src/HOL/Nat.thy
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(*  Title:      HOL/Nat.thy
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    Author:     Tobias Nipkow and Lawrence C Paulson and Markus Wenzel
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Type "nat" is a linear order, and a datatype; arithmetic operators + -
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and * (for div and mod, see theory Divides).
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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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ML_file "~~/src/Tools/rat.ML"
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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" 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:
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  "Nat (Rep_Nat n)"
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  using Rep_Nat by simp
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lemma Nat_Abs_Nat_inverse:
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  "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:
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  "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:
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  "0 = Abs_Nat Zero_Rep"
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instance ..
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end
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definition Suc :: "nat \<Rightarrow> nat" where
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  "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 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 not_sym)
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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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  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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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
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    "0 :: nat"
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  | Suc pred
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where
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  "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'
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     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, 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
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  old.nat.inject[iff del]
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  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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  \<comment> \<open>for backward compatibility -- names of variables differ\<close>
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  "(y = 0 \<Longrightarrow> P) \<Longrightarrow> (\<And>nat. y = Suc nat \<Longrightarrow> P) \<Longrightarrow> P"
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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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  \<comment> \<open>for backward compatibility -- names of variables differ\<close>
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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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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
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  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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  using assms
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  apply (rule_tac x = m in spec)
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  apply (induct n)
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  prefer 2
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  apply (rule allI)
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  apply (induct_tac x, iprover+)
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  done
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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 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" 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 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" 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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hide_fact (open) add_0 add_0_right diff_0
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instantiation nat :: comm_semiring_1_cancel
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begin
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definition
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  One_nat_def [simp]: "1 = Suc 0"
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primrec times_nat 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" 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)" 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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proof
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  fix k n m q :: nat
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  show "0 \<noteq> (1::nat)" unfolding One_nat_def by simp
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  show "1 * n = n" unfolding One_nat_def by simp
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  show "n * m = m * n" by (induct n) simp_all
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  show "(n * m) * q = n * (m * q)" by (induct n) (simp_all add: add_mult_distrib)
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  show "(n + m) * q = n * q + m * q" by (rule add_mult_distrib)
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   290
  show "k * (m - n) = (k * m) - (k * 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
   291
    by (induct m n rule: diff_induct) simp_all
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   292
qed
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25563
diff changeset
   293
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25563
diff changeset
   294
end
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   295
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
   296
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   297
subsubsection \<open>Addition\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   298
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   299
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
   300
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ccbdce905fca misc tuning and modernization;
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diff changeset
   301
lemma add_is_0 [iff]: "(m + n = 0) = (m = 0 \<and> n = 0)" for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   302
  by (cases m) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   303
63110
ccbdce905fca misc tuning and modernization;
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diff changeset
   304
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
   305
  by (cases m) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   306
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   307
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
   308
  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
   309
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   310
lemma add_eq_self_zero: "m + n = m \<Longrightarrow> n = 0" for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   311
  by (induct 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 inj_on_add_nat[simp]: "inj_on (\<lambda>n. n + k) N" for k :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   314
  apply (induct k)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   315
   apply simp
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   316
  apply (drule comp_inj_on[OF _ inj_Suc])
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   317
  apply (simp add: o_def)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   318
  done
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   319
47208
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   320
lemma Suc_eq_plus1: "Suc n = n + 1"
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   321
  unfolding One_nat_def by simp
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   322
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   323
lemma Suc_eq_plus1_left: "Suc n = 1 + n"
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   324
  unfolding One_nat_def by simp
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   325
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   326
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   327
subsubsection \<open>Difference\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   328
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   329
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
   330
  by (simp add: diff_diff_add)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   331
30093
ecb557b021b2 add lemma diff_Suc_1
huffman
parents: 30079
diff changeset
   332
lemma diff_Suc_1 [simp]: "Suc n - 1 = n"
ecb557b021b2 add lemma diff_Suc_1
huffman
parents: 30079
diff changeset
   333
  unfolding One_nat_def by simp
ecb557b021b2 add lemma diff_Suc_1
huffman
parents: 30079
diff changeset
   334
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   335
subsubsection \<open>Multiplication\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   336
63110
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wenzelm
parents: 63099
diff changeset
   337
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
   338
  by (induct m) auto
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   339
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   340
lemma mult_eq_1_iff [simp]: "m * n = Suc 0 \<longleftrightarrow> m = Suc 0 \<and> n = Suc 0"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   341
  apply (induct m)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   342
   apply simp
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   343
  apply (induct n)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   344
   apply auto
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   345
  done
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   346
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   347
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
   348
  apply (rule trans)
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44848
diff changeset
   349
  apply (rule_tac [2] mult_eq_1_iff, fastforce)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   350
  done
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 nat_mult_eq_1_iff [simp]: "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1" 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
   353
  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
   354
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   355
lemma nat_1_eq_mult_iff [simp]: "1 = m * n \<longleftrightarrow> m = 1 \<and> n = 1" 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
   356
  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
   357
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   358
lemma mult_cancel1 [simp]: "k * m = k * n \<longleftrightarrow> m = n \<or> k = 0" for k m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   359
proof -
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   360
  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
   361
  proof (induct n arbitrary: m)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   362
    case 0
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   363
    then show "m = 0" by simp
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   364
  next
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   365
    case (Suc n)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   366
    then show "m = Suc n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   367
      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
   368
  qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   369
  then show ?thesis by auto
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   370
qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   371
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   372
lemma mult_cancel2 [simp]: "m * k = n * k \<longleftrightarrow> m = n \<or> k = 0" for k m n :: nat
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   373
  by (simp add: mult.commute)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   374
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   375
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
   376
  by (subst mult_cancel1) simp
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   377
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   378
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   379
subsection \<open>Orders on @{typ nat}\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   380
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   381
subsubsection \<open>Operation definition\<close>
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   382
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   383
instantiation nat :: linorder
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   384
begin
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   385
55575
a5e33e18fb5c moved 'primrec' up (for real this time) and removed temporary 'old_primrec'
blanchet
parents: 55534
diff changeset
   386
primrec less_eq_nat where
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 61070
diff changeset
   387
  "(0::nat) \<le> n \<longleftrightarrow> True"
44325
84696670feb1 more uniform formatting of specifications
haftmann
parents: 44278
diff changeset
   388
| "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
   389
28514
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
   390
declare less_eq_nat.simps [simp del]
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   391
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   392
lemma le0 [iff]: "0 \<le> n" for n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   393
  by (simp add: less_eq_nat.simps)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   394
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   395
lemma [code]: "0 \<le> n \<longleftrightarrow> True" for n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   396
  by simp
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   397
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   398
definition less_nat where
28514
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
   399
  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
   400
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   401
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
   402
  by (simp add: less_eq_nat.simps(2))
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   403
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   404
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
   405
  unfolding less_eq_Suc_le ..
