule P2_cong) auto
 
 lemma NotInDom_Zero: "H \<turnstile> NotInDom t Zero"
 proof -
   obtain z::name where "atom z \<sharp> t"
     by (metis obtain_fresh)
   hence "{} \<turnstile> NotInDom t Zero"
     by (auto simp: fresh_Pair)
   thus ?thesis
     by (rule thin0)
 qed
 
 lemma NotInDom_Fls: "{HPair d d' IN r, NotInDom d r} \<turnstile> A"
 proof -
   obtain z::name where "atom z \<sharp> (d,r)"
     by (metis obtain_fresh)
   hence "{HPair d d' IN r, NotInDom d r} \<turnstile> Fls"
     by (auto intro!: Ex_I [where x=d'])
   thus ?thesis
     by (metis ExFalso)
 qed
 
 lemma NotInDom_Contra: "H \<turnstile> NotInDom d r \<Longrightarrow> H \<turnstile> HPair x y IN r \<Longrightarrow> insert (x EQ d) H \<turnstile> A"
 by (rule NotInDom_Fls [THEN cut2, THEN ExFalso])
    (auto intro: thin1 NotInDom_cong [OF Assume Refl, THEN Iff_MP2_same])
 
 
-section {*Restriction of a Sequence to a Domain*}
+section \<open>Restriction of a Sequence to a Domain\<close>
 
 nominal_function RestrictedP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom x \<sharp> (y,f,k,g); atom y \<sharp> (f,k,g)\<rbrakk> \<Longrightarrow>
     RestrictedP f k g =
       g SUBS f AND
       All x (All y (HPair (Var x) (Var y) IN g IFF
                     (Var x) IN k AND HPair (Var x) (Var y) IN f))"
 by (auto simp: eqvt_def RestrictedP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma RestrictedP_fresh_iff [simp]: "a \<sharp> RestrictedP f k g \<longleftrightarrow> a \<sharp> f \<and> a \<sharp> k \<and> a \<sharp> g"
 proof -
   obtain x::name and y::name where "atom x \<sharp> (y,f,k,g)" "atom y \<sharp> (f,k,g)"
     by (metis obtain_fresh)
   thus ?thesis
     by auto
 qed
 
 lemma subst_fm_RestrictedP [simp]:
   "(RestrictedP f k g)(i::=u) = RestrictedP (subst i u f) (subst i u k) (subst i u g)"
 proof -
   obtain x::name and y::name  where "atom x \<sharp> (y,f,k,g,i,u)" "atom y \<sharp> (f,k,g,i,u)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: RestrictedP.simps [of x y])
 qed
 
 lemma RestrictedP_cong:
   "\<lbrakk>H \<turnstile> f EQ f'; H \<turnstile> k EQ A'; H \<turnstile> g EQ g'\<rbrakk>
    \<Longrightarrow> H \<turnstile> RestrictedP f k g IFF RestrictedP f' A' g'"
   by (rule P3_cong) auto
 
 lemma RestrictedP_Zero: "H \<turnstile> RestrictedP Zero k Zero"
 proof -
   obtain x::name and y::name  where "atom x \<sharp> (y,k)" "atom y \<sharp> (k)"
     by (metis obtain_fresh)
   hence "{} \<turnstile> RestrictedP Zero k Zero"
     by (auto simp: RestrictedP.simps [of x y])
   thus ?thesis
     by (rule thin0)
 qed
 
 lemma RestrictedP_Mem: "{ RestrictedP s k s', HPair a b IN s, a IN k } \<turnstile> HPair a b IN s'"
 proof -
   obtain x::name and y::name where "atom x \<sharp> (y,s,k,s',a,b)" "atom y \<sharp> (s,k,s',a,b)"
     by (metis obtain_fresh)
   thus ?thesis
     apply (auto simp: RestrictedP.simps [of x y])
     apply (rule All_E [where x=a, THEN rotate2], auto)
     apply (rule All_E [where x=b], auto intro: Iff_E2)
     done
 qed
 
 lemma RestrictedP_imp_Subset: "{RestrictedP s k s'} \<turnstile> s' SUBS s"
 proof -
   obtain x::name and y::name where "atom x \<sharp> (y,s,k,s')" "atom y \<sharp> (s,k,s')"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: RestrictedP.simps [of x y])
 qed
 
 lemma RestrictedP_Mem2:
   "{ RestrictedP s k s', HPair a b IN s' } \<turnstile> HPair a b IN s AND a IN k"
 proof -
   obtain x::name and y::name where "atom x \<sharp> (y,s,k,s',a,b)" "atom y \<sharp> (s,k,s',a,b)"
     by (metis obtain_fresh)
   thus ?thesis
     apply (auto simp: RestrictedP.simps [of x y] intro: Subset_D)
     apply (rule All_E [where x=a, THEN rotate2], auto)
     apply (rule All_E [where x=b], auto intro: Iff_E1)
     done
 qed
 
 lemma RestrictedP_Mem_D: "H \<turnstile> RestrictedP s k t \<Longrightarrow> H \<turnstile> a IN t \<Longrightarrow> insert (a IN s) H \<turnstile> A \<Longrightarrow> H \<turnstile> A"
   by (metis RestrictedP_imp_Subset Subset_E cut1)
 
 lemma RestrictedP_Eats:
   "{ RestrictedP s k s', a IN k } \<turnstile> RestrictedP (Eats s (HPair a b)) k (Eats s' (HPair a b))" (*<*)
 proof -
   obtain x::name and y::name
   where "atom x \<sharp> (y,s,k,s',a,b)" "atom y \<sharp> (s,k,s',a,b)"
     by (metis obtain_fresh)
   thus ?thesis
     apply (auto simp: RestrictedP.simps [of x y])
       apply (metis Assume Subset_Eats_I Subset_trans)
      apply (metis Mem_Eats_I2 Refl)
     apply (rule Swap, auto)
          apply (rule All_E [where x="Var x", THEN rotate2], auto)
          apply (rule All_E [where x="Var y"], simp)
          apply (metis Assume Conj_E Iff_E1)
         apply (blast intro: Subset_D)
        apply (blast intro: Mem_cong [THEN Iff_MP2_same])
       apply (metis Assume AssumeH(2) HPair_cong Mem_Eats_I2)
      apply (rule All_E [where x="Var x", THEN rotate3], auto)
      apply (rule All_E [where x="Var y"], simp)
      apply (metis Assume AssumeH(2) Conj_I Iff_E2 Mem_Eats_I1)
     apply (blast intro: Mem_Eats_I2 HPair_cong)
    done
 qed (*>*)
 
 lemma exists_RestrictedP:
   assumes s: "atom s \<sharp> (f,k)"
   shows  "H \<turnstile> Ex s (RestrictedP f k (Var s))" (*<*)
 proof -
   obtain j::name and x::name and y::name and z::name
     where atoms: "atom j \<sharp> (k,z,s)"  "atom x \<sharp> (j,k,z,s)"  "atom y \<sharp> (x,j,k,z,s)"  "atom z \<sharp> (s,k)"
     by (metis obtain_fresh)
   have "{} \<turnstile> Ex s (RestrictedP (Var z) k (Var s))"
     apply (rule Ind [of j z]) using atoms s
     apply simp_all
     apply (rule Ex_I [where x=Zero], simp add: RestrictedP_Zero)
     apply (rule All_I)+
     apply (auto del: Ex_EH)
     apply (rule thin1)
     apply (rule Ex_E)
     proof (rule Cases [where A="Ex x (Ex y ((Var x) IN k AND Var j EQ HPair (Var x) (Var y)))"], auto)
       show "{Var x IN k, Var j EQ HPair (Var x) (Var y), RestrictedP (Var z) k (Var s)}
             \<turnstile> Ex s (RestrictedP (Eats (Var z) (Var j)) k (Var s))"
         apply (rule Ex_I [where x="Eats (Var s) (HPair (Var x) (Var y))"])
         using atoms s apply auto
         apply (rule RestrictedP_cong [OF _ Refl Refl, THEN Iff_MP2_same])
         apply (blast intro: Eats_cong [OF Refl])
         apply (rule Var_Eq_subst_Iff [THEN rotate2, THEN Iff_MP_same])
         apply (auto intro: RestrictedP_Eats [THEN cut2])
         done
     next
       obtain u::name and v::name
       where uv: "atom u \<sharp> (x,y,z,s,j,k)" "atom v \<sharp> (u,x,y,z,s,j,k)"
         by (metis obtain_fresh)
       show "{Neg (Ex x (Ex y (Var x IN k AND Var j EQ HPair (Var x) (Var y)))),
              RestrictedP (Var z) k (Var s)} \<turnstile>
              Ex s (RestrictedP (Eats (Var z) (Var j)) k (Var s))"
         apply (rule Ex_I [where x="Var s"])
         using uv atoms
         apply (auto simp: RestrictedP.simps [of u v])
          apply (metis Assume Subset_Eats_I Subset_trans)
         apply (rule Swap, auto)
            apply (rule All_E [THEN rotate4, of _ _ "Var u"], auto)
            apply (rule All_E [where x="Var v"], simp)
            apply (metis Assume Conj_E Iff_E1)
           apply (rule Mem_Eats_I1)
           apply (metis Assume AssumeH(3) Subset_D)
          apply (rule All_E [where x="Var u", THEN rotate5], auto)
          apply (rule All_E [where x="Var v"], simp)
          apply (metis Assume AssumeH(2) Conj_I Iff_E2)
         apply (rule ContraProve [THEN rotate3])
        apply (rule Ex_I [where x="Var u"], simp)
        apply (rule Ex_I [where x="Var v"], auto intro: Sym)
        done
     qed
   hence "{} \<turnstile> (Ex s (RestrictedP (Var z) k (Var s)))(z::=f)"
     by (rule Subst) simp
   thus ?thesis using atoms s
     by simp (rule thin0)
 qed (*>*)
 
 lemma cut_RestrictedP:
   assumes s: "atom s \<sharp> (f,k,A)" and "\<forall>C \<in> H. atom s \<sharp> C"
   shows  "insert (RestrictedP f k (Var s)) H \<turnstile> A \<Longrightarrow> H \<turnstile> A"
   apply (rule cut_same [OF exists_RestrictedP [of s]])
   using assms apply auto
   done
 
 lemma RestrictedP_NotInDom: "{ RestrictedP s k s', Neg (j IN k) } \<turnstile> NotInDom j s'"
 proof -
   obtain x::name and y::name and z::name
   where "atom x \<sharp> (y,s,j,k,s')" "atom y \<sharp> (s,j,k,s')" "atom z \<sharp> (s,j,k,s')"
     by (metis obtain_fresh)
   thus ?thesis
     apply (auto simp: RestrictedP.simps [of x y] NotInDom.simps [of z])
     apply (rule All_E [where x=j, THEN rotate3], auto)
     apply (rule All_E, auto intro: Conj_E1 Iff_E1)
     done
 qed
 
 declare RestrictedP.simps [simp del]
 
-section {*Applications to LstSeqP*}
+section \<open>Applications to LstSeqP\<close>
 
 lemma HFun_Sigma_Eats:
   assumes "H \<turnstile> HFun_Sigma r" "H \<turnstile> NotInDom d r" "H \<turnstile> OrdP d"
     shows "H \<turnstile> HFun_Sigma (Eats r (HPair d d'))" (*<*)
 proof -
   obtain x::name and y::name and z::name and x'::name and y'::name and z'::name and z''::name
     where "atom z'' \<sharp> (r,d,d',z,z',x,y,x',y')"
       and "atom z \<sharp> (r,d,d',z',x,y,x',y')" and "atom z' \<sharp> (r,d,d',x,y,x',y')"
       and "atom x \<sharp> (r,d,d',y,x',y')" and "atom y \<sharp> (r,d,d',x',y')"
       and "atom x' \<sharp> (r,d,d',y')" and "atom y' \<sharp> (r,d,d')"
     by (metis obtain_fresh)
   hence "{ HFun_Sigma r, NotInDom d r, OrdP d } \<turnstile> HFun_Sigma (Eats r (HPair d d'))"
     apply (auto simp: HFun_Sigma.simps [of z _ z' x y x' y'])
       \<comment> \<open>case 1\<close>
        apply (rule Ex_I [where x = "Var z"], simp)
        apply (rule Neg_Imp_I, blast)
       apply (rule All_E [where x = "Var z'"], auto)
       \<comment> \<open>case 2\<close>
       apply (rule Ex_I [where x = "Var z"], simp)
       apply (rule Neg_Imp_I, blast)
       apply (rule All_E [where x = "Var z"], simp)
       apply (rule Imp_E, auto del: Disj_EH)
       apply (rule thin1)
       apply (rule thin1)
       apply (rule Ex_I [where x = "Var x"], simp)
       apply (rule Ex_I [where x = "Var y"], simp)
       apply (rule Ex_I [where x = d], simp)
       apply (rule Ex_I [where x = d'], auto)
       apply (blast intro: Disj_I1 OrdNotEqP_I NotInDom_Contra Mem_cong [THEN Iff_MP_same])
     \<comment> \<open>case 3\<close>
     apply (rule Ex_I [where x = "Var z'"])
     apply (subst subst_fm_Ex_with_renaming [where i'=z''] | subst subst_fm.simps)+
     apply (auto simp add: flip_fresh_fresh)
     apply (rule Ex_I [where x = "Var z'", THEN Swap], simp)
     apply (rule Neg_I)
     apply (rule Imp_E, auto del: Disj_EH)
     apply (rule thin1)
     apply (rule thin1)
     apply (rule Ex_I [where x = d], simp)
     apply (rule Ex_I [where x = d'], simp)
     apply (rule Ex_I [where x = "Var x"], simp)
     apply (rule Ex_I [where x = "Var y"], auto)
     apply (blast intro: Disj_I1 Sym_L OrdNotEqP_I NotInDom_Contra Mem_cong [THEN Iff_MP_same])
     \<comment> \<open>case 4\<close>
     apply (rule rotate2 [OF Swap])
     apply (rule Ex_I [where x = d], auto)
     apply (rule Ex_I [where x = d'], auto)
     apply (rule Ex_I [where x = d], auto)
     apply (rule Ex_I [where x = d'], auto intro: Disj_I2)
     done
   thus ?thesis using assms
     by (rule cut3)
 qed (*>*)
 
 lemma HFun_Sigma_single [iff]: "H \<turnstile> OrdP d \<Longrightarrow> H \<turnstile> HFun_Sigma (Eats Zero (HPair d d'))"
   by (metis HFun_Sigma_Eats HFun_Sigma_Zero NotInDom_Zero)
 
 lemma LstSeqP_single [iff]: "H \<turnstile> LstSeqP (Eats Zero (HPair Zero x)) Zero x"
   by (auto simp: LstSeqP.simps intro!: OrdP_SUCC_I HDomain_Incl_Eats_I Mem_Eats_I2)
 
 lemma NotInDom_LstSeqP_Eats:
   "{ NotInDom (SUCC k) s, LstSeqP s k y } \<turnstile> LstSeqP (Eats s (HPair (SUCC k) z)) (SUCC k) z"
 by (auto simp: LstSeqP.simps intro: HDomain_Incl_Eats_I Mem_Eats_I2 OrdP_SUCC_I HFun_Sigma_Eats)
 
 lemma RestrictedP_HDomain_Incl: "{HDomain_Incl s k, RestrictedP s k s'} \<turnstile> HDomain_Incl s' k"
 proof -
   obtain u::name and v::name and x::name and y::name and z::name
     where "atom u \<sharp> (v,s,k,s')" "atom v \<sharp> (s,k,s')"
           "atom x \<sharp> (s,k,s',u,v,y,z)" "atom y \<sharp> (s,k,s',u,v,z)" "atom z \<sharp> (s,k,s',u,v)"
     by (metis obtain_fresh)
   thus ?thesis
     apply (auto simp: HDomain_Incl.simps [of x _ _ y z])
     apply (rule Ex_I [where x="Var x"], auto)
     apply (rule Ex_I [where x="Var y"], auto)
     apply (rule Ex_I [where x="Var z"], simp)
     apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same, THEN rotate2])
     apply (auto simp: RestrictedP.simps [of u v])
     apply (rule All_E [where x="Var x", THEN rotate2], auto)
     apply (rule All_E [where x="Var y"])
     apply (auto intro: Iff_E ContraProve Mem_cong [THEN Iff_MP_same])
     done
 qed
 
 lemma RestrictedP_HFun_Sigma: "{HFun_Sigma s, RestrictedP s k s'} \<turnstile> HFun_Sigma s'"
   by (metis Assume RestrictedP_imp_Subset Subset_HFun_Sigma rcut2)
 
 lemma RestrictedP_LstSeqP:
   "{ RestrictedP s (SUCC k) s', LstSeqP s k y } \<turnstile> LstSeqP s' k y"
   by (auto simp: LstSeqP.simps
            intro: Mem_Neg_refl cut2 [OF RestrictedP_HDomain_Incl]
                                cut2 [OF RestrictedP_HFun_Sigma] cut3 [OF RestrictedP_Mem])
 
 lemma RestrictedP_LstSeqP_Eats:
   "{ RestrictedP s (SUCC k) s', LstSeqP s k y }
    \<turnstile> LstSeqP (Eats s' (HPair (SUCC k) z)) (SUCC k) z"
 by (blast intro: Mem_Neg_refl cut2 [OF NotInDom_LstSeqP_Eats]
                               cut2 [OF RestrictedP_NotInDom] cut2 [OF RestrictedP_LstSeqP])
 
 
-section{*Ordinal Addition*}
-
-subsection{*Predicate form, defined on sequences*}
+section\<open>Ordinal Addition\<close>
+
+subsection\<open>Predicate form, defined on sequences\<close>
 
 nominal_function SeqHaddP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom l \<sharp> (sl,s,k,j); atom sl \<sharp> (s,j)\<rbrakk> \<Longrightarrow>
     SeqHaddP s j k y = LstSeqP s k y AND
           HPair Zero j IN s AND
           All2 l k (Ex sl (HPair (Var l) (Var sl) IN s AND
                            HPair (SUCC (Var l)) (SUCC (Var sl)) IN s))"
 by (auto simp: eqvt_def SeqHaddP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma SeqHaddP_fresh_iff [simp]: "a \<sharp> SeqHaddP s j k y \<longleftrightarrow> a \<sharp> s \<and> a \<sharp> j \<and> a \<sharp> k \<and> a \<sharp> y"
 proof -
   obtain l::name and sl::name where "atom l \<sharp> (sl,s,k,j)" "atom sl \<sharp> (s,j)"
     by (metis obtain_fresh)
   thus ?thesis
     by force
 qed
 
 lemma SeqHaddP_subst [simp]:
   "(SeqHaddP s j k y)(i::=t) = SeqHaddP (subst i t s) (subst i t j) (subst i t k) (subst i t y)"
 proof -
   obtain l::name and sl::name where "atom l \<sharp> (s,k,j,sl,t,i)" "atom sl \<sharp> (s,k,j,t,i)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: SeqHaddP.simps [where l=l and sl=sl])
 qed
 
 declare SeqHaddP.simps [simp del]
 
 nominal_function HaddP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom s \<sharp> (x,y,z)\<rbrakk> \<Longrightarrow>
     HaddP x y z = Ex s (SeqHaddP (Var s) x y z)"
 by (auto simp: eqvt_def HaddP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma HaddP_fresh_iff [simp]: "a \<sharp> HaddP x y z \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> y \<and> a \<sharp> z"
 proof -
   obtain s::name where "atom s \<sharp> (x,y,z)"
     by (metis obtain_fresh)
   thus ?thesis
     by force
 qed
 
 lemma HaddP_subst [simp]: "(HaddP x y z)(i::=t) = HaddP (subst i t x) (subst i t y) (subst i t z)"
 proof -
   obtain s::name where "atom s \<sharp> (x,y,z,t,i)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: HaddP.simps [of s])
 qed
 
 lemma HaddP_cong: "\<lbrakk>H \<turnstile> t EQ t'; H \<turnstile> u EQ u'; H \<turnstile> v EQ v'\<rbrakk> \<Longrightarrow> H \<turnstile> HaddP t u v IFF HaddP t' u' v'"
   by (rule P3_cong) auto
 
 declare HaddP.simps [simp del]
 
 lemma HaddP_Zero2: "H \<turnstile> HaddP x Zero x"
 proof -
   obtain s::name and l::name and sl::name where "atom l \<sharp> (sl,s,x)" "atom sl \<sharp> (s,x)" "atom s \<sharp> x"
     by (metis obtain_fresh)
   hence "{} \<turnstile> HaddP x Zero x"
     by (auto simp: HaddP.simps [of s] SeqHaddP.simps [of l sl]
           intro!: Mem_Eats_I2 Ex_I [where x="Eats Zero (HPair Zero x)"])
   thus ?thesis
     by (rule thin0)
 qed
 
 lemma HaddP_imp_OrdP: "{HaddP x y z} \<turnstile> OrdP y"
 proof -
   obtain s::name and l::name and sl::name
     where "atom l \<sharp> (sl,s,x,y,z)" "atom sl \<sharp> (s,x,y,z)" "atom s \<sharp> (x,y,z)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: HaddP.simps [of s] SeqHaddP.simps [of l sl] LstSeqP.simps)
 qed
 
 lemma HaddP_SUCC2: "{HaddP x y z} \<turnstile> HaddP x (SUCC y) (SUCC z)" (*<*)
 proof -
   obtain s::name and s'::name and l::name and sl::name
     where "atom s' \<sharp> (l,sl,s,x,y,z)" "atom l \<sharp> (sl,s,x,y,z)" "atom sl \<sharp> (s,x,y,z)" "atom s \<sharp> (x,y,z)"
     by (metis obtain_fresh)
   hence "{HaddP x y z, OrdP y} \<turnstile> HaddP x (SUCC y) (SUCC z)"
     apply (auto simp: HaddP.simps [of s] SeqHaddP.simps [of l sl])
     apply (rule cut_RestrictedP [of s' "Var s" "SUCC y"], auto)
     apply (rule Ex_I [where x="Eats (Var s') (HPair (SUCC y) (SUCC z))"])
        apply (auto intro!: Mem_SUCC_EH)
        apply (metis rotate2 RestrictedP_LstSeqP_Eats rotate3 thin1)
       apply (blast intro: Mem_Eats_I1 cut3 [OF RestrictedP_Mem] cut1 [OF Zero_In_SUCC])
      apply (rule Ex_I [where x="Var l"], auto)
      apply (rule Ex_I [where x="Var sl"], auto)
      apply (blast intro: Mem_Eats_I1 cut3 [OF RestrictedP_Mem] Mem_SUCC_I1)
     apply (blast intro: Mem_Eats_I1 cut3 [OF RestrictedP_Mem] OrdP_IN_SUCC)
    apply (rule ContraProve [THEN rotate2])
    apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same], simp add: LstSeqP.simps)
    apply (rule Ex_I [where x=z])
    apply (force intro: Mem_Eats_I1 Mem_Eats_I2 cut3 [OF RestrictedP_Mem] Mem_SUCC_I2)
    done
  thus ?thesis
    by (metis Assume HaddP_imp_OrdP cut2)
 qed (*>*)
 
-subsection{*Proving that these relations are functions*}
+subsection\<open>Proving that these relations are functions\<close>
 
 lemma SeqHaddP_Zero_E: "{SeqHaddP s w Zero z} \<turnstile> w EQ z"
 proof -
   obtain l::name and sl::name where "atom l \<sharp> (s,w,z,sl)" "atom sl \<sharp> (s,w)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: SeqHaddP.simps [of l sl] LstSeqP.simps intro: HFun_Sigma_E)
 qed
 
 lemma SeqHaddP_SUCC_lemma:
   assumes y': "atom y' \<sharp> (s,j,k,y)"
   shows "{SeqHaddP s j (SUCC k) y} \<turnstile> Ex y' (SeqHaddP s j k (Var y') AND y EQ SUCC (Var y'))"
 proof -
   obtain l::name and sl::name  where "atom l \<sharp> (s,j,k,y,y',sl)" "atom sl \<sharp> (s,j,k,y,y')"
     by (metis obtain_fresh)
   thus ?thesis using y'
     apply (auto simp: SeqHaddP.simps [where s=s and l=l and sl=sl])
     apply (rule All2_SUCC_E' [where t=k, THEN rotate2], auto)
     apply (auto intro!: Ex_I [where x="Var sl"])
     apply (blast intro: LstSeqP_SUCC) \<comment> \<open>showing @{term"SeqHaddP s j k (Var sl)"}\<close>
     apply (blast intro: LstSeqP_EQ)
     done
 qed
 
 lemma SeqHaddP_SUCC:
   assumes "H \<turnstile> SeqHaddP s j (SUCC k) y"  "atom y' \<sharp> (s,j,k,y)"
   shows "H \<turnstile> Ex y' (SeqHaddP s j k (Var y') AND y EQ SUCC (Var y'))"
   by (metis SeqHaddP_SUCC_lemma [THEN cut1] assms)
 
 lemma SeqHaddP_unique: "{OrdP x, SeqHaddP s w x y, SeqHaddP s' w x y'} \<turnstile> y' EQ y" (*<*)
 proof -
   obtain i::name and j::name and j'::name and k::name and sl::name and sl'::name
      and l::name and ji::name and ji'::name
    where ij: "atom i \<sharp> (s,s',w,y,y')" "atom j \<sharp> (s,s',w,i,x,y,y')" "atom j' \<sharp> (s,s',w,i,j,x,y,y')"
      and atoms: "atom k \<sharp> (s,s',w,i,j,j')" "atom sl \<sharp> (s,s',w,i,j,j',k)" "atom sl' \<sharp> (s,s',w,i,j,j',k,sl)"
                 "atom ji \<sharp> (s,s',w,i,j,j',k,sl,sl')"  "atom ji' \<sharp> (s,s',w,i,j,j',k,sl,sl',ji)"
     by (metis obtain_fresh)
   have "{OrdP (Var i)}
         \<turnstile> All j (All j' (SeqHaddP s w (Var i) (Var j) IMP (SeqHaddP s' w (Var i) (Var j') IMP Var j' EQ Var j)))"
     apply (rule OrdInd2H)
     using ij atoms apply auto
     apply (metis SeqHaddP_Zero_E [THEN cut1] Assume AssumeH(2) Sym Trans)
     \<comment> \<open>SUCC case\<close>
     apply (rule cut_same [OF SeqHaddP_SUCC [where y' = ji and s=s]], auto)
     apply (rule cut_same [OF SeqHaddP_SUCC [where y' = ji' and s=s']], auto)
     apply (rule Ex_I [where x = "Var ji"], auto)
     apply (rule All_E [where x = "Var ji'"], auto)
     apply (blast intro: Trans [OF Hyp] Sym intro!: SUCC_cong)
     done
   hence "{OrdP (Var i)}
          \<turnstile> (All j' (SeqHaddP s w (Var i) (Var j) IMP (SeqHaddP s' w (Var i) (Var j') IMP Var j' EQ Var j)))(j::=y)"
     by (metis All_D)
   hence "{OrdP (Var i)}
          \<turnstile> All j' (SeqHaddP s w (Var i) y IMP (SeqHaddP s' w (Var i) (Var j') IMP Var j' EQ y))"
     using ij by simp
   hence "{OrdP (Var i)}
          \<turnstile> (SeqHaddP s w (Var i) y IMP (SeqHaddP s' w (Var i) (Var j') IMP Var j' EQ y))(j'::=y')"
     by (metis All_D)
   hence "{OrdP (Var i)} \<turnstile> SeqHaddP s w (Var i) y IMP (SeqHaddP s' w (Var i) y' IMP y' EQ y)"
     using ij by simp
   hence "{} \<turnstile> (OrdP (Var i) IMP SeqHaddP s w (Var i) y IMP (SeqHaddP s' w (Var i) y' IMP y' EQ y))(i::=x)"
     by (metis Imp_I Subst emptyE)
   thus ?thesis
     using ij by simp (metis DisjAssoc2 Disj_commute anti_deduction)
 qed (*>*)
 
 lemma HaddP_unique: "{HaddP w x y, HaddP w x y'} \<turnstile> y' EQ y"
 proof -
   obtain s::name and s'::name  where "atom s \<sharp> (w,x,y,y')" "atom s' \<sharp> (w,x,y,y',s)"
     by (metis obtain_fresh)
   hence "{OrdP x, HaddP w x y, HaddP w x y'} \<turnstile> y' EQ y"
     by (auto simp: HaddP.simps [of s _ _ y]  HaddP.simps [of s' _ _ y']
              intro: SeqHaddP_unique [THEN cut3])
   thus ?thesis
     by (metis HaddP_imp_OrdP cut_same thin1)
 qed
 
 lemma HaddP_Zero1: assumes "H \<turnstile> OrdP x" shows "H \<turnstile> HaddP Zero x x"
 proof -
   fix k::name
   have "{ OrdP (Var k) } \<turnstile> HaddP Zero (Var k) (Var k)"
     by (rule OrdInd2H [where i=k]) (auto intro: HaddP_Zero2 HaddP_SUCC2 [THEN cut1])
   hence "{} \<turnstile> OrdP (Var k) IMP HaddP Zero (Var k) (Var k)"
     by (metis Imp_I)
   hence "{} \<turnstile> (OrdP (Var k) IMP HaddP Zero (Var k) (Var k))(k::=x)"
     by (rule Subst) auto
   hence "{} \<turnstile> OrdP x IMP HaddP Zero x x"
     by simp
   thus ?thesis using assms
     by (metis MP_same thin0)
 qed
 
 lemma HaddP_Zero_D1: "insert (HaddP Zero x y) H \<turnstile> x EQ y"
   by (metis Assume HaddP_imp_OrdP HaddP_Zero1 HaddP_unique [THEN cut2] rcut1)
 
 lemma HaddP_Zero_D2: "insert (HaddP x Zero y) H \<turnstile> x EQ y"
   by (metis Assume HaddP_Zero2 HaddP_unique [THEN cut2])
 
 lemma HaddP_SUCC_Ex2:
   assumes "H \<turnstile> HaddP x (SUCC y) z" "atom z' \<sharp> (x,y,z)"
     shows "H \<turnstile> Ex z' (HaddP x y (Var z') AND z EQ SUCC (Var z'))"
 proof -
   obtain s::name and s'::name  where "atom s \<sharp> (x,y,z,z')" "atom s' \<sharp> (x,y,z,z',s)"
     by (metis obtain_fresh)
   hence "{ HaddP x (SUCC y) z } \<turnstile> Ex z' (HaddP x y (Var z') AND z EQ SUCC (Var z'))"
     using assms
     apply (auto simp: HaddP.simps [of s _ _ ]  HaddP.simps [of s' _ _ ])
     apply (rule cut_same [OF SeqHaddP_SUCC_lemma [of z']], auto)
     apply (rule Ex_I, auto)+
     done
   thus ?thesis
     by (metis assms(1) cut1)
 qed
 
 lemma HaddP_SUCC1: "{ HaddP x y z } \<turnstile> HaddP (SUCC x) y (SUCC z)" (*<*)
 proof -
   obtain i::name and j::name and z'::name
    where atoms: "atom i \<sharp> (x,y,z)"  "atom j \<sharp> (i,x,y,z)"  "atom z' \<sharp> (x,i,j)"
     by (metis obtain_fresh)
   have "{OrdP (Var i)} \<turnstile> All j (HaddP x (Var i) (Var j) IMP HaddP (SUCC x) (Var i) (SUCC (Var j)))"
        (is "_ \<turnstile> ?scheme")
     proof (rule OrdInd2H)
       show "{} \<turnstile> ?scheme(i::=Zero)"
         using atoms apply auto
         apply (rule cut_same [OF HaddP_Zero_D2])
         apply (rule Var_Eq_subst_Iff [THEN Sym_L, THEN Iff_MP_same], auto intro: HaddP_Zero2)
         done
     next
       show "{} \<turnstile> All i (OrdP (Var i) IMP ?scheme IMP ?scheme(i::=SUCC (Var i)))"
         using atoms
         apply auto
         apply (rule cut_same [OF HaddP_SUCC_Ex2 [where z'=z']], auto)
         apply (rule Ex_I [where x="Var z'"], auto)
         apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same, THEN rotate3], simp)
         by (metis Assume HaddP_SUCC2 cut1 thin1)
     qed
   hence "{OrdP (Var i)} \<turnstile> (HaddP x (Var i) (Var j) IMP HaddP (SUCC x) (Var i) (SUCC (Var j)))(j::=z)"
     by (rule All_D)
   hence "{OrdP (Var i)} \<turnstile> HaddP x (Var i) z IMP HaddP (SUCC x) (Var i) (SUCC z)"
     using atoms by auto
   hence "{} \<turnstile> HaddP x (Var i) z IMP HaddP (SUCC x) (Var i) (SUCC z)"
     by (metis HaddP_imp_OrdP Imp_cut)
   hence "{} \<turnstile> (HaddP x (Var i) z IMP HaddP (SUCC x) (Var i) (SUCC z))(i::=y)"
     using atoms by (force intro!: Subst)
   thus ?thesis
     using atoms by simp (metis anti_deduction)
 qed (*>*)
 
 lemma HaddP_commute: "{HaddP x y z, OrdP x} \<turnstile> HaddP y x z" (*<*)
 proof -
   obtain i::name and j::name and z'::name
    where atoms: "atom i \<sharp> (x,y,z)"  "atom j \<sharp> (i,x,y,z)"  "atom z' \<sharp> (x,i,j)"
     by (metis obtain_fresh)
   have "{OrdP (Var i), OrdP x} \<turnstile> All j (HaddP x (Var i) (Var j) IMP HaddP (Var i) x (Var j))"
        (is "_ \<turnstile> ?scheme")
     proof (rule OrdInd2H)
       show "{OrdP x} \<turnstile> ?scheme(i::=Zero)"
         using atoms apply auto
         apply (rule cut_same [OF HaddP_Zero_D2])
         apply (rule Var_Eq_subst_Iff [THEN Sym_L, THEN Iff_MP_same], auto intro: HaddP_Zero1)
         done
     next
       show "{OrdP x} \<turnstile> All i (OrdP (Var i) IMP ?scheme IMP ?scheme(i::=SUCC (Var i)))"
         using atoms
         apply auto
         apply (rule cut_same [OF HaddP_SUCC_Ex2 [where z'=z']], auto)
         apply (rule Ex_I [where x="Var z'"], auto)
         apply (rule rotate3)
         apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same], simp)
         by (metis Assume HaddP_SUCC1 cut1 thin1)
     qed
   hence "{OrdP (Var i), OrdP x} \<turnstile> (HaddP x (Var i) (Var j) IMP HaddP (Var i) x (Var j))(j::=z)"
     by (rule All_D)
   hence "{OrdP (Var i), OrdP x} \<turnstile> HaddP x (Var i) z IMP HaddP (Var i) x z"
     using atoms by auto
   hence "{OrdP x} \<turnstile> HaddP x (Var i) z IMP HaddP (Var i) x z"
     by (metis HaddP_imp_OrdP Imp_cut)
   hence "{OrdP x} \<turnstile> (HaddP x (Var i) z IMP HaddP (Var i) x z)(i::=y)"
     using atoms by (force intro!: Subst)
   thus ?thesis
     using atoms by simp (metis anti_deduction)
 qed (*>*)
 
 lemma HaddP_SUCC_Ex1:
   assumes "atom i \<sharp> (x,y,z)"
     shows "insert (HaddP (SUCC x) y z) (insert (OrdP x) H)
            \<turnstile> Ex i (HaddP x y (Var i) AND z EQ SUCC (Var i))"
 proof -
   have "{ HaddP (SUCC x) y z, OrdP x } \<turnstile> Ex i (HaddP x y (Var i) AND z EQ SUCC (Var i))"
     apply (rule cut_same [OF HaddP_commute [THEN cut2]])
     apply (blast intro: OrdP_SUCC_I)+
     apply (rule cut_same [OF HaddP_SUCC_Ex2 [where z'=i]], blast)
     using assms apply auto
     apply (auto intro!: Ex_I [where x="Var i"])
     by (metis AssumeH(2) HaddP_commute [THEN cut2] HaddP_imp_OrdP rotate2 thin1)
   thus ?thesis
     by (metis Assume AssumeH(2) cut2)
 qed
 
 lemma HaddP_inv2: "{HaddP x y z, HaddP x y' z, OrdP x} \<turnstile> y' EQ y" (*<*)
 proof -
   obtain i::name and j::name and u::name and u'::name
    where atoms: "atom i \<sharp> (x,y,y',z)"  "atom j \<sharp> (i,x,y,y',z)"
                 "atom u \<sharp> (x,y,y',i,j)" "atom u' \<sharp> (x,y,y',u,i,j)"
     by (metis obtain_fresh)
   have "{OrdP (Var i)} \<turnstile> All j (HaddP (Var i) y (Var j) IMP HaddP (Var i) y' (Var j) IMP y' EQ y)"
        (is "_ \<turnstile> ?scheme")
     proof (rule OrdInd2H)
       show "{} \<turnstile> ?scheme(i::=Zero)"
         using atoms
         by auto (metis HaddP_Zero_D1 Sym Trans thin1)
     next
       show "{} \<turnstile> All i (OrdP (Var i) IMP ?scheme IMP ?scheme(i::=SUCC (Var i)))"
         using atoms
         apply auto
         apply (rule cut_same [OF HaddP_SUCC_Ex1 [where y=y and i=u, THEN cut2]], auto)
         apply (rule Ex_I [where x="Var u"], auto)
         apply (rule cut_same [OF HaddP_SUCC_Ex1 [where y=y' and i=u', THEN cut2]], auto)
         apply (rule cut_same [where A="SUCC (Var u) EQ SUCC (Var u')"])
         apply (auto intro: Sym Trans)
         apply (rule rotate4 [OF ContraProve])
         apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same], force)
         done
     qed
   hence "{OrdP (Var i)} \<turnstile> (HaddP (Var i) y (Var j) IMP HaddP (Var i) y' (Var j) IMP y' EQ y)(j::=z)"
     by (rule All_D)
   hence "{OrdP (Var i)} \<turnstile> HaddP (Var i) y z IMP HaddP (Var i) y' z IMP y' EQ y"
     using atoms by auto
   hence "{} \<turnstile> OrdP (Var i) IMP HaddP (Var i) y z IMP HaddP (Var i) y' z IMP y' EQ y"
     by (metis Imp_I)
   hence "{} \<turnstile> (OrdP (Var i) IMP HaddP (Var i) y z IMP HaddP (Var i) y' z IMP y' EQ y)(i::=x)"
     using atoms by (force intro!: Subst)
   thus ?thesis
     using atoms by simp (metis DisjAssoc2 Disj_commute anti_deduction)
 qed (*>*)
 
 lemma Mem_imp_subtract: (*<*)
   assumes "H \<turnstile> x IN y" "H \<turnstile> OrdP y" and k: "atom (k::name) \<sharp> (x,y)"
     shows  "H \<turnstile> Ex k (HaddP x (Var k) y AND Zero IN (Var k))"
 proof -
   obtain i::name
    where atoms: "atom i \<sharp> (x,y,k)"
     by (metis obtain_fresh)
   have "{OrdP (Var i)} \<turnstile> x IN Var i IMP Ex k (HaddP x (Var k) (Var i) AND Zero IN (Var k))"
        (is "_ \<turnstile> ?scheme")
     proof (rule OrdInd2H)
       show "{} \<turnstile> ?scheme(i::=Zero)"
         by auto
     next
       show "{} \<turnstile> All i (OrdP (Var i) IMP ?scheme IMP ?scheme(i::=SUCC (Var i)))"
         using atoms k
         apply (auto intro!: Mem_SUCC_EH)
         apply (rule Ex_I [where x="SUCC (Var k)"], auto)
         apply (metis AssumeH(4) HaddP_SUCC2 cut1 insert_commute)
         apply (blast intro: Mem_SUCC_I1)
         apply (rule Ex_I [where x="SUCC Zero"], auto)
         apply (rule thin1)
         apply (rule Var_Eq_subst_Iff [THEN Sym_L, THEN Iff_MP_same], simp)
         apply (metis HaddP_SUCC2 HaddP_Zero2 cut1)
         apply (rule Ex_I [where x="SUCC (Var k)"], auto intro: Mem_SUCC_I1)
         apply (metis AssumeH(4) HaddP_SUCC2 cut1 insert_commute)
         done
     qed
   hence "{} \<turnstile> OrdP (Var i) IMP x IN Var i IMP Ex k (HaddP x (Var k) (Var i) AND Zero IN (Var k))"
     by (metis Imp_I)
   hence "{} \<turnstile> (OrdP (Var i) IMP x IN Var i IMP Ex k (HaddP x (Var k) (Var i) AND Zero IN (Var k)))(i::=y)"
     by (force intro!: Subst)
   thus ?thesis using assms atoms
     by simp (metis (no_types) anti_deduction cut2)
 qed (*>*)
 
 lemma HaddP_OrdP:
   assumes "H \<turnstile> HaddP x y z" "H \<turnstile> OrdP x"  shows "H \<turnstile> OrdP z"  (*<*)
 proof -
   obtain i::name and j::name and k::name
    where atoms: "atom i \<sharp> (x,y,z)"  "atom j \<sharp> (i,x,y,z)"  "atom k \<sharp> (i,j,x,y,z)"
     by (metis obtain_fresh)
   have "{OrdP (Var i), OrdP x} \<turnstile> All j (HaddP x (Var i) (Var j) IMP OrdP (Var j))"
        (is "_ \<turnstile> ?scheme")
     proof (rule OrdInd2H)
       show "{OrdP x} \<turnstile> ?scheme(i::=Zero)"
         using atoms
         by (auto intro: HaddP_Zero_D2 OrdP_cong [THEN Iff_MP_same])
     next
       show "{OrdP x} \<turnstile> All i (OrdP (Var i) IMP ?scheme IMP ?scheme(i::=SUCC (Var i)))"
         using atoms
         apply auto
         apply (rule cut_same [OF HaddP_SUCC_Ex2 [where z'=k]], auto)
         apply (rule Ex_I [where x="Var k"], auto)
         apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same, THEN rotate3], auto intro: OrdP_SUCC_I)
         done
     qed
   hence "{OrdP (Var i), OrdP x} \<turnstile> (HaddP x (Var i) (Var j) IMP OrdP (Var j))(j::=z)"
     by (rule All_D)
   hence "{OrdP (Var i), OrdP x} \<turnstile> (HaddP x (Var i) z IMP OrdP z)"
     using atoms by simp
   hence "{OrdP x} \<turnstile> HaddP x (Var i) z IMP OrdP z"
     by (metis HaddP_imp_OrdP Imp_cut)
   hence "{OrdP x} \<turnstile> (HaddP x (Var i) z IMP OrdP z)(i::=y)"
     using atoms by (force intro!: Subst)
   thus ?thesis using assms atoms
     by simp (metis anti_deduction cut2)
 qed (*>*)
 
 lemma HaddP_Mem_cancel_left:
   assumes "H \<turnstile> HaddP x y' z'" "H \<turnstile> HaddP x y z" "H \<turnstile> OrdP x"
     shows "H \<turnstile> z' IN z IFF y' IN y" (*<*)
 proof -
   obtain i::name and j::name and j'::name and k::name and k'::name
    where atoms: "atom i \<sharp> (x,y,y',z,z')" "atom j \<sharp> (i,x,y,y',z,z')" "atom j' \<sharp> (i,j,x,y,y',z,z')"
                  "atom k \<sharp> (i,j,j',x,y,y',z,z')" "atom k' \<sharp> (i,j,j',k,x,y,y',z,z')"
     by (metis obtain_fresh)
   have "{OrdP (Var i)}
         \<turnstile> All j (All j' (HaddP (Var i) y' (Var j') IMP (HaddP (Var i) y (Var j) IMP
                          ((Var j') IN (Var j) IFF y' IN y))))"
        (is "_ \<turnstile> ?scheme")
     proof (rule OrdInd2H)
       show "{} \<turnstile> ?scheme(i::=Zero)"
         using atoms apply simp
         apply (rule All_I Imp_I Ex_EH)+
         apply (rule cut_same [where A="Var j EQ y"])
         apply (metis HaddP_Zero_D1 Sym)
         apply (rule cut_same [where A="Var j' EQ y'"])
         apply (metis HaddP_Zero_D1 Sym thin1)
         apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same], simp)
         apply (rule thin1)
         apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same], auto)
         done
     next
       show "{} \<turnstile> All i (OrdP (Var i) IMP ?scheme IMP ?scheme(i::=SUCC (Var i)))"
         using atoms apply simp
         apply (rule All_I Imp_I Ex_EH)+
         apply (rule cut_same [OF HaddP_SUCC_Ex1 [of k "Var i" y "Var j", THEN cut2]], simp_all)
         apply (rule AssumeH Conj_EH Ex_EH)+
         apply (rule cut_same [OF HaddP_SUCC_Ex1 [of k' "Var i" y' "Var j'", THEN cut2]], simp_all)
         apply (rule AssumeH Conj_EH Ex_EH)+
         apply (rule rotate7)
         apply (rule All_E [where x = "Var k"], simp)
         apply (rule All_E [where x = "Var k'"], simp_all)
         apply (rule Imp_E AssumeH)+
         apply (rule Iff_trans)
         prefer 2
         apply (rule AssumeH)
         apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same, THEN rotate3], simp)
         apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same, THEN rotate5], simp)
         apply (blast intro!: HaddP_OrdP OrdP_IN_SUCC_Iff)
         done
     qed
   hence "{OrdP (Var i)}
          \<turnstile> (All j' (HaddP (Var i) y' (Var j') IMP (HaddP (Var i) y (Var j) IMP ((Var j') IN (Var j) IFF y' IN y))))(j::=z)"
     by (metis All_D)
   hence "{OrdP (Var i)}
          \<turnstile> (All j' (HaddP (Var i) y' (Var j') IMP (HaddP (Var i) y z IMP ((Var j') IN z IFF y' IN y))))"
     using atoms by simp
   hence "{OrdP (Var i)}
          \<turnstile> (HaddP (Var i) y' (Var j') IMP (HaddP (Var i) y z IMP ((Var j') IN z IFF y' IN y)))(j'::=z')"
     by (metis All_D)
   hence "{OrdP (Var i)} \<turnstile> HaddP (Var i) y' z' IMP (HaddP (Var i) y z IMP (z' IN z IFF y' IN y))"
     using atoms by simp
   hence "{} \<turnstile> (OrdP (Var i) IMP HaddP (Var i) y' z' IMP (HaddP (Var i) y z IMP (z' IN z IFF y' IN y)))(i::=x)"
     by (metis Imp_I Subst emptyE)
   thus ?thesis
     using atoms by simp (metis assms MP_null MP_same)
 qed (*>*)
 
 lemma HaddP_Mem_cancel_right_Mem:
   assumes "H \<turnstile> HaddP x' y z'" "H \<turnstile> HaddP x y z" "H \<turnstile> x' IN x" "H \<turnstile> OrdP x"
     shows "H \<turnstile> z' IN z"
 proof -
   have "H \<turnstile> OrdP x'"
     by (metis Ord_IN_Ord assms(3) assms(4))
   hence "H \<turnstile> HaddP y x' z'" "H \<turnstile> HaddP y x z"
     by (blast intro: assms HaddP_commute [THEN cut2])+
   thus ?thesis
     by (blast intro: assms HaddP_imp_OrdP [THEN cut1] HaddP_Mem_cancel_left [THEN Iff_MP2_same])
 qed
 
 lemma HaddP_Mem_cases:
   assumes "H \<turnstile> HaddP k1 k2 k" "H \<turnstile> OrdP k1"
           "insert (x IN k1) H \<turnstile> A"
           "insert (Var i IN k2) (insert (HaddP k1 (Var i) x) H) \<turnstile> A"
       and i: "atom (i::name) \<sharp> (k1,k2,k,x,A)" and "\<forall>C \<in> H. atom i \<sharp> C"
     shows "insert (x IN k) H \<turnstile> A" (*<*)
 proof -
   obtain j::name where j: "atom j \<sharp> (k1,k2,k,x)"
     by (metis obtain_fresh)
   have seq: "{HaddP k1 k2 k, x IN k, OrdP k1} \<turnstile> x IN k1 OR (Ex i (HaddP k1 (Var i) x AND Var i IN k2))"
     apply (rule cut_same [OF HaddP_OrdP])
     apply (rule AssumeH)+
     apply (rule cut_same [OF Ord_IN_Ord])
     apply (rule AssumeH)+
     apply (rule OrdP_linear [of _ x k1], (rule AssumeH)+)
     proof -
       show "{x IN k1, OrdP x, OrdP k, HaddP k1 k2 k, x IN k, OrdP k1} \<turnstile> x IN k1 OR Ex i (HaddP k1 (Var i) x AND Var i IN k2)"
         by (blast intro: Disj_I1)
     next
       show "{x EQ k1, OrdP x, OrdP k, HaddP k1 k2 k, x IN k, OrdP k1} \<turnstile> x IN k1 OR Ex i (HaddP k1 (Var i) x AND Var i IN k2)"
         apply (rule cut_same [OF Zero_In_OrdP [of k2, THEN cut1]])
         apply (metis AssumeH(4) HaddP_imp_OrdP cut1)
         apply auto
         apply (rule cut_same [where A="HaddP x Zero k"])
         apply (blast intro: HaddP_cong [THEN Iff_MP_same] Sym)
         apply (rule cut_same [where A="x EQ k"])
         apply (metis HaddP_Zero_D2)
         apply (blast intro: Mem_non_refl Mem_cong [THEN Iff_MP_same])
         apply (rule Disj_I2)
         apply (rule Ex_I [where x=Zero])
         using i apply auto
         apply (rule HaddP_cong [THEN Iff_MP_same])
         apply (rule AssumeH Refl HaddP_Zero2)+
         done
     next
       show "{k1 IN x, OrdP x, OrdP k, HaddP k1 k2 k, x IN k, OrdP k1} \<turnstile> x IN k1 OR Ex i (HaddP k1 (Var i) x AND Var i IN k2)"
         apply (rule Disj_I2)
         apply (rule cut_same [OF Mem_imp_subtract [of _ k1 x j]])
         apply (rule AssumeH)+
         using i j apply auto
         apply (rule Ex_I [where x="Var j"], auto intro: HaddP_Mem_cancel_left [THEN Iff_MP_same])
         done
     qed
   show ?thesis using assms
     by (force intro: cut_same [OF seq [THEN cut3]] thin1 simp: insert_commute)
 qed (*>*)
 
 lemma HaddP_Mem_contra:
   assumes "H \<turnstile> HaddP x y z" "H \<turnstile> z IN x" "H \<turnstile> OrdP x"
     shows "H \<turnstile> A"
 proof -
   obtain i::name and j::name and k::name
    where atoms: "atom i \<sharp> (x,y,z)" "atom j \<sharp> (i,x,y,z)" "atom k \<sharp> (i,j,x,y,z)"
     by (metis obtain_fresh)
   have "{OrdP (Var i)}  \<turnstile> All j (HaddP (Var i) y (Var j) IMP Neg ((Var j) IN (Var i)))"
        (is "_ \<turnstile> ?scheme")
     proof (rule OrdInd2H)
       show "{} \<turnstile> ?scheme(i::=Zero)"
         using atoms by auto
     next
       show "{} \<turnstile> All i (OrdP (Var i) IMP ?scheme IMP ?scheme(i::=SUCC (Var i)))"
         using  atoms apply auto
         apply (rule cut_same [OF HaddP_SUCC_Ex1 [of k "Var i" y "Var j", THEN cut2]], auto)
         apply (rule Ex_I [where x="Var k"], auto)
         apply (blast intro: OrdP_IN_SUCC_D Mem_cong [OF _ Refl, THEN Iff_MP_same])
         done
     qed
   hence "{OrdP (Var i)} \<turnstile> (HaddP (Var i) y (Var j) IMP Neg ((Var j) IN (Var i)))(j::=z)"
     by (metis All_D)
   hence "{} \<turnstile> OrdP (Var i) IMP HaddP (Var i) y z IMP Neg (z IN (Var i))"
     using atoms by simp (metis Imp_I)
   hence "{} \<turnstile> (OrdP (Var i) IMP HaddP (Var i) y z IMP Neg (z IN (Var i)))(i::=x)"
     by (metis Subst emptyE)
   thus ?thesis
     using atoms by simp (metis MP_same MP_null Neg_D assms)
 qed (*>*)
 
 lemma exists_HaddP:
   assumes "H \<turnstile> OrdP y" "atom j \<sharp> (x,y)"
     shows "H \<turnstile> Ex j (HaddP x y (Var j))"
 proof -
   obtain i::name
    where atoms: "atom i \<sharp> (j,x,y)"
     by (metis obtain_fresh)
   have "{OrdP (Var i)} \<turnstile> Ex j (HaddP x (Var i) (Var j))"
        (is "_ \<turnstile> ?scheme")
     proof (rule OrdInd2H)
       show "{} \<turnstile> ?scheme(i::=Zero)"
         using atoms assms
         by (force intro!: Ex_I [where x=x] HaddP_Zero2)
     next
       show "{} \<turnstile> All i (OrdP (Var i) IMP ?scheme IMP ?scheme(i::=SUCC (Var i)))"
         using atoms assms
         apply auto
         apply (auto intro!: Ex_I [where x="SUCC (Var j)"] HaddP_SUCC2)
         apply (metis HaddP_SUCC2 insert_commute thin1)
         done
     qed
   hence "{} \<turnstile> OrdP (Var i) IMP Ex j (HaddP x (Var i) (Var j))"
     by (metis Imp_I)
   hence "{} \<turnstile> (OrdP (Var i) IMP Ex j (HaddP x (Var i) (Var j)))(i::=y)"
     using atoms by (force intro!: Subst)
   thus ?thesis
     using atoms assms by simp (metis MP_null assms(1))
 qed
 
 lemma HaddP_Mem_I:
   assumes "H \<turnstile> HaddP x y z" "H \<turnstile> OrdP x" shows "H \<turnstile> x IN SUCC z"
 proof -
   have "{HaddP x y z, OrdP x} \<turnstile> x IN SUCC z"
     apply (rule OrdP_linear [of _ x "SUCC z"])
     apply (auto intro: OrdP_SUCC_I HaddP_OrdP)
     apply (rule HaddP_Mem_contra, blast)
     apply (metis Assume Mem_SUCC_I2 OrdP_IN_SUCC_D Sym_L thin1 thin2, blast)
     apply (blast intro: HaddP_Mem_contra Mem_SUCC_Refl OrdP_Trans)
     done
   thus ?thesis
     by (rule cut2) (auto intro: assms)
 qed
 
 
-section {*A Shifted Sequence*}
+section \<open>A Shifted Sequence\<close>
 
 nominal_function ShiftP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom x \<sharp> (x',y,z,f,del,k); atom x' \<sharp> (y,z,f,del,k); atom y \<sharp> (z,f,del,k); atom z \<sharp> (f,del,g,k)\<rbrakk> \<Longrightarrow>
     ShiftP f k del g =
       All z (Var z IN g IFF
       (Ex x (Ex x' (Ex y ((Var z) EQ HPair (Var x') (Var y) AND
                           HaddP del (Var x) (Var x') AND
                           HPair (Var x) (Var y) IN f AND Var x IN k)))))"
 by (auto simp: eqvt_def ShiftP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma ShiftP_fresh_iff [simp]: "a \<sharp> ShiftP f k del g \<longleftrightarrow> a \<sharp> f \<and> a \<sharp> k \<and> a \<sharp> del \<and> a \<sharp> g"
 proof -
   obtain x::name and x'::name and y::name and z::name
     where "atom x \<sharp> (x',y,z,f,del,k)" "atom x' \<sharp> (y,z,f,del,k)"
           "atom y \<sharp> (z,f,del,k)" "atom z \<sharp> (f,del,g,k)"
     by (metis obtain_fresh)
   thus ?thesis
     by auto
 qed
 
 lemma subst_fm_ShiftP [simp]:
   "(ShiftP f k del g)(i::=u) = ShiftP (subst i u f) (subst i u k) (subst i u del) (subst i u g)"
 proof -
   obtain x::name and x'::name and y::name and z::name
   where "atom x \<sharp> (x',y,z,f,del,k,i,u)" "atom x' \<sharp> (y,z,f,del,k,i,u)"
         "atom y \<sharp> (z,f,del,k,i,u)" "atom z \<sharp> (f,del,g,k,i,u)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: ShiftP.simps [of x x' y z])
 qed
 
 lemma ShiftP_Zero: "{} \<turnstile> ShiftP Zero k d Zero"
 proof -
   obtain x::name and x'::name and y::name and z::name
   where "atom x \<sharp> (x',y,z,k,d)" "atom x' \<sharp> (y,z,k,d)" "atom y \<sharp> (z,k,d)" "atom z \<sharp> (k,d)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: ShiftP.simps [of x x' y z])
 qed
 
 lemma ShiftP_Mem1:
   "{ShiftP f k del g, HPair a b IN f, HaddP del a a', a IN k} \<turnstile> HPair a' b IN g"
 proof -
   obtain x::name and x'::name and y::name and z::name
     where "atom x \<sharp> (x',y,z,f,del,k,a,a',b)" "atom x' \<sharp> (y,z,f,del,k,a,a',b)"
           "atom y \<sharp> (z,f,del,k,a,a',b)" "atom z \<sharp> (f,del,g,k,a,a',b)"
     by (metis obtain_fresh)
   thus ?thesis
     apply (auto simp: ShiftP.simps [of x x' y z])
     apply (rule All_E [where x="HPair a' b"], auto intro!: Iff_E2)
     apply (rule Ex_I [where x=a], simp)
     apply (rule Ex_I [where x="a'"], simp)
     apply (rule Ex_I [where x=b], auto intro: Mem_Eats_I1)
     done
 qed
 
 lemma ShiftP_Mem2:
   assumes "atom u \<sharp> (f,k,del,a,b)"
   shows  "{ShiftP f k del g, HPair a b IN g} \<turnstile> Ex u ((Var u) IN k AND HaddP del (Var u) a AND HPair (Var u) b IN f)"
 proof -
   obtain x::name and x'::name and y::name and z::name
   where atoms: "atom x \<sharp> (x',y,z,f,del,g,k,a,u,b)" "atom x' \<sharp> (y,z,f,del,g,k,a,u,b)"
                "atom y \<sharp> (z,f,del,g,k,a,u,b)" "atom z \<sharp> (f,del,g,k,a,u,b)"
     by (metis obtain_fresh)
   thus ?thesis using assms
     apply (auto simp: ShiftP.simps [of x x' y z])
     apply (rule All_E [where x="HPair a b"])
     apply (auto intro!: Iff_E1 [OF Assume])
     apply (rule Ex_I [where x="Var x"])
     apply (auto intro: Mem_cong [OF HPair_cong Refl, THEN Iff_MP2_same])
     apply (blast intro: HaddP_cong [OF Refl Refl, THEN Iff_MP2_same])
     done
 qed
 
 lemma ShiftP_Mem_D:
   assumes "H \<turnstile> ShiftP f k del g" "H \<turnstile> a IN g"
           "atom x \<sharp> (x',y,a,f,del,k)" "atom x' \<sharp> (y,a,f,del,k)" "atom y \<sharp> (a,f,del,k)"
   shows  "H \<turnstile> (Ex x (Ex x' (Ex y (a EQ HPair (Var x') (Var y) AND
                                   HaddP del (Var x) (Var x') AND
                                   HPair (Var x) (Var y) IN f AND Var x IN k))))"
          (is "_ \<turnstile> ?concl")
 proof -
   obtain z::name where "atom z \<sharp> (x,x',y,f,del,g,k,a)"
     by (metis obtain_fresh)
   hence "{ShiftP f k del g, a IN g} \<turnstile> ?concl" using assms
     by (auto simp: ShiftP.simps [of x x' y z]) (rule All_E [where x=a], auto intro: Iff_E1)
   thus ?thesis
     by (rule cut2) (rule assms)+
 qed
 
 lemma ShiftP_Eats_Eats:
   "{ShiftP f k del g, HaddP del a a', a IN k}
    \<turnstile> ShiftP (Eats f (HPair a b)) k del (Eats g (HPair a' b))" (*<*)
 proof -
   obtain x::name and x'::name and y::name and z::name
     where "atom x \<sharp> (x',y,z,f,del,g,k,a,a',b)" "atom x' \<sharp> (y,z,f,del,g,k,a,a',b)"
           "atom y \<sharp> (z,f,del,g,k,a,a',b)" "atom z \<sharp> (f,del,g,k,a,a',b)"
     by (metis obtain_fresh)
   thus ?thesis
     apply (auto simp: ShiftP.simps [of x x' y z] intro!: Iff_I [THEN Swap])
     apply (rule All_E [where x="Var z", THEN rotate2], simp)
     apply (rule Iff_E)
     apply auto [1]
     apply (rule Ex_I [where x="Var x"], simp)
     apply (rule Ex_I [where x="Var x'"], simp)
     apply (rule Ex_I [where x="Var y"], simp)
     apply (blast intro: Mem_Eats_I1, blast)
     apply (rule Ex_I [where x=a], simp)
     apply (rule Ex_I [where x="a'"], simp)
     apply (rule Ex_I [where x=b], simp)
     apply (metis Assume AssumeH(3) AssumeH(4) Conj_I Mem_Eats_I2 Refl)
     apply (rule All_E [where x="Var z", THEN rotate5], auto)
     apply (rule Mem_Eats_I1)
     apply (rule Iff_MP2_same [OF Hyp], blast)
     apply (rule Ex_I [where x="Var x"], simp)
     apply (rule Ex_I [where x="Var x'"], simp)
     apply (rule Ex_I [where x="Var y"], auto)
     apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same, THEN rotate5], simp)
     apply (blast intro: Mem_Eats_I2 HaddP_cong [THEN Iff_MP_same] HaddP_unique [THEN cut2] HPair_cong)
     done
 qed (*>*)
 
 lemma ShiftP_Eats_Neg:
   assumes "atom u \<sharp> (u',v,f,k,del,g,c)" "atom u' \<sharp> (v,f,k,del,g,c)" "atom v \<sharp> (f,k,del,g,c)"
   shows
   "{ShiftP f k del g,
     Neg (Ex u (Ex u' (Ex v (c EQ HPair (Var u) (Var v) AND Var u IN k AND HaddP del (Var u) (Var u')))))}
    \<turnstile> ShiftP (Eats f c) k del g"  (*<*)
 proof -
   obtain x::name and x'::name and y::name and z::name
   where atoms: "atom x \<sharp> (x',y,z,u,u',v,f,k,del,g,c)" "atom x' \<sharp> (y,z,u,u',v,f,k,del,g,c)"
                "atom y \<sharp> (z,u,u',v,f,k,del,g,c)" "atom z \<sharp> (u,u',v,f,k,del,g,c)"
     by (metis obtain_fresh)
   thus ?thesis using assms
     apply (auto simp: ShiftP.simps [of x x' y z] intro!: Iff_I [THEN Swap])
     apply (rule All_E [where x="Var z", THEN rotate3])
     apply (auto intro!: Iff_E1 [OF Assume])
     apply (rule Ex_I [where x="Var x"], simp)
     apply (rule Ex_I [where x="Var x'"], simp)
     apply (rule Ex_I [where x="Var y"], simp)
     apply (blast intro: Mem_Eats_I1)
     apply (rule All_E [where x="Var z", THEN rotate6], simp)
     apply (rule Iff_E2)
     apply (rule Ex_I [where x="Var x"], simp)
     apply (rule Ex_I [where x="Var x'"], simp)
     apply (rule Ex_I [where x="Var y"])
     apply (auto intro: Mem_Eats_I1)
     apply (rule Swap [THEN rotate5])
     apply (rule Ex_I [where x="Var x"], simp)
     apply (rule Ex_I [where x="Var x'"], simp)
     apply (rule Ex_I [where x="Var y"], simp)
     apply (blast intro: Sym Mem_Eats_I1)
     done
 qed (*>*)
 
 lemma exists_ShiftP:
   assumes t: "atom t \<sharp> (s,k,del)"
   shows "H \<turnstile> Ex t (ShiftP s k del (Var t))" (*<*)
 proof -
   obtain i::name and j::name
     where i: "atom (i::name) \<sharp> (s,t,k,del)" and j: "atom (j::name) \<sharp> (i,s,t,k,del)"
     by (metis obtain_fresh)
   have "{} \<turnstile> Ex t (ShiftP (Var i) k del (Var t))" (is "{} \<turnstile> ?scheme")
   proof (rule Ind [of j])
     show "atom j \<sharp> (i, ?scheme)" using j
       by simp
   next
     show "{} \<turnstile> ?scheme(i::=Zero)" using i t
       by (auto intro!: Ex_I [where x=Zero] simp: ShiftP_Zero)
   next
     obtain x::name and x'::name and y::name
     where atoms: "atom x \<sharp> (x',y,s,k,del,t,i,j)" "atom x' \<sharp> (y,s,k,del,t,i,j)"
                  "atom y \<sharp> (s,k,del,t,i,j)"
       by (metis obtain_fresh)
     let ?caseA = "Ex x (Ex x' (Ex y ((Var j) EQ HPair (Var x) (Var y) AND Var x IN k AND
                                      HaddP del (Var x) (Var x'))))"
     show "{} \<turnstile> All i (All j (?scheme IMP ?scheme(i::=Var j) IMP ?scheme(i::=Eats (Var i) (Var j))))"
       using i j atoms
       apply (auto del: Ex_EH)
       apply (rule Ex_E)
       apply (auto del: Ex_EH)
       apply (rule Ex_E)
       apply (auto del: Ex_EH)
       apply (rule thin1, auto)
       proof (rule Cases [where A="?caseA"])
         show "{?caseA, ShiftP (Var i) k del (Var t)}
               \<turnstile> Ex t (ShiftP (Eats (Var i) (Var j)) k del (Var t))"
           using i j t atoms
           apply (auto simp del: ShiftP.simps)
           apply (rule Ex_I [where x="Eats (Var t) (HPair (Var x') (Var y))"], auto)
           apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same, THEN rotate3])
           apply (auto intro: ShiftP_Eats_Eats [THEN cut3])
           done
       next
         show "{Neg ?caseA, ShiftP (Var i) k del (Var t)}
               \<turnstile> Ex t (ShiftP (Eats (Var i) (Var j)) k del (Var t))"
           using atoms
           by (auto intro!: Ex_I [where x="Var t"] ShiftP_Eats_Neg [of x x' y, THEN cut2]
                    simp: ShiftP_Zero)
       qed
   qed
   hence "{} \<turnstile> (Ex t (ShiftP (Var i) k del (Var t)))(i::=s)"
     by (blast intro: Subst)
   thus ?thesis using i t
     by (auto intro: thin0)
 qed (*>*)
 
 
-section {*Union of Two Sets*}
+section \<open>Union of Two Sets\<close>
 
 nominal_function UnionP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "atom i \<sharp> (x,y,z) \<Longrightarrow> UnionP x y z = All i (Var i IN z IFF (Var i IN x OR Var i IN y))"
 by (auto simp: eqvt_def UnionP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma UnionP_fresh_iff [simp]: "a \<sharp> UnionP x y z \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> y \<and> a \<sharp> z"
 proof -
   obtain i::name where "atom i \<sharp> (x,y,z)"
     by (metis obtain_fresh)
   thus ?thesis
     by auto
 qed
 
 lemma subst_fm_UnionP [simp]:
   "(UnionP x y z)(i::=u) = UnionP (subst i u x) (subst i u y) (subst i u z)"
 proof -
   obtain j::name where "atom j \<sharp> (x,y,z,i,u)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: UnionP.simps [of j])
 qed
 
 lemma Union_Zero1: "H \<turnstile> UnionP Zero x x"
 proof -
   obtain i::name where "atom i \<sharp> x"
     by (metis obtain_fresh)
   hence "{} \<turnstile> UnionP Zero x x"
     by (auto simp: UnionP.simps [of i] intro: Disj_I2)
   thus ?thesis
     by (metis thin0)
 qed
 
 lemma Union_Eats: "{UnionP x y z} \<turnstile> UnionP (Eats x a) y (Eats z a)"
 proof -
   obtain i::name where "atom i \<sharp> (x,y,z,a)"
     by (metis obtain_fresh)
   thus ?thesis
     apply (auto simp: UnionP.simps [of i])
     apply (rule Ex_I [where x="Var i"])
     apply (auto intro: Iff_E1 [THEN rotate2] Iff_E2 [THEN rotate2] Mem_Eats_I1 Mem_Eats_I2 Disj_I1 Disj_I2)
     done
 qed
 
 lemma exists_Union_lemma:
   assumes z: "atom z \<sharp> (i,y)" and i: "atom i \<sharp> y"
   shows "{} \<turnstile> Ex z (UnionP (Var i) y (Var z))"
 proof -
   obtain j::name where j: "atom j \<sharp> (y,z,i)"
     by (metis obtain_fresh)
   show "{} \<turnstile> Ex z (UnionP (Var i) y (Var z))"
     apply (rule Ind [of j i]) using j z i
     apply simp_all
     apply (rule Ex_I [where x=y], simp add: Union_Zero1)
     apply (auto del: Ex_EH)
     apply (rule Ex_E)
     apply (rule NegNeg_E)
     apply (rule Ex_E)
     apply (auto del: Ex_EH)
     apply (rule thin1, force intro: Ex_I [where x="Eats (Var z) (Var j)"] Union_Eats)
     done
 qed
 
 lemma exists_UnionP:
   assumes z: "atom z \<sharp> (x,y)"  shows "H \<turnstile> Ex z (UnionP x y (Var z))"
 proof -
   obtain i::name  where i: "atom i \<sharp> (y,z)"
     by (metis obtain_fresh)
   hence "{} \<turnstile> Ex z (UnionP (Var i) y (Var z))"
     by (metis exists_Union_lemma fresh_Pair fresh_at_base(2) z)
   hence "{} \<turnstile> (Ex z (UnionP (Var i) y (Var z)))(i::=x)"
     by (metis Subst empty_iff)
   thus ?thesis using i z
     by (simp add: thin0)
 qed
 
 lemma UnionP_Mem1: "{ UnionP x y z, a IN x } \<turnstile> a IN z"
 proof -
   obtain i::name where "atom i \<sharp> (x,y,z,a)"
     by (metis obtain_fresh)
   thus ?thesis
     by (force simp: UnionP.simps [of i] intro: All_E [where x=a] Disj_I1 Iff_E2)
 qed
 
 lemma UnionP_Mem2: "{ UnionP x y z, a IN y } \<turnstile> a IN z"
 proof -
   obtain i::name where "atom i \<sharp> (x,y,z,a)"
     by (metis obtain_fresh)
   thus ?thesis
     by (force simp: UnionP.simps [of i] intro: All_E [where x=a] Disj_I2 Iff_E2)
 qed
 
 lemma UnionP_Mem: "{ UnionP x y z, a IN z } \<turnstile> a IN x OR a IN y"
 proof -
   obtain i::name where "atom i \<sharp> (x,y,z,a)"
     by (metis obtain_fresh)
   thus ?thesis
     by (force simp: UnionP.simps [of i] intro: All_E [where x=a] Iff_E1)
 qed
 
 lemma UnionP_Mem_E:
   assumes "H \<turnstile> UnionP x y z"
       and "insert (a IN x) H \<turnstile> A"
       and "insert (a IN y) H \<turnstile> A"
     shows "insert (a IN z) H \<turnstile> A"
   using assms
   by (blast intro: rotate2 cut_same [OF UnionP_Mem [THEN cut2]] thin1)
 
 
-section {*Append on Sequences*}
+section \<open>Append on Sequences\<close>
 
 nominal_function SeqAppendP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom g1 \<sharp> (g2,f1,k1,f2,k2,g); atom g2 \<sharp> (f1,k1,f2,k2,g)\<rbrakk> \<Longrightarrow>
     SeqAppendP f1 k1 f2 k2 g =
       (Ex g1 (Ex g2 (RestrictedP f1 k1 (Var g1) AND
                      ShiftP f2 k2 k1 (Var g2) AND
                      UnionP (Var g1) (Var g2) g)))"
 by (auto simp: eqvt_def SeqAppendP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma SeqAppendP_fresh_iff [simp]:
   "a \<sharp> SeqAppendP f1 k1 f2 k2 g \<longleftrightarrow> a \<sharp> f1 \<and> a \<sharp> k1 \<and> a \<sharp> f2 \<and> a \<sharp> k2 \<and> a \<sharp> g"
 proof -
   obtain g1::name and g2::name
     where "atom g1 \<sharp> (g2,f1,k1,f2,k2,g)" "atom g2 \<sharp> (f1,k1,f2,k2,g)"
     by (metis obtain_fresh)
   thus ?thesis
     by auto
 qed
 
 lemma subst_fm_SeqAppendP [simp]:
   "(SeqAppendP f1 k1 f2 k2 g)(i::=u) =
    SeqAppendP (subst i u f1) (subst i u k1) (subst i u f2) (subst i u k2) (subst i u g)"
 proof -
   obtain g1::name and g2::name
   where "atom g1 \<sharp> (g2,f1,k1,f2,k2,g,i,u)" "atom g2 \<sharp> (f1,k1,f2,k2,g,i,u)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: SeqAppendP.simps [of g1 g2])
 qed
 
 lemma exists_SeqAppendP:
   assumes "atom g \<sharp> (f1,k1,f2,k2)"
   shows "H \<turnstile> Ex g (SeqAppendP f1 k1 f2 k2 (Var g))"
 proof -
   obtain g1::name and g2::name
   where atoms: "atom g1 \<sharp> (g2,f1,k1,f2,k2,g)" "atom g2 \<sharp> (f1,k1,f2,k2,g)"
     by (metis obtain_fresh)
   hence "{} \<turnstile> Ex g (SeqAppendP f1 k1 f2 k2 (Var g))"
     using assms
     apply (auto simp: SeqAppendP.simps [of g1 g2])
     apply (rule cut_same [OF exists_RestrictedP [of g1 f1 k1]], auto)
     apply (rule cut_same [OF exists_ShiftP [of g2 f2 k2 k1]], auto)
     apply (rule cut_same [OF exists_UnionP [of g "Var g1" "Var g2"]], auto)
     apply (rule Ex_I [where x="Var g"], simp)
     apply (rule Ex_I [where x="Var g1"], simp)
     apply (rule Ex_I [where x="Var g2"], auto)
     done
   thus ?thesis using assms
     by (metis thin0)
 qed
 
 lemma SeqAppendP_Mem1: "{SeqAppendP f1 k1 f2 k2 g, HPair x y IN f1, x IN k1} \<turnstile> HPair x y IN g"
 proof -
   obtain g1::name and g2::name
     where "atom g1 \<sharp> (g2,f1,k1,f2,k2,g,x,y)" "atom g2 \<sharp> (f1,k1,f2,k2,g,x,y)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: SeqAppendP.simps [of g1 g2] intro: UnionP_Mem1 [THEN cut2] RestrictedP_Mem [THEN cut3])
 qed
 
 lemma SeqAppendP_Mem2: "{SeqAppendP f1 k1 f2 k2 g, HaddP k1 x x', x IN k2, HPair x y IN f2} \<turnstile> HPair x' y IN g"
 proof -
   obtain g1::name and g2::name
     where "atom g1 \<sharp> (g2,f1,k1,f2,k2,g,x,x',y)" "atom g2 \<sharp> (f1,k1,f2,k2,g,x,x',y)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: SeqAppendP.simps [of g1 g2] intro: UnionP_Mem2 [THEN cut2] ShiftP_Mem1 [THEN cut4])
 qed
 
 lemma SeqAppendP_Mem_E:
   assumes "H \<turnstile> SeqAppendP f1 k1 f2 k2 g"
       and "insert (HPair x y IN f1) (insert (x IN k1) H) \<turnstile> A"
       and "insert (HPair (Var u) y IN f2) (insert (HaddP k1 (Var u) x) (insert (Var u IN k2) H)) \<turnstile> A"
       and u: "atom u \<sharp> (f1,k1,f2,k2,x,y,g,A)" "\<forall>C \<in> H. atom u \<sharp> C"
   shows "insert (HPair x y IN g) H \<turnstile> A" (*<*)
 proof -
   obtain g1::name and g2::name
   where atoms: "atom g1 \<sharp> (g2,f1,k1,f2,k2,g,x,y,u)" "atom g2 \<sharp> (f1,k1,f2,k2,g,x,y,u)"
     by (metis obtain_fresh)
   hence "{SeqAppendP f1 k1 f2 k2 g, HPair x y IN g}
          \<turnstile> (HPair x y IN f1 AND x IN k1) OR Ex u ((Var u) IN k2 AND HaddP k1 (Var u) x AND HPair (Var u) y IN f2)"
     using u
     apply (auto simp: SeqAppendP.simps [of g1 g2])
     apply (rule UnionP_Mem_E [THEN rotate4])
     apply (rule AssumeH)+
     apply (blast intro: Disj_I1 cut_same [OF RestrictedP_Mem2 [THEN cut2]])
     apply (rule Disj_I2)
     apply (rule cut_same [OF ShiftP_Mem2 [where u=u, THEN cut2]])
     defer 1
     apply force+
     done
   thus ?thesis
     apply (rule cut_same [OF _ [THEN cut2]])
     using assms
     apply (auto intro: thin1 rotate2 thin3 thin4)
     done
 qed (*>*)
 
