src/HOL/Nat.thy
author haftmann
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(*  Title:      HOL/Nat.thy
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    Author:     Tobias Nipkow
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    Author:     Lawrence C Paulson
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    Author:     Markus Wenzel
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*)
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section \<open>Natural numbers\<close>
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theory Nat
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imports Inductive Typedef Fun Rings
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begin
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named_theorems arith "arith facts -- only ground formulas"
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ML_file "Tools/arith_data.ML"
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subsection \<open>Type \<open>ind\<close>\<close>
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typedecl ind
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axiomatization Zero_Rep :: ind and Suc_Rep :: "ind \<Rightarrow> ind"
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  \<comment> \<open>The axiom of infinity in 2 parts:\<close>
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  where Suc_Rep_inject: "Suc_Rep x = Suc_Rep y \<Longrightarrow> x = y"
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    and Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"
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subsection \<open>Type nat\<close>
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text \<open>Type definition\<close>
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inductive Nat :: "ind \<Rightarrow> bool"
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  where
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    Zero_RepI: "Nat Zero_Rep"
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  | Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)"
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typedef nat = "{n. Nat n}"
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  morphisms Rep_Nat Abs_Nat
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  using Nat.Zero_RepI by auto
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lemma Nat_Rep_Nat: "Nat (Rep_Nat n)"
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  using Rep_Nat by simp
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lemma Nat_Abs_Nat_inverse: "Nat n \<Longrightarrow> Rep_Nat (Abs_Nat n) = n"
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  using Abs_Nat_inverse by simp
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lemma Nat_Abs_Nat_inject: "Nat n \<Longrightarrow> Nat m \<Longrightarrow> Abs_Nat n = Abs_Nat m \<longleftrightarrow> n = m"
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  using Abs_Nat_inject by simp
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instantiation nat :: zero
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begin
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definition Zero_nat_def: "0 = Abs_Nat Zero_Rep"
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instance ..
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end
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definition Suc :: "nat \<Rightarrow> nat"
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  where "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"
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lemma Suc_not_Zero: "Suc m \<noteq> 0"
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  by (simp add: Zero_nat_def Suc_def Suc_RepI Zero_RepI
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      Nat_Abs_Nat_inject Suc_Rep_not_Zero_Rep Nat_Rep_Nat)
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lemma Zero_not_Suc: "0 \<noteq> Suc m"
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  by (rule not_sym) (rule Suc_not_Zero)
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lemma Suc_Rep_inject': "Suc_Rep x = Suc_Rep y \<longleftrightarrow> x = y"
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  by (rule iffI, rule Suc_Rep_inject) simp_all
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lemma nat_induct0:
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  assumes "P 0"
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    and "\<And>n. P n \<Longrightarrow> P (Suc n)"
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  shows "P n"
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  using assms
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  apply (unfold Zero_nat_def Suc_def)
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  apply (rule Rep_Nat_inverse [THEN subst]) \<comment> \<open>types force good instantiation\<close>
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  apply (erule Nat_Rep_Nat [THEN Nat.induct])
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  apply (iprover elim: Nat_Abs_Nat_inverse [THEN subst])
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  done
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free_constructors case_nat for "0 :: nat" | Suc pred
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  where "pred (0 :: nat) = (0 :: nat)"
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    apply atomize_elim
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    apply (rename_tac n, induct_tac n rule: nat_induct0, auto)
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   apply (simp add: Suc_def Nat_Abs_Nat_inject Nat_Rep_Nat Suc_RepI Suc_Rep_inject' Rep_Nat_inject)
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  apply (simp only: Suc_not_Zero)
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  done
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\<comment> \<open>Avoid name clashes by prefixing the output of \<open>old_rep_datatype\<close> with \<open>old\<close>.\<close>
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setup \<open>Sign.mandatory_path "old"\<close>
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old_rep_datatype "0 :: nat" Suc
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    apply (erule nat_induct0)
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    apply assumption
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   apply (rule nat.inject)
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  apply (rule nat.distinct(1))
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  done
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setup \<open>Sign.parent_path\<close>
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\<comment> \<open>But erase the prefix for properties that are not generated by \<open>free_constructors\<close>.\<close>
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setup \<open>Sign.mandatory_path "nat"\<close>
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declare old.nat.inject[iff del]
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  and old.nat.distinct(1)[simp del, induct_simp del]
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lemmas induct = old.nat.induct
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lemmas inducts = old.nat.inducts
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lemmas rec = old.nat.rec
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lemmas simps = nat.inject nat.distinct nat.case nat.rec
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setup \<open>Sign.parent_path\<close>
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abbreviation rec_nat :: "'a \<Rightarrow> (nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"
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  where "rec_nat \<equiv> old.rec_nat"
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declare nat.sel[code del]
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hide_const (open) Nat.pred \<comment> \<open>hide everything related to the selector\<close>
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hide_fact
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  nat.case_eq_if
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  nat.collapse
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  nat.expand
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  nat.sel
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  nat.exhaust_sel
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  nat.split_sel
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  nat.split_sel_asm
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lemma nat_exhaust [case_names 0 Suc, cases type: nat]:
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  "(y = 0 \<Longrightarrow> P) \<Longrightarrow> (\<And>nat. y = Suc nat \<Longrightarrow> P) \<Longrightarrow> P"
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  \<comment> \<open>for backward compatibility -- names of variables differ\<close>
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  by (rule old.nat.exhaust)
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lemma nat_induct [case_names 0 Suc, induct type: nat]:
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  fixes n
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  assumes "P 0" and "\<And>n. P n \<Longrightarrow> P (Suc n)"
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  shows "P n"
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  \<comment> \<open>for backward compatibility -- names of variables differ\<close>
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  using assms by (rule nat.induct)
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hide_fact
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  nat_exhaust
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  nat_induct0
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ML \<open>
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val nat_basic_lfp_sugar =
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  let
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    val ctr_sugar = the (Ctr_Sugar.ctr_sugar_of_global @{theory} @{type_name nat});
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    val recx = Logic.varify_types_global @{term rec_nat};
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    val C = body_type (fastype_of recx);
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  in
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    {T = HOLogic.natT, fp_res_index = 0, C = C, fun_arg_Tsss = [[], [[HOLogic.natT, C]]],
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     ctr_sugar = ctr_sugar, recx = recx, rec_thms = @{thms nat.rec}}
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  end;
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\<close>
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setup \<open>
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let
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  fun basic_lfp_sugars_of _ [@{typ nat}] _ _ ctxt =
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      ([], [0], [nat_basic_lfp_sugar], [], [], [], TrueI (*dummy*), [], false, ctxt)
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    | basic_lfp_sugars_of bs arg_Ts callers callssss ctxt =
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      BNF_LFP_Rec_Sugar.default_basic_lfp_sugars_of bs arg_Ts callers callssss ctxt;
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in
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  BNF_LFP_Rec_Sugar.register_lfp_rec_extension
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    {nested_simps = [], is_new_datatype = K (K true), basic_lfp_sugars_of = basic_lfp_sugars_of,
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     rewrite_nested_rec_call = NONE}
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end
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\<close>
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text \<open>Injectiveness and distinctness lemmas\<close>
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lemma inj_Suc[simp]: "inj_on Suc N"
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  by (simp add: inj_on_def)
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lemma Suc_neq_Zero: "Suc m = 0 \<Longrightarrow> R"
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  by (rule notE) (rule Suc_not_Zero)
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lemma Zero_neq_Suc: "0 = Suc m \<Longrightarrow> R"
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  by (rule Suc_neq_Zero) (erule sym)
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lemma Suc_inject: "Suc x = Suc y \<Longrightarrow> x = y"
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  by (rule inj_Suc [THEN injD])
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lemma n_not_Suc_n: "n \<noteq> Suc n"
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  by (induct n) simp_all
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lemma Suc_n_not_n: "Suc n \<noteq> n"
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  by (rule not_sym) (rule n_not_Suc_n)
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text \<open>A special form of induction for reasoning about @{term "m < n"} and @{term "m - n"}.\<close>
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lemma diff_induct:
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  assumes "\<And>x. P x 0"
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    and "\<And>y. P 0 (Suc y)"
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    and "\<And>x y. P x y \<Longrightarrow> P (Suc x) (Suc y)"
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  shows "P m n"
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proof (induct n arbitrary: m)
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  case 0
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  show ?case by (rule assms(1))
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next
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  case (Suc n)
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  show ?case
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  proof (induct m)
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    case 0
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    show ?case by (rule assms(2))
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  next
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    case (Suc m)
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    from \<open>P m n\<close> show ?case by (rule assms(3))
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  qed
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qed
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subsection \<open>Arithmetic operators\<close>
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instantiation nat :: comm_monoid_diff
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begin
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primrec plus_nat
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  where
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    add_0: "0 + n = (n::nat)"
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  | add_Suc: "Suc m + n = Suc (m + n)"
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lemma add_0_right [simp]: "m + 0 = m"
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  for m :: nat
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  by (induct m) simp_all
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lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
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  by (induct m) simp_all