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   406
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   407
lemma le_0_eq [iff]: "n \<le> 0 \<longleftrightarrow> n = 0" for n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   408
  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
   409
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   410
lemma not_less0 [iff]: "\<not> n < 0" for n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   411
  by (simp add: less_eq_Suc_le)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   412
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   413
lemma less_nat_zero_code [code]: "n < 0 \<longleftrightarrow> False" for n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   414
  by simp
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   415
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   416
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
   417
  by (simp add: less_eq_Suc_le)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   418
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   419
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
   420
  by (simp add: less_eq_Suc_le)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   421
56194
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   422
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
   423
  by (cases m) auto
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   424
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   425
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   426
  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
   427
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   428
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
   429
  by (cases n) (auto intro: le_SucI)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   430
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   431
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   432
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   433
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   434
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   435
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   436
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
   437
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
   438
proof
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   439
  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
   440
  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
   441
  proof (induct n arbitrary: m)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   442
    case 0
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   443
    then show ?case 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
   444
  next
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   445
    case (Suc n)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   446
    then show ?case 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
   447
  qed
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   448
  show "n \<le> n" by (induct n) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   449
  then show "n = m" if "n \<le> m" and "m \<le> n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   450
    using that by (induct n arbitrary: m)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   451
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   452
  show "n \<le> q" if "n \<le> m" and "m \<le> q"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   453
    using that
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   454
  proof (induct n arbitrary: m q)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   455
    case 0
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   456
    show ?case by simp
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   457
  next
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   458
    case (Suc n)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   459
    then show ?case
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   460
      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
   461
        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
   462
        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
   463
  qed
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   464
  show "n \<le> m \<or> m \<le> n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   465
    by (induct n arbitrary: m)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   466
      (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
   467
qed
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   468
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   469
end
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   470
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52435
diff changeset
   471
instantiation nat :: order_bot
29652
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   472
begin
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   473
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   474
definition bot_nat :: nat where "bot_nat = 0"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   475
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   476
instance by standard (simp add: bot_nat_def)
29652
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   477
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   478
end
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   479
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
   480
instance nat :: no_top
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 61144
diff changeset
   481
  by standard (auto intro: less_Suc_eq_le [THEN iffD2])
52289
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 51329
diff changeset
   482
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
   483
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   484
subsubsection \<open>Introduction properties\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   485
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   486
lemma lessI [iff]: "n < Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   487
  by (simp add: less_Suc_eq_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   488
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   489
lemma zero_less_Suc [iff]: "0 < Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   490
  by (simp add: less_Suc_eq_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   491
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   492
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   493
subsubsection \<open>Elimination properties\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   494
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   495
lemma less_not_refl: "\<not> n < n" for n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   496
  by (rule order_less_irrefl)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   497
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   498
lemma less_not_refl2: "n < m \<Longrightarrow> m \<noteq> n" 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
   499
  by (rule not_sym) (rule less_imp_neq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   500
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   501
lemma less_not_refl3: "s < t \<Longrightarrow> s \<noteq> t" for s t :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   502
  by (rule less_imp_neq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   503
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   504
lemma less_irrefl_nat: "n < n \<Longrightarrow> R" for n :: nat
26335
961bbcc9d85b removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents: 26315
diff changeset
   505
  by (rule notE, rule less_not_refl)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   506
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   507
lemma less_zeroE: "n < 0 \<Longrightarrow> R" for n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   508
  by (rule notE) (rule not_less0)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   509
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   510
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
   511
  unfolding less_Suc_eq_le le_less ..
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   512
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
   513
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
   514
  by (simp add: less_Suc_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   515
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   516
lemma less_one [iff]: "n < 1 \<longleftrightarrow> n = 0" 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
   517
  unfolding One_nat_def by (rule less_Suc0)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   518
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   519
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
   520
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   521
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   522
text \<open>"Less than" is antisymmetric, sort of\<close>
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   523
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   524
  unfolding not_less less_Suc_eq_le by (rule antisym)
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   525
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   526
lemma nat_neq_iff: "m \<noteq> n \<longleftrightarrow> m < n \<or> n < m" for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   527
  by (rule linorder_neq_iff)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   528
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   529
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   530
subsubsection \<open>Inductive (?) properties\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   531
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   532
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
   533
  unfolding less_eq_Suc_le [of m] le_less by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   534
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   535
lemma lessE:
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   536
  assumes major: "i < k"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   537
    and 1: "k = Suc i \<Longrightarrow> P"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   538
    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
   539
  shows P
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   540
proof -
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   541
  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
   542
    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
   543
  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
   544
    by (auto simp add: less_le)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   545
  with 1 2 show P by auto
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   546
qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   547
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   548
lemma less_SucE:
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   549
  assumes major: "m < Suc n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   550
    and less: "m < n \<Longrightarrow> P"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   551
    and eq: "m = n \<Longrightarrow> P"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   552
  shows P
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   553
  apply (rule major [THEN lessE])
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   554
  apply (rule eq, blast)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   555
  apply (rule less, blast)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   556
  done
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_lessE:
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   559
  assumes major: "Suc i < k"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   560
    and minor: "\<And>j. i < j \<Longrightarrow> k = Suc j \<Longrightarrow> P"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   561
  shows P
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   562
  apply (rule major [THEN lessE])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   563
  apply (erule lessI [THEN minor])
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   564
  apply (erule Suc_lessD [THEN minor], assumption)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   565
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   566
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   567
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
   568
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   569
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   570
lemma less_trans_Suc:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   571
  assumes le: "i < j"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   572
  shows "j < k \<Longrightarrow> Suc i < k"
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   573
  apply (induct k, simp_all)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   574
  using le
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   575
  apply (simp add: less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   576
  apply (blast dest: Suc_lessD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   577
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   578
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   579
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
   580
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   581
  unfolding not_less less_Suc_eq_le ..