-section {*LstSeqP and SeqAppendP*}
+section \<open>LstSeqP and SeqAppendP\<close>
 
 lemma HDomain_Incl_SeqAppendP:  \<comment> \<open>The And eliminates the need to prove @{text cut5}\<close>
   "{SeqAppendP f1 k1 f2 k2 g, HDomain_Incl f1 k1 AND HDomain_Incl f2 k2,
     HaddP k1 k2 k, OrdP k1} \<turnstile> HDomain_Incl g k" (*<*)
 proof -
   obtain x::name and y::name and z::name and i::name
     where "atom x \<sharp> (f1,k1,f2,k2,g,k,y,z,i)"  "atom y \<sharp> (f1,k1,f2,k2,g,k,z,i)"
           "atom z \<sharp> (f1,k1,f2,k2,g,k,i)"  "atom i \<sharp> (f1,k1,f2,k2,g,k)"
     by (metis obtain_fresh)
   thus ?thesis
     apply (auto simp: HDomain_Incl.simps [of x _ _ y z])
     apply (rule HaddP_Mem_cases [where i=i, THEN rotate2], auto)
     \<comment> \<open>case 1\<close>
     apply (rule All_E' [where x = "Var x"], blast, auto)
     apply (rule ContraProve [THEN rotate4])
     apply (rule Ex_I [where x = "Var y"], auto)
     apply (rule Ex_I [where x = "Var z"], auto)
     apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same, THEN rotate2], simp)
     apply (rule SeqAppendP_Mem1 [THEN cut3], auto)
     apply (rule Mem_cong [OF Assume Refl, THEN Iff_MP_same], auto)
     \<comment> \<open>case 2\<close>
     apply (rule Ex_I [where x = "Var i"], auto)
     apply (rule ContraProve [THEN rotate5])
     apply (rule Ex_I [where x = "Var y"], simp)
     apply (rule Ex_I [where x = "HPair (Var x) (Var y)"], auto)
     apply (blast intro: SeqAppendP_Mem2 [THEN cut4] Mem_cong [OF _ Refl, THEN Iff_MP_same])
     done
 qed (*>*)
 
 declare SeqAppendP.simps [simp del]
 
 lemma HFun_Sigma_SeqAppendP:
   "{SeqAppendP f1 k1 f2 k2 g, HFun_Sigma f1, HFun_Sigma f2, OrdP k1} \<turnstile> HFun_Sigma g" (*<*)
 proof -
   obtain x::name and y::name and z::name
      and x'::name and y'::name and z'::name and g1::name and g2::name
      and v::name and v'::name and w::name
     where atoms:
        "atom v \<sharp> (v',w,g1,g2,z,z',x,y,x',y',f1,k1,f2,k2,g)" "atom v' \<sharp> (w,g1,g2,z,z',x,y,x',y',f1,k1,f2,k2,g)"
        "atom w \<sharp> (g1,g2,z,z',x,y,x',y',f1,k1,f2,k2,g)"
        "atom g1 \<sharp> (g2,z,z',x,y,x',y',f1,k1,f2,k2,g)" "atom g2 \<sharp> (z,z',x,y,x',y',f1,k1,f2,k2,g)"
        "atom z \<sharp> (z',x,y,x',y',f1,k1,f2,k2,g)" "atom z' \<sharp> (x,y,x',y',f1,k1,f2,k2,g)"
        "atom x \<sharp> (y,x',y',f1,k1,f2,k2,g)" "atom y \<sharp> (x',y',f1,k1,f2,k2,g)"
        "atom x' \<sharp> (y',f1,k1,f2,k2,g)" "atom y' \<sharp> (f1,k1,f2,k2,g)"
     by (metis obtain_fresh)
   thus ?thesis
     apply (simp add: HFun_Sigma.simps [of z g z' x y x' y'] SeqAppendP.simps [of g1 g2])
     apply (rule Ex_EH Conj_EH All_I Imp_I)+
     apply (rule cut_same [OF UnionP_Mem [where a = "Var z", THEN cut2]])
     apply (rule AssumeH)+
     apply (rule Disj_E)
     apply (rule cut_same [OF UnionP_Mem [where a = "Var z'", THEN cut2]])
     apply (rule AssumeH)+
     apply (rule thin1 [where A="UnionP (Var g1) (Var g2) g", THEN rotate6])
     apply (rule Disj_E)
     \<comment> \<open>case 1/1\<close>
     apply (rule thin1 [where A="ShiftP f2 k2 k1 (Var g2)", THEN rotate5])
     apply (rule RestrictedP_Mem_D [where a = "Var z"])
     apply (rule AssumeH)+
     apply (rule RestrictedP_Mem_D [where a = "Var z'"])
     apply (rule AssumeH)+
     apply (simp add: HFun_Sigma.simps [of z f1 z' x y x' y'])
     apply (rule All2_E [where x = "Var z", THEN rotate8], simp_all, blast)
     apply (rule All2_E [where x = "Var z'"], simp_all, blast)
     apply (rule Ex_EH Conj_EH)+
     apply simp_all
     apply (rule Ex_I [where x="Var x"], simp)
     apply (rule Ex_I [where x="Var y"], simp)
     apply (rule Ex_I [where x="Var x'"], simp)
     apply (rule Ex_I [where x="Var y'"], simp)
     apply (rule Conj_I, blast)+
     apply blast
     \<comment> \<open>case 1/2\<close>
     apply (rule RestrictedP_Mem_D [where a = "Var z"])
     apply (rule AssumeH)+
     apply (rule thin1 [where A="Var z IN g", THEN rotate5])
     apply (rule thin1 [where A="Var z' IN g", THEN rotate4])
     apply (rule cut_same [OF HFun_Sigma_Mem_imp_HPair [of _ f1 "Var z" x y]], simp_all)
     apply (rule AssumeH)+
     apply (rule cut_same [OF ShiftP_Mem_D [where x=v and x'=v' and y=w]])
     apply (rule AssumeH Ex_EH Conj_EH)+
     apply auto [3]
     apply (rule AssumeH Ex_EH Conj_EH)+
     apply simp_all
     apply (rule Ex_I [where x="Var x"], simp)
     apply (rule Ex_I [where x="Var y"], simp)
     apply (rule Ex_I [where x="Var v'"], simp)
     apply (rule Ex_I [where x="Var w"], simp)
     apply auto [1]
     apply (blast intro: Mem_HFun_Sigma_OrdP [THEN cut2] Mem_cong [OF _ Refl, THEN Iff_MP_same])
     apply (blast intro: Hyp HaddP_OrdP)
     apply (rule cut_same [OF RestrictedP_Mem2 [THEN cut2]])
     apply (rule AssumeH)+
     apply (blast intro: Mem_cong [OF _ Refl, THEN Iff_MP_same])
     apply (blast intro: Hyp Mem_cong [OF _ Refl, THEN Iff_MP_same] HaddP_Mem_contra)
     \<comment> \<open>END of case 1/2\<close>
     apply (rule cut_same [OF UnionP_Mem [where a = "Var z'", THEN cut2]])
     apply (rule AssumeH)+
     apply (rule thin1 [where A="UnionP (Var g1) (Var g2) g", THEN rotate6])
     apply (rule Disj_E)
     \<comment> \<open>case 2/1\<close>
     apply (rule RestrictedP_Mem_D [where a = "Var z'"])
     apply (rule AssumeH)+
     apply (rule thin1 [where A="Var z IN g", THEN rotate5])
     apply (rule thin1 [where A="Var z' IN g", THEN rotate4])
     apply (rule cut_same [OF HFun_Sigma_Mem_imp_HPair [of _ f1 "Var z'" x y]], simp_all)
     apply (rule AssumeH)+
     apply (rule cut_same [OF ShiftP_Mem_D [where x=v and x'=v' and y=w]])
     apply (rule AssumeH Ex_EH Conj_EH)+
     apply auto [3]
     apply (rule AssumeH Ex_EH Conj_EH)+
     apply simp_all
     apply (rule Ex_I [where x="Var v'"], simp)
     apply (rule Ex_I [where x="Var w"], simp)
     apply (rule Ex_I [where x="Var x"], simp)
     apply (rule Ex_I [where x="Var y"], simp)
     apply auto [1]
     apply (blast intro: Hyp HaddP_OrdP)
     apply (blast intro: Mem_HFun_Sigma_OrdP [THEN cut2] Mem_cong [OF _ Refl, THEN Iff_MP_same])
     apply (rule cut_same [OF RestrictedP_Mem2 [THEN cut2]])
     apply (rule AssumeH)+
     apply (blast intro: Mem_cong [OF _ Refl, THEN Iff_MP_same])
     apply (blast intro: Mem_cong [OF _ Refl, THEN Iff_MP2_same] HaddP_Mem_contra Hyp)
     \<comment> \<open>case 2/2\<close>
     apply (rule cut_same [OF ShiftP_Mem_D [where x=x and x'=x' and y=y and a = "Var z"]])
     apply (rule AssumeH Ex_EH Conj_EH)+
     apply simp_all
     apply (rule cut_same [OF ShiftP_Mem_D [where x=v and x'=v' and y=w and a = "Var z'"]])
     apply (rule AssumeH Ex_EH Conj_EH)+
     apply simp_all
     apply (rule thin1 [where A="ShiftP f2 k2 k1 (Var g2)", THEN rotate7])
     apply (rule thin1 [where A="RestrictedP f1 k1 (Var g1)", THEN rotate7])
     apply (rule AssumeH Ex_EH Conj_EH)+
     apply simp_all
     apply (rule Ex_I [where x="Var x'"], simp)
     apply (rule Ex_I [where x="Var y"], simp)
     apply (rule Ex_I [where x="Var v'"], simp)
     apply (rule Ex_I [where x="Var w"], auto intro: Hyp HaddP_OrdP)
     apply (rule cut_same [where A="Var x EQ Var v"])
     apply (blast intro: HaddP_inv2 [THEN cut3] HaddP_cong [OF Refl Refl, THEN Iff_MP_same] Hyp)
     apply (rule HFun_Sigma_E [where r=f2])
     apply (auto intro: Hyp Var_Eq_subst_Iff [THEN Iff_MP_same])
     done
 qed (*>*)
 
 lemma LstSeqP_SeqAppendP:
   assumes "H \<turnstile> SeqAppendP f1 (SUCC k1) f2 (SUCC k2) g"
           "H \<turnstile> LstSeqP f1 k1 y1" "H \<turnstile> LstSeqP f2 k2 y2" "H \<turnstile> HaddP k1 k2 k"
   shows "H \<turnstile> LstSeqP g (SUCC k) y2"
 proof -
   have "{SeqAppendP f1 (SUCC k1) f2 (SUCC k2) g, LstSeqP f1 k1 y1, LstSeqP f2 k2 y2, HaddP k1 k2 k}
    \<turnstile> LstSeqP g (SUCC k) y2"
     apply (auto simp: LstSeqP.simps intro: HaddP_OrdP OrdP_SUCC_I)
     apply (rule HDomain_Incl_SeqAppendP [THEN cut4])
     apply (rule AssumeH Conj_I)+
     apply (blast intro: HaddP_SUCC1 [THEN cut1] HaddP_SUCC2 [THEN cut1])
     apply (blast intro: HaddP_OrdP OrdP_SUCC_I)
     apply (rule HFun_Sigma_SeqAppendP [THEN cut4])
     apply (auto intro: HaddP_OrdP OrdP_SUCC_I)
     apply (blast intro: Mem_SUCC_Refl HaddP_SUCC1 [THEN cut1] HaddP_SUCC2 [THEN cut1]
                         SeqAppendP_Mem2 [THEN cut4])
     done
   thus ?thesis using assms
     by (rule cut4)
 qed
 
 lemma SeqAppendP_NotInDom: "{SeqAppendP f1 k1 f2 k2 g, HaddP k1 k2 k, OrdP k1} \<turnstile> NotInDom k g"
 proof -
   obtain x::name and z::name
     where "atom x \<sharp> (z,f1,k1,f2,k2,g,k)"  "atom z \<sharp> (f1,k1,f2,k2,g,k)"
     by (metis obtain_fresh)
   thus ?thesis
     apply (auto simp: NotInDom.simps [of z])
     apply (rule SeqAppendP_Mem_E [where u=x])
     apply (rule AssumeH)+
     apply (blast intro: HaddP_Mem_contra, simp_all)
     apply (rule cut_same [where A="(Var x) EQ k2"])
     apply (blast intro: HaddP_inv2 [THEN cut3])
     apply (blast intro: Mem_non_refl [where x=k2] Mem_cong [OF _ Refl, THEN Iff_MP_same])
     done
 qed
 
 lemma LstSeqP_SeqAppendP_Eats:
   assumes "H \<turnstile> SeqAppendP f1 (SUCC k1) f2 (SUCC k2) g"
           "H \<turnstile> LstSeqP f1 k1 y1" "H \<turnstile> LstSeqP f2 k2 y2" "H \<turnstile> HaddP k1 k2 k"
   shows "H \<turnstile> LstSeqP (Eats g (HPair (SUCC (SUCC k)) z)) (SUCC (SUCC k)) z"
 proof -
   have "{SeqAppendP f1 (SUCC k1) f2 (SUCC k2) g, LstSeqP f1 k1 y1, LstSeqP f2 k2 y2, HaddP k1 k2 k}
         \<turnstile> LstSeqP (Eats g (HPair (SUCC (SUCC k)) z)) (SUCC (SUCC k)) z"
     apply (rule cut2 [OF NotInDom_LstSeqP_Eats])
     apply (rule SeqAppendP_NotInDom [THEN cut3])
     apply (rule AssumeH)
     apply (metis HaddP_SUCC1 HaddP_SUCC2 cut1 thin1)
     apply (metis Assume LstSeqP_OrdP OrdP_SUCC_I insert_commute)
     apply (blast intro: LstSeqP_SeqAppendP)
     done
   thus ?thesis using assms
     by (rule cut4)
 qed
 
-section {*Substitution and Abstraction on Terms*}
-
-subsection {*Atomic cases*}
+section \<open>Substitution and Abstraction on Terms\<close>
+
+subsection \<open>Atomic cases\<close>
 
 lemma SeqStTermP_Var_same:
   assumes "atom s \<sharp> (k,v,i)" "atom k \<sharp> (v,i)"
     shows "{VarP v} \<turnstile> Ex s (Ex k (SeqStTermP v i v i (Var s) (Var k)))"
 proof -
   obtain l::name and sl::name and sl'::name and m::name and sm::name and sm'::name
      and n::name and sn::name and sn'::name
     where "atom l \<sharp> (v,i,s,k,sl,sl',m,n,sm,sm',sn,sn')"
           "atom sl \<sharp> (v,i,s,k,sl',m,n,sm,sm',sn,sn')"
           "atom sl' \<sharp> (v,i,s,k,m,n,sm,sm',sn,sn')"
           "atom m \<sharp> (v,i,s,k,n,sm,sm',sn,sn')" "atom n \<sharp> (v,i,s,k,sm,sm',sn,sn')"
           "atom sm \<sharp> (v,i,s,k,sm',sn,sn')" "atom sm' \<sharp> (v,i,s,k,sn,sn')"
           "atom sn \<sharp> (v,i,s,k,sn')" "atom sn' \<sharp> (v,i,s,k)"
     by (metis obtain_fresh)
   thus ?thesis using assms
     apply (simp add: SeqStTermP.simps [of l _ _ v i sl sl' m n sm sm' sn sn'])
     apply (rule Ex_I [where x = "Eats Zero (HPair Zero (HPair v i))"], simp)
     apply (rule Ex_I [where x = Zero], auto intro!: Mem_SUCC_EH)
     apply (rule Ex_I [where x = v], simp)
     apply (rule Ex_I [where x = i], auto intro: Disj_I1 Mem_Eats_I2 HPair_cong)
     done
 qed
 
 lemma SeqStTermP_Var_diff:
   assumes "atom s \<sharp> (k,v,w,i)" "atom k \<sharp> (v,w,i)"
     shows "{VarP v, VarP w, Neg (v EQ w) } \<turnstile> Ex s (Ex k (SeqStTermP v i w w (Var s) (Var k)))"
 proof -
   obtain l::name and sl::name and sl'::name and m::name and sm::name and sm'::name
      and n::name and sn::name and sn'::name
     where "atom l \<sharp> (v,w,i,s,k,sl,sl',m,n,sm,sm',sn,sn')"
           "atom sl \<sharp> (v,w,i,s,k,sl',m,n,sm,sm',sn,sn')"
           "atom sl' \<sharp> (v,w,i,s,k,m,n,sm,sm',sn,sn')"
           "atom m \<sharp> (v,w,i,s,k,n,sm,sm',sn,sn')" "atom n \<sharp> (v,w,i,s,k,sm,sm',sn,sn')"
           "atom sm \<sharp> (v,w,i,s,k,sm',sn,sn')" "atom sm' \<sharp> (v,w,i,s,k,sn,sn')"
           "atom sn \<sharp> (v,w,i,s,k,sn')" "atom sn' \<sharp> (v,w,i,s,k)"
     by (metis obtain_fresh)
   thus ?thesis using assms
     apply (simp add: SeqStTermP.simps [of l _ _ v i sl sl' m n sm sm' sn sn'])
     apply (rule Ex_I [where x = "Eats Zero (HPair Zero (HPair w w))"], simp)
     apply (rule Ex_I [where x = Zero], auto intro!: Mem_SUCC_EH)
     apply (rule rotate2 [OF Swap])
     apply (rule Ex_I [where x = w], simp)
     apply (rule Ex_I [where x = w], auto simp: VarP_def)
     apply (blast intro: HPair_cong Mem_Eats_I2)
     apply (blast intro: Sym OrdNotEqP_I Disj_I1 Disj_I2)
     done
 qed
 
 lemma SeqStTermP_Zero:
   assumes "atom s \<sharp> (k,v,i)" "atom k \<sharp> (v,i)"
     shows "{VarP v} \<turnstile> Ex s (Ex k (SeqStTermP v i Zero Zero (Var s) (Var k)))" (*<*)
 proof -
   obtain l::name and sl::name and sl'::name and m::name and sm::name and sm'::name
      and n::name and sn::name and sn'::name
     where "atom l \<sharp> (v,i,s,k,sl,sl',m,n,sm,sm',sn,sn')"
           "atom sl \<sharp> (v,i,s,k,sl',m,n,sm,sm',sn,sn')"
           "atom sl' \<sharp> (v,i,s,k,m,n,sm,sm',sn,sn')"
           "atom m \<sharp> (v,i,s,k,n,sm,sm',sn,sn')" "atom n \<sharp> (v,i,s,k,sm,sm',sn,sn')"
           "atom sm \<sharp> (v,i,s,k,sm',sn,sn')" "atom sm' \<sharp> (v,i,s,k,sn,sn')"
           "atom sn \<sharp> (v,i,s,k,sn')" "atom sn' \<sharp> (v,i,s,k)"
     by (metis obtain_fresh)
   thus ?thesis using assms
     apply (simp add: SeqStTermP.simps [of l _ _ v i sl sl' m n sm sm' sn sn'])
     apply (rule Ex_I [where x = "Eats Zero (HPair Zero (HPair Zero Zero))"], simp)
     apply (rule Ex_I [where x = Zero], auto intro!: Mem_SUCC_EH)
     apply (rule Ex_I [where x = Zero], simp)
     apply (rule Ex_I [where x = Zero], simp)
     apply (rule Conj_I)
     apply (force intro: Var_Eq_subst_Iff [THEN Iff_MP_same] Mem_Eats_I2)
     apply (force simp: VarP_def OrdNotEqP.simps intro: Disj_I1 Disj_I2)
     done
 qed (*>*)
 
 corollary SubstTermP_Zero: "{TermP t} \<turnstile> SubstTermP \<lceil>Var v\<rceil> t Zero Zero"
 proof -
   obtain s::name and k::name where "atom s \<sharp> (v,t,k)"  "atom k \<sharp> (v,t)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: SubstTermP.simps [of s _ _ _ _ k] intro: SeqStTermP_Zero [THEN cut1])
 qed
 
 corollary SubstTermP_Var_same: "{VarP v, TermP t} \<turnstile> SubstTermP v t v t"
 proof -
   obtain s::name and k::name where "atom s \<sharp> (v,t,k)" "atom k \<sharp> (v,t)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: SubstTermP.simps [of s _ _ _ _ k] intro: SeqStTermP_Var_same [THEN cut1])
 qed
 
 corollary SubstTermP_Var_diff: "{VarP v, VarP w, Neg (v EQ w), TermP t} \<turnstile> SubstTermP v t w w"
 proof -
   obtain s::name and k::name where "atom s \<sharp> (v,w,t,k)" "atom k \<sharp> (v,w,t)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: SubstTermP.simps [of s _ _ _ _ k] intro: SeqStTermP_Var_diff [THEN cut3])
 qed
 
 lemma SeqStTermP_Ind:
   assumes "atom s \<sharp> (k,v,t,i)" "atom k \<sharp> (v,t,i)"
     shows "{VarP v, IndP t} \<turnstile> Ex s (Ex k (SeqStTermP v i t t (Var s) (Var k)))"
 proof -
   obtain l::name and sl::name and sl'::name and m::name and sm::name and sm'::name
      and n::name and sn::name and sn'::name
     where "atom l \<sharp> (v,t,i,s,k,sl,sl',m,n,sm,sm',sn,sn')"
           "atom sl \<sharp> (v,t,i,s,k,sl',m,n,sm,sm',sn,sn')"
           "atom sl' \<sharp> (v,t,i,s,k,m,n,sm,sm',sn,sn')"
           "atom m \<sharp> (v,t,i,s,k,n,sm,sm',sn,sn')" "atom n \<sharp> (v,t,i,s,k,sm,sm',sn,sn')"
           "atom sm \<sharp> (v,t,i,s,k,sm',sn,sn')" "atom sm' \<sharp> (v,t,i,s,k,sn,sn')"
           "atom sn \<sharp> (v,t,i,s,k,sn')" "atom sn' \<sharp> (v,t,i,s,k)"
     by (metis obtain_fresh)
   thus ?thesis using assms
     apply (simp add: SeqStTermP.simps [of l _ _ v i sl sl' m n sm sm' sn sn'])
     apply (rule Ex_I [where x = "Eats Zero (HPair Zero (HPair t t))"], simp)
     apply (rule Ex_I [where x = Zero], auto intro!: Mem_SUCC_EH)
     apply (rule Ex_I [where x = t], simp)
     apply (rule Ex_I [where x = t], auto intro: HPair_cong Mem_Eats_I2)
     apply (blast intro: Disj_I1 Disj_I2 VarP_neq_IndP)
     done
 qed
 
 corollary SubstTermP_Ind: "{VarP v, IndP w, TermP t} \<turnstile> SubstTermP v t w w"
 proof -
   obtain s::name and k::name where "atom s \<sharp> (v,w,t,k)" "atom k \<sharp> (v,w,t)"
     by (metis obtain_fresh)
   thus ?thesis
     by (force simp: SubstTermP.simps [of s _ _ _ _ k]
               intro: SeqStTermP_Ind [THEN cut2])
 qed
 
-subsection {*Non-atomic cases*}
+subsection \<open>Non-atomic cases\<close>
 
 lemma SeqStTermP_Eats:
   assumes sk: "atom s \<sharp> (k,s1,s2,k1,k2,t1,t2,u1,u2,v,i)"
               "atom k \<sharp> (t1,t2,u1,u2,v,i)"
     shows "{SeqStTermP v i t1 u1 s1 k1, SeqStTermP v i t2 u2 s2 k2}
            \<turnstile> Ex s (Ex k (SeqStTermP v i (Q_Eats t1 t2) (Q_Eats u1 u2) (Var s) (Var k)))" (*<*)
 proof -
   obtain km::name and kn::name and j::name and k'::name and l::name and sl::name and sl'::name
      and m::name and n::name and sm::name and sm'::name and sn::name and sn'::name
     where atoms2: "atom km \<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,v,i,kn,j,k',l,sl,sl',m,n,sm,sm',sn,sn')"
          "atom kn \<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,v,i,j,k',l,sl,sl',m,n,sm,sm',sn,sn')"
          "atom j  \<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,v,i,k',l,sl,sl',m,n,sm,sm',sn,sn')"
       and atoms: "atom k' \<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,v,i,l,sl,sl',m,n,sm,sm',sn,sn')"
          "atom l  \<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,v,i,sl,sl',m,n,sm,sm',sn,sn')"
          "atom sl \<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,v,i,sl',m,n,sm,sm',sn,sn')"
          "atom sl'\<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,v,i,m,n,sm,sm',sn,sn')"
          "atom m  \<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,v,i,n,sm,sm',sn,sn')"
          "atom n  \<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,v,i,sm,sm',sn,sn')"
          "atom sm \<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,v,i,sm',sn,sn')"
          "atom sm'\<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,v,i,sn,sn')"
          "atom sn \<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,v,i,sn')"
          "atom sn'\<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,v,i)"
     by (metis obtain_fresh)
   let ?hyp = "{HaddP k1 k2 (Var k'), OrdP k1, OrdP k2, SeqAppendP s1 (SUCC k1) s2 (SUCC k2) (Var s), SeqStTermP v i t1 u1 s1 k1, SeqStTermP v i t2 u2 s2 k2}"
   show ?thesis
      using sk atoms
      apply (auto simp: SeqStTermP.simps [of l "Var s" _ _ _ sl sl' m n sm sm' sn sn'])
      apply (rule cut_same [where A="OrdP k1 AND OrdP k2"])
      apply (metis Conj_I SeqStTermP_imp_OrdP thin1 thin2)
      apply (rule cut_same [OF exists_SeqAppendP [of s s1 "SUCC k1" s2 "SUCC k2"]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
      apply (rule cut_same [OF exists_HaddP [where j=k' and x=k1 and y=k2]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
      apply (rule Ex_I [where x="Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Q_Eats t1 t2) (Q_Eats u1 u2)))"])
      apply (simp_all (no_asm_simp))
      apply (rule Ex_I [where x="SUCC (SUCC (Var k'))"], simp)
      apply (rule Conj_I [OF _ Conj_I])
      apply (metis SeqStTermP_imp_VarP thin1)
      apply (blast intro: LstSeqP_SeqAppendP_Eats SeqStTermP_imp_LstSeqP [THEN cut1])
      proof (rule All2_SUCC_I, simp_all)
        show "?hyp \<turnstile> Ex sl (Ex sl'
               (HPair (SUCC (SUCC (Var k'))) (HPair (Var sl) (Var sl')) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Q_Eats t1 t2) (Q_Eats u1 u2))) AND
                ((Var sl EQ v AND Var sl' EQ i OR (IndP (Var sl) OR Var sl NEQ v) AND Var sl' EQ Var sl) OR
                 Ex m (Ex n (Ex sm (Ex sm' (Ex sn (Ex sn'
                  (Var m IN SUCC (SUCC (Var k')) AND
                   Var n IN SUCC (SUCC (Var k')) AND
                   HPair (Var m) (HPair (Var sm) (Var sm')) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Q_Eats t1 t2) (Q_Eats u1 u2))) AND
                   HPair (Var n) (HPair (Var sn) (Var sn')) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Q_Eats t1 t2) (Q_Eats u1 u2))) AND
                   Var sl EQ Q_Eats (Var sm) (Var sn) AND Var sl' EQ Q_Eats (Var sm') (Var sn'))))))))))"
        \<comment> \<open>verifying the final values\<close>
        apply (rule Ex_I [where x="Q_Eats t1 t2"])
        using sk atoms apply simp
        apply (rule Ex_I [where x="Q_Eats u1 u2"], simp)
        apply (rule Conj_I, metis Mem_Eats_I2 Refl)
        apply (rule Disj_I2)
        apply (rule Ex_I [where x=k1], simp)
        apply (rule Ex_I [where x="SUCC (Var k')"], simp)
        apply (rule Ex_I [where x=t1], simp)
        apply (rule Ex_I [where x=u1], simp)
        apply (rule Ex_I [where x=t2], simp)
        apply (rule Ex_I [where x=u2], simp)
        apply (rule Conj_I)
        apply (blast intro: HaddP_Mem_I LstSeqP_OrdP Mem_SUCC_I1)
        apply (rule Conj_I [OF Mem_SUCC_Refl Conj_I])
        apply (blast intro: Mem_Eats_I1 SeqAppendP_Mem1 [THEN cut3] Mem_SUCC_Refl SeqStTermP_imp_LstSeqP [THEN cut1]
                            LstSeqP_imp_Mem)
        apply (blast intro: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] Mem_SUCC_Refl SeqStTermP_imp_LstSeqP [THEN cut1]
                            HaddP_SUCC1 [THEN cut1] LstSeqP_imp_Mem)
        done
      next
        show "?hyp \<turnstile> All2 l (SUCC (SUCC (Var k')))
                 (Ex sl (Ex sl'
                   (HPair (Var l) (HPair (Var sl) (Var sl')) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Q_Eats t1 t2) (Q_Eats u1 u2))) AND
                    ((Var sl EQ v AND Var sl' EQ i OR (IndP (Var sl) OR Var sl NEQ v) AND Var sl' EQ Var sl) OR
                     Ex m (Ex n (Ex sm (Ex sm' (Ex sn (Ex sn'
                        (Var m IN Var l AND
                         Var n IN Var l AND
                         HPair (Var m) (HPair (Var sm) (Var sm')) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Q_Eats t1 t2) (Q_Eats u1 u2))) AND
                         HPair (Var n) (HPair (Var sn) (Var sn')) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Q_Eats t1 t2) (Q_Eats u1 u2))) AND
                         Var sl EQ Q_Eats (Var sm) (Var sn) AND Var sl' EQ Q_Eats (Var sm') (Var sn')))))))))))"
      \<comment> \<open>verifying the sequence buildup\<close>
      apply (rule cut_same [where A="HaddP (SUCC k1) (SUCC k2) (SUCC (SUCC (Var k')))"])
      apply (blast intro: HaddP_SUCC1 [THEN cut1] HaddP_SUCC2 [THEN cut1])
      apply (rule All_I Imp_I)+
      apply (rule HaddP_Mem_cases [where i=j])
      using sk atoms atoms2 apply simp_all
      apply (rule AssumeH)
      apply (blast intro: OrdP_SUCC_I LstSeqP_OrdP)
      \<comment> \<open>... the sequence buildup via s1\<close>
      apply (simp add: SeqStTermP.simps [of l s1 _ _ _ sl sl' m n sm sm' sn sn'])
      apply (rule AssumeH Ex_EH Conj_EH)+
      apply (rule All2_E [THEN rotate2])
      apply (simp | rule AssumeH Ex_EH Conj_EH)+
      apply (rule Ex_I [where x="Var sl"], simp)
      apply (rule Ex_I [where x="Var sl'"], simp)
      apply (rule Conj_I)
      apply (metis Mem_Eats_I1 SeqAppendP_Mem1 rotate3 thin2 thin4)
      apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
      apply (rule Ex_I [where x="Var m"], simp)
      apply (rule Ex_I [where x="Var n"], simp)
      apply (rule Ex_I [where x="Var sm"], simp)
      apply (rule Ex_I [where x="Var sm'"], simp)
      apply (rule Ex_I [where x="Var sn"], simp)
      apply (rule Ex_I [where x="Var sn'"], simp_all)
      apply (rule Conj_I, rule AssumeH)+
      apply (blast del: Disj_EH intro: OrdP_Trans [OF OrdP_SUCC_I] Mem_Eats_I1 [OF SeqAppendP_Mem1 [THEN cut3]] Hyp)
      \<comment> \<open>... the sequence buildup via s2\<close>
      apply (simp add: SeqStTermP.simps [of l s2 _ _ _ sl sl' m n sm sm' sn sn'])
      apply (rule AssumeH Ex_EH Conj_EH)+
      apply (rule All2_E [THEN rotate2])
      apply (simp | rule AssumeH Ex_EH Conj_EH)+
      apply (rule Ex_I [where x="Var sl"], simp)
      apply (rule Ex_I [where x="Var sl'"], simp)
      apply (rule cut_same [where A="OrdP (Var j)"])
      apply (metis HaddP_imp_OrdP rotate2 thin2)
      apply (rule Conj_I)
      apply (blast intro: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] del: Disj_EH)
      apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
      apply simp_all
      apply (rule cut_same [OF exists_HaddP [where j=km and x="SUCC k1" and y="Var m"]])
      apply (blast intro!: Ord_IN_Ord, simp)
      apply (rule cut_same [OF exists_HaddP [where j=kn and x="SUCC k1" and y="Var n"]])
      apply (blast intro!: Ord_IN_Ord, simp)
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
      apply (rule Ex_I [where x="Var km"], simp)
      apply (rule Ex_I [where x="Var kn"], simp)
      apply (rule Ex_I [where x="Var sm"], simp)
      apply (rule Ex_I [where x="Var sm'"], simp)
      apply (rule Ex_I [where x="Var sn"], simp)
      apply (rule Ex_I [where x="Var sn'"], simp_all)
      apply (rule Conj_I [OF _ Conj_I])
      apply (blast intro!: HaddP_Mem_cancel_left [THEN Iff_MP2_same] OrdP_SUCC_I intro: LstSeqP_OrdP Hyp)+
      apply (blast intro: OrdP_Trans Hyp Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] HaddP_imp_OrdP [THEN cut1])
      done
    qed
 qed (*>*)
 
 theorem SubstTermP_Eats:
    "{SubstTermP v i t1 u1, SubstTermP v i t2 u2} \<turnstile> SubstTermP v i (Q_Eats t1 t2) (Q_Eats u1 u2)"
 proof -
   obtain k1::name and s1::name and k2::name and s2::name and k::name and s::name
     where "atom s1 \<sharp> (v,i,t1,u1,t2,u2)" "atom k1 \<sharp> (v,i,t1,u1,t2,u2,s1)"
           "atom s2 \<sharp> (v,i,t1,u1,t2,u2,k1,s1)" "atom k2 \<sharp> (v,i,t1,u1,t2,u2,s2,k1,s1)"
           "atom s  \<sharp> (v,i,t1,u1,t2,u2,k2,s2,k1,s1)"
           "atom k  \<sharp> (v,i,t1,u1,t2,u2,s,k2,s2,k1,s1)"
     by (metis obtain_fresh)
    thus ?thesis
      by (auto intro!: SeqStTermP_Eats [THEN cut2]
               simp: SubstTermP.simps [of s _ _ _ "(Q_Eats u1 u2)" k]
                     SubstTermP.simps [of s1 v i t1 u1 k1]
                     SubstTermP.simps [of s2 v i t2 u2 k2])
 qed
 
-subsection {*Substitution over a constant*}
+subsection \<open>Substitution over a constant\<close>
 
 lemma SeqConstP_lemma:
   assumes "atom m \<sharp> (s,k,c,n,sm,sn)"  "atom n \<sharp> (s,k,c,sm,sn)"
           "atom sm \<sharp> (s,k,c,sn)"      "atom sn \<sharp> (s,k,c)"
     shows "{ SeqConstP s k c }
            \<turnstile> c EQ Zero OR
              Ex m (Ex n (Ex sm (Ex sn (Var m IN k AND Var n IN k AND
                    SeqConstP s (Var m) (Var sm) AND
                    SeqConstP s (Var n) (Var sn) AND
                    c EQ Q_Eats (Var sm) (Var sn)))))"  (*<*)
 proof -
   obtain l::name and sl::name where "atom l \<sharp> (s,k,c,sl,m,n,sm,sn)" "atom sl \<sharp> (s,k,c,m,n,sm,sn)"
     by (metis obtain_fresh)
   thus ?thesis using assms
     apply (simp add: SeqCTermP.simps [of l s k sl m n sm sn])
     apply (rule Conj_EH)+
     apply (rule All2_SUCC_E [THEN rotate2], auto del: Disj_EH)
     apply (rule cut_same [where A = "c EQ (Var sl)"])
     apply (metis Assume AssumeH(4) LstSeqP_EQ)
     apply (rule Disj_EH)
     apply (blast intro: Disj_I1 Sym Trans)
     \<comment> \<open>now the quantified case\<close>
     apply (auto intro!: Disj_I2)
     apply (rule Ex_I [where x = "Var m"], simp)
     apply (rule Ex_I [where x = "Var n"], simp)
     apply (rule Ex_I [where x = "Var sm"], simp)
     apply (rule Ex_I [where x = "Var sn"], simp)
     apply (simp_all add: SeqCTermP.simps [of l s _ sl m n sm sn])
     apply ((rule Conj_I)+, blast intro: LstSeqP_Mem)+
     \<comment> \<open>first SeqCTermP subgoal\<close>
     apply (rule All2_Subset [OF Hyp], blast)
     apply (blast intro!: SUCC_Subset_Ord LstSeqP_OrdP, blast, simp)
     \<comment> \<open>next SeqCTermP subgoal\<close>
     apply ((rule Conj_I)+, blast intro: LstSeqP_Mem)+
     apply (rule All2_Subset [OF Hyp], blast)
     apply (blast intro!: SUCC_Subset_Ord LstSeqP_OrdP, blast, simp)
     \<comment> \<open>finally, the equality pair\<close>
     apply (blast intro: Trans)
     done
 qed (*>*)
 
 lemma SeqConstP_imp_SubstTermP: "{SeqConstP s kk c, TermP t} \<turnstile> SubstTermP \<lceil>Var w\<rceil> t c c" (*<*)
 proof -
   obtain j::name and k::name and l::name and sl::name and m::name and n::name and sm::name and sn::name
     where atoms: "atom j \<sharp> (s,kk,c,t,k,l,sl,m,n,sm,sn)"  "atom k \<sharp> (s,kk,c,t,l,sl,m,n,sm,sn)"
                  "atom l \<sharp> (s,kk,c,t,sl,m,n,sm,sn)"   "atom sl \<sharp> (s,kk,c,t,m,n,sm,sn)"
                  "atom m \<sharp> (s,kk,c,t,n,sm,sn)"  "atom n \<sharp> (s,kk,c,t,sm,sn)"
                  "atom sm \<sharp> (s,kk,c,t,sn)"      "atom sn \<sharp> (s,kk,c,t)"
     by (metis obtain_fresh)
   have "{ OrdP (Var k), TermP t } \<turnstile> All j (SeqConstP s (Var k) (Var j) IMP SubstTermP \<lceil>Var w\<rceil> t (Var j) (Var j))"
         (is "_ \<turnstile> ?scheme")
     proof (rule OrdIndH [where j=l])
       show "atom l \<sharp> (k, ?scheme)" using atoms
         by simp
     next
       show "{TermP t} \<turnstile> All k (OrdP (Var k) IMP (All2 l (Var k) (?scheme(k::= Var l)) IMP ?scheme))"
         using atoms apply auto
         apply (rule Swap)
         apply (rule cut_same)
         apply (rule cut1 [OF SeqConstP_lemma [of m s "Var k" "Var j" n sm sn]], auto)
         \<comment> \<open>case 1, @{term Zero}\<close>
         apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same])
         apply (auto intro: SubstTermP_Zero [THEN cut1])
         \<comment> \<open>case 2, @{term Q_Eats}\<close>
         apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same, THEN rotate2], simp)
         apply (rule SubstTermP_Eats [THEN cut2])
         \<comment> \<open>First argument\<close>
         apply (rule All2_E' [OF Hyp, where x="Var m"], blast+, simp_all)
         apply (force intro: All_E [where x="Var sm"])
         \<comment> \<open>Second argument\<close>
         apply (rule All2_E' [OF Hyp, where x="Var n"], blast+, simp_all)
         apply (rule All_E [where x="Var sn"], auto)
         done
     qed
   hence "{OrdP (Var k), TermP t} \<turnstile> (SeqConstP s (Var k) (Var j) IMP SubstTermP \<lceil>Var w\<rceil> t (Var j) (Var j))(j::=c)"
     by (metis All_D)
   hence "{TermP t} \<turnstile> (SeqConstP s (Var k) c IMP SubstTermP \<lceil>Var w\<rceil> t c c)"
     using atoms by simp (metis Imp_cut SeqCTermP_imp_OrdP)
   hence "{TermP t} \<turnstile> (SeqConstP s (Var k) c IMP SubstTermP \<lceil>Var w\<rceil> t c c)(k::=kk)"
     using atoms by (force intro!: Subst)
   thus ?thesis
     using atoms by (simp add: anti_deduction)
 qed (*>*)
 
 theorem SubstTermP_Const: "{ConstP c, TermP t} \<turnstile> SubstTermP \<lceil>Var w\<rceil> t c c"
 proof -
   obtain s::name and k::name where "atom s \<sharp> (c,t,w,k)" "atom k \<sharp> (c,t,w)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: CTermP.simps [of k s c] SeqConstP_imp_SubstTermP)
 qed
 
 
-section {*Substitution on Formulas*}
-
-subsection {*Membership*}
+section \<open>Substitution on Formulas\<close>
+
+subsection \<open>Membership\<close>
 
 lemma SubstAtomicP_Mem:
   "{SubstTermP v i x x', SubstTermP v i y y'} \<turnstile> SubstAtomicP v i (Q_Mem x y) (Q_Mem x' y')"
 proof -
   obtain t::name and u::name and t'::name and u'::name
     where "atom t \<sharp> (v,i,x,x',y,y',t',u,u')" "atom t' \<sharp> (v,i,x,x',y,y',u,u')"
       "atom u \<sharp> (v,i,x,x',y,y',u')" "atom u' \<sharp> (v,i,x,x',y,y')"
     by (metis obtain_fresh)
   thus ?thesis
     apply (simp add: SubstAtomicP.simps [of t _ _ _ _ t' u u'])
     apply (rule Ex_I [where x = x], simp)
     apply (rule Ex_I [where x = y], simp)
     apply (rule Ex_I [where x = x'], simp)
     apply (rule Ex_I [where x = y'], auto intro: Disj_I2)
     done
 qed
 
 lemma SeqSubstFormP_Mem:
   assumes "atom s \<sharp> (k,x,y,x',y',v,i)" "atom k \<sharp> (x,y,x',y',v,i)"
   shows "{SubstTermP v i x x', SubstTermP v i y y'}
            \<turnstile> Ex s (Ex k (SeqSubstFormP v i (Q_Mem x y) (Q_Mem x' y') (Var s) (Var k)))"
 proof -
   let ?vs = "(s,k,x,y,x',y',v,i)"
   obtain l::name and sl::name and sl'::name and m::name and n::name and sm::name and sm'::name and sn::name and sn'::name
     where "atom l \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn')"
       "atom sl \<sharp> (?vs,sl',m,n,sm,sm',sn,sn')" "atom sl' \<sharp> (?vs,m,n,sm,sm',sn,sn')"
       "atom m \<sharp> (?vs,n,sm,sm',sn,sn')" "atom n \<sharp> (?vs,sm,sm',sn,sn')"
       "atom sm \<sharp> (?vs,sm',sn,sn')" "atom sm' \<sharp> (?vs,sn,sn')"
       "atom sn \<sharp> (?vs,sn')" "atom sn' \<sharp> ?vs"
     by (metis obtain_fresh)
   thus ?thesis
     using assms
     apply (auto simp: SeqSubstFormP.simps [of l "Var s" _ _ _ sl sl' m n sm sm' sn sn'])
     apply (rule Ex_I [where x = "Eats Zero (HPair Zero (HPair (Q_Mem x y) (Q_Mem x' y')))"], simp)
     apply (rule Ex_I [where x = Zero], auto intro!: Mem_SUCC_EH)
     apply (rule Ex_I [where x = "Q_Mem x y"], simp)
     apply (rule Ex_I [where x = "Q_Mem x' y'"], auto intro: Mem_Eats_I2 HPair_cong)
     apply (blast intro: SubstAtomicP_Mem [THEN cut2] Disj_I1)
     done
 qed
 
 lemma SubstFormP_Mem:
   "{SubstTermP v i x x', SubstTermP v i y y'} \<turnstile> SubstFormP v i (Q_Mem x y) (Q_Mem x' y')"
 proof -
   obtain k1::name and s1::name and k2::name and s2::name and k::name and s::name
     where "atom s1 \<sharp> (v,i,x,y,x',y')" "atom k1 \<sharp> (v,i,x,y,x',y',s1)"
       "atom s2 \<sharp> (v,i,x,y,x',y',k1,s1)" "atom k2 \<sharp> (v,i,x,y,x',y',s2,k1,s1)"
       "atom s  \<sharp> (v,i,x,y,x',y',k2,s2,k1,s1)" "atom k  \<sharp> (v,i,x,y,x',y',s,k2,s2,k1,s1)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: SubstFormP.simps [of s v i "(Q_Mem x y)" _ k]
         SubstFormP.simps [of s1 v i x x' k1]
         SubstFormP.simps [of s2 v i y y' k2]
         intro: SubstTermP_imp_TermP SubstTermP_imp_VarP SeqSubstFormP_Mem thin1)
 qed
 
-subsection {*Equality*}
+subsection \<open>Equality\<close>
 
 lemma SubstAtomicP_Eq:
   "{SubstTermP v i x x', SubstTermP v i y y'} \<turnstile> SubstAtomicP v i (Q_Eq x y) (Q_Eq x' y')"
 proof -
   obtain t::name and u::name and t'::name  and u'::name
     where "atom t \<sharp> (v,i,x,x',y,y',t',u,u')" "atom t' \<sharp> (v,i,x,x',y,y',u,u')"
       "atom u \<sharp> (v,i,x,x',y,y',u')" "atom u' \<sharp> (v,i,x,x',y,y')"
     by (metis obtain_fresh)
   thus ?thesis
     apply (simp add: SubstAtomicP.simps [of t _ _ _ _ t' u u'])
     apply (rule Ex_I [where x = x], simp)
     apply (rule Ex_I [where x = y], simp)
     apply (rule Ex_I [where x = x'], simp)
     apply (rule Ex_I [where x = y'], auto intro: Disj_I1)
     done
 qed
 
 lemma SeqSubstFormP_Eq:
   assumes sk: "atom s \<sharp> (k,x,y,x',y',v,i)" "atom k \<sharp> (x,y,x',y',v,i)"
   shows "{SubstTermP v i x x', SubstTermP v i y y'}
            \<turnstile> Ex s (Ex k (SeqSubstFormP v i (Q_Eq x y) (Q_Eq x' y') (Var s) (Var k)))"
 proof -
   let ?vs = "(s,k,x,y,x',y',v,i)"
   obtain l::name and sl::name and sl'::name and m::name and n::name and sm::name and sm'::name and sn::name and sn'::name
     where "atom l \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn')"
       "atom sl \<sharp> (?vs,sl',m,n,sm,sm',sn,sn')" "atom sl' \<sharp> (?vs,m,n,sm,sm',sn,sn')"
       "atom m \<sharp> (?vs,n,sm,sm',sn,sn')" "atom n \<sharp> (?vs,sm,sm',sn,sn')"
       "atom sm \<sharp> (?vs,sm',sn,sn')" "atom sm' \<sharp> (?vs,sn,sn')"
       "atom sn \<sharp> (?vs,sn')" "atom sn' \<sharp> ?vs"
     by (metis obtain_fresh)
   thus ?thesis
     using sk
     apply (auto simp: SeqSubstFormP.simps [of l "Var s" _ _ _ sl sl' m n sm sm' sn sn'])
     apply (rule Ex_I [where x = "Eats Zero (HPair Zero (HPair (Q_Eq x y) (Q_Eq x' y')))"], simp)
     apply (rule Ex_I [where x = Zero], auto intro!: Mem_SUCC_EH)
     apply (rule Ex_I [where x = "Q_Eq x y"], simp)
     apply (rule Ex_I [where x = "Q_Eq x' y'"], auto)
      apply (metis Mem_Eats_I2 Assume HPair_cong Refl)
     apply (blast intro: SubstAtomicP_Eq [THEN cut2] Disj_I1)
     done
 qed
 
 lemma SubstFormP_Eq:
   "{SubstTermP v i x x', SubstTermP v i y y'} \<turnstile> SubstFormP v i (Q_Eq x y) (Q_Eq x' y')"
 proof -
   obtain k1::name and s1::name and k2::name and s2::name and k::name and s::name
     where "atom s1 \<sharp> (v,i,x,y,x',y')" "atom k1 \<sharp> (v,i,x,y,x',y',s1)"
       "atom s2 \<sharp> (v,i,x,y,x',y',k1,s1)" "atom k2 \<sharp> (v,i,x,y,x',y',s2,k1,s1)"
       "atom s  \<sharp> (v,i,x,y,x',y',k2,s2,k1,s1)" "atom k  \<sharp> (v,i,x,y,x',y',s,k2,s2,k1,s1)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: SubstFormP.simps [of s v i "(Q_Eq x y)" _ k]
         SubstFormP.simps [of s1 v i x x' k1]
         SubstFormP.simps [of s2 v i y y' k2]
         intro: SeqSubstFormP_Eq SubstTermP_imp_TermP SubstTermP_imp_VarP thin1)
 qed
 
-subsection {*Negation*}
+subsection \<open>Negation\<close>
 
 lemma SeqSubstFormP_Neg:
   assumes "atom s \<sharp> (k,s1,k1,x,x',v,i)" "atom k \<sharp> (s1,k1,x,x',v,i)"
   shows "{SeqSubstFormP v i x x' s1 k1, TermP i, VarP v}
            \<turnstile> Ex s (Ex k (SeqSubstFormP v i (Q_Neg x) (Q_Neg x') (Var s) (Var k)))" (*<*)
 proof -
   let ?vs = "(s1,k1,s,k,x,x',v,i)"
   obtain l::name and sl::name and sl'::name and m::name and n::name and
     sm::name and sm'::name and sn::name and sn'::name
     where atoms:
       "atom l \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn')"
       "atom sl \<sharp> (?vs,sl',m,n,sm,sm',sn,sn')" "atom sl' \<sharp> (?vs,m,n,sm,sm',sn,sn')"
       "atom m \<sharp> (?vs,n,sm,sm',sn,sn')" "atom n \<sharp> (?vs,sm,sm',sn,sn')"
       "atom sm \<sharp> (?vs,sm',sn,sn')" "atom sm' \<sharp> (?vs,sn,sn')"
       "atom sn \<sharp> (?vs,sn')" "atom sn' \<sharp> ?vs"
     by (metis obtain_fresh)
   let ?hyp = "{RestrictedP s1 (SUCC k1) (Var s), OrdP k1, SeqSubstFormP v i x x' s1 k1, TermP i, VarP v}"
   show ?thesis
     using assms atoms
     apply (auto simp: SeqSubstFormP.simps [of l "Var s" _ _ _ sl sl' m n sm sm' sn sn'])
     apply (rule cut_same [where A="OrdP k1"])
      apply (metis SeqSubstFormP_imp_OrdP thin2)
     apply (rule cut_same [OF exists_RestrictedP [of s s1 "SUCC k1"]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
       apply (rule Ex_I [where x="Eats (Var s) (HPair (SUCC k1) (HPair (Q_Neg x) (Q_Neg x')))"])
       apply (simp_all (no_asm_simp))
     apply (rule Ex_I [where x="(SUCC k1)"])
     apply (simp add: flip_fresh_fresh)
     apply (rule Conj_I)
      apply (blast intro: RestrictedP_LstSeqP_Eats [THEN cut2] SeqSubstFormP_imp_LstSeqP [THEN cut1])
   proof (rule All2_SUCC_I, simp_all)
     show "?hyp \<turnstile> Ex sl (Ex sl'
               (HPair (SUCC k1) (HPair (Var sl) (Var sl')) IN Eats (Var s) (HPair (SUCC k1) (HPair (Q_Neg x) (Q_Neg x'))) AND
                (SubstAtomicP v i (Var sl) (Var sl') OR
                 Ex m (Ex n
                    (Ex sm (Ex sm'
                        (Ex sn (Ex sn'
                        (Var m IN SUCC k1 AND
                         Var n IN SUCC k1 AND
                         HPair (Var m) (HPair (Var sm) (Var sm')) IN Eats (Var s) (HPair (SUCC k1) (HPair (Q_Neg x) (Q_Neg x'))) AND
                         HPair (Var n) (HPair (Var sn) (Var sn')) IN Eats (Var s) (HPair (SUCC k1) (HPair (Q_Neg x) (Q_Neg x'))) AND
                         (Var sl EQ Q_Disj (Var sm) (Var sn) AND Var sl' EQ Q_Disj (Var sm') (Var sn') OR
                           Var sl EQ Q_Neg (Var sm) AND Var sl' EQ Q_Neg (Var sm') OR Var sl EQ Q_Ex (Var sm) AND Var sl' EQ Q_Ex (Var sm')))))))))))"
       \<comment> \<open>verifying the final values\<close>
       apply (rule Ex_I [where x="Q_Neg x"])
       using assms atoms apply simp
       apply (rule Ex_I [where x="Q_Neg x'"], simp)
       apply (rule Conj_I, metis Mem_Eats_I2 Refl)
       apply (rule Disj_I2)
       apply (rule Ex_I [where x=k1], simp)
       apply (rule Ex_I [where x=k1], simp)
       apply (rule Ex_I [where x=x], simp)
       apply (rule_tac x=x' in Ex_I, simp)
       apply (rule Ex_I [where x=x], simp)
       apply (rule_tac x=x' in Ex_I, simp)
       apply (rule Conj_I [OF Mem_SUCC_Refl])+
       apply (blast intro: Disj_I1 Disj_I2 Mem_Eats_I1 RestrictedP_Mem [THEN cut3] Mem_SUCC_Refl
           SeqSubstFormP_imp_LstSeqP [THEN cut1] LstSeqP_imp_Mem)
       done
   next
     show "?hyp \<turnstile> All2 l (SUCC k1)
                 (Ex sl (Ex sl'
                     (HPair (Var l) (HPair (Var sl) (Var sl')) IN Eats (Var s) (HPair (SUCC k1) (HPair (Q_Neg x) (Q_Neg x'))) AND
                      (SubstAtomicP v i (Var sl) (Var sl') OR
                       Ex m (Ex n
                          (Ex sm (Ex sm'
                              (Ex sn (Ex sn'
                                  (Var m IN Var l AND
                                   Var n IN Var l AND
                                   HPair (Var m) (HPair (Var sm) (Var sm')) IN Eats (Var s) (HPair (SUCC k1) (HPair (Q_Neg x) (Q_Neg x'))) AND
                                   HPair (Var n) (HPair (Var sn) (Var sn')) IN Eats (Var s) (HPair (SUCC k1) (HPair (Q_Neg x) (Q_Neg x'))) AND
                                   (Var sl EQ Q_Disj (Var sm) (Var sn) AND Var sl' EQ Q_Disj (Var sm') (Var sn') OR
                             Var sl EQ Q_Neg (Var sm) AND Var sl' EQ Q_Neg (Var sm') OR Var sl EQ Q_Ex (Var sm) AND Var sl' EQ Q_Ex (Var sm'))))))))))))"
       \<comment> \<open>verifying the sequence buildup\<close>
       apply (rule All_I Imp_I)+
       using assms atoms apply simp_all
         \<comment> \<open>... the sequence buildup via s1\<close>
       apply (simp add: SeqSubstFormP.simps [of l s1 _ _ _ sl sl' m n sm sm' sn sn'])
       apply (rule AssumeH Ex_EH Conj_EH)+
       apply (rule All2_E [THEN rotate2], auto del: Disj_EH)
       apply (rule Ex_I [where x="Var sl"], simp)
       apply (rule Ex_I [where x="Var sl'"], simp)
       apply (rule Conj_I)
        apply (blast intro: Mem_Eats_I1 [OF RestrictedP_Mem [THEN cut3]] del: Disj_EH)
       apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
                   apply (rule Ex_I [where x="Var m"], simp)
                   apply (rule Ex_I [where x="Var n"], simp)
                   apply (rule Ex_I [where x="Var sm"], simp)
                   apply (rule Ex_I [where x="Var sm'"], simp)
                   apply (rule Ex_I [where x="Var sn"], simp)
                   apply (rule Ex_I [where x="Var sn'"], auto del: Disj_EH)
        apply (blast intro: Mem_Eats_I1 [OF RestrictedP_Mem [THEN cut3]] OrdP_Trans [OF OrdP_SUCC_I])+
       done
   qed
 qed (*>*)
 
 theorem SubstFormP_Neg: "{SubstFormP v i x x'} \<turnstile> SubstFormP v i (Q_Neg x) (Q_Neg x')"
 proof -
   obtain k1::name and s1::name and k::name and s::name
     where "atom s1 \<sharp> (v,i,x,x')"        "atom k1 \<sharp> (v,i,x,x',s1)"
       "atom s  \<sharp> (v,i,x,x',k1,s1)"  "atom k  \<sharp> (v,i,x,x',s,k1,s1)"
     by (metis obtain_fresh)
   thus ?thesis
     by (force simp: SubstFormP.simps [of s v i "Q_Neg x" _ k] SubstFormP.simps [of s1 v i x x' k1]
         intro: SeqSubstFormP_Neg [THEN cut3])
 qed
 
-subsection {*Disjunction*}
+subsection \<open>Disjunction\<close>
 
 lemma SeqSubstFormP_Disj:
   assumes "atom s \<sharp> (k,s1,s2,k1,k2,x,y,x',y',v,i)" "atom k \<sharp> (s1,s2,k1,k2,x,y,x',y',v,i)"
   shows "{SeqSubstFormP v i x x' s1 k1,
             SeqSubstFormP v i y y' s2 k2, TermP i, VarP v}
            \<turnstile> Ex s (Ex k (SeqSubstFormP v i (Q_Disj x y) (Q_Disj x' y') (Var s) (Var k)))" (*<*)
 proof -
   let ?vs = "(s1,s2,s,k1,k2,k,x,y,x',y',v,i)"
   obtain km::name and kn::name and j::name and k'::name
     and l::name and sl::name and sl'::name and m::name and n::name
     and sm::name and sm'::name and sn::name and sn'::name
     where atoms2: "atom km \<sharp> (kn,j,k',l,s1,s2,s,k1,k2,k,x,y,x',y',v,i,sl,sl',m,n,sm,sm',sn,sn')"
       "atom kn \<sharp> (j,k',l,s1,s2,s,k1,k2,k,x,y,x',y',v,i,sl,sl',m,n,sm,sm',sn,sn')"
       "atom j \<sharp> (k',l,s1,s2,s,k1,k2,k,x,y,x',y',v,i,sl,sl',m,n,sm,sm',sn,sn')"
       and atoms: "atom k' \<sharp> (l,s1,s2,s,k1,k2,k,x,y,x',y',v,i,sl,sl',m,n,sm,sm',sn,sn')"
       "atom l \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',v,i,sl,sl',m,n,sm,sm',sn,sn')"
       "atom sl \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',v,i,sl',m,n,sm,sm',sn,sn')"
       "atom sl' \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',v,i,m,n,sm,sm',sn,sn')"
       "atom m \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',v,i,n,sm,sm',sn,sn')"
       "atom n \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',v,i,sm,sm',sn,sn')"
       "atom sm \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',v,i,sm',sn,sn')"
       "atom sm' \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',v,i,sn,sn')"
       "atom sn \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',v,i,sn')"
       "atom sn' \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',v,i)"
     by (metis obtain_fresh)
   let ?hyp = "{HaddP k1 k2 (Var k'), OrdP k1, OrdP k2, SeqAppendP s1 (SUCC k1) s2 (SUCC k2) (Var s),
                SeqSubstFormP v i x x' s1 k1, SeqSubstFormP v i y y' s2 k2, TermP i, VarP v}"
   show ?thesis
     using assms atoms
     apply (auto simp: SeqSubstFormP.simps [of l "Var s" _ _ _ sl sl' m n sm sm' sn sn'])
     apply (rule cut_same [where A="OrdP k1 AND OrdP k2"])
      apply (metis Conj_I SeqSubstFormP_imp_OrdP thin1 thin2)
     apply (rule cut_same [OF exists_SeqAppendP [of s s1 "SUCC k1" s2 "SUCC k2"]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
       apply (rule cut_same [OF exists_HaddP [where j=k' and x=k1 and y=k2]])
         apply (rule AssumeH Ex_EH Conj_EH | simp)+
         apply (rule Ex_I [where x="Eats (Var s) (HPair (SUCC(SUCC(Var k'))) (HPair(Q_Disj x y)(Q_Disj x' y')))"])
         apply (simp_all (no_asm_simp))
     apply (rule Ex_I [where x="SUCC (SUCC (Var k'))"], simp)
     apply (rule Conj_I)
      apply (blast intro: LstSeqP_SeqAppendP_Eats SeqSubstFormP_imp_LstSeqP [THEN cut1])
   proof (rule All2_SUCC_I, simp_all)
     show "?hyp \<turnstile> Ex sl (Ex sl'
                  (HPair (SUCC (SUCC (Var k'))) (HPair (Var sl) (Var sl')) IN
                   Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Q_Disj x y) (Q_Disj x' y'))) AND
                   (SubstAtomicP v i (Var sl) (Var sl') OR
                    Ex m (Ex n (Ex sm (Ex sm' (Ex sn (Ex sn'
                       (Var m IN SUCC (SUCC (Var k')) AND
                        Var n IN SUCC (SUCC (Var k')) AND
                        HPair (Var m) (HPair (Var sm) (Var sm')) IN
                        Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Q_Disj x y) (Q_Disj x' y'))) AND
                        HPair (Var n) (HPair (Var sn) (Var sn')) IN
                        Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Q_Disj x y) (Q_Disj x' y'))) AND
                        (Var sl EQ Q_Disj (Var sm) (Var sn) AND Var sl' EQ Q_Disj (Var sm') (Var sn') OR
                           Var sl EQ Q_Neg (Var sm) AND Var sl' EQ Q_Neg (Var sm') OR Var sl EQ Q_Ex (Var sm) AND Var sl' EQ Q_Ex (Var sm')))))))))))"
       \<comment> \<open>verifying the final values\<close>
       apply (rule Ex_I [where x="Q_Disj x y"])
       using assms atoms apply simp
       apply (rule Ex_I [where x="Q_Disj x' y'"], simp)
       apply (rule Conj_I, metis Mem_Eats_I2 Refl)
       apply (rule Disj_I2)
       apply (rule Ex_I [where x=k1], simp)
       apply (rule Ex_I [where x="SUCC (Var k')"], simp)
       apply (rule Ex_I [where x=x], simp)
       apply (rule_tac x=x' in Ex_I, simp)
       apply (rule Ex_I [where x=y], simp)
       apply (rule_tac x=y' in Ex_I, simp)
       apply (rule Conj_I)
        apply (blast intro: HaddP_Mem_I LstSeqP_OrdP Mem_SUCC_I1)
       apply (rule Conj_I [OF Mem_SUCC_Refl])
       apply (blast intro: Disj_I1 Mem_Eats_I1 Mem_SUCC_Refl SeqSubstFormP_imp_LstSeqP [THEN cut1]
           LstSeqP_imp_Mem SeqAppendP_Mem1 [THEN cut3] SeqAppendP_Mem2 [THEN cut4] HaddP_SUCC1 [THEN cut1])
       done
   next
     show "?hyp \<turnstile>  All2 l (SUCC (SUCC (Var k')))
      (Ex sl
        (Ex sl'
          (HPair (Var l) (HPair (Var sl) (Var sl')) IN
           Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Q_Disj x y) (Q_Disj x' y'))) AND
           (SubstAtomicP v i (Var sl) (Var sl') OR
            Ex m (Ex n
               (Ex sm (Ex sm'
                   (Ex sn (Ex sn'
                       (Var m IN Var l AND
                        Var n IN Var l AND
                        HPair (Var m) (HPair (Var sm) (Var sm')) IN
                        Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Q_Disj x y) (Q_Disj x' y'))) AND
                        HPair (Var n) (HPair (Var sn) (Var sn')) IN
                        Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Q_Disj x y) (Q_Disj x' y'))) AND
                        (Var sl EQ Q_Disj (Var sm) (Var sn) AND Var sl' EQ Q_Disj (Var sm') (Var sn') OR
                             Var sl EQ Q_Neg (Var sm) AND Var sl' EQ Q_Neg (Var sm') OR Var sl EQ Q_Ex (Var sm) AND Var sl' EQ Q_Ex (Var sm'))))))))))))"
       \<comment> \<open>verifying the sequence buildup\<close>
       apply (rule cut_same [where A="HaddP (SUCC k1) (SUCC k2) (SUCC (SUCC (Var k')))"])
        apply (blast intro: HaddP_SUCC1 [THEN cut1] HaddP_SUCC2 [THEN cut1])
       apply (rule All_I Imp_I)+
        apply (rule HaddP_Mem_cases [where i=j])
       using assms atoms atoms2 apply simp_all
           apply (rule AssumeH)
          apply (blast intro: OrdP_SUCC_I LstSeqP_OrdP)
         \<comment> \<open>... the sequence buildup via s1\<close>
         apply (simp add: SeqSubstFormP.simps [of l s1 _ _ _ sl sl' m n sm sm' sn sn'])
         apply (rule AssumeH Ex_EH Conj_EH)+
         apply (rule All2_E [THEN rotate2])
           apply (simp | rule AssumeH Ex_EH Conj_EH)+
             apply (rule Ex_I [where x="Var sl"], simp)
             apply (rule Ex_I [where x="Var sl'"], simp)
             apply (rule Conj_I [OF Mem_Eats_I1])
              apply (metis SeqAppendP_Mem1 rotate3 thin2 thin4)
             apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
                         apply (rule Ex_I [where x="Var m"], simp)
                         apply (rule Ex_I [where x="Var n"], simp)
                         apply (rule Ex_I [where x="Var sm"], simp)
                         apply (rule Ex_I [where x="Var sm'"], simp)
                         apply (rule Ex_I [where x="Var sn"], simp)
                         apply (rule Ex_I [where x="Var sn'"], simp_all (no_asm_simp))
        apply (rule Conj_I, rule AssumeH)+
        apply (rule Conj_I)
         apply (blast intro: OrdP_Trans [OF OrdP_SUCC_I] Mem_Eats_I1 [OF SeqAppendP_Mem1 [THEN cut3]] Hyp)
        apply (blast intro: Disj_I1 Disj_I2 OrdP_Trans [OF OrdP_SUCC_I] Mem_Eats_I1 [OF SeqAppendP_Mem1 [THEN cut3]] Hyp)
         \<comment> \<open>... the sequence buildup via s2\<close>
       apply (simp add: SeqSubstFormP.simps [of l s2 _ _ _ sl sl' m n sm sm' sn sn'])
       apply (rule AssumeH Ex_EH Conj_EH)+
       apply (rule All2_E [THEN rotate2])
         apply (simp | rule AssumeH Ex_EH Conj_EH)+
           apply (rule Ex_I [where x="Var sl"], simp)
           apply (rule Ex_I [where x="Var sl'"], simp)
           apply (rule cut_same [where A="OrdP (Var j)"])
            apply (metis HaddP_imp_OrdP rotate2 thin2)
           apply (rule Conj_I)
            apply (blast intro: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] del: Disj_EH)
           apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
                       apply (rule cut_same [OF exists_HaddP [where j=km and x="SUCC k1" and y="Var m"]])
                         apply (blast intro: Ord_IN_Ord, simp)
                       apply (rule cut_same [OF exists_HaddP [where j=kn and x="SUCC k1" and y="Var n"]])
                         apply (metis AssumeH(6) Ord_IN_Ord0 rotate8, simp)
                       apply (rule AssumeH Ex_EH Conj_EH | simp)+
                           apply (rule Ex_I [where x="Var km"], simp)
                           apply (rule Ex_I [where x="Var kn"], simp)
                           apply (rule Ex_I [where x="Var sm"], simp)
                           apply (rule Ex_I [where x="Var sm'"], simp)
                           apply (rule Ex_I [where x="Var sn"], simp)
                           apply (rule Ex_I [where x="Var sn'"], simp_all (no_asm_simp))
       apply (rule Conj_I [OF _ Conj_I])
         apply (blast intro!: HaddP_Mem_cancel_left [THEN Iff_MP2_same] OrdP_SUCC_I intro: LstSeqP_OrdP Hyp)+
       apply (blast del: Disj_EH  intro: OrdP_Trans Hyp
           intro!: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] HaddP_imp_OrdP [THEN cut1])
       done
   qed
 qed (*>*)
 
 theorem SubstFormP_Disj:
   "{SubstFormP v i x x', SubstFormP v i y y'} \<turnstile> SubstFormP v i (Q_Disj x y) (Q_Disj x' y')"
 proof -
   obtain k1::name and s1::name and k2::name and s2::name and k::name and s::name
     where "atom s1 \<sharp> (v,i,x,y,x',y')"        "atom k1 \<sharp> (v,i,x,y,x',y',s1)"
       "atom s2 \<sharp> (v,i,x,y,x',y',k1,s1)"  "atom k2 \<sharp> (v,i,x,y,x',y',s2,k1,s1)"
       "atom s  \<sharp> (v,i,x,y,x',y',k2,s2,k1,s1)" "atom k  \<sharp> (v,i,x,y,x',y',s,k2,s2,k1,s1)"
     by (metis obtain_fresh)
   thus ?thesis
     by (force simp: SubstFormP.simps [of s v i "Q_Disj x y" _ k]
         SubstFormP.simps [of s1 v i x x' k1]
         SubstFormP.simps [of s2 v i y y' k2]
         intro: SeqSubstFormP_Disj [THEN cut4])
 qed
 
-subsection {*Existential*}
+subsection \<open>Existential\<close>
 
 lemma SeqSubstFormP_Ex:
   assumes "atom s \<sharp> (k,s1,k1,x,x',v,i)" "atom k \<sharp> (s1,k1,x,x',v,i)"
   shows "{SeqSubstFormP v i x x' s1 k1, TermP i, VarP v}
            \<turnstile> Ex s (Ex k (SeqSubstFormP v i (Q_Ex x) (Q_Ex x') (Var s) (Var k)))" (*<*)
 proof -
   obtain l::name and sl::name and sl'::name and m::name and n::name
     and sm::name and sm'::name and sn::name and sn'::name
     where atoms:
       "atom l \<sharp> (s1,k1,s,k,x,x',v,i,sl,sl',m,n,sm,sm',sn,sn')"
       "atom sl \<sharp> (s1,k1,s,k,x,x',v,i,sl',m,n,sm,sm',sn,sn')"
       "atom sl' \<sharp> (s1,k1,s,k,x,x',v,i,m,n,sm,sm',sn,sn')"
       "atom m \<sharp> (s1,k1,s,k,x,x',v,i,n,sm,sm',sn,sn')"
       "atom n \<sharp> (s1,k1,s,k,x,x',v,i,sm,sm',sn,sn')"
       "atom sm \<sharp> (s1,k1,s,k,x,x',v,i,sm',sn,sn')"
       "atom sm' \<sharp> (s1,k1,s,k,x,x',v,i,sn,sn')"
       "atom sn \<sharp> (s1,k1,s,k,x,x',v,i,sn')"
       "atom sn' \<sharp> (s1,k1,s,k,x,x',v,i)"
     by (metis obtain_fresh)
   let ?hyp = "{RestrictedP s1 (SUCC k1) (Var s), OrdP k1, SeqSubstFormP v i x x' s1 k1, TermP i, VarP v}"
   show ?thesis
     using assms atoms
     apply (auto simp: SeqSubstFormP.simps [of l "Var s" _ _ _ sl sl' m n sm sm' sn sn'])
     apply (rule cut_same [where A="OrdP k1"])
      apply (metis SeqSubstFormP_imp_OrdP thin2)
     apply (rule cut_same [OF exists_RestrictedP [of s s1 "SUCC k1"]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
       apply (rule Ex_I [where x="Eats (Var s) (HPair (SUCC k1) (HPair (Q_Ex x) (Q_Ex x')))"], simp)
       apply (rule Ex_I [where x="(SUCC k1)"], simp)
       apply (rule Conj_I)
        apply (blast intro: RestrictedP_LstSeqP_Eats [THEN cut2] SeqSubstFormP_imp_LstSeqP [THEN cut1])
   proof (rule All2_SUCC_I, simp_all)
     show "?hyp \<turnstile> Ex sl (Ex sl'
           (HPair (SUCC k1) (HPair (Var sl) (Var sl')) IN Eats (Var s) (HPair (SUCC k1) (HPair (Q_Ex x) (Q_Ex x'))) AND
            (SubstAtomicP v i (Var sl) (Var sl') OR
             Ex m (Ex n
                (Ex sm (Ex sm'
                    (Ex sn (Ex sn'
                        (Var m IN SUCC k1 AND
                         Var n IN SUCC k1 AND
                         HPair (Var m) (HPair (Var sm) (Var sm')) IN Eats (Var s) (HPair (SUCC k1) (HPair (Q_Ex x) (Q_Ex x'))) AND
                         HPair (Var n) (HPair (Var sn) (Var sn')) IN Eats (Var s) (HPair (SUCC k1) (HPair (Q_Ex x) (Q_Ex x'))) AND
                         (Var sl EQ Q_Disj (Var sm) (Var sn) AND Var sl' EQ Q_Disj (Var sm') (Var sn') OR
                           Var sl EQ Q_Neg (Var sm) AND Var sl' EQ Q_Neg (Var sm') OR Var sl EQ Q_Ex (Var sm) AND Var sl' EQ Q_Ex (Var sm')))))))))))"
       \<comment> \<open>verifying the final values\<close>
       apply (rule Ex_I [where x="Q_Ex x"])
       using assms atoms apply simp
       apply (rule Ex_I [where x="Q_Ex x'"], simp)
       apply (rule Conj_I, metis Mem_Eats_I2 Refl)
       apply (rule Disj_I2)
       apply (rule Ex_I [where x=k1], simp)
       apply (rule Ex_I [where x=k1], simp)
       apply (rule Ex_I [where x=x], simp)
       apply (rule_tac x=x' in Ex_I, simp)
       apply (rule Ex_I [where x=x], simp)
       apply (rule_tac x=x' in Ex_I, simp)
       apply (rule Conj_I [OF Mem_SUCC_Refl])+
       apply (blast intro: Disj_I2 Mem_Eats_I1 RestrictedP_Mem [THEN cut3] Mem_SUCC_Refl
           SeqSubstFormP_imp_LstSeqP [THEN cut1] LstSeqP_imp_Mem)
       done
   next
     show "?hyp
              \<turnstile> All2 l (SUCC k1)
                 (Ex sl (Ex sl'
                     (HPair (Var l) (HPair (Var sl) (Var sl')) IN Eats (Var s) (HPair (SUCC k1) (HPair (Q_Ex x) (Q_Ex x'))) AND
                      (SubstAtomicP v i (Var sl) (Var sl') OR
                       Ex m (Ex n
                          (Ex sm (Ex sm'
                              (Ex sn (Ex sn'
                                  (Var m IN Var l AND
                                   Var n IN Var l AND
                                   HPair (Var m) (HPair (Var sm) (Var sm')) IN Eats (Var s) (HPair (SUCC k1) (HPair (Q_Ex x) (Q_Ex x'))) AND
                                   HPair (Var n) (HPair (Var sn) (Var sn')) IN Eats (Var s) (HPair (SUCC k1) (HPair (Q_Ex x) (Q_Ex x'))) AND
                                   (Var sl EQ Q_Disj (Var sm) (Var sn) AND Var sl' EQ Q_Disj (Var sm') (Var sn') OR
                             Var sl EQ Q_Neg (Var sm) AND Var sl' EQ Q_Neg (Var sm') OR Var sl EQ Q_Ex (Var sm) AND Var sl' EQ Q_Ex (Var sm'))))))))))))"
       \<comment> \<open>verifying the sequence buildup\<close>
       using assms atoms
         \<comment> \<open>... the sequence buildup via s1\<close>
       apply (auto simp add: SeqSubstFormP.simps [of l s1 _ _ _ sl sl' m n sm sm' sn sn'])
       apply (rule Swap)
       apply (rule All2_E, auto del: Disj_EH)
       apply (rule Ex_I [where x="Var sl"], simp)
       apply (rule Ex_I [where x="Var sl'"], simp)
       apply (rule Conj_I)
        apply (blast intro: Mem_Eats_I1 [OF RestrictedP_Mem [THEN cut3]] del: Disj_EH)
       apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
                   apply (rule Ex_I [where x="Var m"], simp)
                   apply (rule Ex_I [where x="Var n"], simp)
                   apply (rule Ex_I [where x="Var sm"], simp)
                   apply (rule Ex_I [where x="Var sm'"], simp)
                   apply (rule Ex_I [where x="Var sn"], simp)
                   apply (rule Ex_I [where x="Var sn'"])
                   apply (auto intro: Mem_Eats_I1 [OF RestrictedP_Mem [THEN cut3]] OrdP_Trans [OF OrdP_SUCC_I] del: Disj_EH)
       done
   qed
 qed (*>*)
 
 theorem SubstFormP_Ex: "{SubstFormP v i x x'} \<turnstile> SubstFormP v i (Q_Ex x) (Q_Ex x')"
 proof -
   obtain k1::name and s1::name and k::name and s::name
     where "atom s1 \<sharp> (v,i,x,x')"        "atom k1 \<sharp> (v,i,x,x',s1)"
       "atom s  \<sharp> (v,i,x,x',k1,s1)"  "atom k  \<sharp> (v,i,x,x',s,k1,s1)"
     by (metis obtain_fresh)
   thus ?thesis
     by (force simp: SubstFormP.simps [of s v i "Q_Ex x" _ k] SubstFormP.simps [of s1 v i x x' k1]
         intro: SeqSubstFormP_Ex [THEN cut3])
 qed
 