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declare add_0 [code]
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lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"
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  by simp
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primrec minus_nat
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  where
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    diff_0 [code]: "m - 0 = (m::nat)"
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  | diff_Suc: "m - Suc n = (case m - n of 0 \<Rightarrow> 0 | Suc k \<Rightarrow> k)"
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declare diff_Suc [simp del]
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lemma diff_0_eq_0 [simp, code]: "0 - n = 0"
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  for n :: nat
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  by (induct n) (simp_all add: diff_Suc)
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lemma diff_Suc_Suc [simp, code]: "Suc m - Suc n = m - n"
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  by (induct n) (simp_all add: diff_Suc)
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instance
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proof
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  fix n m q :: nat
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  show "(n + m) + q = n + (m + q)" by (induct n) simp_all
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  show "n + m = m + n" by (induct n) simp_all
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  show "m + n - m = n" by (induct m) simp_all
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  show "n - m - q = n - (m + q)" by (induct q) (simp_all add: diff_Suc)
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  show "0 + n = n" by simp
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  show "0 - n = 0" by simp
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qed
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end
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3fe7e97ccca8 replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
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hide_fact (open) add_0 add_0_right diff_0
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1b2bae06c796 hide fact Nat.add_0_right; make add_0_right from Groups priority
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instantiation nat :: comm_semiring_1_cancel
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begin
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definition One_nat_def [simp]: "1 = Suc 0"
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primrec times_nat
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  where
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    mult_0: "0 * n = (0::nat)"
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  | mult_Suc: "Suc m * n = n + (m * n)"
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lemma mult_0_right [simp]: "m * 0 = 0"
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  for m :: nat
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  by (induct m) simp_all
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lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
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  by (induct m) (simp_all add: add.left_commute)
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lemma add_mult_distrib: "(m + n) * k = (m * k) + (n * k)"
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  for m n k :: nat
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  by (induct m) (simp_all add: add.assoc)
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instance
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   286
proof
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  fix k n m q :: nat
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  show "0 \<noteq> (1::nat)"
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    by simp
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  show "1 * n = n"
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    by simp
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  show "n * m = m * n"
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    by (induct n) simp_all
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  show "(n * m) * q = n * (m * q)"
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    by (induct n) (simp_all add: add_mult_distrib)
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  show "(n + m) * q = n * q + m * q"
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   297
    by (rule add_mult_distrib)
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  show "k * (m - n) = (k * m) - (k * n)"
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24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
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   299
    by (induct m n rule: diff_induct) simp_all
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qed
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c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25563
diff changeset
   301
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25563
diff changeset
   302
end
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   303
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   304
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   305
subsubsection \<open>Addition\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   306
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   307
text \<open>Reasoning about \<open>m + 0 = 0\<close>, etc.\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   308
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   309
lemma add_is_0 [iff]: "m + n = 0 \<longleftrightarrow> m = 0 \<and> n = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   310
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   311
  by (cases m) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   312
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   313
lemma add_is_1: "m + n = Suc 0 \<longleftrightarrow> m = Suc 0 \<and> n = 0 | m = 0 \<and> n = Suc 0"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   314
  by (cases m) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   315
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   316
lemma one_is_add: "Suc 0 = m + n \<longleftrightarrow> m = Suc 0 \<and> n = 0 | m = 0 \<and> n = Suc 0"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   317
  by (rule trans, rule eq_commute, rule add_is_1)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   318
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   319
lemma add_eq_self_zero: "m + n = m \<Longrightarrow> n = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   320
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   321
  by (induct m) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   322
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   323
lemma inj_on_add_nat [simp]: "inj_on (\<lambda>n. n + k) N"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   324
  for k :: nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   325
proof (induct k)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   326
  case 0
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   327
  then show ?case by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   328
next
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   329
  case (Suc k)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   330
  show ?case
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   331
    using comp_inj_on [OF Suc inj_Suc] by (simp add: o_def)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   332
qed
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   333
47208
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   334
lemma Suc_eq_plus1: "Suc n = n + 1"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   335
  by simp
47208
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   336
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   337
lemma Suc_eq_plus1_left: "Suc n = 1 + n"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   338
  by simp
47208
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   339
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   340
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   341
subsubsection \<open>Difference\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   342
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   343
lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
62365
8a105c235b1f sorted out some duplicate fact bindings
haftmann
parents: 62344
diff changeset
   344
  by (simp add: diff_diff_add)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   345
30093
ecb557b021b2 add lemma diff_Suc_1
huffman
parents: 30079
diff changeset
   346
lemma diff_Suc_1 [simp]: "Suc n - 1 = n"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   347
  by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   348
30093
ecb557b021b2 add lemma diff_Suc_1
huffman
parents: 30079
diff changeset
   349
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   350
subsubsection \<open>Multiplication\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   351
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   352
lemma mult_is_0 [simp]: "m * n = 0 \<longleftrightarrow> m = 0 \<or> n = 0" for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   353
  by (induct m) auto
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   354
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   355
lemma mult_eq_1_iff [simp]: "m * n = Suc 0 \<longleftrightarrow> m = Suc 0 \<and> n = Suc 0"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   356
proof (induct m)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   357
  case 0
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   358
  then show ?case by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   359
next
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   360
  case (Suc m)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   361
  then show ?case by (induct n) auto
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   362
qed
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   363
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   364
lemma one_eq_mult_iff [simp]: "Suc 0 = m * n \<longleftrightarrow> m = Suc 0 \<and> n = Suc 0"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   365
  apply (rule trans)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   366
   apply (rule_tac [2] mult_eq_1_iff)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   367
  apply fastforce
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   368
  done
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   369
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   370
lemma nat_mult_eq_1_iff [simp]: "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   371
  for m n :: nat
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   372
  unfolding One_nat_def by (rule mult_eq_1_iff)
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   373
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   374
lemma nat_1_eq_mult_iff [simp]: "1 = m * n \<longleftrightarrow> m = 1 \<and> n = 1"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   375
  for m n :: nat
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   376
  unfolding One_nat_def by (rule one_eq_mult_iff)
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   377
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   378
lemma mult_cancel1 [simp]: "k * m = k * n \<longleftrightarrow> m = n \<or> k = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   379
  for k m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   380
proof -
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   381
  have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   382
  proof (induct n arbitrary: m)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   383
    case 0
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   384
    then show "m = 0" by simp
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   385
  next
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   386
    case (Suc n)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   387
    then show "m = Suc n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   388
      by (cases m) (simp_all add: eq_commute [of 0])
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   389
  qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   390
  then show ?thesis by auto
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   391
qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   392
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   393
lemma mult_cancel2 [simp]: "m * k = n * k \<longleftrightarrow> m = n \<or> k = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   394
  for k m n :: nat
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   395
  by (simp add: mult.commute)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   396
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   397
lemma Suc_mult_cancel1: "Suc k * m = Suc k * n \<longleftrightarrow> m = n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   398
  by (subst mult_cancel1) simp
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   399
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   400
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   401
subsection \<open>Orders on @{typ nat}\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   402
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   403
subsubsection \<open>Operation definition\<close>
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   404
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   405
instantiation nat :: linorder
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   406
begin
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   407
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   408
primrec less_eq_nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   409
  where
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   410
    "(0::nat) \<le> n \<longleftrightarrow> True"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   411
  | "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   412
28514
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
   413
declare less_eq_nat.simps [simp del]
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   414
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   415
lemma le0 [iff]: "0 \<le> n" for
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   416
  n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   417
  by (simp add: less_eq_nat.simps)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   418
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   419
lemma [code]: "0 \<le> n \<longleftrightarrow> True"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   420
  for n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   421
  by simp
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   422
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   423
definition less_nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   424
  where less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   425
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   426
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   427
  by (simp add: less_eq_nat.simps(2))
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   428
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   429
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   430
  unfolding less_eq_Suc_le ..
