diff --git a/src/HOL/Nat.thy b/src/HOL/Nat.thy
--- a/src/HOL/Nat.thy
+++ b/src/HOL/Nat.thy
@@ -1,2551 +1,2548 @@
 (*  Title:      HOL/Nat.thy
     Author:     Tobias Nipkow
     Author:     Lawrence C Paulson
     Author:     Markus Wenzel
 *)
 
 section \<open>Natural numbers\<close>
 
 theory Nat
 imports Inductive Typedef Fun Rings
 begin
 
 subsection \<open>Type \<open>ind\<close>\<close>
 
 typedecl ind
 
 axiomatization Zero_Rep :: ind and Suc_Rep :: "ind \<Rightarrow> ind"
   \<comment> \<open>The axiom of infinity in 2 parts:\<close>
   where Suc_Rep_inject: "Suc_Rep x = Suc_Rep y \<Longrightarrow> x = y"
     and Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"
 
 
 subsection \<open>Type nat\<close>
 
 text \<open>Type definition\<close>
 
 inductive Nat :: "ind \<Rightarrow> bool"
   where
     Zero_RepI: "Nat Zero_Rep"
   | Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)"
 
 typedef nat = "{n. Nat n}"
   morphisms Rep_Nat Abs_Nat
   using Nat.Zero_RepI by auto
 
 lemma Nat_Rep_Nat: "Nat (Rep_Nat n)"
   using Rep_Nat by simp
 
 lemma Nat_Abs_Nat_inverse: "Nat n \<Longrightarrow> Rep_Nat (Abs_Nat n) = n"
   using Abs_Nat_inverse by simp
 
 lemma Nat_Abs_Nat_inject: "Nat n \<Longrightarrow> Nat m \<Longrightarrow> Abs_Nat n = Abs_Nat m \<longleftrightarrow> n = m"
   using Abs_Nat_inject by simp
 
 instantiation nat :: zero
 begin
 
 definition Zero_nat_def: "0 = Abs_Nat Zero_Rep"
 
 instance ..
 
 end
 
 definition Suc :: "nat \<Rightarrow> nat"
   where "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"
 
 lemma Suc_not_Zero: "Suc m \<noteq> 0"
   by (simp add: Zero_nat_def Suc_def Suc_RepI Zero_RepI
       Nat_Abs_Nat_inject Suc_Rep_not_Zero_Rep Nat_Rep_Nat)
 
 lemma Zero_not_Suc: "0 \<noteq> Suc m"
   by (rule not_sym) (rule Suc_not_Zero)
 
 lemma Suc_Rep_inject': "Suc_Rep x = Suc_Rep y \<longleftrightarrow> x = y"
   by (rule iffI, rule Suc_Rep_inject) simp_all
 
 lemma nat_induct0:
   assumes "P 0" and "\<And>n. P n \<Longrightarrow> P (Suc n)"
   shows "P n"
 proof -
   have "P (Abs_Nat (Rep_Nat n))"
     using assms unfolding Zero_nat_def Suc_def
     by (iprover intro:  Nat_Rep_Nat [THEN Nat.induct] elim: Nat_Abs_Nat_inverse [THEN subst])
   then show ?thesis
     by (simp add: Rep_Nat_inverse)
 qed
 
 free_constructors case_nat for "0 :: nat" | Suc pred
   where "pred (0 :: nat) = (0 :: nat)"
     apply atomize_elim
     apply (rename_tac n, induct_tac n rule: nat_induct0, auto)
    apply (simp add: Suc_def Nat_Abs_Nat_inject Nat_Rep_Nat Suc_RepI Suc_Rep_inject' Rep_Nat_inject)
   apply (simp only: Suc_not_Zero)
   done
 
 \<comment> \<open>Avoid name clashes by prefixing the output of \<open>old_rep_datatype\<close> with \<open>old\<close>.\<close>
 setup \<open>Sign.mandatory_path "old"\<close>
 
 old_rep_datatype "0 :: nat" Suc
   by (erule nat_induct0) auto
 
 setup \<open>Sign.parent_path\<close>
 
 \<comment> \<open>But erase the prefix for properties that are not generated by \<open>free_constructors\<close>.\<close>
 setup \<open>Sign.mandatory_path "nat"\<close>
 
 declare old.nat.inject[iff del]
   and old.nat.distinct(1)[simp del, induct_simp del]
 
 lemmas induct = old.nat.induct
 lemmas inducts = old.nat.inducts
 lemmas rec = old.nat.rec
 lemmas simps = nat.inject nat.distinct nat.case nat.rec
 
 setup \<open>Sign.parent_path\<close>
 
 abbreviation rec_nat :: "'a \<Rightarrow> (nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"
   where "rec_nat \<equiv> old.rec_nat"
 
 declare nat.sel[code del]
 
 hide_const (open) Nat.pred \<comment> \<open>hide everything related to the selector\<close>
 hide_fact
   nat.case_eq_if
   nat.collapse
   nat.expand
   nat.sel
   nat.exhaust_sel
   nat.split_sel
   nat.split_sel_asm
 
 lemma nat_exhaust [case_names 0 Suc, cases type: nat]:
   "(y = 0 \<Longrightarrow> P) \<Longrightarrow> (\<And>nat. y = Suc nat \<Longrightarrow> P) \<Longrightarrow> P"
   \<comment> \<open>for backward compatibility -- names of variables differ\<close>
   by (rule old.nat.exhaust)
 
 lemma nat_induct [case_names 0 Suc, induct type: nat]:
   fixes n
   assumes "P 0" and "\<And>n. P n \<Longrightarrow> P (Suc n)"
   shows "P n"
   \<comment> \<open>for backward compatibility -- names of variables differ\<close>
   using assms by (rule nat.induct)
 
 hide_fact
   nat_exhaust
   nat_induct0
 
 ML \<open>
 val nat_basic_lfp_sugar =
   let
     val ctr_sugar = the (Ctr_Sugar.ctr_sugar_of_global \<^theory> \<^type_name>\<open>nat\<close>);
     val recx = Logic.varify_types_global \<^term>\<open>rec_nat\<close>;
     val C = body_type (fastype_of recx);
   in
     {T = HOLogic.natT, fp_res_index = 0, C = C, fun_arg_Tsss = [[], [[HOLogic.natT, C]]],
      ctr_sugar = ctr_sugar, recx = recx, rec_thms = @{thms nat.rec}}
   end;
 \<close>
 
 setup \<open>
 let
   fun basic_lfp_sugars_of _ [\<^typ>\<open>nat\<close>] _ _ ctxt =
       ([], [0], [nat_basic_lfp_sugar], [], [], [], TrueI (*dummy*), [], false, ctxt)
     | basic_lfp_sugars_of bs arg_Ts callers callssss ctxt =
       BNF_LFP_Rec_Sugar.default_basic_lfp_sugars_of bs arg_Ts callers callssss ctxt;
 in
   BNF_LFP_Rec_Sugar.register_lfp_rec_extension
     {nested_simps = [], special_endgame_tac = K (K (K (K no_tac))), is_new_datatype = K (K true),
      basic_lfp_sugars_of = basic_lfp_sugars_of, rewrite_nested_rec_call = NONE}
 end
 \<close>
 
 text \<open>Injectiveness and distinctness lemmas\<close>
 
 lemma inj_Suc [simp]:
   "inj_on Suc N"
   by (simp add: inj_on_def)
 
 lemma bij_betw_Suc [simp]:
   "bij_betw Suc M N \<longleftrightarrow> Suc ` M = N"
   by (simp add: bij_betw_def)
 
 lemma Suc_neq_Zero: "Suc m = 0 \<Longrightarrow> R"
   by (rule notE) (rule Suc_not_Zero)
 
 lemma Zero_neq_Suc: "0 = Suc m \<Longrightarrow> R"
   by (rule Suc_neq_Zero) (erule sym)
 
 lemma Suc_inject: "Suc x = Suc y \<Longrightarrow> x = y"
   by (rule inj_Suc [THEN injD])
 
 lemma n_not_Suc_n: "n \<noteq> Suc n"
   by (induct n) simp_all
 
 lemma Suc_n_not_n: "Suc n \<noteq> n"
   by (rule not_sym) (rule n_not_Suc_n)
 
 text \<open>A special form of induction for reasoning about \<^term>\<open>m < n\<close> and \<^term>\<open>m - n\<close>.\<close>
 lemma diff_induct:
   assumes "\<And>x. P x 0"
     and "\<And>y. P 0 (Suc y)"
     and "\<And>x y. P x y \<Longrightarrow> P (Suc x) (Suc y)"
   shows "P m n"
 proof (induct n arbitrary: m)
   case 0
   show ?case by (rule assms(1))
 next
   case (Suc n)
   show ?case
   proof (induct m)
     case 0
     show ?case by (rule assms(2))
   next
     case (Suc m)
     from \<open>P m n\<close> show ?case by (rule assms(3))
   qed
 qed
 
 
 subsection \<open>Arithmetic operators\<close>
 
 instantiation nat :: comm_monoid_diff
 begin
 
 primrec plus_nat
   where
     add_0: "0 + n = (n::nat)"
   | add_Suc: "Suc m + n = Suc (m + n)"
 
 lemma add_0_right [simp]: "m + 0 = m"
   for m :: nat
   by (induct m) simp_all
 
 lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
   by (induct m) simp_all
 
 declare add_0 [code]
 
 lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"
   by simp
 
 primrec minus_nat
   where
     diff_0 [code]: "m - 0 = (m::nat)"
   | diff_Suc: "m - Suc n = (case m - n of 0 \<Rightarrow> 0 | Suc k \<Rightarrow> k)"
 
 declare diff_Suc [simp del]
 
 lemma diff_0_eq_0 [simp, code]: "0 - n = 0"
   for n :: nat
   by (induct n) (simp_all add: diff_Suc)
 
 lemma diff_Suc_Suc [simp, code]: "Suc m - Suc n = m - n"
   by (induct n) (simp_all add: diff_Suc)
 
 instance
 proof
   fix n m q :: nat
   show "(n + m) + q = n + (m + q)" by (induct n) simp_all
   show "n + m = m + n" by (induct n) simp_all
   show "m + n - m = n" by (induct m) simp_all
   show "n - m - q = n - (m + q)" by (induct q) (simp_all add: diff_Suc)
   show "0 + n = n" by simp
   show "0 - n = 0" by simp
 qed
 
 end
 
 hide_fact (open) add_0 add_0_right diff_0
 
 instantiation nat :: comm_semiring_1_cancel
 begin
 
 definition One_nat_def [simp]: "1 = Suc 0"
 
 primrec times_nat
   where
     mult_0: "0 * n = (0::nat)"
   | mult_Suc: "Suc m * n = n + (m * n)"
 
 lemma mult_0_right [simp]: "m * 0 = 0"
   for m :: nat
   by (induct m) simp_all
 
 lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
   by (induct m) (simp_all add: add.left_commute)
 
 lemma add_mult_distrib: "(m + n) * k = (m * k) + (n * k)"
   for m n k :: nat
   by (induct m) (simp_all add: add.assoc)
 
 instance
 proof
   fix k n m q :: nat
   show "0 \<noteq> (1::nat)"
     by simp
   show "1 * n = n"
     by simp
   show "n * m = m * n"
     by (induct n) simp_all
   show "(n * m) * q = n * (m * q)"
     by (induct n) (simp_all add: add_mult_distrib)
   show "(n + m) * q = n * q + m * q"
     by (rule add_mult_distrib)
   show "k * (m - n) = (k * m) - (k * n)"
     by (induct m n rule: diff_induct) simp_all
 qed
 
 end
 
 
 subsubsection \<open>Addition\<close>
 
 text \<open>Reasoning about \<open>m + 0 = 0\<close>, etc.\<close>
 
 lemma add_is_0 [iff]: "m + n = 0 \<longleftrightarrow> m = 0 \<and> n = 0"
   for m n :: nat
   by (cases m) simp_all
 
 lemma add_is_1: "m + n = Suc 0 \<longleftrightarrow> m = Suc 0 \<and> n = 0 \<or> m = 0 \<and> n = Suc 0"
   by (cases m) simp_all
 
 lemma one_is_add: "Suc 0 = m + n \<longleftrightarrow> m = Suc 0 \<and> n = 0 \<or> m = 0 \<and> n = Suc 0"
   by (rule trans, rule eq_commute, rule add_is_1)
 
 lemma add_eq_self_zero: "m + n = m \<Longrightarrow> n = 0"
   for m n :: nat
   by (induct m) simp_all
 
 lemma plus_1_eq_Suc:
   "plus 1 = Suc"
   by (simp add: fun_eq_iff)
 