 
-section {*Constant Terms*}
+section \<open>Constant Terms\<close>
 
 lemma ConstP_Zero: "{} \<turnstile> ConstP Zero"
 proof -
   obtain s::name and k::name and l::name and sl::name and m::name and n::name and sm::name and sn::name
     where atoms:
       "atom s \<sharp> (k,l,sl,m,n,sm,sn)"
       "atom k \<sharp> (l,sl,m,n,sm,sn)"
       "atom l \<sharp> (sl,m,n,sm,sn)"
       "atom sl \<sharp> (m,n,sm,sn)"
       "atom m \<sharp> (n,sm,sn)"
       "atom n \<sharp> (sm,sn)"
       "atom sm \<sharp> sn"
     by (metis obtain_fresh)
   then show ?thesis
     apply (subst CTermP.simps[of k s]; auto?)
     apply (rule Ex_I[of _ _ _ "Eats Zero (HPair Zero Zero)"]; auto?)
     apply (rule Ex_I[of _ _ _ "Zero"]; auto?)
     apply (subst SeqCTermP.simps[of l _ _ sl m n sm sn]; auto?)
     apply (rule Ex_I[of _ _ _ "Zero"]; auto?)
      apply (rule Mem_SUCC_E[OF Mem_Zero_E])
      apply (rule Mem_Eats_I2)
      apply (rule HPair_cong[OF Assume Refl])
     apply (rule Disj_I1[OF Refl])
     done
 qed
 
 lemma SeqConstP_Eats:
   assumes "atom s \<sharp> (k,s1,s2,k1,k2,t1,t2)" "atom k \<sharp> (s1,s2,k1,k2,t1,t2)"
     shows "{SeqConstP s1 k1 t1, SeqConstP s2 k2 t2}
            \<turnstile> Ex s (Ex k (SeqConstP (Var s) (Var k) (Q_Eats t1 t2)))" (*<*)
 proof -
   obtain km::name and kn::name and j::name and k'::name
      and l::name and sl::name and m::name and n::name and sm::name and sn::name
     where atoms:
          "atom km \<sharp> (kn,j,k',l,s1,s2,s,k1,k2,k,t1,t2,sl,m,n,sm,sn)"
          "atom kn \<sharp> (j,k',l,s1,s2,s,k1,k2,k,t1,t2,sl,m,n,sm,sn)"
          "atom j \<sharp> (k',l,s1,s2,s,k1,k2,k,t1,t2,sl,m,n,sm,sn)"
          "atom k' \<sharp> (l,s1,s2,s,k1,k2,k,t1,t2,sl,m,n,sm,sn)"
          "atom l \<sharp> (s1,s2,s,k1,k2,k,t1,t2,sl,m,n,sm,sn)"
          "atom sl \<sharp> (s1,s2,s,k1,k2,k,t1,t2,m,n,sm,sn)"
          "atom m \<sharp> (s1,s2,s,k1,k2,k,t1,t2,n,sm,sn)"
          "atom n \<sharp> (s1,s2,s,k1,k2,k,t1,t2,sm,sn)"
          "atom sm \<sharp> (s1,s2,s,k1,k2,k,t1,t2,sn)"
          "atom sn \<sharp> (s1,s2,s,k1,k2,k,t1,t2)"
     by (metis obtain_fresh)
   let ?hyp = "{HaddP k1 k2 (Var k'), OrdP k1, OrdP k2, SeqAppendP s1 (SUCC k1) s2 (SUCC k2) (Var s),
               SeqConstP s1 k1 t1, SeqConstP s2 k2 t2}"
   show ?thesis
      using assms atoms
      apply (auto simp: SeqCTermP.simps [of l "Var s" _ sl m n sm sn])
      apply (rule cut_same [where A="OrdP k1 AND OrdP k2"])
      apply (metis Conj_I SeqCTermP_imp_OrdP thin1 thin2)
      apply (rule cut_same [OF exists_SeqAppendP [of s s1 "SUCC k1" s2 "SUCC k2"]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
      apply (rule cut_same [OF exists_HaddP [where j=k' and x=k1 and y=k2]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
      apply (rule Ex_I [where x="Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (Q_Eats t1 t2))"], simp)
      apply (rule Ex_I [where x="SUCC (SUCC (Var k'))"], simp)
      apply (rule Conj_I)
      apply (blast intro: LstSeqP_SeqAppendP_Eats SeqCTermP_imp_LstSeqP [THEN cut1])
      proof (rule All2_SUCC_I, simp_all)
        show "?hyp \<turnstile> Ex sl (HPair (SUCC (SUCC (Var k'))) (Var sl) IN Eats (Var s)
                           (HPair (SUCC (SUCC (Var k'))) (Q_Eats t1 t2))   AND
                 (Var sl EQ Zero OR Fls OR
                  Ex m (Ex n(Ex sm (Ex sn
                       (Var m IN SUCC (SUCC (Var k')) AND
                        Var n IN SUCC (SUCC (Var k')) AND
                        HPair (Var m) (Var sm) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (Q_Eats t1 t2)) AND
                        HPair (Var n) (Var sn) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (Q_Eats t1 t2)) AND
                        Var sl EQ Q_Eats (Var sm) (Var sn)))))))"
        \<comment> \<open>verifying the final values\<close>
        apply (rule Ex_I [where x="Q_Eats t1 t2"])
        using assms atoms apply simp
        apply (rule Conj_I, metis Mem_Eats_I2 Refl)
        apply (rule Disj_I2)+
        apply (rule Ex_I [where x=k1], simp)
        apply (rule Ex_I [where x="SUCC (Var k')"], simp)
        apply (rule Ex_I [where x=t1], simp)
        apply (rule Ex_I [where x=t2], simp)
        apply (rule Conj_I)
        apply (blast intro: HaddP_Mem_I LstSeqP_OrdP Mem_SUCC_I1)
        apply (blast intro: Mem_Eats_I1 SeqAppendP_Mem1 [THEN cut3] SeqAppendP_Mem2 [THEN cut4]
                  Mem_SUCC_Refl SeqCTermP_imp_LstSeqP [THEN cut1] LstSeqP_imp_Mem HaddP_SUCC1 [THEN cut1])
        done
      next
        show "?hyp \<turnstile> All2 l (SUCC (SUCC (Var k')))
                  (Ex sl
                    (HPair (Var l) (Var sl) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (Q_Eats t1 t2)) AND
                     (Var sl EQ Zero OR Fls OR
                      Ex m (Ex n (Ex sm (Ex sn
                           (Var m IN Var l AND
                            Var n IN Var l AND
                            HPair (Var m) (Var sm) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (Q_Eats t1 t2)) AND
                            HPair (Var n) (Var sn) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (Q_Eats t1 t2)) AND
                            Var sl EQ Q_Eats (Var sm) (Var sn))))))))"
      \<comment> \<open>verifying the sequence buildup\<close>
      apply (rule cut_same [where A="HaddP (SUCC k1) (SUCC k2) (SUCC (SUCC (Var k')))"])
      apply (blast intro: HaddP_SUCC1 [THEN cut1] HaddP_SUCC2 [THEN cut1])
      apply (rule All_I Imp_I)+
      apply (rule HaddP_Mem_cases [where i=j])
      using assms atoms apply simp_all
      apply (rule AssumeH)
      apply (blast intro: OrdP_SUCC_I LstSeqP_OrdP)
      \<comment> \<open>... the sequence buildup via s1\<close>
      apply (simp add: SeqCTermP.simps [of l s1 _ sl m n sm sn])
      apply (rule AssumeH Ex_EH Conj_EH)+
      apply (rule All2_E [THEN rotate2], auto del: Disj_EH)
      apply (rule Ex_I [where x="Var sl"], simp)
      apply (rule Conj_I)
      apply (rule Mem_Eats_I1)
      apply (metis SeqAppendP_Mem1 rotate3 thin2 thin4)
      apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
      apply simp_all
      apply (rule Ex_I [where x="Var m"], simp)
      apply (rule Ex_I [where x="Var n"], simp)
      apply (rule Ex_I [where x="Var sm"], simp)
      apply (rule Ex_I [where x="Var sn"], simp)
      apply (rule Conj_I, rule AssumeH)+
      apply (blast del: Disj_EH intro: OrdP_Trans [OF OrdP_SUCC_I] Mem_Eats_I1 [OF SeqAppendP_Mem1 [THEN cut3]] Hyp)
      \<comment> \<open>... the sequence buildup via s2\<close>
      apply (simp add: SeqCTermP.simps [of l s2 _ sl m n sm sn])
      apply (rule AssumeH Ex_EH Conj_EH)+
      apply (rule All2_E [THEN rotate2], auto del: Disj_EH)
      apply (rule Ex_I [where x="Var sl"], simp)
      apply (rule cut_same [where A="OrdP (Var j)"])
      apply (metis HaddP_imp_OrdP rotate2 thin2)
      apply (rule Conj_I)
      apply (blast intro: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] del: Disj_EH)
      apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
      apply (rule cut_same [OF exists_HaddP [where j=km and x="SUCC k1" and y="Var m"]])
      apply (blast intro: Ord_IN_Ord, simp)
      apply (rule cut_same [OF exists_HaddP [where j=kn and x="SUCC k1" and y="Var n"]])
      apply (metis AssumeH(6) Ord_IN_Ord0 rotate8, simp)
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
      apply (rule Ex_I [where x="Var km"], simp)
      apply (rule Ex_I [where x="Var kn"], simp)
      apply (rule Ex_I [where x="Var sm"], simp)
      apply (rule Ex_I [where x="Var sn"], simp_all)
      apply (rule Conj_I [OF _ Conj_I])
      apply (blast intro!: HaddP_Mem_cancel_left [THEN Iff_MP2_same] OrdP_SUCC_I intro: LstSeqP_OrdP Hyp)+
      apply (blast del: Disj_EH  intro: OrdP_Trans Hyp
                   intro!: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] HaddP_imp_OrdP [THEN cut1])
      done
    qed
  qed (*>*)
 
 theorem ConstP_Eats: "{ConstP t1, ConstP t2} \<turnstile> ConstP (Q_Eats t1 t2)"
 proof -
   obtain k1::name and s1::name and k2::name and s2::name and k::name and s::name
     where "atom s1 \<sharp> (t1,t2)" "atom k1 \<sharp> (t1,t2,s1)"
       "atom s2 \<sharp> (t1,t2,k1,s1)" "atom k2 \<sharp> (t1,t2,s2,k1,s1)"
       "atom s  \<sharp> (t1,t2,k2,s2,k1,s1)" "atom k  \<sharp> (t1,t2,s,k2,s2,k1,s1)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: CTermP.simps [of k s "(Q_Eats t1 t2)"]
         CTermP.simps [of k1 s1 t1]  CTermP.simps [of k2 s2 t2]
         intro!:  SeqConstP_Eats [THEN cut2])
 qed
 
 lemma TermP_Zero: "{} \<turnstile> TermP Zero"
 proof -
   obtain s::name and k::name and l::name and sl::name and m::name and n::name and sm::name and sn::name
     where atoms:
       "atom s \<sharp> (k,l,sl,m,n,sm,sn)"
       "atom k \<sharp> (l,sl,m,n,sm,sn)"
       "atom l \<sharp> (sl,m,n,sm,sn)"
       "atom sl \<sharp> (m,n,sm,sn)"
       "atom m \<sharp> (n,sm,sn)"
       "atom n \<sharp> (sm,sn)"
       "atom sm \<sharp> sn"
     by (metis obtain_fresh)
   then show ?thesis
     apply (subst CTermP.simps[of k s]; auto?)
     apply (rule Ex_I[of _ _ _ "Eats Zero (HPair Zero Zero)"]; auto?)
     apply (rule Ex_I[of _ _ _ "Zero"]; auto?)
     apply (subst SeqCTermP.simps[of l _ _ sl m n sm sn]; auto?)
     apply (rule Ex_I[of _ _ _ "Zero"]; auto?)
      apply (rule Mem_SUCC_E[OF Mem_Zero_E])
      apply (rule Mem_Eats_I2)
      apply (rule HPair_cong[OF Assume Refl])
     apply (rule Disj_I1[OF Refl])
     done
 qed
 
 lemma TermP_Var: "{} \<turnstile> TermP \<lceil>Var x\<rceil>"
 proof -
   obtain s::name and k::name and l::name and sl::name and m::name and n::name and sm::name and sn::name
     where atoms:
       "atom s \<sharp> (k,l,sl,m,n,sm,sn,x)"
       "atom k \<sharp> (l,sl,m,n,sm,sn,x)"
       "atom l \<sharp> (sl,m,n,sm,sn,x)"
       "atom sl \<sharp> (m,n,sm,sn,x)"
       "atom m \<sharp> (n,sm,sn,x)"
       "atom n \<sharp> (sm,sn,x)"
       "atom sm \<sharp> (sn,x)"
       "atom sn \<sharp> x"
     by (metis obtain_fresh)
   then show ?thesis
     apply (subst CTermP.simps[of k s]; auto?)
     apply (rule Ex_I[of _ _ _ "Eats Zero (HPair Zero \<lceil>Var x\<rceil>)"]; auto?)
     apply (rule Ex_I[of _ _ _ "Zero"]; auto?)
     apply (subst SeqCTermP.simps[of l _ _ sl m n sm sn]; auto?)
     apply (rule Ex_I[of _ _ _ "\<lceil>Var x\<rceil>"]; auto?)
      apply (rule Mem_SUCC_E[OF Mem_Zero_E])
      apply (rule Mem_Eats_I2)
      apply (rule HPair_cong[OF Assume Refl])
     apply (rule Disj_I2[OF Disj_I1])
     apply (auto simp: VarP_Var)
     done
 qed
 
 lemma SeqTermP_Eats:
   assumes "atom s \<sharp> (k,s1,s2,k1,k2,t1,t2)" "atom k \<sharp> (s1,s2,k1,k2,t1,t2)"
     shows "{SeqTermP s1 k1 t1, SeqTermP s2 k2 t2}
            \<turnstile> Ex s (Ex k (SeqTermP (Var s) (Var k) (Q_Eats t1 t2)))" (*<*)
 proof -
   obtain km::name and kn::name and j::name and k'::name
      and l::name and sl::name and m::name and n::name and sm::name and sn::name
     where atoms:
          "atom km \<sharp> (kn,j,k',l,s1,s2,s,k1,k2,k,t1,t2,sl,m,n,sm,sn)"
          "atom kn \<sharp> (j,k',l,s1,s2,s,k1,k2,k,t1,t2,sl,m,n,sm,sn)"
          "atom j \<sharp> (k',l,s1,s2,s,k1,k2,k,t1,t2,sl,m,n,sm,sn)"
          "atom k' \<sharp> (l,s1,s2,s,k1,k2,k,t1,t2,sl,m,n,sm,sn)"
          "atom l \<sharp> (s1,s2,s,k1,k2,k,t1,t2,sl,m,n,sm,sn)"
          "atom sl \<sharp> (s1,s2,s,k1,k2,k,t1,t2,m,n,sm,sn)"
          "atom m \<sharp> (s1,s2,s,k1,k2,k,t1,t2,n,sm,sn)"
          "atom n \<sharp> (s1,s2,s,k1,k2,k,t1,t2,sm,sn)"
          "atom sm \<sharp> (s1,s2,s,k1,k2,k,t1,t2,sn)"
          "atom sn \<sharp> (s1,s2,s,k1,k2,k,t1,t2)"
     by (metis obtain_fresh)
   let ?hyp = "{HaddP k1 k2 (Var k'), OrdP k1, OrdP k2, SeqAppendP s1 (SUCC k1) s2 (SUCC k2) (Var s),
               SeqTermP s1 k1 t1, SeqTermP s2 k2 t2}"
   show ?thesis
      using assms atoms
      apply (auto simp: SeqCTermP.simps [of l "Var s" _ sl m n sm sn])
      apply (rule cut_same [where A="OrdP k1 AND OrdP k2"])
      apply (metis Conj_I SeqCTermP_imp_OrdP thin1 thin2)
      apply (rule cut_same [OF exists_SeqAppendP [of s s1 "SUCC k1" s2 "SUCC k2"]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
      apply (rule cut_same [OF exists_HaddP [where j=k' and x=k1 and y=k2]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
      apply (rule Ex_I [where x="Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (Q_Eats t1 t2))"], simp)
      apply (rule Ex_I [where x="SUCC (SUCC (Var k'))"], simp)
      apply (rule Conj_I)
      apply (blast intro: LstSeqP_SeqAppendP_Eats SeqCTermP_imp_LstSeqP [THEN cut1])
      proof (rule All2_SUCC_I, simp_all)
        show "?hyp \<turnstile> Ex sl (HPair (SUCC (SUCC (Var k'))) (Var sl) IN Eats (Var s)
                           (HPair (SUCC (SUCC (Var k'))) (Q_Eats t1 t2))   AND
                 (Var sl EQ Zero OR VarP (Var sl) OR
                  Ex m (Ex n(Ex sm (Ex sn
                       (Var m IN SUCC (SUCC (Var k')) AND
                        Var n IN SUCC (SUCC (Var k')) AND
                        HPair (Var m) (Var sm) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (Q_Eats t1 t2)) AND
                        HPair (Var n) (Var sn) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (Q_Eats t1 t2)) AND
                        Var sl EQ Q_Eats (Var sm) (Var sn)))))))"
        \<comment> \<open>verifying the final values\<close>
        apply (rule Ex_I [where x="Q_Eats t1 t2"])
        using assms atoms apply simp
        apply (rule Conj_I, metis Mem_Eats_I2 Refl)
        apply (rule Disj_I2)+
        apply (rule Ex_I [where x=k1], simp)
        apply (rule Ex_I [where x="SUCC (Var k')"], simp)
        apply (rule Ex_I [where x=t1], simp)
        apply (rule Ex_I [where x=t2], simp)
        apply (rule Conj_I)
        apply (blast intro: HaddP_Mem_I LstSeqP_OrdP Mem_SUCC_I1)
        apply (blast intro: Mem_Eats_I1 SeqAppendP_Mem1 [THEN cut3] SeqAppendP_Mem2 [THEN cut4]
                  Mem_SUCC_Refl SeqCTermP_imp_LstSeqP [THEN cut1] LstSeqP_imp_Mem HaddP_SUCC1 [THEN cut1])
        done
      next
        show "?hyp \<turnstile> All2 l (SUCC (SUCC (Var k')))
                  (Ex sl
                    (HPair (Var l) (Var sl) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (Q_Eats t1 t2)) AND
                     (Var sl EQ Zero OR VarP (Var sl) OR
                      Ex m (Ex n (Ex sm (Ex sn
                           (Var m IN Var l AND
                            Var n IN Var l AND
                            HPair (Var m) (Var sm) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (Q_Eats t1 t2)) AND
                            HPair (Var n) (Var sn) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (Q_Eats t1 t2)) AND
                            Var sl EQ Q_Eats (Var sm) (Var sn))))))))"
      \<comment> \<open>verifying the sequence buildup\<close>
      apply (rule cut_same [where A="HaddP (SUCC k1) (SUCC k2) (SUCC (SUCC (Var k')))"])
      apply (blast intro: HaddP_SUCC1 [THEN cut1] HaddP_SUCC2 [THEN cut1])
      apply (rule All_I Imp_I)+
      apply (rule HaddP_Mem_cases [where i=j])
      using assms atoms apply simp_all
      apply (rule AssumeH)
      apply (blast intro: OrdP_SUCC_I LstSeqP_OrdP)
      \<comment> \<open>... the sequence buildup via s1\<close>
      apply (simp add: SeqCTermP.simps [of l s1 _ sl m n sm sn])
      apply (rule AssumeH Ex_EH Conj_EH)+
      apply (rule All2_E [THEN rotate2], auto del: Disj_EH)
      apply (rule Ex_I [where x="Var sl"], simp)
      apply (rule Conj_I)
      apply (rule Mem_Eats_I1)
      apply (metis SeqAppendP_Mem1 rotate3 thin2 thin4)
      apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
      apply simp_all
      apply (rule Ex_I [where x="Var m"], simp)
      apply (rule Ex_I [where x="Var n"], simp)
      apply (rule Ex_I [where x="Var sm"], simp)
      apply (rule Ex_I [where x="Var sn"], simp)
      apply (rule Conj_I, rule AssumeH)+
      apply (blast del: Disj_EH intro: OrdP_Trans [OF OrdP_SUCC_I] Mem_Eats_I1 [OF SeqAppendP_Mem1 [THEN cut3]] Hyp)
      \<comment> \<open>... the sequence buildup via s2\<close>
      apply (simp add: SeqCTermP.simps [of l s2 _ sl m n sm sn])
      apply (rule AssumeH Ex_EH Conj_EH)+
      apply (rule All2_E [THEN rotate2], auto del: Disj_EH)
      apply (rule Ex_I [where x="Var sl"], simp)
      apply (rule cut_same [where A="OrdP (Var j)"])
      apply (metis HaddP_imp_OrdP rotate2 thin2)
      apply (rule Conj_I)
      apply (blast intro: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] del: Disj_EH)
      apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
      apply (rule cut_same [OF exists_HaddP [where j=km and x="SUCC k1" and y="Var m"]])
      apply (blast intro: Ord_IN_Ord, simp)
      apply (rule cut_same [OF exists_HaddP [where j=kn and x="SUCC k1" and y="Var n"]])
      apply (metis AssumeH(6) Ord_IN_Ord0 rotate8, simp)
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
      apply (rule Ex_I [where x="Var km"], simp)
      apply (rule Ex_I [where x="Var kn"], simp)
      apply (rule Ex_I [where x="Var sm"], simp)
      apply (rule Ex_I [where x="Var sn"], simp_all)
      apply (rule Conj_I [OF _ Conj_I])
      apply (blast intro!: HaddP_Mem_cancel_left [THEN Iff_MP2_same] OrdP_SUCC_I intro: LstSeqP_OrdP Hyp)+
      apply (blast del: Disj_EH  intro: OrdP_Trans Hyp
                   intro!: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] HaddP_imp_OrdP [THEN cut1])
      done
    qed
 qed (*>*)
 
 theorem TermP_Eats: "{TermP t1, TermP t2} \<turnstile> TermP (Q_Eats t1 t2)"
 proof -
   obtain k1::name and s1::name and k2::name and s2::name and k::name and s::name
     where "atom s1 \<sharp> (t1,t2)" "atom k1 \<sharp> (t1,t2,s1)"
       "atom s2 \<sharp> (t1,t2,k1,s1)" "atom k2 \<sharp> (t1,t2,s2,k1,s1)"
       "atom s  \<sharp> (t1,t2,k2,s2,k1,s1)" "atom k  \<sharp> (t1,t2,s,k2,s2,k1,s1)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: CTermP.simps [of k s "(Q_Eats t1 t2)"]
         CTermP.simps [of k1 s1 t1]  CTermP.simps [of k2 s2 t2]
         intro!:  SeqTermP_Eats [THEN cut2])
 qed
 
 
-section {*Proofs*}
+section \<open>Proofs\<close>
 
 lemma PrfP_inference:
   assumes "atom s \<sharp> (k,s1,s2,k1,k2,\<alpha>1,\<alpha>2,\<beta>)" "atom k \<sharp> (s1,s2,k1,k2,\<alpha>1,\<alpha>2,\<beta>)"
     shows "{PrfP s1 k1 \<alpha>1, PrfP s2 k2 \<alpha>2, ModPonP \<alpha>1 \<alpha>2 \<beta> OR ExistsP \<alpha>1 \<beta> OR SubstP \<alpha>1 \<beta>}
            \<turnstile> Ex k (Ex s (PrfP (Var s) (Var k) \<beta>))" (*<*)
 proof -
   obtain km::name and kn::name and j::name and k'::name
      and l::name and sl::name and m::name and n::name and sm::name and sn::name
     where atoms:
          "atom km \<sharp> (kn,j,k',l,s1,s2,s,k1,k2,k,\<alpha>1,\<alpha>2,\<beta>,sl,m,n,sm,sn)"
          "atom kn \<sharp> (j,k',l,s1,s2,s,k1,k2,k,\<alpha>1,\<alpha>2,\<beta>,sl,m,n,sm,sn)"
          "atom j \<sharp> (k',l,s1,s2,s,k1,k2,k,\<alpha>1,\<alpha>2,\<beta>,sl,m,n,sm,sn)"
          "atom k' \<sharp> (l,s1,s2,s,k1,k2,k,\<alpha>1,\<alpha>2,\<beta>,sl,m,n,sm,sn)"
          "atom l \<sharp> (s1,s2,s,k1,k2,k,\<alpha>1,\<alpha>2,\<beta>,sl,m,n,sm,sn)"
          "atom sl \<sharp> (s1,s2,s,k1,k2,k,\<alpha>1,\<alpha>2,\<beta>,m,n,sm,sn)"
          "atom m \<sharp> (s1,s2,s,k1,k2,k,\<alpha>1,\<alpha>2,\<beta>,n,sm,sn)"
          "atom n \<sharp> (s1,s2,s,k1,k2,k,\<alpha>1,\<alpha>2,\<beta>,sm,sn)"
          "atom sm \<sharp> (s1,s2,s,k1,k2,k,\<alpha>1,\<alpha>2,\<beta>,sn)"
          "atom sn \<sharp> (s1,s2,s,k1,k2,k,\<alpha>1,\<alpha>2,\<beta>)"
     by (metis obtain_fresh)
   let ?hyp = "{HaddP k1 k2 (Var k'), OrdP k1, OrdP k2, SeqAppendP s1 (SUCC k1) s2 (SUCC k2) (Var s),
               PrfP s1 k1 \<alpha>1, PrfP s2 k2 \<alpha>2, ModPonP \<alpha>1 \<alpha>2 \<beta> OR ExistsP \<alpha>1 \<beta> OR SubstP \<alpha>1 \<beta>}"
   show ?thesis
      using assms atoms
      apply (simp add: PrfP.simps [of l "Var s" sl m n sm sn])
      apply (rule cut_same [where A="OrdP k1 AND OrdP k2"])
      apply (metis Conj_I PrfP_imp_OrdP thin1 thin2)
      apply (rule cut_same [OF exists_SeqAppendP [of s s1 "SUCC k1" s2 "SUCC k2"]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
      apply (rule cut_same [OF exists_HaddP [where j=k' and x=k1 and y=k2]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
      apply (rule Ex_I [where x="SUCC (SUCC (Var k'))"], simp)
      apply (rule Ex_I [where x="Eats (Var s) (HPair (SUCC (SUCC (Var k'))) \<beta>)"], simp)
      apply (rule Conj_I)
      apply (blast intro: LstSeqP_SeqAppendP_Eats PrfP_imp_LstSeqP [THEN cut1])
      proof (rule All2_SUCC_I, simp_all)
        show "?hyp \<turnstile> Ex sn
                  (HPair (SUCC (SUCC (Var k'))) (Var sn) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) \<beta>) AND
                   (AxiomP (Var sn) OR
                    Ex m (Ex l (Ex sm (Ex sl
                       (Var m IN SUCC (SUCC (Var k')) AND
                        Var l IN SUCC (SUCC (Var k')) AND
                        HPair (Var m) (Var sm) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) \<beta>) AND
                        HPair (Var l) (Var sl) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) \<beta>) AND
                        (ModPonP (Var sm) (Var sl) (Var sn) OR ExistsP (Var sm) (Var sn) OR SubstP (Var sm) (Var sn))))))))"
        \<comment> \<open>verifying the final values\<close>
        apply (rule Ex_I [where x="\<beta>"])
        using assms atoms apply simp
        apply (rule Conj_I, metis Mem_Eats_I2 Refl)
        apply (rule Disj_I2)
        apply (rule Ex_I [where x=k1], simp)
        apply (rule Ex_I [where x="SUCC (Var k')"], simp)
        apply (rule_tac x=\<alpha>1 in Ex_I, simp)
        apply (rule_tac x=\<alpha>2 in Ex_I, simp)
        apply (rule Conj_I)
        apply (blast intro: HaddP_Mem_I LstSeqP_OrdP Mem_SUCC_I1)
        apply (rule Conj_I [OF Mem_SUCC_Refl Conj_I])
        apply (blast intro: Mem_Eats_I1 SeqAppendP_Mem1 [THEN cut3] Mem_SUCC_Refl PrfP_imp_LstSeqP [THEN cut1] LstSeqP_imp_Mem)
        apply (blast del: Disj_EH  intro: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] Mem_SUCC_Refl
                      PrfP_imp_LstSeqP [THEN cut1] HaddP_SUCC1 [THEN cut1] LstSeqP_imp_Mem)
        done
      next
        show "?hyp \<turnstile> All2 n (SUCC (SUCC (Var k')))
                (Ex sn
                  (HPair (Var n) (Var sn) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) \<beta>) AND
                   (AxiomP (Var sn) OR
                    Ex m (Ex l (Ex sm (Ex sl
                       (Var m IN Var n AND
                        Var l IN Var n AND
                        HPair (Var m) (Var sm) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) \<beta>) AND
                        HPair (Var l) (Var sl) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) \<beta>) AND
                        (ModPonP (Var sm) (Var sl) (Var sn) OR ExistsP (Var sm) (Var sn) OR SubstP (Var sm) (Var sn)))))))))"
      \<comment> \<open>verifying the sequence buildup\<close>
      apply (rule cut_same [where A="HaddP (SUCC k1) (SUCC k2) (SUCC (SUCC (Var k')))"])
      apply (blast intro: HaddP_SUCC1 [THEN cut1] HaddP_SUCC2 [THEN cut1])
      apply (rule All_I Imp_I)+
      apply (rule HaddP_Mem_cases [where i=j])
      using assms atoms apply simp_all
      apply (rule AssumeH)
      apply (blast intro: OrdP_SUCC_I LstSeqP_OrdP)
      \<comment> \<open>... the sequence buildup via s1\<close>
      apply (simp add: PrfP.simps [of l s1 sl m n sm sn])
      apply (rule AssumeH Ex_EH Conj_EH)+
      apply (rule All2_E [THEN rotate2])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
      apply (rule Ex_I [where x="Var sn"], simp)
      apply (rule Conj_I)
      apply (rule Mem_Eats_I1)
      apply (metis SeqAppendP_Mem1 rotate3 thin2 thin4)
      apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
      apply (rule Ex_I [where x="Var m"], simp)
      apply (rule Ex_I [where x="Var l"], simp)
      apply (rule Ex_I [where x="Var sm"], simp)
      apply (rule Ex_I [where x="Var sl"], simp_all)
      apply (rule Conj_I, rule AssumeH)+
      apply (blast del: Disj_EH intro: OrdP_Trans [OF OrdP_SUCC_I] Mem_Eats_I1 [OF SeqAppendP_Mem1 [THEN cut3]] Hyp)
      \<comment> \<open>... the sequence buildup via s2\<close>
      apply (simp add: PrfP.simps [of l s2 sl m n sm sn])
      apply (rule AssumeH Ex_EH Conj_EH)+
      apply (rule All2_E [THEN rotate2])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
      apply (rule Ex_I [where x="Var sn"], simp)
      apply (rule cut_same [where A="OrdP (Var j)"])
      apply (metis HaddP_imp_OrdP rotate2 thin2)
      apply (rule Conj_I)
      apply (blast intro!: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] del: Disj_EH)
      apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
      apply (rule cut_same [OF exists_HaddP [where j=km and x="SUCC k1" and y="Var m"]])
      apply (blast intro: Ord_IN_Ord, simp)
      apply (rule cut_same [OF exists_HaddP [where j=kn and x="SUCC k1" and y="Var l"]])
      apply (blast intro!: Ord_IN_Ord)
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
      apply (rule Ex_I [where x="Var km"], simp)
      apply (rule Ex_I [where x="Var kn"], simp)
      apply (rule Ex_I [where x="Var sm"], simp)
      apply (rule Ex_I [where x="Var sl"], simp_all)
      apply (rule Conj_I [OF _ Conj_I])
      apply (blast intro!: HaddP_Mem_cancel_left [THEN Iff_MP2_same] OrdP_SUCC_I intro: LstSeqP_OrdP Hyp)+
      apply (blast del: Disj_EH  intro: OrdP_Trans Hyp
                   intro!: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] HaddP_imp_OrdP [THEN cut1])
      done
    qed
 qed (*>*)
 
 corollary PfP_inference: "{PfP \<alpha>1, PfP \<alpha>2, ModPonP \<alpha>1 \<alpha>2 \<beta> OR ExistsP \<alpha>1 \<beta> OR SubstP \<alpha>1 \<beta>} \<turnstile> PfP \<beta>"
 proof -
   obtain k1::name and s1::name and k2::name and s2::name and k::name and s::name
     where "atom s1 \<sharp> (\<alpha>1,\<alpha>2,\<beta>)" "atom k1 \<sharp> (\<alpha>1,\<alpha>2,\<beta>,s1)"
           "atom s2 \<sharp> (\<alpha>1,\<alpha>2,\<beta>,k1,s1)""atom k2 \<sharp> (\<alpha>1,\<alpha>2,\<beta>,s2,k1,s1)"
           "atom s  \<sharp> (\<alpha>1,\<alpha>2,\<beta>,k2,s2,k1,s1)"
           "atom k  \<sharp> (\<alpha>1,\<alpha>2,\<beta>,s,k2,s2,k1,s1)"
     by (metis obtain_fresh)
    thus ?thesis
      apply (simp add: PfP.simps [of k s \<beta>] PfP.simps [of k1 s1 \<alpha>1]  PfP.simps [of k2 s2 \<alpha>2])
      apply (auto intro!: PrfP_inference [of s k "Var s1" "Var s2", THEN cut3] del: Disj_EH)
      done
 qed
 
 theorem PfP_implies_SubstForm_PfP:
    assumes "H \<turnstile> PfP y" "H \<turnstile> SubstFormP x t y z"
      shows "H \<turnstile> PfP z"
 proof -
   obtain u::name and v::name
     where atoms: "atom u \<sharp> (t,x,y,z,v)"   "atom v \<sharp> (t,x,y,z)"
     by (metis obtain_fresh)
   show ?thesis
     apply (rule PfP_inference [of y, THEN cut3])
     apply (rule assms)+
     using atoms
     apply (auto simp: SubstP.simps [of u _ _ v] intro!: Disj_I2)
     apply (rule Ex_I [where x=x], simp)
     apply (rule Ex_I [where x=t], simp add: assms)
     done
 qed
 
 theorem PfP_implies_ModPon_PfP: "\<lbrakk>H \<turnstile> PfP (Q_Imp x y); H \<turnstile> PfP x\<rbrakk> \<Longrightarrow> H \<turnstile> PfP y"
   by (force intro: PfP_inference [of x, THEN cut3] Disj_I1  simp add: ModPonP_def)
 
 corollary PfP_implies_ModPon_PfP_quot: "\<lbrakk>H \<turnstile> PfP \<lceil>\<alpha> IMP \<beta>\<rceil>; H \<turnstile> PfP \<lceil>\<alpha>\<rceil>\<rbrakk> \<Longrightarrow> H \<turnstile> PfP \<lceil>\<beta>\<rceil>"
   by (auto simp: quot_fm_def intro: PfP_implies_ModPon_PfP)
 
 
 lemma TermP_quot:
   fixes \<alpha> :: tm
   shows "{} \<turnstile> TermP \<lceil>\<alpha>\<rceil>"
   by (induct \<alpha> rule: tm.induct)
     (auto simp: quot_Eats intro: TermP_Zero TermP_Var TermP_Eats[THEN cut2])
 
 lemma TermP_quot_dbtm:
   fixes \<alpha> :: tm
   assumes "wf_dbtm u"
   shows "{} \<turnstile> TermP (quot_dbtm u)"
   using assms
   by (induct u rule: dbtm.induct)
     (auto simp: quot_Eats intro: TermP_Zero
     TermP_Var[unfolded quot_tm_def, simplified] TermP_Eats[THEN cut2])
 
-section {*Formulas*}
-
-section {*Abstraction on Formulas*}
-
-subsection {*Membership*}
+section \<open>Formulas\<close>
+
+section \<open>Abstraction on Formulas\<close>
+
+subsection \<open>Membership\<close>
 
 lemma AbstAtomicP_Mem:
   "{AbstTermP v i x x', AbstTermP v i y y'} \<turnstile> AbstAtomicP v i (Q_Mem x y) (Q_Mem x' y')"
 proof -
   obtain t::name and u::name and t'::name and u'::name
     where "atom t \<sharp> (v,i,x,x',y,y',t',u,u')" "atom t' \<sharp> (v,i,x,x',y,y',u,u')"
       "atom u \<sharp> (v,i,x,x',y,y',u')" "atom u' \<sharp> (v,i,x,x',y,y')"
     by (metis obtain_fresh)
   thus ?thesis
     apply (simp add: AbstAtomicP.simps [of t _ _ _ _ t' u u'])
     apply (rule Ex_I [where x = x], simp)
     apply (rule Ex_I [where x = y], simp)
     apply (rule Ex_I [where x = x'], simp)
     apply (rule Ex_I [where x = y'], auto intro: Disj_I2)
     done
 qed
 
 lemma SeqAbstFormP_Mem:
   assumes "atom s \<sharp> (k,x,y,x',y',v,i)" "atom k \<sharp> (x,y,x',y',v,i)"
   shows "{AbstTermP v i x x', AbstTermP v i y y'}
            \<turnstile> Ex s (Ex k (SeqAbstFormP v i (Q_Mem x y) (Q_Mem x' y') (Var s) (Var k)))"
 proof -
   let ?vs = "(s,k,x,y,x',y',v,i)"
   obtain l::name and sl::name and sl'::name and m::name and n::name and sm::name and sm'::name and sn::name and sn'::name
     and sli smi sni :: name
     where
       "atom sni \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn',l,sli,smi)"
       "atom smi \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn',l,sli)"
       "atom sli \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn',l)"
       "atom l \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn')"
       "atom sl \<sharp> (?vs,sl',m,n,sm,sm',sn,sn')" "atom sl' \<sharp> (?vs,m,n,sm,sm',sn,sn')"
       "atom m \<sharp> (?vs,n,sm,sm',sn,sn')" "atom n \<sharp> (?vs,sm,sm',sn,sn')"
       "atom sm \<sharp> (?vs,sm',sn,sn')" "atom sm' \<sharp> (?vs,sn,sn')"
       "atom sn \<sharp> (?vs,sn')" "atom sn' \<sharp> ?vs"
     by (metis obtain_fresh)
   thus ?thesis
     using assms
     apply (auto simp: SeqAbstFormP.simps [of l "Var s" _ _ sli sl sl' m n smi sm sm' sni sn sn'])
     apply (rule Ex_I [where x = "Eats Zero (HPair Zero (HPair i (HPair (Q_Mem x y) (Q_Mem x' y'))))"], simp)
     apply (rule Ex_I [where x = Zero], auto intro!: Mem_SUCC_EH)
     apply (rule Ex_I [where x = "i"], simp)
     apply (rule Ex_I [where x = "Q_Mem x y"], simp)
     apply (rule Ex_I [where x = "Q_Mem x' y'"], auto intro: Mem_Eats_I2 HPair_cong)
     apply (blast intro: AbstAtomicP_Mem [THEN cut2] Disj_I1)
     done
 qed
 
 lemma AbstFormP_Mem:
   "{AbstTermP v i x x', AbstTermP v i y y'} \<turnstile> AbstFormP v i (Q_Mem x y) (Q_Mem x' y')"
 proof -
   obtain k1::name and s1::name and k2::name and s2::name and k::name and s::name
     where "atom s1 \<sharp> (v,i,x,y,x',y')" "atom k1 \<sharp> (v,i,x,y,x',y',s1)"
       "atom s2 \<sharp> (v,i,x,y,x',y',k1,s1)" "atom k2 \<sharp> (v,i,x,y,x',y',s2,k1,s1)"
       "atom s  \<sharp> (v,i,x,y,x',y',k2,s2,k1,s1)" "atom k  \<sharp> (v,i,x,y,x',y',s,k2,s2,k1,s1)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: AbstFormP.simps [of s v i "(Q_Mem x y)" _ k]
         AbstFormP.simps [of s1 v i x x' k1]
         AbstFormP.simps [of s2 v i y y' k2]
         intro: AbstTermP_imp_VarP AbstTermP_imp_OrdP SeqAbstFormP_Mem thin1)
 qed
 
-subsection {*Equality*}
+subsection \<open>Equality\<close>
 
 lemma AbstAtomicP_Eq:
   "{AbstTermP v i x x', AbstTermP v i y y'} \<turnstile> AbstAtomicP v i (Q_Eq x y) (Q_Eq x' y')"
 proof -
   obtain t::name and u::name and t'::name  and u'::name
     where "atom t \<sharp> (v,i,x,x',y,y',t',u,u')" "atom t' \<sharp> (v,i,x,x',y,y',u,u')"
       "atom u \<sharp> (v,i,x,x',y,y',u')" "atom u' \<sharp> (v,i,x,x',y,y')"
     by (metis obtain_fresh)
   thus ?thesis
     apply (simp add: AbstAtomicP.simps [of t _ _ _ _ t' u u'])
     apply (rule Ex_I [where x = x], simp)
     apply (rule Ex_I [where x = y], simp)
     apply (rule Ex_I [where x = x'], simp)
     apply (rule Ex_I [where x = y'], auto intro: Disj_I1)
     done
 qed
 
 lemma SeqAbstFormP_Eq:
   assumes sk: "atom s \<sharp> (k,x,y,x',y',v,i)" "atom k \<sharp> (x,y,x',y',v,i)"
   shows "{AbstTermP v i x x', AbstTermP v i y y'}
            \<turnstile> Ex s (Ex k (SeqAbstFormP v i (Q_Eq x y) (Q_Eq x' y') (Var s) (Var k)))"
 proof -
   let ?vs = "(s,k,x,y,x',y',v,i)"
   obtain l::name and sl::name and sl'::name and m::name and n::name and sm::name and sm'::name and sn::name and sn'::name
     and sli smi sni :: name
     where
       "atom sni \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn',l,sli,smi)"
       "atom smi \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn',l,sli)"
       "atom sli \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn',l)"
       "atom l \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn')"
       "atom sl \<sharp> (?vs,sl',m,n,sm,sm',sn,sn')" "atom sl' \<sharp> (?vs,m,n,sm,sm',sn,sn')"
       "atom m \<sharp> (?vs,n,sm,sm',sn,sn')" "atom n \<sharp> (?vs,sm,sm',sn,sn')"
       "atom sm \<sharp> (?vs,sm',sn,sn')" "atom sm' \<sharp> (?vs,sn,sn')"
       "atom sn \<sharp> (?vs,sn')" "atom sn' \<sharp> ?vs"
     by (metis obtain_fresh)
   thus ?thesis
     using sk
     apply (auto simp: SeqAbstFormP.simps [of l "Var s" _ _ sli sl sl' m n smi sm sm' sni sn sn'])
     apply (rule Ex_I [where x = "Eats Zero (HPair Zero (HPair i (HPair (Q_Eq x y) (Q_Eq x' y'))))"], simp)
     apply (rule Ex_I [where x = Zero], auto intro!: Mem_SUCC_EH)
     apply (rule Ex_I [where x = "i"], simp)
     apply (rule Ex_I [where x = "Q_Eq x y"], simp)
     apply (rule Ex_I [where x = "Q_Eq x' y'"], auto)
      apply (metis Mem_Eats_I2 Assume HPair_cong Refl)
     apply (blast intro: AbstAtomicP_Eq [THEN cut2] Disj_I1)
     done
 qed
 
 lemma AbstFormP_Eq:
   "{AbstTermP v i x x', AbstTermP v i y y'} \<turnstile> AbstFormP v i (Q_Eq x y) (Q_Eq x' y')"
 proof -
   obtain k1::name and s1::name and k2::name and s2::name and k::name and s::name
     where "atom s1 \<sharp> (v,i,x,y,x',y')" "atom k1 \<sharp> (v,i,x,y,x',y',s1)"
       "atom s2 \<sharp> (v,i,x,y,x',y',k1,s1)" "atom k2 \<sharp> (v,i,x,y,x',y',s2,k1,s1)"
       "atom s  \<sharp> (v,i,x,y,x',y',k2,s2,k1,s1)" "atom k  \<sharp> (v,i,x,y,x',y',s,k2,s2,k1,s1)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: AbstFormP.simps [of s v i "(Q_Eq x y)" _ k]
         AbstFormP.simps [of s1 v i x x' k1]
         AbstFormP.simps [of s2 v i y y' k2]
         intro: SeqAbstFormP_Eq AbstTermP_imp_OrdP AbstTermP_imp_VarP thin1)
 qed
 
-subsection {*Negation*}
+subsection \<open>Negation\<close>
 
 lemma SeqAbstFormP_Neg:
   assumes "atom s \<sharp> (k,s1,k1,x,x',v,i)" "atom k \<sharp> (s1,k1,x,x',v,i)"
   shows "{SeqAbstFormP v i x x' s1 k1, OrdP i, VarP v}
            \<turnstile> Ex s (Ex k (SeqAbstFormP v i (Q_Neg x) (Q_Neg x') (Var s) (Var k)))" (*<*)
 proof -
   let ?vs = "(s1,k1,s,k,x,x',v,i)"
   obtain l::name and sl::name and sl'::name and m::name and n::name and
     sm::name and sm'::name and sn::name and sn'::name
     and sli smi sni :: name
     where atoms:
       "atom sni \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn',l,sli,smi)"
       "atom smi \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn',l,sli)"
       "atom sli \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn',l)"
       "atom l \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn')"
       "atom sl \<sharp> (?vs,sl',m,n,sm,sm',sn,sn')" "atom sl' \<sharp> (?vs,m,n,sm,sm',sn,sn')"
       "atom m \<sharp> (?vs,n,sm,sm',sn,sn')" "atom n \<sharp> (?vs,sm,sm',sn,sn')"
       "atom sm \<sharp> (?vs,sm',sn,sn')" "atom sm' \<sharp> (?vs,sn,sn')"
       "atom sn \<sharp> (?vs,sn')" "atom sn' \<sharp> ?vs"
     by (metis obtain_fresh)
   let ?hyp = "{RestrictedP s1 (SUCC k1) (Var s), OrdP k1, SeqAbstFormP v i x x' s1 k1, OrdP i, VarP v}"
   show ?thesis
     using assms atoms
     apply (auto simp: SeqAbstFormP.simps [of l "Var s" _ _ sli sl sl' m n smi sm sm' sni sn sn'])
     apply (rule cut_same [where A="OrdP k1"])
      apply (metis SeqAbstFormP_imp_OrdP thin2)
     apply (rule cut_same [OF exists_RestrictedP [of s s1 "SUCC k1"]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
       apply (rule Ex_I [where x="Eats (Var s) (HPair (SUCC k1) (HPair i (HPair (Q_Neg x) (Q_Neg x'))))"])
       apply (simp_all (no_asm_simp))
     apply (rule Ex_I [where x="(SUCC k1)"])
     apply (simp add: flip_fresh_fresh)
     apply (rule Conj_I)
      apply (blast intro: RestrictedP_LstSeqP_Eats [THEN cut2] SeqAbstFormP_imp_LstSeqP [THEN cut1])
   proof (rule All2_SUCC_I, simp_all)
     show "?hyp \<turnstile> SyntaxN.Ex sli (SyntaxN.Ex sl (SyntaxN.Ex sl'
             (HPair (SUCC k1) (HPair (Var sli) (HPair (Var sl) (Var sl'))) IN
               Eats (Var s) (HPair (SUCC k1) (HPair i (HPair (Q_Neg x) (Q_Neg x')))) AND
             (AbstAtomicP v (Var sli) (Var sl) (Var sl') OR
               OrdP (Var sli) AND
               SyntaxN.Ex m (SyntaxN.Ex n
               (SyntaxN.Ex smi (SyntaxN.Ex sm (SyntaxN.Ex sm'
               (SyntaxN.Ex sni (SyntaxN.Ex sn (SyntaxN.Ex sn'
               (Var m IN SUCC k1 AND Var n IN SUCC k1 AND
               HPair (Var m) (HPair (Var smi) (HPair (Var sm) (Var sm'))) IN
                 Eats (Var s) (HPair (SUCC k1) (HPair i (HPair (Q_Neg x) (Q_Neg x')))) AND
               HPair (Var n) (HPair (Var sni) (HPair (Var sn) (Var sn'))) IN
                 Eats (Var s) (HPair (SUCC k1) (HPair i (HPair (Q_Neg x) (Q_Neg x')))) AND
               (Var sli EQ Var smi AND Var sli EQ Var sni AND
               Var sl EQ Q_Disj (Var sm) (Var sn) AND  Var sl' EQ Q_Disj (Var sm') (Var sn') OR
               Var sli EQ Var smi AND Var sl EQ Q_Neg (Var sm) AND Var sl' EQ Q_Neg (Var sm') OR
               SUCC (Var sli) EQ Var smi AND Var sl EQ Q_Ex (Var sm) AND Var sl' EQ Q_Ex (Var sm'))))))))))))))"
       \<comment> \<open>verifying the final values\<close>
       apply (rule Ex_I [where x="i"])
       using assms atoms apply simp
       apply (rule Ex_I [where x="Q_Neg x"], simp)
       apply (rule Ex_I [where x="Q_Neg x'"], simp)
       apply (rule Conj_I, metis Mem_Eats_I2 Refl)
       apply (rule Disj_I2)
       apply (rule Conj_I, blast)
       apply (rule Ex_I [where x=k1], simp)
       apply (rule Ex_I [where x=k1], simp)
       apply (rule Ex_I [where x=i], simp)
       apply (rule Ex_I [where x=x], simp)
       apply (rule_tac x=x' in Ex_I, simp)
       apply (rule Ex_I [where x=i], simp)
       apply (rule Ex_I [where x=x], simp)
       apply (rule_tac x=x' in Ex_I, simp)
       apply (rule Conj_I [OF Mem_SUCC_Refl])+
       apply (blast intro: Disj_I1 Disj_I2 Mem_Eats_I1 RestrictedP_Mem [THEN cut3] Mem_SUCC_Refl
           SeqAbstFormP_imp_LstSeqP [THEN cut1] LstSeqP_imp_Mem)
       done
   next
     show "?hyp \<turnstile> All2 l (SUCC k1) (SyntaxN.Ex sli (SyntaxN.Ex sl (SyntaxN.Ex sl'
       (HPair (Var l) (HPair (Var sli) (HPair (Var sl) (Var sl'))) IN
         Eats (Var s) (HPair (SUCC k1) (HPair i (HPair (Q_Neg x) (Q_Neg x')))) AND
       (AbstAtomicP v (Var sli) (Var sl) (Var sl') OR
         OrdP (Var sli) AND
         SyntaxN.Ex m (SyntaxN.Ex n
         (SyntaxN.Ex smi (SyntaxN.Ex sm (SyntaxN.Ex sm'
         (SyntaxN.Ex sni (SyntaxN.Ex sn (SyntaxN.Ex sn'
         (Var m IN Var l AND Var n IN Var l AND
         HPair (Var m) (HPair (Var smi) (HPair (Var sm) (Var sm'))) IN
           Eats (Var s) (HPair (SUCC k1) (HPair i (HPair (Q_Neg x) (Q_Neg x')))) AND
         HPair (Var n) (HPair (Var sni) (HPair (Var sn) (Var sn'))) IN
           Eats (Var s) (HPair (SUCC k1) (HPair i (HPair (Q_Neg x) (Q_Neg x')))) AND
         (Var sli EQ Var smi AND Var sli EQ Var sni AND
         Var sl EQ Q_Disj (Var sm) (Var sn) AND Var sl' EQ Q_Disj (Var sm') (Var sn') OR
         Var sli EQ Var smi AND Var sl EQ Q_Neg (Var sm) AND Var sl' EQ Q_Neg (Var sm') OR
         SUCC (Var sli) EQ Var smi AND Var sl EQ Q_Ex (Var sm) AND Var sl' EQ Q_Ex (Var sm')))))))))))))))"
       \<comment> \<open>verifying the sequence buildup\<close>
       apply (rule All_I Imp_I)+
       using assms atoms apply simp_all
         \<comment> \<open>... the sequence buildup via s1\<close>
        apply (simp add: SeqAbstFormP.simps [of l s1 _ _ sli sl sl' m n smi sm sm' sni sn sn'])
       apply (rule AssumeH Ex_EH Conj_EH)+
       apply (rule All2_E [THEN rotate2], auto del: Disj_EH)
       apply (rule Ex_I [where x="Var sli"], simp)
       apply (rule Ex_I [where x="Var sl"], simp)
       apply (rule Ex_I [where x="Var sl'"], simp)
       apply (rule Conj_I)
        apply (blast intro: Mem_Eats_I1 [OF RestrictedP_Mem [THEN cut3]] del: Disj_EH)
       apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH Conj_I)+
                   apply (rule Ex_I [where x="Var m"], simp)
                   apply (rule Ex_I [where x="Var n"], simp)
                   apply (rule Ex_I [where x="Var smi"], simp)
                   apply (rule Ex_I [where x="Var sm"], simp)
                   apply (rule Ex_I [where x="Var sm'"], simp)
                   apply (rule Ex_I [where x="Var sni"], simp)
                   apply (rule Ex_I [where x="Var sn"], simp)
                   apply (rule Ex_I [where x="Var sn'"], auto del: Disj_EH)
        apply (rule Mem_Eats_I1 [OF RestrictedP_Mem [THEN cut3]] AssumeH OrdP_Trans [OF OrdP_SUCC_I])+
       done
   qed
 qed (*>*)
 
 theorem AbstFormP_Neg: "{AbstFormP v i x x'} \<turnstile> AbstFormP v i (Q_Neg x) (Q_Neg x')"
 proof -
   obtain k1::name and s1::name and k::name and s::name
     where "atom s1 \<sharp> (v,i,x,x')"        "atom k1 \<sharp> (v,i,x,x',s1)"
       "atom s  \<sharp> (v,i,x,x',k1,s1)"  "atom k  \<sharp> (v,i,x,x',s,k1,s1)"
     by (metis obtain_fresh)
   thus ?thesis
     by (force simp: AbstFormP.simps [of s v i "Q_Neg x" _ k] AbstFormP.simps [of s1 v i x x' k1]
         intro: SeqAbstFormP_Neg [THEN cut3])
 qed
 
-subsection {*Disjunction*}
+subsection \<open>Disjunction\<close>
 
 lemma SeqAbstFormP_Disj:
   assumes "atom s \<sharp> (k,s1,s2,k1,k2,x,y,x',y',v,i)" "atom k \<sharp> (s1,s2,k1,k2,x,y,x',y',v,i)"
   shows "{SeqAbstFormP v i x x' s1 k1,
             SeqAbstFormP v i y y' s2 k2, OrdP i, VarP v}
            \<turnstile> Ex s (Ex k (SeqAbstFormP v i (Q_Disj x y) (Q_Disj x' y') (Var s) (Var k)))" (*<*)
 proof -
   let ?vs = "(s1,s2,s,k1,k2,k,x,y,x',y',v,i)"
   obtain km::name and kn::name and j::name and k'::name
     and l::name and sl::name and sl'::name and m::name and n::name
     and sm::name and sm'::name and sn::name and sn'::name and sli sni smi :: name
     where atoms2: "atom km \<sharp> (kn,j,k',sni,smi,sli,l,s1,s2,s,k1,k2,k,x,y,x',y',v,i,sl,sl',m,n,sm,sm',sn,sn')"
       "atom kn \<sharp> (j,k',sni,smi,sli,l,s1,s2,s,k1,k2,k,x,y,x',y',v,i,sl,sl',m,n,sm,sm',sn,sn')"
       "atom j \<sharp> (k',sni,smi,sli,l,s1,s2,s,k1,k2,k,x,y,x',y',v,i,sl,sl',m,n,sm,sm',sn,sn')"
       and atoms: "atom k' \<sharp> (sni,smi,sli,l,s1,s2,s,k1,k2,k,x,y,x',y',v,i,sl,sl',m,n,sm,sm',sn,sn')"
       "atom sni \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn',l,sli,smi)"
       "atom smi \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn',l,sli)"
       "atom sli \<sharp> (?vs,sl,sl',m,n,sm,sm',sn,sn',l)"
       "atom l \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',v,i,sl,sl',m,n,sm,sm',sn,sn')"
       "atom sl \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',v,i,sl',m,n,sm,sm',sn,sn')"
       "atom sl' \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',v,i,m,n,sm,sm',sn,sn')"
       "atom m \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',v,i,n,sm,sm',sn,sn')"
       "atom n \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',v,i,sm,sm',sn,sn')"
       "atom sm \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',v,i,sm',sn,sn')"
       "atom sm' \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',v,i,sn,sn')"
       "atom sn \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',v,i,sn')"
       "atom sn' \<sharp> (s1,s2,s,k1,k2,k,x,y,x',y',v,i)"
     by (metis obtain_fresh)
   let ?hyp = "{HaddP k1 k2 (Var k'), OrdP k1, OrdP k2, SeqAppendP s1 (SUCC k1) s2 (SUCC k2) (Var s),
                SeqAbstFormP v i x x' s1 k1, SeqAbstFormP v i y y' s2 k2, OrdP i, VarP v}"
   show ?thesis
     using assms atoms
     apply (auto simp: SeqAbstFormP.simps [of l "Var s" _ _ sli sl sl' m n smi sm sm' sni sn sn'])
     apply (rule cut_same [where A="OrdP k1 AND OrdP k2"])
      apply (metis Conj_I SeqAbstFormP_imp_OrdP thin1 thin2)
     apply (rule cut_same [OF exists_SeqAppendP [of s s1 "SUCC k1" s2 "SUCC k2"]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
       apply (rule cut_same [OF exists_HaddP [where j=k' and x=k1 and y=k2]])
         apply (rule AssumeH Ex_EH Conj_EH | simp)+
         apply (rule Ex_I [where x="Eats (Var s) (HPair (SUCC(SUCC(Var k'))) (HPair i (HPair(Q_Disj x y)(Q_Disj x' y'))))"])
         apply (simp_all (no_asm_simp))
     apply (rule Ex_I [where x="SUCC (SUCC (Var k'))"], simp)
     apply (rule Conj_I)
      apply (blast intro: LstSeqP_SeqAppendP_Eats SeqAbstFormP_imp_LstSeqP [THEN cut1])
   proof (rule All2_SUCC_I, simp_all)
     show "?hyp \<turnstile> SyntaxN.Ex sli (SyntaxN.Ex sl (SyntaxN.Ex sl'
          (HPair (SUCC (SUCC (Var k'))) (HPair (Var sli) (HPair (Var sl) (Var sl'))) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair i (HPair (Q_Disj x y) (Q_Disj x' y')))) AND
           (AbstAtomicP v (Var sli) (Var sl) (Var sl') OR
            OrdP (Var sli) AND
            SyntaxN.Ex m
             (SyntaxN.Ex n
               (SyntaxN.Ex smi
                 (SyntaxN.Ex sm
                   (SyntaxN.Ex sm'
                     (SyntaxN.Ex sni
                       (SyntaxN.Ex sn
                         (SyntaxN.Ex sn'
                           (Var m IN SUCC (SUCC (Var k')) AND
                            Var n IN SUCC (SUCC (Var k')) AND
                            HPair (Var m) (HPair (Var smi) (HPair (Var sm) (Var sm'))) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair i (HPair (Q_Disj x y) (Q_Disj x' y')))) AND
                            HPair (Var n) (HPair (Var sni) (HPair (Var sn) (Var sn'))) IN Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair i (HPair (Q_Disj x y) (Q_Disj x' y')))) AND
                            (Var sli EQ Var smi AND Var sli EQ Var sni AND Var sl EQ Q_Disj (Var sm) (Var sn) AND Var sl' EQ Q_Disj (Var sm') (Var sn') OR
                             Var sli EQ Var smi AND Var sl EQ Q_Neg (Var sm) AND Var sl' EQ Q_Neg (Var sm') OR
                             SUCC (Var sli) EQ Var smi AND Var sl EQ Q_Ex (Var sm) AND Var sl' EQ Q_Ex (Var sm'))))))))))))))"
       \<comment> \<open>verifying the final values\<close>
       apply (rule Ex_I [where x="i"])
       using assms atoms apply simp
       apply (rule Ex_I [where x="Q_Disj x y"], simp)
       apply (rule Ex_I [where x="Q_Disj x' y'"], simp)
       apply (rule Conj_I, metis Mem_Eats_I2 Refl)
       apply (rule Disj_I2)
       apply (rule Conj_I, blast)
       apply (rule Ex_I [where x=k1], simp)
       apply (rule Ex_I [where x="SUCC (Var k')"], simp)
       apply (rule Ex_I [where x=i], simp)
       apply (rule Ex_I [where x=x], simp)
       apply (rule_tac x=x' in Ex_I, simp)
       apply (rule Ex_I [where x=i], simp)
       apply (rule Ex_I [where x=y], simp)
       apply (rule_tac x=y' in Ex_I, simp)
       apply (rule Conj_I)
        apply (blast intro: HaddP_Mem_I LstSeqP_OrdP Mem_SUCC_I1)
       apply (rule Conj_I [OF Mem_SUCC_Refl])
       apply (blast intro: Disj_I1 Mem_Eats_I1 Mem_SUCC_Refl SeqAbstFormP_imp_LstSeqP [THEN cut1]
           LstSeqP_imp_Mem SeqAppendP_Mem1 [THEN cut3] SeqAppendP_Mem2 [THEN cut4] HaddP_SUCC1 [THEN cut1])
       done
   next
     show "?hyp \<turnstile> All2 l (SUCC (SUCC (Var k')))
      (SyntaxN.Ex sli
        (SyntaxN.Ex sl
          (SyntaxN.Ex sl'
            (HPair (Var l) (HPair (Var sli) (HPair (Var sl) (Var sl'))) IN
             Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair i (HPair (Q_Disj x y) (Q_Disj x' y')))) AND
             (AbstAtomicP v (Var sli) (Var sl) (Var sl') OR
              OrdP (Var sli) AND
              SyntaxN.Ex m
               (SyntaxN.Ex n
                 (SyntaxN.Ex smi
                   (SyntaxN.Ex sm
                     (SyntaxN.Ex sm'
                       (SyntaxN.Ex sni
                         (SyntaxN.Ex sn
                           (SyntaxN.Ex sn'
                             (Var m IN Var l AND
                              Var n IN Var l AND
                              HPair (Var m) (HPair (Var smi) (HPair (Var sm) (Var sm'))) IN
                              Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair i (HPair (Q_Disj x y) (Q_Disj x' y')))) AND
                              HPair (Var n) (HPair (Var sni) (HPair (Var sn) (Var sn'))) IN
                              Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair i (HPair (Q_Disj x y) (Q_Disj x' y')))) AND
                              (Var sli EQ Var smi AND Var sli EQ Var sni AND Var sl EQ Q_Disj (Var sm) (Var sn) AND Var sl' EQ Q_Disj (Var sm') (Var sn') OR
                               Var sli EQ Var smi AND Var sl EQ Q_Neg (Var sm) AND Var sl' EQ Q_Neg (Var sm') OR
                               SUCC (Var sli) EQ Var smi AND Var sl EQ Q_Ex (Var sm) AND Var sl' EQ Q_Ex (Var sm')))))))))))))))"
       \<comment> \<open>verifying the sequence buildup\<close>
       apply (rule cut_same [where A="HaddP (SUCC k1) (SUCC k2) (SUCC (SUCC (Var k')))"])
        apply (blast intro: HaddP_SUCC1 [THEN cut1] HaddP_SUCC2 [THEN cut1])
       apply (rule All_I Imp_I)+
        apply (rule HaddP_Mem_cases [where i=j])
       using assms atoms atoms2 apply simp_all
           apply (rule AssumeH)
          apply (blast intro: OrdP_SUCC_I LstSeqP_OrdP)
         \<comment> \<open>... the sequence buildup via s1\<close>
        apply (simp add: SeqAbstFormP.simps [of l s1 _ _ sli sl sl' m n smi sm sm' sni sn sn'])
         apply (rule AssumeH Ex_EH Conj_EH)+
         apply (rule All2_E [THEN rotate2])
           apply (simp | rule AssumeH Ex_EH Conj_EH)+
             apply (rule Ex_I [where x="Var sli"], simp)
             apply (rule Ex_I [where x="Var sl"], simp)
             apply (rule Ex_I [where x="Var sl'"], simp)
             apply (rule Conj_I [OF Mem_Eats_I1])
              apply (metis SeqAppendP_Mem1 rotate3 thin2 thin4)
             apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
                           apply (rule Conj_I)
                           apply (blast intro: Mem_Eats_I1 [OF RestrictedP_Mem [THEN cut3]] del: Disj_EH)
                         apply (rule Ex_I [where x="Var m"], simp)
                         apply (rule Ex_I [where x="Var n"], simp)
                         apply (rule Ex_I [where x="Var smi"], simp)
                         apply (rule Ex_I [where x="Var sm"], simp)
                         apply (rule Ex_I [where x="Var sm'"], simp)
                         apply (rule Ex_I [where x="Var sni"], simp)
                         apply (rule Ex_I [where x="Var sn"], simp)
                         apply (rule Ex_I [where x="Var sn'"], simp_all (no_asm_simp))
        apply (rule Conj_I, rule AssumeH)+
        apply (rule Conj_I)
         apply (blast intro: OrdP_Trans [OF OrdP_SUCC_I] Mem_Eats_I1 [OF SeqAppendP_Mem1 [THEN cut3]] Hyp)
        apply (blast intro: Disj_I1 Disj_I2 OrdP_Trans [OF OrdP_SUCC_I] Mem_Eats_I1 [OF SeqAppendP_Mem1 [THEN cut3]] Hyp)
         \<comment> \<open>... the sequence buildup via s2\<close>
        apply (simp add: SeqAbstFormP.simps [of l s2 _ _ sli sl sl' m n smi sm sm' sni sn sn'])
       apply (rule AssumeH Ex_EH Conj_EH)+
       apply (rule All2_E [THEN rotate2])
         apply (simp | rule AssumeH Ex_EH Conj_EH)+
           apply (rule Ex_I [where x="Var sli"], simp)
           apply (rule Ex_I [where x="Var sl"], simp)
           apply (rule Ex_I [where x="Var sl'"], simp)
           apply (rule cut_same [where A="OrdP (Var j)"])
            apply (metis HaddP_imp_OrdP rotate2 thin2)
           apply (rule Conj_I)
            apply (blast intro: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] del: Disj_EH)
           apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
                       apply (rule cut_same [OF exists_HaddP [where j=km and x="SUCC k1" and y="Var m"]])
                         apply (blast intro: Ord_IN_Ord, simp)
                       apply (rule cut_same [OF exists_HaddP [where j=kn and x="SUCC k1" and y="Var n"]])
                         apply (metis AssumeH(6) Ord_IN_Ord0 rotate8, simp)
                       apply (rule AssumeH Ex_EH Conj_EH | simp)+
                           apply (rule Conj_I)
                           apply (blast intro: Mem_Eats_I1 [OF RestrictedP_Mem [THEN cut3]] del: Disj_EH)
                           apply (rule Ex_I [where x="Var km"], simp)
                           apply (rule Ex_I [where x="Var kn"], simp)
                           apply (rule Ex_I [where x="Var smi"], simp)
                           apply (rule Ex_I [where x="Var sm"], simp)
                           apply (rule Ex_I [where x="Var sm'"], simp)
                           apply (rule Ex_I [where x="Var sni"], simp)
                           apply (rule Ex_I [where x="Var sn"], simp)
                           apply (rule Ex_I [where x="Var sn'"], simp_all (no_asm_simp))
       apply (rule Conj_I [OF _ Conj_I])
         apply (blast intro!: HaddP_Mem_cancel_left [THEN Iff_MP2_same] OrdP_SUCC_I intro: LstSeqP_OrdP Hyp)+
       apply (blast del: Disj_EH  intro: OrdP_Trans Hyp
           intro!: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] HaddP_imp_OrdP [THEN cut1])
       done
   qed
 qed (*>*)
 
 theorem AbstFormP_Disj:
   "{AbstFormP v i x x', AbstFormP v i y y'} \<turnstile> AbstFormP v i (Q_Disj x y) (Q_Disj x' y')"
 proof -
   obtain k1::name and s1::name and k2::name and s2::name and k::name and s::name
     where "atom s1 \<sharp> (v,i,x,y,x',y')"        "atom k1 \<sharp> (v,i,x,y,x',y',s1)"
       "atom s2 \<sharp> (v,i,x,y,x',y',k1,s1)"  "atom k2 \<sharp> (v,i,x,y,x',y',s2,k1,s1)"
       "atom s  \<sharp> (v,i,x,y,x',y',k2,s2,k1,s1)" "atom k  \<sharp> (v,i,x,y,x',y',s,k2,s2,k1,s1)"
     by (metis obtain_fresh)
   thus ?thesis
     by (force simp: AbstFormP.simps [of s v i "Q_Disj x y" _ k]
         AbstFormP.simps [of s1 v i x x' k1]
         AbstFormP.simps [of s2 v i y y' k2]
         intro: SeqAbstFormP_Disj [THEN cut4])
 qed
 