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   431
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   432
lemma le_0_eq [iff]: "n \<le> 0 \<longleftrightarrow> n = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   433
  for n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   434
  by (induct n) (simp_all add: less_eq_nat.simps(2))
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   435
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   436
lemma not_less0 [iff]: "\<not> n < 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   437
  for n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   438
  by (simp add: less_eq_Suc_le)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   439
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   440
lemma less_nat_zero_code [code]: "n < 0 \<longleftrightarrow> False"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   441
  for n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   442
  by simp
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   443
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   444
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   445
  by (simp add: less_eq_Suc_le)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   446
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   447
lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   448
  by (simp add: less_eq_Suc_le)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   449
56194
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   450
lemma Suc_less_eq2: "Suc n < m \<longleftrightarrow> (\<exists>m'. m = Suc m' \<and> n < m')"
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   451
  by (cases m) auto
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   452
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   453
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   454
  by (induct m arbitrary: n) (simp_all add: less_eq_nat.simps(2) split: nat.splits)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   455
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   456
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   457
  by (cases n) (auto intro: le_SucI)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   458
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   459
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   460
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   461
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   462
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   463
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   464
26315
cb3badaa192e removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents: 26300
diff changeset
   465
instance
cb3badaa192e removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents: 26300
diff changeset
   466
proof
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   467
  fix n m q :: nat
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   468
  show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   469
  proof (induct n arbitrary: m)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   470
    case 0
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   471
    then show ?case
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   472
      by (cases m) (simp_all add: less_eq_Suc_le)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   473
  next
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   474
    case (Suc n)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   475
    then show ?case
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   476
      by (cases m) (simp_all add: less_eq_Suc_le)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   477
  qed
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   478
  show "n \<le> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   479
    by (induct n) simp_all
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   480
  then show "n = m" if "n \<le> m" and "m \<le> n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   481
    using that by (induct n arbitrary: m)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   482
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   483
  show "n \<le> q" if "n \<le> m" and "m \<le> q"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   484
    using that
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   485
  proof (induct n arbitrary: m q)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   486
    case 0
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   487
    show ?case by simp
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   488
  next
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   489
    case (Suc n)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   490
    then show ?case
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   491
      by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   492
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   493
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   494
  qed
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   495
  show "n \<le> m \<or> m \<le> n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   496
    by (induct n arbitrary: m)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   497
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   498
qed
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   499
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   500
end
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   501
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52435
diff changeset
   502
instantiation nat :: order_bot
29652
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   503
begin
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   504
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   505
definition bot_nat :: nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   506
  where "bot_nat = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   507
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   508
instance
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   509
  by standard (simp add: bot_nat_def)
29652
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   510
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   511
end
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   512
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
   513
instance nat :: no_top
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 61144
diff changeset
   514
  by standard (auto intro: less_Suc_eq_le [THEN iffD2])
52289
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 51329
diff changeset
   515
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
   516
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   517
subsubsection \<open>Introduction properties\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   518
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   519
lemma lessI [iff]: "n < Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   520
  by (simp add: less_Suc_eq_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   521
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   522
lemma zero_less_Suc [iff]: "0 < Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   523
  by (simp add: less_Suc_eq_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   524
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   525
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   526
subsubsection \<open>Elimination properties\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   527
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   528
lemma less_not_refl: "\<not> n < n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   529
  for n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   530
  by (rule order_less_irrefl)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   531
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   532
lemma less_not_refl2: "n < m \<Longrightarrow> m \<noteq> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   533
  for m n :: nat
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   534
  by (rule not_sym) (rule less_imp_neq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   535
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   536
lemma less_not_refl3: "s < t \<Longrightarrow> s \<noteq> t"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   537
  for s t :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   538
  by (rule less_imp_neq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   539
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   540
lemma less_irrefl_nat: "n < n \<Longrightarrow> R"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   541
  for n :: nat
26335
961bbcc9d85b removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents: 26315
diff changeset
   542
  by (rule notE, rule less_not_refl)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   543
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   544
lemma less_zeroE: "n < 0 \<Longrightarrow> R"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   545
  for n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   546
  by (rule notE) (rule not_less0)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   547
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   548
lemma less_Suc_eq: "m < Suc n \<longleftrightarrow> m < n \<or> m = n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   549
  unfolding less_Suc_eq_le le_less ..