 lemma Suc_eq_plus1: "Suc n = n + 1"
   by simp
 
 lemma Suc_eq_plus1_left: "Suc n = 1 + n"
   by simp
 
 
 subsubsection \<open>Difference\<close>
 
 lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
   by (simp add: diff_diff_add)
 
 lemma diff_Suc_1 [simp]: "Suc n - 1 = n"
   by simp
 
 
 subsubsection \<open>Multiplication\<close>
 
 lemma mult_is_0 [simp]: "m * n = 0 \<longleftrightarrow> m = 0 \<or> n = 0" for m n :: nat
   by (induct m) auto
 
 lemma mult_eq_1_iff [simp]: "m * n = Suc 0 \<longleftrightarrow> m = Suc 0 \<and> n = Suc 0"
 proof (induct m)
   case 0
   then show ?case by simp
 next
   case (Suc m)
   then show ?case by (induct n) auto
 qed
 
 lemma one_eq_mult_iff [simp]: "Suc 0 = m * n \<longleftrightarrow> m = Suc 0 \<and> n = Suc 0"
   by (simp add: eq_commute flip: mult_eq_1_iff)
 
 lemma nat_mult_eq_1_iff [simp]: "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1" 
   and nat_1_eq_mult_iff [simp]: "1 = m * n \<longleftrightarrow> m = 1 \<and> n = 1" for m n :: nat
   by auto
 
 lemma mult_cancel1 [simp]: "k * m = k * n \<longleftrightarrow> m = n \<or> k = 0"
   for k m n :: nat
 proof -
   have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
   proof (induct n arbitrary: m)
     case 0
     then show "m = 0" by simp
   next
     case (Suc n)
     then show "m = Suc n"
       by (cases m) (simp_all add: eq_commute [of 0])
   qed
   then show ?thesis by auto
 qed
 
 lemma mult_cancel2 [simp]: "m * k = n * k \<longleftrightarrow> m = n \<or> k = 0"
   for k m n :: nat
   by (simp add: mult.commute)
 
 lemma Suc_mult_cancel1: "Suc k * m = Suc k * n \<longleftrightarrow> m = n"
   by (subst mult_cancel1) simp
 
 
 subsection \<open>Orders on \<^typ>\<open>nat\<close>\<close>
 
 subsubsection \<open>Operation definition\<close>
 
 instantiation nat :: linorder
 begin
 
 primrec less_eq_nat
   where
     "(0::nat) \<le> n \<longleftrightarrow> True"
   | "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
 
 declare less_eq_nat.simps [simp del]
 
 lemma le0 [iff]: "0 \<le> n" for
   n :: nat
   by (simp add: less_eq_nat.simps)
 
 lemma [code]: "0 \<le> n \<longleftrightarrow> True"
   for n :: nat
   by simp
 
 definition less_nat
   where less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"
 
 lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
   by (simp add: less_eq_nat.simps(2))
 
 lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
   unfolding less_eq_Suc_le ..
 
 lemma le_0_eq [iff]: "n \<le> 0 \<longleftrightarrow> n = 0"
   for n :: nat
   by (induct n) (simp_all add: less_eq_nat.simps(2))
 
 lemma not_less0 [iff]: "\<not> n < 0"
   for n :: nat
   by (simp add: less_eq_Suc_le)
 
 lemma less_nat_zero_code [code]: "n < 0 \<longleftrightarrow> False"
   for n :: nat
   by simp
 
 lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
   by (simp add: less_eq_Suc_le)
 
 lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
   by (simp add: less_eq_Suc_le)
 
 lemma Suc_less_eq2: "Suc n < m \<longleftrightarrow> (\<exists>m'. m = Suc m' \<and> n < m')"
   by (cases m) auto
 
 lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
   by (induct m arbitrary: n) (simp_all add: less_eq_nat.simps(2) split: nat.splits)
 
 lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
   by (cases n) (auto intro: le_SucI)
 
 lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
   by (simp add: less_eq_Suc_le) (erule Suc_leD)
 
 lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
   by (simp add: less_eq_Suc_le) (erule Suc_leD)
 
 instance
 proof
   fix n m q :: nat
   show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n"
   proof (induct n arbitrary: m)
     case 0
     then show ?case
       by (cases m) (simp_all add: less_eq_Suc_le)
   next
     case (Suc n)
     then show ?case
       by (cases m) (simp_all add: less_eq_Suc_le)
   qed
   show "n \<le> n"
     by (induct n) simp_all
   then show "n = m" if "n \<le> m" and "m \<le> n"
     using that by (induct n arbitrary: m)
       (simp_all add: less_eq_nat.simps(2) split: nat.splits)
   show "n \<le> q" if "n \<le> m" and "m \<le> q"
     using that
   proof (induct n arbitrary: m q)
     case 0
     show ?case by simp
   next
     case (Suc n)
     then show ?case
       by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
         simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
         simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
   qed
   show "n \<le> m \<or> m \<le> n"
     by (induct n arbitrary: m)
       (simp_all add: less_eq_nat.simps(2) split: nat.splits)
 qed
 
 end
 
 instantiation nat :: order_bot
 begin
 
 definition bot_nat :: nat
   where "bot_nat = 0"
 
 instance
   by standard (simp add: bot_nat_def)
 
 end
 
 instance nat :: no_top
   by standard (auto intro: less_Suc_eq_le [THEN iffD2])
 
 
 subsubsection \<open>Introduction properties\<close>
 
 lemma lessI [iff]: "n < Suc n"
   by (simp add: less_Suc_eq_le)
 
 lemma zero_less_Suc [iff]: "0 < Suc n"
   by (simp add: less_Suc_eq_le)
 
 
 subsubsection \<open>Elimination properties\<close>
 
 lemma less_not_refl: "\<not> n < n"
   for n :: nat
   by (rule order_less_irrefl)
 
 lemma less_not_refl2: "n < m \<Longrightarrow> m \<noteq> n"
   for m n :: nat
   by (rule not_sym) (rule less_imp_neq)
 
 lemma less_not_refl3: "s < t \<Longrightarrow> s \<noteq> t"
   for s t :: nat
   by (rule less_imp_neq)
 
 lemma less_irrefl_nat: "n < n \<Longrightarrow> R"
   for n :: nat
   by (rule notE, rule less_not_refl)
 
 lemma less_zeroE: "n < 0 \<Longrightarrow> R"
   for n :: nat
   by (rule notE) (rule not_less0)
 
 lemma less_Suc_eq: "m < Suc n \<longleftrightarrow> m < n \<or> m = n"
   unfolding less_Suc_eq_le le_less ..
 
 lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
   by (simp add: less_Suc_eq)
 
 lemma less_one [iff]: "n < 1 \<longleftrightarrow> n = 0"
   for n :: nat
   unfolding One_nat_def by (rule less_Suc0)
 
 lemma Suc_mono: "m < n \<Longrightarrow> Suc m < Suc n"
   by simp
 
 text \<open>"Less than" is antisymmetric, sort of.\<close>
 lemma less_antisym: "\<not> n < m \<Longrightarrow> n < Suc m \<Longrightarrow> m = n"
   unfolding not_less less_Suc_eq_le by (rule antisym)
 
 lemma nat_neq_iff: "m \<noteq> n \<longleftrightarrow> m < n \<or> n < m"
   for m n :: nat
   by (rule linorder_neq_iff)
 
 
 subsubsection \<open>Inductive (?) properties\<close>
 
 lemma Suc_lessI: "m < n \<Longrightarrow> Suc m \<noteq> n \<Longrightarrow> Suc m < n"
   unfolding less_eq_Suc_le [of m] le_less by simp
 
 lemma lessE:
   assumes major: "i < k"
     and 1: "k = Suc i \<Longrightarrow> P"
     and 2: "\<And>j. i < j \<Longrightarrow> k = Suc j \<Longrightarrow> P"
   shows P
 proof -
   from major have "\<exists>j. i \<le> j \<and> k = Suc j"
     unfolding less_eq_Suc_le by (induct k) simp_all
   then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
     by (auto simp add: less_le)
   with 1 2 show P by auto
 qed
 
 lemma less_SucE:
   assumes major: "m < Suc n"
     and less: "m < n \<Longrightarrow> P"
     and eq: "m = n \<Longrightarrow> P"
   shows P
 proof (rule major [THEN lessE])
   show "Suc n = Suc m \<Longrightarrow> P"
     using eq by blast
   show "\<And>j. \<lbrakk>m < j; Suc n = Suc j\<rbrakk> \<Longrightarrow> P"
     by (blast intro: less)
 qed
 
 lemma Suc_lessE:
   assumes major: "Suc i < k"
     and minor: "\<And>j. i < j \<Longrightarrow> k = Suc j \<Longrightarrow> P"
   shows P
 proof (rule major [THEN lessE])
   show "k = Suc (Suc i) \<Longrightarrow> P"
     using lessI minor by iprover
   show "\<And>j. \<lbrakk>Suc i < j; k = Suc j\<rbrakk> \<Longrightarrow> P"
     using Suc_lessD minor by iprover
 qed
 
 lemma Suc_less_SucD: "Suc m < Suc n \<Longrightarrow> m < n"
   by simp
 
 lemma less_trans_Suc:
   assumes le: "i < j"
   shows "j < k \<Longrightarrow> Suc i < k"
 proof (induct k)
   case 0
   then show ?case by simp
 next
   case (Suc k)
   with le show ?case
     by simp (auto simp add: less_Suc_eq dest: Suc_lessD)
 qed
 
 text \<open>Can be used with \<open>less_Suc_eq\<close> to get \<^prop>\<open>n = m \<or> n < m\<close>.\<close>
 lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
   by (simp only: not_less less_Suc_eq_le)
 
 lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
   by (simp only: not_le Suc_le_eq)
 
 text \<open>Properties of "less than or equal".\<close>
 
 lemma le_imp_less_Suc: "m \<le> n \<Longrightarrow> m < Suc n"
   by (simp only: less_Suc_eq_le)
 
 lemma Suc_n_not_le_n: "\<not> Suc n \<le> n"
   by (simp add: not_le less_Suc_eq_le)
 
 lemma le_Suc_eq: "m \<le> Suc n \<longleftrightarrow> m \<le> n \<or> m = Suc n"
   by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
 
 lemma le_SucE: "m \<le> Suc n \<Longrightarrow> (m \<le> n \<Longrightarrow> R) \<Longrightarrow> (m = Suc n \<Longrightarrow> R) \<Longrightarrow> R"
   by (drule le_Suc_eq [THEN iffD1], iprover+)
 
 lemma Suc_leI: "m < n \<Longrightarrow> Suc m \<le> n"
   by (simp only: Suc_le_eq)
 
 text \<open>Stronger version of \<open>Suc_leD\<close>.\<close>
 lemma Suc_le_lessD: "Suc m \<le> n \<Longrightarrow> m < n"
   by (simp only: Suc_le_eq)
 
 lemma less_imp_le_nat: "m < n \<Longrightarrow> m \<le> n" for m n :: nat
   unfolding less_eq_Suc_le by (rule Suc_leD)
 
 text \<open>For instance, \<open>(Suc m < Suc n) = (Suc m \<le> n) = (m < n)\<close>\<close>
 lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq
 
 
 text \<open>Equivalence of \<open>m \<le> n\<close> and \<open>m < n \<or> m = n\<close>\<close>
 
 lemma less_or_eq_imp_le: "m < n \<or> m = n \<Longrightarrow> m \<le> n"
   for m n :: nat
   unfolding le_less .
 
 lemma le_eq_less_or_eq: "m \<le> n \<longleftrightarrow> m < n \<or> m = n"
   for m n :: nat
   by (rule le_less)
 
 text \<open>Useful with \<open>blast\<close>.\<close>
 lemma eq_imp_le: "m = n \<Longrightarrow> m \<le> n"
   for m n :: nat
   by auto
 
 lemma le_refl: "n \<le> n"
   for n :: nat
   by simp
 
 lemma le_trans: "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k"
   for i j k :: nat
   by (rule order_trans)
 
 lemma le_antisym: "m \<le> n \<Longrightarrow> n \<le> m \<Longrightarrow> m = n"
   for m n :: nat
   by (rule antisym)
 
 lemma nat_less_le: "m < n \<longleftrightarrow> m \<le> n \<and> m \<noteq> n"
   for m n :: nat
   by (rule less_le)
 
 lemma le_neq_implies_less: "m \<le> n \<Longrightarrow> m \<noteq> n \<Longrightarrow> m < n"
   for m n :: nat
   unfolding less_le ..
 