-subsection {*Existential*}
+subsection \<open>Existential\<close>
 
 lemma SeqAbstFormP_Ex:
   assumes "atom s \<sharp> (k,s1,k1,x,x',v,i)" "atom k \<sharp> (s1,k1,x,x',v,i)"
   shows "{SeqAbstFormP v (SUCC i) x x' s1 k1, OrdP i, VarP v}
            \<turnstile> Ex s (Ex k (SeqAbstFormP v i (Q_Ex x) (Q_Ex x') (Var s) (Var k)))" (*<*)
 proof -
   obtain l::name and sl::name and sl'::name and m::name and n::name
     and sm::name and sm'::name and sn::name and sn'::name
     and sli smi sni :: name
     where atoms:
       "atom sni \<sharp> (s1,k1,s,k,x,x',v,i,sl,sl',m,n,sm,sm',sn,sn',l,sli,smi)"
       "atom smi \<sharp> (s1,k1,s,k,x,x',v,i,sl,sl',m,n,sm,sm',sn,sn',l,sli)"
       "atom sli \<sharp> (s1,k1,s,k,x,x',v,i,sl,sl',m,n,sm,sm',sn,sn',l)"
       "atom l \<sharp> (s1,k1,s,k,x,x',v,i,sl,sl',m,n,sm,sm',sn,sn')"
       "atom sl \<sharp> (s1,k1,s,k,x,x',v,i,sl',m,n,sm,sm',sn,sn')"
       "atom sl' \<sharp> (s1,k1,s,k,x,x',v,i,m,n,sm,sm',sn,sn')"
       "atom m \<sharp> (s1,k1,s,k,x,x',v,i,n,sm,sm',sn,sn')"
       "atom n \<sharp> (s1,k1,s,k,x,x',v,i,sm,sm',sn,sn')"
       "atom sm \<sharp> (s1,k1,s,k,x,x',v,i,sm',sn,sn')"
       "atom sm' \<sharp> (s1,k1,s,k,x,x',v,i,sn,sn')"
       "atom sn \<sharp> (s1,k1,s,k,x,x',v,i,sn')"
       "atom sn' \<sharp> (s1,k1,s,k,x,x',v,i)"
     by (metis obtain_fresh)
   let ?hyp = "{RestrictedP s1 (SUCC k1) (Var s), OrdP k1, SeqAbstFormP v (SUCC i) x x' s1 k1, OrdP i, VarP v}"
   show ?thesis
     using assms atoms
     apply (auto simp: SeqAbstFormP.simps [of l "Var s" _ _ sli sl sl' m n smi sm sm' sni sn sn'])
     apply (rule cut_same [where A="OrdP k1"])
      apply (metis SeqAbstFormP_imp_OrdP thin2)
     apply (rule cut_same [OF exists_RestrictedP [of s s1 "SUCC k1"]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
       apply (rule Ex_I [where x="Eats (Var s) (HPair (SUCC k1) (HPair i (HPair (Q_Ex x) (Q_Ex x'))))"], simp)
       apply (rule Ex_I [where x="(SUCC k1)"], simp)
       apply (rule Conj_I)
        apply (blast intro: RestrictedP_LstSeqP_Eats [THEN cut2] SeqAbstFormP_imp_LstSeqP [THEN cut1])
   proof (rule All2_SUCC_I, simp_all)
     show "?hyp \<turnstile> SyntaxN.Ex sli
      (SyntaxN.Ex sl
        (SyntaxN.Ex sl'
          (HPair (SUCC k1) (HPair (Var sli) (HPair (Var sl) (Var sl'))) IN Eats (Var s) (HPair (SUCC k1) (HPair i (HPair (Q_Ex x) (Q_Ex x')))) AND
           (AbstAtomicP v (Var sli) (Var sl) (Var sl') OR
            OrdP (Var sli) AND
            SyntaxN.Ex m
             (SyntaxN.Ex n
               (SyntaxN.Ex smi
                 (SyntaxN.Ex sm
                   (SyntaxN.Ex sm'
                     (SyntaxN.Ex sni
                       (SyntaxN.Ex sn
                         (SyntaxN.Ex sn'
                           (Var m IN SUCC k1 AND
                            Var n IN SUCC k1 AND
                            HPair (Var m) (HPair (Var smi) (HPair (Var sm) (Var sm'))) IN
                            Eats (Var s) (HPair (SUCC k1) (HPair i (HPair (Q_Ex x) (Q_Ex x')))) AND
                            HPair (Var n) (HPair (Var sni) (HPair (Var sn) (Var sn'))) IN
                            Eats (Var s) (HPair (SUCC k1) (HPair i (HPair (Q_Ex x) (Q_Ex x')))) AND
                            (Var sli EQ Var smi AND
                             Var sli EQ Var sni AND Var sl EQ Q_Disj (Var sm) (Var sn) AND Var sl' EQ Q_Disj (Var sm') (Var sn') OR
                             Var sli EQ Var smi AND Var sl EQ Q_Neg (Var sm) AND Var sl' EQ Q_Neg (Var sm') OR
                             SUCC (Var sli) EQ Var smi AND Var sl EQ Q_Ex (Var sm) AND Var sl' EQ Q_Ex (Var sm'))))))))))))))"
       \<comment> \<open>verifying the final values\<close>
       apply (rule Ex_I [where x="i"])
       using assms atoms apply simp
       apply (rule Ex_I [where x="Q_Ex x"], simp)
       apply (rule Ex_I [where x="Q_Ex x'"], simp)
       apply (rule Conj_I, metis Mem_Eats_I2 Refl)
       apply (rule Disj_I2)
       apply (rule Conj_I, blast)
       apply (rule Ex_I [where x=k1], simp)
       apply (rule Ex_I [where x=k1], simp)
       apply (rule Ex_I [where x="SUCC i"], simp)
       apply (rule Ex_I [where x=x], simp)
       apply (rule_tac x=x' in Ex_I, simp)
       apply (rule Ex_I [where x="SUCC i"], simp)
       apply (rule Ex_I [where x=x], simp)
       apply (rule_tac x=x' in Ex_I, simp)
       apply (rule Conj_I [OF Mem_SUCC_Refl])+
       apply (blast intro: Disj_I2 Mem_Eats_I1 RestrictedP_Mem [THEN cut3] Mem_SUCC_Refl
           SeqAbstFormP_imp_LstSeqP [THEN cut1] LstSeqP_imp_Mem)
       done
   next
     show "?hyp
              \<turnstile> All2 l (SUCC k1)
      (SyntaxN.Ex sli
        (SyntaxN.Ex sl
          (SyntaxN.Ex sl'
            (HPair (Var l) (HPair (Var sli) (HPair (Var sl) (Var sl'))) IN Eats (Var s) (HPair (SUCC k1) (HPair i (HPair (Q_Ex x) (Q_Ex x')))) AND
             (AbstAtomicP v (Var sli) (Var sl) (Var sl') OR
              OrdP (Var sli) AND
              SyntaxN.Ex m
               (SyntaxN.Ex n
                 (SyntaxN.Ex smi
                   (SyntaxN.Ex sm
                     (SyntaxN.Ex sm'
                       (SyntaxN.Ex sni
                         (SyntaxN.Ex sn
                           (SyntaxN.Ex sn'
                             (Var m IN Var l AND
                              Var n IN Var l AND
                              HPair (Var m) (HPair (Var smi) (HPair (Var sm) (Var sm'))) IN
                              Eats (Var s) (HPair (SUCC k1) (HPair i (HPair (Q_Ex x) (Q_Ex x')))) AND
                              HPair (Var n) (HPair (Var sni) (HPair (Var sn) (Var sn'))) IN
                              Eats (Var s) (HPair (SUCC k1) (HPair i (HPair (Q_Ex x) (Q_Ex x')))) AND
                              (Var sli EQ Var smi AND Var sli EQ Var sni AND Var sl EQ Q_Disj (Var sm) (Var sn) AND Var sl' EQ Q_Disj (Var sm') (Var sn') OR
                               Var sli EQ Var smi AND Var sl EQ Q_Neg (Var sm) AND Var sl' EQ Q_Neg (Var sm') OR
                               SUCC (Var sli) EQ Var smi AND Var sl EQ Q_Ex (Var sm) AND Var sl' EQ Q_Ex (Var sm')))))))))))))))"
       \<comment> \<open>verifying the sequence buildup\<close>
       using assms atoms
         \<comment> \<open>... the sequence buildup via s1\<close>
       apply (auto simp add: SeqAbstFormP.simps [of l s1 _ _ sli sl sl' m n smi sm sm' sni sn sn'])
       apply (rule Swap)
       apply (rule All2_E, auto del: Disj_EH)
       apply (rule Ex_I [where x="Var sli"], simp)
       apply (rule Ex_I [where x="Var sl"], simp)
       apply (rule Ex_I [where x="Var sl'"], simp)
       apply (rule Conj_I)
        apply (blast intro: Mem_Eats_I1 [OF RestrictedP_Mem [THEN cut3]] del: Disj_EH)
       apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
           apply (rule Conj_I)
            apply (blast intro: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] del: Disj_EH)
                   apply (rule Ex_I [where x="Var m"], simp)
                   apply (rule Ex_I [where x="Var n"], simp)
                   apply (rule Ex_I [where x="Var smi"], simp)
                   apply (rule Ex_I [where x="Var sm"], simp)
                   apply (rule Ex_I [where x="Var sm'"], simp)
                   apply (rule Ex_I [where x="Var sni"], simp)
                   apply (rule Ex_I [where x="Var sn"], simp)
                   apply (rule Ex_I [where x="Var sn'"])
                   apply (auto intro: Mem_Eats_I1 [OF RestrictedP_Mem [THEN cut3]] OrdP_Trans [OF OrdP_SUCC_I] del: Disj_EH)
       done
   qed
 qed (*>*)
 
 theorem AbstFormP_Ex: "{AbstFormP v (SUCC i) x x'} \<turnstile> AbstFormP v i (Q_Ex x) (Q_Ex x')"
 proof -
   obtain k1::name and s1::name and k::name and s::name
     where "atom s1 \<sharp> (v,i,x,x')"        "atom k1 \<sharp> (v,i,x,x',s1)"
       "atom s  \<sharp> (v,i,x,x',k1,s1)"  "atom k  \<sharp> (v,i,x,x',s,k1,s1)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: AbstFormP.simps [of s v i "Q_Ex x" _ k] AbstFormP.simps [of s1 v "SUCC i" x x' k1]
         intro!: SeqAbstFormP_Ex [THEN cut3] Ord_IN_Ord[OF Mem_SUCC_I2[OF Refl], of _ i])
 qed
 
 corollary AbstTermP_Zero: "{OrdP t} \<turnstile> AbstTermP \<lceil>Var v\<rceil> t Zero Zero"
 proof -
   obtain s::name and k::name where "atom s \<sharp> (v,t,k)"  "atom k \<sharp> (v,t)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: AbstTermP.simps [of s _ _ _ _ k] intro: SeqStTermP_Zero [THEN cut1])
 qed
 
 corollary AbstTermP_Var_same: "{VarP v, OrdP t} \<turnstile> AbstTermP v t v (Q_Ind t)"
 proof -
   obtain s::name and k::name where "atom s \<sharp> (v,t,k)" "atom k \<sharp> (v,t)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: AbstTermP.simps [of s _ _ _ _ k] intro: SeqStTermP_Var_same [THEN cut1])
 qed
 
 corollary AbstTermP_Var_diff: "{VarP v, VarP w, Neg (v EQ w), OrdP t} \<turnstile> AbstTermP v t w w"
 proof -
   obtain s::name and k::name where "atom s \<sharp> (v,w,t,k)" "atom k \<sharp> (v,w,t)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: AbstTermP.simps [of s _ _ _ _ k] intro: SeqStTermP_Var_diff [THEN cut3])
 qed
 
 theorem AbstTermP_Eats:
    "{AbstTermP v i t1 u1, AbstTermP v i t2 u2} \<turnstile> AbstTermP v i (Q_Eats t1 t2) (Q_Eats u1 u2)"
 proof -
   obtain k1::name and s1::name and k2::name and s2::name and k::name and s::name
     where "atom s1 \<sharp> (v,i,t1,u1,t2,u2)" "atom k1 \<sharp> (v,i,t1,u1,t2,u2,s1)"
           "atom s2 \<sharp> (v,i,t1,u1,t2,u2,k1,s1)" "atom k2 \<sharp> (v,i,t1,u1,t2,u2,s2,k1,s1)"
           "atom s  \<sharp> (v,i,t1,u1,t2,u2,k2,s2,k1,s1)"
           "atom k  \<sharp> (v,i,t1,u1,t2,u2,s,k2,s2,k1,s1)"
     by (metis obtain_fresh)
    thus ?thesis
      by (auto intro!: SeqStTermP_Eats [THEN cut2]
               simp: AbstTermP.simps [of s _ _ _ "(Q_Eats u1 u2)" k]
                     AbstTermP.simps [of s1 v i t1 u1 k1]
                     AbstTermP.simps [of s2 v i t2 u2 k2])
  qed
 
 corollary AbstTermP_Ind: "{VarP v, IndP w, OrdP t} \<turnstile> AbstTermP v t w w"
 proof -
   obtain s::name and k::name where "atom s \<sharp> (v,w,t,k)" "atom k \<sharp> (v,w,t)"
     by (metis obtain_fresh)
   thus ?thesis
     by (force simp: AbstTermP.simps [of s _ _ _ _ k]
               intro: SeqStTermP_Ind [THEN cut2])
 qed
 
 lemma ORD_OF_EQ_diff: "x \<noteq> y \<Longrightarrow> {ORD_OF x EQ ORD_OF y} \<turnstile> Fls"
 proof (induct x arbitrary: y)
   case (Suc x)
   then show ?case using SUCC_inject_E
     by (cases y) (auto simp: gr0_conv_Suc Eats_EQ_Zero_E SUCC_def)
 qed (auto simp: gr0_conv_Suc SUCC_def)
 
 lemma quot_Var_EQ_diff: "i \<noteq> x \<Longrightarrow> {\<lceil>Var i\<rceil> EQ \<lceil>Var x\<rceil>} \<turnstile> Fls"
   by (auto simp: quot_Var ORD_OF_EQ_diff)
 
 lemma AbstTermP_dbtm: "{} \<turnstile> AbstTermP \<lceil>Var i\<rceil> (ORD_OF n) (quot_dbtm u) (quot_dbtm (abst_dbtm i n u))"
 proof (induct u rule: dbtm.induct)
   case (DBVar x)
   then show ?case
     by (auto simp: quot_Var[symmetric] quot_Var_EQ_diff
       intro!: AbstTermP_Var_same[THEN cut2] AbstTermP_Var_diff[THEN cut4] TermP_Zero)
 qed (auto intro!: AbstTermP_Zero[THEN cut1] AbstTermP_Eats[THEN cut2] AbstTermP_Ind[THEN cut3] IndP_Q_Ind)
 
 lemma AbstFormP_dbfm: "{} \<turnstile> AbstFormP \<lceil>Var i\<rceil> (ORD_OF n) (quot_dbfm db) (quot_dbfm (abst_dbfm i n db))"
   by (induction db arbitrary: n rule: dbfm.induct)
     (auto intro!: AbstTermP_dbtm AbstFormP_Mem[THEN cut2] AbstFormP_Eq[THEN cut2]
         AbstFormP_Disj[THEN cut2] AbstFormP_Neg[THEN cut1] AbstFormP_Ex[THEN cut1]
         dest: meta_spec[of _ "Suc _"])
 
 lemmas AbstFormP = AbstFormP_dbfm[where db="trans_fm [] A" and n = 0 for A,
     simplified, folded quot_fm_def, unfolded abst_trans_fm]
 
 lemma SubstTermP_trivial_dbtm:
   "atom i \<sharp> u \<Longrightarrow> {} \<turnstile> SubstTermP \<lceil>Var i\<rceil> Zero (quot_dbtm u) (quot_dbtm u)"
 proof (induct u rule: dbtm.induct)
   case (DBVar x)
   then show ?case
     by (auto simp: quot_Var[symmetric] quot_Var_EQ_diff
       intro!: SubstTermP_Var_same[THEN cut2] SubstTermP_Var_diff[THEN cut4] TermP_Zero)
 qed (auto intro!: SubstTermP_Zero[THEN cut1] SubstTermP_Eats[THEN cut2] SubstTermP_Ind[THEN cut3]
   TermP_Zero IndP_Q_Ind)
 
 lemma SubstTermP_dbtm: "wf_dbtm t \<Longrightarrow>
    {} \<turnstile> SubstTermP \<lceil>Var i\<rceil> (quot_dbtm t) (quot_dbtm u) (quot_dbtm (subst_dbtm t i u))"
 proof (induct u rule: dbtm.induct)
   case (DBVar x)
   then show ?case
     apply (auto simp: quot_Var[symmetric]
       intro!: SubstTermP_Var_same[THEN cut2] SubstTermP_Var_diff[THEN cut4] TermP_quot_dbtm)
     apply (auto simp: quot_Var ORD_OF_EQ_diff)
     done
 qed (auto intro!: SubstTermP_Zero[THEN cut1] SubstTermP_Ind[THEN cut3] SubstTermP_Eats[THEN cut2]
    TermP_quot_dbtm IndP_Q_Ind)
 
 
 
 lemma SubstFormP_trivial_dbfm:
   fixes X :: fm
   assumes "atom i \<sharp> db"
   shows "{} \<turnstile> SubstFormP \<lceil>Var i\<rceil> Zero (quot_dbfm db) (quot_dbfm db)"
   using assms
   by (induct db rule: dbfm.induct)
     (auto intro!: SubstFormP_Ex[THEN cut1] SubstFormP_Neg[THEN cut1] SubstFormP_Disj[THEN cut2]
           SubstFormP_Eq[THEN cut2] SubstFormP_Mem[THEN cut2] SubstTermP_trivial_dbtm)+
 
 lemma SubstFormP_dbfm:
   assumes "wf_dbtm t"
   shows "{} \<turnstile> SubstFormP \<lceil>Var i\<rceil> (quot_dbtm t) (quot_dbfm db) (quot_dbfm (subst_dbfm t i db))"
   by (induct db rule: dbfm.induct)
     (auto intro!: SubstTermP_dbtm assms SubstFormP_Ex[THEN cut1] SubstFormP_Neg[THEN cut1]
            SubstFormP_Disj[THEN cut2] SubstFormP_Eq[THEN cut2] SubstFormP_Mem[THEN cut2])+
 
 lemmas SubstFormP_trivial = SubstFormP_trivial_dbfm[where db="trans_fm [] A" for A,
     simplified, folded quot_tm_def quot_fm_def quot_subst_eq]
 lemmas SubstFormP = SubstFormP_dbfm[OF wf_dbtm_trans_tm, where db="trans_fm [] A" for A,
     simplified, folded quot_tm_def quot_fm_def quot_subst_eq]
 lemmas SubstFormP_Zero = SubstFormP_dbfm[OF wf_dbtm.Zero, where db="trans_fm [] A" for A,
     simplified, folded  trans_tm.simps[of "[]"], folded quot_tm_def quot_fm_def quot_subst_eq]
 
 lemma AtomicP_Mem:
   "{TermP x, TermP y} \<turnstile> AtomicP (Q_Mem x y)"
 proof -
   obtain t::name and u::name
     where "atom t \<sharp> (x, y)" "atom u \<sharp> (t, x, y)"
     by (metis obtain_fresh)
   thus ?thesis
     apply (simp add: AtomicP.simps [of t u])
     apply (rule Ex_I [where x = x], simp)
     apply (rule Ex_I [where x = y], simp)
     apply (auto intro: Disj_I2)
     done
 qed
 
 lemma AtomicP_Eq:
   "{TermP x, TermP y} \<turnstile> AtomicP (Q_Eq x y)"
 proof -
   obtain t::name and u::name
     where "atom t \<sharp> (x, y)" "atom u \<sharp> (t, x, y)"
     by (metis obtain_fresh)
   thus ?thesis
     apply (simp add: AtomicP.simps [of t u])
     apply (rule Ex_I [where x = x], simp)
     apply (rule Ex_I [where x = y], simp)
     apply (auto intro: Disj_I1)
     done
 qed
 
 lemma SeqFormP_Mem:
   assumes "atom s \<sharp> (k,x,y)" "atom k \<sharp> (x,y)"
   shows "{TermP x, TermP y} \<turnstile> Ex k (Ex s (SeqFormP (Var s) (Var k) (Q_Mem x y)))"
 proof -
   let ?vs = "(x,y,s,k)"
   obtain l::name and sl::name and m::name and n::name and sm::name and sn::name
     where
       "atom l \<sharp> (?vs,sl,m,n,sm,sn)"
       "atom sl \<sharp> (?vs,m,n,sm,sn)"
       "atom m \<sharp> (?vs,n,sm,sn)" "atom n \<sharp> (?vs,sm,sn)"
       "atom sm \<sharp> (?vs,sn)"
       "atom sn \<sharp> (?vs)"
     by (metis obtain_fresh)
   with assms show ?thesis
     apply (auto simp: SeqFormP.simps[of l "Var s" _ _ sl m n sm sn])
     apply (rule Ex_I [where x = Zero], simp)
     apply (rule Ex_I [where x = "Eats Zero (HPair Zero (Q_Mem x y))"], auto intro!: Mem_SUCC_EH)
     apply (rule Ex_I [where x = "Q_Mem x y"], auto intro!: Mem_Eats_I2 HPair_cong Disj_I1 AtomicP_Mem[THEN cut2])
     done
 qed
 
 lemma SeqFormP_Eq:
   assumes "atom s \<sharp> (k,x,y)" "atom k \<sharp> (x,y)"
   shows "{TermP x, TermP y} \<turnstile> Ex k (Ex s (SeqFormP (Var s) (Var k) (Q_Eq x y)))"
 proof -
   let ?vs = "(x,y,s,k)"
   obtain l::name and sl::name and m::name and n::name and sm::name and sn::name
     where
       "atom l \<sharp> (?vs,sl,m,n,sm,sn)"
       "atom sl \<sharp> (?vs,m,n,sm,sn)"
       "atom m \<sharp> (?vs,n,sm,sn)" "atom n \<sharp> (?vs,sm,sn)"
       "atom sm \<sharp> (?vs,sn)"
       "atom sn \<sharp> (?vs)"
     by (metis obtain_fresh)
   with assms show ?thesis
     apply (auto simp: SeqFormP.simps[of l "Var s" _ _ sl m n sm sn])
     apply (rule Ex_I [where x = Zero], simp)
     apply (rule Ex_I [where x = "Eats Zero (HPair Zero (Q_Eq x y))"], auto intro!: Mem_SUCC_EH)
     apply (rule Ex_I [where x = "Q_Eq x y"], auto intro!: Mem_Eats_I2 HPair_cong Disj_I1 AtomicP_Eq[THEN cut2])
     done
 qed
 
 lemma FormP_Mem:
   "{TermP x, TermP y} \<turnstile> FormP (Q_Mem x y)"
 proof -
   obtain s::name and k::name
     where "atom s \<sharp> (x, y)" "atom k \<sharp> (s, x, y)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp add: FormP.simps [of k s] intro!: SeqFormP_Mem)
 qed
 
 lemma FormP_Eq:
   "{TermP x, TermP y} \<turnstile> FormP (Q_Eq x y)"
 proof -
   obtain s::name and k::name
     where "atom s \<sharp> (x, y)" "atom k \<sharp> (s, x, y)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp add: FormP.simps [of k s] intro!: SeqFormP_Eq)
 qed
 
-subsection {*MakeForm*}
+subsection \<open>MakeForm\<close>
 
 lemma MakeFormP_Neg: "{} \<turnstile> MakeFormP (Q_Neg x) x y"
 proof -
   obtain a::name and b::name
     where "atom a \<sharp> (x, y)" "atom b \<sharp> (a, x, y)" by (metis obtain_fresh)
   then show ?thesis
     by (auto simp: MakeFormP.simps[of a _ _ _ b] intro: Disj_I2[OF Disj_I1])
 qed
 
 lemma MakeFormP_Disj: "{} \<turnstile> MakeFormP (Q_Disj x y) x y"
 proof -
   obtain a::name and b::name
     where "atom a \<sharp> (x, y)" "atom b \<sharp> (a, x, y)" by (metis obtain_fresh)
   then show ?thesis
     by (auto simp: MakeFormP.simps[of a _ _ _ b] intro: Disj_I1)
 qed
 
 lemma MakeFormP_Ex: "{AbstFormP v Zero t x} \<turnstile> MakeFormP (Q_Ex x) t y"
 proof -
   obtain a::name and b::name
     where "atom a \<sharp> (v, x, t, y)" "atom b \<sharp> (a, v, x, t, y)" by (metis obtain_fresh)
   then show ?thesis
     by (subst MakeFormP.simps[of a _ _ _ b])
       (force intro!: Disj_I2[OF Disj_I2] intro: Ex_I[of _ _ _ v] Ex_I[of _ _ _ x])+
 qed
 
-subsection {*Negation*}
+subsection \<open>Negation\<close>
 
 lemma SeqFormP_Neg:
   assumes "atom s \<sharp> (k,s1,k1,x)" "atom k \<sharp> (s1,k1,x)"
   shows "{SeqFormP s1 k1 x} \<turnstile> Ex k (Ex s (SeqFormP (Var s) (Var k) (Q_Neg x)))" (*<*)
 proof -
   let ?vs = "(s1,k1,s,k,x)"
   obtain l::name and sl::name and m::name and n::name and
     sm::name and sn::name
     where atoms:
       "atom l \<sharp> (?vs,sl,m,n,sm,sn)"
       "atom sl \<sharp> (?vs,m,n,sm,sn)"
       "atom m \<sharp> (?vs,n,sm,sn)" "atom n \<sharp> (?vs,sm,sn)"
       "atom sm \<sharp> (?vs,sn)"
       "atom sn \<sharp> (?vs)"
     by (metis obtain_fresh)
   let ?hyp = "{RestrictedP s1 (SUCC k1) (Var s), OrdP k1, SeqFormP s1 k1 x}"
   show ?thesis
     using assms atoms
     apply (auto simp: SeqFormP.simps [of l "Var s" _ _ sl m n sm sn])
     apply (rule cut_same [where A="OrdP k1"])
     apply (rule SeqFormP_imp_OrdP)
     apply (rule cut_same [OF exists_RestrictedP [of s s1 "SUCC k1"]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
     apply (rule Ex_I [where x="(SUCC k1)"])
       apply (simp_all (no_asm_simp))
       apply (rule Ex_I [where x="Eats (Var s) (HPair (SUCC k1)  (Q_Neg x))"])
       apply (simp_all (no_asm_simp))
     apply (rule Conj_I)
      apply (blast intro: RestrictedP_LstSeqP_Eats [THEN cut2] SeqFormP_imp_LstSeqP [THEN cut1])
   proof (rule All2_SUCC_I, simp_all)
     show "?hyp \<turnstile> SyntaxN.Ex sn
      (HPair (SUCC k1) (Var sn) IN
       Eats (Var s) (HPair (SUCC k1) (Q_Neg x)) AND
       (AtomicP (Var sn) OR
        SyntaxN.Ex m
         (SyntaxN.Ex l
           (SyntaxN.Ex sm
             (SyntaxN.Ex sl
               (Var m IN SUCC k1 AND
                Var l IN SUCC k1 AND
                HPair (Var m) (Var sm) IN
                Eats (Var s) (HPair (SUCC k1) (Q_Neg x)) AND
                HPair (Var l) (Var sl) IN
                Eats (Var s) (HPair (SUCC k1) (Q_Neg x)) AND
                MakeFormP (Var sn) (Var sm) (Var sl)))))))"
       \<comment> \<open>verifying the final values\<close>
       apply (rule Ex_I [where x="Q_Neg x"])
       using assms atoms apply simp
       apply (rule Conj_I, metis Mem_Eats_I2 Refl)
       apply (rule Disj_I2)
       apply (rule Ex_I [where x=k1], simp)
       apply (rule Ex_I [where x=k1], simp)
       apply (rule Ex_I [where x=x], simp)
       apply (rule Ex_I [where x=x], simp)
       apply (rule Conj_I [OF Mem_SUCC_Refl])+
       apply (blast intro: Disj_I1 Disj_I2 Mem_Eats_I1 RestrictedP_Mem [THEN cut3] Mem_SUCC_Refl
           SeqFormP_imp_LstSeqP [THEN cut1] LstSeqP_imp_Mem MakeFormP_Neg[THEN thin0])
       done
   next
     show "?hyp \<turnstile> All2 n (SUCC k1)
      (SyntaxN.Ex sn
        (HPair (Var n) (Var sn) IN
         Eats (Var s) (HPair (SUCC k1) (Q_Neg x)) AND
         (AtomicP (Var sn) OR
          SyntaxN.Ex m
           (SyntaxN.Ex l
             (SyntaxN.Ex sm
               (SyntaxN.Ex sl
                 (Var m IN Var n AND
                  Var l IN Var n AND
                  HPair (Var m) (Var sm) IN
                  Eats (Var s) (HPair (SUCC k1) (Q_Neg x)) AND
                  HPair (Var l) (Var sl) IN
                  Eats (Var s) (HPair (SUCC k1) (Q_Neg x)) AND
                  MakeFormP (Var sn) (Var sm) (Var sl))))))))"
       \<comment> \<open>verifying the sequence buildup\<close>
       apply (rule All_I Imp_I)+
       using assms atoms apply simp_all
         \<comment> \<open>... the sequence buildup via s1\<close>
     apply (simp add: SeqFormP.simps [of l s1 _ _ sl m n sm sn])
       apply (rule AssumeH Ex_EH Conj_EH)+
       apply (rule All2_E [THEN rotate2], auto del: Disj_EH)
       apply (rule Ex_I [where x="Var sn"], simp)
       apply (rule Conj_I)
        apply (blast intro: Mem_Eats_I1 [OF RestrictedP_Mem [THEN cut3]] del: Disj_EH)
       apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH Conj_I)+
               apply (rule Ex_I [where x="Var m"], simp)
               apply (rule Ex_I [where x="Var l"], simp)
               apply (rule Ex_I [where x="Var sm"], simp)
               apply (rule Ex_I [where x="Var sl"], simp)
               apply auto
        apply (rule Mem_Eats_I1 [OF RestrictedP_Mem [THEN cut3]] AssumeH OrdP_Trans [OF OrdP_SUCC_I])+
       done
   qed
 qed (*>*)
 
 theorem FormP_Neg: "{FormP x} \<turnstile> FormP (Q_Neg x)"
 proof -
   obtain k1::name and s1::name and k::name and s::name
     where "atom s1 \<sharp> x"        "atom k1 \<sharp> (x,s1)"
       "atom s \<sharp> (x,k1,s1)"  "atom k  \<sharp> (x,s,k1,s1)"
     by (metis obtain_fresh)
   thus ?thesis
     by (force simp: FormP.simps [of k s "Q_Neg x"] FormP.simps [of k1 s1 x]
         intro: SeqFormP_Neg [THEN cut1])
 qed
 
-subsection {*Disjunction*}
+subsection \<open>Disjunction\<close>
 
 lemma SeqFormP_Disj:
   assumes "atom s \<sharp> (k,s1,s2,k1,k2,x,y)" "atom k \<sharp> (s1,s2,k1,k2,x,y)"
   shows "{SeqFormP s1 k1 x, SeqFormP s2 k2 y}
            \<turnstile> Ex k (Ex s (SeqFormP (Var s) (Var k) (Q_Disj x y)))" (*<*)
 proof -
   let ?vs = "(s1,s2,s,k1,k2,k,x,y)"
   obtain km::name and kn::name and j::name and k'::name
     and l::name and sl::name and m::name and n::name
     and sm::name and sn::name
     where atoms2: "atom km \<sharp> (kn,j,k',l,s1,s2,s,k1,k2,k,x,y,sl,m,n,sm,sn)"
       "atom kn \<sharp> (j,k',l,s1,s2,s,k1,k2,k,x,y,sl,m,n,sm,sn)"
       "atom j \<sharp> (k',l,s1,s2,s,k1,k2,k,x,y,sl,m,n,sm,sn)"
       and atoms: "atom k' \<sharp> (l,s1,s2,s,k1,k2,k,x,y,sl,m,n,sm,sn)"
       "atom l \<sharp> (s1,s2,s,k1,k2,k,x,y,sl,m,n,sm,sn)"
       "atom sl \<sharp> (s1,s2,s,k1,k2,k,x,y,m,n,sm,sn)"
       "atom m \<sharp> (s1,s2,s,k1,k2,k,x,y,n,sm,sn)"
       "atom n \<sharp> (s1,s2,s,k1,k2,k,x,y,sm,sn)"
       "atom sm \<sharp> (s1,s2,s,k1,k2,k,x,y,sn)"
       "atom sn \<sharp> (s1,s2,s,k1,k2,k,x,y)"
     by (metis obtain_fresh)
   let ?hyp = "{HaddP k1 k2 (Var k'), OrdP k1, OrdP k2, SeqAppendP s1 (SUCC k1) s2 (SUCC k2) (Var s),
                SeqFormP s1 k1 x, SeqFormP s2 k2 y}"
   show ?thesis
     using assms atoms
     apply (auto simp: SeqFormP.simps [of l "Var s" _ _ sl m n sm sn])
     apply (rule cut_same [where A="OrdP k1 AND OrdP k2"])
      apply (metis Conj_I SeqFormP_imp_OrdP thin1 thin2)
     apply (rule cut_same [OF exists_SeqAppendP [of s s1 "SUCC k1" s2 "SUCC k2"]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
       apply (rule cut_same [OF exists_HaddP [where j=k' and x=k1 and y=k2]])
         apply (rule AssumeH Ex_EH Conj_EH | simp)+
         apply (rule Ex_I [where x="SUCC (SUCC (Var k'))"])
         apply (simp_all (no_asm_simp) add: )
     apply (rule Ex_I [where x="Eats (Var s) (HPair (SUCC(SUCC(Var k'))) (Q_Disj x y))"], simp)
     apply (rule Conj_I)
      apply (blast intro: LstSeqP_SeqAppendP_Eats SeqFormP_imp_LstSeqP [THEN cut1])
   proof (rule All2_SUCC_I, simp_all)
     show "?hyp \<turnstile> SyntaxN.Ex sn
      (HPair (SUCC (SUCC (Var k'))) (Var sn) IN
       Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (Q_Disj x y)) AND
       (AtomicP (Var sn) OR
        SyntaxN.Ex m
         (SyntaxN.Ex l
           (SyntaxN.Ex sm
             (SyntaxN.Ex sl
               (Var m IN SUCC (SUCC (Var k')) AND
                Var l IN SUCC (SUCC (Var k')) AND
                HPair (Var m) (Var sm) IN
                Eats (Var s)
                 (HPair (SUCC (SUCC (Var k'))) (Q_Disj x y)) AND
                HPair (Var l) (Var sl) IN
                Eats (Var s)
                 (HPair (SUCC (SUCC (Var k'))) (Q_Disj x y)) AND
                MakeFormP (Var sn) (Var sm) (Var sl)))))))"
       \<comment> \<open>verifying the final values\<close>
       apply (rule Ex_I [where x="Q_Disj x y"])
       using assms atoms apply simp
       apply (rule Conj_I, metis Mem_Eats_I2 Refl)
       apply (rule Disj_I2)
       apply (rule Ex_I [where x=k1], simp)
       apply (rule Ex_I [where x="SUCC (Var k')"], simp)
       apply (rule Ex_I [where x=x], simp)
       apply (rule Ex_I [where x=y], simp)
       apply (rule Conj_I)
        apply (blast intro: HaddP_Mem_I LstSeqP_OrdP Mem_SUCC_I1)
       apply (rule Conj_I [OF Mem_SUCC_Refl])
       apply (blast intro: Disj_I1 Mem_Eats_I1 Mem_SUCC_Refl SeqFormP_imp_LstSeqP [THEN cut1]
           LstSeqP_imp_Mem SeqAppendP_Mem1 [THEN cut3] SeqAppendP_Mem2 [THEN cut4] HaddP_SUCC1 [THEN cut1]
           MakeFormP_Disj[THEN thin0])
       done
   next
     show "?hyp \<turnstile> All2 n (SUCC (SUCC (Var k')))
      (SyntaxN.Ex sn
        (HPair (Var n) (Var sn) IN
         Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (Q_Disj x y)) AND
         (AtomicP (Var sn) OR
          SyntaxN.Ex m
           (SyntaxN.Ex l
             (SyntaxN.Ex sm
               (SyntaxN.Ex sl
                 (Var m IN Var n AND
                  Var l IN Var n AND
                  HPair (Var m) (Var sm) IN
                  Eats (Var s)
                   (HPair (SUCC (SUCC (Var k'))) (Q_Disj x y)) AND
                  HPair (Var l) (Var sl) IN
                  Eats (Var s)
                   (HPair (SUCC (SUCC (Var k'))) (Q_Disj x y)) AND
                  MakeFormP (Var sn) (Var sm) (Var sl))))))))"
       \<comment> \<open>verifying the sequence buildup\<close>
       apply (rule cut_same [where A="HaddP (SUCC k1) (SUCC k2) (SUCC (SUCC (Var k')))"])
        apply (blast intro: HaddP_SUCC1 [THEN cut1] HaddP_SUCC2 [THEN cut1])
       apply (rule All_I Imp_I)+
        apply (rule HaddP_Mem_cases [where i=j])
       using assms atoms atoms2 apply simp_all
           apply (rule AssumeH)
          apply (blast intro: OrdP_SUCC_I LstSeqP_OrdP)
         \<comment> \<open>... the sequence buildup via s1\<close>
         apply (simp add: SeqFormP.simps [of l s1 _ _ sl m n sm sn])
         apply (rule AssumeH Ex_EH Conj_EH)+
         apply (rule All2_E [THEN rotate2])
           apply (simp | rule AssumeH Ex_EH Conj_EH)+
             apply (rule Ex_I [where x="Var sn"], simp)
             apply (rule Conj_I [OF Mem_Eats_I1])
              apply (metis SeqAppendP_Mem1 rotate3 thin2 thin4)
             apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
                         apply (rule Ex_I [where x="Var m"], simp)
                         apply (rule Ex_I [where x="Var l"], simp)
                         apply (rule Ex_I [where x="Var sm"], simp)
                         apply (rule Ex_I [where x="Var sl"], simp_all (no_asm_simp))
        apply (rule Conj_I, rule AssumeH)+
        apply (rule Conj_I)
         apply (blast intro: OrdP_Trans [OF OrdP_SUCC_I] Mem_Eats_I1 [OF SeqAppendP_Mem1 [THEN cut3]] Hyp)
        apply (blast intro: Disj_I1 Disj_I2 OrdP_Trans [OF OrdP_SUCC_I] Mem_Eats_I1 [OF SeqAppendP_Mem1 [THEN cut3]] Hyp)
         \<comment> \<open>... the sequence buildup via s2\<close>
       apply (simp add: SeqFormP.simps [of l s2 _ _ sl m n sm sn])
       apply (rule AssumeH Ex_EH Conj_EH)+
       apply (rule All2_E [THEN rotate2])
         apply (simp | rule AssumeH Ex_EH Conj_EH)+
           apply (rule Ex_I [where x="Var sn"], simp)
           apply (rule cut_same [where A="OrdP (Var j)"])
            apply (metis HaddP_imp_OrdP rotate2 thin2)
           apply (rule Conj_I)
            apply (blast intro: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] del: Disj_EH)
           apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
                       apply (rule cut_same [OF exists_HaddP [where j=km and x="SUCC k1" and y="Var m"]])
                         apply (blast intro: Ord_IN_Ord, simp)
                 apply (rule cut_same [OF exists_HaddP [where j=kn and x="SUCC k1" and y="Var l"]])
                         apply (metis AssumeH(6) Ord_IN_Ord0 rotate8, simp)
                       apply (rule AssumeH Ex_EH Conj_EH | simp)+
                           apply (rule Ex_I [where x="Var km"], simp)
                           apply (rule Ex_I [where x="Var kn"], simp)
                           apply (rule Ex_I [where x="Var sm"], simp)
                           apply (rule Ex_I [where x="Var sl"], simp_all (no_asm_simp))
       apply (rule Conj_I [OF _ Conj_I])
         apply (blast intro!: HaddP_Mem_cancel_left [THEN Iff_MP2_same] OrdP_SUCC_I intro: LstSeqP_OrdP Hyp)+
       apply (blast del: Disj_EH  intro: OrdP_Trans Hyp
           intro!: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] HaddP_imp_OrdP [THEN cut1])
       done
   qed
 qed (*>*)
 
 theorem FormP_Disj:
   "{FormP x, FormP y} \<turnstile> FormP (Q_Disj x y)"
 proof -
   obtain k1::name and s1::name and k2::name and s2::name and k::name and s::name
     where "atom s1 \<sharp> (x,y)"        "atom k1 \<sharp> (x,y,s1)"
       "atom s2 \<sharp> (x,y,k1,s1)"  "atom k2 \<sharp> (x,y,s2,k1,s1)"
       "atom s  \<sharp> (x,y,k2,s2,k1,s1)" "atom k  \<sharp> (x,y,s,k2,s2,k1,s1)"
     by (metis obtain_fresh)
   thus ?thesis
     by (force simp: FormP.simps [of k s "Q_Disj x y"]
         FormP.simps [of k1 s1 x]
         FormP.simps [of k2 s2 y]
         intro: SeqFormP_Disj [THEN cut2])
 qed
 
-subsection {*Existential*}
+subsection \<open>Existential\<close>
 
 lemma SeqFormP_Ex:
   assumes "atom s \<sharp> (k,s1,k1,x,y,v)" "atom k \<sharp> (s1,k1,x,y,v)"
   shows "{SeqFormP s1 k1 x,AbstFormP v Zero x y, VarP v} \<turnstile> Ex k (Ex s (SeqFormP (Var s) (Var k) (Q_Ex y)))"
 proof -
   let ?vs = "(s1,s,k1,k,x,y,v)"
   obtain km::name and kn::name and j::name and k'::name
     and l::name and sl::name and m::name and n::name
     and sm::name and sn::name
     where atoms2: "atom km \<sharp> (kn,j,k',l,s1,s,k1,k,x,y,v,sl,m,n,sm,sn)"
       "atom kn \<sharp> (j,k',l,s1,s,k1,k,x,y,v,sl,m,n,sm,sn)"
       "atom j \<sharp> (k',l,s1,s,k1,k,x,y,v,sl,m,n,sm,sn)"
       and atoms: "atom k' \<sharp> (l,s1,s,k1,k,x,y,v,sl,m,n,sm,sn)"
       "atom l \<sharp> (s1,s,k1,k,x,y,v,sl,m,n,sm,sn)"
       "atom sl \<sharp> (s1,s,k1,k,x,y,v,m,n,sm,sn)"
       "atom m \<sharp> (s1,s,k1,k,x,y,v,n,sm,sn)"
       "atom n \<sharp> (s1,s,k1,k,x,y,v,sm,sn)"
       "atom sm \<sharp> (s1,s,k1,k,x,y,v,sn)"
       "atom sn \<sharp> (s1,s,k1,k,x,y,v)"
     by (metis obtain_fresh)
   let ?hyp = "{RestrictedP s1 (SUCC k1) (Var s), OrdP k1, SeqFormP s1 k1 x,AbstFormP v Zero x y, VarP v}"
   show ?thesis
     using assms atoms
     apply (auto simp: SeqFormP.simps [of l "Var s" _ _ sl m n sm sn])
     apply (rule cut_same [where A="OrdP k1"])
      apply (metis SeqFormP_imp_OrdP thin2)
     apply (rule cut_same [OF exists_RestrictedP [of s s1 "SUCC k1"]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
       apply (rule Ex_I [where x="(SUCC k1)"], simp)
       apply (rule Ex_I [where x="Eats (Var s) (HPair (SUCC k1) (Q_Ex y))"], simp)
       apply (rule Conj_I)
        apply (blast intro: RestrictedP_LstSeqP_Eats [THEN cut2] SeqFormP_imp_LstSeqP [THEN cut1])
   proof (rule All2_SUCC_I, simp_all)
     show "?hyp \<turnstile>  SyntaxN.Ex sn
      (HPair (SUCC k1) (Var sn) IN
       Eats (Var s) (HPair (SUCC k1) (Q_Ex y)) AND
       (AtomicP (Var sn) OR
        SyntaxN.Ex m
         (SyntaxN.Ex l
           (SyntaxN.Ex sm
             (SyntaxN.Ex sl
               (Var m IN SUCC k1 AND
                Var l IN SUCC k1 AND
                HPair (Var m) (Var sm) IN
                Eats (Var s) (HPair (SUCC k1) (Q_Ex y)) AND
                HPair (Var l) (Var sl) IN
                Eats (Var s) (HPair (SUCC k1) (Q_Ex y)) AND
                MakeFormP (Var sn) (Var sm) (Var sl)))))))"
       \<comment> \<open>verifying the final values\<close>
       apply (rule Ex_I [where x="Q_Ex y"])
       using assms atoms apply simp
       apply (rule Conj_I, metis Mem_Eats_I2 Refl)
       apply (rule Disj_I2)
       apply (rule Ex_I [where x=k1], simp)
       apply (rule Ex_I [where x=k1], simp)
       apply (rule Ex_I [where x=x], simp)
       apply (rule Ex_I [where x=x], simp)
       apply (rule Conj_I [OF Mem_SUCC_Refl])+
       apply safe
       apply (blast intro: Disj_I2 Mem_Eats_I1 RestrictedP_Mem [THEN cut3] Mem_SUCC_Refl
           SeqFormP_imp_LstSeqP [THEN cut1] LstSeqP_imp_Mem)
       apply (blast intro: Disj_I2 Mem_Eats_I1 RestrictedP_Mem [THEN cut3] Mem_SUCC_Refl
           SeqFormP_imp_LstSeqP [THEN cut1] LstSeqP_imp_Mem)
       apply (rule MakeFormP_Ex[THEN cut1, of _ v])
       apply blast
       done
   next
     show "?hyp \<turnstile> All2 n (SUCC k1)
      (SyntaxN.Ex sn
        (HPair (Var n) (Var sn) IN
         Eats (Var s) (HPair (SUCC k1) (Q_Ex y)) AND
         (AtomicP (Var sn) OR
          SyntaxN.Ex m
           (SyntaxN.Ex l
             (SyntaxN.Ex sm
               (SyntaxN.Ex sl
                 (Var m IN Var n AND
                  Var l IN Var n AND
                  HPair (Var m) (Var sm) IN
                  Eats (Var s) (HPair (SUCC k1) (Q_Ex y)) AND
                  HPair (Var l) (Var sl) IN
                  Eats (Var s) (HPair (SUCC k1) (Q_Ex y)) AND
                  MakeFormP (Var sn) (Var sm) (Var sl))))))))"
       apply (rule All_I Imp_I)+
       using assms atoms apply simp_all
         \<comment> \<open>... the sequence buildup via s1\<close>
       apply (simp add: SeqFormP.simps [of l s1 _ _ sl m n sm sn])
       apply (rule AssumeH Ex_EH Conj_EH)+
       apply (rule All2_E [THEN rotate2], auto del: Disj_EH)
       apply (rule Ex_I [where x="Var sn"], simp)
       apply (rule Conj_I)
        apply (blast intro: Mem_Eats_I1 [OF RestrictedP_Mem [THEN cut3]] del: Disj_EH)
       apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH Conj_I)+
               apply (rule Ex_I [where x="Var m"], simp)
               apply (rule Ex_I [where x="Var l"], simp)
               apply (rule Ex_I [where x="Var sm"], simp)
               apply (rule Ex_I [where x="Var sl"], simp)
               apply auto
        apply (rule Mem_Eats_I1 [OF RestrictedP_Mem [THEN cut3]] AssumeH OrdP_Trans [OF OrdP_SUCC_I])+
       done
   qed
 qed (*>*)
 
 theorem FormP_Ex: "{FormP t, AbstFormP \<lceil>Var i\<rceil> Zero t x} \<turnstile> FormP (Q_Ex x)"
 proof -
   obtain k1::name and s1::name and k::name and s::name
     where "atom s1 \<sharp> (i,t,x)" "atom k1 \<sharp> (i,t,x,s1)" "atom s  \<sharp> (i,t,x,k1,s1)"  "atom k \<sharp> (i,t,x,s,k1,s1)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: FormP.simps [of k s "Q_Ex x"] FormP.simps [of k1 s1 t]
       intro!: SeqFormP_Ex [THEN cut3])
 qed
 
 lemma FormP_quot_dbfm:
   fixes A :: dbfm
   shows "wf_dbfm A \<Longrightarrow> {} \<turnstile> FormP (quot_dbfm A)"
   by (induct A rule: wf_dbfm.induct)
     (auto simp: intro!: FormP_Mem[THEN cut2] FormP_Eq[THEN cut2] Ex_I
       FormP_Neg[THEN cut1] FormP_Disj[THEN cut2] FormP_Ex[THEN cut2]
       TermP_quot_dbtm AbstFormP_dbfm[where n=0, simplified])
 
 lemma FormP_quot:
   fixes A :: fm
   shows "{} \<turnstile> FormP \<lceil>A\<rceil>"
   unfolding quot_fm_def
   by (rule FormP_quot_dbfm, rule wf_dbfm_trans_fm)
 
 
 lemma PfP_I:
   assumes "{} \<turnstile> PrfP S K A"
   shows "{} \<turnstile> PfP A"
 proof -
   obtain s::name and k::name where "atom s \<sharp> (k,A,S,K)" "atom k \<sharp> (A,S,K)" by (metis obtain_fresh)
   with assms show ?thesis
     apply (subst PfP.simps[of s k]; simp)
     apply (rule Ex_I[of _ _ _ K], auto, rule Ex_I[of _ _ _ S], auto)
     done
 qed
 
 lemmas PfP_Single_I = PfP_I[of "Eats Zero (HPair Zero \<lceil>A\<rceil>)" Zero for A]
 
 lemma PfP_extra: "{} \<turnstile> PfP \<lceil>extra_axiom\<rceil>"
 proof -
   obtain l::name and sl::name and m::name and n::name and sm::name and sn::name
     where atoms:
       "atom l \<sharp> (sl,m,n,sm,sn)"
       "atom sl \<sharp> (m,n,sm,sn)"
       "atom m \<sharp> (n,sm,sn)"
       "atom n \<sharp> (sm,sn)"
       "atom sm \<sharp> sn"
     by (metis obtain_fresh)
   with Extra show ?thesis
     apply (intro PfP_Single_I[of extra_axiom])
     apply (subst PrfP.simps[of l _ sl m n sm sn]; auto?)
     apply (rule Ex_I[of _ _ _ "\<lceil>extra_axiom\<rceil>"]; auto?)
      apply (rule Mem_SUCC_E[OF Mem_Zero_E])
      apply (rule Mem_Eats_I2)
      apply (rule HPair_cong[OF Assume Refl])
     apply (auto simp: AxiomP_def intro!: Disj_I1)
     done
 qed
 
 lemma SentP_I:
   assumes "A \<in> boolean_axioms"
   shows "{} \<turnstile> SentP \<lceil>A\<rceil>"
 proof -
   obtain x y z :: name where "atom z \<sharp> (x,y)" "atom y \<sharp> x" by (metis obtain_fresh)
   with assms show ?thesis
     apply (subst SentP.simps[of x y z]; simp)
     subgoal
     proof (erule boolean_axioms.cases, goal_cases Ident DisjI1 DisjCont DisjAssoc DisjConj)
       case (Ident A)
       then show ?thesis
         by (intro Ex_I[of _ _ _ "\<lceil>A\<rceil>"]; simp)+
           (auto simp: FormP_quot[THEN thin0] quot_simps intro!: Disj_I1)
     next
       case (DisjI1 A B)
       then show ?thesis
         by (intro Ex_I[of _ _ _ "\<lceil>A\<rceil>"]; simp, (intro Ex_I[of _ _ _ "\<lceil>B\<rceil>"]; simp)?)+
           (auto simp: FormP_quot[THEN thin0] quot_simps intro!: Disj_I2[OF Disj_I1])
     next
       case (DisjCont A)
       then show ?thesis
         by (intro Ex_I[of _ _ _ "\<lceil>A\<rceil>"]; simp)+
           (auto simp: FormP_quot[THEN thin0] quot_simps intro!: Disj_I2[OF Disj_I2[OF Disj_I1]])
     next
       case (DisjAssoc A B C)
       then show ?thesis
         by (intro Ex_I[of _ _ _ "\<lceil>A\<rceil>"]; simp, intro Ex_I[of _ _ _ "\<lceil>B\<rceil>"]; simp, intro Ex_I[of _ _ _ "\<lceil>C\<rceil>"]; simp)+
           (auto simp: FormP_quot[THEN thin0] quot_simps intro!: Disj_I2[OF Disj_I2[OF Disj_I2[OF Disj_I1]]])
     next
       case (DisjConj A B C)
       then show ?thesis
         by (intro Ex_I[of _ _ _ "\<lceil>A\<rceil>"]; simp, intro Ex_I[of _ _ _ "\<lceil>B\<rceil>"]; simp, intro Ex_I[of _ _ _ "\<lceil>C\<rceil>"]; simp)+
           (auto simp: FormP_quot[THEN thin0] quot_simps intro!: Disj_I2[OF Disj_I2[OF Disj_I2[OF Disj_I2]]])
     qed
     done
 qed
 
 lemma SentP_subst [simp]: "(SentP A)(j::=w) = SentP (subst j w A)"
 proof -
   obtain x y z ::name where "atom x \<sharp> (y,z,j,w,A)" "atom y \<sharp> (z,j,w,A)"  "atom z \<sharp> (j,w,A)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: SentP.simps [of x y z])
 qed
 
 theorem proved_imp_proved_PfP:
   assumes "{} \<turnstile> \<alpha>"
   shows "{} \<turnstile> PfP \<lceil>\<alpha>\<rceil>"
   using assms
 proof (induct "{} :: fm set" \<alpha> rule: hfthm.induct)
   case (Hyp A)
   then show ?case
     by auto
 next
   case Extra
   then show ?case by (simp add: PfP_extra)
 next
   case (Bool A)
   obtain l::name and sl::name and m::name and n::name and
          sm::name and sn::name and x::name and y::name and z::name
          where atoms:
           "atom l \<sharp> (x,y,z,sl,m,n,sm,sn)"
          "atom sl \<sharp> (x,y,z,m,n,sm,sn)"
           "atom m \<sharp> (x,y,z,n,sm,sn)"
           "atom n \<sharp> (x,y,z,sm,sn)"
          "atom sm \<sharp> (x,y,z,sn)"
          "atom sn \<sharp> (x,y,z)"
           "atom z \<sharp> (x,y)"
           "atom y \<sharp> x"
     by (metis obtain_fresh)
   with Bool show ?case
     apply (intro PfP_Single_I[of A])
     apply (subst PrfP.simps[of l _ sl m n sm sn]; auto?)
     apply (rule Ex_I[of _ _ _ "\<lceil>A\<rceil>"]; auto?)
      apply (rule Mem_SUCC_E[OF Mem_Zero_E])
      apply (rule Mem_Eats_I2)
      apply (rule HPair_cong[OF Assume Refl])
     apply (rule Disj_I1)
     apply (unfold AxiomP_def; simp)
     apply (rule Disj_I2[OF Disj_I1])
     apply (auto elim!: SentP_I[THEN thin0])
     done
 next
   case (Eq A)
   obtain l::name and sl::name and m::name and n::name and sm::name and sn::name and x::name and y::name and z::name
          where atoms:
           "atom l \<sharp> (x,y,z,sl,m,n,sm,sn)"
          "atom sl \<sharp> (x,y,z,m,n,sm,sn)"
           "atom m \<sharp> (x,y,z,n,sm,sn)"
           "atom n \<sharp> (x,y,z,sm,sn)"
          "atom sm \<sharp> (x,y,z,sn)"
          "atom sn \<sharp> (x,y,z)"
           "atom z \<sharp> (x,y)"
           "atom y \<sharp> x"
     by (metis obtain_fresh)
   with Eq show ?case
     apply (intro PfP_Single_I[of A])
     apply (subst PrfP.simps[of l _ sl m n sm sn]; auto?)
     apply (rule Ex_I[of _ _ _ "\<lceil>A\<rceil>"]; auto?)
      apply (rule Mem_SUCC_E[OF Mem_Zero_E])
      apply (rule Mem_Eats_I2)
      apply (rule HPair_cong[OF Assume Refl])
     apply (rule Disj_I1)
     apply (unfold AxiomP_def; simp)
     apply (rule Disj_I2[OF Disj_I2[OF Disj_I1]])
     apply (auto simp: equality_axioms_def
       intro: Disj_I1 Disj_I2[OF Disj_I1] Disj_I2[OF Disj_I2[OF Disj_I1]] Disj_I2[OF Disj_I2[OF Disj_I2]])
     done
 next
   case (Spec A)
   obtain l::name and sl::name and m::name and n::name and
          sm::name and sn::name and x::name and y::name and z::name
          where atoms:
           "atom l \<sharp> (x,y,z,sl,m,n,sm,sn)"
          "atom sl \<sharp> (x,y,z,m,n,sm,sn)"
           "atom m \<sharp> (x,y,z,n,sm,sn)"
           "atom n \<sharp> (x,y,z,sm,sn)"
          "atom sm \<sharp> (x,y,z,sn)"
          "atom sn \<sharp> (x,y,z)"
           "atom z \<sharp> (x,y)"
           "atom y \<sharp> x"
     by (metis obtain_fresh)
   let ?vs = "(x,y,z,l,sl,m,n,sm,sn)"
   from Spec atoms show ?case
     apply (intro PfP_Single_I[of A])
     apply (subst PrfP.simps[of l _ sl m n sm sn]; auto?)
     apply (rule Ex_I[of _ _ _ "\<lceil>A\<rceil>"]; auto?)
      apply (rule Mem_SUCC_E[OF Mem_Zero_E])
      apply (rule Mem_Eats_I2)
      apply (rule HPair_cong[OF Assume Refl])
     apply (rule Disj_I1)
     apply (unfold AxiomP_def; simp)
     apply (rule Disj_I2[OF Disj_I2[OF Disj_I2[OF Disj_I2[OF Disj_I1]]]])
     subgoal premises prems
     using prems proof (cases A rule: special_axioms.cases)
     case (I X i t)
     let ?vs' = "(?vs, X, i, t)"
       obtain AA XX ii tt res :: name
         where atoms:
           "atom AA \<sharp> (?vs', res, tt, ii, XX)"
           "atom XX \<sharp> (?vs', res, tt, ii)"
           "atom ii \<sharp> (?vs', res, tt)"
           "atom tt \<sharp> (?vs', res)"
           "atom res \<sharp> ?vs'"
         by (metis obtain_fresh)
       with I show ?thesis
         apply (subst Special_axP.simps[of ii _ res tt AA XX]; simp?)
         apply (rule Ex_I[of _ _ _ "\<lceil>Var i\<rceil>"]; auto?)
         apply (rule Ex_I[of _ _ _ "\<lceil>X\<rceil>"]; auto?)
         apply (rule Ex_I[of _ _ _ "quot_dbfm (trans_fm [i] X)"]; auto?)
         apply (rule Ex_I[of _ _ _ "\<lceil>t\<rceil>"]; auto?)
         apply (rule Ex_I[of _ _ _ "\<lceil>X(i::=t)\<rceil>"]; auto?)
             apply (auto simp: TermP_quot[THEN thin0] FormP_quot[THEN thin0]
                SubstFormP[THEN thin0] AbstFormP[THEN thin0]
                quot_Ex quot_Disj quot_Neg vquot_fm_def)
         done
     qed
     done
 next
   case (HF A)
   obtain l::name and sl::name and m::name and n::name and
          sm::name and sn::name and x::name and y::name and z::name
          where atoms:
           "atom l \<sharp> (x,y,z,sl,m,n,sm,sn)"
          "atom sl \<sharp> (x,y,z,m,n,sm,sn)"
           "atom m \<sharp> (x,y,z,n,sm,sn)"
           "atom n \<sharp> (x,y,z,sm,sn)"
          "atom sm \<sharp> (x,y,z,sn)"
          "atom sn \<sharp> (x,y,z)"
           "atom z \<sharp> (x,y)"
           "atom y \<sharp> x"
     by (metis obtain_fresh)
   with HF show ?case
     apply (intro PfP_Single_I[of A])
     apply (subst PrfP.simps[of l _ sl m n sm sn]; auto?)
     apply (rule Ex_I[of _ _ _ "\<lceil>A\<rceil>"]; auto?)
      apply (rule Mem_SUCC_E[OF Mem_Zero_E])
      apply (rule Mem_Eats_I2)
      apply (rule HPair_cong[OF Assume Refl])
     apply (rule Disj_I1)
     apply (unfold AxiomP_def; simp)
     apply (rule Disj_I2[OF Disj_I2[OF Disj_I2[OF Disj_I1]]])
     apply (auto simp: HF_axioms_def intro: Disj_I1 Disj_I2)
     done
 next
   case (Ind A)
   obtain l::name and sl::name and m::name and n::name and
          sm::name and sn::name and x::name and y::name and z::name
          where atoms:
           "atom l \<sharp> (x,y,z,sl,m,n,sm,sn)"
          "atom sl \<sharp> (x,y,z,m,n,sm,sn)"
           "atom m \<sharp> (x,y,z,n,sm,sn)"
           "atom n \<sharp> (x,y,z,sm,sn)"
          "atom sm \<sharp> (x,y,z,sn)"
          "atom sn \<sharp> (x,y,z)"
           "atom z \<sharp> (x,y)"
           "atom y \<sharp> x"
     by (metis obtain_fresh)
   let ?vs = "(x,y,z,l,sl,m,n,sm,sn)"
   from Ind atoms show ?case
     apply (intro PfP_Single_I[of A])
     apply (subst PrfP.simps[of l _ sl m n sm sn]; auto?)
     apply (rule Ex_I[of _ _ _ "\<lceil>A\<rceil>"]; auto?)
      apply (rule Mem_SUCC_E[OF Mem_Zero_E])
      apply (rule Mem_Eats_I2)
      apply (rule HPair_cong[OF Assume Refl])
     apply (rule Disj_I1)
     apply (unfold AxiomP_def; simp)
     apply (rule Disj_I2[OF Disj_I2[OF Disj_I2[OF  Disj_I2[OF Disj_I2]]]])
     subgoal premises prems
     using prems proof (cases A rule: induction_axioms.cases)
       case (ind j i X)
       let ?vs' = "(?vs, X, i, j)"
       obtain ax allvw allw xevw xw x0 xa w v :: name
         where atoms:
           "atom ax \<sharp> (?vs', v, w, xa, x0, xw, xevw, allw, allvw)"
           "atom allvw \<sharp> (?vs', v, w, xa, x0, xw, xevw, allw)"
           "atom allw \<sharp> (?vs', v, w, xa, x0, xw, xevw)"
           "atom xevw \<sharp> (?vs', v, w, xa, x0, xw)"
           "atom xw \<sharp> (?vs', v, w, xa, x0)"
           "atom x0 \<sharp> (?vs', v, w, xa)"
           "atom xa \<sharp> (?vs', v, w)"
           "atom w \<sharp> (?vs', v)"
           "atom v \<sharp> (?vs')"
         by (metis obtain_fresh)
       with ind(2) show ?thesis
         unfolding ind(1)
         apply (subst Induction_axP.simps[of ax _ allvw allw xevw xw x0 xa w v])
                  apply simp_all
         apply (rule Ex_I[of _ _ _ "\<lceil>Var i\<rceil>"]; auto?)
         apply (rule Ex_I[of _ _ _ "\<lceil>Var j\<rceil>"]; auto?)
         apply (rule Ex_I[of _ _ _ "\<lceil>X\<rceil>"]; auto?)
         apply (rule Ex_I[of _ _ _ "\<lceil>X(i::=Zero)\<rceil>"]; auto?)
         apply (rule Ex_I[of _ _ _ "\<lceil>X(i::=Var j)\<rceil>"]; auto?)
         apply (rule Ex_I[of _ _ _ "\<lceil>X(i::=Eats (Var i) (Var j))\<rceil>"]; auto?)
         apply (rule Ex_I[of _ _ _ "quot_dbfm (trans_fm [j] (X IMP (X(i::= Var j) IMP X(i::= Eats(Var i)(Var j)))))"]; auto?)
         apply (rule Ex_I[of _ _ _ "Q_All (quot_dbfm (trans_fm [j,i] (X IMP (X(i::= Var j) IMP X(i::= Eats(Var i)(Var j))))))"]; auto?)
         apply (rule Ex_I[of _ _ _ "quot_dbfm (trans_fm [i] X)"]; auto?)
         subgoal
           apply (rule thin0)
           apply (rule OrdNotEqP_I)
             apply (auto simp: quot_Var ORD_OF_EQ_diff intro!: OrdP_SUCC_I0[THEN cut1])
           done
         subgoal
           by (auto simp: VarNonOccFormP.simps FormP_quot[THEN thin0] SubstFormP_trivial[THEN thin0])
         subgoal
           by (rule SubstFormP_Zero[THEN thin0])
         subgoal
           by (rule SubstFormP[THEN thin0])
         subgoal
           unfolding quot_Eats[symmetric] One_nat_def[symmetric]
           by (rule SubstFormP[THEN thin0])
         subgoal
           unfolding quot_simps[symmetric] quot_dbfm.simps[symmetric] trans_fm.simps[symmetric]
           by (rule AbstFormP[THEN thin0])
         subgoal
           by (auto simp only: quot_simps[symmetric] quot_dbfm.simps[symmetric] trans_fm.simps[symmetric]
             fresh_Cons fresh_Nil fresh_Pair trans_fm.simps(5)[symmetric, of j "[]"]
             quot_fm_def[symmetric] intro!: AbstFormP[THEN thin0])
         subgoal
           unfolding quot_simps[symmetric] quot_dbfm.simps[symmetric] trans_fm.simps[symmetric]
           by (rule AbstFormP[THEN thin0])
         subgoal
           by (auto simp: quot_simps trans_fm.simps(5)[of j "[i]"]
             fresh_Cons fresh_Pair)
         done
     qed
     done
 next
   case (MP H A B H')
   then show ?case
     by (auto elim!: PfP_implies_ModPon_PfP_quot)
 next
   case (Exists A B i)
   obtain a x y z::name
     where atoms:
       "atom a \<sharp> (i,x,y,z)"
       "atom z \<sharp> (i,x,y)"
       "atom y \<sharp> (i,x)"
       "atom x \<sharp> i"
     by (metis obtain_fresh)
   with Exists show ?case
     apply (auto elim!: PfP_inference [THEN cut3] intro!: PfP_extra Disj_I2[OF Disj_I1])
     apply (subst ExistsP.simps[of x _ _ a y z]; (auto simp: VarNonOccFormP.simps)?)
     apply (rule Ex_I[of _ _ _ "\<lceil>A\<rceil>"]; auto?)
     apply (rule Ex_I[of _ _ _ "quot_dbfm (trans_fm [i] A)"]; auto?)
     apply (rule Ex_I[of _ _ _ "\<lceil>B\<rceil>"]; auto?)
     apply (rule Ex_I[of _ _ _ "\<lceil>Var i\<rceil>"]; auto?)
          apply (auto simp: FormP_quot quot_Disj quot_Neg quot_Ex SubstFormP_trivial AbstFormP)
     done
 qed
 
 end
 
diff --git a/thys/Goedel_HFSet_Semanticless/Pf_Predicates.thy b/thys/Goedel_HFSet_Semanticless/Pf_Predicates.thy
--- a/thys/Goedel_HFSet_Semanticless/Pf_Predicates.thy
+++ b/thys/Goedel_HFSet_Semanticless/Pf_Predicates.thy
@@ -1,331 +1,331 @@
-chapter{*Formalizing Provability*}
+chapter\<open>Formalizing Provability\<close>
 
 theory Pf_Predicates
 imports Coding_Predicates
 begin
 
-section {* Section 4 Predicates (Leading up to Pf)*}
+section \<open>Section 4 Predicates (Leading up to Pf)\<close>
 
-subsection {*The predicate @{text SentP}, for the Sentiential (Boolean) Axioms*}
+subsection \<open>The predicate \<open>SentP\<close>, for the Sentiential (Boolean) Axioms\<close>
 
 nominal_function SentP :: "tm \<Rightarrow> fm"
   where "\<lbrakk>atom y \<sharp> (z,w,x); atom z \<sharp> (w,x); atom w \<sharp> x\<rbrakk> \<Longrightarrow>
     SentP x = Ex y (Ex z (Ex w (FormP (Var y) AND FormP (Var z) AND FormP (Var w) AND
                ( (x EQ Q_Imp (Var y) (Var y)) OR
                  (x EQ Q_Imp (Var y) (Q_Disj (Var y) (Var z)) OR
                  (x EQ Q_Imp (Q_Disj (Var y) (Var y)) (Var y)) OR
                  (x EQ Q_Imp (Q_Disj (Var y) (Q_Disj (Var z) (Var w)))
                              (Q_Disj (Q_Disj (Var y) (Var z)) (Var w))) OR
                  (x EQ Q_Imp (Q_Disj (Var y) (Var z))
                              (Q_Imp (Q_Disj (Q_Neg (Var y)) (Var w)) (Q_Disj (Var z) (Var w)))))))))"
   by (auto simp: eqvt_def SentP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
  shows SentP_fresh_iff [simp]: "a \<sharp> SentP x \<longleftrightarrow> a \<sharp> x"                  (is ?thesis1)
    and SentP_sf [iff]:         "Sigma_fm (SentP x)"                     (is ?thsf)
 proof -
   obtain y::name and z::name and w::name  where "atom y \<sharp> (z,w,x)" "atom z \<sharp> (w,x)" "atom w \<sharp> x"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf
     by auto
 qed
 
-subsection {*The predicate @{text Equality_axP}, for the Equality Axioms*}
+subsection \<open>The predicate \<open>Equality_axP\<close>, for the Equality Axioms\<close>
 
 function Equality_axP :: "tm \<Rightarrow> fm"
   where "Equality_axP x =
     x EQ \<lceil>refl_ax\<rceil> OR x EQ \<lceil>eq_cong_ax\<rceil> OR x EQ \<lceil>mem_cong_ax\<rceil> OR x EQ \<lceil>eats_cong_ax\<rceil>"
 by auto
 
 termination
   by lexicographic_order
 
-subsection {*The predicate @{text HF_axP}, for the HF Axioms*}
+subsection \<open>The predicate \<open>HF_axP\<close>, for the HF Axioms\<close>
 
 function HF_axP :: "tm \<Rightarrow> fm"
   where "HF_axP x = x EQ \<lceil>HF1\<rceil> OR x EQ \<lceil>HF2\<rceil>"
 by auto
 
 termination
   by lexicographic_order
 
 lemma HF_axP_sf [iff]: "Sigma_fm (HF_axP t)"
   by auto
 
 
-subsection {*The specialisation axioms*}
+subsection \<open>The specialisation axioms\<close>
 
-subsubsection {*Defining the syntax*}
+subsubsection \<open>Defining the syntax\<close>
 
 nominal_function Special_axP :: "tm \<Rightarrow> fm" where
   "\<lbrakk>atom v \<sharp> (p,sx,y,ax,x); atom x \<sharp> (p,sx,y,ax);
     atom ax \<sharp> (p,sx,y); atom y \<sharp> (p,sx); atom sx \<sharp> p\<rbrakk> \<Longrightarrow>
   Special_axP p = Ex v (Ex x (Ex ax (Ex y (Ex sx
                    (FormP (Var x) AND VarP (Var v) AND TermP (Var y) AND
                     AbstFormP (Var v) Zero (Var x) (Var ax) AND
                     SubstFormP (Var v) (Var y) (Var x) (Var sx) AND
                     p EQ Q_Imp (Var sx) (Q_Ex (Var ax)))))))"
   by (auto simp: eqvt_def Special_axP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
  shows Special_axP_fresh_iff [simp]: "a \<sharp> Special_axP p \<longleftrightarrow> a \<sharp> p" (is ?thesis1)
    and Special_axP_sf [iff]:       "Sigma_fm (Special_axP p)" (is ?thesis3)
 proof -
   obtain v::name and x::name and ax::name and y::name and sx::name
     where "atom v \<sharp> (p,sx,y,ax,x)" "atom x \<sharp> (p,sx,y,ax)"
           "atom ax \<sharp> (p,sx,y)" "atom y \<sharp> (p,sx)" "atom sx \<sharp> p"
     by (metis obtain_fresh)
   thus ?thesis1 ?thesis3
     by auto
 qed
 
-subsection {*The induction axioms*}
+subsection \<open>The induction axioms\<close>
 
-subsubsection {*Defining the syntax*}
+subsubsection \<open>Defining the syntax\<close>
 
 nominal_function Induction_axP :: "tm \<Rightarrow> fm" where
   "\<lbrakk>atom ax \<sharp> (p,v,w,x,x0,xw,xevw,allw,allvw);
     atom allvw \<sharp> (p,v,w,x,x0,xw,xevw,allw); atom allw \<sharp> (p,v,w,x,x0,xw,xevw);
     atom xevw \<sharp> (p,v,w,x,x0,xw); atom xw \<sharp> (p,v,w,x,x0);
     atom x0 \<sharp> (p,v,w,x); atom x \<sharp> (p,v,w);
     atom w \<sharp> (p,v); atom v \<sharp> p\<rbrakk> \<Longrightarrow>
   Induction_axP p = Ex v (Ex w (Ex x (Ex x0 (Ex xw (Ex xevw (Ex allw (Ex allvw (Ex ax
                ((Var v NEQ Var w) AND VarNonOccFormP (Var w) (Var x) AND
                 SubstFormP (Var v) Zero (Var x) (Var x0) AND
                 SubstFormP (Var v) (Var w) (Var x) (Var xw) AND
                 SubstFormP (Var v) (Q_Eats (Var v) (Var w)) (Var x) (Var xevw) AND
                 AbstFormP (Var w) Zero (Q_Imp (Var x) (Q_Imp (Var xw) (Var xevw))) (Var allw) AND
                 AbstFormP (Var v) Zero (Q_All (Var allw)) (Var allvw) AND
                 AbstFormP (Var v) Zero (Var x) (Var ax) AND
                 p EQ Q_Imp (Var x0) (Q_Imp (Q_All (Var allvw)) (Q_All (Var ax))))))))))))"
   by (auto simp: eqvt_def Induction_axP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
  shows Induction_axP_fresh_iff [simp]: "a \<sharp> Induction_axP p \<longleftrightarrow> a \<sharp> p" (is ?thesis1)
    and Induction_axP_sf [iff]: "Sigma_fm (Induction_axP p)" (is ?thesis3)
 proof -
   obtain v::name and w::name and x::name and x0::name and xw::name and xevw::name
                  and allw::name and allvw::name and ax::name
     where atoms: "atom ax \<sharp> (p,v,w,x,x0,xw,xevw,allw,allvw)"
                  "atom allvw \<sharp> (p,v,w,x,x0,xw,xevw,allw)" "atom allw \<sharp> (p,v,w,x,x0,xw,xevw)"
                  "atom xevw \<sharp> (p,v,w,x,x0,xw)" "atom xw \<sharp> (p,v,w,x,x0)" "atom x0 \<sharp> (p,v,w,x)"
                  "atom x \<sharp> (p,v,w)" "atom w \<sharp> (p,v)" "atom v \<sharp> p"
     by (metis obtain_fresh)
   thus ?thesis1 ?thesis3
     by auto
 qed
 
-subsection {*The predicate @{text AxiomP}, for any Axioms*}
+subsection \<open>The predicate \<open>AxiomP\<close>, for any Axioms\<close>
 
 definition AxiomP :: "tm \<Rightarrow> fm"
   where "AxiomP x \<equiv> x EQ \<lceil>extra_axiom\<rceil> OR SentP x OR Equality_axP x OR
                     HF_axP x OR Special_axP x OR Induction_axP x"
 
 lemma AxiomP_I:
   "{} \<turnstile> AxiomP \<lceil>extra_axiom\<rceil>"
   "{} \<turnstile> SentP x \<Longrightarrow> {} \<turnstile> AxiomP x"
   "{} \<turnstile> Equality_axP x \<Longrightarrow> {} \<turnstile> AxiomP x"
   "{} \<turnstile> HF_axP x \<Longrightarrow> {} \<turnstile> AxiomP x"
   "{} \<turnstile> Special_axP x \<Longrightarrow> {} \<turnstile> AxiomP x"
   "{} \<turnstile> Induction_axP x \<Longrightarrow> {} \<turnstile> AxiomP x"
   unfolding AxiomP_def
   by (rule Disj_I1, rule Refl,
       rule Disj_I2, rule Disj_I1, assumption,
       rule Disj_I2, rule Disj_I2, rule Disj_I1, assumption,
       rule Disj_I2, rule Disj_I2, rule Disj_I2, rule Disj_I1, assumption,
       rule Disj_I2, rule Disj_I2, rule Disj_I2, rule Disj_I2, rule Disj_I1, assumption,
       rule Disj_I2, rule Disj_I2, rule Disj_I2, rule Disj_I2, rule Disj_I2, assumption)
 
 lemma AxiomP_eqvt [eqvt]: "(p \<bullet> AxiomP x) = AxiomP (p \<bullet> x)"
   by (simp add: AxiomP_def)
 
 lemma AxiomP_fresh_iff [simp]: "a \<sharp> AxiomP x \<longleftrightarrow> a \<sharp> x"
   by (auto simp: AxiomP_def)
 
 lemma AxiomP_sf [iff]: "Sigma_fm (AxiomP t)"
   by (auto simp: AxiomP_def)
 
 
-subsection {*The predicate @{text ModPonP}, for the inference rule Modus Ponens*}
+subsection \<open>The predicate \<open>ModPonP\<close>, for the inference rule Modus Ponens\<close>
 
 definition ModPonP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "ModPonP x y z = (y EQ Q_Imp x z)"
 
 lemma ModPonP_eqvt [eqvt]: "(p \<bullet> ModPonP x y z) = ModPonP (p \<bullet> x) (p \<bullet> y) (p \<bullet> z)"
   by (simp add: ModPonP_def)
 
 lemma ModPonP_fresh_iff [simp]: "a \<sharp> ModPonP x y z \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> y \<and> a \<sharp> z"
   by (auto simp: ModPonP_def)
 
 lemma ModPonP_sf [iff]: "Sigma_fm (ModPonP t u v)"
   by (auto simp: ModPonP_def)
 
 lemma ModPonP_subst [simp]:
   "(ModPonP t u v)(i::=w) = ModPonP (subst i w t) (subst i w u) (subst i w v)"
   by (auto simp: ModPonP_def)
 
 
-subsection {*The predicate @{text ExistsP}, for the existential rule *}
-subsubsection {*Definition*}
+subsection \<open>The predicate \<open>ExistsP\<close>, for the existential rule\<close>
+subsubsection \<open>Definition\<close>
 
 (*  "\<turnstile> A IMP B \<Longrightarrow> atom i \<sharp> B \<Longrightarrow>  \<turnstile> (Ex i A) IMP B" *)
 nominal_function ExistsP :: "tm \<Rightarrow> tm \<Rightarrow> fm" where
   "\<lbrakk>atom x \<sharp> (p,q,v,y,x'); atom x' \<sharp> (p,q,v,y);
     atom y \<sharp> (p,q,v); atom v \<sharp> (p,q)\<rbrakk> \<Longrightarrow>
   ExistsP p q = Ex x (Ex x' (Ex y (Ex v (FormP (Var x) AND
                     VarNonOccFormP (Var v) (Var y) AND
                     AbstFormP (Var v) Zero (Var x) (Var x') AND
                     p EQ Q_Imp (Var x) (Var y) AND
                     q EQ Q_Imp (Q_Ex (Var x')) (Var y)))))"
   by (auto simp: eqvt_def ExistsP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
  shows ExistsP_fresh_iff [simp]: "a \<sharp> ExistsP p q \<longleftrightarrow> a \<sharp> p \<and> a \<sharp> q"    (is ?thesis1)
    and ExistsP_sf [iff]:       "Sigma_fm (ExistsP p q)"   (is ?thesis3)
 proof -
   obtain x::name and x'::name and y::name and v::name
     where "atom x \<sharp> (p,q,v,y,x')"  "atom x' \<sharp> (p,q,v,y)" "atom y \<sharp> (p,q,v)"  "atom v \<sharp> (p,q)"
     by (metis obtain_fresh)
   thus ?thesis1 ?thesis3
     by auto
 qed
 
 lemma ExistsP_subst [simp]: "(ExistsP p q)(j::=w) = ExistsP (subst j w p) (subst j w q)"
 proof -
   obtain x::name and x'::name and y::name and v::name
     where "atom x \<sharp> (j,w,p,q,v,y,x')"   "atom x' \<sharp> (j,w,p,q,v,y)"
           "atom y \<sharp> (j,w,p,q,v)"   "atom v \<sharp> (j,w,p,q)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: ExistsP.simps [of x _ _ x' y v])
 qed
 
-subsection {*The predicate @{text SubstP}, for the substitution rule *}
+subsection \<open>The predicate \<open>SubstP\<close>, for the substitution rule\<close>
 
-text{*Although the substitution rule is derivable in the calculus, the derivation is
+text\<open>Although the substitution rule is derivable in the calculus, the derivation is
 too complicated to reproduce within the proof function. It is much easier to
 provide it as an immediate inference step, justifying its soundness in terms
-of other inference rules.*}
+of other inference rules.\<close>
 
-subsubsection {*Definition*}
+subsubsection \<open>Definition\<close>
 
 nominal_function SubstP :: "tm \<Rightarrow> tm \<Rightarrow> fm" where
   "\<lbrakk>atom u \<sharp> (p,q,v); atom v \<sharp> (p,q)\<rbrakk> \<Longrightarrow>
    SubstP p q = Ex v (Ex u (SubstFormP (Var v) (Var u) p q))"
   by (auto simp: eqvt_def SubstP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
  shows SubstP_fresh_iff [simp]: "a \<sharp> SubstP p q \<longleftrightarrow> a \<sharp> p \<and> a \<sharp> q"        (is ?thesis1)
    and SubstP_sf [iff]: "Sigma_fm (SubstP p q)"                           (is ?thesis3)
 proof -
   obtain u::name and v::name  where "atom u \<sharp> (p,q,v)" "atom v \<sharp> (p,q)"
     by (metis obtain_fresh)
   thus ?thesis1 ?thesis3
     by auto
 qed
 
 lemma SubstP_subst [simp]: "(SubstP p q)(j::=w) = SubstP (subst j w p) (subst j w q)"
 proof -
   obtain u::name and v::name  where "atom u \<sharp> (j,w,p,q,v)"  "atom v \<sharp> (j,w,p,q)"
     by (metis obtain_fresh)
   thus ?thesis
     by (simp add: SubstP.simps [of u _ _ v])
 qed
 
 
-subsection {*The predicate @{text PrfP}*}
+subsection \<open>The predicate \<open>PrfP\<close>\<close>
 