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   550
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   551
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   552
  by (simp add: less_Suc_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   553
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   554
lemma less_one [iff]: "n < 1 \<longleftrightarrow> n = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   555
  for n :: nat
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   556
  unfolding One_nat_def by (rule less_Suc0)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   557
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   558
lemma Suc_mono: "m < n \<Longrightarrow> Suc m < Suc n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   559
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   560
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   561
text \<open>"Less than" is antisymmetric, sort of.\<close>
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   562
lemma less_antisym: "\<not> n < m \<Longrightarrow> n < Suc m \<Longrightarrow> m = n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   563
  unfolding not_less less_Suc_eq_le by (rule antisym)
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   564
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   565
lemma nat_neq_iff: "m \<noteq> n \<longleftrightarrow> m < n \<or> n < m"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   566
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   567
  by (rule linorder_neq_iff)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   568
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   569
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   570
subsubsection \<open>Inductive (?) properties\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   571
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   572
lemma Suc_lessI: "m < n \<Longrightarrow> Suc m \<noteq> n \<Longrightarrow> Suc m < n"
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   573
  unfolding less_eq_Suc_le [of m] le_less by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   574
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   575
lemma lessE:
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   576
  assumes major: "i < k"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   577
    and 1: "k = Suc i \<Longrightarrow> P"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   578
    and 2: "\<And>j. i < j \<Longrightarrow> k = Suc j \<Longrightarrow> P"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   579
  shows P
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   580
proof -
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   581
  from major have "\<exists>j. i \<le> j \<and> k = Suc j"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   582
    unfolding less_eq_Suc_le by (induct k) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   583
  then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   584
    by (auto simp add: less_le)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   585
  with 1 2 show P by auto
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   586
qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   587
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   588
lemma less_SucE:
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   589
  assumes major: "m < Suc n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   590
    and less: "m < n \<Longrightarrow> P"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   591
    and eq: "m = n \<Longrightarrow> P"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   592
  shows P
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   593
  apply (rule major [THEN lessE])
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   594
   apply (rule eq)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   595
   apply blast
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   596
  apply (rule less)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   597
  apply blast
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   598
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   599
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   600
lemma Suc_lessE:
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   601
  assumes major: "Suc i < k"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   602
    and minor: "\<And>j. i < j \<Longrightarrow> k = Suc j \<Longrightarrow> P"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   603
  shows P
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   604
  apply (rule major [THEN lessE])
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   605
   apply (erule lessI [THEN minor])
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   606
  apply (erule Suc_lessD [THEN minor])
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   607
  apply assumption
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   608
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   609
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   610
lemma Suc_less_SucD: "Suc m < Suc n \<Longrightarrow> m < n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   611
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   612
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   613
lemma less_trans_Suc:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   614
  assumes le: "i < j"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   615
  shows "j < k \<Longrightarrow> Suc i < k"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   616
proof (induct k)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   617
  case 0
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   618
  then show ?case by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   619
next
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   620
  case (Suc k)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   621
  with le show ?case
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   622
    by simp (auto simp add: less_Suc_eq dest: Suc_lessD)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   623
qed
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   624
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   625
text \<open>Can be used with \<open>less_Suc_eq\<close> to get @{prop "n = m \<or> n < m"}.\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   626
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   627
  by (simp only: not_less less_Suc_eq_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   628
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   629
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   630
  by (simp only: not_le Suc_le_eq)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   631
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   632
text \<open>Properties of "less than or equal".\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   633
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   634
lemma le_imp_less_Suc: "m \<le> n \<Longrightarrow> m < Suc n"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   635
  by (simp only: less_Suc_eq_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   636
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   637
lemma Suc_n_not_le_n: "\<not> Suc n \<le> n"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   638
  by (simp add: not_le less_Suc_eq_le)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   639
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   640
lemma le_Suc_eq: "m \<le> Suc n \<longleftrightarrow> m \<le> n \<or> m = Suc n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   641
  by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   642
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   643
lemma le_SucE: "m \<le> Suc n \<Longrightarrow> (m \<le> n \<Longrightarrow> R) \<Longrightarrow> (m = Suc n \<Longrightarrow> R) \<Longrightarrow> R"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   644
  by (drule le_Suc_eq [THEN iffD1], iprover+)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   645
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   646
lemma Suc_leI: "m < n \<Longrightarrow> Suc m \<le> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   647
  by (simp only: Suc_le_eq)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   648
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   649
text \<open>Stronger version of \<open>Suc_leD\<close>.\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   650
lemma Suc_le_lessD: "Suc m \<le> n \<Longrightarrow> m < n"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   651
  by (simp only: Suc_le_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   652
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   653
lemma less_imp_le_nat: "m < n \<Longrightarrow> m \<le> n" for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   654
  unfolding less_eq_Suc_le by (rule Suc_leD)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   655
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   656
text \<open>For instance, \<open>(Suc m < Suc n) = (Suc m \<le> n) = (m < n)\<close>\<close>
26315
cb3badaa192e removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents: 26300
diff changeset
   657
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   658
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   659
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   660
text \<open>Equivalence of \<open>m \<le> n\<close> and \<open>m < n \<or> m = n\<close>\<close>
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   661
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   662
lemma less_or_eq_imp_le: "m < n \<or> m = n \<Longrightarrow> m \<le> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   663
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   664
  unfolding le_less .
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   665
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   666
lemma le_eq_less_or_eq: "m \<le> n \<longleftrightarrow> m < n \<or> m = n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   667
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   668
  by (rule le_less)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   669
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   670
text \<open>Useful with \<open>blast\<close>.\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   671
lemma eq_imp_le: "m = n \<Longrightarrow> m \<le> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   672
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   673
  by auto
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   674
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   675
lemma le_refl: "n \<le> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   676
  for n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   677
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   678
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   679
lemma le_trans: "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   680
  for i j k :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   681
  by (rule order_trans)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   682
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   683
lemma le_antisym: "m \<le> n \<Longrightarrow> n \<le> m \<Longrightarrow> m = n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   684
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   685
  by (rule antisym)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   686
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   687
lemma nat_less_le: "m < n \<longleftrightarrow> m \<le> n \<and> m \<noteq> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   688
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   689
  by (rule less_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   690
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   691
lemma le_neq_implies_less: "m \<le> n \<Longrightarrow> m \<noteq> n \<Longrightarrow> m < n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   692
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   693
  unfolding less_le ..
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   694
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   695
lemma nat_le_linear: "m \<le> n | n \<le> m"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   696
  for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   697
  by (rule linear)
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   698
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   699
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
15921
b6e345548913 Fixing a problem with lin.arith.