 lemma nat_le_linear: "m \<le> n \<or> n \<le> m"
   for m n :: nat
   by (rule linear)
 
 lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
 
 lemma le_less_Suc_eq: "m \<le> n \<Longrightarrow> n < Suc m \<longleftrightarrow> n = m"
   unfolding less_Suc_eq_le by auto
 
 lemma not_less_less_Suc_eq: "\<not> n < m \<Longrightarrow> n < Suc m \<longleftrightarrow> n = m"
   unfolding not_less by (rule le_less_Suc_eq)
 
 lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
 
 lemma not0_implies_Suc: "n \<noteq> 0 \<Longrightarrow> \<exists>m. n = Suc m"
   by (cases n) simp_all
 
 lemma gr0_implies_Suc: "n > 0 \<Longrightarrow> \<exists>m. n = Suc m"
   by (cases n) simp_all
 
 lemma gr_implies_not0: "m < n \<Longrightarrow> n \<noteq> 0"
   for m n :: nat
   by (cases n) simp_all
 
 lemma neq0_conv[iff]: "n \<noteq> 0 \<longleftrightarrow> 0 < n"
   for n :: nat
   by (cases n) simp_all
 
 text \<open>This theorem is useful with \<open>blast\<close>\<close>
 lemma gr0I: "(n = 0 \<Longrightarrow> False) \<Longrightarrow> 0 < n"
   for n :: nat
   by (rule neq0_conv[THEN iffD1]) iprover
 
 lemma gr0_conv_Suc: "0 < n \<longleftrightarrow> (\<exists>m. n = Suc m)"
   by (fast intro: not0_implies_Suc)
 
 lemma not_gr0 [iff]: "\<not> 0 < n \<longleftrightarrow> n = 0"
   for n :: nat
   using neq0_conv by blast
 
 lemma Suc_le_D: "Suc n \<le> m' \<Longrightarrow> \<exists>m. m' = Suc m"
   by (induct m') simp_all
 
 text \<open>Useful in certain inductive arguments\<close>
 lemma less_Suc_eq_0_disj: "m < Suc n \<longleftrightarrow> m = 0 \<or> (\<exists>j. m = Suc j \<and> j < n)"
   by (cases m) simp_all
 
 lemma All_less_Suc: "(\<forall>i < Suc n. P i) = (P n \<and> (\<forall>i < n. P i))"
   by (auto simp: less_Suc_eq)
 
 lemma All_less_Suc2: "(\<forall>i < Suc n. P i) = (P 0 \<and> (\<forall>i < n. P(Suc i)))"
   by (auto simp: less_Suc_eq_0_disj)
 
 lemma Ex_less_Suc: "(\<exists>i < Suc n. P i) = (P n \<or> (\<exists>i < n. P i))"
   by (auto simp: less_Suc_eq)
 
 lemma Ex_less_Suc2: "(\<exists>i < Suc n. P i) = (P 0 \<or> (\<exists>i < n. P(Suc i)))"
   by (auto simp: less_Suc_eq_0_disj)
 
 text \<open>@{term mono} (non-strict) doesn't imply increasing, as the function could be constant\<close>
 lemma strict_mono_imp_increasing:
   fixes n::nat
   assumes "strict_mono f" shows "f n \<ge> n"
 proof (induction n)
   case 0
   then show ?case
     by auto
 next
   case (Suc n)
   then show ?case
     unfolding not_less_eq_eq [symmetric]
     using Suc_n_not_le_n assms order_trans strict_mono_less_eq by blast
 qed
 
 subsubsection \<open>Monotonicity of Addition\<close>
 
 lemma Suc_pred [simp]: "n > 0 \<Longrightarrow> Suc (n - Suc 0) = n"
   by (simp add: diff_Suc split: nat.split)
 
 lemma Suc_diff_1 [simp]: "0 < n \<Longrightarrow> Suc (n - 1) = n"
   unfolding One_nat_def by (rule Suc_pred)
 
 lemma nat_add_left_cancel_le [simp]: "k + m \<le> k + n \<longleftrightarrow> m \<le> n"
   for k m n :: nat
   by (induct k) simp_all
 
 lemma nat_add_left_cancel_less [simp]: "k + m < k + n \<longleftrightarrow> m < n"
   for k m n :: nat
   by (induct k) simp_all
 
 lemma add_gr_0 [iff]: "m + n > 0 \<longleftrightarrow> m > 0 \<or> n > 0"
   for m n :: nat
   by (auto dest: gr0_implies_Suc)
 
 text \<open>strict, in 1st argument\<close>
 lemma add_less_mono1: "i < j \<Longrightarrow> i + k < j + k"
   for i j k :: nat
   by (induct k) simp_all
 
 text \<open>strict, in both arguments\<close>
 lemma add_less_mono: 
   fixes i j k l :: nat
   assumes "i < j" "k < l" shows "i + k < j + l"
 proof -
   have "i + k < j + k"
     by (simp add: add_less_mono1 assms)
   also have "...  < j + l"
     using \<open>i < j\<close> by (induction j) (auto simp: assms)
   finally show ?thesis .
 qed
 
 lemma less_imp_Suc_add: "m < n \<Longrightarrow> \<exists>k. n = Suc (m + k)"
 proof (induct n)
   case 0
   then show ?case by simp
 next
   case Suc
   then show ?case
     by (simp add: order_le_less)
       (blast elim!: less_SucE intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])
 qed
 
 lemma le_Suc_ex: "k \<le> l \<Longrightarrow> (\<exists>n. l = k + n)"
   for k l :: nat
   by (auto simp: less_Suc_eq_le[symmetric] dest: less_imp_Suc_add)
 
 lemma less_natE:
   assumes \<open>m < n\<close>
   obtains q where \<open>n = Suc (m + q)\<close>
   using assms by (auto dest: less_imp_Suc_add intro: that)
 
 text \<open>strict, in 1st argument; proof is by induction on \<open>k > 0\<close>\<close>
 lemma mult_less_mono2:
   fixes i j :: nat
   assumes "i < j" and "0 < k"
   shows "k * i < k * j"
   using \<open>0 < k\<close>
 proof (induct k)
   case 0
   then show ?case by simp
 next
   case (Suc k)
   with \<open>i < j\<close> show ?case
     by (cases k) (simp_all add: add_less_mono)
 qed
 
 text \<open>Addition is the inverse of subtraction:
   if \<^term>\<open>n \<le> m\<close> then \<^term>\<open>n + (m - n) = m\<close>.\<close>
 lemma add_diff_inverse_nat: "\<not> m < n \<Longrightarrow> n + (m - n) = m"
   for m n :: nat
   by (induct m n rule: diff_induct) simp_all
 
 lemma nat_le_iff_add: "m \<le> n \<longleftrightarrow> (\<exists>k. n = m + k)"
   for m n :: nat
   using nat_add_left_cancel_le[of m 0] by (auto dest: le_Suc_ex)
 
 text \<open>The naturals form an ordered \<open>semidom\<close> and a \<open>dioid\<close>.\<close>
 
 instance nat :: linordered_semidom
 proof
   fix m n q :: nat
   show "0 < (1::nat)"
     by simp
   show "m \<le> n \<Longrightarrow> q + m \<le> q + n"
     by simp
   show "m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n"
     by (simp add: mult_less_mono2)
   show "m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> m * n \<noteq> 0"
     by simp
   show "n \<le> m \<Longrightarrow> (m - n) + n = m"
     by (simp add: add_diff_inverse_nat add.commute linorder_not_less)
 qed
 
 instance nat :: dioid
   by standard (rule nat_le_iff_add)
 
 declare le0[simp del] \<comment> \<open>This is now @{thm zero_le}\<close>
 declare le_0_eq[simp del] \<comment> \<open>This is now @{thm le_zero_eq}\<close>
 declare not_less0[simp del] \<comment> \<open>This is now @{thm not_less_zero}\<close>
 declare not_gr0[simp del] \<comment> \<open>This is now @{thm not_gr_zero}\<close>
 
 instance nat :: ordered_cancel_comm_monoid_add ..
 instance nat :: ordered_cancel_comm_monoid_diff ..
 
 
 subsubsection \<open>\<^term>\<open>min\<close> and \<^term>\<open>max\<close>\<close>
 
 global_interpretation bot_nat_0: ordering_top \<open>(\<ge>)\<close> \<open>(>)\<close> \<open>0::nat\<close>
   by standard simp
 
 global_interpretation max_nat: semilattice_neutr_order max \<open>0::nat\<close> \<open>(\<ge>)\<close> \<open>(>)\<close>
   by standard (simp add: max_def)
 
 lemma mono_Suc: "mono Suc"
   by (rule monoI) simp
 
 lemma min_0L [simp]: "min 0 n = 0"
   for n :: nat
   by (rule min_absorb1) simp
 
 lemma min_0R [simp]: "min n 0 = 0"
   for n :: nat
   by (rule min_absorb2) simp
 
 lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
   by (simp add: mono_Suc min_of_mono)
 
 lemma min_Suc1: "min (Suc n) m = (case m of 0 \<Rightarrow> 0 | Suc m' \<Rightarrow> Suc(min n m'))"
   by (simp split: nat.split)
 
 lemma min_Suc2: "min m (Suc n) = (case m of 0 \<Rightarrow> 0 | Suc m' \<Rightarrow> Suc(min m' n))"
   by (simp split: nat.split)
 
 lemma max_0L [simp]: "max 0 n = n"
   for n :: nat
   by (fact max_nat.left_neutral)
 
 lemma max_0R [simp]: "max n 0 = n"
   for n :: nat
   by (fact max_nat.right_neutral)
 
 lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc (max m n)"
   by (simp add: mono_Suc max_of_mono)
 
 lemma max_Suc1: "max (Suc n) m = (case m of 0 \<Rightarrow> Suc n | Suc m' \<Rightarrow> Suc (max n m'))"
   by (simp split: nat.split)
 
 lemma max_Suc2: "max m (Suc n) = (case m of 0 \<Rightarrow> Suc n | Suc m' \<Rightarrow> Suc (max m' n))"
   by (simp split: nat.split)
 
-lemma max_0_iff: "max m n = (0::nat) \<longleftrightarrow> m = 0 \<and> n = 0"
-  by (fact max_nat.eq_neutr_iff)
-
 lemma nat_mult_min_left: "min m n * q = min (m * q) (n * q)"
   for m n q :: nat
   by (simp add: min_def not_le)
     (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
 
 lemma nat_mult_min_right: "m * min n q = min (m * n) (m * q)"
   for m n q :: nat
   by (simp add: min_def not_le)
     (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
 
 lemma nat_add_max_left: "max m n + q = max (m + q) (n + q)"
   for m n q :: nat
   by (simp add: max_def)
 
 lemma nat_add_max_right: "m + max n q = max (m + n) (m + q)"
   for m n q :: nat
   by (simp add: max_def)
 
 lemma nat_mult_max_left: "max m n * q = max (m * q) (n * q)"
   for m n q :: nat
   by (simp add: max_def not_le)
     (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
 
 lemma nat_mult_max_right: "m * max n q = max (m * n) (m * q)"
   for m n q :: nat
   by (simp add: max_def not_le)
     (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
 
 
 subsubsection \<open>Additional theorems about \<^term>\<open>(\<le>)\<close>\<close>
 
 text \<open>Complete induction, aka course-of-values induction\<close>
 
 instance nat :: wellorder
 proof
   fix P and n :: nat
   assume step: "(\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n" for n :: nat
   have "\<And>q. q \<le> n \<Longrightarrow> P q"
   proof (induct n)
     case (0 n)
     have "P 0" by (rule step) auto
     with 0 show ?case by auto
   next
     case (Suc m n)
     then have "n \<le> m \<or> n = Suc m"
       by (simp add: le_Suc_eq)
     then show ?case
     proof
       assume "n \<le> m"
       then show "P n" by (rule Suc(1))
     next
       assume n: "n = Suc m"
       show "P n" by (rule step) (rule Suc(1), simp add: n le_simps)
     qed
   qed
   then show "P n" by auto
 qed
 
 
 lemma Least_eq_0[simp]: "P 0 \<Longrightarrow> Least P = 0"
   for P :: "nat \<Rightarrow> bool"
   by (rule Least_equality[OF _ le0])
 
 lemma Least_Suc:
   assumes "P n" "\<not> P 0" 
   shows "(LEAST n. P n) = Suc (LEAST m. P (Suc m))"
 proof (cases n)
   case (Suc m)
   show ?thesis
   proof (rule antisym)
     show "(LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))"
       using assms Suc by (force intro: LeastI Least_le)
     have \<section>: "P (LEAST x. P x)"
       by (blast intro: LeastI assms)
     show "Suc (LEAST m. P (Suc m)) \<le> (LEAST n. P n)"
     proof (cases "(LEAST n. P n)")
       case 0
       then show ?thesis
         using \<section> by (simp add: assms)
     next
       case Suc
       with \<section> show ?thesis
         by (auto simp: Least_le)
     qed
   qed
 qed (use assms in auto)
 