 (*Prf(s,k,t) \<equiv> LstSeq(s,k,t) \<and> (\<forall>n\<in>k)[Sent (s n) \<or> (\<exists>m,l\<in>n)[ModPon (s m) (s l) (s n)]]*)
 nominal_function PrfP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom l \<sharp> (s,sl,m,n,sm,sn); atom sl \<sharp> (s,m,n,sm,sn);
           atom m \<sharp> (s,n,sm,sn); atom n \<sharp> (s,k,sm,sn);
           atom sm \<sharp> (s,sn); atom sn \<sharp> (s)\<rbrakk> \<Longrightarrow>
     PrfP s k t =
       LstSeqP s k t AND
       All2 n (SUCC k) (Ex sn (HPair (Var n) (Var sn) IN s AND (AxiomP (Var sn) OR
                 Ex m (Ex l (Ex sm (Ex sl (Var m IN Var n AND Var l IN Var n AND
                        HPair (Var m) (Var sm) IN s AND HPair (Var l) (Var sl) IN s AND
                        (ModPonP (Var sm) (Var sl) (Var sn) OR
                         ExistsP (Var sm) (Var sn) OR
                         SubstP (Var sm) (Var sn)))))))))"
   by (auto simp: eqvt_def PrfP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows PrfP_fresh_iff [simp]: "a \<sharp> PrfP s k t \<longleftrightarrow> a \<sharp> s \<and> a \<sharp> k \<and> a \<sharp> t"      (is ?thesis1)
   and PrfP_imp_OrdP [simp]:    "{PrfP s k t} \<turnstile> OrdP k"         (is ?thord)
   and PrfP_imp_LstSeqP [simp]: "{PrfP s k t} \<turnstile> LstSeqP s k t"  (is ?thlstseq)
   and PrfP_sf [iff]:           "Sigma_fm (PrfP s k t)"         (is ?thsf)
 proof -
   obtain l::name and sl::name and m::name and n::name and sm::name and sn::name
     where atoms: "atom l \<sharp> (s,sl,m,n,sm,sn)"   "atom sl \<sharp> (s,m,n,sm,sn)"
         "atom m \<sharp> (s,n,sm,sn)"   "atom n \<sharp> (s,k,sm,sn)"
         "atom sm \<sharp> (s,sn)"   "atom sn \<sharp> (s)"
     by (metis obtain_fresh)
   thus ?thesis1 ?thord ?thlstseq ?thsf
     by (auto intro: LstSeqP_OrdP)
 qed
 
 lemma PrfP_subst [simp]:
      "(PrfP t u v)(j::=w) = PrfP (subst j w t) (subst j w u) (subst j w v)"
 proof -
   obtain l::name and sl::name and m::name and n::name and sm::name and sn::name
     where "atom l \<sharp> (t,u,v,j,w,sl,m,n,sm,sn)"   "atom sl \<sharp> (t,u,v,j,w,m,n,sm,sn)"
           "atom m \<sharp> (t,u,v,j,w,n,sm,sn)"   "atom n \<sharp> (t,u,v,j,w,sm,sn)"
           "atom sm \<sharp> (t,u,v,j,w,sn)"   "atom sn \<sharp> (t,u,v,j,w)"
     by (metis obtain_fresh)
   thus ?thesis
     by (simp add: PrfP.simps [of l _ sl m n sm sn])
 qed
 
 
-subsection {*The predicate @{text PfP}*}
+subsection \<open>The predicate \<open>PfP\<close>\<close>
 
 nominal_function PfP :: "tm \<Rightarrow> fm"
   where "\<lbrakk>atom k \<sharp> (s,y); atom s \<sharp> y\<rbrakk> \<Longrightarrow>
     PfP y = Ex k (Ex s (PrfP (Var s) (Var k) y))"
   by (auto simp: eqvt_def PfP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
  shows PfP_fresh_iff [simp]: "a \<sharp> PfP y \<longleftrightarrow> a \<sharp> y"           (is ?thesis1)
    and PfP_sf [iff]: "Sigma_fm (PfP y)"                      (is ?thsf)
 proof -
   obtain k::name and s::name where "atom k \<sharp> (s,y)" "atom s \<sharp> y"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf
     by auto
 qed
 
 lemma PfP_subst [simp]: "(PfP t)(j::=w) = PfP (subst j w t)"
 proof -
   obtain k::name and s::name where "atom k \<sharp> (s,t,j,w)" "atom s \<sharp> (t,j,w)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: PfP.simps [of k s])
 qed
 
 lemma ground_PfP [simp]: "ground_fm (PfP y) = ground y"
   by (simp add: ground_aux_def ground_fm_aux_def supp_conv_fresh)
 
 end
 
diff --git a/thys/Goedel_HFSet_Semanticless/Predicates.thy b/thys/Goedel_HFSet_Semanticless/Predicates.thy
--- a/thys/Goedel_HFSet_Semanticless/Predicates.thy
+++ b/thys/Goedel_HFSet_Semanticless/Predicates.thy
@@ -1,1273 +1,1273 @@
-chapter{*Basic Predicates*}
+chapter\<open>Basic Predicates\<close>
 
 theory Predicates
 imports SyntaxN
 begin
 
-section {*The Subset Relation*}
+section \<open>The Subset Relation\<close>
 
 nominal_function Subset :: "tm \<Rightarrow> tm \<Rightarrow> fm"   (infixr "SUBS" 150)
   where "atom z \<sharp> (t, u) \<Longrightarrow> t SUBS u = All2 z t ((Var z) IN u)"
   by (auto simp: eqvt_def Subset_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 declare Subset.simps [simp del]
 
 lemma Subset_fresh_iff [simp]: "a \<sharp> t SUBS u \<longleftrightarrow> a \<sharp> t \<and> a \<sharp> u"
 apply (rule obtain_fresh [where x="(t, u)"])
 apply (subst Subset.simps, auto)
   done
 
 lemma subst_fm_Subset [simp]: "(t SUBS u)(i::=x) = (subst i x t) SUBS (subst i x u)"
 proof -
   obtain j::name where "atom j \<sharp> (i,x,t,u)"
     by (rule obtain_fresh)
   thus ?thesis
     by (auto simp: Subset.simps [of j])
 qed
 
 lemma Subset_I:
   assumes "insert ((Var i) IN t) H \<turnstile> (Var i) IN u" "atom i \<sharp> (t,u)" "\<forall>B \<in> H. atom i \<sharp> B"
   shows "H \<turnstile> t SUBS u"
 by (subst Subset.simps [of i]) (auto simp: assms)
 
 lemma Subset_D:
   assumes major: "H \<turnstile> t SUBS u" and minor: "H \<turnstile> a IN t" shows "H \<turnstile> a IN u"
 proof -
   obtain i::name where i: "atom i \<sharp> (t, u)"
     by (rule obtain_fresh)
   hence "H \<turnstile> (Var i IN t IMP Var i IN u) (i::=a)"
     by (metis Subset.simps major All_D)
   thus ?thesis
     using i  by simp (metis MP_same minor)
 qed
 
 lemma Subset_E: "H \<turnstile> t SUBS u \<Longrightarrow> H \<turnstile> a IN t \<Longrightarrow> insert (a IN u) H \<turnstile> A \<Longrightarrow> H \<turnstile> A"
   by (metis Subset_D cut_same)
 
 lemma Subset_cong: "H \<turnstile> t EQ t' \<Longrightarrow> H \<turnstile> u EQ u' \<Longrightarrow> H \<turnstile> t SUBS u IFF t' SUBS u'"
   by (rule P2_cong) auto
 
 lemma Set_MP: "x SUBS y \<in> H \<Longrightarrow> z IN x \<in> H \<Longrightarrow> insert (z IN y) H \<turnstile> A \<Longrightarrow> H \<turnstile> A"
   by (metis Assume Subset_D cut_same insert_absorb)
 
 lemma Zero_Subset_I [intro!]: "H \<turnstile> Zero SUBS t"
 proof -
   have "{} \<turnstile> Zero SUBS t"
     by (rule obtain_fresh [where x="(Zero,t)"]) (auto intro: Subset_I)
   thus ?thesis
     by (auto intro: thin)
 qed
 
 lemma Zero_SubsetE: "H \<turnstile> A \<Longrightarrow> insert (Zero SUBS X) H \<turnstile> A"
   by (rule thin1)
 
 lemma Subset_Zero_D:
    assumes "H \<turnstile> t SUBS Zero" shows "H \<turnstile> t EQ Zero"
 proof -
   obtain i::name where i [iff]: "atom i \<sharp> t"
     by (rule obtain_fresh)
   have "{t SUBS Zero} \<turnstile> t EQ Zero"
   proof (rule Eq_Zero_I)
     fix A
     show "{Var i IN t, t SUBS Zero} \<turnstile> A"
       by (metis Hyp Subset_D insertI1 thin1 Mem_Zero_E cut1)
   qed auto
   thus ?thesis
     by (metis assms cut1)
 qed
 
 lemma Subset_refl: "H \<turnstile> t SUBS t"
 proof -
   obtain i::name where "atom i \<sharp> t"
     by (rule obtain_fresh)
   thus ?thesis
     by (metis Assume Subset_I empty_iff fresh_Pair thin0)
 qed
 
 lemma Eats_Subset_Iff: "H \<turnstile> Eats x y SUBS z IFF (x SUBS z) AND (y IN z)"
 proof -
   obtain i::name where i: "atom i \<sharp> (x,y,z)"
     by (rule obtain_fresh)
   have "{} \<turnstile> (Eats x y SUBS z) IFF (x SUBS z AND y IN z)"
   proof (rule Iff_I)
     show "{Eats x y SUBS z} \<turnstile> x SUBS z AND y IN z"
     proof (rule Conj_I)
       show "{Eats x y SUBS z} \<turnstile> x SUBS z"
         apply (rule Subset_I [where i=i]) using i
         apply (auto intro: Subset_D Mem_Eats_I1)
         done
     next
       show "{Eats x y SUBS z} \<turnstile> y IN z"
         by (metis Subset_D Assume Mem_Eats_I2 Refl)
     qed
   next
     show "{x SUBS z AND y IN z} \<turnstile> Eats x y SUBS z" using i
       by (auto intro!: Subset_I [where i=i] intro: Subset_D Mem_cong [THEN Iff_MP2_same])
   qed
   thus ?thesis
     by (rule thin0)
 qed
 
 lemma Eats_Subset_I [intro!]: "H \<turnstile> x SUBS z \<Longrightarrow> H \<turnstile> y IN z \<Longrightarrow> H \<turnstile> Eats x y SUBS z"
   by (metis Conj_I Eats_Subset_Iff Iff_MP2_same)
 
 lemma Eats_Subset_E [intro!]:
   "insert (x SUBS z) (insert (y IN z) H) \<turnstile> C \<Longrightarrow> insert (Eats x y SUBS z) H \<turnstile> C"
   by (metis Conj_E Eats_Subset_Iff Iff_MP_left')
 
-text{*A surprising proof: a consequence of @{thm Eats_Subset_Iff} and reflexivity!*}
+text\<open>A surprising proof: a consequence of @{thm Eats_Subset_Iff} and reflexivity!\<close>
 lemma Subset_Eats_I [intro!]: "H \<turnstile> x SUBS Eats x y"
   by (metis Conj_E1 Eats_Subset_Iff Iff_MP_same Subset_refl)
 
 lemma SUCC_Subset_I [intro!]: "H \<turnstile> x SUBS z \<Longrightarrow> H \<turnstile> x IN z \<Longrightarrow> H \<turnstile> SUCC x SUBS z"
   by (metis Eats_Subset_I SUCC_def)
 
 lemma SUCC_Subset_E [intro!]:
   "insert (x SUBS z) (insert (x IN z) H) \<turnstile> C \<Longrightarrow> insert (SUCC x SUBS z) H \<turnstile> C"
   by (metis Eats_Subset_E SUCC_def)
 
 lemma Subset_trans0: "{ a SUBS b, b SUBS c } \<turnstile> a SUBS c"
 proof -
   obtain i::name where [simp]: "atom i \<sharp> (a,b,c)"
     by (rule obtain_fresh)
   show ?thesis
     by (rule Subset_I [of i]) (auto intro: Subset_D)
 qed
 
 lemma Subset_trans: "H \<turnstile> a SUBS b \<Longrightarrow> H \<turnstile> b SUBS c \<Longrightarrow> H \<turnstile> a SUBS c"
   by (metis Subset_trans0 cut2)
 
 lemma Subset_SUCC: "H \<turnstile> a SUBS (SUCC a)"
   by (metis SUCC_def Subset_Eats_I)
 
 lemma All2_Subset_lemma: "atom l \<sharp> (k',k) \<Longrightarrow> {P} \<turnstile> P' \<Longrightarrow> {All2 l k P, k' SUBS k} \<turnstile> All2 l k' P'"
   apply auto
   apply (rule Ex_I [where x = "Var l"])
   apply (auto intro: ContraProve Set_MP cut1)
   done
 
 lemma All2_Subset: "\<lbrakk>H \<turnstile> All2 l k P; H \<turnstile> k' SUBS k; {P} \<turnstile> P'; atom l \<sharp> (k', k)\<rbrakk> \<Longrightarrow> H \<turnstile> All2 l k' P'"
   by (rule cut2 [OF All2_Subset_lemma]) auto
 
 
-section{*Extensionality*}
+section\<open>Extensionality\<close>
 
 lemma Extensionality: "H \<turnstile> x EQ y IFF x SUBS y AND y SUBS x"
 proof -
   obtain i::name and j::name and k::name
     where atoms: "atom i \<sharp> (x,y)"  "atom j \<sharp> (i,x,y)"  "atom k \<sharp> (i,j,y)"
     by (metis obtain_fresh)
   have "{} \<turnstile> (Var i EQ y IFF Var i SUBS y AND y SUBS Var i)" (is "{} \<turnstile> ?scheme")
   proof (rule Ind [of j])
     show "atom j \<sharp> (i, ?scheme)" using atoms
       by simp
   next
     show "{} \<turnstile> ?scheme(i::=Zero)" using atoms
     proof auto
       show "{Zero EQ y} \<turnstile> y SUBS Zero"
         by (rule Subset_cong [OF Assume Refl, THEN Iff_MP_same]) (rule Subset_refl)
     next
       show "{Zero SUBS y, y SUBS Zero} \<turnstile> Zero EQ y"
         by (metis AssumeH(2) Subset_Zero_D Sym)
     qed
   next
     show "{} \<turnstile> All i (All j (?scheme IMP ?scheme(i::=Var j) IMP ?scheme(i::=Eats (Var i) (Var j))))"
       using atoms
       apply auto
       apply (metis Subset_cong [OF Refl Assume, THEN Iff_MP_same] Subset_Eats_I)
       apply (metis Mem_cong [OF Refl Assume, THEN Iff_MP_same] Mem_Eats_I2 Refl)
       apply (metis Subset_cong [OF Assume Refl, THEN Iff_MP_same] Subset_refl)
       apply (rule Eq_Eats_I [of _ k, THEN Sym])
       apply (auto intro: Set_MP [where x=y] Subset_D [where t = "Var i"] Disj_I1 Disj_I2)
       apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same], auto)
       done
   qed
   hence "{} \<turnstile> (Var i EQ y IFF Var i SUBS y AND y SUBS Var i)(i::=x)"
     by (metis Subst emptyE)
   thus ?thesis using atoms
     by (simp add: thin0)
 qed
 
 lemma Equality_I: "H \<turnstile> y SUBS x \<Longrightarrow> H \<turnstile> x SUBS y \<Longrightarrow> H \<turnstile> x EQ y"
   by (metis Conj_I Extensionality Iff_MP2_same)
 
 lemma EQ_imp_SUBS: "insert (t EQ u) H \<turnstile> (t SUBS u)"
 proof -
   have "{t EQ u} \<turnstile> (t SUBS u)"
     by (metis Assume Conj_E Extensionality Iff_MP_left')
 thus ?thesis
   by (metis Assume cut1)
 qed
 
 lemma EQ_imp_SUBS2: "insert (u EQ t) H \<turnstile> (t SUBS u)"
   by (metis EQ_imp_SUBS Sym_L)
 
 lemma Equality_E: "insert (t SUBS u) (insert (u SUBS t) H) \<turnstile> A \<Longrightarrow> insert (t EQ u) H \<turnstile> A"
   by (metis Conj_E Extensionality Iff_MP_left')
 
 
-section {*The Disjointness Relation*}
+section \<open>The Disjointness Relation\<close>
 
-text{*The following predicate is defined in order to prove Lemma 2.3, Foundation*}
+text\<open>The following predicate is defined in order to prove Lemma 2.3, Foundation\<close>
 
 nominal_function Disjoint :: "tm \<Rightarrow> tm \<Rightarrow> fm"
   where "atom z \<sharp> (t, u) \<Longrightarrow> Disjoint t u = All2 z t (Neg ((Var z) IN u))"
   by (auto simp: eqvt_def Disjoint_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 declare Disjoint.simps [simp del]
 
 lemma Disjoint_fresh_iff [simp]: "a \<sharp> Disjoint t u \<longleftrightarrow> a \<sharp> t \<and> a \<sharp> u"
 proof -
   obtain j::name where j: "atom j \<sharp> (a,t,u)"
     by (rule obtain_fresh)
   thus ?thesis
     by (auto simp: Disjoint.simps [of j])
 qed
 
 lemma subst_fm_Disjoint [simp]:
   "(Disjoint t u)(i::=x) = Disjoint (subst i x t) (subst i x u)"
 proof -
   obtain j::name where j: "atom j \<sharp> (i,x,t,u)"
     by (rule obtain_fresh)
   thus ?thesis
     by (auto simp: Disjoint.simps [of j])
 qed
 
 lemma Disjoint_cong: "H \<turnstile> t EQ t' \<Longrightarrow> H \<turnstile> u EQ u' \<Longrightarrow> H \<turnstile> Disjoint t u IFF Disjoint t' u'"
   by (rule P2_cong) auto
 
 lemma Disjoint_I:
   assumes "insert ((Var i) IN t) (insert ((Var i) IN u) H) \<turnstile> Fls"
           "atom i \<sharp> (t,u)" "\<forall>B \<in> H. atom i \<sharp> B"
   shows "H \<turnstile> Disjoint t u"
 by (subst Disjoint.simps [of i]) (auto simp: assms insert_commute)
 
 lemma Disjoint_E:
   assumes major: "H \<turnstile> Disjoint t u" and minor: "H \<turnstile> a IN t" "H \<turnstile> a IN u" shows "H \<turnstile> A"
 proof -
   obtain i::name where i: "atom i \<sharp> (t, u)"
     by (rule obtain_fresh)
   hence "H \<turnstile> (Var i IN t IMP Neg (Var i IN u)) (i::=a)"
     by (metis Disjoint.simps major All_D)
   thus ?thesis using i
     by simp (metis MP_same Neg_D minor)
 qed
 
 lemma Disjoint_commute: "{ Disjoint t u } \<turnstile> Disjoint u t"
 proof -
   obtain i::name where "atom i \<sharp> (t,u)"
     by (rule obtain_fresh)
   thus ?thesis
     by (auto simp: fresh_Pair intro: Disjoint_I Disjoint_E)
 qed
 
 lemma Disjoint_commute_I: "H \<turnstile> Disjoint t u \<Longrightarrow> H \<turnstile> Disjoint u t"
   by (metis Disjoint_commute cut1)
 
 lemma Disjoint_commute_D: "insert (Disjoint t u) H \<turnstile> A \<Longrightarrow> insert (Disjoint u t) H \<turnstile> A"
   by (metis Assume Disjoint_commute_I cut_same insert_commute thin1)
 
 lemma Zero_Disjoint_I1 [iff]: "H \<turnstile> Disjoint Zero t"
 proof -
   obtain i::name where i: "atom i \<sharp> t"
     by (rule obtain_fresh)
   hence "{} \<turnstile> Disjoint Zero t"
     by (auto intro: Disjoint_I [of i])
   thus ?thesis
     by (metis thin0)
 qed
 
 lemma Zero_Disjoint_I2 [iff]: "H \<turnstile> Disjoint t Zero"
   by (metis Disjoint_commute Zero_Disjoint_I1 cut1)
 
 lemma Disjoint_Eats_D1: "{ Disjoint (Eats x y) z } \<turnstile> Disjoint x z"
 proof -
   obtain i::name where i: "atom i \<sharp> (x,y,z)"
     by (rule obtain_fresh)
   show ?thesis
     apply (rule Disjoint_I [of i])
     apply (blast intro: Disjoint_E Mem_Eats_I1)
     using i apply auto
     done
 qed
 
 lemma Disjoint_Eats_D2: "{ Disjoint (Eats x y) z } \<turnstile> Neg(y IN z)"
 proof -
   obtain i::name where i: "atom i \<sharp> (x,y,z)"
     by (rule obtain_fresh)
   show ?thesis
     by (force intro: Disjoint_E [THEN rotate2] Mem_Eats_I2)
 qed
 
 lemma Disjoint_Eats_E:
   "insert (Disjoint x z) (insert (Neg(y IN z)) H) \<turnstile> A \<Longrightarrow> insert (Disjoint (Eats x y) z) H \<turnstile> A"
   apply (rule cut_same [OF cut1 [OF Disjoint_Eats_D2, OF Assume]])
   apply (rule cut_same [OF cut1 [OF Disjoint_Eats_D1, OF Hyp]])
   apply (auto intro: thin)
   done
 
 lemma Disjoint_Eats_E2:
   "insert (Disjoint z x) (insert (Neg(y IN z)) H) \<turnstile> A \<Longrightarrow> insert (Disjoint z (Eats x y)) H \<turnstile> A"
   by (metis Disjoint_Eats_E Disjoint_commute_D)
 
 lemma Disjoint_Eats_Imp: "{ Disjoint x z, Neg(y IN z) } \<turnstile> Disjoint (Eats x y) z"
 proof -
   obtain i::name where"atom i \<sharp> (x,y,z)"
     by (rule obtain_fresh)
   then show ?thesis
     by (auto intro: Disjoint_I [of i]  Disjoint_E [THEN rotate3]
                     Mem_cong [OF Assume Refl, THEN Iff_MP_same])
 qed
 
 lemma Disjoint_Eats_I [intro!]: "H \<turnstile> Disjoint x z \<Longrightarrow> insert (y IN z) H \<turnstile> Fls \<Longrightarrow> H \<turnstile> Disjoint (Eats x y) z"
   by (metis Neg_I cut2 [OF Disjoint_Eats_Imp])
 
 lemma Disjoint_Eats_I2 [intro!]: "H \<turnstile> Disjoint z x \<Longrightarrow> insert (y IN z) H \<turnstile> Fls \<Longrightarrow> H \<turnstile> Disjoint z (Eats x y)"
   by (metis Disjoint_Eats_I Disjoint_commute cut1)
 
 
-section {*The Foundation Theorem*}
+section \<open>The Foundation Theorem\<close>
 
 lemma Foundation_lemma:
   assumes i: "atom i \<sharp> z"
   shows "{ All2 i z (Neg (Disjoint (Var i) z)) } \<turnstile> Neg (Var i IN z) AND Disjoint (Var i) z"
 proof -
   obtain j::name  where j: "atom j \<sharp> (z,i)"
     by (metis obtain_fresh)
   show ?thesis
     apply (rule Ind [of j]) using i j
     apply auto
     apply (rule Ex_I [where x=Zero], auto)
     apply (rule Ex_I [where x="Eats (Var i) (Var j)"], auto)
     apply (metis ContraAssume insertI1 insert_commute)
     apply (metis ContraProve Disjoint_Eats_Imp rotate2 thin1)
     apply (metis Assume Disj_I1 anti_deduction rotate3)
     done
 qed
 
 theorem Foundation: "atom i \<sharp> z \<Longrightarrow> {} \<turnstile> All2 i z (Neg (Disjoint (Var i) z)) IMP z EQ Zero"
   apply auto
   apply (rule Eq_Zero_I)
   apply (rule cut_same [where A = "(Neg ((Var i) IN z) AND Disjoint (Var i) z)"])
   apply (rule Foundation_lemma [THEN cut1], auto)
   done
 
 lemma Mem_Neg_refl: "{} \<turnstile> Neg (x IN x)"
 proof -
   obtain i::name where i: "atom i \<sharp> x"
     by (metis obtain_fresh)
   have "{} \<turnstile> Disjoint x (Eats Zero x)"
     apply (rule cut_same [OF Foundation [where z = "Eats Zero x"]]) using i
     apply auto
     apply (rule cut_same [where A = "Disjoint x (Eats Zero x)"])
     apply (metis Assume thin1 Disjoint_cong [OF Assume Refl, THEN Iff_MP_same])
     apply (metis Assume AssumeH(4) Disjoint_E Mem_Eats_I2 Refl)
     done
   thus ?thesis
     by (metis Disjoint_Eats_D2 Disjoint_commute cut_same)
 qed
 
 lemma Mem_refl_E [intro!]: "insert (x IN x) H \<turnstile> A"
   by (metis Disj_I1 Mem_Neg_refl anti_deduction thin0)
 
 lemma Mem_non_refl: assumes "H \<turnstile> x IN x" shows "H \<turnstile> A"
   by (metis Mem_refl_E assms cut_same)
 
 lemma Mem_Neg_sym: "{ x IN y, y IN x } \<turnstile> Fls"
 proof -
   obtain i::name where i: "atom i \<sharp> (x,y)"
     by (metis obtain_fresh)
   have "{} \<turnstile> Disjoint x (Eats Zero y) OR Disjoint y (Eats Zero x)"
     apply (rule cut_same [OF Foundation [where i=i and z = "Eats (Eats Zero y) x"]]) using i
     apply (auto intro!: Disjoint_Eats_E2 [THEN rotate2])
     apply (rule Disj_I2, auto)
     apply (metis Assume EQ_imp_SUBS2 Subset_D insert_commute)
     apply (blast intro!: Disj_I1 Disjoint_cong [OF Hyp Refl, THEN Iff_MP_same])
     done
   thus ?thesis
     by (auto intro: cut0 Disjoint_Eats_E2)
 qed
 
 lemma Mem_not_sym: "insert (x IN y) (insert (y IN x) H) \<turnstile> A"
   by (rule cut_thin [OF Mem_Neg_sym]) auto
 
 
-section {*The Ordinal Property*}
+section \<open>The Ordinal Property\<close>
 
 nominal_function OrdP :: "tm \<Rightarrow> fm"
   where "\<lbrakk>atom y \<sharp> (x, z); atom z \<sharp> x\<rbrakk> \<Longrightarrow>
     OrdP x = All2 y x ((Var y) SUBS x AND  All2 z (Var y) ((Var z) SUBS (Var y)))"
   by (auto simp: eqvt_def OrdP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
  shows OrdP_fresh_iff [simp]: "a \<sharp> OrdP x \<longleftrightarrow> a \<sharp> x"       (is ?thesis1)
 proof -
   obtain z::name and y::name where "atom z \<sharp> x" "atom y \<sharp> (x, z)"
     by (metis obtain_fresh)
   thus ?thesis1
     by (auto simp: OrdP.simps [of y _ z] Ord_def Transset_def)
 qed
 
 lemma subst_fm_OrdP [simp]: "(OrdP t)(i::=x) = OrdP (subst i x t)"
 proof -
   obtain z::name and y::name where "atom z \<sharp> (t,i,x)" "atom y \<sharp> (t,i,x,z)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: OrdP.simps [of y _ z])
 qed
 
 lemma OrdP_cong: "H \<turnstile> x EQ x' \<Longrightarrow> H \<turnstile> OrdP x IFF OrdP x'"
   by (rule P1_cong) auto
 
 lemma OrdP_Mem_lemma:
   assumes z: "atom z \<sharp> (k,l)" and l: "insert (OrdP k) H \<turnstile> l IN k"
   shows "insert (OrdP k) H \<turnstile> l SUBS k AND All2 z l (Var z SUBS l)"
 proof -
   obtain y::name where y: "atom y \<sharp> (k,l,z)"
     by (metis obtain_fresh)
   have "insert (OrdP k) H
         \<turnstile> (Var y IN k IMP (Var y SUBS k AND All2 z (Var y) (Var z SUBS Var y)))(y::=l)"
     by (rule All_D) (simp add: OrdP.simps [of y _ z] y z Assume)
   also have "... = l IN k IMP (l SUBS k AND All2 z l (Var z SUBS l))"
     using y z  by simp
   finally show ?thesis
     by (metis MP_same l)
 qed
 
 lemma OrdP_Mem_E:
   assumes "atom z \<sharp> (k,l)"
           "insert (OrdP k) H \<turnstile> l IN k"
           "insert (l SUBS k) (insert (All2 z l (Var z SUBS l)) H) \<turnstile> A"
   shows "insert (OrdP k) H \<turnstile> A"
   apply (rule OrdP_Mem_lemma [THEN cut_same])
   apply (auto simp: insert_commute)
   apply (blast intro: assms thin1)+
   done
 
 lemma OrdP_Mem_imp_Subset:
   assumes k: "H \<turnstile> k IN l" and l: "H \<turnstile> OrdP l" shows "H \<turnstile> k SUBS l"
   apply (rule obtain_fresh [of "(l,k)"])
   apply (rule cut_same [OF l])
   using k  apply (auto intro: OrdP_Mem_E thin1)
   done
 
 lemma SUCC_Subset_Ord_lemma: "{ k' IN k, OrdP k } \<turnstile> SUCC k' SUBS k"
   by auto (metis Assume thin1 OrdP_Mem_imp_Subset)
 
 lemma SUCC_Subset_Ord: "H \<turnstile> k' IN k \<Longrightarrow> H \<turnstile> OrdP k \<Longrightarrow> H \<turnstile> SUCC k' SUBS k"
   by (blast intro!: cut2 [OF SUCC_Subset_Ord_lemma])
 
 lemma OrdP_Trans_lemma: "{ OrdP k, i IN j, j IN k } \<turnstile> i IN k"
 proof -
   obtain m::name where "atom m \<sharp> (i,j,k)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto intro: OrdP_Mem_E [of m k j] Subset_D [THEN rotate3])
 qed
 
 lemma OrdP_Trans: "H \<turnstile>  OrdP k \<Longrightarrow> H \<turnstile> i IN j \<Longrightarrow> H \<turnstile> j IN k \<Longrightarrow> H  \<turnstile> i IN k"
   by (blast intro: cut3 [OF OrdP_Trans_lemma])
 
 lemma Ord_IN_Ord0:
   assumes l: "H \<turnstile> l IN k"
   shows "insert (OrdP k) H \<turnstile> OrdP l"
 proof -
   obtain z::name and y::name where z: "atom z \<sharp> (k,l)" and y: "atom y \<sharp> (k,l,z)"
     by (metis obtain_fresh)
   have "{Var y IN l, OrdP k, l IN k} \<turnstile> All2 z (Var y) (Var z SUBS Var y)"  using y z
     apply (simp add: insert_commute [of _ "OrdP k"])
     apply (auto intro: OrdP_Mem_E [of z k "Var y"] OrdP_Trans_lemma del: All_I Neg_I)
     done
   hence "{OrdP k, l IN k} \<turnstile> OrdP l" using z y
     apply (auto simp: OrdP.simps [of y l z])
     apply (simp add: insert_commute [of _ "OrdP k"])
     apply (rule OrdP_Mem_E [of y k l], simp_all)
     apply (metis Assume thin1)
     apply (rule All_E [where x= "Var y", THEN thin1], simp)
     apply (metis Assume anti_deduction insert_commute)
     done
   thus ?thesis
     by (metis (full_types) Assume l cut2 thin1)
 qed
 
 lemma Ord_IN_Ord: "H \<turnstile> l IN k \<Longrightarrow> H \<turnstile> OrdP k \<Longrightarrow> H \<turnstile> OrdP l"
   by (metis Ord_IN_Ord0 cut_same)
 
 lemma OrdP_I:
   assumes "insert (Var y IN x) H \<turnstile> (Var y) SUBS x"
       and "insert (Var z IN Var y) (insert (Var y IN x) H) \<turnstile> (Var z) SUBS (Var y)"
       and "atom y \<sharp> (x, z)" "\<forall>B \<in> H. atom y \<sharp> B"  "atom z \<sharp> x" "\<forall>B \<in> H. atom z \<sharp> B"
     shows "H \<turnstile> OrdP x"
   using assms by auto
 
 lemma OrdP_Zero [simp]: "H \<turnstile> OrdP Zero"
 proof -
   obtain y::name and z::name where "atom y \<sharp> z"
     by (rule obtain_fresh)
   hence "{} \<turnstile> OrdP Zero"
     by (auto intro: OrdP_I [of y _ _ z])
   thus ?thesis
     by (metis thin0)
 qed
 
 lemma OrdP_SUCC_I0: "{ OrdP k } \<turnstile> OrdP (SUCC k)"
 proof -
   obtain w::name and y::name and z::name where atoms: "atom w \<sharp> (k,y,z)" "atom y \<sharp> (k,z)" "atom z \<sharp> k"
     by (metis obtain_fresh)
   have 1: "{Var y IN SUCC k, OrdP k} \<turnstile> Var y SUBS SUCC k"
     apply (rule Mem_SUCC_E)
     apply (rule OrdP_Mem_E [of w _ "Var y", THEN rotate2]) using atoms
     apply auto
     apply (metis Assume Subset_SUCC Subset_trans)
     apply (metis EQ_imp_SUBS Subset_SUCC Subset_trans)
     done
   have in_case: "{Var y IN k, Var z IN Var y, OrdP k} \<turnstile> Var z SUBS Var y"
     apply (rule OrdP_Mem_E [of w _ "Var y", THEN rotate3]) using atoms
     apply (auto intro:  All2_E [THEN thin1])
     done
   have "{Var y EQ k, Var z IN k, OrdP k} \<turnstile> Var z SUBS Var y"
     by (metis AssumeH(2) AssumeH(3) EQ_imp_SUBS2 OrdP_Mem_imp_Subset Subset_trans)
   hence eq_case: "{Var y EQ k, Var z IN Var y, OrdP k} \<turnstile> Var z SUBS Var y"
     by (rule cut3) (auto intro: EQ_imp_SUBS [THEN cut1] Subset_D)
   have 2: "{Var z IN Var y, Var y IN SUCC k, OrdP k} \<turnstile> Var z SUBS Var y"
     by (metis rotate2 Mem_SUCC_E in_case eq_case)
   show ?thesis
     apply (rule OrdP_I [OF 1 2])
     using atoms apply auto
     done
 qed
 
 lemma OrdP_SUCC_I: "H \<turnstile> OrdP k \<Longrightarrow> H \<turnstile> OrdP (SUCC k)"
   by (metis OrdP_SUCC_I0 cut1)
 
 lemma Zero_In_OrdP: "{ OrdP x } \<turnstile> x EQ Zero OR Zero IN x"
 proof -
   obtain i::name and j::name
     where i: "atom i \<sharp> x" and j: "atom j \<sharp> (x,i)"
     by (metis obtain_fresh)
   show ?thesis
     apply (rule cut_thin [where HB = "{OrdP x}", OF Foundation [where i=i and z = x]])
     using i j apply auto
     prefer 2 apply (metis Assume Disj_I1)
     apply (rule Disj_I2)
     apply (rule cut_same [where A = "Var i EQ Zero"])
     prefer 2 apply (blast intro: Iff_MP_same [OF Mem_cong [OF Assume Refl]])
     apply (auto intro!: Eq_Zero_I [where i=j] Ex_I [where x="Var i"])
     apply (blast intro: Disjoint_E Subset_D)
     done
 qed
 
 lemma OrdP_HPairE: "insert (OrdP (HPair x y)) H \<turnstile> A"
 proof -
   have "{ OrdP (HPair x y) } \<turnstile> A"
     by (rule cut_same [OF Zero_In_OrdP]) (auto simp: HPair_def)
   thus ?thesis
     by (metis Assume cut1)
 qed
 
 lemmas OrdP_HPairEH = OrdP_HPairE OrdP_HPairE [THEN rotate2] OrdP_HPairE [THEN rotate3] OrdP_HPairE [THEN rotate4] OrdP_HPairE [THEN rotate5]
                  OrdP_HPairE [THEN rotate6] OrdP_HPairE [THEN rotate7] OrdP_HPairE [THEN rotate8] OrdP_HPairE [THEN rotate9] OrdP_HPairE [THEN rotate10]
 declare OrdP_HPairEH [intro!]
 
 lemma Zero_Eq_HPairE: "insert (Zero EQ HPair x y) H \<turnstile> A"
   by (metis Eats_EQ_Zero_E2 HPair_def)
 
 lemmas Zero_Eq_HPairEH = Zero_Eq_HPairE Zero_Eq_HPairE [THEN rotate2] Zero_Eq_HPairE [THEN rotate3] Zero_Eq_HPairE [THEN rotate4] Zero_Eq_HPairE [THEN rotate5]
                  Zero_Eq_HPairE [THEN rotate6] Zero_Eq_HPairE [THEN rotate7] Zero_Eq_HPairE [THEN rotate8] Zero_Eq_HPairE [THEN rotate9] Zero_Eq_HPairE [THEN rotate10]
 declare Zero_Eq_HPairEH [intro!]
 
 lemma HPair_Eq_ZeroE: "insert (HPair x y EQ Zero) H \<turnstile> A"
   by (metis Sym_L Zero_Eq_HPairE)
 
 lemmas HPair_Eq_ZeroEH = HPair_Eq_ZeroE HPair_Eq_ZeroE [THEN rotate2] HPair_Eq_ZeroE [THEN rotate3] HPair_Eq_ZeroE [THEN rotate4] HPair_Eq_ZeroE [THEN rotate5]
                  HPair_Eq_ZeroE [THEN rotate6] HPair_Eq_ZeroE [THEN rotate7] HPair_Eq_ZeroE [THEN rotate8] HPair_Eq_ZeroE [THEN rotate9] HPair_Eq_ZeroE [THEN rotate10]
 declare HPair_Eq_ZeroEH [intro!]
 
-section {*Induction on Ordinals*}
+section \<open>Induction on Ordinals\<close>
 
 lemma OrdInd_lemma:
   assumes j: "atom (j::name) \<sharp> (i,A)"
   shows "{ OrdP (Var i) } \<turnstile> (All i (OrdP (Var i) IMP ((All2 j (Var i) (A(i::= Var j))) IMP A))) IMP A"
 proof -
   obtain l::name and k::name
       where l: "atom l \<sharp> (i,j,A)" and k: "atom k \<sharp> (i,j,l,A)"
       by (metis obtain_fresh)
   have "{ (All i (OrdP (Var i) IMP ((All2 j (Var i) (A(i::= Var j))) IMP A))) }
         \<turnstile> (All2 l (Var i) (OrdP (Var l) IMP A(i::= Var l)))"
     apply (rule Ind [of k])
     using j k l apply auto
     apply (rule All_E [where x="Var l", THEN rotate5], auto)
     apply (metis Assume Disj_I1 anti_deduction thin1)
     apply (rule Ex_I [where x="Var l"], auto)
     apply (rule All_E [where x="Var j", THEN rotate6], auto)
     apply (blast intro: ContraProve Iff_MP_same [OF Mem_cong [OF Refl]])
     apply (metis Assume Ord_IN_Ord0 ContraProve insert_commute)
     apply (metis Assume Neg_D thin1)+
     done
   hence "{ (All i (OrdP (Var i) IMP ((All2 j (Var i) (A(i::= Var j))) IMP A))) }
           \<turnstile> (All2 l (Var i) (OrdP (Var l) IMP A(i::= Var l)))(i::= Eats Zero (Var i))"
     by (rule Subst, auto)
   hence indlem: "{ All i (OrdP (Var i) IMP ((All2 j (Var i) (A(i::= Var j))) IMP A)) }
              \<turnstile> All2 l (Eats Zero (Var i)) (OrdP (Var l) IMP A(i::=Var l))"
     using j l by simp
   show ?thesis
     apply (rule Imp_I)
     apply (rule cut_thin [OF indlem, where HB = "{OrdP (Var i)}"])
     apply (rule All2_Eats_E) using j l
     apply auto
     done
 qed
 
 lemma OrdInd:
   assumes j: "atom (j::name) \<sharp> (i,A)"
   and x: "H \<turnstile> OrdP (Var i)" and step: "H \<turnstile> All i (OrdP (Var i) IMP (All2 j (Var i) (A(i::= Var j)) IMP A))"
   shows "H \<turnstile> A"
   apply (rule cut_thin [OF x, where HB=H])
   apply (rule MP_thin [OF OrdInd_lemma step])
   apply (auto simp: j)
   done
 
 lemma OrdIndH:
   assumes "atom (j::name) \<sharp> (i,A)"
       and "H \<turnstile> All i (OrdP (Var i) IMP (All2 j (Var i) (A(i::= Var j)) IMP A))"
     shows "insert (OrdP (Var i)) H \<turnstile> A"
   by (metis assms thin1 Assume OrdInd)
 
 
-section {*Linearity of Ordinals*}
+section \<open>Linearity of Ordinals\<close>
 
 lemma OrdP_linear_lemma:
   assumes j: "atom j \<sharp> i"
   shows "{ OrdP (Var i) } \<turnstile> All j (OrdP (Var j) IMP (Var i IN Var j OR Var i EQ Var j OR Var j IN Var i))"
          (is "_ \<turnstile> ?scheme")
 proof -
   obtain k::name and l::name and m::name
     where k: "atom k \<sharp> (i,j)" and l: "atom l \<sharp> (i,j,k)" and m: "atom m \<sharp> (i,j)"
     by (metis obtain_fresh)
   show ?thesis
   proof (rule OrdIndH [where i=i and j=k])
     show "atom k \<sharp> (i, ?scheme)"
       using k  by (force simp add: fresh_Pair)
   next
     show "{} \<turnstile> All i (OrdP (Var i) IMP (All2 k (Var i) (?scheme(i::= Var k)) IMP ?scheme))"
       using j k
       apply simp
       apply (rule All_I Imp_I)+
       defer 1
       apply auto [2]
       apply (rule OrdIndH [where i=j and j=l]) using l
       \<comment> \<open>nested induction\<close>
       apply (force simp add: fresh_Pair)
       apply simp
       apply (rule All_I Imp_I)+
       prefer 2  apply force
       apply (rule Disj_3I)
       apply (rule Equality_I)
       \<comment> \<open>Now the opposite inclusion, @{term"Var j SUBS Var i"}\<close>
       apply (rule Subset_I [where i=m])
       apply (rule All2_E [THEN rotate4]) using l m
       apply auto
       apply (blast intro: ContraProve [THEN rotate3] OrdP_Trans)
       apply (blast intro: ContraProve [THEN rotate3] Mem_cong [OF Hyp Refl, THEN Iff_MP2_same])
       \<comment> \<open>Now the opposite inclusion, @{term"Var i SUBS Var j"}\<close>
       apply (rule Subset_I [where i=m])
       apply (rule All2_E [THEN rotate6], auto)
       apply (rule All_E [where x = "Var j"], auto)
       apply (blast intro: ContraProve [THEN rotate4] Mem_cong [OF Hyp Refl, THEN Iff_MP_same])
       apply (blast intro: ContraProve [THEN rotate4] OrdP_Trans)
       done
   qed
 qed
 
 lemma OrdP_linear_imp: "{} \<turnstile> OrdP x IMP OrdP y IMP x IN y OR x EQ y OR y IN x"
 proof -
   obtain i::name and j::name
     where atoms: "atom i \<sharp> (x,y)" "atom j \<sharp> (x,y,i)"
     by (metis obtain_fresh)
   have "{ OrdP (Var i) } \<turnstile> (OrdP (Var j) IMP (Var i IN Var j OR Var i EQ Var j OR Var j IN Var i))(j::=y)"
     using atoms  by (metis All_D OrdP_linear_lemma fresh_Pair)
   hence "{} \<turnstile> OrdP (Var i) IMP OrdP y IMP (Var i IN y OR Var i EQ y OR y IN Var i)"
     using atoms by auto
   hence "{} \<turnstile> (OrdP (Var i) IMP OrdP y IMP (Var i IN y OR Var i EQ y OR y IN Var i))(i::=x)"
     by (metis Subst empty_iff)
   thus ?thesis
     using atoms by auto
 qed
 
 lemma OrdP_linear:
   assumes "H \<turnstile> OrdP x" "H \<turnstile> OrdP y"
           "insert (x IN y) H \<turnstile> A" "insert (x EQ y) H \<turnstile> A" "insert (y IN x) H \<turnstile> A"
     shows "H \<turnstile> A"
 proof -
   have "{ OrdP x, OrdP y } \<turnstile> x IN y OR x EQ y OR y IN x"
     by (metis OrdP_linear_imp Imp_Imp_commute anti_deduction)
   thus ?thesis
     using assms  by (metis cut2 Disj_E cut_same)
 qed
 
 lemma Zero_In_SUCC: "{OrdP k} \<turnstile> Zero IN SUCC k"
   by (rule OrdP_linear [OF OrdP_Zero OrdP_SUCC_I]) (force simp: SUCC_def)+
 
 
-section {*The predicate @{text OrdNotEqP}*}
+section \<open>The predicate \<open>OrdNotEqP\<close>\<close>
 
 nominal_function OrdNotEqP :: "tm \<Rightarrow> tm \<Rightarrow> fm"  (infixr "NEQ" 150)
   where "OrdNotEqP x y = OrdP x AND OrdP y AND (x IN y OR y IN x)"
   by (auto simp: eqvt_def OrdNotEqP_graph_aux_def)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma OrdNotEqP_fresh_iff [simp]: "a \<sharp> OrdNotEqP x y \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> y"
   by auto
 
 lemma OrdNotEqP_subst [simp]: "(OrdNotEqP x y)(i::=t) = OrdNotEqP (subst i t x) (subst i t y)"
   by simp
 
 lemma OrdNotEqP_cong: "H \<turnstile> x EQ x' \<Longrightarrow> H \<turnstile> y EQ y' \<Longrightarrow> H \<turnstile> OrdNotEqP x y IFF OrdNotEqP x' y'"
   by (rule P2_cong) auto
 
 lemma OrdNotEqP_self_contra: "{x NEQ x} \<turnstile> Fls"
   by auto
 
 lemma OrdNotEqP_OrdP_E: "insert (OrdP x) (insert (OrdP y) H) \<turnstile> A \<Longrightarrow> insert (x NEQ y) H \<turnstile> A"
   by (auto intro: thin1 rotate2)
 
 lemma OrdNotEqP_I: "insert (x EQ y) H \<turnstile> Fls \<Longrightarrow> H \<turnstile> OrdP x \<Longrightarrow> H \<turnstile> OrdP y \<Longrightarrow> H \<turnstile> x NEQ y"
   by (rule OrdP_linear [of _ x y]) (auto intro: ExFalso thin1 Disj_I1 Disj_I2)
 
 declare OrdNotEqP.simps [simp del]
 
 lemma OrdNotEqP_imp_Neg_Eq: "{x NEQ y} \<turnstile> Neg (x EQ y)"
   by (blast intro: OrdNotEqP_cong [THEN Iff_MP2_same]  OrdNotEqP_self_contra [of x, THEN cut1])
 
 lemma OrdNotEqP_E: "H \<turnstile> x EQ y \<Longrightarrow> insert (x NEQ y) H \<turnstile> A"
   by (metis ContraProve OrdNotEqP_imp_Neg_Eq rcut1)
 
 
-section {*Predecessor of an Ordinal*}
+section \<open>Predecessor of an Ordinal\<close>
 
 lemma OrdP_set_max_lemma:
   assumes j: "atom (j::name) \<sharp> i" and k: "atom (k::name) \<sharp> (i,j)"
   shows "{} \<turnstile> (Neg (Var i EQ Zero) AND (All2 j (Var i) (OrdP (Var j)))) IMP
               (Ex j (Var j IN Var i AND (All2 k (Var i) (Var k SUBS Var j))))"
 proof -
   obtain l::name where l: "atom l \<sharp> (i,j,k)"
     by (metis obtain_fresh)
   show ?thesis
     apply (rule Ind [of l i]) using j k l
       apply simp_all
      apply (metis Conj_E Refl Swap Imp_I)
     apply (rule All_I Imp_I)+
       apply simp_all
     apply clarify
     apply (rule thin1)
     apply (rule thin1 [THEN rotate2])
     apply (rule Disj_EH)
      apply (rule Neg_Conj_E)
       apply (auto simp: All2_Eats_E1)
      apply (rule Ex_I [where x="Var l"], auto intro: Mem_Eats_I2)
       apply (metis Assume Eq_Zero_D rotate3)
      apply (metis Assume EQ_imp_SUBS Neg_D thin1)
     apply (rule Cases [where A = "Var j IN Var l"])
     apply (rule Ex_I [where x="Var l"], auto intro: Mem_Eats_I2)
     apply (rule Ex_I [where x="Var l"], auto intro: Mem_Eats_I2 ContraProve)
     apply (rule Ex_I [where x="Var k"], auto)
     apply (metis Assume Subset_trans OrdP_Mem_imp_Subset thin1)
     apply (rule Ex_I [where x="Var l"], auto intro: Mem_Eats_I2 ContraProve)
     apply (metis ContraProve EQ_imp_SUBS rotate3)
     \<comment> \<open>final case\<close>
     apply (rule All2_Eats_E [THEN rotate4], simp_all)
     apply (rule Ex_I [where x="Var j"], auto intro: Mem_Eats_I1)
     apply (rule All2_E [where x = "Var k", THEN rotate3], auto)
     apply (rule Ex_I [where x="Var k"], simp)
     apply (metis Assume NegNeg_I Neg_Disj_I rotate3)
     apply (rule cut_same [where A = "OrdP (Var j)"])
     apply (rule All2_E [where x = "Var j", THEN rotate3], auto)
     apply (rule cut_same [where A = "Var l EQ Var j OR Var l IN Var j"])
     apply (rule OrdP_linear [of _ "Var l" "Var j"], auto intro: Disj_CI)
     apply (metis Assume ContraProve rotate7)
     apply (metis ContraProve [THEN rotate4] EQ_imp_SUBS Subset_trans rotate3)
     apply (blast intro: ContraProve [THEN rotate4] OrdP_Mem_imp_Subset Iff_MP2_same [OF Mem_cong])
     done
 qed
 
 lemma OrdP_max_imp:
   assumes j: "atom j \<sharp> (x)" and k: "atom k \<sharp> (x,j)"
   shows  "{ OrdP x, Neg (x EQ Zero) } \<turnstile> Ex j (Var j IN x AND (All2 k x (Var k SUBS Var j)))"
 proof -
   obtain i::name where i: "atom i \<sharp> (x,j,k)"
     by (metis obtain_fresh)
   have "{} \<turnstile> ((Neg (Var i EQ Zero) AND (All2 j (Var i) (OrdP (Var j)))) IMP
               (Ex j (Var j IN Var i AND (All2 k (Var i) (Var k SUBS Var j)))))(i::=x)"
     apply (rule Subst [OF OrdP_set_max_lemma])
     using i k apply auto
     done
   hence "{ Neg (x EQ Zero) AND (All2 j x (OrdP (Var j))) }
          \<turnstile> Ex j (Var j IN x AND (All2 k x (Var k SUBS Var j)))"
    using i j k by simp (metis anti_deduction)
   hence "{ All2 j x (OrdP (Var j)), Neg (x EQ Zero) }
              \<turnstile> Ex j (Var j IN x AND (All2 k x (Var k SUBS Var j)))"
     by (rule cut1) (metis Assume Conj_I thin1)
   moreover have "{ OrdP x } \<turnstile> All2 j x (OrdP (Var j))" using j
     by auto (metis Assume Ord_IN_Ord thin1)
   ultimately show ?thesis
    by (metis rcut1)
 qed
 
 declare OrdP.simps [simp del]
 
 
-section {*Case Analysis and Zero/SUCC Induction*}
+section \<open>Case Analysis and Zero/SUCC Induction\<close>
 
 lemma OrdP_cases_lemma:
   assumes p: "atom p \<sharp> x"
   shows  "{ OrdP x, Neg (x EQ Zero) } \<turnstile> Ex p (OrdP (Var p) AND x EQ SUCC (Var p))"
 proof -
   obtain j::name and k::name where j: "atom j \<sharp> (x,p)" and k: "atom k \<sharp> (x,j,p)"
     by (metis obtain_fresh)
   show ?thesis
     apply (rule cut_same [OF OrdP_max_imp [of j x k]])
     using p j k apply auto
     apply (rule Ex_I [where x="Var j"], auto)
      apply (metis Assume Ord_IN_Ord thin1)
     apply (rule cut_same [where A = "OrdP (SUCC (Var j))"])
     apply (metis Assume Ord_IN_Ord0 OrdP_SUCC_I rotate2 thin1)
     apply (rule OrdP_linear [where x = x, OF _ Assume], auto intro!: Mem_SUCC_EH)
     apply (metis Mem_not_sym rotate3)
     apply (rule Mem_non_refl, blast intro: Mem_cong [OF Assume Refl, THEN Iff_MP2_same])
     apply (force intro: thin1 All2_E [where x = "SUCC (Var j)", THEN rotate4])
     done
 qed
 
 lemma OrdP_cases_disj:
   assumes p: "atom p \<sharp> x"
   shows  "insert (OrdP x) H \<turnstile> x EQ Zero OR Ex p (OrdP (Var p) AND x EQ SUCC (Var p))"
   by (metis Disj_CI Assume cut2 [OF OrdP_cases_lemma [OF p]] Swap thin1)
 
 lemma OrdP_cases_E:
   "\<lbrakk>insert (x EQ Zero) H \<turnstile> A;
     insert (x EQ SUCC (Var k)) (insert (OrdP (Var k)) H) \<turnstile> A;
     atom k \<sharp> (x,A);   \<forall>C \<in> H. atom k \<sharp> C\<rbrakk>
    \<Longrightarrow> insert (OrdP x) H \<turnstile> A"
   by (rule cut_same [OF OrdP_cases_disj [of k]]) (auto simp: insert_commute intro: thin1)
 
 lemma OrdInd2_lemma:
   "{ OrdP (Var i), A(i::= Zero), (All i (OrdP (Var i) IMP A IMP (A(i::= SUCC (Var i))))) } \<turnstile> A"
 proof -
   obtain j::name and k::name  where atoms: "atom j \<sharp> (i,A)" "atom k \<sharp> (i,j,A)"
     by (metis obtain_fresh)
   show ?thesis
   apply (rule OrdIndH [where i=i and j=j])
   using atoms  apply auto
   apply (rule OrdP_cases_E [where k=k, THEN rotate3])
   apply (rule ContraProve [THEN rotate2]) using Var_Eq_imp_subst_Iff
   apply (metis Assume AssumeH(3) Iff_MP_same)
   apply (rule Ex_I [where x="Var k"], simp)
   apply (rule Neg_Imp_I, blast)
   apply (rule cut_same [where A = "A(i::=Var k)"])
   apply (rule All2_E [where x = "Var k", THEN rotate5])
   apply (auto intro: Mem_SUCC_I2 Mem_cong [OF Refl, THEN Iff_MP2_same])
   apply (rule ContraProve [THEN rotate5])
   by (metis Assume Iff_MP_left' Var_Eq_subst_Iff thin1)
 qed
 
 lemma OrdInd2:
   assumes "H \<turnstile> OrdP (Var i)"
       and "H \<turnstile> A(i::= Zero)"
       and "H \<turnstile> All i (OrdP (Var i) IMP A IMP (A(i::= SUCC (Var i))))"
     shows "H \<turnstile> A"
   by (metis cut3 [OF OrdInd2_lemma] assms)
 
 lemma OrdInd2H:
   assumes "H \<turnstile> A(i::= Zero)"
       and "H \<turnstile> All i (OrdP (Var i) IMP A IMP (A(i::= SUCC (Var i))))"
     shows "insert (OrdP (Var i)) H \<turnstile> A"
   by (metis assms thin1 Assume OrdInd2)
 
 
-section {*The predicate @{text HFun_Sigma}*}
+section \<open>The predicate \<open>HFun_Sigma\<close>\<close>
 
-text{*To characterise the concept of a function using only bounded universal quantifiers.*}
+text\<open>To characterise the concept of a function using only bounded universal quantifiers.\<close>
 
-text{*See the note after the proof of Lemma 2.3.*}
+text\<open>See the note after the proof of Lemma 2.3.\<close>
 
 definition hfun_sigma where
  "hfun_sigma r \<equiv> \<forall>z \<^bold>\<in> r. \<forall>z' \<^bold>\<in> r. \<exists>x y x' y'. z = \<langle>x,y\<rangle> \<and> z' = \<langle>x',y'\<rangle> \<and> (x=x' \<longrightarrow> y=y')"
 
 definition hfun_sigma_ord where
  "hfun_sigma_ord r \<equiv> \<forall>z \<^bold>\<in> r. \<forall>z' \<^bold>\<in> r. \<exists>x y x' y'. z = \<langle>x,y\<rangle> \<and> z' = \<langle>x',y'\<rangle> \<and> Ord x \<and> Ord x' \<and> (x=x' \<longrightarrow> y=y')"
 
 nominal_function HFun_Sigma :: "tm \<Rightarrow> fm"
   where "\<lbrakk>atom z \<sharp> (r,z',x,y,x',y'); atom z' \<sharp> (r,x,y,x',y');
           atom x \<sharp> (r,y,x',y'); atom y \<sharp> (r,x',y'); atom x' \<sharp> (r,y'); atom y' \<sharp> (r) \<rbrakk> \<Longrightarrow>
     HFun_Sigma r =
          All2 z r (All2 z' r (Ex x (Ex y (Ex x' (Ex y'
              (Var z EQ HPair (Var x) (Var y) AND Var z' EQ HPair (Var x') (Var y')
               AND OrdP (Var x) AND OrdP (Var x') AND
               ((Var x EQ Var x') IMP (Var y EQ Var y'))))))))"
 by (auto simp: eqvt_def HFun_Sigma_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows HFun_Sigma_fresh_iff [simp]: "a \<sharp> HFun_Sigma r \<longleftrightarrow> a \<sharp> r" (is ?thesis1)
 proof -
   obtain x::name and y::name and z::name and x'::name and y'::name and z'::name
     where "atom z \<sharp> (r,z',x,y,x',y')"  "atom z' \<sharp> (r,x,y,x',y')"
           "atom x \<sharp> (r,y,x',y')"  "atom y \<sharp> (r,x',y')"
           "atom x' \<sharp> (r,y')"  "atom y' \<sharp> (r)"
     by (metis obtain_fresh)
   thus ?thesis1
     by (auto simp: HBall_def hfun_sigma_ord_def)
 qed
 
 lemma HFun_Sigma_subst [simp]: "(HFun_Sigma r)(i::=t) = HFun_Sigma (subst i t r)"
 proof -
   obtain x::name and y::name and z::name and x'::name and y'::name and z'::name
     where "atom z \<sharp> (r,t,i,z',x,y,x',y')" "atom z' \<sharp> (r,t,i,x,y,x',y')"
           "atom x \<sharp> (r,t,i,y,x',y')" "atom y \<sharp> (r,t,i,x',y')"
           "atom x' \<sharp> (r,t,i,y')" "atom y' \<sharp> (r,t,i)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: HFun_Sigma.simps [of z _ z' x y x' y'])
 qed
 
 lemma HFun_Sigma_Zero: "H \<turnstile> HFun_Sigma Zero"
 proof -
   obtain x::name and y::name and z::name and x'::name and y'::name and z'::name and z''::name
     where "atom z'' \<sharp> (z,z',x,y,x',y')" "atom z \<sharp> (z',x,y,x',y')" "atom z' \<sharp> (x,y,x',y')"
       "atom x \<sharp> (y,x',y')" "atom y \<sharp> (x',y')" "atom x' \<sharp> y'"
     by (metis obtain_fresh)
   hence "{} \<turnstile> HFun_Sigma Zero"
     by (auto simp: HFun_Sigma.simps [of z _ z' x y x' y'])
   thus ?thesis
     by (metis thin0)
 qed
 
 lemma Subset_HFun_Sigma: "{HFun_Sigma s, s' SUBS s} \<turnstile> HFun_Sigma s'"
 proof -
   obtain x::name and y::name and z::name and x'::name and y'::name and z'::name and z''::name
     where "atom z'' \<sharp> (z,z',x,y,x',y',s,s')"
       "atom z \<sharp> (z',x,y,x',y',s,s')" "atom z' \<sharp> (x,y,x',y',s,s')"
       "atom x \<sharp> (y,x',y',s,s')" "atom y \<sharp> (x',y',s,s')"
       "atom x' \<sharp> (y',s,s')" "atom y' \<sharp> (s,s')"
     by (metis obtain_fresh)
   thus ?thesis
     apply (auto simp: HFun_Sigma.simps [of z _ z' x y x' y'])
     apply (rule Ex_I [where x="Var z"], auto)
     apply (blast intro: Subset_D ContraProve)
     apply (rule All_E [where x="Var z'"], auto intro: Subset_D ContraProve)
     done
 qed
 
-text{*Captures the property of being a relation, using fewer variables than the full definition*}
+text\<open>Captures the property of being a relation, using fewer variables than the full definition\<close>
 lemma HFun_Sigma_Mem_imp_HPair:
   assumes "H \<turnstile> HFun_Sigma r" "H \<turnstile> a IN r"
       and xy: "atom x \<sharp> (y,a,r)" "atom y \<sharp> (a,r)"
     shows "H \<turnstile> (Ex x (Ex y (a EQ HPair (Var x) (Var y))))"  (is "_ \<turnstile> ?concl")
 proof -
   obtain x'::name and y'::name and z::name and z'::name
     where atoms: "atom z \<sharp> (z',x',y',x,y,a,r)" "atom z' \<sharp> (x',y',x,y,a,r)"
                  "atom x' \<sharp> (y',x,y,a,r)" "atom y' \<sharp> (x,y,a,r)"
     by (metis obtain_fresh)
   hence "{HFun_Sigma r, a IN r} \<turnstile> ?concl" using xy
     apply (auto simp: HFun_Sigma.simps [of z r z' x y x' y'])
     apply (rule All_E [where x=a], auto)
     apply (rule All_E [where x=a], simp)
     apply (rule Imp_E, blast)
     apply (rule Ex_EH Conj_EH)+
     apply simp_all
     apply (rule Ex_I [where x="Var x"], simp)
     apply (rule Ex_I [where x="Var y"], auto)
     done
   thus ?thesis
     by (rule cut2) (rule assms)+
 qed
 
 
-section {*The predicate @{text HDomain_Incl}*}
+section \<open>The predicate \<open>HDomain_Incl\<close>\<close>
 
-text {*This is an internal version of @{term "\<forall>x \<^bold>\<in> d. \<exists>y z. z \<^bold>\<in> r \<and> z = \<langle>x,y\<rangle>"}. *}
+text \<open>This is an internal version of @{term "\<forall>x \<^bold>\<in> d. \<exists>y z. z \<^bold>\<in> r \<and> z = \<langle>x,y\<rangle>"}.\<close>
 
 nominal_function HDomain_Incl :: "tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom x \<sharp> (r,d,y,z); atom y \<sharp> (r,d,z); atom z \<sharp> (r,d)\<rbrakk> \<Longrightarrow>
     HDomain_Incl r d = All2 x d (Ex y (Ex z (Var z IN r AND Var z EQ HPair (Var x) (Var y))))"
   by (auto simp: eqvt_def HDomain_Incl_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows HDomain_Incl_fresh_iff [simp]:
       "a \<sharp> HDomain_Incl r d \<longleftrightarrow> a \<sharp> r \<and> a \<sharp> d" (is ?thesis1)
 proof -
   obtain x::name and y::name and z::name
     where "atom x \<sharp> (r,d,y,z)" "atom y \<sharp> (r,d,z)" "atom z \<sharp> (r,d)"
     by (metis obtain_fresh)
   thus ?thesis1
     by (auto simp: HDomain_Incl.simps [of x _ _ y z] hdomain_def)
 qed
 
 lemma HDomain_Incl_subst [simp]:
       "(HDomain_Incl r d)(i::=t) = HDomain_Incl (subst i t r) (subst i t d)"
 proof -
   obtain x::name and y::name and z::name
     where "atom x \<sharp> (r,d,y,z,t,i)"  "atom y \<sharp> (r,d,z,t,i)" "atom z \<sharp> (r,d,t,i)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: HDomain_Incl.simps [of x _ _ y z])
 qed
 
 lemma HDomain_Incl_Subset_lemma: "{ HDomain_Incl r k, k' SUBS k } \<turnstile> HDomain_Incl r k'"
 proof -
   obtain x::name and y::name and z::name
     where "atom x \<sharp> (r,k,k',y,z)" "atom y \<sharp> (r,k,k',z)" "atom z \<sharp> (r,k,k')"
     by (metis obtain_fresh)
   thus ?thesis
     apply (simp add: HDomain_Incl.simps [of x _ _ y z], auto)
     apply (rule Ex_I [where x = "Var x"], auto intro: ContraProve Subset_D)
     done
 qed
 
 lemma HDomain_Incl_Subset: "H \<turnstile> HDomain_Incl r k \<Longrightarrow> H \<turnstile> k' SUBS k \<Longrightarrow> H \<turnstile> HDomain_Incl r k'"
   by (metis HDomain_Incl_Subset_lemma cut2)
 
 lemma HDomain_Incl_Mem_Ord: "H \<turnstile> HDomain_Incl r k \<Longrightarrow> H \<turnstile> k' IN k \<Longrightarrow> H \<turnstile> OrdP k \<Longrightarrow> H \<turnstile> HDomain_Incl r k'"
   by (metis HDomain_Incl_Subset OrdP_Mem_imp_Subset)
 
 lemma HDomain_Incl_Zero [simp]: "H \<turnstile> HDomain_Incl r Zero"
 proof -
   obtain x::name and y::name and z::name
     where "atom x \<sharp> (r,y,z)" "atom y \<sharp> (r,z)" "atom z \<sharp> r"
     by (metis obtain_fresh)
   hence "{} \<turnstile> HDomain_Incl r Zero"
     by (auto simp: HDomain_Incl.simps [of x _ _ y z])
   thus ?thesis
     by (metis thin0)
 qed
 
 lemma HDomain_Incl_Eats: "{ HDomain_Incl r d } \<turnstile> HDomain_Incl (Eats r (HPair d d')) (SUCC d)"
 proof -
   obtain x::name and y::name and z::name
     where x: "atom x \<sharp> (r,d,d',y,z)" and y: "atom y \<sharp> (r,d,d',z)" and z: "atom z \<sharp> (r,d,d')"
     by (metis obtain_fresh)
   thus ?thesis
     apply (auto simp: HDomain_Incl.simps [of x _ _ y z] intro!: Mem_SUCC_EH)
     apply (rule Ex_I [where x = "Var x"], auto)
     apply (rule Ex_I [where x = "Var y"], auto)
     apply (rule Ex_I [where x = "Var z"], auto intro: Mem_Eats_I1)
     apply (rule rotate2 [OF Swap])
     apply (rule Ex_I [where x = d'], auto)
     apply (rule Ex_I [where x = "HPair d d'"], auto intro: Mem_Eats_I2 HPair_cong Sym)
     done
 qed
 
 lemma HDomain_Incl_Eats_I: "H \<turnstile> HDomain_Incl r d \<Longrightarrow> H \<turnstile> HDomain_Incl (Eats r (HPair d d')) (SUCC d)"
   by (metis HDomain_Incl_Eats cut1)
 
 
-section {*@{term HPair} is Provably Injective*}
+section \<open>@{term HPair} is Provably Injective\<close>
 
 lemma Doubleton_E:
   assumes "insert (a EQ c) (insert (b EQ d) H) \<turnstile> A"
           "insert (a EQ d) (insert (b EQ c) H) \<turnstile> A"
     shows    "insert ((Eats (Eats Zero b) a) EQ (Eats (Eats Zero d) c)) H \<turnstile> A"
 apply (rule Equality_E) using assms
 apply (auto intro!: Zero_SubsetE rotate2 [of "a IN b"])
 apply (rule_tac [!] rotate3)
 apply (auto intro!: Zero_SubsetE rotate2 [of "a IN b"])
 apply (metis Sym_L insert_commute thin1)+
 done
 
 lemma HFST: "{HPair a b EQ HPair c d} \<turnstile> a EQ c"
   unfolding HPair_def  by (metis Assume Doubleton_E thin1)
 
 lemma b_EQ_d_1: "{a EQ c, a EQ d, b EQ c} \<turnstile> b EQ d"
   by (metis Assume thin1 Sym Trans)
 
 lemma HSND: "{HPair a b EQ HPair c d} \<turnstile> b EQ d"
   unfolding HPair_def
   by (metis  AssumeH(2) Doubleton_E b_EQ_d_1 rotate3 thin2)
 
 lemma HPair_E [intro!]:
   assumes "insert (a EQ c) (insert (b EQ d) H) \<turnstile> A"
     shows "insert (HPair a b EQ HPair c d) H \<turnstile> A"
   by (metis Conj_E [OF assms] Conj_I [OF HFST HSND] rcut1)
 
 declare HPair_E [THEN rotate2, intro!]
 declare HPair_E [THEN rotate3, intro!]
 declare HPair_E [THEN rotate4, intro!]
 declare HPair_E [THEN rotate5, intro!]
 declare HPair_E [THEN rotate6, intro!]
 declare HPair_E [THEN rotate7, intro!]
 declare HPair_E [THEN rotate8, intro!]
 
 lemma HFun_Sigma_E:
   assumes r: "H \<turnstile> HFun_Sigma r"
       and b: "H \<turnstile> HPair a b IN r"
       and b': "H \<turnstile> HPair a b' IN r"
     shows "H \<turnstile> b EQ b'"
 proof -
   obtain x::name and y::name and z::name and x'::name and y'::name and z'::name
     where atoms: "atom z \<sharp> (r,a,b,b',z',x,y,x',y')"  "atom z' \<sharp> (r,a,b,b',x,y,x',y')"
        "atom x \<sharp> (r,a,b,b',y,x',y')"  "atom y \<sharp> (r,a,b,b',x',y')"
        "atom x' \<sharp> (r,a,b,b',y')"  "atom y' \<sharp> (r,a,b,b')"
     by (metis obtain_fresh)
   hence d1: "H \<turnstile> All2 z r (All2 z' r (Ex x (Ex y (Ex x' (Ex y'
                   (Var z EQ HPair (Var x) (Var y) AND Var z' EQ HPair (Var x') (Var y')
                    AND OrdP (Var x) AND OrdP (Var x') AND ((Var x EQ Var x') IMP (Var y EQ Var y'))))))))"
       using r HFun_Sigma.simps [of z r z' x y x' y']
     by simp
   have d2: "H \<turnstile> All2 z' r (Ex x (Ex y (Ex x' (Ex y'
              (HPair a b EQ HPair (Var x) (Var y) AND Var z' EQ HPair (Var x') (Var y')
                    AND OrdP (Var x) AND OrdP (Var x') AND ((Var x EQ Var x') IMP (Var y EQ Var y')))))))"
     using All_D [where x = "HPair a b", OF d1]  atoms
     by simp (metis MP_same b)
   have d4: "H \<turnstile> Ex x (Ex y (Ex x' (Ex y'
              (HPair a b EQ HPair (Var x) (Var y) AND HPair a b' EQ HPair (Var x') (Var y')
                AND OrdP (Var x) AND OrdP (Var x') AND ((Var x EQ Var x') IMP (Var y EQ Var y'))))))"
     using All_D [where x = "HPair a b'", OF d2]  atoms
     by simp (metis MP_same b')
   have d': "{ Ex x (Ex y (Ex x' (Ex y'
              (HPair a b EQ HPair (Var x) (Var y) AND HPair a b' EQ HPair (Var x') (Var y')
               AND OrdP (Var x) AND OrdP (Var x') AND ((Var x EQ Var x') IMP (Var y EQ Var y')))))) } \<turnstile> b EQ b'"
      using atoms
      by (auto intro: ContraProve Trans Sym)
   thus ?thesis
     by (rule cut_thin [OF d4], auto)
 qed
 
 
-section {*@{term SUCC} is Provably Injective*}
+section \<open>@{term SUCC} is Provably Injective\<close>
 
 lemma SUCC_SUBS_lemma: "{SUCC x SUBS SUCC y} \<turnstile> x SUBS y"
   apply (rule obtain_fresh [where x="(x,y)"])
   apply (auto simp: SUCC_def)
   prefer 2  apply (metis Assume Conj_E1 Extensionality Iff_MP_same)
   apply (auto intro!: Subset_I)
   apply (blast intro: Set_MP cut_same [OF Mem_cong [OF Refl Assume, THEN Iff_MP2_same]]
             Mem_not_sym thin2)
   done
 
 lemma SUCC_SUBS: "insert (SUCC x SUBS SUCC y) H \<turnstile> x SUBS y"
   by (metis Assume SUCC_SUBS_lemma cut1)
 
 lemma SUCC_inject: "insert (SUCC x EQ SUCC y) H \<turnstile> x EQ y"
   by (metis Equality_I EQ_imp_SUBS SUCC_SUBS Sym_L cut1)
 
 lemma SUCC_inject_E [intro!]: "insert (x EQ y) H \<turnstile> A \<Longrightarrow> insert (SUCC x EQ SUCC y) H \<turnstile> A"
   by (metis SUCC_inject cut_same insert_commute thin1)
 
 declare SUCC_inject_E [THEN rotate2, intro!]
 declare SUCC_inject_E [THEN rotate3, intro!]
 declare SUCC_inject_E [THEN rotate4, intro!]
 declare SUCC_inject_E [THEN rotate5, intro!]
 declare SUCC_inject_E [THEN rotate6, intro!]
 declare SUCC_inject_E [THEN rotate7, intro!]
 declare SUCC_inject_E [THEN rotate8, intro!]
 
 lemma OrdP_IN_SUCC_lemma: "{OrdP x, y IN x} \<turnstile> SUCC y IN SUCC x"
   apply (rule OrdP_linear [of _ "SUCC x" "SUCC y"])
   apply (auto intro!: Mem_SUCC_EH  intro: OrdP_SUCC_I Ord_IN_Ord0)
   apply (metis Hyp Mem_SUCC_I1 Mem_not_sym cut_same insertCI)
   apply (metis Assume EQ_imp_SUBS Mem_SUCC_I1 Mem_non_refl Subset_D thin1)
   apply (blast intro: cut_same [OF Mem_cong [THEN Iff_MP2_same]])
   done
 
 lemma OrdP_IN_SUCC: "H \<turnstile> OrdP x \<Longrightarrow> H \<turnstile> y IN x \<Longrightarrow> H \<turnstile> SUCC y IN SUCC x"
   by (rule cut2 [OF OrdP_IN_SUCC_lemma])
 
 lemma OrdP_IN_SUCC_D_lemma: "{OrdP x, SUCC y IN SUCC x} \<turnstile> y IN x"
   apply (rule OrdP_linear [of _ "x" "y"], auto)
   apply (metis Assume AssumeH(2) Mem_SUCC_Refl OrdP_SUCC_I Ord_IN_Ord)
   apply (rule Mem_SUCC_E [THEN rotate3])
   apply (blast intro: Mem_SUCC_Refl OrdP_Trans)
   apply (metis AssumeH(2) EQ_imp_SUBS Mem_SUCC_I1 Mem_non_refl Subset_D)
   apply (metis EQ_imp_SUBS Mem_SUCC_I2 Mem_SUCC_EH(2) Mem_SUCC_I1 Refl SUCC_Subset_Ord_lemma Subset_D thin1)
   done
 
 lemma OrdP_IN_SUCC_D: "H \<turnstile> OrdP x \<Longrightarrow> H \<turnstile> SUCC y IN SUCC x \<Longrightarrow> H \<turnstile> y IN x"
   by (rule cut2 [OF OrdP_IN_SUCC_D_lemma])
 
 lemma OrdP_IN_SUCC_Iff: "H \<turnstile> OrdP y \<Longrightarrow> H \<turnstile> SUCC x IN SUCC y IFF x IN y"
   by (metis Assume Iff_I OrdP_IN_SUCC OrdP_IN_SUCC_D thin1)
 