nipkow
parents: 15539
diff changeset
   700
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   701
lemma le_less_Suc_eq: "m \<le> n \<Longrightarrow> n < Suc m \<longleftrightarrow> n = m"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   702
  unfolding less_Suc_eq_le by auto
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   703
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   704
lemma not_less_less_Suc_eq: "\<not> n < m \<Longrightarrow> n < Suc m \<longleftrightarrow> n = m"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   705
  unfolding not_less by (rule le_less_Suc_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   706
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   707
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   708
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   709
lemma not0_implies_Suc: "n \<noteq> 0 \<Longrightarrow> \<exists>m. n = Suc m"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   710
  by (cases n) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   711
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   712
lemma gr0_implies_Suc: "n > 0 \<Longrightarrow> \<exists>m. n = Suc m"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   713
  by (cases n) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   714
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   715
lemma gr_implies_not0: "m < n \<Longrightarrow> n \<noteq> 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   716
  for m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   717
  by (cases n) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   718
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   719
lemma neq0_conv[iff]: "n \<noteq> 0 \<longleftrightarrow> 0 < n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   720
  for n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   721
  by (cases n) simp_all
25140
273772abbea2 More changes from >0 to ~=0::nat
nipkow
parents: 25134
diff changeset
   722
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   723
text \<open>This theorem is useful with \<open>blast\<close>\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   724
lemma gr0I: "(n = 0 \<Longrightarrow> False) \<Longrightarrow> 0 < n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   725
  for n :: nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   726
  by (rule neq0_conv[THEN iffD1]) iprover
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   727
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   728
lemma gr0_conv_Suc: "0 < n \<longleftrightarrow> (\<exists>m. n = Suc m)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   729
  by (fast intro: not0_implies_Suc)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   730
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   731
lemma not_gr0 [iff]: "\<not> 0 < n \<longleftrightarrow> n = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   732
  for n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   733
  using neq0_conv by blast
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   734
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   735
lemma Suc_le_D: "Suc n \<le> m' \<Longrightarrow> \<exists>m. m' = Suc m"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   736
  by (induct m') simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   737
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   738
text \<open>Useful in certain inductive arguments\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   739
lemma less_Suc_eq_0_disj: "m < Suc n \<longleftrightarrow> m = 0 \<or> (\<exists>j. m = Suc j \<and> j < n)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   740
  by (cases m) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   741
64447
e44f5c123f26 added lemma
nipkow
parents: 63979
diff changeset
   742
lemma All_less_Suc: "(\<forall>i < Suc n. P i) = (P n \<and> (\<forall>i < n. P i))"
e44f5c123f26 added lemma
nipkow
parents: 63979
diff changeset
   743
by (auto simp: less_Suc_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   744
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   745
subsubsection \<open>Monotonicity of Addition\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   746
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   747
lemma Suc_pred [simp]: "n > 0 \<Longrightarrow> Suc (n - Suc 0) = n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   748
  by (simp add: diff_Suc split: nat.split)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   749
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   750
lemma Suc_diff_1 [simp]: "0 < n \<Longrightarrow> Suc (n - 1) = n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   751
  unfolding One_nat_def by (rule Suc_pred)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   752
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   753
lemma nat_add_left_cancel_le [simp]: "k + m \<le> k + n \<longleftrightarrow> m \<le> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   754
  for k m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   755
  by (induct k) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   756
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   757
lemma nat_add_left_cancel_less [simp]: "k + m < k + n \<longleftrightarrow> m < n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   758
  for k m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   759
  by (induct k) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   760
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   761
lemma add_gr_0 [iff]: "m + n > 0 \<longleftrightarrow> m > 0 \<or> n > 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   762
  for m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   763
  by (auto dest: gr0_implies_Suc)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   764
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   765
text \<open>strict, in 1st argument\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   766
lemma add_less_mono1: "i < j \<Longrightarrow> i + k < j + k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   767
  for i j k :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   768
  by (induct k) simp_all
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   769
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   770
text \<open>strict, in both arguments\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   771
lemma add_less_mono: "i < j \<Longrightarrow> k < l \<Longrightarrow> i + k < j + l"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   772
  for i j k l :: nat
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   773
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   774
  apply (induct j)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   775
   apply simp_all
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   776
  done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   777
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   778
text \<open>Deleted \<open>less_natE\<close>; use \<open>less_imp_Suc_add RS exE\<close>\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   779
lemma less_imp_Suc_add: "m < n \<Longrightarrow> \<exists>k. n = Suc (m + k)"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   780
proof (induct n)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   781
  case 0
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   782
  then show ?case by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   783
next
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   784
  case Suc
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   785
  then show ?case
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   786
    by (simp add: order_le_less)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   787
      (blast elim!: less_SucE intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   788
qed
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   789
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   790
lemma le_Suc_ex: "k \<le> l \<Longrightarrow> (\<exists>n. l = k + n)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   791
  for k l :: nat
56194
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   792
  by (auto simp: less_Suc_eq_le[symmetric] dest: less_imp_Suc_add)
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   793
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   794
text \<open>strict, in 1st argument; proof is by induction on \<open>k > 0\<close>\<close>
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   795
lemma mult_less_mono2:
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   796
  fixes i j :: nat
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   797
  assumes "i < j" and "0 < k"
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   798
  shows "k * i < k * j"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   799
  using \<open>0 < k\<close>
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   800
proof (induct k)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   801
  case 0
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   802
  then show ?case by simp
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   803
next
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   804
  case (Suc k)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   805
  with \<open>i < j\<close> show ?case
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   806
    by (cases k) (simp_all add: add_less_mono)
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   807
qed
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   808
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   809
text \<open>Addition is the inverse of subtraction:
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   810
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}.\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   811
lemma add_diff_inverse_nat: "\<not> m < n \<Longrightarrow> n + (m - n) = m"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   812
  for m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   813
  by (induct m n rule: diff_induct) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   814
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   815
lemma nat_le_iff_add: "m \<le> n \<longleftrightarrow> (\<exists>k. n = m + k)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   816
  for m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   817
  using nat_add_left_cancel_le[of m 0] by (auto dest: le_Suc_ex)
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62365
diff changeset
   818
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   819
text \<open>The naturals form an ordered \<open>semidom\<close> and a \<open>dioid\<close>.\<close>
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62365
diff changeset
   820
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34208
diff changeset
   821
instance nat :: linordered_semidom
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   822
proof
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   823
  fix m n q :: nat
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   824
  show "0 < (1::nat)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   825
    by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   826
  show "m \<le> n \<Longrightarrow> q + m \<le> q + n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   827
    by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   828
  show "m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   829
    by (simp add: mult_less_mono2)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   830
  show "m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> m * n \<noteq> 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   831
    by simp
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   832
  show "n \<le> m \<Longrightarrow> (m - n) + n = m"
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   833
    by (simp add: add_diff_inverse_nat add.commute linorder_not_less)
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62365
diff changeset
   834
qed
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62365
diff changeset
   835
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62365
diff changeset
   836
instance nat :: dioid
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   837
  by standard (rule nat_le_iff_add)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   838
63145
703edebd1d92 isabelle update_cartouches -c -t;
wenzelm
parents: 63113
diff changeset
   839
declare le0[simp del] \<comment> \<open>This is now @{thm zero_le}\<close>
703edebd1d92 isabelle update_cartouches -c -t;
wenzelm
parents: 63113
diff changeset
   840
declare le_0_eq[simp del] \<comment> \<open>This is now @{thm le_zero_eq}\<close>
703edebd1d92 isabelle update_cartouches -c -t;
wenzelm
parents: 63113
diff changeset
   841
declare not_less0[simp del] \<comment> \<open>This is now @{thm not_less_zero}\<close>
703edebd1d92 isabelle update_cartouches -c -t;
wenzelm
parents: 63113
diff changeset
   842
declare not_gr0[simp del] \<comment> \<open>This is now @{thm not_gr_zero}\<close>
62376
85f38d5f8807 Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents: 62365
diff changeset
   843
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   844
instance nat :: ordered_cancel_comm_monoid_add ..
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   845
instance nat :: ordered_cancel_comm_monoid_diff ..