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   582
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   583
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   584
  unfolding not_le Suc_le_eq ..
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
   585
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   586
text \<open>Properties of "less than or equal"\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   587
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   588
lemma le_imp_less_Suc: "m \<le> n \<Longrightarrow> m < Suc n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   589
  unfolding less_Suc_eq_le .
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   590
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   591
lemma Suc_n_not_le_n: "\<not> Suc n \<le> n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   592
  unfolding not_le less_Suc_eq_le ..
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   593
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   594
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   595
  by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   596
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   597
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
   598
  by (drule le_Suc_eq [THEN iffD1], iprover+)
13449
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_leI: "m < n \<Longrightarrow> Suc(m) \<le> n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   601
  unfolding Suc_le_eq .
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   602
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   603
text \<open>Stronger version of \<open>Suc_leD\<close>\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   604
lemma Suc_le_lessD: "Suc m \<le> n \<Longrightarrow> m < n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   605
  unfolding Suc_le_eq .
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   606
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   607
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
   608
  unfolding less_eq_Suc_le by (rule Suc_leD)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   609
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   610
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
   611
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
   612
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   613
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   614
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
   615
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   616
lemma less_or_eq_imp_le: "m < n \<or> 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
   617
  unfolding le_less .
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   618
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   619
lemma le_eq_less_or_eq: "m \<le> n \<longleftrightarrow> m < n \<or> m = n" for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   620
  by (rule le_less)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   621
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   622
text \<open>Useful with \<open>blast\<close>.\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   623
lemma eq_imp_le: "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
   624
  by auto
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   625
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   626
lemma le_refl: "n \<le> n" for n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   627
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   628
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   629
lemma le_trans: "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k" for i j k :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   630
  by (rule order_trans)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   631
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   632
lemma le_antisym: "m \<le> n \<Longrightarrow> n \<le> m \<Longrightarrow> m = n" for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   633
  by (rule antisym)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   634
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   635
lemma nat_less_le: "m < n \<longleftrightarrow> m \<le> n \<and> m \<noteq> n" for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   636
  by (rule less_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   637
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   638
lemma le_neq_implies_less: "m \<le> n \<Longrightarrow> m \<noteq> n \<Longrightarrow> m < n" for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   639
  unfolding less_le ..
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   640
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   641
lemma nat_le_linear: "m \<le> n | n \<le> m" for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   642
  by (rule linear)
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   643
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   644
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
15921
b6e345548913 Fixing a problem with lin.arith.
nipkow
parents: 15539
diff changeset
   645
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   646
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
   647
  unfolding less_Suc_eq_le by auto
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   648
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   649
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
   650
  unfolding not_less by (rule le_less_Suc_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   651
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   652
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   653
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   654
lemma not0_implies_Suc: "n \<noteq> 0 \<Longrightarrow> \<exists>m. n = Suc m"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   655
  by (cases n) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   656
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   657
lemma gr0_implies_Suc: "n > 0 \<Longrightarrow> \<exists>m. n = Suc m"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   658
  by (cases n) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   659
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   660
lemma gr_implies_not0: "m < n \<Longrightarrow> n \<noteq> 0" for m n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   661
  by (cases n) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   662
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   663
lemma neq0_conv[iff]: "n \<noteq> 0 \<longleftrightarrow> 0 < n" for n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   664
  by (cases n) simp_all
25140
273772abbea2 More changes from >0 to ~=0::nat
nipkow
parents: 25134
diff changeset
   665
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   666
text \<open>This theorem is useful with \<open>blast\<close>\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   667
lemma gr0I: "(n = 0 \<Longrightarrow> False) \<Longrightarrow> 0 < n" for n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   668
  by (rule neq0_conv[THEN iffD1], iprover)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   669
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   670
lemma gr0_conv_Suc: "0 < n \<longleftrightarrow> (\<exists>m. n = Suc m)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   671
  by (fast intro: not0_implies_Suc)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   672
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   673
lemma not_gr0 [iff]: "\<not> 0 < n \<longleftrightarrow> n = 0" for n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   674
  using neq0_conv by blast
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   675
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   676
lemma Suc_le_D: "Suc n \<le> m' \<Longrightarrow> \<exists>m. m' = Suc m"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   677
  by (induct m') simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   678
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   679
text \<open>Useful in certain inductive arguments\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   680
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
   681
  by (cases m) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   682
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   683
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   684
subsubsection \<open>Monotonicity of Addition\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   685
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   686
lemma Suc_pred [simp]: "n > 0 \<Longrightarrow> Suc (n - Suc 0) = n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   687
  by (simp add: diff_Suc split: nat.split)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   688
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   689
lemma Suc_diff_1 [simp]: "0 < n \<Longrightarrow> Suc (n - 1) = n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   690
  unfolding One_nat_def by (rule Suc_pred)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   691
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   692
lemma nat_add_left_cancel_le [simp]: "k + m \<le> k + n \<longleftrightarrow> m \<le> n" for k m n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   693
  by (induct k) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   694
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   695
lemma nat_add_left_cancel_less [simp]: "k + m < k + n \<longleftrightarrow> m < n" for k m n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   696
  by (induct k) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   697
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   698
lemma add_gr_0 [iff]: "m + n > 0 \<longleftrightarrow> m > 0 \<or> n > 0" for m n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   699
  by (auto dest: gr0_implies_Suc)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   700
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   701
text \<open>strict, in 1st argument\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   702
lemma add_less_mono1: "i < j \<Longrightarrow> i + k < j + k" for i j k :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   703
  by (induct k) simp_all
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   704
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   705