 lemma Least_Suc2: "P n \<Longrightarrow> Q m \<Longrightarrow> \<not> P 0 \<Longrightarrow> \<forall>k. P (Suc k) = Q k \<Longrightarrow> Least P = Suc (Least Q)"
   by (erule (1) Least_Suc [THEN ssubst]) simp
 
 lemma ex_least_nat_le:
   fixes P :: "nat \<Rightarrow> bool"
   assumes "P n" "\<not> P 0" 
   shows "\<exists>k\<le>n. (\<forall>i<k. \<not> P i) \<and> P k"
 proof (cases n)
   case (Suc m)
   with assms show ?thesis
     by (blast intro: Least_le LeastI_ex dest: not_less_Least)
 qed (use assms in auto)
 
 lemma ex_least_nat_less:
   fixes P :: "nat \<Rightarrow> bool"
   assumes "P n" "\<not> P 0" 
   shows "\<exists>k<n. (\<forall>i\<le>k. \<not> P i) \<and> P (Suc k)"
 proof (cases n)
   case (Suc m)
   then obtain k where k: "k \<le> n" "\<forall>i<k. \<not> P i" "P k"
     using ex_least_nat_le [OF assms] by blast
   show ?thesis 
     by (cases k) (use assms k less_eq_Suc_le in auto)
 qed (use assms in auto)
 
 
 lemma nat_less_induct:
   fixes P :: "nat \<Rightarrow> bool"
   assumes "\<And>n. \<forall>m. m < n \<longrightarrow> P m \<Longrightarrow> P n"
   shows "P n"
   using assms less_induct by blast
 
 lemma measure_induct_rule [case_names less]:
   fixes f :: "'a \<Rightarrow> 'b::wellorder"
   assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
   shows "P a"
   by (induct m \<equiv> "f a" arbitrary: a rule: less_induct) (auto intro: step)
 
 text \<open>old style induction rules:\<close>
 lemma measure_induct:
   fixes f :: "'a \<Rightarrow> 'b::wellorder"
   shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
   by (rule measure_induct_rule [of f P a]) iprover
 
 lemma full_nat_induct:
   assumes step: "\<And>n. (\<forall>m. Suc m \<le> n \<longrightarrow> P m) \<Longrightarrow> P n"
   shows "P n"
   by (rule less_induct) (auto intro: step simp:le_simps)
 
 text\<open>An induction rule for establishing binary relations\<close>
 lemma less_Suc_induct [consumes 1]:
   assumes less: "i < j"
     and step: "\<And>i. P i (Suc i)"
     and trans: "\<And>i j k. i < j \<Longrightarrow> j < k \<Longrightarrow> P i j \<Longrightarrow> P j k \<Longrightarrow> P i k"
   shows "P i j"
 proof -
   from less obtain k where j: "j = Suc (i + k)"
     by (auto dest: less_imp_Suc_add)
   have "P i (Suc (i + k))"
   proof (induct k)
     case 0
     show ?case by (simp add: step)
   next
     case (Suc k)
     have "0 + i < Suc k + i" by (rule add_less_mono1) simp
     then have "i < Suc (i + k)" by (simp add: add.commute)
     from trans[OF this lessI Suc step]
     show ?case by simp
   qed
   then show "P i j" by (simp add: j)
 qed
 
 text \<open>
   The method of infinite descent, frequently used in number theory.
   Provided by Roelof Oosterhuis.
   \<open>P n\<close> is true for all natural numbers if
   \<^item> case ``0'': given \<open>n = 0\<close> prove \<open>P n\<close>
   \<^item> case ``smaller'': given \<open>n > 0\<close> and \<open>\<not> P n\<close> prove there exists
     a smaller natural number \<open>m\<close> such that \<open>\<not> P m\<close>.
 \<close>
 
 lemma infinite_descent: "(\<And>n. \<not> P n \<Longrightarrow> \<exists>m<n. \<not> P m) \<Longrightarrow> P n" for P :: "nat \<Rightarrow> bool"
   \<comment> \<open>compact version without explicit base case\<close>
   by (induct n rule: less_induct) auto
 
 lemma infinite_descent0 [case_names 0 smaller]:
   fixes P :: "nat \<Rightarrow> bool"
   assumes "P 0"
     and "\<And>n. n > 0 \<Longrightarrow> \<not> P n \<Longrightarrow> \<exists>m. m < n \<and> \<not> P m"
   shows "P n"
 proof (rule infinite_descent)
   show "\<And>n. \<not> P n \<Longrightarrow> \<exists>m<n. \<not> P m"
   using assms by (case_tac "n > 0") auto
 qed
 
 text \<open>
   Infinite descent using a mapping to \<open>nat\<close>:
   \<open>P x\<close> is true for all \<open>x \<in> D\<close> if there exists a \<open>V \<in> D \<Rightarrow> nat\<close> and
   \<^item> case ``0'': given \<open>V x = 0\<close> prove \<open>P x\<close>
   \<^item> ``smaller'': given \<open>V x > 0\<close> and \<open>\<not> P x\<close> prove
   there exists a \<open>y \<in> D\<close> such that \<open>V y < V x\<close> and \<open>\<not> P y\<close>.
 \<close>
 corollary infinite_descent0_measure [case_names 0 smaller]:
   fixes V :: "'a \<Rightarrow> nat"
   assumes 1: "\<And>x. V x = 0 \<Longrightarrow> P x"
     and 2: "\<And>x. V x > 0 \<Longrightarrow> \<not> P x \<Longrightarrow> \<exists>y. V y < V x \<and> \<not> P y"
   shows "P x"
 proof -
   obtain n where "n = V x" by auto
   moreover have "\<And>x. V x = n \<Longrightarrow> P x"
   proof (induct n rule: infinite_descent0)
     case 0
     with 1 show "P x" by auto
   next
     case (smaller n)
     then obtain x where *: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
     with 2 obtain y where "V y < V x \<and> \<not> P y" by auto
     with * obtain m where "m = V y \<and> m < n \<and> \<not> P y" by auto
     then show ?case by auto
   qed
   ultimately show "P x" by auto
 qed
 
 text \<open>Again, without explicit base case:\<close>
 lemma infinite_descent_measure:
   fixes V :: "'a \<Rightarrow> nat"
   assumes "\<And>x. \<not> P x \<Longrightarrow> \<exists>y. V y < V x \<and> \<not> P y"
   shows "P x"
 proof -
   from assms obtain n where "n = V x" by auto
   moreover have "\<And>x. V x = n \<Longrightarrow> P x"
   proof (induct n rule: infinite_descent, auto)
     show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" if "\<not> P x" for x
       using assms and that by auto
   qed
   ultimately show "P x" by auto
 qed
 
 text \<open>A (clumsy) way of lifting \<open><\<close> monotonicity to \<open>\<le>\<close> monotonicity\<close>
 lemma less_mono_imp_le_mono:
   fixes f :: "nat \<Rightarrow> nat"
     and i j :: nat
   assumes "\<And>i j::nat. i < j \<Longrightarrow> f i < f j"
     and "i \<le> j"
   shows "f i \<le> f j"
   using assms by (auto simp add: order_le_less)
 
 
 text \<open>non-strict, in 1st argument\<close>
 lemma add_le_mono1: "i \<le> j \<Longrightarrow> i + k \<le> j + k"
   for i j k :: nat
   by (rule add_right_mono)
 
 text \<open>non-strict, in both arguments\<close>
 lemma add_le_mono: "i \<le> j \<Longrightarrow> k \<le> l \<Longrightarrow> i + k \<le> j + l"
   for i j k l :: nat
   by (rule add_mono)
 
 lemma le_add2: "n \<le> m + n"
   for m n :: nat
   by simp
 
 lemma le_add1: "n \<le> n + m"
   for m n :: nat
   by simp
 
 lemma less_add_Suc1: "i < Suc (i + m)"
   by (rule le_less_trans, rule le_add1, rule lessI)
 
 lemma less_add_Suc2: "i < Suc (m + i)"
   by (rule le_less_trans, rule le_add2, rule lessI)
 
 lemma less_iff_Suc_add: "m < n \<longleftrightarrow> (\<exists>k. n = Suc (m + k))"
   by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
 
 lemma trans_le_add1: "i \<le> j \<Longrightarrow> i \<le> j + m"
   for i j m :: nat
   by (rule le_trans, assumption, rule le_add1)
 
 lemma trans_le_add2: "i \<le> j \<Longrightarrow> i \<le> m + j"
   for i j m :: nat
   by (rule le_trans, assumption, rule le_add2)
 
 lemma trans_less_add1: "i < j \<Longrightarrow> i < j + m"
   for i j m :: nat
   by (rule less_le_trans, assumption, rule le_add1)
 
 lemma trans_less_add2: "i < j \<Longrightarrow> i < m + j"
   for i j m :: nat
   by (rule less_le_trans, assumption, rule le_add2)
 
 lemma add_lessD1: "i + j < k \<Longrightarrow> i < k"
   for i j k :: nat
   by (rule le_less_trans [of _ "i+j"]) (simp_all add: le_add1)
 
 lemma not_add_less1 [iff]: "\<not> i + j < i"
   for i j :: nat
   by simp
 
 lemma not_add_less2 [iff]: "\<not> j + i < i"
   for i j :: nat
   by simp
 
 lemma add_leD1: "m + k \<le> n \<Longrightarrow> m \<le> n"
   for k m n :: nat
   by (rule order_trans [of _ "m + k"]) (simp_all add: le_add1)
 
 lemma add_leD2: "m + k \<le> n \<Longrightarrow> k \<le> n"
   for k m n :: nat
   by (force simp add: add.commute dest: add_leD1)
 
 lemma add_leE: "m + k \<le> n \<Longrightarrow> (m \<le> n \<Longrightarrow> k \<le> n \<Longrightarrow> R) \<Longrightarrow> R"
   for k m n :: nat
   by (blast dest: add_leD1 add_leD2)
 
 text \<open>needs \<open>\<And>k\<close> for \<open>ac_simps\<close> to work\<close>
 lemma less_add_eq_less: "\<And>k. k < l \<Longrightarrow> m + l = k + n \<Longrightarrow> m < n"
   for l m n :: nat
   by (force simp del: add_Suc_right simp add: less_iff_Suc_add add_Suc_right [symmetric] ac_simps)
 
 
 subsubsection \<open>More results about difference\<close>
 
 lemma Suc_diff_le: "n \<le> m \<Longrightarrow> Suc m - n = Suc (m - n)"
   by (induct m n rule: diff_induct) simp_all
 
 lemma diff_less_Suc: "m - n < Suc m"
   by (induct m n rule: diff_induct) (auto simp: less_Suc_eq)
 
 lemma diff_le_self [simp]: "m - n \<le> m"
   for m n :: nat
   by (induct m n rule: diff_induct) (simp_all add: le_SucI)
 
 lemma less_imp_diff_less: "j < k \<Longrightarrow> j - n < k"
   for j k n :: nat
   by (rule le_less_trans, rule diff_le_self)
 
 lemma diff_Suc_less [simp]: "0 < n \<Longrightarrow> n - Suc i < n"
   by (cases n) (auto simp add: le_simps)
 
 lemma diff_add_assoc: "k \<le> j \<Longrightarrow> (i + j) - k = i + (j - k)"
   for i j k :: nat
   by (fact ordered_cancel_comm_monoid_diff_class.diff_add_assoc) 
 
 lemma add_diff_assoc [simp]: "k \<le> j \<Longrightarrow> i + (j - k) = i + j - k"
   for i j k :: nat
   by (fact ordered_cancel_comm_monoid_diff_class.add_diff_assoc)
 
 lemma diff_add_assoc2: "k \<le> j \<Longrightarrow> (j + i) - k = (j - k) + i"
   for i j k :: nat
   by (fact ordered_cancel_comm_monoid_diff_class.diff_add_assoc2)
 
 lemma add_diff_assoc2 [simp]: "k \<le> j \<Longrightarrow> j - k + i = j + i - k"
   for i j k :: nat
   by (fact ordered_cancel_comm_monoid_diff_class.add_diff_assoc2)
 
 lemma le_imp_diff_is_add: "i \<le> j \<Longrightarrow> (j - i = k) = (j = k + i)"
   for i j k :: nat
   by auto
 
 lemma diff_is_0_eq [simp]: "m - n = 0 \<longleftrightarrow> m \<le> n"
   for m n :: nat
   by (induct m n rule: diff_induct) simp_all
 
 lemma diff_is_0_eq' [simp]: "m \<le> n \<Longrightarrow> m - n = 0"
   for m n :: nat
   by (rule iffD2, rule diff_is_0_eq)
 