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   846
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   847
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   848
subsubsection \<open>@{term min} and @{term max}\<close>
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   849
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   850
lemma mono_Suc: "mono Suc"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   851
  by (rule monoI) simp
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   852
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   853
lemma min_0L [simp]: "min 0 n = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   854
  for n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   855
  by (rule min_absorb1) simp
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   856
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   857
lemma min_0R [simp]: "min n 0 = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   858
  for n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   859
  by (rule min_absorb2) simp
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   860
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   861
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   862
  by (simp add: mono_Suc min_of_mono)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   863
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   864
lemma min_Suc1: "min (Suc n) m = (case m of 0 \<Rightarrow> 0 | Suc m' \<Rightarrow> Suc(min n m'))"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   865
  by (simp split: nat.split)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   866
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   867
lemma min_Suc2: "min m (Suc n) = (case m of 0 \<Rightarrow> 0 | Suc m' \<Rightarrow> Suc(min m' n))"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   868
  by (simp split: nat.split)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   869
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   870
lemma max_0L [simp]: "max 0 n = n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   871
  for n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   872
  by (rule max_absorb2) simp
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   873
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   874
lemma max_0R [simp]: "max n 0 = n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   875
  for n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   876
  by (rule max_absorb1) simp
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   877
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   878
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc (max m n)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   879
  by (simp add: mono_Suc max_of_mono)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   880
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   881
lemma max_Suc1: "max (Suc n) m = (case m of 0 \<Rightarrow> Suc n | Suc m' \<Rightarrow> Suc (max n m'))"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   882
  by (simp split: nat.split)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   883
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   884
lemma max_Suc2: "max m (Suc n) = (case m of 0 \<Rightarrow> Suc n | Suc m' \<Rightarrow> Suc (max m' n))"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   885
  by (simp split: nat.split)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   886
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   887
lemma nat_mult_min_left: "min m n * q = min (m * q) (n * q)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   888
  for m n q :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   889
  by (simp add: min_def not_le)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   890
    (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   891
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   892
lemma nat_mult_min_right: "m * min n q = min (m * n) (m * q)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   893
  for m n q :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   894
  by (simp add: min_def not_le)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   895
    (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   896
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   897
lemma nat_add_max_left: "max m n + q = max (m + q) (n + q)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   898
  for m n q :: nat
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   899
  by (simp add: max_def)
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   900
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   901
lemma nat_add_max_right: "m + max n q = max (m + n) (m + q)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   902
  for m n q :: nat
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   903
  by (simp add: max_def)
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   904
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   905
lemma nat_mult_max_left: "max m n * q = max (m * q) (n * q)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   906
  for m n q :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   907
  by (simp add: max_def not_le)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   908
    (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   909
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   910
lemma nat_mult_max_right: "m * max n q = max (m * n) (m * q)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   911
  for m n q :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   912
  by (simp add: max_def not_le)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   913
    (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   914
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   915
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   916
subsubsection \<open>Additional theorems about @{term "op \<le>"}\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   917
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   918
text \<open>Complete induction, aka course-of-values induction\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   919
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   920
instance nat :: wellorder
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   921
proof
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   922
  fix P and n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   923
  assume step: "(\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n" for n :: nat
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   924
  have "\<And>q. q \<le> n \<Longrightarrow> P q"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   925
  proof (induct n)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   926
    case (0 n)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   927
    have "P 0" by (rule step) auto
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   928
    with 0 show ?case by auto
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   929
  next
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   930
    case (Suc m n)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   931
    then have "n \<le> m \<or> n = Suc m"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   932
      by (simp add: le_Suc_eq)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   933
    then show ?case
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   934
    proof
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   935
      assume "n \<le> m"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   936
      then show "P n" by (rule Suc(1))
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   937
    next
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   938
      assume n: "n = Suc m"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   939
      show "P n" by (rule step) (rule Suc(1), simp add: n le_simps)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   940
    qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   941
  qed
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   942
  then show "P n" by auto
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   943
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   944
57015
842bb6d36263 added lemma
nipkow
parents: 56194
diff changeset
   945
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   946
lemma Least_eq_0[simp]: "P 0 \<Longrightarrow> Least P = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   947
  for P :: "nat \<Rightarrow> bool"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   948
  by (rule Least_equality[OF _ le0])
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   949
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   950
lemma Least_Suc: "P n \<Longrightarrow> \<not> P 0 \<Longrightarrow> (LEAST n. P n) = Suc (LEAST m. P (Suc m))"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   951
  apply (cases n)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   952
   apply auto
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   953
  apply (frule LeastI)
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   954
  apply (drule_tac P = "\<lambda>x. P (Suc x)" in LeastI)
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   955
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   956
   apply (erule_tac [2] Least_le)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   957
  apply (cases "LEAST x. P x")
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   958
   apply auto
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   959
  apply (drule_tac P = "\<lambda>x. P (Suc x)" in Least_le)
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   960
  apply (blast intro: order_antisym)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   961
  done
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   962
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   963
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)"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   964
  by (erule (1) Least_Suc [THEN ssubst]) simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   965
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   966
lemma ex_least_nat_le: "\<not> P 0 \<Longrightarrow> P n \<Longrightarrow> \<exists>k\<le>n. (\<forall>i<k. \<not> P i) \<and> P k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   967
  for P :: "nat \<Rightarrow> bool"
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   968
  apply (cases n)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   969
   apply blast
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   970
  apply (rule_tac x="LEAST k. P k" in exI)
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   971
  apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   972
  done
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   973
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   974
lemma ex_least_nat_less: "\<not> P 0 \<Longrightarrow> P n \<Longrightarrow> \<exists>k<n. (\<forall>i\<le>k. \<not> P i) \<and> P (k + 1)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
   975
  for P :: "nat \<Rightarrow> bool"
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   976
  apply (cases n)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   977
   apply blast
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   978
  apply (frule (1) ex_least_nat_le)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   979
  apply (erule exE)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   980
  apply (case_tac k)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   981
   apply simp
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   982
  apply (rename_tac k1)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   983
  apply (rule_tac x=k1 in exI)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   984
  apply (auto simp add: less_eq_Suc_le)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   985
  done
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   986
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   987
lemma nat_less_induct:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   988
  fixes P :: "nat \<Rightarrow> bool"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   989
  assumes "\<And>n. \<forall>m. m < n \<longrightarrow> P m \<Longrightarrow> P n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   990
  shows "P n"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   991
  using assms less_induct by blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   992
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   993
lemma measure_induct_rule [case_names less]:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   994
  fixes f :: "'a \<Rightarrow> nat"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   995
  assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   996
  shows "P a"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   997
  by (induct m \<equiv> "f a" arbitrary: a rule: less_induct) (auto intro: step)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   998
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   999
text \<open>old style induction rules:\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1000
lemma measure_induct:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1001
  fixes f :: "'a \<Rightarrow> nat"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1002
  shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1003
  by (rule measure_induct_rule [of f P a]) iprover
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1004
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1005
lemma full_nat_induct:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1006
  assumes step: "\<And>n. (\<forall>m. Suc m \<le> n \<longrightarrow> P m) \<Longrightarrow> P n"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1007
  shows "P n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1008
  by (rule less_induct) (auto intro: step simp:le_simps)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1009
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1010
text\<open>An induction rule for establishing binary relations\<close>
62683
ddd1c864408b clarified rule structure;
wenzelm
parents: 62608
diff changeset
  1011
lemma less_Suc_induct [consumes 1]:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1012
  assumes less: "i < j"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1013
    and step: "\<And>i. P i (Suc i)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1014
    and trans: "\<And>i j k. i < j \<Longrightarrow> j < k \<Longrightarrow> P i j \<Longrightarrow> P j k \<Longrightarrow> P i k"
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
  1015
  shows "P i j"
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
  1016
proof -
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1017
  from less obtain k where j: "j = Suc (i + k)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1018
    by (auto dest: less_imp_Suc_add)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1019
  have "P i (Suc (i + k))"
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
  1020
  proof (induct k)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1021
    case 0
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1022
    show ?case by (simp add: step)
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
  1023
  next
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
  1024
    case (Suc k)
31714
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
  1025
    have "0 + i < Suc k + i" by (rule add_less_mono1) simp
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1026
    then have "i < Suc (i + k)" by (simp add: add.commute)
31714
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
  1027
    from trans[OF this lessI Suc step]
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
  1028
    show ?case by simp
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
  1029
  qed
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1030
  then show "P i j" by (simp add: j)
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
  1031
qed
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
  1032
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1033
text \<open>
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1034
  The method of infinite descent, frequently used in number theory.