text \<open>strict, in both arguments\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   706
lemma add_less_mono: "i < j \<Longrightarrow> k < l \<Longrightarrow> i + k < j + l" 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
   707
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   708
  apply (induct j, simp_all)
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   709
  done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   710
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   711
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
   712
lemma less_imp_Suc_add: "m < n \<Longrightarrow> \<exists>k. n = Suc (m + k)"
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   713
  apply (induct n)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   714
  apply (simp_all add: order_le_less)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   715
  apply (blast elim!: less_SucE
35047
1b2bae06c796 hide fact Nat.add_0_right; make add_0_right from Groups priority
haftmann
parents: 35028
diff changeset
   716
               intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   717
  done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   718
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   719
lemma le_Suc_ex: "k \<le> l \<Longrightarrow> (\<exists>n. l = k + n)" for k l :: nat
56194
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   720
  by (auto simp: less_Suc_eq_le[symmetric] dest: less_imp_Suc_add)
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   721
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   722
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
   723
lemma mult_less_mono2:
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   724
  fixes i j :: nat
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   725
  assumes "i < j" and "0 < k"
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   726
  shows "k * i < k * j"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   727
  using \<open>0 < k\<close>
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   728
proof (induct k)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   729
  case 0
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   730
  then show ?case by simp
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   731
next
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   732
  case (Suc k)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   733
  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
   734
    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
   735
qed
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   736
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   737
text \<open>Addition is the inverse of subtraction:
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   738
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}.\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   739
lemma add_diff_inverse_nat: "\<not> m < n \<Longrightarrow> n + (m - n) = m" for m n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   740
  by (induct m n rule: diff_induct) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   741
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   742
lemma nat_le_iff_add: "m \<le> n \<longleftrightarrow> (\<exists>k. n = m + k)" for m n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   743
  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
   744
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
   745
text\<open>The naturals form an ordered \<open>semidom\<close> and a \<open>dioid\<close>\<close>
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
   746
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34208
diff changeset
   747
instance nat :: linordered_semidom
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   748
proof
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   749
  fix m n q :: nat
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   750
  show "0 < (1::nat)" by simp
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   751
  show "m \<le> n \<Longrightarrow> q + m \<le> q + n" by simp
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   752
  show "m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n" by (simp add: mult_less_mono2)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   753
  show "m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> m * n \<noteq> 0" by simp
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   754
  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
   755
    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
   756
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
   757
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
   758
instance nat :: dioid
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   759
  by standard (rule nat_le_iff_add)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62376
diff changeset
   760
declare le0[simp del] -- \<open>This is now @{thm zero_le}\<close>
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62376
diff changeset
   761
declare le_0_eq[simp del] -- \<open>This is now @{thm le_zero_eq}\<close>
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62376
diff changeset
   762
declare not_less0[simp del] -- \<open>This is now @{thm not_less_zero}\<close>
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62376
diff changeset
   763
declare not_gr0[simp del] -- \<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
   764
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   765
instance nat :: ordered_cancel_comm_monoid_add ..
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   766
instance nat :: ordered_cancel_comm_monoid_diff ..
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   767
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   768
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   769
subsubsection \<open>@{term min} and @{term max}\<close>
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   770
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   771
lemma mono_Suc: "mono Suc"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   772
  by (rule monoI) simp
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   773
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   774
lemma min_0L [simp]: "min 0 n = 0" for n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   775
  by (rule min_absorb1) simp
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   776
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   777
lemma min_0R [simp]: "min n 0 = 0" for n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   778
  by (rule min_absorb2) simp
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   779
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   780
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
   781
  by (simp add: mono_Suc min_of_mono)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   782
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   783
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
   784
  by (simp split: nat.split)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   785
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   786
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
   787
  by (simp split: nat.split)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   788
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   789
lemma max_0L [simp]: "max 0 n = n" for n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   790
  by (rule max_absorb2) simp
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   791
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   792
lemma max_0R [simp]: "max n 0 = n" for n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   793
  by (rule max_absorb1) simp
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   794
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   795
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc (max m n)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   796
  by (simp add: mono_Suc max_of_mono)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   797
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   798
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
   799
  by (simp split: nat.split)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   800
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   801
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
   802
  by (simp split: nat.split)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   803
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   804
lemma nat_mult_min_left: "min m n * q = min (m * q) (n * q)" for m n q :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   805
  by (simp add: min_def not_le)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   806
    (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
   807
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   808
lemma nat_mult_min_right: "m * min n q = min (m * n) (m * q)" for m n q :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   809
  by (simp add: min_def not_le)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   810
    (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
   811
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   812
lemma nat_add_max_left: "max m n + q = max (m + q) (n + q)" for m n q :: nat
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   813
  by (simp add: max_def)
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   814
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   815
lemma nat_add_max_right: "m + max n q = max (m + n) (m + q)" for m n q :: nat
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   816
  by (simp add: max_def)
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   817
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   818
lemma nat_mult_max_left: "max m n * q = max (m * q) (n * q)" for m n q :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   819
  by (simp add: max_def not_le)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   820
    (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
   821
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   822
lemma nat_mult_max_right: "m * max n q = max (m * n) (m * q)" for m n q :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   823
  by (simp add: max_def not_le)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   824