 
-section {*The predicate @{text LstSeqP}*}
+section \<open>The predicate \<open>LstSeqP\<close>\<close>
 
 lemma hfun_sigma_ord_iff: "hfun_sigma_ord s \<longleftrightarrow> OrdDom s \<and> hfun_sigma s"
   by (auto simp: hfun_sigma_ord_def OrdDom_def hfun_sigma_def HBall_def, metis+)
 
 lemma hfun_sigma_iff: "hfun_sigma r \<longleftrightarrow> hfunction r \<and> hrelation r"
   by (auto simp add: HBall_def hfun_sigma_def hfunction_def hrelation_def is_hpair_def, metis+)
 
 lemma Seq_iff: "Seq r d \<longleftrightarrow> d \<le> hdomain r \<and> hfun_sigma r"
   by (auto simp: Seq_def hfun_sigma_iff)
 
 lemma LstSeq_iff: "LstSeq s k y \<longleftrightarrow> succ k \<le> hdomain s \<and> \<langle>k,y\<rangle> \<^bold>\<in> s \<and> hfun_sigma_ord s"
   by (auto simp: OrdDom_def LstSeq_def Seq_iff hfun_sigma_ord_iff)
 
 nominal_function LstSeqP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where
     "LstSeqP s k y = OrdP k AND HDomain_Incl s (SUCC k) AND HFun_Sigma s AND HPair k y IN s"
   by (auto simp: eqvt_def LstSeqP_graph_aux_def)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
  shows LstSeqP_fresh_iff [simp]:
       "a \<sharp> LstSeqP s k y \<longleftrightarrow> a \<sharp> s \<and> a \<sharp> k \<and> a \<sharp> y"         (is ?thesis1)
 proof -
   show ?thesis1
     by (auto simp: LstSeq_iff OrdDom_def hfun_sigma_ord_iff)
 qed
 
 lemma LstSeqP_subst [simp]:
   "(LstSeqP s k y)(i::=t) = LstSeqP (subst i t s) (subst i t k) (subst i t y)"
   by (auto simp: fresh_Pair fresh_at_base)
 
 lemma LstSeqP_E:
   assumes "insert (HDomain_Incl s (SUCC k))
             (insert (OrdP k) (insert (HFun_Sigma s)
               (insert (HPair k y IN s) H))) \<turnstile> B"
     shows "insert (LstSeqP s k y) H \<turnstile> B"
   using assms by (auto simp: insert_commute)
 
 declare LstSeqP.simps [simp del]
 
 lemma LstSeqP_cong:
   assumes "H \<turnstile> s EQ s'" "H \<turnstile> k EQ k'" "H \<turnstile> y EQ y'"
   shows "H \<turnstile> LstSeqP s k y IFF LstSeqP s' k' y'"
   by (rule P3_cong [OF _ assms], auto)
 
 lemma LstSeqP_OrdP: "H \<turnstile> LstSeqP r k y \<Longrightarrow> H \<turnstile> OrdP k"
   by (metis Conj_E1 LstSeqP.simps)
 
 lemma LstSeqP_Mem_lemma: "{ LstSeqP r k y, HPair k' z IN r, k' IN k } \<turnstile> LstSeqP r k' z"
   by (auto simp: LstSeqP.simps intro: Ord_IN_Ord OrdP_SUCC_I OrdP_IN_SUCC HDomain_Incl_Mem_Ord)
 
 lemma LstSeqP_Mem: "H \<turnstile> LstSeqP r k y \<Longrightarrow> H \<turnstile> HPair k' z IN r \<Longrightarrow> H \<turnstile> k' IN k \<Longrightarrow> H \<turnstile> LstSeqP r k' z"
   by (rule cut3 [OF LstSeqP_Mem_lemma])
 
 lemma LstSeqP_imp_Mem: "H \<turnstile> LstSeqP s k y \<Longrightarrow> H \<turnstile> HPair k y IN s"
   by (auto simp: LstSeqP.simps) (metis Conj_E2)
 
 lemma LstSeqP_SUCC: "H \<turnstile> LstSeqP r (SUCC d) y \<Longrightarrow> H \<turnstile> HPair d z IN r \<Longrightarrow> H \<turnstile> LstSeqP r d z"
   by (metis LstSeqP_Mem Mem_SUCC_I2 Refl)
 
 lemma LstSeqP_EQ: "\<lbrakk>H \<turnstile> LstSeqP s k y; H \<turnstile> HPair k y' IN s\<rbrakk> \<Longrightarrow> H \<turnstile> y EQ y'"
   by (metis AssumeH(2) HFun_Sigma_E LstSeqP_E cut1 insert_commute)
 
 end
diff --git a/thys/Goedel_HFSet_Semanticless/Pseudo_Coding.thy b/thys/Goedel_HFSet_Semanticless/Pseudo_Coding.thy
--- a/thys/Goedel_HFSet_Semanticless/Pseudo_Coding.thy
+++ b/thys/Goedel_HFSet_Semanticless/Pseudo_Coding.thy
@@ -1,328 +1,328 @@
-chapter{*Pseudo-Coding: Section 7 Material*}
+chapter\<open>Pseudo-Coding: Section 7 Material\<close>
 
 theory Pseudo_Coding
 imports II_Prelims
 begin
 
-section{*General Lemmas*}
+section\<open>General Lemmas\<close>
 
 lemma Collect_disj_Un: "{f i |i. P i \<or> Q i} = {f i |i. P i} \<union> {f i |i. Q i}"
 by auto
 
 abbreviation Q_Subset :: "tm \<Rightarrow> tm \<Rightarrow> tm"
   where "Q_Subset t u \<equiv> (Q_All (Q_Imp (Q_Mem (Q_Ind Zero) t) (Q_Mem (Q_Ind Zero) u)))"
 
 lemma NEQ_quot_tm: "i\<noteq>j \<Longrightarrow> {} \<turnstile> \<lceil>Var i\<rceil> NEQ \<lceil>Var j\<rceil>"
   using VarP_Var[of "{}" i] VarP_Var[of "{}" j]
   by (intro OrdNotEqP_I) (auto simp: VarP_def quot_Var ORD_OF_EQ_diff dest!: Conj_E1)
 
 lemma EQ_quot_tm_Fls: "i\<noteq>j \<Longrightarrow> insert (\<lceil>Var i\<rceil> EQ \<lceil>Var j\<rceil>) H \<turnstile> Fls"
   by (metis (full_types) NEQ_quot_tm Assume OrdNotEqP_E cut2 thin0)
 
 lemma perm_commute: "a \<sharp> p \<Longrightarrow> a' \<sharp> p \<Longrightarrow> (a \<rightleftharpoons> a') + p = p + (a \<rightleftharpoons> a')"
   by (rule plus_perm_eq) (simp add: supp_swap fresh_def)
 
 lemma perm_self_inverseI: "\<lbrakk>-p = q; a \<sharp> p; a' \<sharp> p\<rbrakk> \<Longrightarrow> - ((a \<rightleftharpoons> a') + p) = (a \<rightleftharpoons> a') + q"
   by (simp_all add: perm_commute fresh_plus_perm minus_add)
 
 lemma fresh_image:
   fixes f :: "'a \<Rightarrow> 'b::fs"  shows "finite A \<Longrightarrow> i \<sharp> f ` A \<longleftrightarrow> (\<forall>x\<in>A. i \<sharp> f x)"
   by (induct rule: finite_induct) (auto simp: fresh_finite_insert)
 
 lemma atom_in_atom_image [simp]: "atom j \<in> atom ` V \<longleftrightarrow> j \<in> V"
   by auto
 
 lemma fresh_star_empty [simp]: "{} \<sharp>* bs"
   by (simp add: fresh_star_def)
 
 declare fresh_star_insert [simp]
 
 lemma fresh_star_finite_insert:
   fixes S :: "('a::fs) set"  shows "finite S \<Longrightarrow> a \<sharp>* insert x S \<longleftrightarrow> a \<sharp>* x \<and> a \<sharp>* S"
   by (auto simp: fresh_star_def fresh_finite_insert)
 
 lemma fresh_finite_Diff_single [simp]:
   fixes V :: "name set"   shows "finite V \<Longrightarrow> a \<sharp> (V - {j}) \<longleftrightarrow> (a \<sharp> j \<longrightarrow> a \<sharp> V)"
 apply (auto simp: fresh_finite_insert)
 apply (metis finite_Diff fresh_finite_insert insert_Diff_single)
 apply (metis Diff_iff finite_Diff fresh_atom fresh_atom_at_base fresh_finite_set_at_base insertI1)
 apply (metis Diff_idemp Diff_insert_absorb finite_Diff fresh_finite_insert insert_Diff_single insert_absorb)
 done
 
 lemma fresh_image_atom [simp]: "finite A \<Longrightarrow> i \<sharp> atom ` A \<longleftrightarrow> i \<sharp> A"
   by (induct rule: finite_induct) (auto simp: fresh_finite_insert)
 
 lemma atom_fresh_star_atom_set_conv: "\<lbrakk>atom i \<sharp> bs; finite bs\<rbrakk> \<Longrightarrow> bs \<sharp>* i"
 by (metis fresh_finite_atom_set fresh_ineq_at_base fresh_star_def)
 
 lemma notin_V:
   assumes p: "atom i \<sharp> p" and V: "finite V" "atom ` (p \<bullet> V) \<sharp>* V"
   shows "i \<notin> V" "i \<notin> p \<bullet> V"
   using V
   apply (auto simp: fresh_def fresh_star_def supp_finite_set_at_base)
   apply (metis p mem_permute_iff fresh_at_base_permI)+
   done
 
-section{*Simultaneous Substitution*}
+section\<open>Simultaneous Substitution\<close>
 
 definition ssubst :: "tm \<Rightarrow> name set \<Rightarrow> (name \<Rightarrow> tm) \<Rightarrow> tm"
   where "ssubst t V F = Finite_Set.fold (\<lambda>i. subst i (F i)) t V"
 
 definition make_F :: "name set \<Rightarrow> perm \<Rightarrow> name \<Rightarrow> tm"
   where "make_F Vs p \<equiv> \<lambda>i. if i \<in> Vs then Var (p \<bullet> i) else Var i"
 
 lemma ssubst_empty [simp]: "ssubst t {} F = t"
   by (simp add: ssubst_def)
 
-text{*Renaming a finite set of variables. Based on the theorem @{text at_set_avoiding}*}
+text\<open>Renaming a finite set of variables. Based on the theorem \<open>at_set_avoiding\<close>\<close>
 locale quote_perm =
   fixes p :: perm and Vs :: "name set" and F :: "name \<Rightarrow> tm"
   assumes p: "atom ` (p \<bullet> Vs) \<sharp>* Vs"
       and pinv: "-p = p"
       and Vs:  "finite Vs"
   defines "F \<equiv> make_F Vs p"
 begin
 
 lemma F_unfold: "F i = (if i \<in> Vs then Var (p \<bullet> i) else Var i)"
   by (simp add: F_def make_F_def)
 
 lemma finite_V [simp]: "V \<subseteq> Vs \<Longrightarrow> finite V"
   by (metis Vs finite_subset)
 
 lemma perm_exits_Vs: "i \<in> Vs \<Longrightarrow> (p \<bullet> i) \<notin> Vs"
   by (metis Vs fresh_finite_set_at_base imageI fresh_star_def mem_permute_iff p)
 
 lemma atom_fresh_perm: "\<lbrakk>x \<in> Vs; y \<in> Vs\<rbrakk> \<Longrightarrow> atom x \<sharp> p \<bullet> y"
   by (metis imageI Vs p fresh_finite_set_at_base fresh_star_def mem_permute_iff fresh_at_base(2))
 
 lemma fresh_pj: "\<lbrakk>a \<sharp> p; j \<in> Vs\<rbrakk> \<Longrightarrow> a \<sharp> p \<bullet> j"
   by (metis atom_fresh_perm fresh_at_base(2) fresh_perm fresh_permute_left pinv)
 
 lemma fresh_Vs: "a \<sharp> p \<Longrightarrow> a \<sharp> Vs"
   by (metis Vs fresh_def fresh_perm fresh_permute_iff fresh_star_def p permute_finite supp_finite_set_at_base)
 
 lemma fresh_pVs: "a \<sharp> p \<Longrightarrow> a \<sharp> p \<bullet> Vs"
   by (metis fresh_Vs fresh_perm fresh_permute_left pinv)
 
 lemma assumes "V \<subseteq> Vs" "a \<sharp> p"
       shows fresh_pV [simp]: "a \<sharp> p \<bullet> V" and fresh_V [simp]: "a \<sharp> V"
   using fresh_pVs fresh_Vs assms
   apply (auto simp: fresh_def)
   apply (metis (full_types) Vs finite_V permute_finite set_mp subset_Un_eq supp_of_finite_union union_eqvt)
   by (metis Vs finite_V set_mp subset_Un_eq supp_of_finite_union)
 
 lemma qp_insert:
   fixes i::name and i'::name
   assumes "atom i \<sharp> p" "atom i' \<sharp> (i,p)"
     shows  "quote_perm ((atom i \<rightleftharpoons> atom i') + p) (insert i Vs)"
 using p pinv Vs assms
   by (auto simp: quote_perm_def fresh_at_base_permI atom_fresh_star_atom_set_conv swap_fresh_fresh
                  fresh_star_finite_insert fresh_finite_insert perm_self_inverseI)
 
 lemma subst_F_left_commute: "subst x (F x) (subst y (F y) t) = subst y (F y) (subst x (F x) t)"
   by (metis subst_tm_commute2 F_unfold subst_tm_id F_unfold atom_fresh_perm tm.fresh(2))
 
 lemma
   assumes "finite V" "i \<notin> V"
   shows ssubst_insert:  "ssubst t (insert i V) F = subst i (F i) (ssubst t V F)"  (is ?thesis1)
     and ssubst_insert2: "ssubst t (insert i V) F = ssubst (subst i (F i) t) V F"  (is ?thesis2)
 proof -
   interpret comp_fun_commute "(\<lambda>i. subst i (F i))"
   proof qed (simp add: subst_F_left_commute fun_eq_iff)
   show ?thesis1 using assms Vs
     by (simp add: ssubst_def)
   show ?thesis2 using assms Vs
     by (simp add: ssubst_def fold_insert2 del: fold_insert)
 qed
 
 lemma ssubst_insert_if:
   "finite V \<Longrightarrow>
      ssubst t (insert i V) F = (if i \<in> V then ssubst t V F
                                          else subst i (F i) (ssubst t V F))"
   by (simp add: ssubst_insert insert_absorb)
 
 lemma ssubst_single [simp]: "ssubst t {i} F = subst i (F i) t"
   by (simp add: ssubst_insert)
 
 lemma ssubst_Var_if [simp]:
   assumes "finite V"
     shows "ssubst (Var i) V F = (if i \<in> V then F i else Var i)"
 using assms
   apply (induction V, auto)
   apply (metis ssubst_insert subst.simps(2))
   apply (metis ssubst_insert2 subst.simps(2))+
   done
 
 lemma ssubst_Zero [simp]: "finite V \<Longrightarrow> ssubst Zero V F = Zero"
   by (induct V rule: finite_induct) (auto simp: ssubst_insert)
 
 lemma ssubst_Eats [simp]: "finite V \<Longrightarrow> ssubst (Eats t u) V F = Eats (ssubst t V F) (ssubst u V F)"
   by (induct V rule: finite_induct) (auto simp: ssubst_insert)
 
 lemma ssubst_SUCC [simp]: "finite V \<Longrightarrow> ssubst (SUCC t) V F = SUCC (ssubst t V F)"
   by (metis SUCC_def ssubst_Eats)
 
 lemma ssubst_ORD_OF [simp]: "finite V \<Longrightarrow> ssubst (ORD_OF n) V F = ORD_OF n"
   by (induction n) auto
 
 lemma ssubst_HPair [simp]:
   "finite V \<Longrightarrow> ssubst (HPair t u) V F = HPair (ssubst t V F) (ssubst u V F)"
   by (simp add: HPair_def)
 
 lemma ssubst_HTuple [simp]: "finite V \<Longrightarrow> ssubst (HTuple n) V F = (HTuple n)"
   by (induction n) (auto simp: HTuple.simps)
 
 lemma ssubst_Subset:
   assumes "finite V" shows "ssubst \<lfloor>t SUBS u\<rfloor>V V F = Q_Subset (ssubst \<lfloor>t\<rfloor>V V F) (ssubst \<lfloor>u\<rfloor>V V F)"
 proof -
   obtain i::name where "atom i \<sharp> (t,u)"
     by (rule obtain_fresh)
   thus ?thesis using assms
     by (auto simp: Subset.simps [of i] vquot_fm_def vquot_tm_def trans_tm_forget)
 qed
 
 lemma fresh_ssubst:
   assumes "finite V" "a \<sharp> p \<bullet> V" "a \<sharp> t"
     shows "a \<sharp> ssubst t V F"
 using assms
   by (induct V)
      (auto simp: ssubst_insert_if fresh_finite_insert F_unfold intro: fresh_ineq_at_base)
 
 lemma fresh_ssubst':
   assumes "finite V" "atom i \<sharp> t" "atom (p \<bullet> i) \<sharp> t"
     shows "atom i \<sharp> ssubst t V F"
 using assms
   by (induct t rule: tm.induct) (auto simp: F_unfold fresh_permute_left pinv)
 
 lemma ssubst_vquot_Ex:
   "\<lbrakk>finite V; atom i \<sharp> p \<bullet> V\<rbrakk>
    \<Longrightarrow> ssubst \<lfloor>Ex i A\<rfloor>(insert i V) (insert i V) F = ssubst \<lfloor>Ex i A\<rfloor>V V F"
    by (simp add: ssubst_insert_if insert_absorb vquot_fm_insert fresh_ssubst)
 
 lemma ground_ssubst_eq: "\<lbrakk>finite V; supp t = {}\<rbrakk> \<Longrightarrow> ssubst t V F = t"
   by (induct V rule: finite_induct) (auto simp: ssubst_insert fresh_def)
 
 lemma ssubst_quot_tm [simp]:
   fixes t::tm shows "finite V \<Longrightarrow> ssubst \<lceil>t\<rceil> V F = \<lceil>t\<rceil>"
   by (simp add: ground_ssubst_eq supp_conv_fresh)
 
 lemma ssubst_quot_fm [simp]:
   fixes A::fm shows "finite V \<Longrightarrow> ssubst \<lceil>A\<rceil> V F = \<lceil>A\<rceil>"
   by (simp add: ground_ssubst_eq supp_conv_fresh)
 
 lemma atom_in_p_Vs: "\<lbrakk>i \<in> p \<bullet> V; V \<subseteq> Vs\<rbrakk> \<Longrightarrow> i \<in> p \<bullet> Vs"
   by (metis (full_types) True_eqvt set_mp subset_eqvt)
 
 
-section{*The Main Theorems of Section 7*}
+section\<open>The Main Theorems of Section 7\<close>
 
 lemma SubstTermP_vquot_dbtm:
   assumes w: "w \<in> Vs - V" and V: "V \<subseteq> Vs" "V' = p \<bullet> V"
       and s: "supp dbtm \<subseteq> atom ` Vs"
   shows
   "insert (ConstP (F w)) {ConstP (F i) | i. i \<in> V}
    \<turnstile> SubstTermP \<lceil>Var w\<rceil> (F w)
                 (ssubst (vquot_dbtm V dbtm) V F)
                 (subst w (F w) (ssubst (vquot_dbtm (insert w V) dbtm) V F))"
 using s
 proof (induct dbtm rule: dbtm.induct)
   case DBZero thus ?case using V w
     by (auto intro: SubstTermP_Zero [THEN cut1] ConstP_imp_TermP [THEN cut1])
 next
   case (DBInd n) thus ?case using V
     apply auto
     apply (rule thin [of "{ConstP (F w)}"])
     apply (rule SubstTermP_Ind [THEN cut3])
     apply (auto simp: IndP_Q_Ind OrdP_ORD_OF ConstP_imp_TermP)
     done
 next
   case (DBVar i) show ?case
   proof (cases "i \<in> V'")
     case True hence "i \<notin> Vs" using assms
       by (metis p Vs atom_in_atom_image atom_in_p_Vs fresh_finite_set_at_base fresh_star_def)
     thus ?thesis using DBVar True V
       by auto
   next
     case False thus ?thesis using DBVar V w
       apply (auto simp: quot_Var [symmetric])
         apply (blast intro: thin [of "{ConstP (F w)}"] ConstP_imp_TermP
                             SubstTermP_Var_same [THEN cut2])
        apply (subst forget_subst_tm, metis F_unfold atom_fresh_perm tm.fresh(2))
        apply (blast intro: Hyp thin [of "{ConstP (F w)}"] ConstP_imp_TermP
                            SubstTermP_Const [THEN cut2])
        apply (blast intro: Hyp thin [of "{ConstP (F w)}"] ConstP_imp_TermP EQ_quot_tm_Fls
                            SubstTermP_Var_diff [THEN cut4])
       done
   qed
 next
   case (DBEats tm1 tm2) thus ?case using V
     by (auto simp: SubstTermP_Eats [THEN cut2])
 qed
 
 lemma SubstFormP_vquot_dbfm:
   assumes w: "w \<in> Vs - V" and V: "V \<subseteq> Vs" "V' = p \<bullet> V"
       and s: "supp dbfm \<subseteq> atom ` Vs"
   shows
   "insert (ConstP (F w)) {ConstP (F i) | i. i \<in> V}
    \<turnstile> SubstFormP \<lceil>Var w\<rceil> (F w)
                 (ssubst (vquot_dbfm V dbfm) V F)
                 (subst w (F w) (ssubst (vquot_dbfm (insert w V) dbfm) V F))"
 using w s
 proof (induct dbfm rule: dbfm.induct)
   case (DBMem t u) thus ?case using V
     by (auto intro: SubstTermP_vquot_dbtm SubstFormP_Mem [THEN cut2])
 next
   case (DBEq t u) thus ?case using V
     by (auto intro: SubstTermP_vquot_dbtm SubstFormP_Eq [THEN cut2])
 next
   case (DBDisj A B) thus ?case using V
     by (auto intro: SubstFormP_Disj [THEN cut2])
 next
   case (DBNeg A) thus ?case using V
     by (auto intro: SubstFormP_Neg [THEN cut1])
 next
   case (DBEx A) thus ?case using V
     by (auto intro: SubstFormP_Ex [THEN cut1])
 qed
 
-text{*Lemmas 7.5 and 7.6*}
+text\<open>Lemmas 7.5 and 7.6\<close>
 lemma ssubst_SubstFormP:
   fixes A::fm
   assumes w: "w \<in> Vs - V" and V: "V \<subseteq> Vs" "V' = p \<bullet> V"
       and s: "supp A \<subseteq> atom ` Vs"
   shows
   "insert (ConstP (F w)) {ConstP (F i) | i. i \<in> V}
    \<turnstile> SubstFormP \<lceil>Var w\<rceil> (F w)
                 (ssubst \<lfloor>A\<rfloor>V V F)
                 (ssubst \<lfloor>A\<rfloor>(insert w V) (insert w V) F)"
 proof -
   have "w \<notin> V" using assms
     by auto
   thus ?thesis using assms
     by (simp add: vquot_fm_def supp_conv_fresh ssubst_insert_if SubstFormP_vquot_dbfm)
 qed
 
-text{*Theorem 7.3*}
+text\<open>Theorem 7.3\<close>
 theorem PfP_implies_PfP_ssubst:
   fixes \<beta>::fm
   assumes \<beta>: "{} \<turnstile> PfP \<lceil>\<beta>\<rceil>"
       and V: "V \<subseteq> Vs"
       and s: "supp \<beta> \<subseteq> atom ` Vs"
     shows    "{ConstP (F i) | i. i \<in> V} \<turnstile> PfP (ssubst \<lfloor>\<beta>\<rfloor>V V F)"
 proof -
   show ?thesis using finite_V [OF V] V
   proof induction
     case empty thus ?case
       by (auto simp: \<beta>)
   next
     case (insert i V)
     thus ?case using assms
       by (auto simp: Collect_disj_Un fresh_finite_set_at_base
                intro: PfP_implies_SubstForm_PfP thin1 ssubst_SubstFormP)
   qed
 qed
 
 end
 
 end
 
diff --git a/thys/Goedel_HFSet_Semanticless/Quote.thy b/thys/Goedel_HFSet_Semanticless/Quote.thy
--- a/thys/Goedel_HFSet_Semanticless/Quote.thy
+++ b/thys/Goedel_HFSet_Semanticless/Quote.thy
@@ -1,1437 +1,1437 @@
-chapter{*Quotations of the Free Variables*}
+chapter\<open>Quotations of the Free Variables\<close>
 
 theory Quote
 imports Pseudo_Coding
 begin
 
-section {*Sequence version of the ``Special p-Function, F*''*}
+section \<open>Sequence version of the ``Special p-Function, F*''\<close>
 
-text{*The definition below describes a relation, not a function.
-      This material relates to Section 8, but omits the ordering of the universe.*}
+text\<open>The definition below describes a relation, not a function.
+      This material relates to Section 8, but omits the ordering of the universe.\<close>
 
-subsection {*Defining the syntax: quantified body*}
+subsection \<open>Defining the syntax: quantified body\<close>
 
 nominal_function SeqQuoteP :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom l \<sharp> (s,k,sl,sl',m,n,sm,sm',sn,sn');
           atom sl \<sharp> (s,sl',m,n,sm,sm',sn,sn'); atom sl' \<sharp> (s,m,n,sm,sm',sn,sn');
           atom m \<sharp> (s,n,sm,sm',sn,sn');  atom n \<sharp> (s,sm,sm',sn,sn');
           atom sm \<sharp> (s,sm',sn,sn');  atom sm' \<sharp> (s,sn,sn');
           atom sn \<sharp> (s,sn');  atom sn' \<sharp> s\<rbrakk> \<Longrightarrow>
     SeqQuoteP t u s k =
       LstSeqP s k (HPair t u) AND
       All2 l (SUCC k) (Ex sl (Ex sl' (HPair (Var l) (HPair (Var sl) (Var sl')) IN s AND
                 ((Var sl EQ Zero AND Var sl' EQ Zero) OR
                 Ex m (Ex n (Ex sm (Ex sm' (Ex sn (Ex sn' (Var m IN Var l AND Var n IN Var l AND
                        HPair (Var m) (HPair (Var sm) (Var sm')) IN s AND
                        HPair (Var n) (HPair (Var sn) (Var sn')) IN s AND
                        Var sl EQ Eats (Var sm) (Var sn) AND
                        Var sl' EQ Q_Eats (Var sm') (Var sn')))))))))))"
 by (auto simp: eqvt_def SeqQuoteP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma
   shows SeqQuoteP_fresh_iff [simp]:
       "a \<sharp> SeqQuoteP t u s k \<longleftrightarrow> a \<sharp> t \<and> a \<sharp> u \<and> a \<sharp> s \<and> a \<sharp> k"  (is ?thesis1)
   and SeqQuoteP_sf [iff]:
       "Sigma_fm (SeqQuoteP t u s k)"    (is ?thsf)
   and SeqQuoteP_imp_OrdP:
       "{ SeqQuoteP t u s k } \<turnstile> OrdP k"  (is ?thord)
   and SeqQuoteP_imp_LstSeqP:
       "{ SeqQuoteP t u s k } \<turnstile> LstSeqP s k (HPair t u)"  (is ?thlstseq)
 proof -
   obtain l::name and sl::name and sl'::name and m::name and n::name and
          sm::name and sm'::name and sn::name and sn'::name
     where atoms:
          "atom l \<sharp> (s,k,sl,sl',m,n,sm,sm',sn,sn')"
          "atom sl \<sharp> (s,sl',m,n,sm,sm',sn,sn')"  "atom sl' \<sharp> (s,m,n,sm,sm',sn,sn')"
          "atom m \<sharp> (s,n,sm,sm',sn,sn')"  "atom n \<sharp> (s,sm,sm',sn,sn')"
          "atom sm \<sharp> (s,sm',sn,sn')"  "atom sm' \<sharp> (s,sn,sn')"
          "atom sn \<sharp> (s,sn')"  "atom sn' \<sharp> s"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf ?thord ?thlstseq
     by auto (auto simp: LstSeqP.simps)
 qed
 
 lemma SeqQuoteP_subst [simp]:
       "(SeqQuoteP t u s k)(j::=w) =
        SeqQuoteP (subst j w t) (subst j w u) (subst j w s) (subst j w k)"
 proof -
   obtain l::name and sl::name and sl'::name and m::name and n::name and
          sm::name and sm'::name and sn::name and sn'::name
     where "atom l \<sharp> (s,k,w,j,sl,sl',m,n,sm,sm',sn,sn')"
           "atom sl \<sharp> (s,w,j,sl',m,n,sm,sm',sn,sn')"  "atom sl' \<sharp> (s,w,j,m,n,sm,sm',sn,sn')"
           "atom m \<sharp> (s,w,j,n,sm,sm',sn,sn')"  "atom n \<sharp> (s,w,j,sm,sm',sn,sn')"
           "atom sm \<sharp> (s,w,j,sm',sn,sn')"  "atom sm' \<sharp> (s,w,j,sn,sn')"
           "atom sn \<sharp> (s,w,j,sn')"  "atom sn' \<sharp> (s,w,j)"
     by (metis obtain_fresh)
   thus ?thesis
     by (force simp add: SeqQuoteP.simps [of l _ _ sl sl' m n sm sm' sn sn'])
 qed
 
 declare SeqQuoteP.simps [simp del]
 
-subsection {*Correctness properties*}
+subsection \<open>Correctness properties\<close>
 
 lemma SeqQuoteP_lemma:
   fixes m::name and sm::name and sm'::name and n::name and sn::name and sn'::name
   assumes "atom m \<sharp> (t,u,s,k,n,sm,sm',sn,sn')"  "atom n \<sharp> (t,u,s,k,sm,sm',sn,sn')"
           "atom sm \<sharp> (t,u,s,k,sm',sn,sn')"  "atom sm' \<sharp> (t,u,s,k,sn,sn')"
           "atom sn \<sharp> (t,u,s,k,sn')"  "atom sn' \<sharp> (t,u,s,k)"
     shows "{ SeqQuoteP t u s k }
            \<turnstile> (t EQ Zero AND u EQ Zero) OR
              Ex m (Ex n (Ex sm (Ex sm' (Ex sn (Ex sn' (Var m IN k AND Var n IN k AND
                  SeqQuoteP (Var sm) (Var sm') s (Var m) AND
                  SeqQuoteP (Var sn) (Var sn') s (Var n) AND
                  t EQ Eats (Var sm) (Var sn) AND
                  u EQ Q_Eats (Var sm') (Var sn')))))))"
 proof -
   obtain l::name and sl::name and sl'::name
     where "atom l \<sharp> (t,u,s,k,sl,sl',m,n,sm,sm',sn,sn')"
           "atom sl \<sharp> (t,u,s,k,sl',m,n,sm,sm',sn,sn')"
           "atom sl' \<sharp> (t,u,s,k,m,n,sm,sm',sn,sn')"
     by (metis obtain_fresh)
   thus ?thesis using assms
     apply (simp add: SeqQuoteP.simps [of l s k sl sl' m n sm sm' sn sn'])
     apply (rule Conj_EH Ex_EH All2_SUCC_E [THEN rotate2] | simp)+
     apply (rule cut_same [where A = "HPair t u EQ HPair (Var sl) (Var sl')"])
     apply (metis Assume AssumeH(4) LstSeqP_EQ)
     apply clarify
     apply (rule Disj_EH)
     apply (rule Disj_I1)
     apply (rule anti_deduction)
     apply (rule Var_Eq_subst_Iff [THEN Sym_L, THEN Iff_MP_same])
     apply (rule rotate2)
     apply (rule Var_Eq_subst_Iff [THEN Sym_L, THEN Iff_MP_same], force)
     \<comment> \<open>now the quantified case\<close>
     apply (rule Ex_EH Conj_EH)+
     apply simp_all
     apply (rule Disj_I2)
     apply (rule Ex_I [where x = "Var m"], simp)
     apply (rule Ex_I [where x = "Var n"], simp)
     apply (rule Ex_I [where x = "Var sm"], simp)
     apply (rule Ex_I [where x = "Var sm'"], simp)
     apply (rule Ex_I [where x = "Var sn"], simp)
     apply (rule Ex_I [where x = "Var sn'"], simp)
     apply (simp_all add: SeqQuoteP.simps [of l s _ sl sl' m n sm sm' sn sn'])
     apply ((rule Conj_I)+, blast intro: LstSeqP_Mem)+
     \<comment> \<open>first SeqQuoteP subgoal\<close>
     apply (rule All2_Subset [OF Hyp])
     apply (blast intro!: SUCC_Subset_Ord LstSeqP_OrdP)+
     apply simp
     \<comment> \<open>next SeqQuoteP subgoal\<close>
     apply ((rule Conj_I)+, blast intro: LstSeqP_Mem)+
     apply (rule All2_Subset [OF Hyp], blast)
     apply (auto intro!: SUCC_Subset_Ord LstSeqP_OrdP intro: Trans)
     done
 qed
 
-section {*The ``special function'' itself*}
+section \<open>The ``special function'' itself\<close>
 
 nominal_function QuoteP :: "tm \<Rightarrow> tm \<Rightarrow> fm"
   where "\<lbrakk>atom s \<sharp> (t,u,k); atom k \<sharp> (t,u)\<rbrakk> \<Longrightarrow>
     QuoteP t u = Ex s (Ex k (SeqQuoteP t u (Var s) (Var k)))"
 by (auto simp: eqvt_def QuoteP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 
 lemma
   shows QuoteP_fresh_iff [simp]: "a \<sharp> QuoteP t u \<longleftrightarrow> a \<sharp> t \<and> a \<sharp> u"  (is ?thesis1)
   and QuoteP_sf [iff]: "Sigma_fm (QuoteP t u)"  (is ?thsf)
 proof -
   obtain s::name and k::name  where "atom s \<sharp> (t,u,k)"  "atom k \<sharp> (t,u)"
     by (metis obtain_fresh)
   thus ?thesis1 ?thsf
     by auto
 qed
 
 lemma QuoteP_subst [simp]:
   "(QuoteP t u)(j::=w) = QuoteP (subst j w t) (subst j w u)"
 proof -
   obtain s::name and k::name  where "atom s \<sharp> (t,u,w,j,k)"  "atom k \<sharp> (t,u,w,j)"
     by (metis obtain_fresh)
   thus ?thesis
     by (simp add: QuoteP.simps [of s _ _ k])
 qed
 
 declare QuoteP.simps [simp del]
 
-subsection {*Correctness properties*}
+subsection \<open>Correctness properties\<close>
 
 lemma QuoteP_Zero: "{} \<turnstile> QuoteP Zero Zero"
 proof -
  obtain l :: "name"
     and sl :: "name"
     and sl' :: "name"
     and m :: "name"
     and n :: "name"
     and sm :: "name"
     and sm' :: "name"
     and sn :: "name"
     and sn' :: "name"
     and s :: "name"
     and k :: "name"
   where "atom l \<sharp> (s, k, sl, sl', m, n, sm, sm', sn, sn')"
     and "atom sl \<sharp> (s, k, sl', m, n, sm, sm', sn, sn')"
     and "atom sl' \<sharp> (s, k, m, n, sm, sm', sn, sn')"
     and "atom m \<sharp> (s, k, n, sm, sm', sn, sn')"
     and "atom n \<sharp> (s, k, sm, sm', sn, sn')"
     and "atom sm \<sharp> (s, k, sm', sn, sn')"
     and "atom sm' \<sharp> (s, k, sn, sn')"
     and "atom sn \<sharp> (s, k, sn')"
     and "atom sn' \<sharp> (s, k)"
     and "atom k \<sharp> s"
     by (metis obtain_fresh)
   then show ?thesis
     apply (subst QuoteP.simps[of s _ _ k]; simp)
     apply (rule Ex_I[of _ _ _ "Eats Zero (HPair Zero (HPair Zero Zero))"]; simp)
     apply (rule Ex_I[of _ _ _ "Zero"]; simp)
     apply (subst SeqQuoteP.simps[of l _ _ sl sl' m n sm sm' sn sn']; simp?)
     apply (rule Conj_I)
      apply (rule LstSeqP_single)
     apply (auto intro!: Ex_I[of _ _ _ Zero])
      apply (rule Mem_SUCC_E[OF Mem_Zero_E])
      apply (rule Mem_Eats_I2)
      apply (rule HPair_cong[OF Assume Refl])
     apply (auto intro!: Disj_I1)
     done
 qed
 
 lemma SeqQuoteP_Eats:
   assumes "atom s \<sharp> (k,s1,s2,k1,k2,t1,t2,u1,u2)" "atom k \<sharp> (s1,s2,k1,k2,t1,t2,u1,u2)"
     shows "{SeqQuoteP t1 u1 s1 k1, SeqQuoteP t2 u2 s2 k2} \<turnstile>
            Ex s (Ex k (SeqQuoteP (Eats t1 t2) (Q_Eats u1 u2) (Var s) (Var k)))"
 proof -
   obtain km::name and kn::name and j::name and k'::name and l::name
      and sl::name and sl'::name and m::name and n::name and sm::name
      and sm'::name and sn::name and sn'::name
    where atoms2:
          "atom km \<sharp> (kn,j,k',l,s1,s2,s,k1,k2,k,t1,t2,u1,u2,sl,sl',m,n,sm,sm',sn,sn')"
          "atom kn \<sharp> (j,k',l,s1,s2,s,k1,k2,k,t1,t2,u1,u2,sl,sl',m,n,sm,sm',sn,sn')"
          "atom j \<sharp> (k',l,s1,s2,s,k1,k2,k,t1,t2,u1,u2,sl,sl',m,n,sm,sm',sn,sn')"
      and atoms: "atom k' \<sharp> (l,s1,s2,s,k1,k2,k,t1,t2,u1,u2,sl,sl',m,n,sm,sm',sn,sn')"
          "atom l \<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,sl,sl',m,n,sm,sm',sn,sn')"
          "atom sl \<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,sl',m,n,sm,sm',sn,sn')"
          "atom sl' \<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,m,n,sm,sm',sn,sn')"
          "atom m \<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,n,sm,sm',sn,sn')"
          "atom n \<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,sm,sm',sn,sn')"
          "atom sm \<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,sm',sn,sn')"
          "atom sm' \<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,sn,sn')"
          "atom sn \<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2,sn')"
          "atom sn' \<sharp> (s1,s2,s,k1,k2,k,t1,t2,u1,u2)"
     by (metis obtain_fresh)
   show ?thesis
      using assms atoms
      apply (auto simp: SeqQuoteP.simps [of l "Var s" _ sl sl' m n sm sm' sn sn'])
      apply (rule cut_same [where A="OrdP k1 AND OrdP k2"])
      apply (metis Conj_I SeqQuoteP_imp_OrdP thin1 thin2)
      apply (rule cut_same [OF exists_SeqAppendP [of s s1 "SUCC k1" s2 "SUCC k2"]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
      apply (rule cut_same [OF exists_HaddP [where j=k' and x=k1 and y=k2]])
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
      apply (rule Ex_I [where x="Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Eats t1 t2) (Q_Eats u1 u2)))"])
      apply (simp_all (no_asm_simp))
      apply (rule Ex_I [where x="SUCC (SUCC (Var k'))"])
      apply simp
      apply (rule Conj_I [OF LstSeqP_SeqAppendP_Eats])
      apply (blast intro: SeqQuoteP_imp_LstSeqP [THEN cut1])+
      proof (rule All2_SUCC_I, simp_all)
        show "{HaddP k1 k2 (Var k'), OrdP k1, OrdP k2, SeqAppendP s1 (SUCC k1) s2 (SUCC k2) (Var s),
               SeqQuoteP t1 u1 s1 k1, SeqQuoteP t2 u2 s2 k2}
              \<turnstile> Ex sl (Ex sl'
                  (HPair (SUCC (SUCC (Var k'))) (HPair (Var sl) (Var sl')) IN
                   Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Eats t1 t2) (Q_Eats u1 u2))) AND
                   (Var sl EQ Zero AND Var sl' EQ Zero OR
                    Ex m (Ex n (Ex sm (Ex sm' (Ex sn (Ex sn'
                     (Var m IN SUCC (SUCC (Var k')) AND
                      Var n IN SUCC (SUCC (Var k')) AND
                      HPair (Var m) (HPair (Var sm) (Var sm')) IN
                      Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Eats t1 t2) (Q_Eats u1 u2))) AND
                      HPair (Var n) (HPair (Var sn) (Var sn')) IN
                      Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Eats t1 t2) (Q_Eats u1 u2))) AND
                      Var sl EQ Eats (Var sm) (Var sn) AND Var sl' EQ Q_Eats (Var sm') (Var sn'))))))))))"
        \<comment> \<open>verifying the final values\<close>
        apply (rule Ex_I [where x="Eats t1 t2"])
        using assms atoms apply simp
        apply (rule Ex_I [where x="Q_Eats u1 u2"], simp)
        apply (rule Conj_I [OF Mem_Eats_I2 [OF Refl]])
        apply (rule Disj_I2)
        apply (rule Ex_I [where x=k1], simp)
        apply (rule Ex_I [where x="SUCC (Var k')"], simp)
        apply (rule Ex_I [where x=t1], simp)
        apply (rule Ex_I [where x=u1], simp)
        apply (rule Ex_I [where x=t2], simp)
        apply (rule Ex_I [where x=u2], simp)
        apply (rule Conj_I)
        apply (blast intro: HaddP_Mem_I Mem_SUCC_I1)
        apply (rule Conj_I [OF Mem_SUCC_Refl])
        apply (rule Conj_I)
        apply (blast intro: Mem_Eats_I1 SeqAppendP_Mem1 [THEN cut3] Mem_SUCC_Refl
                 SeqQuoteP_imp_LstSeqP [THEN cut1] LstSeqP_imp_Mem)
        apply (blast intro: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] Mem_SUCC_Refl
                 SeqQuoteP_imp_LstSeqP [THEN cut1] LstSeqP_imp_Mem  HaddP_SUCC1 [THEN cut1])
        done
      next
        show "{HaddP k1 k2 (Var k'), OrdP k1, OrdP k2, SeqAppendP s1 (SUCC k1) s2 (SUCC k2) (Var s),
               SeqQuoteP t1 u1 s1 k1, SeqQuoteP t2 u2 s2 k2}
              \<turnstile> All2 l (SUCC (SUCC (Var k')))
                  (Ex sl (Ex sl'
                    (HPair (Var l) (HPair (Var sl) (Var sl')) IN
                     Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Eats t1 t2) (Q_Eats u1 u2))) AND
                     (Var sl EQ Zero AND Var sl' EQ Zero OR
                      Ex m (Ex n (Ex sm (Ex sm' (Ex sn (Ex sn'
                       (Var m IN Var l AND
                        Var n IN Var l AND
                        HPair (Var m) (HPair (Var sm) (Var sm')) IN
                        Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Eats t1 t2) (Q_Eats u1 u2))) AND
                        HPair (Var n) (HPair (Var sn) (Var sn')) IN
                        Eats (Var s) (HPair (SUCC (SUCC (Var k'))) (HPair (Eats t1 t2) (Q_Eats u1 u2))) AND
                        Var sl EQ Eats (Var sm) (Var sn) AND Var sl' EQ Q_Eats (Var sm') (Var sn')))))))))))"
      \<comment> \<open>verifying the sequence buildup\<close>
      apply (rule cut_same [where A="HaddP (SUCC k1) (SUCC k2) (SUCC (SUCC (Var k')))"])
      apply (blast intro: HaddP_SUCC1 [THEN cut1] HaddP_SUCC2 [THEN cut1])
      apply (rule All_I Imp_I)+
      apply (rule HaddP_Mem_cases [where i=j])
      using assms atoms atoms2 apply simp_all
      apply (rule AssumeH)
      apply (blast intro: OrdP_SUCC_I)
      \<comment> \<open>... the sequence buildup via s1\<close>
      apply (simp add: SeqQuoteP.simps [of l s1 _ sl sl' m n sm sm' sn sn'])
      apply (rule AssumeH Ex_EH Conj_EH)+
      apply (rule All2_E [THEN rotate2])
      apply (simp | rule AssumeH Ex_EH Conj_EH)+
      apply (rule Ex_I [where x="Var sl"], simp)
      apply (rule Ex_I [where x="Var sl'"], simp)
      apply (rule Conj_I)
      apply (rule Mem_Eats_I1)
      apply (metis SeqAppendP_Mem1 rotate3 thin2 thin4)
      apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
      apply (rule Ex_I [where x="Var m"], simp)
      apply (rule Ex_I [where x="Var n"], simp)
      apply (rule Ex_I [where x="Var sm"], simp)
      apply (rule Ex_I [where x="Var sm'"], simp)
      apply (rule Ex_I [where x="Var sn"], simp)
      apply (rule Ex_I [where x="Var sn'"], simp_all)
      apply (rule Conj_I, rule AssumeH)+
      apply (blast intro: OrdP_Trans [OF OrdP_SUCC_I] Mem_Eats_I1 [OF SeqAppendP_Mem1 [THEN cut3]] Hyp)
      \<comment> \<open>... the sequence buildup via s2\<close>
      apply (simp add: SeqQuoteP.simps [of l s2 _ sl sl' m n sm sm' sn sn'])
      apply (rule AssumeH Ex_EH Conj_EH)+
      apply (rule All2_E [THEN rotate2])
      apply (simp | rule AssumeH Ex_EH Conj_EH)+
      apply (rule Ex_I [where x="Var sl"], simp)
      apply (rule Ex_I [where x="Var sl'"], simp)
      apply (rule cut_same [where A="OrdP (Var j)"])
      apply (metis HaddP_imp_OrdP rotate2 thin2)
      apply (rule Conj_I)
      apply (blast intro: Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4]  del: Disj_EH)
      apply (rule AssumeH Disj_IE1H Ex_EH Conj_EH)+
      apply (rule cut_same [OF exists_HaddP [where j=km and x="SUCC k1" and y="Var m"]])
      apply (blast intro: Ord_IN_Ord, simp)
      apply (rule cut_same [OF exists_HaddP [where j=kn and x="SUCC k1" and y="Var n"]])
      apply (metis AssumeH(6) Ord_IN_Ord0 rotate8, simp)
      apply (rule AssumeH Ex_EH Conj_EH | simp)+
      apply (rule Ex_I [where x="Var km"], simp)
      apply (rule Ex_I [where x="Var kn"], simp)
      apply (rule Ex_I [where x="Var sm"], simp)
      apply (rule Ex_I [where x="Var sm'"], simp)
      apply (rule Ex_I [where x="Var sn"], simp)
      apply (rule Ex_I [where x="Var sn'"], simp_all)
      apply (rule Conj_I [OF _ Conj_I])
      apply (blast intro: Hyp OrdP_SUCC_I HaddP_Mem_cancel_left [THEN Iff_MP2_same])
      apply (blast intro: Hyp OrdP_SUCC_I HaddP_Mem_cancel_left [THEN Iff_MP2_same])
      apply (blast intro: Hyp Mem_Eats_I1 SeqAppendP_Mem2 [THEN cut4] OrdP_Trans HaddP_imp_OrdP [THEN cut1])
      done
    qed
 qed
 
 
 lemma QuoteP_Eats: "{QuoteP t1 u1, QuoteP t2 u2} \<turnstile> QuoteP (Eats t1 t2) (Q_Eats u1 u2)"
 proof -
   obtain k1::name and s1::name and k2::name and s2::name and k::name and s::name
     where "atom s1 \<sharp> (t1,u1,t2,u2)"             "atom k1 \<sharp> (t1,u1,t2,u2,s1)"
           "atom s2 \<sharp> (t1,u1,t2,u2,k1,s1)"       "atom k2 \<sharp> (t1,u1,t2,u2,s2,k1,s1)"
           "atom s  \<sharp> (t1,u1,t2,u2,k2,s2,k1,s1)" "atom k  \<sharp> (t1,u1,t2,u2,s,k2,s2,k1,s1)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: QuoteP.simps [of s _ "(Q_Eats u1 u2)" k]
                    QuoteP.simps [of s1 t1 u1 k1] QuoteP.simps [of s2 t2 u2 k2]
              intro!: SeqQuoteP_Eats [THEN cut2])
 qed
 
 lemma exists_QuoteP:
   assumes j: "atom j \<sharp> x"  shows "{} \<turnstile> Ex j (QuoteP x (Var j))"
 proof -
   obtain i::name and j'::name and k::name
     where atoms: "atom i \<sharp> (j,x)"  "atom j' \<sharp> (i,j,x)"  "atom (k::name) \<sharp> (i,j,j',x)"
     by (metis obtain_fresh)
   have "{} \<turnstile> Ex j (QuoteP (Var i) (Var j))" (is "{} \<turnstile> ?scheme")
   proof (rule Ind [of k])
     show "atom k \<sharp> (i, ?scheme)" using atoms
       by simp
   next
     show "{} \<turnstile> ?scheme(i::=Zero)" using j atoms
       by (auto intro: Ex_I [where x=Zero] simp add: QuoteP_Zero)
   next
     show "{} \<turnstile> All i (All k (?scheme IMP ?scheme(i::=Var k) IMP ?scheme(i::=Eats (Var i) (Var k))))"
       apply (rule All_I Imp_I)+
       using atoms assms
       apply simp_all
       apply (rule Ex_E)
       apply (rule Ex_E_with_renaming [where i'=j', THEN rotate2], auto)
       apply (rule Ex_I [where x= "Q_Eats (Var j') (Var j)"], auto intro: QuoteP_Eats)
       done
   qed
   hence "{} \<turnstile> (Ex j (QuoteP (Var i) (Var j))) (i::= x)"
     by (rule Subst) auto
   thus ?thesis
     using atoms j by auto
 qed
 
 lemma QuoteP_imp_ConstP: "{ QuoteP x y } \<turnstile> ConstP y"
 proof -
   obtain j::name and j'::name and l::name and s::name and k::name
      and m::name and n::name and sm::name and sn::name and sm'::name and sn'::name
     where atoms: "atom j \<sharp> (x,y,s,k,j',l,m,n,sm,sm',sn,sn')"
              "atom j' \<sharp> (x,y,s,k,l,m,n,sm,sm',sn,sn')"
              "atom l \<sharp> (s,k,m,n,sm,sm',sn,sn')"
              "atom m \<sharp> (s,k,n,sm,sm',sn,sn')" "atom n \<sharp> (s,k,sm,sm',sn,sn')"
              "atom sm \<sharp> (s,k,sm',sn,sn')" "atom sm' \<sharp> (s,k,sn,sn')"
              "atom sn \<sharp> (s,k,sn')" "atom sn' \<sharp> (s,k)" "atom s \<sharp> (k,x,y)" "atom k \<sharp> (x,y)"
     by (metis obtain_fresh)
   have "{OrdP (Var k)}
         \<turnstile> All j (All j' (SeqQuoteP (Var j) (Var j') (Var s) (Var k) IMP ConstP (Var j')))"
         (is "_ \<turnstile> ?scheme")
     proof (rule OrdIndH [where j=l])
       show "atom l \<sharp> (k, ?scheme)" using atoms
         by simp
     next
       show "{} \<turnstile> All k (OrdP (Var k) IMP (All2 l (Var k) (?scheme(k::= Var l)) IMP ?scheme))"
         apply (rule All_I Imp_I)+
         using atoms
         apply (simp_all add: fresh_at_base fresh_finite_set_at_base)
         \<comment> \<open>freshness finally proved!\<close>
         apply (rule cut_same)
         apply (rule cut1 [OF SeqQuoteP_lemma [of m "Var j" "Var j'" "Var s" "Var k" n sm sm' sn sn']], simp_all, blast)
         apply (rule Imp_I Disj_EH Conj_EH)+
         \<comment> \<open>case 1, Var j EQ Zero\<close>
         apply (rule thin1)
         apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same], simp)
         apply (metis thin0 ConstP_Zero)
         \<comment> \<open>case 2, @{term "Var j EQ Eats (Var sm) (Var sn)"}\<close>
         apply (rule Imp_I Conj_EH Ex_EH)+
         apply simp_all
         apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same, THEN rotate2], simp)
         apply (rule ConstP_Eats [THEN cut2])
         \<comment> \<open>Operand 1. IH for sm\<close>
         apply (rule All2_E [where x="Var m", THEN rotate8], auto)
         apply (rule All_E [where x="Var sm"], simp)
         apply (rule All_E [where x="Var sm'"], auto)
         \<comment> \<open>Operand 2. IH for sm\<close>
         apply (rule All2_E [where x="Var n", THEN rotate8], auto)
         apply (rule All_E [where x="Var sn"], simp)
         apply (rule All_E [where x="Var sn'"], auto)
         done
     qed
   hence "{OrdP(Var k)}
          \<turnstile> (All j' (SeqQuoteP (Var j) (Var j') (Var s) (Var k) IMP ConstP (Var j'))) (j::=x)"
     by (metis All_D)
   hence "{OrdP(Var k)} \<turnstile> All j' (SeqQuoteP x (Var j') (Var s) (Var k) IMP ConstP (Var j'))"
     using atoms by simp
   hence "{OrdP(Var k)} \<turnstile> (SeqQuoteP x (Var j') (Var s) (Var k) IMP ConstP (Var j')) (j'::=y)"
     by (metis All_D)
   hence "{OrdP(Var k)} \<turnstile> SeqQuoteP x y (Var s) (Var k) IMP ConstP y"
     using atoms by simp
   hence "{ SeqQuoteP x y (Var s) (Var k) } \<turnstile> ConstP y"
     by (metis Imp_cut SeqQuoteP_imp_OrdP anti_deduction)
   thus "{ QuoteP x y } \<turnstile> ConstP y" using atoms
     by (auto simp: QuoteP.simps [of s _ _ k])
 qed
 
 lemma SeqQuoteP_imp_QuoteP: "{SeqQuoteP t u s k} \<turnstile> QuoteP t u"
 proof -
   obtain s'::name and k'::name  where "atom s' \<sharp> (k',t,u,s,k)"  "atom k' \<sharp> (t,u,s,k)"
     by (metis obtain_fresh)
   thus ?thesis
     apply (simp add: QuoteP.simps [of s' _ _ k'])
     apply (rule Ex_I [where x = s], simp)
     apply (rule Ex_I [where x = k], auto)
     done
 qed
 
 lemmas QuoteP_I = SeqQuoteP_imp_QuoteP [THEN cut1]
 
 
-section {*The Operator @{term quote_all}*}
+section \<open>The Operator @{term quote_all}\<close>
 
-subsection {*Definition and basic properties*}
+subsection \<open>Definition and basic properties\<close>
 
 definition quote_all :: "[perm, name set] \<Rightarrow> fm set"
   where "quote_all p V = {QuoteP (Var i) (Var (p \<bullet> i)) | i. i \<in> V}"
 
 lemma quote_all_empty [simp]: "quote_all p {} = {}"
   by (simp add: quote_all_def)
 
 lemma quote_all_insert [simp]:
   "quote_all p (insert i V) = insert (QuoteP (Var i) (Var (p \<bullet> i))) (quote_all p V)"
   by (auto simp: quote_all_def)
 
 lemma finite_quote_all [simp]: "finite V \<Longrightarrow> finite (quote_all p V)"
   by (induct rule: finite_induct) auto
 
 lemma fresh_quote_all [simp]: "finite V \<Longrightarrow> i \<sharp> quote_all p V \<longleftrightarrow> i \<sharp> V \<and> i \<sharp> p\<bullet>V"
   by (induct rule: finite_induct) (auto simp: fresh_finite_insert)
 
 lemma fresh_quote_all_mem: "\<lbrakk>A \<in> quote_all p V; finite V; i \<sharp> V; i \<sharp> p \<bullet> V\<rbrakk> \<Longrightarrow> i \<sharp> A"
   by (metis Set.set_insert finite_insert finite_quote_all fresh_finite_insert fresh_quote_all)
 
 lemma quote_all_perm_eq:
   assumes "finite V" "atom i \<sharp> (p,V)" "atom i' \<sharp> (p,V)"
     shows "quote_all ((atom i \<rightleftharpoons> atom i') + p) V = quote_all p V"
 proof -
   { fix W
     assume w: "W \<subseteq> V"
     have "finite W"
-      by (metis `finite V` finite_subset w)
+      by (metis \<open>finite V\<close> finite_subset w)
     hence "quote_all ((atom i \<rightleftharpoons> atom i') + p) W = quote_all p W" using w
         apply induction  using assms
         apply (auto simp: fresh_Pair perm_commute)
         apply (metis fresh_finite_set_at_base swap_at_base_simps(3))+
         done}
   thus ?thesis
     by (metis order_refl)
 qed
 
 
-subsection {*Transferring theorems to the level of derivability*}
+subsection \<open>Transferring theorems to the level of derivability\<close>
 
 context quote_perm
 begin
 
 lemma QuoteP_imp_ConstP_F_hyps:
   assumes "Us \<subseteq> Vs" "{ConstP (F i) | i. i \<in> Us} \<turnstile> A"  shows "quote_all p Us \<turnstile> A"
 proof -
-  show ?thesis using finite_V [OF `Us \<subseteq> Vs`]  assms
+  show ?thesis using finite_V [OF \<open>Us \<subseteq> Vs\<close>]  assms
   proof (induction arbitrary: A rule: finite_induct)
     case empty thus ?case by simp
   next
     case (insert v Us) thus ?case
       by (auto simp: Collect_disj_Un)
          (metis (lifting) anti_deduction Imp_cut [OF _ QuoteP_imp_ConstP] Disj_I2 F_unfold)
   qed
 qed
 
-text{*Lemma 8.3*}
+text\<open>Lemma 8.3\<close>
 theorem quote_all_PfP_ssubst:
   assumes \<beta>: "{} \<turnstile> \<beta>"
       and V: "V \<subseteq> Vs"
       and s: "supp \<beta> \<subseteq> atom ` Vs"
     shows    "quote_all p V \<turnstile> PfP (ssubst \<lfloor>\<beta>\<rfloor>V V F)"
 proof -
   have "{} \<turnstile> PfP \<lceil>\<beta>\<rceil>"
     by (metis \<beta> proved_imp_proved_PfP)
   hence "{ConstP (F i) | i. i \<in> V} \<turnstile> PfP (ssubst \<lfloor>\<beta>\<rfloor>V V F)"
     by (simp add: PfP_implies_PfP_ssubst V s)
   thus ?thesis
     by (rule QuoteP_imp_ConstP_F_hyps [OF V])
 qed
 
-text{*Lemma 8.4*}
+text\<open>Lemma 8.4\<close>
 corollary quote_all_MonPon_PfP_ssubst:
   assumes A: "{} \<turnstile> \<alpha> IMP \<beta>"
       and V: "V \<subseteq> Vs"
       and s: "supp \<alpha> \<subseteq> atom ` Vs" "supp \<beta> \<subseteq> atom ` Vs"
     shows    "quote_all p V \<turnstile> PfP (ssubst \<lfloor>\<alpha>\<rfloor>V V F) IMP PfP (ssubst \<lfloor>\<beta>\<rfloor>V V F)"
 using quote_all_PfP_ssubst [OF A V] s
   by (auto simp: V vquot_fm_def intro: PfP_implies_ModPon_PfP thin1)
 
-text{*Lemma 8.4b*}
+text\<open>Lemma 8.4b\<close>
 corollary quote_all_MonPon2_PfP_ssubst:
   assumes A: "{} \<turnstile> \<alpha>1 IMP \<alpha>2 IMP \<beta>"
       and V: "V \<subseteq> Vs"
       and s: "supp \<alpha>1 \<subseteq> atom ` Vs" "supp \<alpha>2 \<subseteq> atom ` Vs" "supp \<beta> \<subseteq> atom ` Vs"
     shows "quote_all p V \<turnstile> PfP (ssubst \<lfloor>\<alpha>1\<rfloor>V V F) IMP PfP (ssubst \<lfloor>\<alpha>2\<rfloor>V V F) IMP PfP (ssubst \<lfloor>\<beta>\<rfloor>V V F)"
 using quote_all_PfP_ssubst [OF A V] s
   by (force simp: V vquot_fm_def intro: PfP_implies_ModPon_PfP [OF PfP_implies_ModPon_PfP] thin1)
 
 lemma quote_all_Disj_I1_PfP_ssubst:
   assumes "V \<subseteq> Vs" "supp \<alpha> \<subseteq> atom ` Vs" "supp \<beta> \<subseteq> atom ` Vs"
       and prems: "H \<turnstile> PfP (ssubst \<lfloor>\<alpha>\<rfloor>V V F)" "quote_all p V \<subseteq> H"
     shows "H \<turnstile> PfP (ssubst \<lfloor>\<alpha> OR \<beta>\<rfloor>V V F)"
 proof -
   have "{} \<turnstile> \<alpha> IMP (\<alpha> OR \<beta>)"
     by (blast intro: Disj_I1)
   hence "quote_all p V \<turnstile> PfP (ssubst \<lfloor>\<alpha>\<rfloor>V V F) IMP PfP (ssubst \<lfloor>\<alpha> OR \<beta>\<rfloor>V V F)"
     using assms by (auto simp: quote_all_MonPon_PfP_ssubst)
   thus ?thesis
     by (metis MP_same prems thin)
 qed
 
 lemma quote_all_Disj_I2_PfP_ssubst:
   assumes "V \<subseteq> Vs" "supp \<alpha> \<subseteq> atom ` Vs" "supp \<beta> \<subseteq> atom ` Vs"
       and prems: "H \<turnstile> PfP (ssubst \<lfloor>\<beta>\<rfloor>V V F)" "quote_all p V \<subseteq> H"
     shows "H \<turnstile> PfP (ssubst \<lfloor>\<alpha> OR \<beta>\<rfloor>V V F)"
 proof -
   have "{} \<turnstile> \<beta> IMP (\<alpha> OR \<beta>)"
     by (blast intro: Disj_I2)
   hence "quote_all p V \<turnstile> PfP (ssubst \<lfloor>\<beta>\<rfloor>V V F) IMP PfP (ssubst \<lfloor>\<alpha> OR \<beta>\<rfloor>V V F)"
     using assms by (auto simp: quote_all_MonPon_PfP_ssubst)
   thus ?thesis
     by (metis MP_same prems thin)
 qed
 
 lemma quote_all_Conj_I_PfP_ssubst:
   assumes "V \<subseteq> Vs" "supp \<alpha> \<subseteq> atom ` Vs" "supp \<beta> \<subseteq> atom ` Vs"
       and prems: "H \<turnstile> PfP (ssubst \<lfloor>\<alpha>\<rfloor>V V F)" "H \<turnstile> PfP (ssubst \<lfloor>\<beta>\<rfloor>V V F)" "quote_all p V \<subseteq> H"
     shows "H \<turnstile> PfP (ssubst \<lfloor>\<alpha> AND \<beta>\<rfloor>V V F)"
 proof -
   have "{} \<turnstile> \<alpha> IMP \<beta> IMP (\<alpha> AND \<beta>)"
     by blast
   hence "quote_all p V
          \<turnstile> PfP (ssubst \<lfloor>\<alpha>\<rfloor>V V F) IMP PfP (ssubst \<lfloor>\<beta>\<rfloor>V V F) IMP PfP (ssubst \<lfloor>\<alpha> AND \<beta>\<rfloor>V V F)"
     using assms by (auto simp: quote_all_MonPon2_PfP_ssubst)
   thus ?thesis
     by (metis MP_same prems thin)
 qed
 
 lemma quote_all_Contra_PfP_ssubst:
   assumes "V \<subseteq> Vs" "supp \<alpha> \<subseteq> atom ` Vs"
   shows "quote_all p V
          \<turnstile> PfP (ssubst \<lfloor>\<alpha>\<rfloor>V V F) IMP PfP (ssubst \<lfloor>Neg \<alpha>\<rfloor>V V F) IMP PfP (ssubst \<lfloor>Fls\<rfloor>V V F)"
 proof -
   have "{} \<turnstile> \<alpha> IMP Neg \<alpha> IMP Fls"
     by blast
   thus ?thesis
     using assms by (auto simp: quote_all_MonPon2_PfP_ssubst supp_conv_fresh)
 qed
 
 lemma fresh_ssubst_dbtm: "\<lbrakk>atom i \<sharp> p\<bullet>V; V \<subseteq> Vs\<rbrakk> \<Longrightarrow> atom i \<sharp> ssubst (vquot_dbtm V t) V F"
   by (induct t rule: dbtm.induct) (auto simp: F_unfold fresh_image permute_set_eq_image)
 
 lemma fresh_ssubst_dbfm: "\<lbrakk>atom i \<sharp> p\<bullet>V; V \<subseteq> Vs\<rbrakk> \<Longrightarrow> atom i \<sharp> ssubst (vquot_dbfm V A) V F"
   by (nominal_induct A rule: dbfm.strong_induct) (auto simp: fresh_ssubst_dbtm)
 
 lemma fresh_ssubst_fm:
   fixes A::fm shows "\<lbrakk>atom i \<sharp> p\<bullet>V; V \<subseteq> Vs\<rbrakk> \<Longrightarrow> atom i \<sharp> ssubst (\<lfloor>A\<rfloor>V) V F"
   by (simp add: fresh_ssubst_dbfm vquot_fm_def)
 
 end
 
 
-section {*Star Property. Equality and Membership: Lemmas 9.3 and 9.4*}
+section \<open>Star Property. Equality and Membership: Lemmas 9.3 and 9.4\<close>
 
 lemma SeqQuoteP_Mem_imp_QMem_and_Subset:
   assumes "atom i \<sharp> (j,j',i',si,ki,sj,kj)" "atom i' \<sharp> (j,j',si,ki,sj,kj)"
           "atom j \<sharp> (j',si,ki,sj,kj)" "atom j' \<sharp> (si,ki,sj,kj)"
           "atom si \<sharp> (ki,sj,kj)" "atom sj \<sharp> (ki,kj)"
   shows "{SeqQuoteP (Var i) (Var i') (Var si) ki, SeqQuoteP (Var j) (Var j') (Var sj) kj}
          \<turnstile> (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j')))  AND
            (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j')))"
 proof -
   obtain k::name and l::name and li::name and lj::name
      and m::name and n::name and sm::name and sn::name and sm'::name and sn'::name
    where atoms: "atom lj \<sharp> (li,l,i,j,j',i',si,ki,sj,kj,i,i',k,m,n,sm,sm',sn,sn')"
                 "atom li \<sharp> (l,j,j',i,i',si,ki,sj,kj,i,i',k,m,n,sm,sm',sn,sn')"
                 "atom l \<sharp> (j,j',i,i',si,ki,sj,kj,i,i',k,m,n,sm,sm',sn,sn')"
                 "atom k \<sharp> (j,j',i,i',si,ki,sj,kj,m,n,sm,sm',sn,sn')"
                 "atom m \<sharp> (j,j',i,i',si,ki,sj,kj,n,sm,sm',sn,sn')"
                 "atom n \<sharp> (j,j',i,i',si,ki,sj,kj,sm,sm',sn,sn')"
                 "atom sm \<sharp> (j,j',i,i',si,ki,sj,kj,sm',sn,sn')"
                 "atom sm' \<sharp> (j,j',i,i',si,ki,sj,kj,sn,sn')"
                 "atom sn \<sharp> (j,j',i,i',si,ki,sj,kj,sn')"
                 "atom sn' \<sharp> (j,j',i,i',si,ki,sj,kj)"
     by (metis obtain_fresh)
   have "{OrdP(Var k)}
        \<turnstile> All i (All i' (All si (All li (All j (All j' (All sj (All lj
           (SeqQuoteP (Var i) (Var i') (Var si) (Var li) IMP
            SeqQuoteP (Var j) (Var j') (Var sj) (Var lj) IMP
            HaddP (Var li) (Var lj) (Var k) IMP
              ( (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j')))  AND
                (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j'))))))))))))"
         (is "_ \<turnstile> ?scheme")
     proof (rule OrdIndH [where j=l])
       show "atom l \<sharp> (k, ?scheme)" using atoms
         by simp
     next
       define V p where "V = {i,j,sm,sn}"
         and "p = (atom i \<rightleftharpoons> atom i') + (atom j \<rightleftharpoons> atom j') +
                  (atom sm \<rightleftharpoons> atom sm') + (atom sn \<rightleftharpoons> atom sn')"
       define F where "F \<equiv> make_F V p"
       interpret qp: quote_perm p V F
         proof unfold_locales
           show "finite V" by (simp add: V_def)
           show "atom ` (p \<bullet> V) \<sharp>* V"
             using atoms assms
             by (auto simp: p_def V_def F_def make_F_def fresh_star_def fresh_finite_insert)
           show "-p = p"  using assms atoms
             by (simp add: p_def add.assoc perm_self_inverseI fresh_swap fresh_plus_perm)
           show "F \<equiv> make_F V p"
             by (rule F_def)
         qed
       have V_mem: "i \<in> V" "j \<in> V" "sm \<in> V" "sn \<in> V"
         by (auto simp: V_def)  \<comment> \<open>Part of (2) from page 32\<close>
       have Mem1: "{} \<turnstile> (Var i IN Var sm) IMP (Var i IN Eats (Var sm) (Var sn))"
         by (blast intro: Mem_Eats_I1)
       have Q_Mem1: "quote_all p V
                      \<turnstile> PfP (Q_Mem (Var i') (Var sm')) IMP
                        PfP (Q_Mem (Var i') (Q_Eats (Var sm') (Var sn')))"
          using qp.quote_all_MonPon_PfP_ssubst [OF Mem1 subset_refl] assms atoms V_mem
          by (simp add: vquot_fm_def qp.Vs) (simp add: qp.F_unfold p_def)
       have Mem2: "{} \<turnstile> (Var i EQ Var sn) IMP (Var i IN Eats (Var sm) (Var sn))"
         by (blast intro: Mem_Eats_I2)
       have Q_Mem2: "quote_all p V
                  \<turnstile> PfP (Q_Eq (Var i') (Var sn')) IMP
                    PfP (Q_Mem (Var i') (Q_Eats (Var sm') (Var sn')))"
          using qp.quote_all_MonPon_PfP_ssubst [OF Mem2 subset_refl] assms atoms V_mem
          by (simp add: vquot_fm_def qp.Vs) (simp add: qp.F_unfold p_def)
       have Subs1: "{} \<turnstile> Zero SUBS Var j"
         by blast
       have Q_Subs1: "{QuoteP (Var j) (Var j')} \<turnstile> PfP (Q_Subset Zero (Var j'))"
          using qp.quote_all_PfP_ssubst [OF Subs1, of "{j}"] assms atoms
          by (simp add: qp.ssubst_Subset vquot_tm_def supp_conv_fresh fresh_at_base del: qp.ssubst_single)
             (simp add: qp.F_unfold p_def V_def)
       have Subs2: "{} \<turnstile> Var sm SUBS Var j IMP Var sn IN Var j IMP Eats (Var sm) (Var sn) SUBS Var j"
         by blast
       have Q_Subs2: "quote_all p V
                      \<turnstile> PfP (Q_Subset (Var sm') (Var j')) IMP
                        PfP (Q_Mem (Var sn') (Var j')) IMP
                        PfP (Q_Subset (Q_Eats (Var sm') (Var sn')) (Var j'))"
          using qp.quote_all_MonPon2_PfP_ssubst [OF Subs2 subset_refl] assms atoms V_mem
          by (simp add: qp.ssubst_Subset vquot_tm_def supp_conv_fresh subset_eq fresh_at_base)
             (simp add: vquot_fm_def qp.F_unfold p_def V_def)
       have Ext: "{} \<turnstile> Var i SUBS Var sn IMP Var sn SUBS Var i IMP Var i EQ Var sn"
         by (blast intro: Equality_I)
       have Q_Ext: "{QuoteP (Var i) (Var i'), QuoteP (Var sn) (Var sn')}
                    \<turnstile> PfP (Q_Subset (Var i') (Var sn')) IMP
                      PfP (Q_Subset (Var sn') (Var i')) IMP
                      PfP (Q_Eq (Var i') (Var sn'))"
          using qp.quote_all_MonPon2_PfP_ssubst [OF Ext, of "{i,sn}"] assms atoms
          by (simp add: qp.ssubst_Subset vquot_tm_def supp_conv_fresh subset_eq fresh_at_base
                   del: qp.ssubst_single)
             (simp add: vquot_fm_def qp.F_unfold p_def V_def)
       show "{} \<turnstile> All k (OrdP (Var k) IMP (All2 l (Var k) (?scheme(k::= Var l)) IMP ?scheme))"
         apply (rule All_I Imp_I)+
         using atoms assms
         apply simp_all
         apply (rule cut_same [where A = "QuoteP (Var i) (Var i')"])
         apply (blast intro: QuoteP_I)
         apply (rule cut_same [where A = "QuoteP (Var j) (Var j')"])
         apply (blast intro: QuoteP_I)
         apply (rule rotate6)
         apply (rule Conj_I)
         \<comment> \<open>@{term"Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))"}\<close>
         apply (rule cut_same)
         apply (rule cut1 [OF SeqQuoteP_lemma [of m "Var j" "Var j'" "Var sj" "Var lj" n sm sm' sn sn']], simp_all, blast)
         apply (rule Imp_I Disj_EH Conj_EH)+
         \<comment> \<open>case 1, @{term "Var j EQ Zero"}\<close>
         apply (rule cut_same [where A = "Var i IN Zero"])
         apply (blast intro: Mem_cong [THEN Iff_MP_same], blast)
         \<comment> \<open>case 2, @{term "Var j EQ Eats (Var sm) (Var sn)"}\<close>
         apply (rule Imp_I Conj_EH Ex_EH)+
         apply simp_all
         apply (rule Var_Eq_subst_Iff [THEN rotate2, THEN Iff_MP_same], simp)
         apply (rule cut_same [where A = "QuoteP (Var sm) (Var sm')"])
         apply (blast intro: QuoteP_I)
         apply (rule cut_same [where A = "QuoteP (Var sn) (Var sn')"])
         apply (blast intro: QuoteP_I)
         apply (rule cut_same [where A = "Var i IN Eats (Var sm) (Var sn)"])
         apply (rule Mem_cong [OF Refl, THEN Iff_MP_same])
         apply (rule AssumeH Mem_Eats_E)+
         \<comment> \<open>Eats case 1. IH for sm\<close>
         apply (rule cut_same [where A = "OrdP (Var m)"])
         apply (blast intro: Hyp Ord_IN_Ord SeqQuoteP_imp_OrdP [THEN cut1])
         apply (rule cut_same [OF exists_HaddP [where j=l and x="Var li" and y="Var m"]])
         apply auto
         apply (rule All2_E [where x="Var l", THEN rotate13], simp_all)
         apply (blast intro: Hyp HaddP_Mem_cancel_left [THEN Iff_MP2_same] SeqQuoteP_imp_OrdP [THEN cut1])
         apply (rule All_E [where x="Var i"], simp)
         apply (rule All_E [where x="Var i'"], simp)
         apply (rule All_E [where x="Var si"], simp)
         apply (rule All_E [where x="Var li"], simp)
         apply (rule All_E [where x="Var sm"], simp)
         apply (rule All_E [where x="Var sm'"], simp)
         apply (rule All_E [where x="Var sj"], simp)
         apply (rule All_E [where x="Var m"], simp)
         apply (force intro: MP_thin [OF Q_Mem1] simp add: V_def p_def)
         \<comment> \<open>Eats case 2\<close>
         apply (rule rotate13)
         apply (rule cut_same [where A = "OrdP (Var n)"])
         apply (blast intro: Hyp Ord_IN_Ord SeqQuoteP_imp_OrdP [THEN cut1])
         apply (rule cut_same [OF exists_HaddP [where j=l and x="Var li" and y="Var n"]])
         apply auto
         apply (rule MP_same)
         apply (rule Q_Mem2 [THEN thin])
         apply (simp add: V_def p_def)
         apply (rule MP_same)
         apply (rule MP_same)
         apply (rule Q_Ext [THEN thin])
         apply (simp add: V_def p_def)
         \<comment> \<open>@{term"PfP (Q_Subset (Var i') (Var sn'))"}\<close>
         apply (rule All2_E [where x="Var l", THEN rotate14], simp_all)
         apply (blast intro: Hyp HaddP_Mem_cancel_left [THEN Iff_MP2_same] SeqQuoteP_imp_OrdP [THEN cut1])
         apply (rule All_E [where x="Var i"], simp)
         apply (rule All_E [where x="Var i'"], simp)
         apply (rule All_E [where x="Var si"], simp)
         apply (rule All_E [where x="Var li"], simp)
         apply (rule All_E [where x="Var sn"], simp)
         apply (rule All_E [where x="Var sn'"], simp)
         apply (rule All_E [where x="Var sj"], simp)
         apply (rule All_E [where x="Var n"], simp)
         apply (rule Imp_E, blast intro: Hyp)+
         apply (rule Conj_E)
         apply (rule thin1)
         apply (blast intro!: Imp_E EQ_imp_SUBS [THEN cut1])
         \<comment> \<open>@{term"PfP (Q_Subset (Var sn') (Var i'))"}\<close>
         apply (rule All2_E [where x="Var l", THEN rotate14], simp_all)
         apply (blast intro: Hyp HaddP_Mem_cancel_left [THEN Iff_MP2_same] SeqQuoteP_imp_OrdP [THEN cut1])
         apply (rule All_E [where x="Var sn"], simp)
         apply (rule All_E [where x="Var sn'"], simp)
         apply (rule All_E [where x="Var sj"], simp)
         apply (rule All_E [where x="Var n"], simp)
         apply (rule All_E [where x="Var i"], simp)
         apply (rule All_E [where x="Var i'"], simp)
         apply (rule All_E [where x="Var si"], simp)
         apply (rule All_E [where x="Var li"], simp)
         apply (rule Imp_E, blast intro: Hyp)+
         apply (rule Imp_E)
         apply (blast intro: Hyp HaddP_commute [THEN cut2] SeqQuoteP_imp_OrdP [THEN cut1])
         apply (rule Conj_E)
         apply (rule thin1)
         apply (blast intro!: Imp_E EQ_imp_SUBS2 [THEN cut1])
         \<comment> \<open>@{term"Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j'))"}\<close>
         apply (rule cut_same)
         apply (rule cut1 [OF SeqQuoteP_lemma [of m "Var i" "Var i'" "Var si" "Var li" n sm sm' sn sn']], simp_all, blast)
         apply (rule Imp_I Disj_EH Conj_EH)+
         \<comment> \<open>case 1, Var i EQ Zero\<close>
         apply (rule cut_same [where A = "PfP (Q_Subset Zero (Var j'))"])
         apply (blast intro: Q_Subs1 [THEN cut1] SeqQuoteP_imp_QuoteP [THEN cut1])
         apply (force intro: Var_Eq_subst_Iff [THEN Iff_MP_same, THEN rotate3])
         \<comment> \<open>case 2, @{term "Var i EQ Eats (Var sm) (Var sn)"}\<close>
         apply (rule Conj_EH Ex_EH)+
         apply simp_all
         apply (rule cut_same [where A = "OrdP (Var lj)"])
         apply (blast intro: Hyp SeqQuoteP_imp_OrdP [THEN cut1])
         apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same, THEN rotate3], simp)
         apply (rule cut_same [where A = "QuoteP (Var sm) (Var sm')"])
         apply (blast intro: QuoteP_I)
         apply (rule cut_same [where A = "QuoteP (Var sn) (Var sn')"])
         apply (blast intro: QuoteP_I)
         apply (rule cut_same [where A = "Eats (Var sm) (Var sn) SUBS Var j"])
         apply (rule Subset_cong [OF _ Refl, THEN Iff_MP_same])
         apply (rule AssumeH Mem_Eats_E)+
         \<comment> \<open>Eats case split\<close>
         apply (rule Eats_Subset_E)
         apply (rule rotate15)
         apply (rule MP_same [THEN MP_same])
         apply (rule Q_Subs2 [THEN thin])
         apply (simp add: V_def p_def)
         \<comment> \<open>Eats case 1: @{term "PfP (Q_Subset (Var sm') (Var j'))"}\<close>
         apply (rule cut_same [OF exists_HaddP [where j=l and x="Var m" and y="Var lj"]])
         apply (rule AssumeH Ex_EH Conj_EH | simp)+
         \<comment> \<open>IH for sm\<close>
         apply (rule All2_E [where x="Var l", THEN rotate15], simp_all)
         apply (blast intro: Hyp HaddP_Mem_cancel_right_Mem SeqQuoteP_imp_OrdP [THEN cut1])
         apply (rule All_E [where x="Var sm"], simp)
         apply (rule All_E [where x="Var sm'"], simp)
         apply (rule All_E [where x="Var si"], simp)
         apply (rule All_E [where x="Var m"], simp)
         apply (rule All_E [where x="Var j"], simp)
         apply (rule All_E [where x="Var j'"], simp)
         apply (rule All_E [where x="Var sj"], simp)
         apply (rule All_E [where x="Var lj"], simp)
         apply (blast intro: thin1 Imp_E)
         \<comment> \<open>Eats case 2: @{term "PfP (Q_Mem (Var sn') (Var j'))"}\<close>
         apply (rule cut_same [OF exists_HaddP [where j=l and x="Var n" and y="Var lj"]])
         apply (rule AssumeH Ex_EH Conj_EH | simp)+
         \<comment> \<open>IH for sn\<close>
         apply (rule All2_E [where x="Var l", THEN rotate15], simp_all)
         apply (blast intro: Hyp HaddP_Mem_cancel_right_Mem SeqQuoteP_imp_OrdP [THEN cut1])
         apply (rule All_E [where x="Var sn"], simp)
         apply (rule All_E [where x="Var sn'"], simp)
         apply (rule All_E [where x="Var si"], simp)
         apply (rule All_E [where x="Var n"], simp)
         apply (rule All_E [where x="Var j"], simp)
         apply (rule All_E [where x="Var j'"], simp)
         apply (rule All_E [where x="Var sj"], simp)
         apply (rule All_E [where x="Var lj"], simp)
         apply (blast intro: Hyp Imp_E)
         done
     qed
   hence p1: "{OrdP(Var k)}
              \<turnstile> (All i' (All si (All li
                   (All j (All j' (All sj (All lj
                         (SeqQuoteP (Var i) (Var i') (Var si) (Var li) IMP
                          SeqQuoteP (Var j) (Var j') (Var sj) (Var lj) IMP
                          HaddP (Var li) (Var lj) (Var k) IMP
                          (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))) AND
                          (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j'))))))))))) (i::= Var i)"
         by (metis All_D)
   have p2: "{OrdP(Var k)}
             \<turnstile> (All si (All li
                 (All j (All j' (All sj (All lj
                       (SeqQuoteP (Var i) (Var i') (Var si) (Var li) IMP
                        SeqQuoteP (Var j) (Var j') (Var sj) (Var lj) IMP
                        HaddP (Var li) (Var lj) (Var k) IMP
                        (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))) AND
                        (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j'))))))))))(i'::= Var i')"
     apply (rule All_D)
     using atoms p1 by simp
   have p3: "{OrdP(Var k)}
             \<turnstile> (All li
               (All j (All j' (All sj (All lj
                     (SeqQuoteP (Var i) (Var i') (Var si) (Var li) IMP
                      SeqQuoteP (Var j) (Var j') (Var sj) (Var lj) IMP
                      HaddP (Var li) (Var lj) (Var k) IMP
                      (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))) AND
                      (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j'))))))))) (si::= Var si)"
     apply (rule All_D)
     using atoms p2 by simp
   have p4: "{OrdP(Var k)}
             \<turnstile> (All j (All j' (All sj (All lj
                     (SeqQuoteP (Var i) (Var i') (Var si) (Var li) IMP
                      SeqQuoteP (Var j) (Var j') (Var sj) (Var lj) IMP
                      HaddP (Var li) (Var lj) (Var k) IMP
                      (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))) AND
                      (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j')))))))) (li::= ki)"
     apply (rule All_D)
     using atoms p3 by simp
   have p5: "{OrdP(Var k)}
             \<turnstile> (All j' (All sj (All lj
                 (SeqQuoteP (Var i) (Var i') (Var si) ki IMP
                  SeqQuoteP (Var j) (Var j') (Var sj) (Var lj) IMP
                  HaddP ki (Var lj) (Var k) IMP
                  (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))) AND
                  (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j'))))))) (j::= Var j)"
     apply (rule All_D)
     using atoms assms p4 by simp
   have p6: "{OrdP(Var k)}
             \<turnstile> (All sj (All lj
                   (SeqQuoteP (Var i) (Var i') (Var si) ki IMP
                    SeqQuoteP (Var j) (Var j') (Var sj) (Var lj) IMP
                    HaddP ki (Var lj) (Var k) IMP
                    (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))) AND
                    (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j')))))) (j'::= Var j')"
     apply (rule All_D)
     using atoms p5 by simp
   have p7: "{OrdP(Var k)}
             \<turnstile> (All lj (SeqQuoteP (Var i) (Var i') (Var si) ki IMP
                        SeqQuoteP (Var j) (Var j') (Var sj) (Var lj) IMP
                        HaddP ki (Var lj) (Var k) IMP
                        (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))) AND
                        (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j'))))) (sj::= Var sj)"
     apply (rule All_D)
     using atoms p6 by simp
   have p8: "{OrdP(Var k)}
             \<turnstile> (SeqQuoteP (Var i) (Var i') (Var si) ki IMP
                SeqQuoteP (Var j) (Var j') (Var sj) (Var lj) IMP
                HaddP ki (Var lj) (Var k) IMP
                (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))) AND
                (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j')))) (lj::= kj)"
     apply (rule All_D)
     using atoms p7 by simp
   hence p9: "{OrdP(Var k)}
              \<turnstile> SeqQuoteP (Var i) (Var i') (Var si) ki IMP
                SeqQuoteP (Var j) (Var j') (Var sj) kj IMP
                HaddP ki kj (Var k) IMP
                (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))) AND
                (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j')))"
     using assms atoms by simp
   have p10:  "{ HaddP ki kj (Var k),
                SeqQuoteP (Var i) (Var i') (Var si) ki,
                SeqQuoteP (Var j) (Var j') (Var sj) kj, OrdP (Var k) }
              \<turnstile> (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))) AND
                (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j')))"
     apply (rule MP_same [THEN MP_same [THEN MP_same]])
     apply (rule p9 [THEN thin])
     apply (auto intro: MP_same)
     done
   show ?thesis
     apply (rule cut_same [OF exists_HaddP [where j=k and x=ki and y=kj]])
     apply (metis SeqQuoteP_imp_OrdP thin1)
     prefer 2
     apply (rule Ex_E)
     apply (rule p10 [THEN cut4])
     using assms atoms
     apply (auto intro: HaddP_OrdP SeqQuoteP_imp_OrdP [THEN cut1])
     done
 qed
 
 
 lemma
   assumes "atom i \<sharp> (j,j',i')" "atom i' \<sharp> (j,j')" "atom j \<sharp> (j')"
   shows QuoteP_Mem_imp_QMem:
         "{QuoteP (Var i) (Var i'), QuoteP (Var j) (Var j'), Var i IN Var j}
          \<turnstile> PfP (Q_Mem (Var i') (Var j'))"     (is ?thesis1)
     and QuoteP_Mem_imp_QSubset:
         "{QuoteP (Var i) (Var i'), QuoteP (Var j) (Var j'), Var i SUBS Var j}
          \<turnstile> PfP (Q_Subset (Var i') (Var j'))"  (is ?thesis2)
 proof -
   obtain si::name and ki::name and sj::name and kj::name
     where atoms: "atom si \<sharp> (ki,sj,kj,i,j,j',i')"  "atom ki \<sharp> (sj,kj,i,j,j',i')"
                  "atom sj \<sharp> (kj,i,j,j',i')"  "atom kj \<sharp> (i,j,j',i')"
     by (metis obtain_fresh)
   hence C:  "{QuoteP (Var i) (Var i'), QuoteP (Var j) (Var j')}
              \<turnstile> (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j')))  AND
                (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j')))"
     using assms
     by (auto simp: QuoteP.simps [of si "Var i" _ ki] QuoteP.simps [of sj "Var j" _ kj]
              intro!: SeqQuoteP_Mem_imp_QMem_and_Subset del: Conj_I)
   show ?thesis1
     by (best intro: Conj_E1 [OF C, THEN MP_thin])
   show ?thesis2
     by (best intro: Conj_E2 [OF C, THEN MP_thin])
 qed
 