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1035
  Provided by Roelof Oosterhuis.
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1036
  \<open>P n\<close> is true for all natural numbers if
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1037
  \<^item> case ``0'': given \<open>n = 0\<close> prove \<open>P n\<close>
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1038
  \<^item> case ``smaller'': given \<open>n > 0\<close> and \<open>\<not> P n\<close> prove there exists
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1039
    a smaller natural number \<open>m\<close> such that \<open>\<not> P m\<close>.
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1040
\<close>
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1041
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1042
lemma infinite_descent: "(\<And>n. \<not> P n \<Longrightarrow> \<exists>m<n. \<not> P m) \<Longrightarrow> P n" for P :: "nat \<Rightarrow> bool"
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1043
  \<comment> \<open>compact version without explicit base case\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1044
  by (induct n rule: less_induct) auto
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1045
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1046
lemma infinite_descent0 [case_names 0 smaller]:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1047
  fixes P :: "nat \<Rightarrow> bool"
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1048
  assumes "P 0"
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1049
    and "\<And>n. n > 0 \<Longrightarrow> \<not> P n \<Longrightarrow> \<exists>m. m < n \<and> \<not> P m"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1050
  shows "P n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1051
  apply (rule infinite_descent)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1052
  using assms
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1053
  apply (case_tac "n > 0")
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1054
   apply auto
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1055
  done
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1056
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1057
text \<open>
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1058
  Infinite descent using a mapping to \<open>nat\<close>:
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1059
  \<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
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1060
  \<^item> case ``0'': given \<open>V x = 0\<close> prove \<open>P x\<close>
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1061
  \<^item> ``smaller'': given \<open>V x > 0\<close> and \<open>\<not> P x\<close> prove
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1062
  there exists a \<open>y \<in> D\<close> such that \<open>V y < V x\<close> and \<open>\<not> P y\<close>.
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1063
\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1064
corollary infinite_descent0_measure [case_names 0 smaller]:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1065
  fixes V :: "'a \<Rightarrow> nat"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1066
  assumes 1: "\<And>x. V x = 0 \<Longrightarrow> P x"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1067
    and 2: "\<And>x. V x > 0 \<Longrightarrow> \<not> P x \<Longrightarrow> \<exists>y. V y < V x \<and> \<not> P y"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1068
  shows "P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1069
proof -
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1070
  obtain n where "n = V x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1071
  moreover have "\<And>x. V x = n \<Longrightarrow> P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1072
  proof (induct n rule: infinite_descent0)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1073
    case 0
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1074
    with 1 show "P x" by auto
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1075
  next
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1076
    case (smaller n)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1077
    then obtain x where *: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1078
    with 2 obtain y where "V y < V x \<and> \<not> P y" by auto
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1079
    with * obtain m where "m = V y \<and> m < n \<and> \<not> P y" by auto
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1080
    then show ?case by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1081
  qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1082
  ultimately show "P x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1083
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1084
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1085
text \<open>Again, without explicit base case:\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1086
lemma infinite_descent_measure:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1087
  fixes V :: "'a \<Rightarrow> nat"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1088
  assumes "\<And>x. \<not> P x \<Longrightarrow> \<exists>y. V y < V x \<and> \<not> P y"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1089
  shows "P x"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1090
proof -
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1091
  from assms obtain n where "n = V x" by auto
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1092
  moreover have "\<And>x. V x = n \<Longrightarrow> P x"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1093
  proof (induct n rule: infinite_descent, auto)
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1094
    show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" if "\<not> P x" for x
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1095
      using assms and that by auto
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1096
  qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1097
  ultimately show "P x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1098
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1099
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1100
text \<open>A (clumsy) way of lifting \<open><\<close> monotonicity to \<open>\<le>\<close> monotonicity\<close>
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1101
lemma less_mono_imp_le_mono:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1102
  fixes f :: "nat \<Rightarrow> nat"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1103
    and i j :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1104
  assumes "\<And>i j::nat. i < j \<Longrightarrow> f i < f j"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1105
    and "i \<le> j"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1106
  shows "f i \<le> f j"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1107
  using assms by (auto simp add: order_le_less)
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1108
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1109
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1110
text \<open>non-strict, in 1st argument\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1111
lemma add_le_mono1: "i \<le> j \<Longrightarrow> i + k \<le> j + k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1112
  for i j k :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1113
  by (rule add_right_mono)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1114
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1115
text \<open>non-strict, in both arguments\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1116
lemma add_le_mono: "i \<le> j \<Longrightarrow> k \<le> l \<Longrightarrow> i + k \<le> j + l"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1117
  for i j k l :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1118
  by (rule add_mono)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1119
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1120
lemma le_add2: "n \<le> m + n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1121
  for m n :: nat
62608
19f87fa0cfcb more theorems on orderings
haftmann
parents: 62502
diff changeset
  1122
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1123
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1124
lemma le_add1: "n \<le> n + m"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1125
  for m n :: nat
62608
19f87fa0cfcb more theorems on orderings
haftmann
parents: 62502
diff changeset
  1126
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1127
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1128
lemma less_add_Suc1: "i < Suc (i + m)"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1129
  by (rule le_less_trans, rule le_add1, rule lessI)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1130
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1131
lemma less_add_Suc2: "i < Suc (m + i)"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1132
  by (rule le_less_trans, rule le_add2, rule lessI)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1133
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1134
lemma less_iff_Suc_add: "m < n \<longleftrightarrow> (\<exists>k. n = Suc (m + k))"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1135
  by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1136
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1137
lemma trans_le_add1: "i \<le> j \<Longrightarrow> i \<le> j + m"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1138
  for i j m :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1139
  by (rule le_trans, assumption, rule le_add1)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1140
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1141
lemma trans_le_add2: "i \<le> j \<Longrightarrow> i \<le> m + j"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1142
  for i j m :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1143
  by (rule le_trans, assumption, rule le_add2)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1144
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1145
lemma trans_less_add1: "i < j \<Longrightarrow> i < j + m"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1146
  for i j m :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1147
  by (rule less_le_trans, assumption, rule le_add1)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1148
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1149
lemma trans_less_add2: "i < j \<Longrightarrow> i < m + j"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1150
  for i j m :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1151
  by (rule less_le_trans, assumption, rule le_add2)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1152
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1153
lemma add_lessD1: "i + j < k \<Longrightarrow> i < k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1154
  for i j k :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1155
  by (rule le_less_trans [of _ "i+j"]) (simp_all add: le_add1)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1156
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1157
lemma not_add_less1 [iff]: "\<not> i + j < i"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1158
  for i j :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1159
  apply (rule notI)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1160
  apply (drule add_lessD1)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1161
  apply (erule less_irrefl [THEN notE])
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1162
  done
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1163
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1164
lemma not_add_less2 [iff]: "\<not> j + i < i"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1165
  for i j :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1166
  by (simp add: add.commute)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1167
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1168
lemma add_leD1: "m + k \<le> n \<Longrightarrow> m \<le> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1169
  for k m n :: nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1170
  by (rule order_trans [of _ "m + k"]) (simp_all add: le_add1)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1171
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1172
lemma add_leD2: "m + k \<le> n \<Longrightarrow> k \<le> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1173
  for k m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1174
  apply (simp add: add.commute)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1175