    (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
   825
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   826
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   827
subsubsection \<open>Additional theorems about @{term "op \<le>"}\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   828
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   829
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
   830
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   831
instance nat :: wellorder
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   832
proof
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   833
  fix P and n :: nat
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   834
  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
   835
  have "\<And>q. q \<le> n \<Longrightarrow> P q"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   836
  proof (induct n)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   837
    case (0 n)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   838
    have "P 0" by (rule step) auto
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   839
    then show ?case using 0 by auto
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   840
  next
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   841
    case (Suc m n)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   842
    then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   843
    then show ?case
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   844
    proof
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   845
      assume "n \<le> m"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   846
      then show "P n" by (rule Suc(1))
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   847
    next
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   848
      assume n: "n = Suc m"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   849
      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
   850
    qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   851
  qed
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   852
  then show "P n" by auto
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   853
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   854
57015
842bb6d36263 added lemma
nipkow
parents: 56194
diff changeset
   855
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   856
lemma Least_eq_0[simp]: "P 0 \<Longrightarrow> Least P = 0" for P :: "nat \<Rightarrow> bool"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   857
  by (rule Least_equality[OF _ le0])
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   858
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   859
lemma Least_Suc: "P n \<Longrightarrow> \<not> P 0 \<Longrightarrow> (LEAST n. P n) = Suc (LEAST m. P (Suc m))"
47988
e4b69e10b990 tuned proofs;
wenzelm
parents: 47489
diff changeset
   860
  apply (cases n, auto)
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   861
  apply (frule LeastI)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   862
  apply (drule_tac P = "\<lambda>x. P (Suc x) " in LeastI)
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   863
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   864
  apply (erule_tac [2] Least_le)
47988
e4b69e10b990 tuned proofs;
wenzelm
parents: 47489
diff changeset
   865
  apply (cases "LEAST x. P x", auto)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   866
  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
   867
  apply (blast intro: order_antisym)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   868
  done
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   869
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   870
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)"
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   871
  apply (erule (1) Least_Suc [THEN ssubst])
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   872
  apply simp
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   873
  done
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   874
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   875
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" for P :: "nat \<Rightarrow> bool"
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   876
  apply (cases n)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   877
   apply blast
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   878
  apply (rule_tac x="LEAST k. P k" in exI)
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   879
  apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   880
  done
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   881
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   882
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)" for P :: "nat \<Rightarrow> bool"
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
   883
  unfolding One_nat_def
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   884
  apply (cases n)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   885
   apply blast
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   886
  apply (frule (1) ex_least_nat_le)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   887
  apply (erule exE)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   888
  apply (case_tac k)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   889
   apply simp
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   890
  apply (rename_tac k1)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   891
  apply (rule_tac x=k1 in exI)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   892
  apply (auto simp add: less_eq_Suc_le)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   893
  done
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   894
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   895
lemma nat_less_induct:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   896
  fixes P :: "nat \<Rightarrow> bool"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   897
  assumes "\<And>n. \<forall>m. m < n \<longrightarrow> P m \<Longrightarrow> P n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   898
  shows "P n"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   899
  using assms less_induct by blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   900
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   901
lemma measure_induct_rule [case_names less]:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   902
  fixes f :: "'a \<Rightarrow> nat"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   903
  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
   904
  shows "P a"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   905
  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
   906
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   907
text \<open>old style induction rules:\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   908
lemma measure_induct:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   909
  fixes f :: "'a \<Rightarrow> nat"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   910
  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
   911
  by (rule measure_induct_rule [of f P a]) iprover
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   912
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   913
lemma full_nat_induct:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   914
  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
   915
  shows "P n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   916
  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
   917
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   918
text\<open>An induction rule for establishing binary relations\<close>
62683
ddd1c864408b clarified rule structure;
wenzelm
parents: 62608
diff changeset
   919
lemma less_Suc_induct [consumes 1]:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   920
  assumes less: "i < j"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   921
    and step: "\<And>i. P i (Suc i)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   922
    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
   923
  shows "P i j"
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   924
proof -
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   925
  from less obtain k where j: "j = Suc (i + k)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   926
    by (auto dest: less_imp_Suc_add)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   927
  have "P i (Suc (i + k))"
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   928
  proof (induct k)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   929
    case 0
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   930
    show ?case by (simp add: step)
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   931
  next
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   932
    case (Suc k)
31714
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
   933
    have "0 + i < Suc k + i" by (rule add_less_mono1) simp
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   934
    then have "i < Suc (i + k)" by (simp add: add.commute)
31714
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
   935
    from trans[OF this lessI Suc step]
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
   936
    show ?case by simp
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   937
  qed
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   938
  then show "P i j" by (simp add: j)
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   939
qed
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   940
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
   941
text \<open>
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
   942
  The method of infinite descent, frequently used in number theory.
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
   943
  Provided by Roelof Oosterhuis.
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
   944
  \<open>P n\<close> is true for all natural numbers if
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
   945
  \<^item> case ``0'': given \<open>n = 0\<close> prove \<open>P n\<close>
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
   946
  \<^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
   947
    a smaller natural number \<open>m\<close> such that \<open>\<not> P m\<close>.