 
-section {*Star Property. Universal Quantifier: Lemma 9.7*}
+section \<open>Star Property. Universal Quantifier: Lemma 9.7\<close>
 
 lemma (in quote_perm) SeqQuoteP_Mem_imp_All2:
   assumes IH: "insert (QuoteP (Var i) (Var i')) (quote_all p Vs)
                \<turnstile> \<alpha> IMP PfP (ssubst \<lfloor>\<alpha>\<rfloor>(insert i Vs) (insert i Vs) Fi)"
       and sp: "supp \<alpha> - {atom i} \<subseteq> atom ` Vs"
       and j: "j \<in> Vs" and j': "p \<bullet> j = j'"
       and pi: "pi = (atom i \<rightleftharpoons> atom i') + p"
       and Fi: "Fi = make_F (insert i Vs) pi"
       and atoms: "atom i \<sharp> (j,j',s,k,p)" "atom i' \<sharp> (i,p,\<alpha>)"
                  "atom j \<sharp> (j',s,k,\<alpha>)" "atom j' \<sharp> (s,k,\<alpha>)"
                  "atom s \<sharp> (k,\<alpha>)" "atom k \<sharp> (\<alpha>,p)"
   shows "insert (SeqQuoteP (Var j) (Var j') (Var s) (Var k)) (quote_all p (Vs-{j}))
          \<turnstile> All2 i (Var j) \<alpha> IMP PfP (ssubst \<lfloor>All2 i (Var j) \<alpha>\<rfloor>Vs Vs F)"
 proof -
   have pj' [simp]: "p \<bullet> j' = j" using pinv j'
     by (metis permute_minus_cancel(2))
   have [simp]: "F j = Var j'" using j j'
     by (auto simp: F_unfold)
   hence i': "atom i' \<sharp> Vs" using atoms
     by (auto simp: Vs)
   have fresh_ss [simp]: "\<And>i A::fm. atom i \<sharp> p \<Longrightarrow> atom i \<sharp> ssubst (\<lfloor>A\<rfloor>Vs) Vs F"
     by (simp add: vquot_fm_def fresh_ssubst_dbfm)
   obtain l::name and m::name and n::name and sm::name and sn::name and sm'::name and sn'::name
     where atoms': "atom l \<sharp> (p,\<alpha>,i,j,j',s,k,m,n,sm,sm',sn,sn')"
        "atom m \<sharp> (p,\<alpha>,i,j,j',s,k,n,sm,sm',sn,sn')" "atom n \<sharp> (p,\<alpha>,i,j,j',s,k,sm,sm',sn,sn')"
        "atom sm \<sharp> (p,\<alpha>,i,j,j',s,k,sm',sn,sn')" "atom sm' \<sharp> (p,\<alpha>,i,j,j',s,k,sn,sn')"
        "atom sn \<sharp> (p,\<alpha>,i,j,j',s,k,sn')" "atom sn' \<sharp> (p,\<alpha>,i,j,j',s,k)"
     by (metis obtain_fresh)
   define V' p'
     where "V' = {sm,sn} \<union> Vs"
       and "p' = (atom sm \<rightleftharpoons> atom sm') + (atom sn \<rightleftharpoons> atom sn') + p"
   define F' where "F' \<equiv> make_F V' p'"
   interpret qp': quote_perm p' V' F'
     proof unfold_locales
       show "finite V'" by (simp add: V'_def)
       show "atom ` (p' \<bullet> V') \<sharp>* V'"
         using atoms atoms' p
         by (auto simp: p'_def V'_def swap_fresh_fresh fresh_at_base_permI
                 fresh_star_finite_insert fresh_finite_insert atom_fresh_star_atom_set_conv)
       show "F' \<equiv> make_F V' p'"
         by (rule F'_def)
       show "- p' = p'" using atoms atoms' pinv
         by (simp add: p'_def add.assoc perm_self_inverseI fresh_swap fresh_plus_perm)
     qed
   have All2_Zero: "{} \<turnstile> All2 i Zero \<alpha>"
     by auto
   have Q_All2_Zero:
     "quote_all p Vs \<turnstile> PfP (Q_All (Q_Imp (Q_Mem (Q_Ind Zero) Zero)
                                 (ssubst (vquot_dbfm Vs (trans_fm [i] \<alpha>)) Vs F)))"
          using quote_all_PfP_ssubst [OF All2_Zero] assms
          by (force simp add: vquot_fm_def supp_conv_fresh)
   have All2_Eats: "{} \<turnstile> All2 i (Var sm) \<alpha> IMP \<alpha>(i::=Var sn) IMP All2 i (Eats (Var sm) (Var sn)) \<alpha>"
     using atoms'  apply auto
     apply (rule Ex_I [where x = "Var i"], auto)
     apply (rule rotate2)
     apply (blast intro: ContraProve Var_Eq_imp_subst_Iff [THEN Iff_MP_same])
     done
   have [simp]: "F' sm = Var sm'" "F' sn = Var sn'" using atoms'
     by (auto simp: V'_def p'_def qp'.F_unfold swap_fresh_fresh fresh_at_base_permI)
   have smn' [simp]: "sm \<in> V'" "sn \<in> V'" "sm \<notin> Vs" "sn \<notin> Vs" using atoms'
     by (auto simp: V'_def fresh_finite_set_at_base [symmetric])
   hence Q_All2_Eats: "quote_all p' V'
                       \<turnstile> PfP (ssubst \<lfloor>All2 i (Var sm) \<alpha>\<rfloor>V' V' F') IMP
                         PfP (ssubst \<lfloor>\<alpha>(i::=Var sn)\<rfloor>V' V' F') IMP
                         PfP (ssubst \<lfloor>All2 i (Eats (Var sm) (Var sn)) \<alpha>\<rfloor>V' V' F')"
      using sp  qp'.quote_all_MonPon2_PfP_ssubst [OF All2_Eats subset_refl]
      by (simp add: supp_conv_fresh subset_eq V'_def)
         (metis Diff_iff empty_iff fresh_ineq_at_base insertE mem_Collect_eq)
   interpret qpi: quote_perm pi "insert i Vs" Fi
     unfolding pi
     apply (rule qp_insert) using atoms
     apply (auto simp: Fi pi)
     done
   have F'_eq_F: "\<And>name. name \<in> Vs \<Longrightarrow> F' name = F name"
     using atoms'
     by (auto simp: F_unfold qp'.F_unfold p'_def swap_fresh_fresh V'_def fresh_pj)
   { fix t::dbtm
     assume "supp t \<subseteq> atom ` V'" "supp t \<subseteq> atom ` Vs"
     hence "ssubst (vquot_dbtm V' t) V' F' = ssubst (vquot_dbtm Vs t) Vs F"
       by (induction t rule: dbtm.induct) (auto simp: F'_eq_F)
   } note ssubst_v_tm = this
   { fix A::dbfm
     assume "supp A \<subseteq> atom ` V'" "supp A \<subseteq> atom ` Vs"
     hence "ssubst (vquot_dbfm V' A) V' F' = ssubst (vquot_dbfm Vs A) Vs F"
       by (induction A rule: dbfm.induct) (auto simp: ssubst_v_tm F'_eq_F)
   } note ssubst_v_fm = this
   have ss_noprimes: "ssubst (vquot_dbfm V' (trans_fm [i] \<alpha>)) V' F' =
                      ssubst (vquot_dbfm Vs (trans_fm [i] \<alpha>)) Vs F"
     apply (rule ssubst_v_fm)
     using sp  apply (auto simp: V'_def supp_conv_fresh)
     done
   { fix t::dbtm
     assume "supp t - {atom i} \<subseteq> atom ` Vs"
     hence "subst i' (Var sn') (ssubst (vquot_dbtm (insert i Vs) t) (insert i Vs) Fi) =
            ssubst (vquot_dbtm V' (subst_dbtm (DBVar sn) i t)) V' F'"
       apply (induction t rule: dbtm.induct)
       using atoms atoms'
       apply (auto simp: vquot_tm_def pi V'_def qpi.F_unfold qp'.F_unfold p'_def fresh_pj swap_fresh_fresh fresh_at_base_permI)
       done
   } note perm_v_tm = this
   { fix A::dbfm
     assume "supp A - {atom i} \<subseteq> atom ` Vs"
     hence "subst i' (Var sn') (ssubst (vquot_dbfm (insert i Vs) A) (insert i Vs) Fi) =
            ssubst (vquot_dbfm V' (subst_dbfm (DBVar sn) i A)) V' F'"
       by (induct A rule: dbfm.induct) (auto simp: Un_Diff perm_v_tm)
   } note perm_v_fm = this
   have "quote_all p Vs \<turnstile> QuoteP (Var i) (Var i') IMP
                  (\<alpha> IMP PfP (ssubst \<lfloor>\<alpha>\<rfloor>(insert i Vs) (insert i Vs) Fi))"
     using IH by auto
   hence "quote_all p Vs
          \<turnstile> (QuoteP (Var i) (Var i') IMP
                  (\<alpha> IMP PfP (ssubst \<lfloor>\<alpha>\<rfloor>(insert i Vs) (insert i Vs) Fi))) (i'::=Var sn')"
     using atoms IH
     by (force intro!: Subst elim!: fresh_quote_all_mem)
   hence "quote_all p Vs
          \<turnstile> QuoteP (Var i) (Var sn') IMP
            (\<alpha> IMP PfP (subst i' (Var sn') (ssubst \<lfloor>\<alpha>\<rfloor>(insert i Vs) (insert i Vs) Fi)))"
     using atoms by simp
   moreover have "subst i' (Var sn') (ssubst \<lfloor>\<alpha>\<rfloor>(insert i Vs) (insert i Vs) Fi)
                  = ssubst \<lfloor>\<alpha>(i::=Var sn)\<rfloor>V' V' F'"
     using sp
     by (auto simp: vquot_fm_def perm_v_fm supp_conv_fresh subst_fm_trans_commute [symmetric])
   ultimately
   have "quote_all p Vs
          \<turnstile> QuoteP (Var i) (Var sn') IMP (\<alpha> IMP PfP (ssubst \<lfloor>\<alpha>(i::=Var sn)\<rfloor>V' V' F'))"
     by simp
   hence "quote_all p Vs
          \<turnstile> (QuoteP (Var i) (Var sn') IMP (\<alpha> IMP PfP (ssubst \<lfloor>\<alpha>(i::=Var sn)\<rfloor>V' V' F'))) (i::=Var sn)"
-    using `atom i \<sharp> _`
+    using \<open>atom i \<sharp> _\<close>
     by (force intro!: Subst elim!: fresh_quote_all_mem)
   hence "quote_all p Vs
          \<turnstile> (QuoteP (Var sn) (Var sn') IMP
            (\<alpha>(i::=Var sn) IMP PfP (subst i (Var sn) (ssubst \<lfloor>\<alpha>(i::=Var sn)\<rfloor>V' V' F'))))"
     using atoms atoms' by simp
   moreover have "subst i (Var sn) (ssubst \<lfloor>\<alpha>(i::=Var sn)\<rfloor>V' V' F')
                  = ssubst \<lfloor>\<alpha>(i::=Var sn)\<rfloor>V' V' F'"
     using atoms atoms' i'
     by (auto simp: swap_fresh_fresh fresh_at_base_permI p'_def
              intro!: forget_subst_tm [OF qp'.fresh_ssubst'])
   ultimately
   have "quote_all p Vs
          \<turnstile> QuoteP (Var sn) (Var sn') IMP (\<alpha>(i::=Var sn) IMP PfP (ssubst \<lfloor>\<alpha>(i::=Var sn)\<rfloor>V' V' F'))"
     using atoms atoms' by simp
   hence star0: "insert (QuoteP (Var sn) (Var sn')) (quote_all p Vs)
                 \<turnstile> \<alpha>(i::=Var sn) IMP PfP (ssubst \<lfloor>\<alpha>(i::=Var sn)\<rfloor>V' V' F')"
     by (rule anti_deduction)
   have subst_i_star: "quote_all p' V' \<turnstile> \<alpha>(i::=Var sn) IMP PfP (ssubst \<lfloor>\<alpha>(i::=Var sn)\<rfloor>V' V' F')"
     apply (rule thin [OF star0])
     using atoms'
     apply (force simp: V'_def p'_def fresh_swap fresh_plus_perm fresh_at_base_permI add.assoc
                        quote_all_perm_eq)
     done
   have "insert (OrdP (Var k)) (quote_all p (Vs-{j}))
         \<turnstile> All j (All j' (SeqQuoteP (Var j) (Var j') (Var s) (Var k) IMP
                          All2 i (Var j) \<alpha> IMP PfP (ssubst \<lfloor>All2 i (Var j) \<alpha>\<rfloor>Vs Vs F)))"
         (is "_ \<turnstile> ?scheme")
     proof (rule OrdIndH [where j=l])
       show "atom l \<sharp> (k, ?scheme)" using atoms atoms' j j' fresh_pVs
         by (simp add: fresh_Pair F_unfold)
     next
       have substj: "\<And>t j. atom j \<sharp> \<alpha> \<Longrightarrow> atom (p \<bullet> j) \<sharp> \<alpha> \<Longrightarrow>
                            subst j t (ssubst (vquot_dbfm Vs (trans_fm [i] \<alpha>)) Vs F) =
                            ssubst (vquot_dbfm Vs (trans_fm [i] \<alpha>)) Vs F"
         by (auto simp: fresh_ssubst')
       { fix W
         assume W: "W \<subseteq> Vs"
         hence "finite W" by (metis Vs infinite_super)
         hence "quote_all p' W = quote_all p W" using W
         proof (induction)
           case empty thus ?case
             by simp
         next
           case (insert w W)
           hence "w \<in> Vs" "atom sm \<sharp> p \<bullet> Vs" "atom sm' \<sharp> p \<bullet> Vs" "atom sn \<sharp> p \<bullet> Vs" "atom sn' \<sharp> p \<bullet> Vs"
             using atoms' Vs by (auto simp: fresh_pVs)
           hence "atom sm \<sharp> p \<bullet> w" "atom sm' \<sharp> p \<bullet> w" "atom sn \<sharp> p \<bullet> w" "atom sn' \<sharp> p \<bullet> w"
             by (metis Vs fresh_at_base(2) fresh_finite_set_at_base fresh_permute_left)+
           thus ?case using insert
           by (simp add: p'_def swap_fresh_fresh)
         qed
       }
       hence "quote_all p' Vs = quote_all p Vs"
         by (metis subset_refl)
       also have "... = insert (QuoteP (Var j) (Var j')) (quote_all p (Vs - {j}))"
         using j j' by (auto simp: quote_all_def)
       finally have "quote_all p' V' =
                     {QuoteP (Var sn) (Var sn'), QuoteP (Var sm) (Var sm')} \<union>
                     insert (QuoteP (Var j) (Var j')) (quote_all p (Vs - {j}))"
         using atoms'
         by (auto simp: p'_def V'_def fresh_at_base_permI Collect_disj_Un)
       also have "... = {QuoteP (Var sn) (Var sn'), QuoteP (Var sm) (Var sm'), QuoteP (Var j) (Var j')}
                        \<union> quote_all p (Vs - {j})"
         by blast
       finally have quote_all'_eq:
             "quote_all p' V' =
               {QuoteP (Var sn) (Var sn'), QuoteP (Var sm) (Var sm'), QuoteP (Var j) (Var j')}
               \<union> quote_all p (Vs - {j})" .
       have pjV: "p \<bullet> j \<notin> Vs"
         by (metis j perm_exits_Vs)
       hence jpV: "atom j \<sharp> p \<bullet> Vs"
         by (simp add: fresh_permute_left pinv fresh_finite_set_at_base)
       show "quote_all p (Vs-{j}) \<turnstile> All k (OrdP (Var k) IMP (All2 l (Var k) (?scheme(k::= Var l)) IMP ?scheme))"
         apply (rule All_I Imp_I)+
         using atoms atoms' j jpV pjV
         apply (auto simp: fresh_at_base fresh_finite_set_at_base j' elim!: fresh_quote_all_mem)
         apply (rule cut_same [where A = "QuoteP (Var j) (Var j')"])
         apply (blast intro: QuoteP_I)
         apply (rule cut_same)
         apply (rule cut1 [OF SeqQuoteP_lemma [of m "Var j" "Var j'" "Var s" "Var k" n sm sm' sn sn']], simp_all, blast)
         apply (rule Imp_I Disj_EH Conj_EH)+
         \<comment> \<open>case 1, Var j EQ Zero\<close>
         apply (simp add: vquot_fm_def)
         apply (rule thin1)
         apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same], simp)
         apply (simp add: substj)
         apply (rule Q_All2_Zero [THEN thin])
         using assms
         apply (simp add: quote_all_def, blast)
         \<comment> \<open>case 2, @{term "Var j EQ Eats (Var sm) (Var sn)"}\<close>
         apply (rule Imp_I Conj_EH Ex_EH)+
         using atoms  apply (auto elim!: fresh_quote_all_mem)
         apply (rule cut_same [where A = "QuoteP (Var sm) (Var sm')"])
         apply (blast intro: QuoteP_I)
         apply (rule cut_same [where A = "QuoteP (Var sn) (Var sn')"])
         apply (blast intro: QuoteP_I)
         \<comment> \<open>Eats case. IH for sm\<close>
         apply (rule All2_E [where x="Var m", THEN rotate12], simp_all, blast)
         apply (rule All_E [where x="Var sm"], simp)
         apply (rule All_E [where x="Var sm'"], simp)
         apply (rule Imp_E, blast)
         \<comment> \<open>Setting up the subgoal\<close>
         apply (rule cut_same [where A = "PfP (ssubst \<lfloor>All2 i (Eats (Var sm) (Var sn)) \<alpha>\<rfloor>V' V' F')"])
          defer 1
          apply (rule rotate6)
          apply (simp add: vquot_fm_def)
          apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same], force simp add: substj ss_noprimes j')
         apply (rule cut_same [where A = "All2 i (Eats (Var sm) (Var sn)) \<alpha>"])
          apply (rule All2_cong [OF Hyp Iff_refl, THEN Iff_MP_same], blast)
           apply (force elim!: fresh_quote_all_mem
                        simp add: fresh_at_base fresh_finite_set_at_base, blast)
         apply (rule All2_Eats_E, simp)
         apply (rule MP_same [THEN MP_same])
         apply (rule Q_All2_Eats [THEN thin])
         apply (force simp add: quote_all'_eq)
         \<comment> \<open>Proving @{term "PfP (ssubst \<lfloor>All2 i (Var sm) \<alpha>\<rfloor>V' V' F')"}\<close>
         apply (force intro!: Imp_E [THEN rotate3] simp add: vquot_fm_def substj j' ss_noprimes)
         \<comment> \<open>Proving @{term "PfP (ssubst \<lfloor>\<alpha>(i::=Var sn)\<rfloor>V' V' F')"}\<close>
         apply (rule MP_same [OF subst_i_star [THEN thin]])
         apply (force simp add: quote_all'_eq, blast)
         done
     qed
   hence p1: "insert (OrdP (Var k)) (quote_all p (Vs-{j}))
              \<turnstile> (All j' (SeqQuoteP (Var j) (Var j') (Var s) (Var k) IMP
                 All2 i (Var j) \<alpha> IMP PfP (ssubst \<lfloor>All2 i (Var j) \<alpha>\<rfloor>Vs Vs F))) (j::=Var j)"
     by (metis All_D)
   have "insert (OrdP (Var k)) (quote_all p (Vs-{j}))
         \<turnstile> (SeqQuoteP (Var j) (Var j') (Var s) (Var k) IMP
             All2 i (Var j) \<alpha> IMP PfP (ssubst \<lfloor>All2 i (Var j) \<alpha>\<rfloor>Vs Vs F)) (j'::=Var j')"
     apply (rule All_D)
     using p1 atoms by simp
   thus ?thesis
     using atoms
     by simp (metis SeqQuoteP_imp_OrdP Imp_cut anti_deduction)
 qed
 
 lemma (in quote_perm) quote_all_Mem_imp_All2:
   assumes IH: "insert (QuoteP (Var i) (Var i')) (quote_all p Vs)
                \<turnstile> \<alpha> IMP PfP (ssubst \<lfloor>\<alpha>\<rfloor>(insert i Vs) (insert i Vs) Fi)"
       and "supp (All2 i (Var j) \<alpha>) \<subseteq> atom ` Vs"
       and j: "atom j \<sharp> (i,\<alpha>)" and i: "atom i \<sharp> p" and i': "atom i' \<sharp> (i,p,\<alpha>)"
       and pi: "pi = (atom i \<rightleftharpoons> atom i') + p"
       and Fi: "Fi = make_F (insert i Vs) pi"
   shows "insert (All2 i (Var j) \<alpha>) (quote_all p Vs) \<turnstile> PfP (ssubst \<lfloor>All2 i (Var j) \<alpha>\<rfloor>Vs Vs F)"
 proof -
   have sp: "supp \<alpha> - {atom i} \<subseteq> atom ` Vs" and jV: "j \<in> Vs"
     using assms
     by (auto simp: fresh_def supp_Pair)
   obtain s::name and k::name
     where atoms: "atom s \<sharp> (k,i,j,p\<bullet>j,\<alpha>,p)" "atom k \<sharp> (i,j,p\<bullet>j,\<alpha>,p)"
     by (metis obtain_fresh)
   hence ii: "atom i \<sharp> (j, p \<bullet> j, s, k, p)" using i j
     by (simp add: fresh_Pair) (metis fresh_at_base(2) fresh_perm fresh_permute_left pinv)
   have jj: "atom j \<sharp> (p \<bullet> j, s, k, \<alpha>)" using atoms j
     by (auto simp: fresh_Pair) (metis atom_fresh_perm jV)
   have pj: "atom (p \<bullet> j) \<sharp> (s, k, \<alpha>)" using atoms ii sp jV
     by (simp add: fresh_Pair) (auto simp: fresh_def perm_exits_Vs dest!: subsetD)
   show ?thesis
     apply (rule cut_same [where A = "QuoteP (Var j) (Var (p \<bullet> j))"])
     apply (force intro: jV Hyp simp add: quote_all_def)
     using atoms
     apply (auto simp: QuoteP.simps [of s _ _ k] elim!: fresh_quote_all_mem)
     apply (rule MP_same)
     apply (rule SeqQuoteP_Mem_imp_All2 [OF IH sp jV refl pi Fi ii i' jj pj, THEN thin])
     apply (auto simp: fresh_at_base_permI quote_all_def intro!: fresh_ssubst')
     done
 qed
 
 
-section {*The Derivability Condition, Theorem 9.1*}
+section \<open>The Derivability Condition, Theorem 9.1\<close>
 
 lemma SpecI: "H \<turnstile> A IMP Ex i A"
   by (metis Imp_I Assume Ex_I subst_fm_id)
 
 lemma star:
   fixes p :: perm and F :: "name \<Rightarrow> tm"
   assumes C: "ss_fm \<alpha>"
       and p: "atom ` (p \<bullet> V) \<sharp>* V" "-p = p"
       and V: "finite V" "supp \<alpha> \<subseteq> atom ` V"
       and F: "F = make_F V p"
     shows "insert \<alpha> (quote_all p V) \<turnstile> PfP (ssubst \<lfloor>\<alpha>\<rfloor>V V F)"
 using C V p F
 proof (nominal_induct avoiding: p arbitrary: V F rule: ss_fm.strong_induct)
     case (MemI i j) show ?case
     proof (cases "i=j")
       case True thus ?thesis
         by auto
     next
       case False
       hence ij: "atom i \<sharp> j" "{i, j} \<subseteq> V" using MemI
         by auto
       interpret qp: quote_perm p V F
         by unfold_locales (auto simp: image_iff F make_F_def p MemI)
       have "insert (Var i IN Var j) (quote_all p V) \<turnstile> PfP (Q_Mem (Var (p \<bullet> i)) (Var (p \<bullet> j)))"
         apply (rule QuoteP_Mem_imp_QMem [of i j, THEN cut3])
         using ij  apply (auto simp: quote_all_def qp.atom_fresh_perm intro: Hyp)
         apply (metis atom_eqvt fresh_Pair fresh_at_base(2) fresh_permute_iff qp.atom_fresh_perm)
         done
       thus ?thesis
         apply (simp add: vquot_fm_def)
         using MemI  apply (auto simp: make_F_def)
         done
     qed
   next
     case (DisjI A B)
       interpret qp: quote_perm p V F
         by unfold_locales (auto simp: image_iff DisjI)
     show ?case
       apply auto
       apply (rule_tac [2] qp.quote_all_Disj_I2_PfP_ssubst)
       apply (rule qp.quote_all_Disj_I1_PfP_ssubst)
       using DisjI by auto
   next
     case (ConjI A B)
       interpret qp: quote_perm p V F
         by unfold_locales (auto simp: image_iff ConjI)
     show ?case
       apply (rule qp.quote_all_Conj_I_PfP_ssubst)
       using ConjI  by (auto intro: thin1 thin2)
   next
     case (ExI A i)
     interpret qp: quote_perm p V F
       by unfold_locales (auto simp: image_iff ExI)
     obtain i'::name where i': "atom i' \<sharp> (i,p,A)"
       by (metis obtain_fresh)
     define p' where "p' = (atom i \<rightleftharpoons> atom i') + p"
     define F' where "F' = make_F (insert i V) p'"
     have p'_apply [simp]: "!!v. p' \<bullet> v = (if v=i then i' else if v=i' then i else p \<bullet> v)"
-      using  `atom i \<sharp> p`  i'
+      using  \<open>atom i \<sharp> p\<close>  i'
       by (auto simp: p'_def fresh_Pair fresh_at_base_permI)
          (metis atom_eq_iff fresh_at_base_permI permute_eq_iff swap_at_base_simps(3))
     have p'V: "p' \<bullet> V = p \<bullet> V"
-      by (metis i' p'_def permute_plus fresh_Pair qp.fresh_pVs swap_fresh_fresh `atom i \<sharp> p`)
+      by (metis i' p'_def permute_plus fresh_Pair qp.fresh_pVs swap_fresh_fresh \<open>atom i \<sharp> p\<close>)
     have i: "i \<notin> V" "i \<notin> p \<bullet> V"  "atom i \<sharp> V" "atom i \<sharp> p \<bullet> V"  "atom i \<sharp> p' \<bullet> V" using ExI
       by (auto simp: p'V fresh_finite_set_at_base notin_V)
     interpret qp': quote_perm p' "insert i V" F'
-      by (auto simp: qp.qp_insert i' p'_def F'_def  `atom i \<sharp> p`)
+      by (auto simp: qp.qp_insert i' p'_def F'_def  \<open>atom i \<sharp> p\<close>)
     { fix W t assume W: "W \<subseteq> V" "i\<notin>W" "i'\<notin>W"
-      hence "finite W" by (metis `finite V` infinite_super)
+      hence "finite W" by (metis \<open>finite V\<close> infinite_super)
       hence "ssubst t W F' = ssubst t W F" using W
         by induct (auto simp: qp.ssubst_insert_if qp'.ssubst_insert_if qp.F_unfold qp'.F_unfold)
     }
     hence ss_simp: "ssubst \<lfloor>Ex i A\<rfloor>(insert i V) (insert i V) F' = ssubst \<lfloor>Ex i A\<rfloor>V V F" using i
       by (metis equalityE insertCI p'_apply qp'.perm_exits_Vs qp'.ssubst_vquot_Ex qp.Vs)
     have qa_p': "quote_all p' V = quote_all p V" using i i' ExI.hyps(1)
       by (auto simp: p'_def quote_all_perm_eq)
     have ss: "(quote_all p' (insert i V))
               \<turnstile> PfP (ssubst \<lfloor>A\<rfloor>(insert i V) (insert i V) F') IMP
                 PfP (ssubst \<lfloor>Ex i A\<rfloor>(insert i V) (insert i V) F')"
       apply (rule qp'.quote_all_MonPon_PfP_ssubst [OF SpecI])
       using ExI  apply auto
       done
     hence "insert A (quote_all p' (insert i V))
            \<turnstile> PfP (ssubst \<lfloor>Ex i A\<rfloor>(insert i V) (insert i V) F')"
       apply (rule MP_thin)
       apply (rule ExI(3) [of "insert i V" p' F'])
-      apply (metis `finite V` finite_insert)
-      using `supp (Ex i A) \<subseteq> _` qp'.p qp'.pinv i'
+      apply (metis \<open>finite V\<close> finite_insert)
+      using \<open>supp (Ex i A) \<subseteq> _\<close> qp'.p qp'.pinv i'
       apply (auto simp: F'_def fresh_finite_insert)
       done
     hence "insert (QuoteP (Var i) (Var i')) (insert A (quote_all p V))
            \<turnstile> PfP (ssubst \<lfloor>Ex i A\<rfloor>V V F)"
       by (auto simp: insert_commute ss_simp qa_p')
     hence Exi': "insert (Ex i' (QuoteP (Var i) (Var i'))) (insert A (quote_all p V))
                  \<turnstile> PfP (ssubst \<lfloor>Ex i A\<rfloor>V V F)"
       by (auto intro!: qp.fresh_ssubst_fm) (auto simp: ExI i' fresh_quote_all_mem)
     have "insert A (quote_all p V) \<turnstile> PfP (ssubst \<lfloor>Ex i A\<rfloor>V V F)"
       using i'  by (auto intro: cut0 [OF exists_QuoteP Exi'])
     thus "insert (Ex i A) (quote_all p V) \<turnstile> PfP (ssubst \<lfloor>Ex i A\<rfloor>V V F)"
       apply (rule Ex_E, simp)
       apply (rule qp.fresh_ssubst_fm)  using i ExI
       apply (auto simp: fresh_quote_all_mem)
       done
    next
     case (All2I A j i p V F)
     interpret qp: quote_perm p V F
       by unfold_locales (auto simp: image_iff All2I)
     obtain i'::name where i': "atom i' \<sharp> (i,p,A)"
       by (metis obtain_fresh)
     define p' where "p' = (atom i \<rightleftharpoons> atom i') + p"
     define F' where "F' = make_F (insert i V) p'"
     interpret qp': quote_perm p' "insert i V" F'
-      using `atom i \<sharp> p` i'
+      using \<open>atom i \<sharp> p\<close> i'
       by (auto simp: qp.qp_insert p'_def F'_def)
     have p'_apply [simp]: "p' \<bullet> i = i'"
-      using `atom i \<sharp> p` by (auto simp: p'_def fresh_at_base_permI)
+      using \<open>atom i \<sharp> p\<close> by (auto simp: p'_def fresh_at_base_permI)
     have qa_p': "quote_all p' V = quote_all p V" using i' All2I
       by (auto simp: p'_def quote_all_perm_eq)
     have "insert A (quote_all p' (insert i V))
           \<turnstile> PfP (ssubst \<lfloor>A\<rfloor>(insert i V) (insert i V) F')"
       apply (rule All2I.hyps)
-      using `supp (All2 i _ A) \<subseteq> _`   qp'.p qp'.pinv
+      using \<open>supp (All2 i _ A) \<subseteq> _\<close>   qp'.p qp'.pinv
       apply (auto simp: F'_def fresh_finite_insert)
       done
     hence "insert (QuoteP (Var i) (Var i')) (quote_all p V)
            \<turnstile> A IMP PfP (ssubst \<lfloor>A\<rfloor>(insert i V) (insert i V) (make_F (insert i V) p'))"
       by (auto simp: insert_commute qa_p' F'_def)
     thus "insert (All2 i (Var j) A) (quote_all p V) \<turnstile> PfP (ssubst \<lfloor>All2 i (Var j) A\<rfloor>V V F)"
       using All2I i' qp.quote_all_Mem_imp_All2 by (simp add: p'_def)
 qed
 
 theorem Provability:
   assumes "Sigma_fm \<alpha>" "ground_fm \<alpha>"
     shows "{\<alpha>} \<turnstile> PfP \<lceil>\<alpha>\<rceil>"
 proof -
   obtain \<beta> where \<beta>: "ss_fm \<beta>" "ground_fm \<beta>" "{} \<turnstile> \<alpha> IFF \<beta>" using assms
     by (auto simp: Sigma_fm_def ground_fm_aux_def)
   hence "{\<beta>} \<turnstile> PfP \<lceil>\<beta>\<rceil>" using star [of \<beta> 0 "{}"]
     by (auto simp: ground_fm_aux_def fresh_star_def)
   then have "{\<alpha>} \<turnstile> PfP \<lceil>\<beta>\<rceil>" using \<beta>
     by (metis Iff_MP_left')
   moreover have "{} \<turnstile> PfP \<lceil>\<beta> IMP \<alpha>\<rceil>" using \<beta>
     by (metis Conj_E2 Iff_def proved_imp_proved_PfP)
   ultimately show ?thesis
     by (metis PfP_implies_ModPon_PfP_quot thin0)
 qed
 
 end
 
diff --git a/thys/Goedel_HFSet_Semanticless/Sigma.thy b/thys/Goedel_HFSet_Semanticless/Sigma.thy
--- a/thys/Goedel_HFSet_Semanticless/Sigma.thy
+++ b/thys/Goedel_HFSet_Semanticless/Sigma.thy
@@ -1,424 +1,424 @@
-chapter {* Sigma-Formulas and Theorem 2.5 *}
+chapter \<open>Sigma-Formulas and Theorem 2.5\<close>
 
 theory Sigma
 imports Predicates
 begin
 
-section{*Ground Terms and Formulas*}
+section\<open>Ground Terms and Formulas\<close>
 
 definition ground_aux :: "tm \<Rightarrow> atom set \<Rightarrow> bool"
   where "ground_aux t S \<equiv> (supp t \<subseteq> S)"
 
 abbreviation ground :: "tm \<Rightarrow> bool"
   where "ground t \<equiv> ground_aux t {}"
 
 definition ground_fm_aux :: "fm \<Rightarrow> atom set \<Rightarrow> bool"
   where "ground_fm_aux A S \<equiv> (supp A \<subseteq> S)"
 
 abbreviation ground_fm :: "fm \<Rightarrow> bool"
   where "ground_fm A \<equiv> ground_fm_aux A {}"
 
 lemma ground_aux_simps[simp]:
   "ground_aux Zero S = True"
   "ground_aux (Var k) S = (if atom k \<in> S then True else False)"
   "ground_aux (Eats t u) S = (ground_aux t S \<and> ground_aux u S)"
 unfolding ground_aux_def
 by (simp_all add: supp_at_base)
 
 lemma ground_fm_aux_simps[simp]:
   "ground_fm_aux Fls S = True"
   "ground_fm_aux (t IN u) S = (ground_aux t S \<and> ground_aux u S)"
   "ground_fm_aux (t EQ u) S = (ground_aux t S \<and> ground_aux u S)"
   "ground_fm_aux (A OR B) S = (ground_fm_aux A S \<and> ground_fm_aux B S)"
   "ground_fm_aux (A AND B) S = (ground_fm_aux A S \<and> ground_fm_aux B S)"
   "ground_fm_aux (A IFF B) S = (ground_fm_aux A S \<and> ground_fm_aux B S)"
   "ground_fm_aux (Neg A) S =  (ground_fm_aux A S)"
   "ground_fm_aux (Ex x A) S = (ground_fm_aux A (S \<union> {atom x}))"
 by (auto simp: ground_fm_aux_def ground_aux_def supp_conv_fresh)
 
 lemma ground_fresh[simp]:
   "ground t \<Longrightarrow> atom i \<sharp> t"
   "ground_fm A \<Longrightarrow> atom i \<sharp> A"
 unfolding ground_aux_def ground_fm_aux_def fresh_def
 by simp_all
 
 
-section{*Sigma Formulas*}
+section\<open>Sigma Formulas\<close>
 
-text{*Section 2 material*}
+text\<open>Section 2 material\<close>
 
-subsection {* Strict Sigma Formulas *}
+subsection \<open>Strict Sigma Formulas\<close>
 
-text{*Definition 2.1*}
+text\<open>Definition 2.1\<close>
 inductive ss_fm :: "fm \<Rightarrow> bool" where
     MemI:  "ss_fm (Var i IN Var j)"
   | DisjI: "ss_fm A \<Longrightarrow> ss_fm B \<Longrightarrow> ss_fm (A OR B)"
   | ConjI: "ss_fm A \<Longrightarrow> ss_fm B \<Longrightarrow> ss_fm (A AND B)"
   | ExI:   "ss_fm A \<Longrightarrow> ss_fm (Ex i A)"
   | All2I: "ss_fm A \<Longrightarrow> atom j \<sharp> (i,A) \<Longrightarrow> ss_fm (All2 i (Var j) A)"
 
 equivariance ss_fm
 
 nominal_inductive ss_fm
   avoids ExI: "i" | All2I: "i"
 by (simp_all add: fresh_star_def)
 
 declare ss_fm.intros [intro]
 
 
 definition Sigma_fm :: "fm \<Rightarrow> bool"
   where "Sigma_fm A \<longleftrightarrow> (\<exists>B. ss_fm B \<and> supp B \<subseteq> supp A \<and> {} \<turnstile> A IFF B)"
 
 lemma Sigma_fm_Iff: "\<lbrakk>{} \<turnstile> B IFF A; supp A \<subseteq> supp B; Sigma_fm A\<rbrakk> \<Longrightarrow> Sigma_fm B"
   by (metis Sigma_fm_def Iff_trans order_trans)
 
 lemma ss_fm_imp_Sigma_fm [intro]: "ss_fm A \<Longrightarrow> Sigma_fm A"
   by (metis Iff_refl Sigma_fm_def order_refl)
 
 lemma Sigma_fm_Fls [iff]: "Sigma_fm Fls"
   by (rule Sigma_fm_Iff [of _ "Ex i (Var i IN Var i)"]) auto
 
-subsection{* Closure properties for Sigma-formulas *}
+subsection\<open>Closure properties for Sigma-formulas\<close>
 
 lemma
   assumes "Sigma_fm A" "Sigma_fm B"
     shows Sigma_fm_AND [intro!]: "Sigma_fm (A AND B)"
       and Sigma_fm_OR [intro!]:  "Sigma_fm (A OR B)"
       and Sigma_fm_Ex [intro!]:  "Sigma_fm (Ex i A)"
 proof -
   obtain SA SB where "ss_fm SA" "{} \<turnstile> A IFF SA" "supp SA \<subseteq> supp A"
                  and "ss_fm SB" "{} \<turnstile> B IFF SB" "supp SB \<subseteq> supp B"
     using assms by (auto simp add: Sigma_fm_def)
   then show "Sigma_fm (A AND B)"  "Sigma_fm (A OR B)"  "Sigma_fm (Ex i A)"
     apply (auto simp: Sigma_fm_def)
     apply (metis ss_fm.ConjI Conj_cong Un_mono supp_Conj)
     apply (metis ss_fm.DisjI Disj_cong Un_mono fm.supp(3))
     apply (rule exI [where x = "Ex i SA"])
     apply (auto intro!: Ex_cong)
     done
 qed
 
 lemma Sigma_fm_All2_Var:
   assumes H0: "Sigma_fm A" and ij: "atom j \<sharp> (i,A)"
   shows "Sigma_fm (All2 i (Var j) A)"
 proof -
   obtain SA where SA: "ss_fm SA" "{} \<turnstile> A IFF SA" "supp SA \<subseteq> supp A"
     using H0 by (auto simp add: Sigma_fm_def)
   show "Sigma_fm (All2 i (Var j) A)"
     apply (rule Sigma_fm_Iff [of _ "All2 i (Var j) SA"])
     apply (metis All2_cong Refl SA(2) emptyE)
     using SA ij
     apply (auto simp: supp_conv_fresh subset_iff)
     apply (metis ss_fm.All2I fresh_Pair ss_fm_imp_Sigma_fm)
     done
 qed
 
 
-section{* Lemma 2.2: Atomic formulas are Sigma-formulas *}
+section\<open>Lemma 2.2: Atomic formulas are Sigma-formulas\<close>
 
 lemma Eq_Eats_Iff:
    assumes [unfolded fresh_Pair, simp]: "atom i \<sharp> (z,x,y)"
    shows "{} \<turnstile> z EQ Eats x y IFF (All2 i z (Var i IN x OR Var i EQ y)) AND x SUBS z AND y IN z"
 proof (rule Iff_I, auto)
   have "{Var i IN z, z EQ Eats x y} \<turnstile> Var i IN Eats x y"
     by (metis Assume Iff_MP_left Iff_sym Mem_cong Refl)
   then show "{Var i IN z, z EQ Eats x y} \<turnstile> Var i IN x OR Var i EQ y"
     by (metis Iff_MP_same Mem_Eats_Iff)
 next
   show "{z EQ Eats x y} \<turnstile> x SUBS z"
     by (metis Iff_MP2_same Subset_cong [OF Refl Assume] Subset_Eats_I)
 next
   show "{z EQ Eats x y} \<turnstile> y IN z"
     by (metis Iff_MP2_same Mem_cong Assume Refl Mem_Eats_I2)
 next
   show "{x SUBS z, y IN z, All2 i z (Var i IN x OR Var i EQ y)} \<turnstile> z EQ Eats x y"
        (is "{_, _, ?allHyp} \<turnstile> _")
     apply (rule Eq_Eats_iff [OF assms, THEN Iff_MP2_same], auto)
     apply (rule Ex_I [where x="Var i"])
     apply (auto intro: Subset_D  Mem_cong [OF Assume Refl, THEN Iff_MP2_same])
     done
 qed
 
 lemma Subset_Zero_sf: "Sigma_fm (Var i SUBS Zero)"
 proof -
   obtain j::name where j: "atom j \<sharp> i"
     by (rule obtain_fresh)
   hence Subset_Zero_Iff: "{} \<turnstile> Var i SUBS Zero IFF (All2 j (Var i) Fls)"
     by (auto intro!: Subset_I [of j] intro: Eq_Zero_D Subset_Zero_D All2_E [THEN rotate2])
   thus ?thesis using j
     by (auto simp: supp_conv_fresh
              intro!: Sigma_fm_Iff [OF Subset_Zero_Iff] Sigma_fm_All2_Var)
 qed
 
 lemma Eq_Zero_sf: "Sigma_fm (Var i EQ Zero)"
 proof -
   obtain j::name where "atom j \<sharp> i"
     by (rule obtain_fresh)
   thus ?thesis
     by (auto simp add: supp_conv_fresh
              intro!: Sigma_fm_Iff [OF _ _ Subset_Zero_sf] Subset_Zero_D EQ_imp_SUBS)
 qed
 
 lemma theorem_sf: assumes "{} \<turnstile> A" shows "Sigma_fm A"
 proof -
   obtain i::name and j::name
     where ij: "atom i \<sharp> (j,A)" "atom j \<sharp> A"
     by (metis obtain_fresh)
   show ?thesis
     apply (rule Sigma_fm_Iff [where A = "Ex i (Ex j (Var i IN Var j))"])
     using ij
     apply (auto simp: )
     apply (rule Ex_I [where x=Zero], simp)
     apply (rule Ex_I [where x="Eats Zero Zero"])
     apply (auto intro: Mem_Eats_I2 assms thin0)
     done
 qed
 
-text {*The subset relation *}
+text \<open>The subset relation\<close>
 lemma Var_Subset_sf: "Sigma_fm (Var i SUBS Var j)"
 proof -
   obtain k::name where k: "atom (k::name) \<sharp> (i,j)"
     by (metis obtain_fresh)
   thus ?thesis
   proof (cases "i=j")
     case True thus ?thesis using k
       by (auto intro!: theorem_sf Subset_I [where i=k])
   next
     case False thus ?thesis using k
       by (auto simp: ss_fm_imp_Sigma_fm Subset.simps [of k] ss_fm.intros)
   qed
 qed
 
 lemma Zero_Mem_sf: "Sigma_fm (Zero IN Var i)"
 proof -
   obtain j::name where "atom j \<sharp> i"
     by (rule obtain_fresh)
   hence Zero_Mem_Iff: "{} \<turnstile> Zero IN Var i IFF (Ex j (Var j  EQ Zero AND Var j  IN Var i))"
     by (auto intro: Ex_I [where x = Zero]  Mem_cong [OF Assume Refl, THEN Iff_MP_same])
   show ?thesis
     by (auto intro!: Sigma_fm_Iff [OF Zero_Mem_Iff] Eq_Zero_sf)
 qed
 
 lemma ijk: "i + k < Suc (i + j + k)"
   by arith
 
 lemma All2_term_Iff_fresh: "i\<noteq>j \<Longrightarrow> atom j' \<sharp> (i,j,A) \<Longrightarrow>
    {} \<turnstile> (All2 i (Var j) A) IFF Ex j' (Var j EQ Var j' AND All2 i (Var j') A)"
 apply auto
 apply (rule Ex_I [where x="Var j"], auto)
 apply (rule Ex_I [where x="Var i"], auto intro: ContraProve Mem_cong [THEN Iff_MP_same])
 done
 
 lemma Sigma_fm_All2_fresh:
   assumes "Sigma_fm A" "i\<noteq>j"
     shows "Sigma_fm (All2 i (Var j) A)"
 proof -
   obtain j'::name where j': "atom j' \<sharp> (i,j,A)"
     by (metis obtain_fresh)
   show "Sigma_fm (All2 i (Var j) A)"
     apply (rule Sigma_fm_Iff [OF All2_term_Iff_fresh [OF _ j']])
     using assms j'
     apply (auto simp: supp_conv_fresh Var_Subset_sf
                 intro!: Sigma_fm_All2_Var Sigma_fm_Iff [OF Extensionality _ _])
     done
 qed
 
 lemma Subset_Eats_sf:
   assumes "\<And>j::name. Sigma_fm (Var j IN t)"
       and "\<And>k::name. Sigma_fm (Var k EQ u)"
   shows "Sigma_fm (Var i SUBS Eats t u)"
 proof -
   obtain k::name where k: "atom k \<sharp> (t,u,Var i)"
     by (metis obtain_fresh)
   hence "{} \<turnstile> Var i SUBS Eats t u IFF All2 k (Var i) (Var k IN t OR Var k EQ u)"
     apply (auto simp: fresh_Pair intro: Set_MP Disj_I1 Disj_I2)
     apply (force intro!: Subset_I [where i=k] intro: All2_E' [OF Hyp] Mem_Eats_I1 Mem_Eats_I2)
     done
   thus ?thesis
     apply (rule Sigma_fm_Iff)
     using k
     apply (auto intro!: Sigma_fm_All2_fresh simp add: assms fresh_Pair supp_conv_fresh fresh_at_base)
     done
 qed
 
 lemma Eq_Eats_sf:
   assumes "\<And>j::name. Sigma_fm (Var j EQ t)"
       and "\<And>k::name. Sigma_fm (Var k EQ u)"
   shows "Sigma_fm (Var i EQ Eats t u)"
 proof -
   obtain j::name and k::name and l::name
     where atoms: "atom j \<sharp> (t,u,i)" "atom k \<sharp> (t,u,i,j)" "atom l \<sharp> (t,u,i,j,k)"
     by (metis obtain_fresh)
   hence "{} \<turnstile> Var i EQ Eats t u IFF
               Ex j (Ex k (Var i EQ Eats (Var j) (Var k) AND Var j EQ t AND Var k EQ u))"
     apply auto
     apply (rule Ex_I [where x=t], simp)
     apply (rule Ex_I [where x=u], auto intro: Trans Eats_cong)
     done
   thus ?thesis
     apply (rule Sigma_fm_Iff)
     apply (auto simp: assms supp_at_base)
     apply (rule Sigma_fm_Iff [OF Eq_Eats_Iff [of l]])
     using atoms
     apply (auto simp: supp_conv_fresh fresh_at_base Var_Subset_sf
                 intro!: Sigma_fm_All2_Var Sigma_fm_Iff [OF Extensionality _ _])
     done
 qed
 
 lemma Eats_Mem_sf:
   assumes "\<And>j::name. Sigma_fm (Var j EQ t)"
       and "\<And>k::name. Sigma_fm (Var k EQ u)"
   shows "Sigma_fm (Eats t u IN Var i)"
 proof -
   obtain j::name where j: "atom j \<sharp> (t,u,Var i)"
     by (metis obtain_fresh)
   hence "{} \<turnstile> Eats t u IN Var i IFF
               Ex j (Var j IN Var i AND Var j EQ Eats t u)"
     apply (auto simp: fresh_Pair intro: Ex_I [where x="Eats t u"])
     apply (metis Assume Mem_cong [OF _ Refl, THEN Iff_MP_same] rotate2)
     done
   thus ?thesis
     by (rule Sigma_fm_Iff) (auto simp: assms supp_conv_fresh Eq_Eats_sf)
 qed
 
 lemma Subset_Mem_sf_lemma:
   "size t + size u < n \<Longrightarrow> Sigma_fm (t SUBS u) \<and> Sigma_fm (t IN u)"
 proof (induction n arbitrary: t u rule: less_induct)
   case (less n t u)
   show ?case
   proof
     show "Sigma_fm (t SUBS u)"
       proof (cases t rule: tm.exhaust)
         case Zero thus ?thesis
           by (auto intro: theorem_sf)
       next
         case (Var i) thus ?thesis using less.prems
           apply (cases u rule: tm.exhaust)
           apply (auto simp: Subset_Zero_sf Var_Subset_sf)
           apply (force simp: supp_conv_fresh less.IH
                        intro: Subset_Eats_sf Sigma_fm_Iff [OF Extensionality])
           done
       next
         case (Eats t1 t2) thus ?thesis using less.IH [OF _ ijk] less.prems
           by (auto intro!: Sigma_fm_Iff [OF Eats_Subset_Iff]  simp: supp_conv_fresh)
              (metis add.commute)
       qed
   next
     show "Sigma_fm (t IN u)"
       proof (cases u rule: tm.exhaust)
         case Zero show ?thesis
           by (rule Sigma_fm_Iff [where A=Fls]) (auto simp: supp_conv_fresh Zero)
       next
         case (Var i) show ?thesis
         proof (cases t rule: tm.exhaust)
-          case Zero thus ?thesis using `u = Var i`
+          case Zero thus ?thesis using \<open>u = Var i\<close>
             by (auto intro: Zero_Mem_sf)
         next
           case (Var j)
-          thus ?thesis using `u = Var i`
+          thus ?thesis using \<open>u = Var i\<close>
             by auto
         next
-          case (Eats t1 t2) thus ?thesis using `u = Var i` less.prems
+          case (Eats t1 t2) thus ?thesis using \<open>u = Var i\<close> less.prems
             by (force intro: Eats_Mem_sf Sigma_fm_Iff [OF Extensionality _ _]
                       simp: supp_conv_fresh less.IH [THEN conjunct1])
         qed
       next
         case (Eats t1 t2) thus ?thesis  using  less.prems
           by (force intro: Sigma_fm_Iff [OF Mem_Eats_Iff] Sigma_fm_Iff [OF Extensionality _ _]
                     simp: supp_conv_fresh less.IH)
       qed
   qed
 qed
 
 lemma Subset_sf [iff]: "Sigma_fm (t SUBS u)"
   by (metis Subset_Mem_sf_lemma [OF lessI])
 
 lemma Mem_sf [iff]: "Sigma_fm (t IN u)"
   by (metis Subset_Mem_sf_lemma [OF lessI])
 
-text {*The equality relation is a Sigma-Formula *}
+text \<open>The equality relation is a Sigma-Formula\<close>
 lemma Equality_sf [iff]: "Sigma_fm (t EQ u)"
   by (auto intro: Sigma_fm_Iff [OF Extensionality] simp: supp_conv_fresh)
 
 
-section{*Universal Quantification Bounded by an Arbitrary Term*}
+section\<open>Universal Quantification Bounded by an Arbitrary Term\<close>
 
 lemma All2_term_Iff: "atom i \<sharp> t \<Longrightarrow> atom j \<sharp> (i,t,A) \<Longrightarrow>
                   {} \<turnstile> (All2 i t A) IFF Ex j (Var j EQ t AND All2 i (Var j) A)"
 apply auto
 apply (rule Ex_I [where x=t], auto)
 apply (rule Ex_I [where x="Var i"])
 apply (auto intro: ContraProve Mem_cong [THEN Iff_MP2_same])
 done
 
 lemma Sigma_fm_All2 [intro!]:
   assumes "Sigma_fm A" "atom i \<sharp> t"
     shows "Sigma_fm (All2 i t A)"
 proof -
   obtain j::name where j: "atom j \<sharp> (i,t,A)"
     by (metis obtain_fresh)
   show "Sigma_fm (All2 i t A)"
     apply (rule Sigma_fm_Iff [OF All2_term_Iff [of i t j]])
     using assms j
     apply (auto simp: supp_conv_fresh Sigma_fm_All2_Var)
     done
 qed
 
 
-section {* Lemma 2.3: Sequence-related concepts are Sigma-formulas *}
+section \<open>Lemma 2.3: Sequence-related concepts are Sigma-formulas\<close>
 
 lemma OrdP_sf [iff]: "Sigma_fm (OrdP t)"
 proof -
   obtain z::name and y::name where "atom z \<sharp> t" "atom y \<sharp> (t, z)"
     by (metis obtain_fresh)
   thus ?thesis
     by (auto simp: OrdP.simps)
 qed
 
 lemma OrdNotEqP_sf [iff]: "Sigma_fm (OrdNotEqP t u)"
   by (auto simp: OrdNotEqP.simps)
 
 lemma HDomain_Incl_sf [iff]: "Sigma_fm (HDomain_Incl t u)"
 proof -
   obtain x::name and y::name and z::name
     where "atom x \<sharp> (t,u,y,z)" "atom y \<sharp> (t,u,z)" "atom z \<sharp> (t,u)"
     by (metis obtain_fresh)
   thus ?thesis
     by auto
 qed
 
 lemma HFun_Sigma_Iff:
   assumes "atom z \<sharp> (r,z',x,y,x',y')"  "atom z' \<sharp> (r,x,y,x',y')"
        "atom x \<sharp> (r,y,x',y')"  "atom y \<sharp> (r,x',y')"
        "atom x' \<sharp> (r,y')"  "atom y' \<sharp> (r)"
   shows
   "{} \<turnstile>HFun_Sigma r IFF
          All2 z r (All2 z' r (Ex x (Ex y (Ex x' (Ex y'
              (Var z EQ HPair (Var x) (Var y) AND Var z' EQ HPair (Var x') (Var y')
               AND OrdP (Var x) AND OrdP (Var x') AND
               ((Var x NEQ Var x') OR (Var y EQ Var y'))))))))"
   apply (simp add: HFun_Sigma.simps [OF assms])
   apply (rule Iff_refl All_cong Imp_cong Ex_cong)+
   apply (rule Conj_cong [OF Iff_refl])
   apply (rule Conj_cong [OF Iff_refl], auto)
   apply (blast intro: Disj_I1 Neg_D OrdNotEqP_I)
   apply (blast intro: Disj_I2)
   apply (blast intro: OrdNotEqP_E rotate2)
   done
 
 lemma HFun_Sigma_sf [iff]: "Sigma_fm (HFun_Sigma t)"
 proof -
   obtain x::name and y::name and z::name and x'::name and y'::name and z'::name
     where atoms: "atom z \<sharp> (t,z',x,y,x',y')"  "atom z' \<sharp> (t,x,y,x',y')"
        "atom x \<sharp> (t,y,x',y')"  "atom y \<sharp> (t,x',y')"
        "atom x' \<sharp> (t,y')"  "atom y' \<sharp> (t)"
     by (metis obtain_fresh)
   show ?thesis
     by (auto intro!: Sigma_fm_Iff [OF HFun_Sigma_Iff [OF atoms]] simp: supp_conv_fresh atoms)
 qed
 
 lemma LstSeqP_sf [iff]: "Sigma_fm (LstSeqP t u v)"
   by (auto simp: LstSeqP.simps)
 
 end
 
diff --git a/thys/Goedel_HFSet_Semanticless/SyntaxN.thy b/thys/Goedel_HFSet_Semanticless/SyntaxN.thy
--- a/thys/Goedel_HFSet_Semanticless/SyntaxN.thy
+++ b/thys/Goedel_HFSet_Semanticless/SyntaxN.thy
@@ -1,1409 +1,1409 @@
-chapter{*Syntax of Terms and Formulas using Nominal Logic*}
+chapter\<open>Syntax of Terms and Formulas using Nominal Logic\<close>
 
 theory SyntaxN
 imports Nominal2.Nominal2 HereditarilyFinite.OrdArith
 begin
 
-section{*Terms and Formulas*}
+section\<open>Terms and Formulas\<close>
 
-subsection{* Hf is a pure permutation type *}
+subsection\<open>Hf is a pure permutation type\<close>
 
 instantiation hf :: pt
 begin
   definition "p \<bullet> (s::hf) = s"
   instance
     by standard (simp_all add: permute_hf_def)
 end
 
 instance hf :: pure
   proof qed (rule permute_hf_def)
 
 atom_decl name
 
 declare fresh_set_empty [simp]
 
 lemma supp_name [simp]: fixes i::name shows "supp i = {atom i}"
   by (rule supp_at_base)
 
-subsection{*The datatypes*}
+subsection\<open>The datatypes\<close>
 
 nominal_datatype tm = Zero | Var name | Eats tm tm
 
 nominal_datatype fm =
     Mem tm tm    (infixr "IN" 150)
   | Eq tm tm     (infixr "EQ" 150)
   | Disj fm fm   (infixr "OR" 130)
   | Neg fm
   | Ex x::name f::fm binds x in f
 
-text {*Mem, Eq are atomic formulas; Disj, Neg, Ex are non-atomic*}
+text \<open>Mem, Eq are atomic formulas; Disj, Neg, Ex are non-atomic\<close>
 
 declare tm.supp [simp] fm.supp [simp]
 
-subsection{*Substitution*}
+subsection\<open>Substitution\<close>
 
 nominal_function subst :: "name \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm"
   where
    "subst i x Zero       = Zero"
  | "subst i x (Var k)    = (if i=k then x else Var k)"
  | "subst i x (Eats t u) = Eats (subst i x t) (subst i x u)"
 by (auto simp: eqvt_def subst_graph_aux_def) (metis tm.strong_exhaust)
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma fresh_subst_if [simp]:
   "j \<sharp> subst i x t \<longleftrightarrow> (atom i \<sharp> t \<and> j \<sharp> t) \<or> (j \<sharp> x \<and> (j \<sharp> t \<or> j = atom i))"
   by (induct t rule: tm.induct) (auto simp: fresh_at_base)
 
 lemma forget_subst_tm [simp]: "atom a \<sharp> tm \<Longrightarrow> subst a x tm = tm"
   by (induct tm rule: tm.induct) (simp_all add: fresh_at_base)
 
 lemma subst_tm_id [simp]: "subst a (Var a) tm = tm"
   by (induct tm rule: tm.induct) simp_all
 
 lemma subst_tm_commute [simp]:
   "atom j \<sharp> tm \<Longrightarrow> subst j u (subst i t tm) = subst i (subst j u t) tm"
   by (induct tm rule: tm.induct) (auto simp: fresh_Pair)
 
 lemma subst_tm_commute2 [simp]:
   "atom j \<sharp> t \<Longrightarrow> atom i \<sharp> u \<Longrightarrow> i \<noteq> j \<Longrightarrow> subst j u (subst i t tm) = subst i t (subst j u tm)"
   by (induct tm rule: tm.induct) auto
 
 lemma repeat_subst_tm [simp]: "subst i u (subst i t tm) = subst i (subst i u t) tm"
   by (induct tm rule: tm.induct) auto
 
 nominal_function  subst_fm :: "fm \<Rightarrow> name \<Rightarrow> tm \<Rightarrow> fm" ("_'(_::=_')" [1000, 0, 0] 200)
   where
     Mem:  "(Mem t u)(i::=x)  = Mem (subst i x t) (subst i x u)"
   | Eq:   "(Eq t u)(i::=x)   = Eq  (subst i x t) (subst i x u)"
   | Disj: "(Disj A B)(i::=x) = Disj (A(i::=x)) (B(i::=x))"
   | Neg:  "(Neg A)(i::=x)    = Neg  (A(i::=x))"
   | Ex:   "atom j \<sharp> (i, x) \<Longrightarrow> (Ex j A)(i::=x) = Ex j (A(i::=x))"
 apply (simp add: eqvt_def subst_fm_graph_aux_def)
 apply auto [16]
 apply (rule_tac y=a and c="(aa, b)" in fm.strong_exhaust)
 apply (auto simp: eqvt_at_def fresh_star_def fresh_Pair fresh_at_base)
 apply (metis flip_at_base_simps(3) flip_fresh_fresh)
 done
 
 nominal_termination (eqvt)
   by lexicographic_order
 
 lemma size_subst_fm [simp]: "size (A(i::=x)) = size A"
   by (nominal_induct A avoiding: i x rule: fm.strong_induct) auto
 
 lemma forget_subst_fm [simp]: "atom a \<sharp> A \<Longrightarrow> A(a::=x) = A"
   by (nominal_induct A avoiding: a x rule: fm.strong_induct) (auto simp: fresh_at_base)
 
 lemma subst_fm_id [simp]: "A(a::=Var a) = A"
   by (nominal_induct A avoiding: a rule: fm.strong_induct) (auto simp: fresh_at_base)
 
 lemma fresh_subst_fm_if [simp]:
   "j \<sharp> (A(i::=x)) \<longleftrightarrow> (atom i \<sharp> A \<and> j \<sharp> A) \<or> (j \<sharp> x \<and> (j \<sharp> A \<or> j = atom i))"
   by (nominal_induct A avoiding: i x rule: fm.strong_induct) (auto simp: fresh_at_base)
 
 lemma subst_fm_commute [simp]:
   "atom j \<sharp> A \<Longrightarrow> (A(i::=t))(j::=u) = A(i ::= subst j u t)"
   by (nominal_induct A avoiding: i j t u rule: fm.strong_induct) (auto simp: fresh_at_base)
 
 lemma repeat_subst_fm [simp]: "(A(i::=t))(i::=u) = A(i ::= subst i u t)"
   by (nominal_induct A avoiding: i t u rule: fm.strong_induct) auto
 
 lemma subst_fm_Ex_with_renaming:
   "atom i' \<sharp> (A, i, j, t) \<Longrightarrow> (Ex i A)(j ::= t) = Ex i' (((i \<leftrightarrow> i') \<bullet> A)(j ::= t))"
   by (rule subst [of "Ex i' ((i \<leftrightarrow> i') \<bullet> A)" "Ex i A"])
      (auto simp: Abs1_eq_iff flip_def swap_commute)
 
-text {* the simplifier cannot apply the rule above, because
-it introduces a new variable at the right hand side. *}
+text \<open>the simplifier cannot apply the rule above, because
+it introduces a new variable at the right hand side.\<close>
 
-simproc_setup subst_fm_renaming ("(Ex i A)(j ::= t)") = {* fn _ => fn ctxt => fn ctrm =>
+simproc_setup subst_fm_renaming ("(Ex i A)(j ::= t)") = \<open>fn _ => fn ctxt => fn ctrm =>
   let
      val _ $ (_ $ i $ A) $ j $ t = Thm.term_of ctrm
 
      val atoms = Simplifier.prems_of ctxt
       |> map_filter (fn thm => case Thm.prop_of thm of
            _ $ (Const (@{const_name fresh}, _) $ atm $ _) => SOME (atm) | _ => NONE)
       |> distinct ((=))
 
      fun get_thm atm =
        let
          val goal = HOLogic.mk_Trueprop (mk_fresh atm (HOLogic.mk_tuple [A, i, j, t]))
        in
          SOME ((Goal.prove ctxt [] [] goal (K (asm_full_simp_tac ctxt 1)))
            RS @{thm subst_fm_Ex_with_renaming} RS eq_reflection)
          handle ERROR _ => NONE
        end
   in
     get_first get_thm atoms
   end
-*}
+\<close>
 
-subsection{*Derived syntax*}
+subsection\<open>Derived syntax\<close>
 
-subsubsection{*Ordered pairs*}
+subsubsection\<open>Ordered pairs\<close>
 
 definition HPair :: "tm \<Rightarrow> tm \<Rightarrow> tm"
   where "HPair a b = Eats (Eats Zero (Eats (Eats Zero b) a)) (Eats (Eats Zero a) a)"
 
 lemma HPair_eqvt [eqvt]: "(p \<bullet> HPair a b) = HPair (p \<bullet> a) (p \<bullet> b)"
   by (auto simp: HPair_def)
 
 lemma fresh_HPair [simp]: "x \<sharp> HPair a b \<longleftrightarrow> (x \<sharp> a \<and> x \<sharp> b)"
   by (auto simp: HPair_def)
 
 lemma HPair_injective_iff [iff]: "HPair a b = HPair a' b' \<longleftrightarrow> (a = a' \<and> b = b')"
   by (auto simp: HPair_def)
 
 lemma subst_tm_HPair [simp]: "subst i x (HPair a b) = HPair (subst i x a) (subst i x b)"
   by (auto simp: HPair_def)
 
-subsubsection{*Ordinals*}
+subsubsection\<open>Ordinals\<close>
 
 definition
   SUCC  :: "tm \<Rightarrow> tm"  where
     "SUCC x \<equiv> Eats x x"
 
 fun ORD_OF :: "nat \<Rightarrow> tm"
   where
    "ORD_OF 0 = Zero"
  | "ORD_OF (Suc k) = SUCC (ORD_OF k)"
 
 lemma SUCC_fresh_iff [simp]: "a \<sharp> SUCC t \<longleftrightarrow> a \<sharp> t"
   by (simp add: SUCC_def)
 
 lemma SUCC_eqvt [eqvt]: "(p \<bullet> SUCC a) = SUCC (p \<bullet> a)"
   by (simp add: SUCC_def)
 
 lemma SUCC_subst [simp]: "subst i t (SUCC k) = SUCC (subst i t k)"
   by (simp add: SUCC_def)
 
 lemma ORD_OF_fresh [simp]: "a \<sharp> ORD_OF n"
   by (induct n) (auto simp: SUCC_def)
 
 lemma ORD_OF_eqvt [eqvt]: "(p \<bullet> ORD_OF n) = ORD_OF (p \<bullet> n)"
   by (induct n) (auto simp: permute_pure SUCC_eqvt)
 
-subsection{*Derived logical connectives*}
+subsection\<open>Derived logical connectives\<close>
 
 abbreviation Imp :: "fm \<Rightarrow> fm \<Rightarrow> fm"   (infixr "IMP" 125)
   where "Imp A B \<equiv> Disj (Neg A) B"
 
 abbreviation All :: "name \<Rightarrow> fm \<Rightarrow> fm"
   where "All i A \<equiv> Neg (Ex i (Neg A))"
 
 abbreviation All2 :: "name \<Rightarrow> tm \<Rightarrow> fm \<Rightarrow> fm" \<comment> \<open>bounded universal quantifier, for Sigma formulas\<close>
   where "All2 i t A \<equiv> All i ((Var i IN t) IMP A)"
 
-subsubsection{*Conjunction*}
+subsubsection\<open>Conjunction\<close>
 
 definition Conj :: "fm \<Rightarrow> fm \<Rightarrow> fm"   (infixr "AND" 135)
   where "Conj A B \<equiv> Neg (Disj (Neg A) (Neg B))"
 
 lemma Conj_eqvt [eqvt]: "p \<bullet> (A AND B) = (p \<bullet> A) AND (p \<bullet> B)"
   by (simp add: Conj_def)
 
 lemma fresh_Conj [simp]: "a \<sharp> A AND B \<longleftrightarrow> (a \<sharp> A \<and> a \<sharp> B)"
   by (auto simp: Conj_def)
 
 lemma supp_Conj [simp]: "supp (A AND B) = supp A \<union> supp B"
   by (auto simp: Conj_def)
 
 lemma size_Conj [simp]: "size (A AND B) = size A + size B + 4"
   by (simp add: Conj_def)
 
 lemma Conj_injective_iff [iff]: "(A AND B) = (A' AND B') \<longleftrightarrow> (A = A' \<and> B = B')"
   by (auto simp: Conj_def)
 
 lemma subst_fm_Conj [simp]: "(A AND B)(i::=x) = (A(i::=x)) AND (B(i::=x))"
   by (auto simp: Conj_def)
 