  apply (erule add_leD1)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1176
  done
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1177
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1178
lemma add_leE: "m + k \<le> n \<Longrightarrow> (m \<le> n \<Longrightarrow> k \<le> n \<Longrightarrow> R) \<Longrightarrow> R"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1179
  for k m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1180
  by (blast dest: add_leD1 add_leD2)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1181
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1182
text \<open>needs \<open>\<And>k\<close> for \<open>ac_simps\<close> to work\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1183
lemma less_add_eq_less: "\<And>k. k < l \<Longrightarrow> m + l = k + n \<Longrightarrow> m < n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1184
  for l m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1185
  by (force simp del: add_Suc_right simp add: less_iff_Suc_add add_Suc_right [symmetric] ac_simps)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1186
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1187
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1188
subsubsection \<open>More results about difference\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1189
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1190
lemma Suc_diff_le: "n \<le> m \<Longrightarrow> Suc m - n = Suc (m - n)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1191
  by (induct m n rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1192
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1193
lemma diff_less_Suc: "m - n < Suc m"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1194
  apply (induct m n rule: diff_induct)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1195
    apply (erule_tac [3] less_SucE)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1196
     apply (simp_all add: less_Suc_eq)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1197
  done
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1198
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1199
lemma diff_le_self [simp]: "m - n \<le> m"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1200
  for m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1201
  by (induct m n rule: diff_induct) (simp_all add: le_SucI)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1202
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1203
lemma less_imp_diff_less: "j < k \<Longrightarrow> j - n < k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1204
  for j k n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1205
  by (rule le_less_trans, rule diff_le_self)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1206
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1207
lemma diff_Suc_less [simp]: "0 < n \<Longrightarrow> n - Suc i < n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1208
  by (cases n) (auto simp add: le_simps)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1209
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1210
lemma diff_add_assoc: "k \<le> j \<Longrightarrow> (i + j) - k = i + (j - k)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1211
  for i j k :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1212
  by (induct j k rule: diff_induct) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1213
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1214
lemma add_diff_assoc [simp]: "k \<le> j \<Longrightarrow> i + (j - k) = i + j - k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1215
  for i j k :: nat
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
  1216
  by (fact diff_add_assoc [symmetric])
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
  1217
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1218
lemma diff_add_assoc2: "k \<le> j \<Longrightarrow> (j + i) - k = (j - k) + i"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1219
  for i j k :: nat
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
  1220
  by (simp add: ac_simps)
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
  1221
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1222
lemma add_diff_assoc2 [simp]: "k \<le> j \<Longrightarrow> j - k + i = j + i - k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1223
  for i j k :: nat
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
  1224
  by (fact diff_add_assoc2 [symmetric])
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1225
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1226
lemma le_imp_diff_is_add: "i \<le> j \<Longrightarrow> (j - i = k) = (j = k + i)"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1227
  for i j k :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1228
  by auto
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1229
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1230
lemma diff_is_0_eq [simp]: "m - n = 0 \<longleftrightarrow> m \<le> n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1231
  for m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1232
  by (induct m n rule: diff_induct) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1233
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1234
lemma diff_is_0_eq' [simp]: "m \<le> n \<Longrightarrow> m - n = 0"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1235
  for m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1236
  by (rule iffD2, rule diff_is_0_eq)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1237
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1238
lemma zero_less_diff [simp]: "0 < n - m \<longleftrightarrow> m < n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1239
  for m n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1240
  by (induct m n rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1241
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1242
lemma less_imp_add_positive:
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1243
  assumes "i < j"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1244
  shows "\<exists>k::nat. 0 < k \<and> i + k = j"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1245
proof
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1246
  from assms show "0 < j - i \<and> i + (j - i) = j"
23476
839db6346cc8 fix looping simp rule
huffman
parents: 23438
diff changeset
  1247
    by (simp add: order_less_imp_le)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1248
qed
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
  1249
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1250
text \<open>a nice rewrite for bounded subtraction\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1251
lemma nat_minus_add_max: "n - m + m = max n m"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1252
  for m n :: nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1253
  by (simp add: max_def not_le order_less_imp_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1254
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1255
lemma nat_diff_split: "P (a - b) \<longleftrightarrow> (a < b \<longrightarrow> P 0) \<and> (\<forall>d. a = b + d \<longrightarrow> P d)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1256
  for a b :: nat
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1257
  \<comment> \<open>elimination of \<open>-\<close> on \<open>nat\<close>\<close>
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1258
  by (cases "a < b") (auto simp add: not_less le_less dest!: add_eq_self_zero [OF sym])
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1259
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1260
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))"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1261
  for a b :: nat
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1262
  \<comment> \<open>elimination of \<open>-\<close> on \<open>nat\<close> in assumptions\<close>
62365
8a105c235b1f sorted out some duplicate fact bindings
haftmann
parents: 62344
diff changeset
  1263
  by (auto split: nat_diff_split)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1264
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1265
lemma Suc_pred': "0 < n \<Longrightarrow> n = Suc(n - 1)"
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1266
  by simp
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1267
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1268
lemma add_eq_if: "m + n = (if m = 0 then n else Suc ((m - 1) + n))"
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1269
  unfolding One_nat_def by (cases m) simp_all
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1270
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1271
lemma mult_eq_if: "m * n = (if m = 0 then 0 else n + ((m - 1) * n))"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1272
  for m n :: nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1273
  by (cases m) simp_all
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1274
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1275
lemma Suc_diff_eq_diff_pred: "0 < n \<Longrightarrow> Suc m - n = m - (n - 1)"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1276
  by (cases n) simp_all
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1277
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1278
lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1279
  by (cases m) simp_all
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1280
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1281
lemma Let_Suc [simp]: "Let (Suc n) f \<equiv> f (Suc n)"
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1282
  by (fact Let_def)
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1283
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1284
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1285
subsubsection \<open>Monotonicity of multiplication\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1286
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1287
lemma mult_le_mono1: "i \<le> j \<Longrightarrow> i * k \<le> j * k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1288
  for i j k :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1289
  by (simp add: mult_right_mono)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1290
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1291
lemma mult_le_mono2: "i \<le> j \<Longrightarrow> k * i \<le> k * j"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1292
  for i j k :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1293
  by (simp add: mult_left_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1294
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1295
text \<open>\<open>\<le>\<close> monotonicity, BOTH arguments\<close>
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1296
lemma mult_le_mono: "i \<le> j \<Longrightarrow> k \<le> l \<Longrightarrow> i * k \<le> j * l"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1297
  for i j k l :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1298
  by (simp add: mult_mono)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1299
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1300
lemma mult_less_mono1: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> i * k < j * k"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1301
  for i j k :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1302
  by (simp add: mult_strict_right_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1303
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1304
text \<open>Differs from the standard \<open>zero_less_mult_iff\<close> in that there are no negative numbers.\<close>
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1305
lemma nat_0_less_mult_iff [simp]: "0 < m * n \<longleftrightarrow> 0 < m \<and> 0 < n"
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1306
  for m n :: nat
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1307
proof (induct m)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1308
  case 0
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1309
  then show ?case by simp
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1310
next
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1311
  case (Suc m)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1312
  then show ?case by (cases n) simp_all
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1313
qed
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1314
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1315
lemma one_le_mult_iff [simp]: "Suc 0 \<le> m * n \<longleftrightarrow> Suc 0 \<le> m \<and> Suc 0 \<le> n"
63588
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff changeset
  1316
proof (induct m)
d0e2bad67bd4 misc tuning and modernization;
wenzelm
parents: 63561
diff <