 lemma zero_less_diff [simp]: "0 < n - m \<longleftrightarrow> m < n"
   for m n :: nat
   by (induct m n rule: diff_induct) simp_all
 
 lemma less_imp_add_positive:
   assumes "i < j"
   shows "\<exists>k::nat. 0 < k \<and> i + k = j"
 proof
   from assms show "0 < j - i \<and> i + (j - i) = j"
     by (simp add: order_less_imp_le)
 qed
 
 text \<open>a nice rewrite for bounded subtraction\<close>
 lemma nat_minus_add_max: "n - m + m = max n m"
   for m n :: nat
   by (simp add: max_def not_le order_less_imp_le)
 
 lemma nat_diff_split: "P (a - b) \<longleftrightarrow> (a < b \<longrightarrow> P 0) \<and> (\<forall>d. a = b + d \<longrightarrow> P d)"
   for a b :: nat
   \<comment> \<open>elimination of \<open>-\<close> on \<open>nat\<close>\<close>
   by (cases "a < b") (auto simp add: not_less le_less dest!: add_eq_self_zero [OF sym])
 
 lemma nat_diff_split_asm: "P (a - b) \<longleftrightarrow> \<not> (a < b \<and> \<not> P 0 \<or> (\<exists>d. a = b + d \<and> \<not> P d))"
   for a b :: nat
   \<comment> \<open>elimination of \<open>-\<close> on \<open>nat\<close> in assumptions\<close>
   by (auto split: nat_diff_split)
 
 lemma Suc_pred': "0 < n \<Longrightarrow> n = Suc(n - 1)"
   by simp
 
 lemma add_eq_if: "m + n = (if m = 0 then n else Suc ((m - 1) + n))"
   unfolding One_nat_def by (cases m) simp_all
 
 lemma mult_eq_if: "m * n = (if m = 0 then 0 else n + ((m - 1) * n))"
   for m n :: nat
   by (cases m) simp_all
 
 lemma Suc_diff_eq_diff_pred: "0 < n \<Longrightarrow> Suc m - n = m - (n - 1)"
   by (cases n) simp_all
 
 lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
   by (cases m) simp_all
 
 lemma Let_Suc [simp]: "Let (Suc n) f \<equiv> f (Suc n)"
   by (fact Let_def)
 
 
 subsubsection \<open>Monotonicity of multiplication\<close>
 
 lemma mult_le_mono1: "i \<le> j \<Longrightarrow> i * k \<le> j * k"
   for i j k :: nat
   by (simp add: mult_right_mono)
 
 lemma mult_le_mono2: "i \<le> j \<Longrightarrow> k * i \<le> k * j"
   for i j k :: nat
   by (simp add: mult_left_mono)
 
 text \<open>\<open>\<le>\<close> monotonicity, BOTH arguments\<close>
 lemma mult_le_mono: "i \<le> j \<Longrightarrow> k \<le> l \<Longrightarrow> i * k \<le> j * l"
   for i j k l :: nat
   by (simp add: mult_mono)
 
 lemma mult_less_mono1: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> i * k < j * k"
   for i j k :: nat
   by (simp add: mult_strict_right_mono)
 
 text \<open>Differs from the standard \<open>zero_less_mult_iff\<close> in that there are no negative numbers.\<close>
 lemma nat_0_less_mult_iff [simp]: "0 < m * n \<longleftrightarrow> 0 < m \<and> 0 < n"
   for m n :: nat
 proof (induct m)
   case 0
   then show ?case by simp
 next
   case (Suc m)
   then show ?case by (cases n) simp_all
 qed
 
 lemma one_le_mult_iff [simp]: "Suc 0 \<le> m * n \<longleftrightarrow> Suc 0 \<le> m \<and> Suc 0 \<le> n"
 proof (induct m)
   case 0
   then show ?case by simp
 next
   case (Suc m)
   then show ?case by (cases n) simp_all
 qed
 
 lemma mult_less_cancel2 [simp]: "m * k < n * k \<longleftrightarrow> 0 < k \<and> m < n"
   for k m n :: nat
 proof (intro iffI conjI)
   assume m: "m * k < n * k"
   then show "0 < k"
     by (cases k) auto
   show "m < n"
   proof (cases k)
     case 0
     then show ?thesis
       using m by auto
   next
     case (Suc k')
     then show ?thesis
       using m
       by (simp flip: linorder_not_le) (blast intro: add_mono mult_le_mono1)
   qed
 next
   assume "0 < k \<and> m < n"
   then show "m * k < n * k"
     by (blast intro: mult_less_mono1)
 qed
 
 lemma mult_less_cancel1 [simp]: "k * m < k * n \<longleftrightarrow> 0 < k \<and> m < n"
   for k m n :: nat
   by (simp add: mult.commute [of k])
 
 lemma mult_le_cancel1 [simp]: "k * m \<le> k * n \<longleftrightarrow> (0 < k \<longrightarrow> m \<le> n)"
   for k m n :: nat
   by (simp add: linorder_not_less [symmetric], auto)
 
 lemma mult_le_cancel2 [simp]: "m * k \<le> n * k \<longleftrightarrow> (0 < k \<longrightarrow> m \<le> n)"
   for k m n :: nat
   by (simp add: linorder_not_less [symmetric], auto)
 
 lemma Suc_mult_less_cancel1: "Suc k * m < Suc k * n \<longleftrightarrow> m < n"
   by (subst mult_less_cancel1) simp
 
 lemma Suc_mult_le_cancel1: "Suc k * m \<le> Suc k * n \<longleftrightarrow> m \<le> n"
   by (subst mult_le_cancel1) simp
 
 lemma le_square: "m \<le> m * m"
   for m :: nat
   by (cases m) (auto intro: le_add1)
 
 lemma le_cube: "m \<le> m * (m * m)"
   for m :: nat
   by (cases m) (auto intro: le_add1)
 
 text \<open>Lemma for \<open>gcd\<close>\<close>
 lemma mult_eq_self_implies_10: 
   fixes m n :: nat
   assumes "m = m * n" shows "n = 1 \<or> m = 0"
 proof (rule disjCI)
   assume "m \<noteq> 0"
   show "n = 1"
   proof (cases n "1::nat" rule: linorder_cases)
     case greater
     show ?thesis
       using assms mult_less_mono2 [OF greater, of m] \<open>m \<noteq> 0\<close> by auto
   qed (use assms \<open>m \<noteq> 0\<close> in auto)
 qed
 
 lemma mono_times_nat:
   fixes n :: nat
   assumes "n > 0"
   shows "mono (times n)"
 proof
   fix m q :: nat
   assume "m \<le> q"
   with assms show "n * m \<le> n * q" by simp
 qed
 
 text \<open>The lattice order on \<^typ>\<open>nat\<close>.\<close>
 
 instantiation nat :: distrib_lattice
 begin
 
 definition "(inf :: nat \<Rightarrow> nat \<Rightarrow> nat) = min"
 
 definition "(sup :: nat \<Rightarrow> nat \<Rightarrow> nat) = max"
 
 instance
   by intro_classes
     (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def
       intro: order_less_imp_le antisym elim!: order_trans order_less_trans)
 
 end
 
 
 subsection \<open>Natural operation of natural numbers on functions\<close>
 
 text \<open>
   We use the same logical constant for the power operations on
   functions and relations, in order to share the same syntax.
 \<close>
 
 consts compow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
 
 abbreviation compower :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^^" 80)
   where "f ^^ n \<equiv> compow n f"
 
 notation (latex output)
   compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
 
 text \<open>\<open>f ^^ n = f \<circ> \<dots> \<circ> f\<close>, the \<open>n\<close>-fold composition of \<open>f\<close>\<close>
 
 overloading
   funpow \<equiv> "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
 begin
 
 primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
   where
     "funpow 0 f = id"
   | "funpow (Suc n) f = f \<circ> funpow n f"
 
 end
 
 lemma funpow_0 [simp]: "(f ^^ 0) x = x"
   by simp
 
 lemma funpow_Suc_right: "f ^^ Suc n = f ^^ n \<circ> f"
 proof (induct n)
   case 0
   then show ?case by simp
 next
   fix n
   assume "f ^^ Suc n = f ^^ n \<circ> f"
   then show "f ^^ Suc (Suc n) = f ^^ Suc n \<circ> f"
     by (simp add: o_assoc)
 qed
 
 lemmas funpow_simps_right = funpow.simps(1) funpow_Suc_right
 
 text \<open>For code generation.\<close>
 
 definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
   where funpow_code_def [code_abbrev]: "funpow = compow"
 
 lemma [code]:
   "funpow (Suc n) f = f \<circ> funpow n f"
   "funpow 0 f = id"
   by (simp_all add: funpow_code_def)
 
 hide_const (open) funpow
 
 lemma funpow_add: "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
   by (induct m) simp_all
 
 lemma funpow_mult: "(f ^^ m) ^^ n = f ^^ (m * n)"
   for f :: "'a \<Rightarrow> 'a"
   by (induct n) (simp_all add: funpow_add)
 
 lemma funpow_swap1: "f ((f ^^ n) x) = (f ^^ n) (f x)"
 proof -
   have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp
   also have "\<dots>  = (f ^^ n \<circ> f ^^ 1) x" by (simp only: funpow_add)
   also have "\<dots> = (f ^^ n) (f x)" by simp
   finally show ?thesis .
 qed
 
 lemma comp_funpow: "comp f ^^ n = comp (f ^^ n)"
   for f :: "'a \<Rightarrow> 'a"
   by (induct n) simp_all
 
 lemma Suc_funpow[simp]: "Suc ^^ n = ((+) n)"
   by (induct n) simp_all
 
 lemma id_funpow[simp]: "id ^^ n = id"
   by (induct n) simp_all
 
 lemma funpow_mono: "mono f \<Longrightarrow> A \<le> B \<Longrightarrow> (f ^^ n) A \<le> (f ^^ n) B"
   for f :: "'a \<Rightarrow> ('a::order)"
   by (induct n arbitrary: A B)
      (auto simp del: funpow.simps(2) simp add: funpow_Suc_right mono_def)
 
 lemma funpow_mono2:
   assumes "mono f"
     and "i \<le> j"
     and "x \<le> y"
     and "x \<le> f x"
   shows "(f ^^ i) x \<le> (f ^^ j) y"
   using assms(2,3)
 proof (induct j arbitrary: y)
   case 0
   then show ?case by simp
 next
   case (Suc j)
   show ?case
   proof(cases "i = Suc j")
     case True
     with assms(1) Suc show ?thesis
       by (simp del: funpow.simps add: funpow_simps_right monoD funpow_mono)
   next
     case False
     with assms(1,4) Suc show ?thesis
       by (simp del: funpow.simps add: funpow_simps_right le_eq_less_or_eq less_Suc_eq_le)
         (simp add: Suc.hyps monoD order_subst1)
   qed
 qed
 
 lemma inj_fn[simp]:
   fixes f::"'a \<Rightarrow> 'a"
   assumes "inj f"
   shows "inj (f^^n)"
 proof (induction n)
   case Suc thus ?case using inj_compose[OF assms Suc.IH] by (simp del: comp_apply)
 qed simp
 
 lemma surj_fn[simp]:
   fixes f::"'a \<Rightarrow> 'a"
   assumes "surj f"
   shows "surj (f^^n)"
 proof (induction n)
   case Suc thus ?case by (simp add: comp_surj[OF Suc.IH assms] del: comp_apply)
 qed simp
 
 lemma bij_fn[simp]:
   fixes f::"'a \<Rightarrow> 'a"
   assumes "bij f"
   shows "bij (f^^n)"
 by (rule bijI[OF inj_fn[OF bij_is_inj[OF assms]] surj_fn[OF bij_is_surj[OF assms]]])
 
 
 subsection \<open>Kleene iteration\<close>
 
 lemma Kleene_iter_lpfp:
   fixes f :: "'a::order_bot \<Rightarrow> 'a"
   assumes "mono f"
     and "f p \<le> p"
   shows "(f ^^ k) bot \<le> p"
 proof (induct k)
   case 0
   show ?case by simp
 next
   case Suc
   show ?case
     using monoD[OF assms(1) Suc] assms(2) by simp
 qed
 
 lemma lfp_Kleene_iter:
   assumes "mono f"
     and "(f ^^ Suc k) bot = (f ^^ k) bot"
   shows "lfp f = (f ^^ k) bot"
 proof (rule antisym)
   show "lfp f \<le> (f ^^ k) bot"
   proof (rule lfp_lowerbound)
     show "f ((f ^^ k) bot) \<le> (f ^^ k) bot"
       using assms(2) by simp
   qed
   show "(f ^^ k) bot \<le> lfp f"
     using Kleene_iter_lpfp[OF assms(1)] lfp_unfold[OF assms(1)] by simp
 qed
 