-subsubsection{*If and only if*}
+subsubsection\<open>If and only if\<close>
 
 definition Iff :: "fm \<Rightarrow> fm \<Rightarrow> fm"   (infixr "IFF" 125)
   where "Iff A B = Conj (Imp A B) (Imp B A)"
 
 lemma Iff_eqvt [eqvt]: "p \<bullet> (A IFF B) = (p \<bullet> A) IFF (p \<bullet> B)"
   by (simp add: Iff_def)
 
 lemma fresh_Iff [simp]: "a \<sharp> A IFF B \<longleftrightarrow> (a \<sharp> A \<and> a \<sharp> B)"
   by (auto simp: Conj_def Iff_def)
 
 lemma size_Iff [simp]: "size (A IFF B) = 2*(size A + size B) + 8"
   by (simp add: Iff_def)
 
 lemma Iff_injective_iff [iff]: "(A IFF B) = (A' IFF B') \<longleftrightarrow> (A = A' \<and> B = B')"
   by (auto simp: Iff_def)
 
 lemma subst_fm_Iff [simp]: "(A IFF B)(i::=x) = (A(i::=x)) IFF (B(i::=x))"
   by (auto simp: Iff_def)
 
 
-section{*Axioms and Theorems*}
+section\<open>Axioms and Theorems\<close>
 
-subsection{*Logical axioms*}
+subsection\<open>Logical axioms\<close>
 
 inductive_set boolean_axioms :: "fm set"
   where
     Ident:     "A IMP A \<in> boolean_axioms"
   | DisjI1:    "A IMP (A OR B) \<in> boolean_axioms"
   | DisjCont:  "(A OR A) IMP A \<in> boolean_axioms"
   | DisjAssoc: "(A OR (B OR C)) IMP ((A OR B) OR C) \<in> boolean_axioms"
   | DisjConj:  "(C OR A) IMP (((Neg C) OR B) IMP (A OR B)) \<in> boolean_axioms"
 
 inductive_set special_axioms :: "fm set" where
   I: "A(i::=x) IMP (Ex i A) \<in> special_axioms"
 
 inductive_set induction_axioms :: "fm set" where
   ind:
   "atom (j::name) \<sharp> (i,A)
    \<Longrightarrow> A(i::=Zero) IMP ((All i (All j (A IMP (A(i::= Var j) IMP A(i::= Eats(Var i)(Var j))))))
       IMP (All i A))
     \<in> induction_axioms"
 
-subsection {* Concrete variables *}
+subsection \<open>Concrete variables\<close>
 
 declare Abs_name_inject[simp]
 
 abbreviation
   "X0 \<equiv> Abs_name (Atom (Sort ''SyntaxN.name'' []) 0)"
 
 abbreviation
   "X1 \<equiv> Abs_name (Atom (Sort ''SyntaxN.name'' []) (Suc 0))"
    \<comment> \<open>We prefer @{term "Suc 0"} because simplification will transform 1 to that form anyway.\<close>
 
 abbreviation
   "X2 \<equiv> Abs_name (Atom (Sort ''SyntaxN.name'' []) 2)"
 
 abbreviation
   "X3 \<equiv> Abs_name (Atom (Sort ''SyntaxN.name'' []) 3)"
 
 abbreviation
   "X4 \<equiv> Abs_name (Atom (Sort ''SyntaxN.name'' []) 4)"
 
 
-subsection{*The HF axioms*}
+subsection\<open>The HF axioms\<close>
 
 definition HF1 :: fm where  \<comment> \<open>the axiom @{term"z=0 \<longleftrightarrow> (\<forall>x. \<not> x \<^bold>\<in> z)"}\<close>
   "HF1 = (Var X0 EQ Zero) IFF (All X1 (Neg (Var X1 IN Var X0)))"
 
 definition HF2 :: fm where  \<comment> \<open>the axiom @{term"z = x \<triangleleft> y \<longleftrightarrow> (\<forall>u. u \<^bold>\<in> z \<longleftrightarrow> u \<^bold>\<in> x | u=y)"}\<close>
   "HF2 \<equiv> Var X0 EQ Eats (Var X1) (Var X2) IFF
           All X3 (Var X3 IN Var X0 IFF Var X3 IN Var X1 OR Var X3 EQ Var X2)"
 
 definition HF_axioms where "HF_axioms = {HF1, HF2}"
 
-subsection{*Equality axioms*}
+subsection\<open>Equality axioms\<close>
 
 definition refl_ax :: fm where
   "refl_ax = Var X1 EQ Var X1"
 
 definition eq_cong_ax :: fm where
   "eq_cong_ax = ((Var X1 EQ Var X2) AND (Var X3 EQ Var X4)) IMP
                 ((Var X1 EQ Var X3) IMP (Var X2 EQ Var X4))"
 
 definition mem_cong_ax :: fm where
   "mem_cong_ax = ((Var X1 EQ Var X2) AND (Var X3 EQ Var X4)) IMP
                  ((Var X1 IN Var X3) IMP (Var X2 IN Var X4))"
 
 definition eats_cong_ax :: fm where
   "eats_cong_ax = ((Var X1 EQ Var X2) AND (Var X3 EQ Var X4)) IMP
                   ((Eats (Var X1) (Var X3)) EQ (Eats (Var X2) (Var X4)))"
 
 definition equality_axioms :: "fm set" where
   "equality_axioms = {refl_ax, eq_cong_ax, mem_cong_ax, eats_cong_ax}"
 
-subsection{*The proof system*}
+subsection\<open>The proof system\<close>
 
-text{*This arbitrary additional axiom generalises the statements of the incompleteness theorems
+text\<open>This arbitrary additional axiom generalises the statements of the incompleteness theorems
   and other results to any formal system stronger than the HF theory. The additional axiom
   could be the conjunction of any finite number of assertions. Any more general extension
-  must be a form that can be formalised for the proof predicate.*}
+  must be a form that can be formalised for the proof predicate.\<close>
 consts  extra_axiom :: fm
 
 inductive hfthm :: "fm set \<Rightarrow> fm \<Rightarrow> bool" (infixl "\<turnstile>" 55)
   where
     Hyp:    "A \<in> H \<Longrightarrow> H \<turnstile> A"
   | Extra:  "H \<turnstile> extra_axiom"
   | Bool:   "A \<in> boolean_axioms \<Longrightarrow> H \<turnstile> A"
   | Eq:     "A \<in> equality_axioms \<Longrightarrow> H \<turnstile> A"
   | Spec:   "A \<in> special_axioms \<Longrightarrow> H \<turnstile> A"
   | HF:     "A \<in> HF_axioms \<Longrightarrow> H \<turnstile> A"
   | Ind:    "A \<in> induction_axioms \<Longrightarrow> H \<turnstile> A"
   | MP:     "H \<turnstile> A IMP B \<Longrightarrow> H' \<turnstile> A \<Longrightarrow> H \<union> H' \<turnstile> B"
   | Exists: "H \<turnstile> A IMP B \<Longrightarrow> atom i \<sharp> B \<Longrightarrow> \<forall>C \<in> H. atom i \<sharp> C \<Longrightarrow> H \<turnstile> (Ex i A) IMP B"
 
-subsection{*Derived rules of inference*}
+subsection\<open>Derived rules of inference\<close>
 
 lemma contraction: "insert A (insert A H) \<turnstile> B \<Longrightarrow> insert A H \<turnstile> B"
   by (metis insert_absorb2)
 
 lemma thin_Un: "H \<turnstile> A \<Longrightarrow> H \<union> H' \<turnstile> A"
   by (metis Bool MP boolean_axioms.Ident sup_commute)
 
 lemma thin: "H \<turnstile> A \<Longrightarrow> H \<subseteq> H' \<Longrightarrow> H' \<turnstile> A"
   by (metis Un_absorb1 thin_Un)
 
 lemma thin0: "{} \<turnstile> A \<Longrightarrow> H \<turnstile> A"
   by (metis sup_bot_left thin_Un)
 
 lemma thin1: "H \<turnstile> B \<Longrightarrow> insert A H \<turnstile> B"
   by (metis subset_insertI thin)
 
 lemma thin2: "insert A1 H \<turnstile> B \<Longrightarrow> insert A1 (insert A2 H) \<turnstile> B"
   by (blast intro: thin)
 
 lemma thin3: "insert A1 (insert A2 H) \<turnstile> B \<Longrightarrow> insert A1 (insert A2 (insert A3 H)) \<turnstile> B"
   by (blast intro: thin)
 
 lemma thin4:
   "insert A1 (insert A2 (insert A3 H)) \<turnstile> B
    \<Longrightarrow> insert A1 (insert A2 (insert A3 (insert A4 H))) \<turnstile> B"
   by (blast intro: thin)
 
 lemma rotate2: "insert A2 (insert A1 H) \<turnstile> B \<Longrightarrow> insert A1 (insert A2 H) \<turnstile> B"
   by (blast intro: thin)
 
 lemma rotate3: "insert A3 (insert A1 (insert A2 H)) \<turnstile> B \<Longrightarrow> insert A1 (insert A2 (insert A3 H)) \<turnstile> B"
   by (blast intro: thin)
 
 lemma rotate4:
   "insert A4 (insert A1 (insert A2 (insert A3 H))) \<turnstile> B
    \<Longrightarrow> insert A1 (insert A2 (insert A3 (insert A4 H))) \<turnstile> B"
   by (blast intro: thin)
 
 lemma rotate5:
   "insert A5 (insert A1 (insert A2 (insert A3 (insert A4 H)))) \<turnstile> B
    \<Longrightarrow> insert A1 (insert A2 (insert A3 (insert A4 (insert A5 H)))) \<turnstile> B"
   by (blast intro: thin)
 
 lemma rotate6:
   "insert A6 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 H))))) \<turnstile> B
    \<Longrightarrow> insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 H))))) \<turnstile> B"
   by (blast intro: thin)
 
 lemma rotate7:
   "insert A7 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 H)))))) \<turnstile> B
    \<Longrightarrow> insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 H)))))) \<turnstile> B"
   by (blast intro: thin)
 
 lemma rotate8:
   "insert A8 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 H))))))) \<turnstile> B
    \<Longrightarrow> insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 H))))))) \<turnstile> B"
   by (blast intro: thin)
 
 lemma rotate9:
   "insert A9 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 H)))))))) \<turnstile> B
    \<Longrightarrow> insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 H)))))))) \<turnstile> B"
   by (blast intro: thin)
 
 lemma rotate10:
   "insert A10 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 H))))))))) \<turnstile> B
    \<Longrightarrow> insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 H))))))))) \<turnstile> B"
   by (blast intro: thin)
 
 lemma rotate11:
   "insert A11 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 H)))))))))) \<turnstile> B
    \<Longrightarrow> insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 H)))))))))) \<turnstile> B"
   by (blast intro: thin)
 
 lemma rotate12:
   "insert A12 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 H))))))))))) \<turnstile> B
    \<Longrightarrow> insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 H))))))))))) \<turnstile> B"
   by (blast intro: thin)
 
 lemma rotate13:
   "insert A13 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 H)))))))))))) \<turnstile> B
    \<Longrightarrow> insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 (insert A13 H)))))))))))) \<turnstile> B"
   by (blast intro: thin)
 
 lemma rotate14:
   "insert A14 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 (insert A13 H))))))))))))) \<turnstile> B
    \<Longrightarrow> insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 (insert A13 (insert A14 H))))))))))))) \<turnstile> B"
   by (blast intro: thin)
 
 lemma rotate15:
   "insert A15 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 (insert A13 (insert A14 H)))))))))))))) \<turnstile> B
    \<Longrightarrow> insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 (insert A13 (insert A14 (insert A15 H)))))))))))))) \<turnstile> B"
   by (blast intro: thin)
 
 lemma MP_same: "H \<turnstile> A IMP B \<Longrightarrow> H \<turnstile> A \<Longrightarrow> H \<turnstile> B"
   by (metis MP Un_absorb)
 
 lemma MP_thin: "HA \<turnstile> A IMP B \<Longrightarrow> HB \<turnstile> A \<Longrightarrow> HA \<union> HB \<subseteq> H \<Longrightarrow> H \<turnstile> B"
   by (metis MP_same le_sup_iff thin)
 
 lemma MP_null: "{} \<turnstile> A IMP B \<Longrightarrow> H \<turnstile> A \<Longrightarrow> H \<turnstile> B"
   by (metis MP_same thin0)
 
 lemma Disj_commute: "H \<turnstile> B OR A \<Longrightarrow> H \<turnstile> A OR B"
   using DisjConj [of B A B] Ident [of B]
 by (metis Bool MP_same)
 
 lemma S: assumes  "H \<turnstile> A IMP (B IMP C)" "H' \<turnstile> A IMP B" shows "H \<union> H' \<turnstile> A IMP C"
 proof -
   have "H' \<union> H \<turnstile> (Neg A) OR (C OR (Neg A))"
     by (metis Bool MP MP_same boolean_axioms.DisjConj Disj_commute DisjAssoc assms)
   thus ?thesis
     by (metis Bool Disj_commute Un_commute MP_same DisjAssoc DisjCont DisjI1)
 qed
 
 lemma Assume: "insert A H \<turnstile> A"
   by (metis Hyp insertI1)
 
 lemmas AssumeH = Assume Assume [THEN rotate2] Assume [THEN rotate3] Assume [THEN rotate4] Assume [THEN rotate5]
                  Assume [THEN rotate6] Assume [THEN rotate7] Assume [THEN rotate8] Assume [THEN rotate9] Assume [THEN rotate10]
                  Assume [THEN rotate11] Assume [THEN rotate12]
 declare AssumeH [intro!]
 
 lemma Imp_triv_I: "H \<turnstile> B \<Longrightarrow> H \<turnstile> A IMP B"
   by (metis Bool Disj_commute MP_same boolean_axioms.DisjI1)
 
 lemma DisjAssoc1: "H \<turnstile> A OR (B OR C) \<Longrightarrow> H \<turnstile> (A OR B) OR C"
   by (metis Bool MP_same boolean_axioms.DisjAssoc)
 
 lemma DisjAssoc2: "H \<turnstile> (A OR B) OR C \<Longrightarrow> H \<turnstile> A OR (B OR C)"
   by (metis DisjAssoc1 Disj_commute)
 
 lemma Disj_commute_Imp: "H \<turnstile> (B OR A) IMP (A OR B)"
   using DisjConj [of B A B] Ident [of B]
   by (metis Bool DisjAssoc2 Disj_commute MP_same)
 
 lemma Disj_Semicong_1: "H \<turnstile> A OR C \<Longrightarrow> H \<turnstile> A IMP B \<Longrightarrow> H \<turnstile> B OR C"
   using DisjConj [of A C B]
   by (metis Bool Disj_commute MP_same)
 
 lemma Imp_Imp_commute: "H \<turnstile> B IMP (A IMP C) \<Longrightarrow> H \<turnstile> A IMP (B IMP C)"
   by (metis DisjAssoc1 DisjAssoc2 Disj_Semicong_1 Disj_commute_Imp)
 
 
-subsection{*The Deduction Theorem*}
+subsection\<open>The Deduction Theorem\<close>
 
 lemma deduction_Diff: assumes "H \<turnstile> B" shows "H - {C} \<turnstile> C IMP B"
 using assms
 proof (induct)
   case (Hyp A H) thus ?case
     by (metis Bool Imp_triv_I boolean_axioms.Ident hfthm.Hyp member_remove remove_def)
 next
   case (Extra H) thus ?case
     by (metis Imp_triv_I hfthm.Extra)
 next
   case (Bool A H) thus ?case
     by (metis Imp_triv_I hfthm.Bool)
 next
   case (Eq A H) thus ?case
     by (metis Imp_triv_I hfthm.Eq)
 next
   case (Spec A H) thus ?case
     by (metis Imp_triv_I hfthm.Spec)
 next
   case (HF A H) thus ?case
     by (metis Imp_triv_I hfthm.HF)
 next
   case (Ind A H) thus ?case
     by (metis Imp_triv_I hfthm.Ind)
 next
   case (MP H A B H')
   hence "(H-{C}) \<union> (H'-{C}) \<turnstile> Imp C B"
     by (simp add: S)
   thus ?case
     by (metis Un_Diff)
 next
   case (Exists H A B i) show ?case
   proof (cases "C \<in> H")
     case True
     hence "atom i \<sharp> C" using Exists by auto
     moreover have "H - {C} \<turnstile> A IMP C IMP B" using Exists
       by (metis Imp_Imp_commute)
     ultimately have "H - {C} \<turnstile> (Ex i A) IMP C IMP B" using Exists
       by (metis fm.fresh(3) fm.fresh(4) hfthm.Exists member_remove remove_def)
     thus ?thesis
       by (metis Imp_Imp_commute)
   next
     case False
     hence "H - {C} = H" by auto
     thus ?thesis using Exists
       by (metis Imp_triv_I hfthm.Exists)
   qed
 qed
 
 theorem Imp_I [intro!]: "insert A H \<turnstile> B \<Longrightarrow> H \<turnstile> A IMP B"
   by (metis Diff_insert_absorb Imp_triv_I deduction_Diff insert_absorb)
 
 lemma anti_deduction: "H \<turnstile> A IMP B \<Longrightarrow> insert A H \<turnstile> B"
    by (metis Assume MP_same thin1)
 
-subsection{*Cut rules*}
+subsection\<open>Cut rules\<close>
 
 lemma cut:  "H \<turnstile> A \<Longrightarrow> insert A H' \<turnstile> B \<Longrightarrow> H \<union> H' \<turnstile> B"
   by (metis MP Un_commute Imp_I)
 
 lemma cut_same: "H \<turnstile> A \<Longrightarrow> insert A H \<turnstile> B \<Longrightarrow> H \<turnstile> B"
   by (metis Un_absorb cut)
 
 lemma cut_thin: "HA \<turnstile> A \<Longrightarrow> insert A HB \<turnstile> B \<Longrightarrow> HA \<union> HB \<subseteq> H \<Longrightarrow> H \<turnstile> B"
   by (metis thin cut)
 
 lemma cut0: "{} \<turnstile> A \<Longrightarrow> insert A H \<turnstile> B \<Longrightarrow> H \<turnstile> B"
   by (metis cut_same thin0)
 
 lemma cut1: "{A} \<turnstile> B \<Longrightarrow> H \<turnstile> A \<Longrightarrow> H \<turnstile> B"
   by (metis cut sup_bot_right)
 
 lemma rcut1: "{A} \<turnstile> B \<Longrightarrow> insert B H \<turnstile> C \<Longrightarrow> insert A H \<turnstile> C"
   by (metis Assume cut1 cut_same rotate2 thin1)
 
 lemma cut2: "\<lbrakk>{A,B} \<turnstile> C; H \<turnstile> A; H \<turnstile> B\<rbrakk> \<Longrightarrow> H \<turnstile> C"
   by (metis Un_empty_right Un_insert_right cut cut_same)
 
 lemma rcut2: "{A,B} \<turnstile> C \<Longrightarrow> insert C H \<turnstile> D \<Longrightarrow> H \<turnstile> B \<Longrightarrow> insert A H \<turnstile> D"
   by (metis Assume cut2 cut_same insert_commute thin1)
 
 lemma cut3: "\<lbrakk>{A,B,C} \<turnstile> D; H \<turnstile> A; H \<turnstile> B; H \<turnstile> C\<rbrakk> \<Longrightarrow> H \<turnstile> D"
   by (metis MP_same cut2 Imp_I)
 
 lemma cut4: "\<lbrakk>{A,B,C,D} \<turnstile> E; H \<turnstile> A; H \<turnstile> B; H \<turnstile> C; H \<turnstile> D\<rbrakk> \<Longrightarrow> H \<turnstile> E"
   by (metis MP_same cut3 [of B C D] Imp_I)
 
 
-section{*Miscellaneous logical rules*}
+section\<open>Miscellaneous logical rules\<close>
 
 lemma Disj_I1: "H \<turnstile> A \<Longrightarrow> H \<turnstile> A OR B"
   by (metis Bool MP_same boolean_axioms.DisjI1)
 
 lemma Disj_I2: "H \<turnstile> B \<Longrightarrow> H \<turnstile> A OR B"
   by (metis Disj_commute Disj_I1)
 
 lemma Peirce: "H \<turnstile> (Neg A) IMP A \<Longrightarrow> H \<turnstile> A"
   using DisjConj [of "Neg A" A A]  DisjCont [of A]
   by (metis Bool MP_same boolean_axioms.Ident)
 
 lemma Contra: "insert (Neg A) H \<turnstile> A \<Longrightarrow> H \<turnstile> A"
   by (metis Peirce Imp_I)
 
 lemma Imp_Neg_I: "H \<turnstile> A IMP B \<Longrightarrow> H \<turnstile> A IMP (Neg B) \<Longrightarrow> H \<turnstile> Neg A"
   by (metis DisjConj [of B "Neg A" "Neg A"] DisjCont Bool Disj_commute MP_same)
 
 lemma NegNeg_I: "H \<turnstile> A \<Longrightarrow> H \<turnstile> Neg (Neg A)"
   using DisjConj [of "Neg (Neg A)" "Neg A" "Neg (Neg A)"]
   by (metis Bool Ident MP_same)
 
 lemma NegNeg_D: "H \<turnstile> Neg (Neg A) \<Longrightarrow> H \<turnstile> A"
   by (metis Disj_I1 Peirce)
 
 lemma Neg_D: "H \<turnstile> Neg A \<Longrightarrow> H \<turnstile> A \<Longrightarrow> H \<turnstile> B"
   by (metis Imp_Neg_I Imp_triv_I NegNeg_D)
 
 lemma Disj_Neg_1: "H \<turnstile> A OR B \<Longrightarrow> H \<turnstile> Neg B \<Longrightarrow> H \<turnstile> A"
   by (metis Disj_I1 Disj_Semicong_1 Disj_commute Peirce)
 
 lemma Disj_Neg_2: "H \<turnstile> A OR B \<Longrightarrow> H \<turnstile> Neg A \<Longrightarrow> H \<turnstile> B"
   by (metis Disj_Neg_1 Disj_commute)
 
 lemma Neg_Disj_I: "H \<turnstile> Neg A \<Longrightarrow> H \<turnstile> Neg B \<Longrightarrow> H \<turnstile> Neg (A OR B)"
 by (metis Bool Disj_Neg_1 MP_same boolean_axioms.Ident DisjAssoc)
 
 lemma Conj_I [intro!]: "H \<turnstile> A \<Longrightarrow> H \<turnstile> B \<Longrightarrow> H \<turnstile> A AND B"
   by (metis Conj_def NegNeg_I Neg_Disj_I)
 
 lemma Conj_E1: "H \<turnstile> A AND B \<Longrightarrow> H \<turnstile> A"
   by (metis Conj_def Bool Disj_Neg_1 NegNeg_D boolean_axioms.DisjI1)
 
 lemma Conj_E2: "H \<turnstile> A AND B \<Longrightarrow> H \<turnstile> B"
   by (metis Conj_def Bool Disj_I2 Disj_Neg_2 MP_same DisjAssoc Ident)
 
 lemma Conj_commute: "H \<turnstile> B AND A \<Longrightarrow> H \<turnstile> A AND B"
   by (metis Conj_E1 Conj_E2 Conj_I)
 
 lemma Conj_E: assumes "insert A (insert B H) \<turnstile> C" shows "insert (A AND B) H \<turnstile> C"
 apply (rule cut_same [where A=A], metis Conj_E1 Hyp insertI1)
 by (metis (full_types) AssumeH(2) Conj_E2 assms cut_same [where A=B] insert_commute thin2)
 
 lemmas Conj_EH = Conj_E Conj_E [THEN rotate2] Conj_E [THEN rotate3] Conj_E [THEN rotate4] Conj_E [THEN rotate5]
                  Conj_E [THEN rotate6] Conj_E [THEN rotate7] Conj_E [THEN rotate8] Conj_E [THEN rotate9] Conj_E [THEN rotate10]
 declare Conj_EH [intro!]
 
 lemma Neg_I0: assumes "(\<And>B. atom i \<sharp> B \<Longrightarrow> insert A H \<turnstile> B)" shows "H \<turnstile> Neg A"
   by (rule Imp_Neg_I [where B = "Zero IN Zero"]) (auto simp: assms)
 
 lemma Neg_mono: "insert A H \<turnstile> B \<Longrightarrow> insert (Neg B) H \<turnstile> Neg A"
   by (rule Neg_I0) (metis Hyp Neg_D insert_commute insertI1 thin1)
 
 lemma Conj_mono: "insert A H \<turnstile> B \<Longrightarrow> insert C H \<turnstile> D \<Longrightarrow> insert (A AND C) H \<turnstile> B AND D"
   by (metis Conj_E1 Conj_E2 Conj_I Hyp Un_absorb2 cut insertI1 subset_insertI)
 
 lemma Disj_mono:
   assumes "insert A H \<turnstile> B" "insert C H \<turnstile> D" shows "insert (A OR C) H \<turnstile> B OR D"
 proof -
   { fix A B C H
     have "insert (A OR C) H \<turnstile> (A IMP B) IMP C OR B"
       by (metis Bool Hyp MP_same boolean_axioms.DisjConj insertI1)
     hence "insert A H \<turnstile> B \<Longrightarrow> insert (A OR C) H \<turnstile> C OR B"
       by (metis MP_same Un_absorb Un_insert_right Imp_I thin_Un)
   }
   thus ?thesis
     by (metis cut_same assms thin2)
 qed
 
 lemma Disj_E:
   assumes A: "insert A H \<turnstile> C" and B: "insert B H \<turnstile> C" shows "insert (A OR B) H \<turnstile> C"
   by (metis A B Disj_mono NegNeg_I Peirce)
 
 lemmas Disj_EH = Disj_E Disj_E [THEN rotate2] Disj_E [THEN rotate3] Disj_E [THEN rotate4] Disj_E [THEN rotate5]
                  Disj_E [THEN rotate6] Disj_E [THEN rotate7] Disj_E [THEN rotate8] Disj_E [THEN rotate9] Disj_E [THEN rotate10]
 declare Disj_EH [intro!]
 
 lemma Contra': "insert A H \<turnstile> Neg A \<Longrightarrow> H \<turnstile> Neg A"
   by (metis Contra Neg_mono)
 
 lemma NegNeg_E [intro!]: "insert A H \<turnstile> B \<Longrightarrow> insert (Neg (Neg A)) H \<turnstile> B"
   by (metis NegNeg_D Neg_mono)
 
 declare NegNeg_E [THEN rotate2, intro!]
 declare NegNeg_E [THEN rotate3, intro!]
 declare NegNeg_E [THEN rotate4, intro!]
 declare NegNeg_E [THEN rotate5, intro!]
 declare NegNeg_E [THEN rotate6, intro!]
 declare NegNeg_E [THEN rotate7, intro!]
 declare NegNeg_E [THEN rotate8, intro!]
 
 lemma Imp_E:
   assumes A: "H \<turnstile> A" and B: "insert B H \<turnstile> C" shows "insert (A IMP B) H \<turnstile> C"
 proof -
   have "insert (A IMP B) H \<turnstile> B"
     by (metis Hyp A thin1 MP_same insertI1)
   thus ?thesis
     by (metis cut [where B=C] Un_insert_right sup_commute sup_idem B)
 qed
 
 lemma Imp_cut:
   assumes "insert C H \<turnstile> A IMP B" "{A} \<turnstile> C"
     shows "H \<turnstile> A IMP B"
   by (metis Contra Disj_I1 Neg_mono assms rcut1)
 
 lemma Iff_I [intro!]: "insert A H \<turnstile> B \<Longrightarrow> insert B H \<turnstile> A \<Longrightarrow> H \<turnstile> A IFF B"
   by (metis Iff_def Conj_I Imp_I)
 
 lemma Iff_MP_same: "H \<turnstile> A IFF B \<Longrightarrow> H \<turnstile> A \<Longrightarrow> H \<turnstile> B"
   by (metis Iff_def Conj_E1 MP_same)
 
 lemma Iff_MP2_same: "H \<turnstile> A IFF B \<Longrightarrow> H \<turnstile> B \<Longrightarrow> H \<turnstile> A"
   by (metis Iff_def Conj_E2 MP_same)
 
 lemma Iff_refl [intro!]: "H \<turnstile> A IFF A"
   by (metis Hyp Iff_I insertI1)
 
 lemma Iff_sym: "H \<turnstile> A IFF B \<Longrightarrow> H \<turnstile> B IFF A"
   by (metis Iff_def Conj_commute)
 
 lemma Iff_trans: "H \<turnstile> A IFF B \<Longrightarrow> H \<turnstile> B IFF C \<Longrightarrow> H \<turnstile> A IFF C"
   unfolding Iff_def
   by (metis Conj_E1 Conj_E2 Conj_I Disj_Semicong_1 Disj_commute)
 
 lemma Iff_E:
   "insert A (insert B H) \<turnstile> C \<Longrightarrow> insert (Neg A) (insert (Neg B) H) \<turnstile> C \<Longrightarrow> insert (A IFF B) H \<turnstile> C"
   apply (auto simp: Iff_def insert_commute)
   apply (metis Disj_I1 Hyp anti_deduction insertCI)
   apply (metis Assume Disj_I1 anti_deduction)
   done
 
 lemma Iff_E1:
   assumes A: "H \<turnstile> A" and B: "insert B H \<turnstile> C" shows "insert (A IFF B) H \<turnstile> C"
   by (metis Iff_def A B Conj_E Imp_E insert_commute thin1)
 
 lemma Iff_E2:
   assumes A: "H \<turnstile> A" and B: "insert B H \<turnstile> C" shows "insert (B IFF A) H \<turnstile> C"
   by (metis Iff_def A B Bool Conj_E2 Conj_mono Imp_E boolean_axioms.Ident)
 
 lemma Iff_MP_left: "H \<turnstile> A IFF B \<Longrightarrow> insert A H \<turnstile> C \<Longrightarrow> insert B H \<turnstile> C"
   by (metis Hyp Iff_E2 cut_same insertI1 insert_commute thin1)
 
 lemma Iff_MP_left': "H \<turnstile> A IFF B \<Longrightarrow> insert B H \<turnstile> C \<Longrightarrow> insert A H \<turnstile> C"
   by (metis Iff_MP_left Iff_sym)
 
 lemma Swap: "insert (Neg B) H \<turnstile> A \<Longrightarrow> insert (Neg A) H \<turnstile> B"
   by (metis NegNeg_D Neg_mono)
 
 lemma Cases: "insert A H \<turnstile> B \<Longrightarrow> insert (Neg A) H \<turnstile> B \<Longrightarrow> H \<turnstile> B"
   by (metis Contra Neg_D Neg_mono)
 
 lemma Neg_Conj_E: "H \<turnstile> B \<Longrightarrow> insert (Neg A) H \<turnstile> C \<Longrightarrow> insert (Neg (A AND B)) H \<turnstile> C"
   by (metis Conj_I Swap thin1)
 
 lemma Disj_CI: "insert (Neg B) H \<turnstile> A \<Longrightarrow> H \<turnstile> A OR B"
   by (metis Contra Disj_I1 Disj_I2 Swap)
 
 lemma Disj_3I: "insert (Neg A) (insert (Neg C) H) \<turnstile> B \<Longrightarrow> H \<turnstile> A OR B OR C"
   by (metis Disj_CI Disj_commute insert_commute)
 
 lemma Contrapos1: "H \<turnstile> A IMP B \<Longrightarrow> H \<turnstile> Neg B IMP Neg A"
   by (metis Bool MP_same boolean_axioms.DisjConj boolean_axioms.Ident)
 
 lemma Contrapos2: "H \<turnstile> (Neg B) IMP (Neg A) \<Longrightarrow> H \<turnstile> A IMP B"
   by (metis Bool MP_same boolean_axioms.DisjConj boolean_axioms.Ident)
 
 lemma ContraAssumeN [intro]: "B \<in> H \<Longrightarrow> insert (Neg B) H \<turnstile> A"
   by (metis Hyp Swap thin1)
 
 lemma ContraAssume: "Neg B \<in> H \<Longrightarrow> insert B H \<turnstile> A"
   by (metis Disj_I1 Hyp anti_deduction)
 
 lemma ContraProve: "H \<turnstile> B \<Longrightarrow> insert (Neg B) H \<turnstile> A"
   by (metis Swap thin1)
 
 lemma Disj_IE1: "insert B H \<turnstile> C \<Longrightarrow> insert (A OR B) H \<turnstile> A OR C"
   by (metis Assume Disj_mono)
 
 lemmas Disj_IE1H = Disj_IE1 Disj_IE1 [THEN rotate2] Disj_IE1 [THEN rotate3] Disj_IE1 [THEN rotate4] Disj_IE1 [THEN rotate5]
                  Disj_IE1 [THEN rotate6] Disj_IE1 [THEN rotate7] Disj_IE1 [THEN rotate8]
 declare Disj_IE1H [intro!]
 
-subsection{*Quantifier reasoning*}
+subsection\<open>Quantifier reasoning\<close>
 
 lemma Ex_I: "H \<turnstile> A(i::=x) \<Longrightarrow> H \<turnstile> Ex i A"
   by (metis MP_same Spec special_axioms.intros)
 
 lemma Ex_E:
   assumes "insert A H \<turnstile> B" "atom i \<sharp> B" "\<forall>C \<in> H. atom i \<sharp> C"
   shows "insert (Ex i A) H \<turnstile> B"
   by (metis Exists Imp_I anti_deduction assms)
 
 lemma Ex_E_with_renaming:
   assumes "insert ((i \<leftrightarrow> i') \<bullet> A) H \<turnstile> B" "atom i' \<sharp> (A,i,B)" "\<forall>C \<in> H. atom i' \<sharp> C"
   shows "insert (Ex i A) H \<turnstile> B"
 proof -
   have "Ex i A = Ex i' ((i \<leftrightarrow> i') \<bullet> A)" using assms
     apply (auto simp: Abs1_eq_iff fresh_Pair)
     apply (metis flip_at_simps(2) fresh_at_base_permute_iff)+
     done
   thus ?thesis
     by (metis Ex_E assms fresh_Pair)
 qed
 
 lemmas Ex_EH = Ex_E Ex_E [THEN rotate2] Ex_E [THEN rotate3] Ex_E [THEN rotate4] Ex_E [THEN rotate5]
                  Ex_E [THEN rotate6] Ex_E [THEN rotate7] Ex_E [THEN rotate8] Ex_E [THEN rotate9] Ex_E [THEN rotate10]
 declare Ex_EH [intro!]
 
 lemma Ex_mono: "insert A H \<turnstile> B \<Longrightarrow> \<forall>C \<in> H. atom i \<sharp> C \<Longrightarrow> insert (Ex i A) H \<turnstile> (Ex i B)"
   by (auto simp add: intro: Ex_I [where x="Var i"])
 
 lemma All_I [intro!]: "H \<turnstile> A \<Longrightarrow> \<forall>C \<in> H. atom i \<sharp> C \<Longrightarrow> H \<turnstile> All i A"
   by (auto intro: ContraProve Neg_I0)
 
 lemma All_D: "H \<turnstile> All i A \<Longrightarrow> H \<turnstile> A(i::=x)"
   by (metis Assume Ex_I NegNeg_D Neg_mono SyntaxN.Neg cut_same)
 
 lemma All_E: "insert (A(i::=x)) H \<turnstile> B \<Longrightarrow> insert (All i A) H \<turnstile> B"
   by (metis Ex_I NegNeg_D Neg_mono SyntaxN.Neg)
 
 lemma All_E': "H \<turnstile> All i A \<Longrightarrow> insert (A(i::=x)) H \<turnstile> B \<Longrightarrow> H \<turnstile> B"
   by (metis All_D cut_same)
 
 lemma All2_E: "\<lbrakk>atom i \<sharp> t; H \<turnstile> x IN t; insert (A(i::=x)) H \<turnstile> B\<rbrakk> \<Longrightarrow> insert (All2 i t A) H \<turnstile> B"
   apply (rule All_E [where x=x], auto)
   by (metis Swap thin1)
 
 lemma All2_E': "\<lbrakk>H \<turnstile> All2 i t A; H \<turnstile> x IN t; insert (A(i::=x)) H \<turnstile> B; atom i \<sharp> t\<rbrakk> \<Longrightarrow> H \<turnstile> B"
   by (metis All2_E cut_same)
 
 
-subsection{*Congruence rules*}
+subsection\<open>Congruence rules\<close>
 
 lemma Neg_cong: "H \<turnstile> A IFF A' \<Longrightarrow> H \<turnstile> Neg A IFF Neg A'"
   by (metis Iff_def Conj_E1 Conj_E2 Conj_I Contrapos1)
 
 lemma Disj_cong: "H \<turnstile> A IFF A' \<Longrightarrow> H \<turnstile> B IFF B' \<Longrightarrow> H \<turnstile> A OR B IFF A' OR B'"
   by (metis Conj_E1 Conj_E2 Disj_mono Iff_I Iff_def anti_deduction)
 
 lemma Conj_cong: "H \<turnstile> A IFF A' \<Longrightarrow> H \<turnstile> B IFF B' \<Longrightarrow> H \<turnstile> A AND B IFF A' AND B'"
   by (metis Conj_def Disj_cong Neg_cong)
 
 lemma Imp_cong: "H \<turnstile> A IFF A' \<Longrightarrow> H \<turnstile> B IFF B' \<Longrightarrow> H \<turnstile> (A IMP B) IFF (A' IMP B')"
   by (metis Disj_cong Neg_cong)
 
 lemma Iff_cong: "H \<turnstile> A IFF A' \<Longrightarrow> H \<turnstile> B IFF B' \<Longrightarrow> H \<turnstile> (A IFF B) IFF (A' IFF B')"
   by (metis Iff_def Conj_cong Imp_cong)
 
 lemma Ex_cong: "H \<turnstile> A IFF A' \<Longrightarrow> \<forall>C \<in> H. atom i \<sharp> C \<Longrightarrow> H \<turnstile> (Ex i A) IFF (Ex i A')"
   apply (rule Iff_I)
    apply (metis Ex_mono Hyp Iff_MP_same Un_absorb Un_insert_right insertI1 thin_Un)
   apply (metis Ex_mono Hyp Iff_MP2_same Un_absorb Un_insert_right insertI1 thin_Un)
   done
 
 lemma All_cong: "H \<turnstile> A IFF A' \<Longrightarrow> \<forall>C \<in> H. atom i \<sharp> C \<Longrightarrow> H \<turnstile> (All i A) IFF (All i A')"
   by (metis Ex_cong Neg_cong)
 
 lemma Subst: "H \<turnstile> A \<Longrightarrow> \<forall>B \<in> H. atom i \<sharp> B \<Longrightarrow> H \<turnstile> A (i::=x)"
   by (metis All_D All_I)
 
 
-section{*Equality reasoning*}
+section\<open>Equality reasoning\<close>
 
-subsection{*The congruence property for @{term Eq}, and other basic properties of equality*}
+subsection\<open>The congruence property for @{term Eq}, and other basic properties of equality\<close>
 
 lemma Eq_cong1: "{} \<turnstile> (t EQ t' AND u EQ u') IMP (t EQ u IMP t' EQ u')"
 proof -
   obtain v2::name and v3::name and v4::name
     where v2: "atom v2 \<sharp> (t,X1,X3,X4)"
       and v3: "atom v3 \<sharp> (t,t',X1,v2,X4)"
       and v4: "atom v4 \<sharp> (t,t',u,X1,v2,v3)"
     by (metis obtain_fresh)
   have "{} \<turnstile> (Var X1 EQ Var X2 AND Var X3 EQ Var X4) IMP (Var X1 EQ Var X3 IMP Var X2 EQ Var X4)"
     by (rule Eq) (simp add: eq_cong_ax_def equality_axioms_def)
   hence "{} \<turnstile> (Var X1 EQ Var X2 AND Var X3 EQ Var X4) IMP (Var X1 EQ Var X3 IMP Var X2 EQ Var X4)"
     by (drule_tac i=X1 and x="Var X1" in Subst) simp_all
   hence "{} \<turnstile> (Var X1 EQ Var v2 AND Var X3 EQ Var X4) IMP (Var X1 EQ Var X3 IMP Var v2 EQ Var X4)"
     by (drule_tac i=X2 and x="Var v2" in Subst) simp_all
   hence "{} \<turnstile> (Var X1 EQ Var v2 AND Var v3 EQ Var X4) IMP (Var X1 EQ Var v3 IMP Var v2 EQ Var X4)"
     using v2
     by (drule_tac i=X3 and x="Var v3" in Subst) simp_all
   hence "{} \<turnstile> (Var X1 EQ Var v2 AND Var v3 EQ Var v4) IMP (Var X1 EQ Var v3 IMP Var v2 EQ Var v4)"
     using v2 v3
     by (drule_tac i=X4 and x="Var v4" in Subst) simp_all
   hence "{} \<turnstile> (t EQ Var v2 AND Var v3 EQ Var v4) IMP (t EQ Var v3 IMP Var v2 EQ Var v4)"
     using v2 v3 v4
     by (drule_tac i=X1 and x=t in Subst) simp_all
   hence "{} \<turnstile> (t EQ t' AND Var v3 EQ Var v4) IMP (t EQ Var v3 IMP t' EQ Var v4)"
     using v2 v3 v4
     by (drule_tac i=v2 and x="t'" in Subst) simp_all
   hence "{} \<turnstile> (t EQ t' AND u EQ Var v4) IMP (t EQ u IMP t' EQ Var v4)"
     using v3 v4
     by (drule_tac i=v3 and x=u in Subst) simp_all
   thus ?thesis
     using v4
     by (drule_tac i=v4 and x="u'" in Subst) simp_all
 qed
 
 lemma Refl [iff]: "H \<turnstile> t EQ t"
 proof -
   have "{} \<turnstile> Var X1 EQ Var X1"
     by (rule Eq) (simp add: equality_axioms_def refl_ax_def)
   hence "{} \<turnstile> t EQ t"
     by (drule_tac i=X1 and x=t in Subst) simp_all
   thus ?thesis
     by (metis empty_subsetI thin)
 qed
 
-text{*Apparently necessary in order to prove the congruence property.*}
+text\<open>Apparently necessary in order to prove the congruence property.\<close>
 lemma Sym: assumes "H \<turnstile> t EQ u" shows "H \<turnstile> u EQ t"
 proof -
   have "{} \<turnstile>  (t EQ u AND t EQ t) IMP (t EQ t IMP u EQ t)"
     by (rule Eq_cong1)
    moreover have "{t EQ u} \<turnstile> t EQ u AND t EQ t"
     by (metis Assume Conj_I Refl)
   ultimately have "{t EQ u} \<turnstile> u EQ t"
     by (metis MP_same MP Refl sup_bot_left)
   thus "H \<turnstile> u EQ t" by (metis assms cut1)
 qed
 
 lemma Sym_L: "insert (t EQ u) H \<turnstile> A \<Longrightarrow> insert (u EQ t) H \<turnstile> A"
   by (metis Assume Sym Un_empty_left Un_insert_left cut)
 
 lemma Trans: assumes "H \<turnstile> x EQ y" "H \<turnstile> y EQ z" shows "H \<turnstile> x EQ z"
 proof -
   have "\<And>H. H \<turnstile> (x EQ x AND y EQ z) IMP (x EQ y IMP x EQ z)"
     by (metis Eq_cong1 bot_least thin)
   moreover have "{x EQ y, y EQ z} \<turnstile> x EQ x AND y EQ z"
     by (metis Assume Conj_I Refl thin1)
   ultimately have "{x EQ y, y EQ z} \<turnstile> x EQ z"
     by (metis Hyp MP_same insertI1)
   thus ?thesis
     by (metis assms cut2)
 qed
 
 lemma Eq_cong:
   assumes "H \<turnstile> t EQ t'" "H \<turnstile> u EQ u'" shows "H \<turnstile> t EQ u IFF t' EQ u'"
 proof -
   { fix t t' u u'
     assume  "H \<turnstile> t EQ t'" "H \<turnstile> u EQ u'"
     moreover have "{t EQ t', u EQ u'} \<turnstile> t EQ u IMP t' EQ u'" using Eq_cong1
       by (metis Assume Conj_I MP_null insert_commute)
     ultimately have "H \<turnstile> t EQ u IMP t' EQ u'"
       by (metis cut2)
   }
   thus ?thesis
     by (metis Iff_def Conj_I assms Sym)
 qed
 
 lemma Eq_Trans_E: "H \<turnstile> x EQ u \<Longrightarrow> insert (t EQ u) H \<turnstile> A \<Longrightarrow> insert (x EQ t) H \<turnstile> A"
   by (metis Assume Sym_L Trans cut_same thin1 thin2)
 
-subsection{*The congruence property for @{term Mem}*}
+subsection\<open>The congruence property for @{term Mem}\<close>
 
 lemma Mem_cong1: "{} \<turnstile> (t EQ t' AND u EQ u') IMP (t IN u IMP t' IN u')"
 proof -
   obtain v2::name and v3::name and v4::name
     where v2: "atom v2 \<sharp> (t,X1,X3,X4)"
       and v3: "atom v3 \<sharp> (t,t',X1,v2,X4)"
       and v4: "atom v4 \<sharp> (t,t',u,X1,v2,v3)"
     by (metis obtain_fresh)
   have "{} \<turnstile> (Var X1 EQ Var X2 AND Var X3 EQ Var X4) IMP (Var X1 IN Var X3 IMP Var X2 IN Var X4)"
     by (metis mem_cong_ax_def equality_axioms_def insert_iff Eq)
   hence "{} \<turnstile> (Var X1 EQ Var v2 AND Var X3 EQ Var X4) IMP (Var X1 IN Var X3 IMP Var v2 IN Var X4)"
     by (drule_tac i=X2 and x="Var v2" in Subst) simp_all
   hence "{} \<turnstile> (Var X1 EQ Var v2 AND Var v3 EQ Var X4) IMP (Var X1 IN Var v3 IMP Var v2 IN Var X4)"
     using v2
     by (drule_tac i=X3 and x="Var v3" in Subst) simp_all
   hence "{} \<turnstile> (Var X1 EQ Var v2 AND Var v3 EQ Var v4) IMP (Var X1 IN Var v3 IMP Var v2 IN Var v4)"
     using v2 v3
     by (drule_tac i=X4 and x="Var v4" in Subst) simp_all
   hence "{} \<turnstile> (t EQ Var v2 AND Var v3 EQ Var v4) IMP (t IN Var v3 IMP Var v2 IN Var v4)"
     using v2 v3 v4
     by (drule_tac i=X1 and x=t in Subst) simp_all
   hence "{} \<turnstile> (t EQ t' AND Var v3 EQ Var v4) IMP (t IN Var v3 IMP t' IN Var v4)"
     using v2 v3 v4
     by (drule_tac i=v2 and x=t' in Subst) simp_all
   hence "{} \<turnstile> (t EQ t' AND u EQ Var v4) IMP (t IN u IMP t' IN Var v4)"
     using v3 v4
     by (drule_tac i=v3 and x=u in Subst) simp_all
   thus ?thesis
     using v4
     by (drule_tac i=v4 and x=u' in Subst) simp_all
 qed
 
 lemma Mem_cong:
   assumes "H \<turnstile> t EQ t'" "H \<turnstile> u EQ u'" shows "H \<turnstile> t IN u IFF t' IN u'"
 proof -
   { fix t t' u u'
     have cong: "{t EQ t', u EQ u'} \<turnstile> t IN u IMP t' IN u'"
       by (metis AssumeH(2) Conj_I MP_null Mem_cong1 insert_commute)
   }
   thus ?thesis
     by (metis Iff_def Conj_I cut2 assms Sym)
 qed
 
-subsection{*The congruence properties for @{term Eats} and @{term HPair}*}
+subsection\<open>The congruence properties for @{term Eats} and @{term HPair}\<close>
 
 lemma Eats_cong1: "{} \<turnstile> (t EQ t' AND u EQ u') IMP (Eats t u EQ Eats t' u')"
 proof -
   obtain v2::name and v3::name and v4::name
     where v2: "atom v2 \<sharp> (t,X1,X3,X4)"
       and v3: "atom v3 \<sharp> (t,t',X1,v2,X4)"
       and v4: "atom v4 \<sharp> (t,t',u,X1,v2,v3)"
     by (metis obtain_fresh)
   have "{} \<turnstile> (Var X1 EQ Var X2 AND Var X3 EQ Var X4) IMP (Eats (Var X1) (Var X3) EQ Eats (Var X2) (Var X4))"
     by (metis eats_cong_ax_def equality_axioms_def insert_iff Eq)
   hence "{} \<turnstile> (Var X1 EQ Var v2 AND Var X3 EQ Var X4) IMP (Eats (Var X1) (Var X3) EQ Eats (Var v2) (Var X4))"
     by (drule_tac i=X2 and x="Var v2" in Subst) simp_all
   hence "{} \<turnstile> (Var X1 EQ Var v2 AND Var v3 EQ Var X4) IMP (Eats (Var X1) (Var v3) EQ Eats (Var v2) (Var X4))"
     using v2
     by (drule_tac i=X3 and x="Var v3" in Subst) simp_all
   hence "{} \<turnstile> (Var X1 EQ Var v2 AND Var v3 EQ Var v4) IMP (Eats (Var X1) (Var v3) EQ Eats (Var v2) (Var v4))"
     using v2 v3
     by (drule_tac i=X4 and x="Var v4" in Subst) simp_all
   hence "{} \<turnstile> (t EQ Var v2 AND Var v3 EQ Var v4) IMP (Eats t (Var v3) EQ Eats (Var v2) (Var v4))"
     using v2 v3 v4
     by (drule_tac i=X1 and x=t in Subst) simp_all
   hence "{} \<turnstile> (t EQ t' AND Var v3 EQ Var v4) IMP (Eats t (Var v3) EQ Eats t' (Var v4))"
     using v2 v3 v4
     by (drule_tac i=v2 and x=t' in Subst) simp_all
   hence "{} \<turnstile> (t EQ t' AND u EQ Var v4) IMP (Eats t u EQ Eats t' (Var v4))"
     using v3 v4
     by (drule_tac i=v3 and x=u in Subst) simp_all
   thus ?thesis
     using v4
     by (drule_tac i=v4 and x=u' in Subst) simp_all
 qed
 
 lemma Eats_cong: "\<lbrakk>H \<turnstile> t EQ t'; H \<turnstile> u EQ u'\<rbrakk> \<Longrightarrow> H \<turnstile> Eats t u EQ Eats t' u'"
   by (metis Conj_I anti_deduction Eats_cong1 cut1)
 
 lemma HPair_cong: "\<lbrakk>H \<turnstile> t EQ t'; H \<turnstile> u EQ u'\<rbrakk> \<Longrightarrow> H \<turnstile> HPair t u EQ HPair t' u'"
   by (metis HPair_def Eats_cong Refl)
 
 lemma SUCC_cong: "H \<turnstile> t EQ t' \<Longrightarrow> H \<turnstile> SUCC t EQ SUCC t'"
   by (metis Eats_cong SUCC_def)
 
-subsection{*Substitution for Equalities*}
+subsection\<open>Substitution for Equalities\<close>
 
 lemma Eq_subst_tm_Iff: "{t EQ u} \<turnstile> subst i t tm EQ subst i u tm"
   by (induct tm rule: tm.induct) (auto simp: Eats_cong)
 
 lemma Eq_subst_fm_Iff: "insert (t EQ u) H \<turnstile> A(i::=t) IFF A(i::=u)"
 proof -
   have "{ t EQ u } \<turnstile> A(i::=t) IFF A(i::=u)"
     by (nominal_induct A avoiding: i t u rule: fm.strong_induct)
        (auto simp: Disj_cong Neg_cong Ex_cong Mem_cong Eq_cong Eq_subst_tm_Iff)
   thus ?thesis
     by (metis Assume cut1)
 qed
 
 lemma Var_Eq_subst_Iff: "insert (Var i EQ t) H \<turnstile> A(i::=t) IFF A"
   by (metis Eq_subst_fm_Iff Iff_sym subst_fm_id)
 
 lemma Var_Eq_imp_subst_Iff: "H \<turnstile> Var i EQ t \<Longrightarrow> H \<turnstile> A(i::=t) IFF A"
   by (metis Var_Eq_subst_Iff cut_same)
 
-subsection{*Congruence Rules for Predicates*}
+subsection\<open>Congruence Rules for Predicates\<close>
 
 lemma P1_cong:
   fixes tms :: "tm list"
   assumes "\<And>i t x. atom i \<sharp> tms \<Longrightarrow> (P t)(i::=x) = P (subst i x t)" and "H \<turnstile> x EQ x'"
   shows "H \<turnstile> P x IFF P x'"
 proof -
   obtain i::name where i: "atom i \<sharp> tms"
     by (metis obtain_fresh)
   have "insert (x EQ x') H  \<turnstile> (P (Var i))(i::=x) IFF (P(Var i))(i::=x')"
     by (rule Eq_subst_fm_Iff)
   thus ?thesis using assms i
     by (metis cut_same subst.simps(2))
 qed
 
 lemma P2_cong:
   fixes tms :: "tm list"
   assumes sub: "\<And>i t u x. atom i \<sharp> tms \<Longrightarrow> (P t u)(i::=x) = P (subst i x t) (subst i x u)"
       and eq:  "H \<turnstile> x EQ x'" "H \<turnstile> y EQ y'"
   shows "H \<turnstile> P x y IFF P x' y'"
 proof -
   have yy': "{ y EQ y' } \<turnstile> P x' y IFF P x' y'"
     by (rule P1_cong [where tms="[y,x']@tms"]) (auto simp: fresh_Cons sub)
   have "{ x EQ x' } \<turnstile> P x y IFF P x' y"
     by (rule P1_cong [where tms="[y,x']@tms"]) (auto simp: fresh_Cons sub)
   hence "{x EQ x', y EQ y'} \<turnstile> P x y IFF P x' y'"
     by (metis Assume Iff_trans cut1 rotate2 yy')
   thus ?thesis
     by (metis cut2 eq)
  qed
 
 lemma P3_cong:
   fixes tms :: "tm list"
   assumes sub: "\<And>i t u v x. atom i \<sharp> tms \<Longrightarrow>
                    (P t u v)(i::=x) = P (subst i x t) (subst i x u) (subst i x v)"
       and eq:  "H \<turnstile> x EQ x'" "H \<turnstile> y EQ y'" "H \<turnstile> z EQ z'"
   shows "H \<turnstile> P x y z IFF P x' y' z'"
 proof -
   obtain i::name where i: "atom i \<sharp> (z,z',y,y',x,x')"
     by (metis obtain_fresh)
   have tl: "{ y EQ y', z EQ z' } \<turnstile> P x' y z IFF P x' y' z'"
     by (rule P2_cong [where tms="[z,z',y,y',x,x']@tms"]) (auto simp: fresh_Cons sub)
   have hd: "{ x EQ x' } \<turnstile> P x y z IFF P x' y z"
     by (rule P1_cong [where tms="[z,y,x']@tms"]) (auto simp: fresh_Cons sub)
   have "{x EQ x', y EQ y', z EQ z'} \<turnstile> P x y z IFF P x' y' z'"
     by (metis Assume thin1 hd [THEN cut1] tl Iff_trans)
   thus ?thesis
     by (rule cut3) (rule eq)+
 qed
 
 lemma P4_cong:
   fixes tms :: "tm list"
   assumes sub: "\<And>i t1 t2 t3 t4 x. atom i \<sharp> tms \<Longrightarrow>
                  (P t1 t2 t3 t4)(i::=x) = P (subst i x t1) (subst i x t2) (subst i x t3) (subst i x t4)"
       and eq:  "H \<turnstile> x1 EQ x1'" "H \<turnstile> x2 EQ x2'" "H \<turnstile> x3 EQ x3'" "H \<turnstile> x4 EQ x4'"
   shows "H \<turnstile> P x1 x2 x3 x4 IFF P x1' x2' x3' x4'"
 proof -
   obtain i::name where i: "atom i \<sharp> (x4,x4',x3,x3',x2,x2',x1,x1')"
     by (metis obtain_fresh)
   have tl: "{ x2 EQ x2', x3 EQ x3', x4 EQ x4' } \<turnstile> P x1' x2 x3 x4 IFF P x1' x2' x3' x4'"
     by (rule P3_cong [where tms="[x4,x4',x3,x3',x2,x2',x1,x1']@tms"]) (auto simp: fresh_Cons sub)
   have hd: "{ x1 EQ x1' } \<turnstile> P x1 x2 x3 x4 IFF P x1' x2 x3 x4"
     by (auto simp: fresh_Cons sub intro!: P1_cong [where tms="[x4,x3,x2,x1']@tms"])
   have "{x1 EQ x1', x2 EQ x2', x3 EQ x3', x4 EQ x4'} \<turnstile> P x1 x2 x3 x4 IFF P x1' x2' x3' x4'"
     by (metis Assume thin1 hd [THEN cut1] tl Iff_trans)
   thus ?thesis
     by (rule cut4) (rule eq)+
 qed
 
 
-section{*Zero and Falsity*}
+section\<open>Zero and Falsity\<close>
 
 lemma Mem_Zero_iff:
   assumes "atom i \<sharp> t" shows "H \<turnstile> (t EQ Zero) IFF (All i (Neg ((Var i) IN t)))"
 proof -
   obtain i'::name where i': "atom i' \<sharp> (t, X0, X1, i)"
     by (rule obtain_fresh)
   have "{} \<turnstile> ((Var X0) EQ Zero) IFF (All X1 (Neg ((Var X1) IN (Var X0))))"
     by (simp add: HF HF_axioms_def HF1_def)
   then have "{} \<turnstile> (((Var X0) EQ Zero) IFF (All X1 (Neg ((Var X1) IN (Var X0)))))(X0 ::= t)"
      by (rule Subst) simp
   hence "{} \<turnstile> (t EQ Zero) IFF (All i' (Neg ((Var i') IN t)))" using i'
     by simp
   also have "... = (FRESH i'. (t EQ Zero) IFF (All i' (Neg ((Var i') IN t))))"
     using i' by simp
   also have "... = (t EQ Zero) IFF (All i (Neg ((Var i) IN t)))"
     using assms by simp
   finally show ?thesis
     by (metis empty_subsetI thin)
 qed
 
 lemma Mem_Zero_E [intro!]: "insert (x IN Zero) H \<turnstile> A"
 proof -
   obtain i::name where "atom i \<sharp> Zero"
     by (rule obtain_fresh)
   hence "{} \<turnstile> All i (Neg ((Var i) IN Zero))"
     by (metis Mem_Zero_iff Iff_MP_same Refl)
   hence "{} \<turnstile> Neg (x IN Zero)"
     by (drule_tac x=x in All_D) simp
   thus ?thesis
     by (metis Contrapos2 Hyp Imp_triv_I MP_same empty_subsetI insertI1 thin)
 qed
 
 declare Mem_Zero_E [THEN rotate2, intro!]
 declare Mem_Zero_E [THEN rotate3, intro!]
 declare Mem_Zero_E [THEN rotate4, intro!]
 declare Mem_Zero_E [THEN rotate5, intro!]
 declare Mem_Zero_E [THEN rotate6, intro!]
 declare Mem_Zero_E [THEN rotate7, intro!]
 declare Mem_Zero_E [THEN rotate8, intro!]
 
 
-subsection{*The Formula @{term Fls}*}
+subsection\<open>The Formula @{term Fls}\<close>
 
 definition Fls  where "Fls \<equiv> Zero IN Zero"
 
 lemma Fls_eqvt [eqvt]: "(p \<bullet> Fls) = Fls"
   by (simp add: Fls_def)
 
 lemma Fls_fresh [simp]: "a \<sharp> Fls"
   by (simp add: Fls_def)
 
 lemma Neg_I [intro!]: "insert A H \<turnstile> Fls \<Longrightarrow> H \<turnstile> Neg A"
   unfolding Fls_def
   by (rule Neg_I0) (metis Mem_Zero_E cut_same)
 
 lemma Neg_E [intro!]: "H \<turnstile> A \<Longrightarrow> insert (Neg A) H \<turnstile> Fls"
   by (rule ContraProve)
 
 declare Neg_E [THEN rotate2, intro!]
 declare Neg_E [THEN rotate3, intro!]
 declare Neg_E [THEN rotate4, intro!]
 declare Neg_E [THEN rotate5, intro!]
 declare Neg_E [THEN rotate6, intro!]
 declare Neg_E [THEN rotate7, intro!]
 declare Neg_E [THEN rotate8, intro!]
 
-text{*We need these because Neg (A IMP B) doesn't have to be syntactically a conjunction.*}
+text\<open>We need these because Neg (A IMP B) doesn't have to be syntactically a conjunction.\<close>
 lemma Neg_Imp_I [intro!]: "H \<turnstile> A \<Longrightarrow> insert B H \<turnstile> Fls \<Longrightarrow> H \<turnstile> Neg (A IMP B)"
   by (metis NegNeg_I Neg_Disj_I Neg_I)
 
 lemma Neg_Imp_E [intro!]: "insert (Neg B) (insert A H) \<turnstile> C \<Longrightarrow> insert (Neg (A IMP B)) H \<turnstile> C"
 apply (rule cut_same [where A=A])
  apply (metis Assume Disj_I1 NegNeg_D Neg_mono)
 apply (metis Swap Imp_I rotate2 thin1)
 done
 
 declare Neg_Imp_E [THEN rotate2, intro!]
 declare Neg_Imp_E [THEN rotate3, intro!]
 declare Neg_Imp_E [THEN rotate4, intro!]
 declare Neg_Imp_E [THEN rotate5, intro!]
 declare Neg_Imp_E [THEN rotate6, intro!]
 declare Neg_Imp_E [THEN rotate7, intro!]
 declare Neg_Imp_E [THEN rotate8, intro!]
 
 lemma Fls_E [intro!]: "insert Fls H \<turnstile> A"
   by (metis Mem_Zero_E Fls_def)
 
 declare Fls_E [THEN rotate2, intro!]
 declare Fls_E [THEN rotate3, intro!]
 declare Fls_E [THEN rotate4, intro!]
 declare Fls_E [THEN rotate5, intro!]
 declare Fls_E [THEN rotate6, intro!]
 declare Fls_E [THEN rotate7, intro!]
 declare Fls_E [THEN rotate8, intro!]
 
 lemma truth_provable: "H \<turnstile> (Neg Fls)"
   by (metis Fls_E Neg_I)
 
 lemma ExFalso: "H \<turnstile> Fls \<Longrightarrow> H \<turnstile> A"
   by (metis Neg_D truth_provable)
 
-subsection{*More properties of @{term Zero}*}
+subsection\<open>More properties of @{term Zero}\<close>
 
 lemma Eq_Zero_D:
   assumes "H \<turnstile> t EQ Zero" "H \<turnstile> u IN t" shows "H \<turnstile> A"
 proof -
   obtain i::name where  i: "atom i \<sharp> t"
     by (rule obtain_fresh)
   with assms have an: "H \<turnstile> (All i (Neg ((Var i) IN t)))"
     by (metis Iff_MP_same Mem_Zero_iff)
   have "H \<turnstile> Neg (u IN t)" using All_D [OF an, of u] i
     by simp
   thus ?thesis using assms
     by (metis Neg_D)
 qed
 
 lemma Eq_Zero_thm:
   assumes "atom i \<sharp> t" shows "{All i (Neg ((Var i) IN t))} \<turnstile> t EQ Zero"
 by (metis Assume Iff_MP2_same Mem_Zero_iff assms)
 
 lemma Eq_Zero_I:
   assumes insi: "insert ((Var i) IN t) H \<turnstile> Fls" and i1: "atom i \<sharp> t" and i2: "\<forall>B \<in> H. atom i \<sharp> B"
   shows "H \<turnstile> t EQ Zero"
 proof -
   have "H \<turnstile> All i (Neg ((Var i) IN t))"
     by (metis All_I Neg_I i2 insi)
   thus ?thesis
     by (metis cut_same  cut [OF Eq_Zero_thm [OF i1] Hyp] insertCI insert_is_Un)
 qed
 
-subsection{*Basic properties of @{term Eats}*}
+subsection\<open>Basic properties of @{term Eats}\<close>
 
 lemma Eq_Eats_iff:
   assumes "atom i \<sharp> (z,t,u)"
   shows "H \<turnstile> (z EQ Eats t u) IFF (All i (Var i IN z IFF Var i IN t OR Var i EQ u))"
 proof -
   obtain v1::name and v2::name and i'::name
     where v1: "atom v1 \<sharp> (z,X0,X2,X3)"
       and v2: "atom v2 \<sharp> (t,z,X0,v1,X3)"
       and i': "atom i' \<sharp> (t,u,z,X0,v1,v2,X3)"
     by (metis obtain_fresh)
   have "{} \<turnstile> ((Var X0) EQ (Eats (Var X1) (Var X2))) IFF
              (All X3 (Var X3 IN Var X0 IFF Var X3 IN Var X1 OR Var X3 EQ Var X2))"
     by (simp add: HF HF_axioms_def HF2_def)
   hence "{} \<turnstile> ((Var X0) EQ (Eats (Var X1) (Var X2))) IFF
               (All X3 (Var X3 IN Var X0 IFF Var X3 IN Var X1 OR Var X3 EQ Var X2))"
     by (drule_tac i=X0 and x="Var X0" in Subst) simp_all
   hence "{} \<turnstile> ((Var X0) EQ (Eats (Var v1) (Var X2))) IFF
                  (All X3 (Var X3 IN Var X0 IFF Var X3 IN Var v1 OR Var X3 EQ Var X2))"
     using v1 by (drule_tac i=X1 and x="Var v1" in Subst) simp_all
   hence "{} \<turnstile> ((Var X0) EQ (Eats (Var v1) (Var v2))) IFF
                  (All X3 (Var X3 IN Var X0 IFF Var X3 IN Var v1 OR Var X3 EQ Var v2))"
     using v1 v2 by (drule_tac i=X2 and x="Var v2" in Subst) simp_all
   hence "{} \<turnstile> (((Var X0) EQ (Eats (Var v1) (Var v2))) IFF
              (All X3 (Var X3 IN Var X0 IFF Var X3 IN Var v1 OR Var X3 EQ Var v2)))(X0 ::= z)"
      by (rule Subst) simp
   hence "{} \<turnstile> ((z EQ (Eats (Var v1) (Var v2))) IFF
               (All i' (Var i' IN z IFF Var i' IN Var v1 OR Var i' EQ Var v2)))"
     using v1 v2 i' by (simp add: Conj_def Iff_def)
   hence "{} \<turnstile> (z EQ (Eats t (Var v2))) IFF
                  (All i' (Var i' IN z IFF Var i' IN t OR Var i' EQ Var v2))"
     using v1 v2 i' by (drule_tac i=v1 and x=t in Subst) simp_all
   hence "{} \<turnstile> (z EQ Eats t u) IFF
                  (All i' (Var i' IN z IFF Var i' IN t OR Var i' EQ u))"
     using v1 v2 i' by (drule_tac i=v2 and x=u in Subst) simp_all
   also have "... = (FRESH i'. (z EQ Eats t u) IFF (All i' (Var i' IN z IFF Var i' IN t OR Var i' EQ u)))"
     using i' by simp
   also have "... = (z EQ Eats t u) IFF (All i (Var i IN z IFF Var i IN t OR Var i EQ u))"
     using assms i' by simp
   finally show ?thesis
     by (rule thin0)
 qed
 
 lemma Eq_Eats_I:
   "H \<turnstile> All i (Var i IN z IFF Var i IN t OR Var i EQ u) \<Longrightarrow> atom i \<sharp> (z,t,u) \<Longrightarrow> H \<turnstile> z EQ Eats t u"
   by (metis Iff_MP2_same Eq_Eats_iff)
 
 lemma Mem_Eats_Iff:
   "H \<turnstile> x IN (Eats t u) IFF x IN t OR x EQ u"
 proof -
   obtain i::name where "atom i \<sharp> (Eats t u, t, u)"
     by (rule obtain_fresh)
   thus ?thesis
     using Iff_MP_same [OF Eq_Eats_iff, THEN All_D]
     by auto
 qed
 
 lemma Mem_Eats_I1: "H \<turnstile> u IN t \<Longrightarrow> H \<turnstile> u IN Eats t z"
   by (metis Disj_I1 Iff_MP2_same Mem_Eats_Iff)
 
 lemma Mem_Eats_I2: "H \<turnstile> u EQ z \<Longrightarrow> H \<turnstile> u IN Eats t z"
   by (metis Disj_I2 Iff_MP2_same Mem_Eats_Iff)
 
 lemma Mem_Eats_E:
   assumes A: "insert (u IN t) H \<turnstile> C" and B: "insert (u EQ z) H \<turnstile> C"
     shows "insert (u IN Eats t z) H \<turnstile> C"
   by (rule Mem_Eats_Iff [of _ u t z, THEN Iff_MP_left']) (metis A B Disj_E)
 
 lemmas Mem_Eats_EH = Mem_Eats_E Mem_Eats_E [THEN rotate2] Mem_Eats_E [THEN rotate3] Mem_Eats_E [THEN rotate4] Mem_Eats_E [THEN rotate5]
                  Mem_Eats_E [THEN rotate6] Mem_Eats_E [THEN rotate7] Mem_Eats_E [THEN rotate8]
 declare Mem_Eats_EH [intro!]
 
 lemma Mem_SUCC_I1: "H \<turnstile> u IN t \<Longrightarrow> H \<turnstile> u IN SUCC t"
   by (metis Mem_Eats_I1 SUCC_def)
 
 lemma Mem_SUCC_I2: "H \<turnstile> u EQ t \<Longrightarrow> H \<turnstile> u IN SUCC t"
   by (metis Mem_Eats_I2 SUCC_def)
 
 lemma Mem_SUCC_Refl [simp]: "H \<turnstile> k IN SUCC k"
   by (metis Mem_SUCC_I2 Refl)
 
 lemma Mem_SUCC_E:
   assumes "insert (u IN t) H \<turnstile> C" "insert (u EQ t) H \<turnstile> C" shows "insert (u IN SUCC t) H \<turnstile> C"
   by (metis assms Mem_Eats_E SUCC_def)
 
 lemmas Mem_SUCC_EH = Mem_SUCC_E Mem_SUCC_E [THEN rotate2] Mem_SUCC_E [THEN rotate3] Mem_SUCC_E [THEN rotate4] Mem_SUCC_E [THEN rotate5]
                  Mem_SUCC_E [THEN rotate6] Mem_SUCC_E [THEN rotate7] Mem_SUCC_E [THEN rotate8]
 
 lemma Eats_EQ_Zero_E: "insert (Eats t u EQ Zero) H \<turnstile> A"
   by (metis Assume Eq_Zero_D Mem_Eats_I2 Refl)
 
 lemmas Eats_EQ_Zero_EH = Eats_EQ_Zero_E Eats_EQ_Zero_E [THEN rotate2] Eats_EQ_Zero_E [THEN rotate3] Eats_EQ_Zero_E [THEN rotate4] Eats_EQ_Zero_E [THEN rotate5]
                  Eats_EQ_Zero_E [THEN rotate6] Eats_EQ_Zero_E [THEN rotate7] Eats_EQ_Zero_E [THEN rotate8]
 declare Eats_EQ_Zero_EH [intro!]
 
 lemma Eats_EQ_Zero_E2: "insert (Zero EQ Eats t u) H \<turnstile> A"
   by (metis Eats_EQ_Zero_E Sym_L)
 
 lemmas Eats_EQ_Zero_E2H = Eats_EQ_Zero_E2 Eats_EQ_Zero_E2 [THEN rotate2] Eats_EQ_Zero_E2 [THEN rotate3] Eats_EQ_Zero_E2 [THEN rotate4] Eats_EQ_Zero_E2 [THEN rotate5]
                  Eats_EQ_Zero_E2 [THEN rotate6] Eats_EQ_Zero_E2 [THEN rotate7] Eats_EQ_Zero_E2 [THEN rotate8]
 declare Eats_EQ_Zero_E2H [intro!]
 
 
-section{*Bounded Quantification involving @{term Eats}*}
+section\<open>Bounded Quantification involving @{term Eats}\<close>
 
 lemma All2_cong: "H \<turnstile> t EQ t' \<Longrightarrow> H \<turnstile> A IFF A' \<Longrightarrow> \<forall>C \<in> H. atom i \<sharp> C \<Longrightarrow> H \<turnstile> (All2 i t A) IFF (All2 i t' A')"
   by (metis All_cong Imp_cong Mem_cong Refl)
 
 lemma All2_Zero_E [intro!]: "H  \<turnstile> B \<Longrightarrow> insert (All2 i Zero A) H \<turnstile> B"
   by (rule thin1)
 
 lemma All2_Eats_I_D:
   "atom i \<sharp> (t,u) \<Longrightarrow> { All2 i t A, A(i::=u)} \<turnstile> (All2 i (Eats t u) A)"
     apply (auto, auto intro!: Ex_I [where x="Var i"])
     apply (metis Assume thin1 Var_Eq_subst_Iff [THEN Iff_MP_same])
     done
 
 lemma All2_Eats_I:
   "\<lbrakk>atom i \<sharp> (t,u); H \<turnstile> All2 i t A; H \<turnstile> A(i::=u)\<rbrakk> \<Longrightarrow> H \<turnstile> (All2 i (Eats t u) A)"
   by (rule cut2 [OF All2_Eats_I_D], auto)
 
 lemma All2_Eats_E1:
   "\<lbrakk>atom i \<sharp> (t,u); \<forall>C \<in> H. atom i \<sharp> C\<rbrakk> \<Longrightarrow> insert (All2 i (Eats t u) A) H \<turnstile> All2 i t A"
   by auto (metis Assume Ex_I Imp_E Mem_Eats_I1 Neg_mono subst_fm_id)
 
 lemma All2_Eats_E2:
   "\<lbrakk>atom i \<sharp> (t,u); \<forall>C \<in> H. atom i \<sharp> C\<rbrakk> \<Longrightarrow> insert (All2 i (Eats t u) A) H \<turnstile> A(i::=u)"
   by (rule All_E [where x=u]) (auto intro: ContraProve Mem_Eats_I2)
 
 lemma All2_Eats_E:
   assumes i: "atom i \<sharp> (t,u)"
       and B: "insert (All2 i t A) (insert (A(i::=u)) H) \<turnstile> B"
     shows "insert (All2 i (Eats t u) A) H \<turnstile> B"
   using i
   apply (rule cut_thin [OF All2_Eats_E2, where HB = "insert (All2 i (Eats t u) A) H"], auto)
   apply (rule cut_thin [OF All2_Eats_E1 B], auto)
   done
 
 lemma All2_SUCC_I:
   "atom i \<sharp> t \<Longrightarrow> H \<turnstile> All2 i t A \<Longrightarrow> H \<turnstile> A(i::=t) \<Longrightarrow> H \<turnstile> (All2 i (SUCC t) A)"
   by (simp add: SUCC_def All2_Eats_I)
 
 lemma All2_SUCC_E:
   assumes "atom i \<sharp> t"
       and "insert (All2 i t A) (insert (A(i::=t)) H) \<turnstile> B"
     shows "insert (All2 i (SUCC t) A) H \<turnstile> B"
   by (simp add: SUCC_def All2_Eats_E assms)
 
 lemma All2_SUCC_E':
   assumes "H \<turnstile> u EQ SUCC t"
       and "atom i \<sharp> t" "\<forall>C \<in> H. atom i \<sharp> C"
       and "insert (All2 i t A) (insert (A(i::=t)) H) \<turnstile> B"
     shows "insert (All2 i u A) H \<turnstile> B"
   by (metis All2_SUCC_E Iff_MP_left' Iff_refl All2_cong assms)
 
-section{*Induction*}
+section\<open>Induction\<close>
 
 lemma Ind:
   assumes j: "atom (j::name) \<sharp> (i,A)"
   and prems: "H \<turnstile> A(i::=Zero)" "H \<turnstile> All i (All j (A IMP (A(i::= Var j) IMP A(i::= Eats(Var i)(Var j)))))"
   shows "H \<turnstile> A"
 proof -
   have  "{A(i::=Zero), All i (All j (A IMP (A(i::= Var j) IMP A(i::= Eats(Var i)(Var j)))))} \<turnstile> All i A"
     by (metis j hfthm.Ind ind anti_deduction insert_commute)
   hence "H \<turnstile> (All i A)"
     by (metis cut2 prems)
   thus ?thesis
     by (metis All_E' Assume subst_fm_id)
 qed
 
 end