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
   948
\<close>
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
   949
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   950
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
   951
  \<comment> \<open>compact version without explicit base case\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   952
  by (induct n rule: less_induct) auto
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   953
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
   954
lemma infinite_descent0 [case_names 0 smaller]:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   955
  fixes P :: "nat \<Rightarrow> bool"
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
   956
  assumes "P 0"
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
   957
    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
   958
  shows "P n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   959
  apply (rule infinite_descent)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   960
  using assms
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   961
  apply (case_tac "n > 0")
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   962
  apply auto
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   963
  done
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   964
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   965
text \<open>
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
   966
  Infinite descent using a mapping to \<open>nat\<close>:
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
   967
  \<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
   968
  \<^item> case ``0'': given \<open>V x = 0\<close> prove \<open>P x\<close>
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
   969
  \<^item> ``smaller'': given \<open>V x > 0\<close> and \<open>\<not> P x\<close> prove
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
   970
  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
   971
\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   972
corollary infinite_descent0_measure [case_names 0 smaller]:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   973
  fixes V :: "'a \<Rightarrow> nat"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   974
  assumes 1: "\<And>x. V x = 0 \<Longrightarrow> P x"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   975
    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
   976
  shows "P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   977
proof -
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   978
  obtain n where "n = V x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   979
  moreover have "\<And>x. V x = n \<Longrightarrow> P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   980
  proof (induct n rule: infinite_descent0)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   981
    case 0
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   982
    with 1 show "P x" by auto
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   983
  next
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   984
    case (smaller n)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   985
    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
   986
    with 2 obtain y where "V y < V x \<and> \<not> P y" by auto
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
   987
    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
   988
    then show ?case by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   989
  qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   990
  ultimately show "P x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   991
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   992
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   993
text\<open>Again, without explicit base case:\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   994
lemma infinite_descent_measure:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   995
  fixes V :: "'a \<Rightarrow> nat"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
   996
  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
   997
  shows "P x"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   998
proof -
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   999
  from assms obtain n where "n = V x" by auto
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1000
  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
  1001
  proof (induct n rule: infinite_descent, auto)
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1002
    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
  1003
      using assms and that by auto
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1004
  qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1005
  ultimately show "P x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1006
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1007
63111
caa0c513bbca tuned document;
wenzelm
parents: 63110
diff changeset
  1008
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
  1009
lemma less_mono_imp_le_mono:
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1010
  fixes f :: "nat \<Rightarrow> nat"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1011
    and i j :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1012
  assumes "\<And>i j::nat. i < j \<Longrightarrow> f i < f j"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1013
    and "i \<le> j"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1014
  shows "f i \<le> f j"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1015
  using assms by (auto simp add: order_le_less)
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1016
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1017
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1018
text \<open>non-strict, in 1st argument\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1019
lemma add_le_mono1: "i \<le> j \<Longrightarrow> i + k \<le> j + k" for i j k :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1020
  by (rule add_right_mono)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1021
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1022
text \<open>non-strict, in both arguments\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1023
lemma add_le_mono: "i \<le> j \<Longrightarrow> k \<le> l \<Longrightarrow> i + k \<le> j + l" for i j k l :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1024
  by (rule add_mono)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1025
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1026
lemma le_add2: "n \<le> m + n" for m n :: nat
62608
19f87fa0cfcb more theorems on orderings
haftmann
parents: 62502
diff changeset
  1027
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1028
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1029
lemma le_add1: "n \<le> n + m" for m n :: nat
62608
19f87fa0cfcb more theorems on orderings
haftmann
parents: 62502
diff changeset
  1030
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1031
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1032
lemma less_add_Suc1: "i < Suc (i + m)"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1033
  by (rule le_less_trans, rule le_add1, rule lessI)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1034
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1035
lemma less_add_Suc2: "i < Suc (m + i)"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1036
  by (rule le_less_trans, rule le_add2, rule lessI)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1037
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1038
lemma less_iff_Suc_add: "m < n \<longleftrightarrow> (\<exists>k. n = Suc (m + k))"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1039
  by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1040
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1041
lemma trans_le_add1: "i \<le> j \<Longrightarrow> i \<le> j + m" for i j m :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1042
  by (rule le_trans, assumption, rule le_add1)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1043
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1044
lemma trans_le_add2: "i \<le> j \<Longrightarrow> i \<le> m + j" for i j m :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1045
  by (rule le_trans, assumption, rule le_add2)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1046
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1047
lemma trans_less_add1: "i < j \<Longrightarrow> i < j + m" for i j m :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1048
  by (rule less_le_trans, assumption, rule le_add1)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1049
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1050
lemma trans_less_add2: "i < j \<Longrightarrow> i < m + j" for i j m :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1051
  by (rule less_le_trans, assumption, rule le_add2)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1052
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1053
lemma add_lessD1: "i + j < k \<Longrightarrow> i < k" for i j k :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1054
  by (rule le_less_trans [of _ "i+j"]) (simp_all add: le_add1)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1055
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1056
lemma not_add_less1 [iff]: "\<not> i + j < i" for i j :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1057
  apply (rule notI)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1058
  apply (drule add_lessD1)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1059
  apply (erule less_irrefl [THEN notE])
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1060
  done
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1061
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1062
lemma not_add_less2 [iff]: "\<not> j + i < i" for i j :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1063
  by (simp add: add.commute)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1064
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1065
lemma add_leD1: "m + k \<le> n \<Longrightarrow> m \<le> n" for k m n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1066
  by (rule order_trans [of _ "m+k"]) (simp_all add: le_add1)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1067
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1068
lemma add_leD2: "m + k \<le> n \<Longrightarrow> k \<le> n" for k m n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1069
  apply (simp add: add.commute)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1070
  apply (erule add_leD1)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1071