 lemma mono_pow: "mono f \<Longrightarrow> mono (f ^^ n)"
   for f :: "'a \<Rightarrow> 'a::complete_lattice"
   by (induct n) (auto simp: mono_def)
 
 lemma lfp_funpow:
   assumes f: "mono f"
   shows "lfp (f ^^ Suc n) = lfp f"
 proof (rule antisym)
   show "lfp f \<le> lfp (f ^^ Suc n)"
   proof (rule lfp_lowerbound)
     have "f (lfp (f ^^ Suc n)) = lfp (\<lambda>x. f ((f ^^ n) x))"
       unfolding funpow_Suc_right by (simp add: lfp_rolling f mono_pow comp_def)
     then show "f (lfp (f ^^ Suc n)) \<le> lfp (f ^^ Suc n)"
       by (simp add: comp_def)
   qed
   have "(f ^^ n) (lfp f) = lfp f" for n
     by (induct n) (auto intro: f lfp_fixpoint)
   then show "lfp (f ^^ Suc n) \<le> lfp f"
     by (intro lfp_lowerbound) (simp del: funpow.simps)
 qed
 
 lemma gfp_funpow:
   assumes f: "mono f"
   shows "gfp (f ^^ Suc n) = gfp f"
 proof (rule antisym)
   show "gfp f \<ge> gfp (f ^^ Suc n)"
   proof (rule gfp_upperbound)
     have "f (gfp (f ^^ Suc n)) = gfp (\<lambda>x. f ((f ^^ n) x))"
       unfolding funpow_Suc_right by (simp add: gfp_rolling f mono_pow comp_def)
     then show "f (gfp (f ^^ Suc n)) \<ge> gfp (f ^^ Suc n)"
       by (simp add: comp_def)
   qed
   have "(f ^^ n) (gfp f) = gfp f" for n
     by (induct n) (auto intro: f gfp_fixpoint)
   then show "gfp (f ^^ Suc n) \<ge> gfp f"
     by (intro gfp_upperbound) (simp del: funpow.simps)
 qed
 
 lemma Kleene_iter_gpfp:
   fixes f :: "'a::order_top \<Rightarrow> 'a"
   assumes "mono f"
     and "p \<le> f p"
   shows "p \<le> (f ^^ k) top"
 proof (induct k)
   case 0
   show ?case by simp
 next
   case Suc
   show ?case
     using monoD[OF assms(1) Suc] assms(2) by simp
 qed
 
 lemma gfp_Kleene_iter:
   assumes "mono f"
     and "(f ^^ Suc k) top = (f ^^ k) top"
   shows "gfp f = (f ^^ k) top"
     (is "?lhs = ?rhs")
 proof (rule antisym)
   have "?rhs \<le> f ?rhs"
     using assms(2) by simp
   then show "?rhs \<le> ?lhs"
     by (rule gfp_upperbound)
   show "?lhs \<le> ?rhs"
     using Kleene_iter_gpfp[OF assms(1)] gfp_unfold[OF assms(1)] by simp
 qed
 
 
 subsection \<open>Embedding of the naturals into any \<open>semiring_1\<close>: \<^term>\<open>of_nat\<close>\<close>
 
 context semiring_1
 begin
 
 definition of_nat :: "nat \<Rightarrow> 'a"
   where "of_nat n = (plus 1 ^^ n) 0"
 
 lemma of_nat_simps [simp]:
   shows of_nat_0: "of_nat 0 = 0"
     and of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"
   by (simp_all add: of_nat_def)
 
 lemma of_nat_1 [simp]: "of_nat 1 = 1"
   by (simp add: of_nat_def)
 
 lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"
   by (induct m) (simp_all add: ac_simps)
 
 lemma of_nat_mult [simp]: "of_nat (m * n) = of_nat m * of_nat n"
   by (induct m) (simp_all add: ac_simps distrib_right)
 
 lemma mult_of_nat_commute: "of_nat x * y = y * of_nat x"
   by (induct x) (simp_all add: algebra_simps)
 
 primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"
   where
     "of_nat_aux inc 0 i = i"
   | "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" \<comment> \<open>tail recursive\<close>
 
 lemma of_nat_code: "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"
 proof (induct n)
   case 0
   then show ?case by simp
 next
   case (Suc n)
   have "\<And>i. of_nat_aux (\<lambda>i. i + 1) n (i + 1) = of_nat_aux (\<lambda>i. i + 1) n i + 1"
     by (induct n) simp_all
   from this [of 0] have "of_nat_aux (\<lambda>i. i + 1) n 1 = of_nat_aux (\<lambda>i. i + 1) n 0 + 1"
     by simp
   with Suc show ?case
     by (simp add: add.commute)
 qed
 
 lemma of_nat_of_bool [simp]:
   "of_nat (of_bool P) = of_bool P"
   by auto
 
 end
 
 declare of_nat_code [code]
 
 context semiring_1_cancel
 begin
 
 lemma of_nat_diff:
   \<open>of_nat (m - n) = of_nat m - of_nat n\<close> if \<open>n \<le> m\<close>
 proof -
   from that obtain q where \<open>m = n + q\<close>
     by (blast dest: le_Suc_ex)
   then show ?thesis
     by simp
 qed
 
 end
 
 text \<open>Class for unital semirings with characteristic zero.
  Includes non-ordered rings like the complex numbers.\<close>
 
 class semiring_char_0 = semiring_1 +
   assumes inj_of_nat: "inj of_nat"
 begin
 
 lemma of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"
   by (auto intro: inj_of_nat injD)
 
 text \<open>Special cases where either operand is zero\<close>
 
 lemma of_nat_0_eq_iff [simp]: "0 = of_nat n \<longleftrightarrow> 0 = n"
   by (fact of_nat_eq_iff [of 0 n, unfolded of_nat_0])
 
 lemma of_nat_eq_0_iff [simp]: "of_nat m = 0 \<longleftrightarrow> m = 0"
   by (fact of_nat_eq_iff [of m 0, unfolded of_nat_0])
 
 lemma of_nat_1_eq_iff [simp]: "1 = of_nat n \<longleftrightarrow> n=1"
   using of_nat_eq_iff by fastforce
 
 lemma of_nat_eq_1_iff [simp]: "of_nat n = 1 \<longleftrightarrow> n=1"
   using of_nat_eq_iff by fastforce
 
 lemma of_nat_neq_0 [simp]: "of_nat (Suc n) \<noteq> 0"
   unfolding of_nat_eq_0_iff by simp
 
 lemma of_nat_0_neq [simp]: "0 \<noteq> of_nat (Suc n)"
   unfolding of_nat_0_eq_iff by simp
 
 end
 
 class ring_char_0 = ring_1 + semiring_char_0
 
 context linordered_nonzero_semiring
 begin
 
 lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"
   by (induct n) simp_all
 
 lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"
   by (simp add: not_less)
 
 lemma of_nat_mono[simp]: "i \<le> j \<Longrightarrow> of_nat i \<le> of_nat j"
   by (auto simp: le_iff_add intro!: add_increasing2)
 
 lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"
 proof(induct m n rule: diff_induct)
   case (1 m) then show ?case
     by auto
 next
   case (2 n) then show ?case
     by (simp add: add_pos_nonneg)
 next
   case (3 m n)
   then show ?case
     by (auto simp: add_commute [of 1] add_mono1 not_less add_right_mono leD)
 qed
 
 lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"
   by (simp add: not_less [symmetric] linorder_not_less [symmetric])
 
 lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"
   by simp
 
 lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"
   by simp
 
 text \<open>Every \<open>linordered_nonzero_semiring\<close> has characteristic zero.\<close>
 
 subclass semiring_char_0
   by standard (auto intro!: injI simp add: eq_iff)
 
 text \<open>Special cases where either operand is zero\<close>
 
 lemma of_nat_le_0_iff [simp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"
   by (rule of_nat_le_iff [of _ 0, simplified])
 
 lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"
   by (rule of_nat_less_iff [of 0, simplified])
 
 end
 
 context linordered_nonzero_semiring
 begin
 
 lemma of_nat_max: "of_nat (max x y) = max (of_nat x) (of_nat y)"
   by (auto simp: max_def ord_class.max_def)
 
 lemma of_nat_min: "of_nat (min x y) = min (of_nat x) (of_nat y)"
   by (auto simp: min_def ord_class.min_def)
 
 end
 
 context linordered_semidom
 begin
 
 subclass linordered_nonzero_semiring ..
 
 subclass semiring_char_0 ..
 
 end
 
 context linordered_idom
 begin
 
 lemma abs_of_nat [simp]:
   "\<bar>of_nat n\<bar> = of_nat n"
   by (simp add: abs_if)
 
 lemma sgn_of_nat [simp]:
   "sgn (of_nat n) = of_bool (n > 0)"
   by simp
 
 end
 
 lemma of_nat_id [simp]: "of_nat n = n"
   by (induct n) simp_all
 
 lemma of_nat_eq_id [simp]: "of_nat = id"
   by (auto simp add: fun_eq_iff)
 
 
 subsection \<open>The set of natural numbers\<close>
 
 context semiring_1
 begin
 
 definition Nats :: "'a set"  ("\<nat>")
   where "\<nat> = range of_nat"
 
 lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"
   by (simp add: Nats_def)
 
 lemma Nats_0 [simp]: "0 \<in> \<nat>"
   using of_nat_0 [symmetric] unfolding Nats_def
   by (rule range_eqI)
 
 lemma Nats_1 [simp]: "1 \<in> \<nat>"
   using of_nat_1 [symmetric] unfolding Nats_def
   by (rule range_eqI)
 
 lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"
   unfolding Nats_def using of_nat_add [symmetric]
   by (blast intro: range_eqI)
 
 lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"
   unfolding Nats_def using of_nat_mult [symmetric]
   by (blast intro: range_eqI)
 
 lemma Nats_cases [cases set: Nats]:
   assumes "x \<in> \<nat>"
   obtains (of_nat) n where "x = of_nat n"
   unfolding Nats_def
 proof -
   from \<open>x \<in> \<nat>\<close> have "x \<in> range of_nat" unfolding Nats_def .
   then obtain n where "x = of_nat n" ..
   then show thesis ..
 qed
 
 lemma Nats_induct [case_names of_nat, induct set: Nats]: "x \<in> \<nat> \<Longrightarrow> (\<And>n. P (of_nat n)) \<Longrightarrow> P x"
   by (rule Nats_cases) auto
 
 end
 
 lemma Nats_diff [simp]:
   fixes a:: "'a::linordered_idom"
   assumes "a \<in> \<nat>" "b \<in> \<nat>" "b \<le> a" shows "a - b \<in> \<nat>"
 proof -
   obtain i where i: "a = of_nat i"
     using Nats_cases assms by blast
   obtain j where j: "b = of_nat j"
     using Nats_cases assms by blast
   have "j \<le> i"
     using \<open>b \<le> a\<close> i j of_nat_le_iff by blast
   then have *: "of_nat i - of_nat j = (of_nat (i-j) :: 'a)"
     by (simp add: of_nat_diff)
   then show ?thesis
     by (simp add: * i j)
 qed
 
 
 subsection \<open>Further arithmetic facts concerning the natural numbers\<close>
 
 lemma subst_equals:
   assumes "t = s" and "u = t"
   shows "u = s"
   using assms(2,1) by (rule trans)
 
 locale nat_arith
 begin
 
 lemma add1: "(A::'a::comm_monoid_add) \<equiv> k + a \<Longrightarrow> A + b \<equiv> k + (a + b)"
   by (simp only: ac_simps)
 
 lemma add2: "(B::'a::comm_monoid_add) \<equiv> k + b \<Longrightarrow> a + B \<equiv> k + (a + b)"
   by (simp only: ac_simps)
 
 lemma suc1: "A == k + a \<Longrightarrow> Suc A \<equiv> k + Suc a"
   by (simp only: add_Suc_right)
 
 lemma rule0: "(a::'a::comm_monoid_add) \<equiv> a + 0"
   by (simp only: add_0_right)
 
 end
 
 ML_file \<open>Tools/nat_arith.ML\<close>
 
 simproc_setup nateq_cancel_sums
   ("(l::nat) + m = n" | "(l::nat) = m + n" | "Suc m = n" | "m = Suc n") =
   \<open>fn phi => try o Nat_Arith.cancel_eq_conv\<close>
 
 simproc_setup natless_cancel_sums
   ("(l::nat) + m < n" | "(l::nat) < m + n" | "Suc m < n" | "m < Suc n") =
   \<open>fn phi => try o Nat_Arith.cancel_less_conv\<close>
 
 simproc_setup natle_cancel_sums
   ("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" | "Suc m \<le> n" | "m \<le> Suc n") =
   \<open>fn phi => try o Nat_Arith.cancel_le_conv\<close>
 