  done
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1072
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1073
lemma add_leE: "m + k \<le> n \<Longrightarrow> (m \<le> n \<Longrightarrow> k \<le> n \<Longrightarrow> R) \<Longrightarrow> R" for k m n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1074
  by (blast dest: add_leD1 add_leD2)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1075
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1076
text \<open>needs \<open>\<And>k\<close> for \<open>ac_simps\<close> to work\<close>
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1077
lemma less_add_eq_less: "\<And>k::nat. k < l \<Longrightarrow> m + l = k + n \<Longrightarrow> m < n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1078
  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
  1079
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1080
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1081
subsubsection \<open>More results about difference\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1082
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1083
lemma Suc_diff_le: "n \<le> m \<Longrightarrow> Suc m - n = Suc (m - n)"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1084
  by (induct m n rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1085
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1086
lemma diff_less_Suc: "m - n < Suc m"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1087
apply (induct m n rule: diff_induct)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1088
apply (erule_tac [3] less_SucE)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1089
apply (simp_all add: less_Suc_eq)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1090
done
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1091
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1092
lemma diff_le_self [simp]: "m - n \<le> m" for m n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1093
  by (induct m n rule: diff_induct) (simp_all add: le_SucI)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1094
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1095
lemma less_imp_diff_less: "j < k \<Longrightarrow> j - n < k" for j k n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1096
  by (rule le_less_trans, rule diff_le_self)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1097
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1098
lemma diff_Suc_less [simp]: "0 < n \<Longrightarrow> n - Suc i < n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1099
  by (cases n) (auto simp add: le_simps)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1100
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1101
lemma diff_add_assoc: "k \<le> j \<Longrightarrow> (i + j) - k = i + (j - k)" for i j k :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1102
  by (induct j k rule: diff_induct) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1103
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1104
lemma add_diff_assoc [simp]: "k \<le> j \<Longrightarrow> i + (j - k) = i + j - k" for i j k :: nat
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
  1105
  by (fact diff_add_assoc [symmetric])
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
  1106
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1107
lemma diff_add_assoc2: "k \<le> j \<Longrightarrow> (j + i) - k = (j - k) + i" for i j k :: nat
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
  1108
  by (simp add: ac_simps)
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
  1109
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1110
lemma add_diff_assoc2 [simp]: "k \<le> j \<Longrightarrow> j - k + i = j + i - k" for i j k :: nat
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
  1111
  by (fact diff_add_assoc2 [symmetric])
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1112
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1113
lemma le_imp_diff_is_add: "i \<le> j \<Longrightarrow> (j - i = k) = (j = k + i)" for i j k :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1114
  by auto
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1115
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1116
lemma diff_is_0_eq [simp]: "m - n = 0 \<longleftrightarrow> m \<le> n" for m n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1117
  by (induct m n rule: diff_induct) simp_all
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1118
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1119
lemma diff_is_0_eq' [simp]: "m \<le> n \<Longrightarrow> m - n = 0" for m n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1120
  by (rule iffD2, rule diff_is_0_eq)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1121
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1122
lemma zero_less_diff [simp]: "0 < n - m \<longleftrightarrow> m < n" for m n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1123
  by (induct m n rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1124
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1125
lemma less_imp_add_positive:
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1126
  assumes "i < j"
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1127
  shows "\<exists>k::nat. 0 < k \<and> i + k = j"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1128
proof
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1129
  from assms show "0 < j - i \<and> i + (j - i) = j"
23476
839db6346cc8 fix looping simp rule
huffman
parents: 23438
diff changeset
  1130
    by (simp add: order_less_imp_le)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1131
qed
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
  1132
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1133
text \<open>a nice rewrite for bounded subtraction\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1134
lemma nat_minus_add_max: "n - m + m = max n m" for m n :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1135
    by (simp add: max_def not_le order_less_imp_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1136
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1137
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
  1138
  for a b :: nat
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1139
    \<comment> \<open>elimination of \<open>-\<close> on \<open>nat\<close>\<close>
62365
8a105c235b1f sorted out some duplicate fact bindings
haftmann
parents: 62344
diff changeset
  1140
  by (cases "a < b")
8a105c235b1f sorted out some duplicate fact bindings
haftmann
parents: 62344
diff changeset
  1141
    (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
  1142
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1143
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
  1144
  for a b :: nat
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1145
    \<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
  1146
  by (auto split: nat_diff_split)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1147
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1148
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
  1149
  by simp
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1150
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1151
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
  1152
  unfolding One_nat_def by (cases m) simp_all
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1153
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1154
lemma mult_eq_if: "m * n = (if m = 0 then 0 else n + ((m - 1) * n))" for m n :: nat
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1155
  unfolding One_nat_def by (cases m) simp_all
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1156
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1157
lemma Suc_diff_eq_diff_pred: "0 < n \<Longrightarrow> Suc m - n = m - (n - 1)"
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1158
  unfolding One_nat_def by (cases n) simp_all
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1159
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1160
lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1161
  unfolding One_nat_def by (cases m) simp_all
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1162
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1163
lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1164
  by (fact Let_def)
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1165
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1166
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1167
subsubsection \<open>Monotonicity of multiplication\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1168
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1169
lemma mult_le_mono1: "i \<le> j \<Longrightarrow> i * k \<le> j * k" for i j k :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1170
  by (simp add: mult_right_mono)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1171
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1172
lemma mult_le_mono2: "i \<le> j \<Longrightarrow> k * i \<le> k * j" for i j k :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1173
  by (simp add: mult_left_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1174
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1175
text \<open>\<open>\<le>\<close> monotonicity, BOTH arguments\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1176
lemma mult_le_mono: "i \<le> j \<Longrightarrow> k \<le> l \<Longrightarrow> i * k \<le> j * l" for i j k l :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1177
  by (simp add: mult_mono)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1178
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1179
lemma mult_less_mono1: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> i * k < j * k" for i j k :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1180
  by (simp add: mult_strict_right_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1181
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1182
text\<open>Differs from the standard \<open>zero_less_mult_iff\<close> in that
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1183
      there are no negative numbers.\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1184
lemma nat_0_less_mult_iff [simp]: "0 < m * n \<longleftrightarrow> 0 < m \<and> 0 < n" for m n :: nat
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1185
  apply (induct m)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1186
   apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1187
  apply (case_tac n)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1188