 simproc_setup natdiff_cancel_sums
   ("(l::nat) + m - n" | "(l::nat) - (m + n)" | "Suc m - n" | "m - Suc n") =
   \<open>fn phi => try o Nat_Arith.cancel_diff_conv\<close>
 
 context order
 begin
 
 lemma lift_Suc_mono_le:
   assumes mono: "\<And>n. f n \<le> f (Suc n)"
     and "n \<le> n'"
   shows "f n \<le> f n'"
 proof (cases "n < n'")
   case True
   then show ?thesis
     by (induct n n' rule: less_Suc_induct) (auto intro: mono)
 next
   case False
   with \<open>n \<le> n'\<close> show ?thesis by auto
 qed
 
 lemma lift_Suc_antimono_le:
   assumes mono: "\<And>n. f n \<ge> f (Suc n)"
     and "n \<le> n'"
   shows "f n \<ge> f n'"
 proof (cases "n < n'")
   case True
   then show ?thesis
     by (induct n n' rule: less_Suc_induct) (auto intro: mono)
 next
   case False
   with \<open>n \<le> n'\<close> show ?thesis by auto
 qed
 
 lemma lift_Suc_mono_less:
   assumes mono: "\<And>n. f n < f (Suc n)"
     and "n < n'"
   shows "f n < f n'"
   using \<open>n < n'\<close> by (induct n n' rule: less_Suc_induct) (auto intro: mono)
 
 lemma lift_Suc_mono_less_iff: "(\<And>n. f n < f (Suc n)) \<Longrightarrow> f n < f m \<longleftrightarrow> n < m"
   by (blast intro: less_asym' lift_Suc_mono_less [of f]
     dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq [THEN iffD1])
 
 end
 
 lemma mono_iff_le_Suc: "mono f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
   unfolding mono_def by (auto intro: lift_Suc_mono_le [of f])
 
 lemma antimono_iff_le_Suc: "antimono f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
   unfolding antimono_def by (auto intro: lift_Suc_antimono_le [of f])
 
 lemma mono_nat_linear_lb:
   fixes f :: "nat \<Rightarrow> nat"
   assumes "\<And>m n. m < n \<Longrightarrow> f m < f n"
   shows "f m + k \<le> f (m + k)"
 proof (induct k)
   case 0
   then show ?case by simp
 next
   case (Suc k)
   then have "Suc (f m + k) \<le> Suc (f (m + k))" by simp
   also from assms [of "m + k" "Suc (m + k)"] have "Suc (f (m + k)) \<le> f (Suc (m + k))"
     by (simp add: Suc_le_eq)
   finally show ?case by simp
 qed
 
 
 text \<open>Subtraction laws, mostly by Clemens Ballarin\<close>
 
 lemma diff_less_mono:
   fixes a b c :: nat
   assumes "a < b" and "c \<le> a"
   shows "a - c < b - c"
 proof -
   from assms obtain d e where "b = c + (d + e)" and "a = c + e" and "d > 0"
     by (auto dest!: le_Suc_ex less_imp_Suc_add simp add: ac_simps)
   then show ?thesis by simp
 qed
 
 lemma less_diff_conv: "i < j - k \<longleftrightarrow> i + k < j"
   for i j k :: nat
   by (cases "k \<le> j") (auto simp add: not_le dest: less_imp_Suc_add le_Suc_ex)
 
 lemma less_diff_conv2: "k \<le> j \<Longrightarrow> j - k < i \<longleftrightarrow> j < i + k"
   for j k i :: nat
   by (auto dest: le_Suc_ex)
 
 lemma le_diff_conv: "j - k \<le> i \<longleftrightarrow> j \<le> i + k"
   for j k i :: nat
   by (cases "k \<le> j") (auto simp add: not_le dest!: less_imp_Suc_add le_Suc_ex)
 
 lemma diff_diff_cancel [simp]: "i \<le> n \<Longrightarrow> n - (n - i) = i"
   for i n :: nat
   by (auto dest: le_Suc_ex)
 
 lemma diff_less [simp]: "0 < n \<Longrightarrow> 0 < m \<Longrightarrow> m - n < m"
   for i n :: nat
   by (auto dest: less_imp_Suc_add)
 
 text \<open>Simplification of relational expressions involving subtraction\<close>
 
 lemma diff_diff_eq: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k - (n - k) = m - n"
   for m n k :: nat
   by (auto dest!: le_Suc_ex)
 
 hide_fact (open) diff_diff_eq
 
 lemma eq_diff_iff: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k = n - k \<longleftrightarrow> m = n"
   for m n k :: nat
   by (auto dest: le_Suc_ex)
 
 lemma less_diff_iff: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k < n - k \<longleftrightarrow> m < n"
   for m n k :: nat
   by (auto dest!: le_Suc_ex)
 
 lemma le_diff_iff: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k \<le> n - k \<longleftrightarrow> m \<le> n"
   for m n k :: nat
   by (auto dest!: le_Suc_ex)
 
 lemma le_diff_iff': "a \<le> c \<Longrightarrow> b \<le> c \<Longrightarrow> c - a \<le> c - b \<longleftrightarrow> b \<le> a"
   for a b c :: nat
   by (force dest: le_Suc_ex)
 
 
 text \<open>(Anti)Monotonicity of subtraction -- by Stephan Merz\<close>
 
 lemma diff_le_mono: "m \<le> n \<Longrightarrow> m - l \<le> n - l"
   for m n l :: nat
   by (auto dest: less_imp_le less_imp_Suc_add split: nat_diff_split)
 
 lemma diff_le_mono2: "m \<le> n \<Longrightarrow> l - n \<le> l - m"
   for m n l :: nat
   by (auto dest: less_imp_le le_Suc_ex less_imp_Suc_add less_le_trans split: nat_diff_split)
 
 lemma diff_less_mono2: "m < n \<Longrightarrow> m < l \<Longrightarrow> l - n < l - m"
   for m n l :: nat
   by (auto dest: less_imp_Suc_add split: nat_diff_split)
 
 lemma diffs0_imp_equal: "m - n = 0 \<Longrightarrow> n - m = 0 \<Longrightarrow> m = n"
   for m n :: nat
   by (simp split: nat_diff_split)
 
 lemma min_diff: "min (m - i) (n - i) = min m n - i"
   for m n i :: nat
   by (cases m n rule: le_cases)
     (auto simp add: not_le min.absorb1 min.absorb2 min.absorb_iff1 [symmetric] diff_le_mono)
 
 lemma inj_on_diff_nat:
   fixes k :: nat
   assumes "\<And>n. n \<in> N \<Longrightarrow> k \<le> n"
   shows "inj_on (\<lambda>n. n - k) N"
 proof (rule inj_onI)
   fix x y
   assume a: "x \<in> N" "y \<in> N" "x - k = y - k"
   with assms have "x - k + k = y - k + k" by auto
   with a assms show "x = y" by (auto simp add: eq_diff_iff)
 qed
 
 text \<open>Rewriting to pull differences out\<close>
 
 lemma diff_diff_right [simp]: "k \<le> j \<Longrightarrow> i - (j - k) = i + k - j"
   for i j k :: nat
   by (fact diff_diff_right)
 
 lemma diff_Suc_diff_eq1 [simp]:
   assumes "k \<le> j"
   shows "i - Suc (j - k) = i + k - Suc j"
 proof -
   from assms have *: "Suc (j - k) = Suc j - k"
     by (simp add: Suc_diff_le)
   from assms have "k \<le> Suc j"
     by (rule order_trans) simp
   with diff_diff_right [of k "Suc j" i] * show ?thesis
     by simp
 qed
 
 lemma diff_Suc_diff_eq2 [simp]:
   assumes "k \<le> j"
   shows "Suc (j - k) - i = Suc j - (k + i)"
 proof -
   from assms obtain n where "j = k + n"
     by (auto dest: le_Suc_ex)
   moreover have "Suc n - i = (k + Suc n) - (k + i)"
     using add_diff_cancel_left [of k "Suc n" i] by simp
   ultimately show ?thesis by simp
 qed
 
 lemma Suc_diff_Suc:
   assumes "n < m"
   shows "Suc (m - Suc n) = m - n"
 proof -
   from assms obtain q where "m = n + Suc q"
     by (auto dest: less_imp_Suc_add)
   moreover define r where "r = Suc q"
   ultimately have "Suc (m - Suc n) = r" and "m = n + r"
     by simp_all
   then show ?thesis by simp
 qed
 
 lemma one_less_mult: "Suc 0 < n \<Longrightarrow> Suc 0 < m \<Longrightarrow> Suc 0 < m * n"
   using less_1_mult [of n m] by (simp add: ac_simps)
 
 lemma n_less_m_mult_n: "0 < n \<Longrightarrow> Suc 0 < m \<Longrightarrow> n < m * n"
   using mult_strict_right_mono [of 1 m n] by simp
 
 lemma n_less_n_mult_m: "0 < n \<Longrightarrow> Suc 0 < m \<Longrightarrow> n < n * m"
   using mult_strict_left_mono [of 1 m n] by simp
 
 
 text \<open>Induction starting beyond zero\<close>
 
 lemma nat_induct_at_least [consumes 1, case_names base Suc]:
   "P n" if "n \<ge> m" "P m" "\<And>n. n \<ge> m \<Longrightarrow> P n \<Longrightarrow> P (Suc n)"
 proof -
   define q where "q = n - m"
   with \<open>n \<ge> m\<close> have "n = m + q"
     by simp
   moreover have "P (m + q)"
     by (induction q) (use that in simp_all)
   ultimately show "P n"
     by simp
 qed
 
 lemma nat_induct_non_zero [consumes 1, case_names 1 Suc]:
   "P n" if "n > 0" "P 1" "\<And>n. n > 0 \<Longrightarrow> P n \<Longrightarrow> P (Suc n)"
 proof -
   from \<open>n > 0\<close> have "n \<ge> 1"
     by (cases n) simp_all
   moreover note \<open>P 1\<close>
   moreover have "\<And>n. n \<ge> 1 \<Longrightarrow> P n \<Longrightarrow> P (Suc n)"
     using \<open>\<And>n. n > 0 \<Longrightarrow> P n \<Longrightarrow> P (Suc n)\<close>
     by (simp add: Suc_le_eq)
   ultimately show "P n"
     by (rule nat_induct_at_least)
 qed
 
 
 text \<open>Specialized induction principles that work "backwards":\<close>
 
 lemma inc_induct [consumes 1, case_names base step]:
   assumes less: "i \<le> j"
     and base: "P j"
     and step: "\<And>n. i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P (Suc n) \<Longrightarrow> P n"
   shows "P i"
   using less step
 proof (induct "j - i" arbitrary: i)
   case (0 i)
   then have "i = j" by simp
   with base show ?case by simp
 next
   case (Suc d n)
   from Suc.hyps have "n \<noteq> j" by auto
   with Suc have "n < j" by (simp add: less_le)
   from \<open>Suc d = j - n\<close> have "d + 1 = j - n" by simp
   then have "d + 1 - 1 = j - n - 1" by simp
   then have "d = j - n - 1" by simp
   then have "d = j - (n + 1)" by (simp add: diff_diff_eq)
   then have "d = j - Suc n" by simp
   moreover from \<open>n < j\<close> have "Suc n \<le> j" by (simp add: Suc_le_eq)
   ultimately have "P (Suc n)"
   proof (rule Suc.hyps)
     fix q
     assume "Suc n \<le> q"
     then have "n \<le> q" by (simp add: Suc_le_eq less_imp_le)
     moreover assume "q < j"
     moreover assume "P (Suc q)"
     ultimately show "P q" by (rule Suc.prems)
   qed
   with order_refl \<open>n < j\<close> show "P n" by (rule Suc.prems)
 qed
 
 lemma strict_inc_induct [consumes 1, case_names base step]:
   assumes less: "i < j"
     and base: "\<And>i. j = Suc i \<Longrightarrow> P i"
     and step: "\<And>i. i < j \<Longrightarrow> P (Suc i) \<Longrightarrow> P i"
   shows "P i"
 using less proof (induct "j - i - 1" arbitrary: i)
   case (0 i)
   from \<open>i < j\<close> obtain n where "j = i + n" and "n > 0"
     by (auto dest!: less_imp_Suc_add)
   with 0 have "j = Suc i"
     by (auto intro: order_antisym simp add: Suc_le_eq)
   with base show ?case by simp
 next
   case (Suc d i)
   from \<open>Suc d = j - i - 1\<close> have *: "Suc d = j - Suc i"
     by (simp add: diff_diff_add)
   then have "Suc d - 1 = j - Suc i - 1" by simp
   then have "d = j - Suc i - 1" by simp
   moreover from * have "j - Suc i \<noteq> 0" by auto
   then have "Suc i < j" by (simp add: not_le)
   ultimately have "P (Suc i)" by (rule Suc.hyps)
   with \<open>i < j\<close> show "P i" by (rule step)
 qed
 
 lemma zero_induct_lemma: "P k \<Longrightarrow> (\<And>n. P (Suc n) \<Longrightarrow> P n) \<Longrightarrow> P (k - i)"
   using inc_induct[of "k - i" k P, simplified] by blast
 
 lemma zero_induct: "P k \<Longrightarrow> (\<And>n. P (Suc n) \<Longrightarrow> P n) \<Longrightarrow> P 0"
   using inc_induct[of 0 k P] by blast
 
 text \<open>Further induction rule similar to @{thm inc_induct}.\<close>
 
 lemma dec_induct [consumes 1, case_names base step]:
   "i \<le> j \<Longrightarrow> P i \<Longrightarrow> (\<And>n. i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P n \<Longrightarrow> P (Suc n)) \<Longrightarrow> P j"
 proof (induct j arbitrary: i)
   case 0
   then show ?case by simp
 next
   case (Suc j)
   from Suc.prems consider "i \<le> j" | "i = Suc j"
     by (auto simp add: le_Suc_eq)
   then show ?case
   proof cases
     case 1
     moreover have "j < Suc j" by simp
     moreover have "P j" using \<open>i \<le> j\<close> \<open>P i\<close>
     proof (rule Suc.hyps)
       fix q
       assume "i \<le> q"
       moreover assume "q < j" then have "q < Suc j"
         by (simp add: less_Suc_eq)
       moreover assume "P q"
       ultimately show "P (Suc q)" by (rule Suc.prems)
     qed
     ultimately show "P (Suc j)" by (rule Suc.prems)
   next
     case 2
     with \<open>P i\<close> show "P (Suc j)" by simp
   qed
 qed
 
 lemma transitive_stepwise_le:
   assumes "m \<le> n" "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" and "\<And>n. R n (Suc n)"
   shows "R m n"
 using \<open>m \<le> n\<close>  
   by (induction rule: dec_induct) (use assms in blast)+
 
 
 subsubsection \<open>Greatest operator\<close>
 
 lemma ex_has_greatest_nat:
   "P (k::nat) \<Longrightarrow> \<forall>y. P y \<longrightarrow> y \<le> b \<Longrightarrow> \<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> y \<le> x)"
 proof (induction "b-k" arbitrary: b k rule: less_induct)
   case less
   show ?case
   proof cases
     assume "\<exists>n>k. P n"
     then obtain n where "n>k" "P n" by blast
     have "n \<le> b" using \<open>P n\<close> less.prems(2) by auto
     hence "b-n < b-k"
       by(rule diff_less_mono2[OF \<open>k<n\<close> less_le_trans[OF \<open>k<n\<close>]])
     from less.hyps[OF this \<open>P n\<close> less.prems(2)]
     show ?thesis .
   next
     assume "\<not> (\<exists>n>k. P n)"
     hence "\<forall>y. P y \<longrightarrow> y \<le> k" by (auto simp: not_less)
     thus ?thesis using less.prems(1) by auto
   qed
 qed
 
 lemma
   fixes k::nat
   assumes "P k" and minor: "\<And>y. P y \<Longrightarrow> y \<le> b"
   shows GreatestI_nat: "P (Greatest P)" 
     and Greatest_le_nat: "k \<le> Greatest P"
 proof -
   obtain x where "P x" "\<And>y. P y \<Longrightarrow> y \<le> x"
     using assms ex_has_greatest_nat by blast
   with \<open>P k\<close> show "P (Greatest P)" "k \<le> Greatest P"
     using GreatestI2_order by blast+
 qed
 
 lemma GreatestI_ex_nat:
   "\<lbrakk> \<exists>k::nat. P k;  \<And>y. P y \<Longrightarrow> y \<le> b \<rbrakk> \<Longrightarrow> P (Greatest P)"
   by (blast intro: GreatestI_nat)
 
 
 subsection \<open>Monotonicity of \<open>funpow\<close>\<close>
 
 lemma funpow_increasing: "m \<le> n \<Longrightarrow> mono f \<Longrightarrow> (f ^^ n) \<top> \<le> (f ^^ m) \<top>"
   for f :: "'a::{lattice,order_top} \<Rightarrow> 'a"
   by (induct rule: inc_induct)
     (auto simp del: funpow.simps(2) simp add: funpow_Suc_right
       intro: order_trans[OF _ funpow_mono])
 
 lemma funpow_decreasing: "m \<le> n \<Longrightarrow> mono f \<Longrightarrow> (f ^^ m) \<bottom> \<le> (f ^^ n) \<bottom>"
   for f :: "'a::{lattice,order_bot} \<Rightarrow> 'a"
   by (induct rule: dec_induct)
     (auto simp del: funpow.simps(2) simp add: funpow_Suc_right
       intro: order_trans[OF _ funpow_mono])
 
 lemma mono_funpow: "mono Q \<Longrightarrow> mono (\<lambda>i. (Q ^^ i) \<bottom>)"
   for Q :: "'a::{lattice,order_bot} \<Rightarrow> 'a"
   by (auto intro!: funpow_decreasing simp: mono_def)
 
 lemma antimono_funpow: "mono Q \<Longrightarrow> antimono (\<lambda>i. (Q ^^ i) \<top>)"
   for Q :: "'a::{lattice,order_top} \<Rightarrow> 'a"
   by (auto intro!: funpow_increasing simp: antimono_def)
 
 
 subsection \<open>The divides relation on \<^typ>\<open>nat\<close>\<close>
 
 lemma dvd_1_left [iff]: "Suc 0 dvd k"
   by (simp add: dvd_def)
 
 lemma dvd_1_iff_1 [simp]: "m dvd Suc 0 \<longleftrightarrow> m = Suc 0"
   by (simp add: dvd_def)
 
 lemma nat_dvd_1_iff_1 [simp]: "m dvd 1 \<longleftrightarrow> m = 1"
   for m :: nat
   by (simp add: dvd_def)
 
 lemma dvd_antisym: "m dvd n \<Longrightarrow> n dvd m \<Longrightarrow> m = n"
   for m n :: nat
   unfolding dvd_def by (force dest: mult_eq_self_implies_10 simp add: mult.assoc)
 
 lemma dvd_diff_nat [simp]: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m - n)"
   for k m n :: nat
   unfolding dvd_def by (blast intro: right_diff_distrib' [symmetric])
 
 lemma dvd_diffD: 
   fixes k m n :: nat
   assumes "k dvd m - n" "k dvd n" "n \<le> m"
   shows "k dvd m"
 proof -
   have "k dvd n + (m - n)"
     using assms by (blast intro: dvd_add)
   with assms show ?thesis
     by simp
 qed
 
 lemma dvd_diffD1: "k dvd m - n \<Longrightarrow> k dvd m \<Longrightarrow> n \<le> m \<Longrightarrow> k dvd n"
   for k m n :: nat
   by (drule_tac m = m in dvd_diff_nat) auto
 
 lemma dvd_mult_cancel:
   fixes m n k :: nat
   assumes "k * m dvd k * n" and "0 < k"
   shows "m dvd n"
 proof -
   from assms(1) obtain q where "k * n = (k * m) * q" ..
   then have "k * n = k * (m * q)" by (simp add: ac_simps)
   with \<open>0 < k\<close> have "n = m * q" by (auto simp add: mult_left_cancel)
   then show ?thesis ..
 qed
 
 lemma dvd_mult_cancel1:
   fixes m n :: nat
   assumes "0 < m"
   shows "m * n dvd m \<longleftrightarrow> n = 1"
 proof 
   assume "m * n dvd m"
   then have "m * n dvd m * 1"
     by simp
   then have "n dvd 1"
     by (iprover intro: assms dvd_mult_cancel)
   then show "n = 1"
     by auto
 qed auto
 
 lemma dvd_mult_cancel2: "0 < m \<Longrightarrow> n * m dvd m \<longleftrightarrow> n = 1"
   for m n :: nat
   using dvd_mult_cancel1 [of m n] by (simp add: ac_simps)
 
 lemma dvd_imp_le: "k dvd n \<Longrightarrow> 0 < n \<Longrightarrow> k \<le> n"
   for k n :: nat
   by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
 
 lemma nat_dvd_not_less: "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"
   for m n :: nat
   by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
 
 lemma less_eq_dvd_minus:
   fixes m n :: nat
   assumes "m \<le> n"
   shows "m dvd n \<longleftrightarrow> m dvd n - m"
 proof -
   from assms have "n = m + (n - m)" by simp
   then obtain q where "n = m + q" ..
   then show ?thesis by (simp add: add.commute [of m])
 qed
 
 lemma dvd_minus_self: "m dvd n - m \<longleftrightarrow> n < m \<or> m dvd n"
   for m n :: nat
   by (cases "n < m") (auto elim!: dvdE simp add: not_less le_imp_diff_is_add dest: less_imp_le)
 
 lemma dvd_minus_add:
   fixes m n q r :: nat
   assumes "q \<le> n" "q \<le> r * m"
   shows "m dvd n - q \<longleftrightarrow> m dvd n + (r * m - q)"
 proof -
   have "m dvd n - q \<longleftrightarrow> m dvd r * m + (n - q)"
     using dvd_add_times_triv_left_iff [of m r] by simp
   also from assms have "\<dots> \<longleftrightarrow> m dvd r * m + n - q" by simp
   also from assms have "\<dots> \<longleftrightarrow> m dvd (r * m - q) + n" by simp
   also have "\<dots> \<longleftrightarrow> m dvd n + (r * m - q)" by (simp add: add.commute)
   finally show ?thesis .
 qed
 
 
 subsection \<open>Aliasses\<close>
 
 lemma nat_mult_1: "1 * n = n"
   for n :: nat
   by (fact mult_1_left)
 
 lemma nat_mult_1_right: "n * 1 = n"
   for n :: nat
   by (fact mult_1_right)
 
 lemma diff_mult_distrib: "(m - n) * k = (m * k) - (n * k)"
   for k m n :: nat
   by (fact left_diff_distrib')
 
 lemma diff_mult_distrib2: "k * (m - n) = (k * m) - (k * n)"
   for k m n :: nat
   by (fact right_diff_distrib')
 
 (*Used in AUTO2 and Groups.le_diff_conv2 (with variables renamed) doesn't work for some reason*)
 lemma le_diff_conv2: "k \<le> j \<Longrightarrow> (i \<le> j - k) = (i + k \<le> j)"
   for i j k :: nat
   by (fact le_diff_conv2) 
 
 lemma diff_self_eq_0 [simp]: "m - m = 0"
   for m :: nat
   by (fact diff_cancel)
 
 lemma diff_diff_left [simp]: "i - j - k = i - (j + k)"
   for i j k :: nat
   by (fact diff_diff_add)
 
 lemma diff_commute: "i - j - k = i - k - j"
   for i j k :: nat
   by (fact diff_right_commute)
 
 lemma diff_add_inverse: "(n + m) - n = m"
   for m n :: nat
   by (fact add_diff_cancel_left')
 
 lemma diff_add_inverse2: "(m + n) - n = m"
   for m n :: nat
   by (fact add_diff_cancel_right')
 
 lemma diff_cancel: "(k + m) - (k + n) = m - n"
   for k m n :: nat
   by (fact add_diff_cancel_left)
 
 lemma diff_cancel2: "(m + k) - (n + k) = m - n"
   for k m n :: nat
   by (fact add_diff_cancel_right)
 
 lemma diff_add_0: "n - (n + m) = 0"
   for m n :: nat
   by (fact diff_add_zero)
 
 lemma add_mult_distrib2: "k * (m + n) = (k * m) + (k * n)"
   for k m n :: nat
   by (fact distrib_left)
 
 lemmas nat_distrib =
   add_mult_distrib distrib_left diff_mult_distrib diff_mult_distrib2
 
 
 subsection \<open>Size of a datatype value\<close>
 
 class size =
   fixes size :: "'a \<Rightarrow> nat" \<comment> \<open>see further theory \<open>Wellfounded\<close>\<close>
 
 instantiation nat :: size
 begin
 
 definition size_nat where [simp, code]: "size (n::nat) = n"
 
 instance ..
 
 end
 
 lemmas size_nat = size_nat_def
 
 lemma size_neq_size_imp_neq: "size x \<noteq> size y \<Longrightarrow> x \<noteq> y"
   by (erule contrapos_nn) (rule arg_cong)
 
 
 subsection \<open>Code module namespace\<close>
 
 code_identifier
   code_module Nat \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
 
 hide_const (open) of_nat_aux
 
 end