   apply simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1189
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1190
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1191
lemma one_le_mult_iff [simp]: "Suc 0 \<le> m * n \<longleftrightarrow> Suc 0 \<le> m \<and> Suc 0 \<le> n"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1192
  apply (induct m)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1193
   apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1194
  apply (case_tac n)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1195
   apply simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1196
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1197
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1198
lemma mult_less_cancel2 [simp]: "m * k < n * k \<longleftrightarrow> 0 < k \<and> m < n" for k m n :: nat
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1199
  apply (safe intro!: mult_less_mono1)
47988
e4b69e10b990 tuned proofs;
wenzelm
parents: 47489
diff changeset
  1200
  apply (cases k, auto)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1201
  apply (simp add: linorder_not_le [symmetric])
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1202
  apply (blast intro: mult_le_mono1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1203
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1204
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1205
lemma mult_less_cancel1 [simp]: "k * m < k * n \<longleftrightarrow> 0 < k \<and> m < n" for k m n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1206
  by (simp add: mult.commute [of k])
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1207
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1208
lemma mult_le_cancel1 [simp]: "k * m \<le> k * n \<longleftrightarrow> (0 < k \<longrightarrow> m \<le> n)" for k m n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1209
  by (simp add: linorder_not_less [symmetric], auto)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1210
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1211
lemma mult_le_cancel2 [simp]: "m * k \<le> n * k \<longleftrightarrow> (0 < k \<longrightarrow> m \<le> n)" for k m n :: nat
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1212
  by (simp add: linorder_not_less [symmetric], auto)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1213
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1214
lemma Suc_mult_less_cancel1: "Suc k * m < Suc k * n \<longleftrightarrow> m < n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1215
  by (subst mult_less_cancel1) simp
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1216
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1217
lemma Suc_mult_le_cancel1: "Suc k * m \<le> Suc k * n \<longleftrightarrow> m \<le> n"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1218
  by (subst mult_le_cancel1) simp
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1219
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1220
lemma le_square: "m \<le> m * m" for m :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1221
  by (cases m) (auto intro: le_add1)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1222
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1223
lemma le_cube: "m \<le> m * (m * m)" for m :: nat
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1224
  by (cases m) (auto intro: le_add1)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1225
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1226
text \<open>Lemma for \<open>gcd\<close>\<close>
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1227
lemma mult_eq_self_implies_10: "m = m * n \<Longrightarrow> n = 1 \<or> m = 0" for m n :: nat
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1228
  apply (drule sym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1229
  apply (rule disjCI)
63113
fe31996e3898 removed odd cases rule (see also 8cb42cd97579);
wenzelm
parents: 63111
diff changeset
  1230
  apply (rule linorder_cases, erule_tac [2] _)
25157
8b80535cd017 random tidying of proofs
paulson
parents: 25145
diff changeset
  1231
   apply (drule_tac [2] mult_less_mono2)
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
  1232
    apply (auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1233
  done
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
  1234
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1235
lemma mono_times_nat:
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1236
  fixes n :: nat
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1237
  assumes "n > 0"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1238
  shows "mono (times n)"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1239
proof
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1240
  fix m q :: nat
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1241
  assume "m \<le> q"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1242
  with assms show "n * m \<le> n * q" by simp
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1243
qed
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1244
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1245
text \<open>the lattice order on @{typ nat}\<close>
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1246
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1247
instantiation nat :: distrib_lattice
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1248
begin
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1249
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1250
definition "(inf :: nat \<Rightarrow> nat \<Rightarrow> nat) = min"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1251
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1252
definition "(sup :: nat \<Rightarrow> nat \<Rightarrow> nat) = max"
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1253
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1254
instance
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1255
  by intro_classes
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1256
    (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1257
      intro: order_less_imp_le antisym elim!: order_trans order_less_trans)
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1258
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1259
end
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1260
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1261
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1262
subsection \<open>Natural operation of natural numbers on functions\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1263
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1264
text \<open>
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1265
  We use the same logical constant for the power operations on
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1266
  functions and relations, in order to share the same syntax.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1267
\<close>
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1268
45965
2af982715e5c generalized type signature to permit overloading on `set`
haftmann
parents: 45933
diff changeset
  1269
consts compow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1270
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1271
abbreviation compower :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^^" 80)
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1272
  where "f ^^ n \<equiv> compow n f"
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1273
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1274
notation (latex output)
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1275
  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1276
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1277
text \<open>\<open>f ^^ n = f o ... o f\<close>, the n-fold composition of \<open>f\<close>\<close>
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1278
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1279
overloading
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1280
  funpow \<equiv> "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1281
begin
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1282
55575
a5e33e18fb5c moved 'primrec' up (for real this time) and removed temporary 'old_primrec'
blanchet
parents: 55534
diff changeset
  1283
primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
44325
84696670feb1 more uniform formatting of specifications
haftmann
parents: 44278
diff changeset
  1284
  "funpow 0 f = id"
84696670feb1 more uniform formatting of specifications
haftmann
parents: 44278
diff changeset
  1285
| "funpow (Suc n) f = f o funpow n f"
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30686
diff changeset
  1286
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1287
end
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30966
diff changeset
  1288
62217
527488dc8b90 Reorganised a huge proof
paulson <lp15@cam.ac.uk>
parents: 61799
diff changeset
  1289
lemma funpow_0 [simp]: "(f ^^ 0) x = x"
527488dc8b90 Reorganised a huge proof
paulson <lp15@cam.ac.uk>
parents: 61799
diff changeset
  1290
  by simp
527488dc8b90 Reorganised a huge proof
paulson <lp15@cam.ac.uk>
parents: 61799
diff changeset
  1291
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1292
lemma funpow_Suc_right: "f ^^ Suc n = f ^^ n \<circ> f"
49723
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 49388
diff changeset
  1293
proof (induct n)
63110
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1294
  case 0
ccbdce905fca misc tuning and modernization;
wenzelm
parents: 63099
diff changeset
  1295
  then show ?case by simp
49723
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 49388
diff changeset
  1296
next
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 49388
diff changeset
  1297
  fix n
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 49388
diff changeset
  1298
  assume "f ^^ Suc n = f ^^ n \<circ> f"
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 49388
diff changeset
  1299
  then show "f ^^ Suc (Suc n) = f ^^ Suc n \<circ> f"
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 49388
diff changeset
  1300
    by (simp add: o_assoc)
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 49388
diff changeset
  1301
qed
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 49388
diff changeset
  1302
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 49388
diff changeset
  1303
lemmas funpow_simps_right = funpow.simps(1) funpow_Suc_right
bbc2942ba09f alternative simplification of ^^ to the righthand side;
haftmann
parents: 49388
diff changeset
  1304
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset