src/HOL/Nat.thy
author hoelzl
Wed Feb 10 18:43:19 2016 +0100 (2016-02-10)
changeset 62376 85f38d5f8807
parent 62365 8a105c235b1f
child 62378 85ed00c1fe7c
permissions -rw-r--r--
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.
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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 Fields
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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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ML_file "~~/src/Provers/Arith/fast_lin_arith.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 => ind" where
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  \<comment> \<open>the axiom of infinity in 2 parts\<close>
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  Suc_Rep_inject:       "Suc_Rep x = Suc_Rep y ==> x = y" and
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  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" where
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  "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: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==>
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    (!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n"
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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::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 => 0 | Suc k => k)"
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declare diff_Suc [simp del]
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lemma diff_0_eq_0 [simp, code]: "0 - n = (0::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 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::nat) * 0 = 0"
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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)::nat)"
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  by (induct m) (simp_all add: add.assoc)
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instance proof
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  fix 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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next
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  fix k m n :: nat
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  show "k * ((m::nat) - n) = (k * m) - (k * n)"
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    by (induct m n rule: diff_induct) simp_all
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qed
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end
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subsubsection \<open>Addition\<close>
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text \<open>Reasoning about \<open>m + 0 = 0\<close>, etc.\<close>
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lemma add_is_0 [iff]:
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  fixes m n :: nat
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  shows "(m + n = 0) = (m = 0 & n = 0)"
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  by (cases m) simp_all
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lemma add_is_1:
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  "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
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  by (cases m) simp_all
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lemma one_is_add:
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  "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"
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  by (rule trans, rule eq_commute, rule add_is_1)
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lemma add_eq_self_zero:
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  fixes m n :: nat
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  shows "m + n = m \<Longrightarrow> n = 0"
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  by (induct m) simp_all
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lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N"
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  apply (induct k)
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   apply simp
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  apply(drule comp_inj_on[OF _ inj_Suc])
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  apply (simp add:o_def)
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  done
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lemma Suc_eq_plus1: "Suc n = n + 1"
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  unfolding One_nat_def by simp
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lemma Suc_eq_plus1_left: "Suc n = 1 + n"
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  unfolding One_nat_def by simp
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subsubsection \<open>Difference\<close>
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lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
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  by (simp add: diff_diff_add)
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lemma diff_Suc_1 [simp]: "Suc n - 1 = n"
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  unfolding One_nat_def by simp
huffman@30093
   337
wenzelm@60758
   338
subsubsection \<open>Multiplication\<close>
haftmann@26072
   339
haftmann@26072
   340
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
haftmann@26072
   341
  by (induct m) auto
haftmann@26072
   342
huffman@30079
   343
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = Suc 0 & n = Suc 0)"
haftmann@26072
   344
  apply (induct m)
haftmann@26072
   345
   apply simp
haftmann@26072
   346
  apply (induct n)
haftmann@26072
   347
   apply auto
haftmann@26072
   348
  done
haftmann@26072
   349
blanchet@54147
   350
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = Suc 0 & n = Suc 0)"
haftmann@26072
   351
  apply (rule trans)
nipkow@44890
   352
  apply (rule_tac [2] mult_eq_1_iff, fastforce)
haftmann@26072
   353
  done
haftmann@26072
   354
huffman@30079
   355
lemma nat_mult_eq_1_iff [simp]: "m * n = (1::nat) \<longleftrightarrow> m = 1 \<and> n = 1"
huffman@30079
   356
  unfolding One_nat_def by (rule mult_eq_1_iff)
huffman@30079
   357
huffman@30079
   358
lemma nat_1_eq_mult_iff [simp]: "(1::nat) = m * n \<longleftrightarrow> m = 1 \<and> n = 1"
huffman@30079
   359
  unfolding One_nat_def by (rule one_eq_mult_iff)
huffman@30079
   360
haftmann@26072
   361
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
haftmann@26072
   362
proof -
haftmann@26072
   363
  have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
haftmann@26072
   364
  proof (induct n arbitrary: m)
haftmann@26072
   365
    case 0 then show "m = 0" by simp
haftmann@26072
   366
  next
haftmann@26072
   367
    case (Suc n) then show "m = Suc n"
haftmann@26072
   368
      by (cases m) (simp_all add: eq_commute [of "0"])
haftmann@26072
   369
  qed
haftmann@26072
   370
  then show ?thesis by auto
haftmann@26072
   371
qed
haftmann@26072
   372
haftmann@26072
   373
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
haftmann@57512
   374
  by (simp add: mult.commute)
haftmann@26072
   375
haftmann@26072
   376
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
haftmann@26072
   377
  by (subst mult_cancel1) simp
haftmann@26072
   378
haftmann@24995
   379
wenzelm@60758
   380
subsection \<open>Orders on @{typ nat}\<close>
wenzelm@60758
   381
wenzelm@60758
   382
subsubsection \<open>Operation definition\<close>
haftmann@24995
   383
haftmann@26072
   384
instantiation nat :: linorder
haftmann@25510
   385
begin
haftmann@25510
   386
blanchet@55575
   387
primrec less_eq_nat where
wenzelm@61076
   388
  "(0::nat) \<le> n \<longleftrightarrow> True"
haftmann@44325
   389
| "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
haftmann@26072
   390
haftmann@28514
   391
declare less_eq_nat.simps [simp del]
wenzelm@61076
   392
lemma le0 [iff]: "0 \<le> (n::nat)" by (simp add: less_eq_nat.simps)
wenzelm@61076
   393
lemma [code]: "(0::nat) \<le> n \<longleftrightarrow> True" by simp
haftmann@26072
   394
haftmann@26072
   395
definition less_nat where
haftmann@28514
   396
  less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   397
haftmann@26072
   398
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
haftmann@26072
   399
  by (simp add: less_eq_nat.simps(2))
haftmann@26072
   400
haftmann@26072
   401
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
haftmann@26072
   402
  unfolding less_eq_Suc_le ..
haftmann@26072
   403
wenzelm@61076
   404
lemma le_0_eq [iff]: "(n::nat) \<le> 0 \<longleftrightarrow> n = 0"
haftmann@26072
   405
  by (induct n) (simp_all add: less_eq_nat.simps(2))
haftmann@26072
   406
wenzelm@61076
   407
lemma not_less0 [iff]: "\<not> n < (0::nat)"
haftmann@26072
   408
  by (simp add: less_eq_Suc_le)
haftmann@26072
   409
wenzelm@61076
   410
lemma less_nat_zero_code [code]: "n < (0::nat) \<longleftrightarrow> False"
haftmann@26072
   411
  by simp
haftmann@26072
   412
haftmann@26072
   413
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
haftmann@26072
   414
  by (simp add: less_eq_Suc_le)
haftmann@26072
   415
haftmann@26072
   416
lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
haftmann@26072
   417
  by (simp add: less_eq_Suc_le)
haftmann@26072
   418
hoelzl@56194
   419
lemma Suc_less_eq2: "Suc n < m \<longleftrightarrow> (\<exists>m'. m = Suc m' \<and> n < m')"
hoelzl@56194
   420
  by (cases m) auto
hoelzl@56194
   421
haftmann@26072
   422
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
haftmann@26072
   423
  by (induct m arbitrary: n)
haftmann@26072
   424
    (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   425
haftmann@26072
   426
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
haftmann@26072
   427
  by (cases n) (auto intro: le_SucI)
haftmann@26072
   428
haftmann@26072
   429
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
haftmann@26072
   430
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@24995
   431
haftmann@26072
   432
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
haftmann@26072
   433
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@25510
   434
wenzelm@26315
   435
instance
wenzelm@26315
   436
proof
haftmann@26072
   437
  fix n m :: nat
lp15@60562
   438
  show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n"
haftmann@26072
   439
  proof (induct n arbitrary: m)
haftmann@27679
   440
    case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   441
  next
haftmann@27679
   442
    case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   443
  qed
haftmann@26072
   444
next
haftmann@26072
   445
  fix n :: nat show "n \<le> n" by (induct n) simp_all
haftmann@26072
   446
next
haftmann@26072
   447
  fix n m :: nat assume "n \<le> m" and "m \<le> n"
haftmann@26072
   448
  then show "n = m"
haftmann@26072
   449
    by (induct n arbitrary: m)
haftmann@26072
   450
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   451
next
haftmann@26072
   452
  fix n m q :: nat assume "n \<le> m" and "m \<le> q"
haftmann@26072
   453
  then show "n \<le> q"
haftmann@26072
   454
  proof (induct n arbitrary: m q)
haftmann@26072
   455
    case 0 show ?case by simp
haftmann@26072
   456
  next
haftmann@26072
   457
    case (Suc n) then show ?case
haftmann@26072
   458
      by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   459
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   460
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   461
  qed
haftmann@26072
   462
next
haftmann@26072
   463
  fix n m :: nat show "n \<le> m \<or> m \<le> n"
haftmann@26072
   464
    by (induct n arbitrary: m)
haftmann@26072
   465
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   466
qed
haftmann@25510
   467
haftmann@25510
   468
end
berghofe@13449
   469
haftmann@52729
   470
instantiation nat :: order_bot
haftmann@29652
   471
begin
haftmann@29652
   472
haftmann@29652
   473
definition bot_nat :: nat where
haftmann@29652
   474
  "bot_nat = 0"
haftmann@29652
   475
haftmann@29652
   476
instance proof
haftmann@29652
   477
qed (simp add: bot_nat_def)
haftmann@29652
   478
haftmann@29652
   479
end
haftmann@29652
   480
hoelzl@51329
   481
instance nat :: no_top
wenzelm@61169
   482
  by standard (auto intro: less_Suc_eq_le [THEN iffD2])
haftmann@52289
   483
hoelzl@51329
   484
wenzelm@60758
   485
subsubsection \<open>Introduction properties\<close>
berghofe@13449
   486
haftmann@26072
   487
lemma lessI [iff]: "n < Suc n"
haftmann@26072
   488
  by (simp add: less_Suc_eq_le)
berghofe@13449
   489
haftmann@26072
   490
lemma zero_less_Suc [iff]: "0 < Suc n"
haftmann@26072
   491
  by (simp add: less_Suc_eq_le)
berghofe@13449
   492
berghofe@13449
   493
wenzelm@60758
   494
subsubsection \<open>Elimination properties\<close>
berghofe@13449
   495
berghofe@13449
   496
lemma less_not_refl: "~ n < (n::nat)"
haftmann@26072
   497
  by (rule order_less_irrefl)
berghofe@13449
   498
wenzelm@26335
   499
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)"
lp15@60562
   500
  by (rule not_sym) (rule less_imp_neq)
berghofe@13449
   501
paulson@14267
   502
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
haftmann@26072
   503
  by (rule less_imp_neq)
berghofe@13449
   504
wenzelm@26335
   505
lemma less_irrefl_nat: "(n::nat) < n ==> R"
wenzelm@26335
   506
  by (rule notE, rule less_not_refl)
berghofe@13449
   507
berghofe@13449
   508
lemma less_zeroE: "(n::nat) < 0 ==> R"
haftmann@26072
   509
  by (rule notE) (rule not_less0)
berghofe@13449
   510
berghofe@13449
   511
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
haftmann@26072
   512
  unfolding less_Suc_eq_le le_less ..
berghofe@13449
   513
huffman@30079
   514
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
haftmann@26072
   515
  by (simp add: less_Suc_eq)
berghofe@13449
   516
blanchet@54147
   517
lemma less_one [iff]: "(n < (1::nat)) = (n = 0)"
huffman@30079
   518
  unfolding One_nat_def by (rule less_Suc0)
berghofe@13449
   519
berghofe@13449
   520
lemma Suc_mono: "m < n ==> Suc m < Suc n"
haftmann@26072
   521
  by simp
berghofe@13449
   522
wenzelm@60758
   523
text \<open>"Less than" is antisymmetric, sort of\<close>
nipkow@14302
   524
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
haftmann@26072
   525
  unfolding not_less less_Suc_eq_le by (rule antisym)
nipkow@14302
   526
paulson@14267
   527
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
haftmann@26072
   528
  by (rule linorder_neq_iff)
berghofe@13449
   529
berghofe@13449
   530
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
berghofe@13449
   531
  and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
berghofe@13449
   532
  shows "P n m"
berghofe@13449
   533
  apply (rule less_linear [THEN disjE])
berghofe@13449
   534
  apply (erule_tac [2] disjE)
berghofe@13449
   535
  apply (erule lessCase)
berghofe@13449
   536
  apply (erule sym [THEN eqCase])
berghofe@13449
   537
  apply (erule major)
berghofe@13449
   538
  done
berghofe@13449
   539
berghofe@13449
   540
wenzelm@60758
   541
subsubsection \<open>Inductive (?) properties\<close>
berghofe@13449
   542
paulson@14267
   543
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
lp15@60562
   544
  unfolding less_eq_Suc_le [of m] le_less by simp
berghofe@13449
   545
haftmann@26072
   546
lemma lessE:
haftmann@26072
   547
  assumes major: "i < k"
haftmann@26072
   548
  and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
haftmann@26072
   549
  shows P
haftmann@26072
   550
proof -
haftmann@26072
   551
  from major have "\<exists>j. i \<le> j \<and> k = Suc j"
haftmann@26072
   552
    unfolding less_eq_Suc_le by (induct k) simp_all
haftmann@26072
   553
  then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
haftmann@26072
   554
    by (clarsimp simp add: less_le)
haftmann@26072
   555
  with p1 p2 show P by auto
haftmann@26072
   556
qed
haftmann@26072
   557
haftmann@26072
   558
lemma less_SucE: assumes major: "m < Suc n"
haftmann@26072
   559
  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
haftmann@26072
   560
  apply (rule major [THEN lessE])
haftmann@26072
   561
  apply (rule eq, blast)
haftmann@26072
   562
  apply (rule less, blast)
berghofe@13449
   563
  done
berghofe@13449
   564
berghofe@13449
   565
lemma Suc_lessE: assumes major: "Suc i < k"
berghofe@13449
   566
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
berghofe@13449
   567
  apply (rule major [THEN lessE])
berghofe@13449
   568
  apply (erule lessI [THEN minor])
paulson@14208
   569
  apply (erule Suc_lessD [THEN minor], assumption)
berghofe@13449
   570
  done
berghofe@13449
   571
berghofe@13449
   572
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
haftmann@26072
   573
  by simp
berghofe@13449
   574
berghofe@13449
   575
lemma less_trans_Suc:
berghofe@13449
   576
  assumes le: "i < j" shows "j < k ==> Suc i < k"
paulson@14208
   577
  apply (induct k, simp_all)
berghofe@13449
   578
  apply (insert le)
berghofe@13449
   579
  apply (simp add: less_Suc_eq)
berghofe@13449
   580
  apply (blast dest: Suc_lessD)
berghofe@13449
   581
  done
berghofe@13449
   582
wenzelm@61799
   583
text \<open>Can be used with \<open>less_Suc_eq\<close> to get @{term "n = m | n < m"}\<close>
haftmann@26072
   584
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
haftmann@26072
   585
  unfolding not_less less_Suc_eq_le ..
berghofe@13449
   586
haftmann@26072
   587
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   588
  unfolding not_le Suc_le_eq ..
wenzelm@21243
   589
wenzelm@60758
   590
text \<open>Properties of "less than or equal"\<close>
berghofe@13449
   591
paulson@14267
   592
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
haftmann@26072
   593
  unfolding less_Suc_eq_le .
berghofe@13449
   594
paulson@14267
   595
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
haftmann@26072
   596
  unfolding not_le less_Suc_eq_le ..
berghofe@13449
   597
paulson@14267
   598
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
haftmann@26072
   599
  by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
berghofe@13449
   600
paulson@14267
   601
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
haftmann@26072
   602
  by (drule le_Suc_eq [THEN iffD1], iprover+)
berghofe@13449
   603
paulson@14267
   604
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
haftmann@26072
   605
  unfolding Suc_le_eq .
berghofe@13449
   606
wenzelm@61799
   607
text \<open>Stronger version of \<open>Suc_leD\<close>\<close>
paulson@14267
   608
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
haftmann@26072
   609
  unfolding Suc_le_eq .
berghofe@13449
   610
wenzelm@26315
   611
lemma less_imp_le_nat: "m < n ==> m \<le> (n::nat)"
haftmann@26072
   612
  unfolding less_eq_Suc_le by (rule Suc_leD)
berghofe@13449
   613
wenzelm@61799
   614
text \<open>For instance, \<open>(Suc m < Suc n) = (Suc m \<le> n) = (m < n)\<close>\<close>
wenzelm@26315
   615
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq
berghofe@13449
   616
berghofe@13449
   617
wenzelm@60758
   618
text \<open>Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"}\<close>
berghofe@13449
   619
paulson@14267
   620
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
haftmann@26072
   621
  unfolding le_less .
berghofe@13449
   622
paulson@14267
   623
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
haftmann@26072
   624
  by (rule le_less)
berghofe@13449
   625
wenzelm@61799
   626
text \<open>Useful with \<open>blast\<close>.\<close>
paulson@14267
   627
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
haftmann@26072
   628
  by auto
berghofe@13449
   629
paulson@14267
   630
lemma le_refl: "n \<le> (n::nat)"
haftmann@26072
   631
  by simp
berghofe@13449
   632
paulson@14267
   633
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
haftmann@26072
   634
  by (rule order_trans)
berghofe@13449
   635
nipkow@33657
   636
lemma le_antisym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
haftmann@26072
   637
  by (rule antisym)
berghofe@13449
   638
paulson@14267
   639
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
haftmann@26072
   640
  by (rule less_le)
berghofe@13449
   641
paulson@14267
   642
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
haftmann@26072
   643
  unfolding less_le ..
berghofe@13449
   644
haftmann@26072
   645
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
haftmann@26072
   646
  by (rule linear)
paulson@14341
   647
wenzelm@22718
   648
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
nipkow@15921
   649
haftmann@26072
   650
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
haftmann@26072
   651
  unfolding less_Suc_eq_le by auto
berghofe@13449
   652
haftmann@26072
   653
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
haftmann@26072
   654
  unfolding not_less by (rule le_less_Suc_eq)
berghofe@13449
   655
berghofe@13449
   656
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
berghofe@13449
   657
paulson@14267
   658
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   659
by (cases n) simp_all
nipkow@25162
   660
nipkow@25162
   661
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   662
by (cases n) simp_all
berghofe@13449
   663
wenzelm@22718
   664
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
nipkow@25162
   665
by (cases n) simp_all
berghofe@13449
   666
nipkow@25162
   667
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
nipkow@25162
   668
by (cases n) simp_all
nipkow@25140
   669
wenzelm@61799
   670
text \<open>This theorem is useful with \<open>blast\<close>\<close>
berghofe@13449
   671
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
nipkow@25162
   672
by (rule neq0_conv[THEN iffD1], iprover)
berghofe@13449
   673
paulson@14267
   674
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
nipkow@25162
   675
by (fast intro: not0_implies_Suc)
berghofe@13449
   676
blanchet@54147
   677
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"
nipkow@25134
   678
using neq0_conv by blast
berghofe@13449
   679
paulson@14267
   680
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
nipkow@25162
   681
by (induct m') simp_all
berghofe@13449
   682
wenzelm@60758
   683
text \<open>Useful in certain inductive arguments\<close>
paulson@14267
   684
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
nipkow@25162
   685
by (cases m) simp_all
berghofe@13449
   686
berghofe@13449
   687
wenzelm@60758
   688
subsubsection \<open>Monotonicity of Addition\<close>
berghofe@13449
   689
haftmann@26072
   690
lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n"
haftmann@26072
   691
by (simp add: diff_Suc split: nat.split)
berghofe@13449
   692
huffman@30128
   693
lemma Suc_diff_1 [simp]: "0 < n ==> Suc (n - 1) = n"
huffman@30128
   694
unfolding One_nat_def by (rule Suc_pred)
huffman@30128
   695
paulson@14331
   696
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
nipkow@25162
   697
by (induct k) simp_all
berghofe@13449
   698
paulson@14331
   699
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
nipkow@25162
   700
by (induct k) simp_all
berghofe@13449
   701
nipkow@25162
   702
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0 | n>0)"
nipkow@25162
   703
by(auto dest:gr0_implies_Suc)
berghofe@13449
   704
wenzelm@60758
   705
text \<open>strict, in 1st argument\<close>
paulson@14341
   706
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
nipkow@25162
   707
by (induct k) simp_all
paulson@14341
   708
wenzelm@60758
   709
text \<open>strict, in both arguments\<close>
paulson@14341
   710
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
paulson@14341
   711
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
paulson@15251
   712
  apply (induct j, simp_all)
paulson@14341
   713
  done
paulson@14341
   714
wenzelm@61799
   715
text \<open>Deleted \<open>less_natE\<close>; use \<open>less_imp_Suc_add RS exE\<close>\<close>
paulson@14341
   716
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
paulson@14341
   717
  apply (induct n)
paulson@14341
   718
  apply (simp_all add: order_le_less)
wenzelm@22718
   719
  apply (blast elim!: less_SucE
haftmann@35047
   720
               intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])
paulson@14341
   721
  done
paulson@14341
   722
hoelzl@56194
   723
lemma le_Suc_ex: "(k::nat) \<le> l \<Longrightarrow> (\<exists>n. l = k + n)"
hoelzl@56194
   724
  by (auto simp: less_Suc_eq_le[symmetric] dest: less_imp_Suc_add)
hoelzl@56194
   725
wenzelm@61799
   726
text \<open>strict, in 1st argument; proof is by induction on \<open>k > 0\<close>\<close>
nipkow@25134
   727
lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j"
nipkow@25134
   728
apply(auto simp: gr0_conv_Suc)
nipkow@25134
   729
apply (induct_tac m)
nipkow@25134
   730
apply (simp_all add: add_less_mono)
nipkow@25134
   731
done
paulson@14341
   732
wenzelm@60758
   733
text \<open>Addition is the inverse of subtraction:
wenzelm@60758
   734
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}.\<close>
lp15@60562
   735
lemma add_diff_inverse_nat: "~  m < n ==> n + (m - n) = (m::nat)"
lp15@60562
   736
by (induct m n rule: diff_induct) simp_all
lp15@60562
   737
hoelzl@62376
   738
lemma nat_le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
hoelzl@62376
   739
using nat_add_left_cancel_le[of m 0] by (auto dest: le_Suc_ex)
hoelzl@62376
   740
hoelzl@62376
   741
text\<open>The naturals form an ordered \<open>semidom\<close> and a \<open>dioid\<close>\<close>
hoelzl@62376
   742
haftmann@35028
   743
instance nat :: linordered_semidom
paulson@14341
   744
proof
paulson@14348
   745
  show "0 < (1::nat)" by simp
haftmann@52289
   746
  show "\<And>m n q :: nat. m \<le> n \<Longrightarrow> q + m \<le> q + n" by simp
haftmann@52289
   747
  show "\<And>m n q :: nat. m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n" by (simp add: mult_less_mono2)
haftmann@59833
   748
  show "\<And>m n :: nat. m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> m * n \<noteq> 0" by simp
lp15@60562
   749
  show "\<And>m n :: nat. n \<le> m ==> (m - n) + n = (m::nat)"
lp15@60562
   750
    by (simp add: add_diff_inverse_nat add.commute linorder_not_less)
hoelzl@62376
   751
qed
hoelzl@62376
   752
hoelzl@62376
   753
instance nat :: dioid
hoelzl@62376
   754
  proof qed (rule nat_le_iff_add)
hoelzl@62376
   755
hoelzl@62376
   756
instance nat :: ordered_cancel_comm_monoid_add
hoelzl@62376
   757
  proof qed
hoelzl@62376
   758
hoelzl@62376
   759
instance nat :: ordered_cancel_comm_monoid_diff
hoelzl@62376
   760
  proof qed
haftmann@44817
   761
wenzelm@60758
   762
subsubsection \<open>@{term min} and @{term max}\<close>
haftmann@44817
   763
haftmann@44817
   764
lemma mono_Suc: "mono Suc"
haftmann@44817
   765
by (rule monoI) simp
haftmann@44817
   766
haftmann@44817
   767
lemma min_0L [simp]: "min 0 n = (0::nat)"
noschinl@45931
   768
by (rule min_absorb1) simp
haftmann@44817
   769
haftmann@44817
   770
lemma min_0R [simp]: "min n 0 = (0::nat)"
noschinl@45931
   771
by (rule min_absorb2) simp
haftmann@44817
   772
haftmann@44817
   773
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
haftmann@44817
   774
by (simp add: mono_Suc min_of_mono)
haftmann@44817
   775
haftmann@44817
   776
lemma min_Suc1:
haftmann@44817
   777
   "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
haftmann@44817
   778
by (simp split: nat.split)
haftmann@44817
   779
haftmann@44817
   780
lemma min_Suc2:
haftmann@44817
   781
   "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
haftmann@44817
   782
by (simp split: nat.split)
haftmann@44817
   783
haftmann@44817
   784
lemma max_0L [simp]: "max 0 n = (n::nat)"
noschinl@45931
   785
by (rule max_absorb2) simp
haftmann@44817
   786
haftmann@44817
   787
lemma max_0R [simp]: "max n 0 = (n::nat)"
noschinl@45931
   788
by (rule max_absorb1) simp
haftmann@44817
   789
haftmann@44817
   790
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
haftmann@44817
   791
by (simp add: mono_Suc max_of_mono)
haftmann@44817
   792
haftmann@44817
   793
lemma max_Suc1:
haftmann@44817
   794
   "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
haftmann@44817
   795
by (simp split: nat.split)
haftmann@44817
   796
haftmann@44817
   797
lemma max_Suc2:
haftmann@44817
   798
   "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
haftmann@44817
   799
by (simp split: nat.split)
paulson@14267
   800
haftmann@44817
   801
lemma nat_mult_min_left:
haftmann@44817
   802
  fixes m n q :: nat
haftmann@44817
   803
  shows "min m n * q = min (m * q) (n * q)"
haftmann@44817
   804
  by (simp add: min_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
haftmann@44817
   805
haftmann@44817
   806
lemma nat_mult_min_right:
haftmann@44817
   807
  fixes m n q :: nat
haftmann@44817
   808
  shows "m * min n q = min (m * n) (m * q)"
haftmann@44817
   809
  by (simp add: min_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
haftmann@44817
   810
haftmann@44817
   811
lemma nat_add_max_left:
haftmann@44817
   812
  fixes m n q :: nat
haftmann@44817
   813
  shows "max m n + q = max (m + q) (n + q)"
haftmann@44817
   814
  by (simp add: max_def)
haftmann@44817
   815
haftmann@44817
   816
lemma nat_add_max_right:
haftmann@44817
   817
  fixes m n q :: nat
haftmann@44817
   818
  shows "m + max n q = max (m + n) (m + q)"
haftmann@44817
   819
  by (simp add: max_def)
haftmann@44817
   820
haftmann@44817
   821
lemma nat_mult_max_left:
haftmann@44817
   822
  fixes m n q :: nat
haftmann@44817
   823
  shows "max m n * q = max (m * q) (n * q)"
haftmann@44817
   824
  by (simp add: max_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
haftmann@44817
   825
haftmann@44817
   826
lemma nat_mult_max_right:
haftmann@44817
   827
  fixes m n q :: nat
haftmann@44817
   828
  shows "m * max n q = max (m * n) (m * q)"
haftmann@44817
   829
  by (simp add: max_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
paulson@14267
   830
paulson@14267
   831
wenzelm@60758
   832
subsubsection \<open>Additional theorems about @{term "op \<le>"}\<close>
wenzelm@60758
   833
wenzelm@60758
   834
text \<open>Complete induction, aka course-of-values induction\<close>
krauss@26748
   835
haftmann@27823
   836
instance nat :: wellorder proof
haftmann@27823
   837
  fix P and n :: nat
haftmann@27823
   838
  assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n"
haftmann@27823
   839
  have "\<And>q. q \<le> n \<Longrightarrow> P q"
haftmann@27823
   840
  proof (induct n)
haftmann@27823
   841
    case (0 n)
krauss@26748
   842
    have "P 0" by (rule step) auto
krauss@26748
   843
    thus ?case using 0 by auto
krauss@26748
   844
  next
haftmann@27823
   845
    case (Suc m n)
haftmann@27823
   846
    then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq)
krauss@26748
   847
    thus ?case
krauss@26748
   848
    proof
haftmann@27823
   849
      assume "n \<le> m" thus "P n" by (rule Suc(1))
krauss@26748
   850
    next
haftmann@27823
   851
      assume n: "n = Suc m"
haftmann@27823
   852
      show "P n"
haftmann@27823
   853
        by (rule step) (rule Suc(1), simp add: n le_simps)
krauss@26748
   854
    qed
krauss@26748
   855
  qed
haftmann@27823
   856
  then show "P n" by auto
krauss@26748
   857
qed
krauss@26748
   858
nipkow@57015
   859
nipkow@57015
   860
lemma Least_eq_0[simp]: "P(0::nat) \<Longrightarrow> Least P = 0"
nipkow@57015
   861
by (rule Least_equality[OF _ le0])
nipkow@57015
   862
haftmann@27823
   863
lemma Least_Suc:
haftmann@27823
   864
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
wenzelm@47988
   865
  apply (cases n, auto)
haftmann@27823
   866
  apply (frule LeastI)
haftmann@27823
   867
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
haftmann@27823
   868
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
haftmann@27823
   869
  apply (erule_tac [2] Least_le)
wenzelm@47988
   870
  apply (cases "LEAST x. P x", auto)
haftmann@27823
   871
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
haftmann@27823
   872
  apply (blast intro: order_antisym)
haftmann@27823
   873
  done
haftmann@27823
   874
haftmann@27823
   875
lemma Least_Suc2:
haftmann@27823
   876
   "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
haftmann@27823
   877
  apply (erule (1) Least_Suc [THEN ssubst])
haftmann@27823
   878
  apply simp
haftmann@27823
   879
  done
haftmann@27823
   880
haftmann@27823
   881
lemma ex_least_nat_le: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k\<le>n. (\<forall>i<k. \<not>P i) & P(k)"
haftmann@27823
   882
  apply (cases n)
haftmann@27823
   883
   apply blast
haftmann@27823
   884
  apply (rule_tac x="LEAST k. P(k)" in exI)
haftmann@27823
   885
  apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
haftmann@27823
   886
  done
haftmann@27823
   887
haftmann@27823
   888
lemma ex_least_nat_less: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k<n. (\<forall>i\<le>k. \<not>P i) & P(k+1)"
huffman@30079
   889
  unfolding One_nat_def
haftmann@27823
   890
  apply (cases n)
haftmann@27823
   891
   apply blast
haftmann@27823
   892
  apply (frule (1) ex_least_nat_le)
haftmann@27823
   893
  apply (erule exE)
haftmann@27823
   894
  apply (case_tac k)
haftmann@27823
   895
   apply simp
haftmann@27823
   896
  apply (rename_tac k1)
haftmann@27823
   897
  apply (rule_tac x=k1 in exI)
haftmann@27823
   898
  apply (auto simp add: less_eq_Suc_le)
haftmann@27823
   899
  done
haftmann@27823
   900
krauss@26748
   901
lemma nat_less_induct:
krauss@26748
   902
  assumes "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
krauss@26748
   903
  using assms less_induct by blast
krauss@26748
   904
krauss@26748
   905
lemma measure_induct_rule [case_names less]:
krauss@26748
   906
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   907
  assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
krauss@26748
   908
  shows "P a"
krauss@26748
   909
by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step)
krauss@26748
   910
wenzelm@60758
   911
text \<open>old style induction rules:\<close>
krauss@26748
   912
lemma measure_induct:
krauss@26748
   913
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   914
  shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
krauss@26748
   915
  by (rule measure_induct_rule [of f P a]) iprover
krauss@26748
   916
krauss@26748
   917
lemma full_nat_induct:
krauss@26748
   918
  assumes step: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"
krauss@26748
   919
  shows "P n"
krauss@26748
   920
  by (rule less_induct) (auto intro: step simp:le_simps)
paulson@14267
   921
wenzelm@60758
   922
text\<open>An induction rule for estabilishing binary relations\<close>
wenzelm@22718
   923
lemma less_Suc_induct:
paulson@19870
   924
  assumes less:  "i < j"
paulson@19870
   925
     and  step:  "!!i. P i (Suc i)"
krauss@31714
   926
     and  trans: "!!i j k. i < j ==> j < k ==>  P i j ==> P j k ==> P i k"
paulson@19870
   927
  shows "P i j"
paulson@19870
   928
proof -
krauss@31714
   929
  from less obtain k where j: "j = Suc (i + k)" by (auto dest: less_imp_Suc_add)
wenzelm@22718
   930
  have "P i (Suc (i + k))"
paulson@19870
   931
  proof (induct k)
wenzelm@22718
   932
    case 0
wenzelm@22718
   933
    show ?case by (simp add: step)
paulson@19870
   934
  next
paulson@19870
   935
    case (Suc k)
krauss@31714
   936
    have "0 + i < Suc k + i" by (rule add_less_mono1) simp
haftmann@57512
   937
    hence "i < Suc (i + k)" by (simp add: add.commute)
krauss@31714
   938
    from trans[OF this lessI Suc step]
krauss@31714
   939
    show ?case by simp
paulson@19870
   940
  qed
wenzelm@22718
   941
  thus "P i j" by (simp add: j)
paulson@19870
   942
qed
paulson@19870
   943
wenzelm@60758
   944
text \<open>The method of infinite descent, frequently used in number theory.
krauss@26748
   945
Provided by Roelof Oosterhuis.
krauss@26748
   946
$P(n)$ is true for all $n\in\mathbb{N}$ if
krauss@26748
   947
\begin{itemize}
krauss@26748
   948
  \item case ``0'': given $n=0$ prove $P(n)$,
krauss@26748
   949
  \item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists
krauss@26748
   950
        a smaller integer $m$ such that $\neg P(m)$.
wenzelm@60758
   951
\end{itemize}\<close>
wenzelm@60758
   952
wenzelm@60758
   953
text\<open>A compact version without explicit base case:\<close>
krauss@26748
   954
lemma infinite_descent:
krauss@26748
   955
  "\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow>  \<exists>m<n. \<not>  P m \<rbrakk> \<Longrightarrow>  P n"
wenzelm@47988
   956
by (induct n rule: less_induct) auto
krauss@26748
   957
lp15@60562
   958
lemma infinite_descent0[case_names 0 smaller]:
krauss@26748
   959
  "\<lbrakk> P 0; !!n. n>0 \<Longrightarrow> \<not> P n \<Longrightarrow> (\<exists>m::nat. m < n \<and> \<not>P m) \<rbrakk> \<Longrightarrow> P n"
krauss@26748
   960
by (rule infinite_descent) (case_tac "n>0", auto)
krauss@26748
   961
wenzelm@60758
   962
text \<open>
krauss@26748
   963
Infinite descent using a mapping to $\mathbb{N}$:
krauss@26748
   964
$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
krauss@26748
   965
\begin{itemize}
krauss@26748
   966
\item case ``0'': given $V(x)=0$ prove $P(x)$,
krauss@26748
   967
\item case ``smaller'': given $V(x)>0$ and $\neg P(x)$ prove there exists a $y \in D$ such that $V(y)<V(x)$ and $~\neg P(y)$.
krauss@26748
   968
\end{itemize}
wenzelm@60758
   969
NB: the proof also shows how to use the previous lemma.\<close>
krauss@26748
   970
krauss@26748
   971
corollary infinite_descent0_measure [case_names 0 smaller]:
krauss@26748
   972
  assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"
krauss@26748
   973
    and   A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
krauss@26748
   974
  shows "P x"
krauss@26748
   975
proof -
krauss@26748
   976
  obtain n where "n = V x" by auto
krauss@26748
   977
  moreover have "\<And>x. V x = n \<Longrightarrow> P x"
krauss@26748
   978
  proof (induct n rule: infinite_descent0)
wenzelm@61799
   979
    case 0 \<comment> "i.e. $V(x) = 0$"
krauss@26748
   980
    with A0 show "P x" by auto
wenzelm@61799
   981
  next \<comment> "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
krauss@26748
   982
    case (smaller n)
krauss@26748
   983
    then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
krauss@26748
   984
    with A1 obtain y where "V y < V x \<and> \<not> P y" by auto
krauss@26748
   985
    with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
krauss@26748
   986
    then show ?case by auto
krauss@26748
   987
  qed
krauss@26748
   988
  ultimately show "P x" by auto
krauss@26748
   989
qed
krauss@26748
   990
wenzelm@60758
   991
text\<open>Again, without explicit base case:\<close>
krauss@26748
   992
lemma infinite_descent_measure:
krauss@26748
   993
assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"
krauss@26748
   994
proof -
krauss@26748
   995
  from assms obtain n where "n = V x" by auto
krauss@26748
   996
  moreover have "!!x. V x = n \<Longrightarrow> P x"
krauss@26748
   997
  proof (induct n rule: infinite_descent, auto)
krauss@26748
   998
    fix x assume "\<not> P x"
krauss@26748
   999
    with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto
krauss@26748
  1000
  qed
krauss@26748
  1001
  ultimately show "P x" by auto
krauss@26748
  1002
qed
krauss@26748
  1003
wenzelm@61799
  1004
text \<open>A [clumsy] way of lifting \<open><\<close>
wenzelm@61799
  1005
  monotonicity to \<open>\<le>\<close> monotonicity\<close>
paulson@14267
  1006
lemma less_mono_imp_le_mono:
nipkow@24438
  1007
  "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"
nipkow@24438
  1008
by (simp add: order_le_less) (blast)
nipkow@24438
  1009
paulson@14267
  1010
wenzelm@60758
  1011
text \<open>non-strict, in 1st argument\<close>
paulson@14267
  1012
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
nipkow@24438
  1013
by (rule add_right_mono)
paulson@14267
  1014
wenzelm@60758
  1015
text \<open>non-strict, in both arguments\<close>
paulson@14267
  1016
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
nipkow@24438
  1017
by (rule add_mono)
paulson@14267
  1018
paulson@14267
  1019
lemma le_add2: "n \<le> ((m + n)::nat)"
nipkow@24438
  1020
by (insert add_right_mono [of 0 m n], simp)
berghofe@13449
  1021
paulson@14267
  1022
lemma le_add1: "n \<le> ((n + m)::nat)"
haftmann@57512
  1023
by (simp add: add.commute, rule le_add2)
berghofe@13449
  1024
berghofe@13449
  1025
lemma less_add_Suc1: "i < Suc (i + m)"
nipkow@24438
  1026
by (rule le_less_trans, rule le_add1, rule lessI)
berghofe@13449
  1027
berghofe@13449
  1028
lemma less_add_Suc2: "i < Suc (m + i)"
nipkow@24438
  1029
by (rule le_less_trans, rule le_add2, rule lessI)
berghofe@13449
  1030
paulson@14267
  1031
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
nipkow@24438
  1032
by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
berghofe@13449
  1033
paulson@14267
  1034
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
nipkow@24438
  1035
by (rule le_trans, assumption, rule le_add1)
berghofe@13449
  1036
paulson@14267
  1037
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
nipkow@24438
  1038
by (rule le_trans, assumption, rule le_add2)
berghofe@13449
  1039
berghofe@13449
  1040
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
nipkow@24438
  1041
by (rule less_le_trans, assumption, rule le_add1)
berghofe@13449
  1042
berghofe@13449
  1043
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
nipkow@24438
  1044
by (rule less_le_trans, assumption, rule le_add2)
berghofe@13449
  1045
berghofe@13449
  1046
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
nipkow@24438
  1047
apply (rule le_less_trans [of _ "i+j"])
nipkow@24438
  1048
apply (simp_all add: le_add1)
nipkow@24438
  1049
done
berghofe@13449
  1050
berghofe@13449
  1051
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
nipkow@24438
  1052
apply (rule notI)
wenzelm@26335
  1053
apply (drule add_lessD1)
wenzelm@26335
  1054
apply (erule less_irrefl [THEN notE])
nipkow@24438
  1055
done
berghofe@13449
  1056
berghofe@13449
  1057
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
haftmann@57512
  1058
by (simp add: add.commute)
berghofe@13449
  1059
paulson@14267
  1060
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
nipkow@24438
  1061
apply (rule order_trans [of _ "m+k"])
nipkow@24438
  1062
apply (simp_all add: le_add1)
nipkow@24438
  1063
done
berghofe@13449
  1064
paulson@14267
  1065
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
haftmann@57512
  1066
apply (simp add: add.commute)
nipkow@24438
  1067
apply (erule add_leD1)
nipkow@24438
  1068
done
berghofe@13449
  1069
paulson@14267
  1070
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
nipkow@24438
  1071
by (blast dest: add_leD1 add_leD2)
berghofe@13449
  1072
wenzelm@61799
  1073
text \<open>needs \<open>!!k\<close> for \<open>ac_simps\<close> to work\<close>
berghofe@13449
  1074
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
nipkow@24438
  1075
by (force simp del: add_Suc_right
haftmann@57514
  1076
    simp add: less_iff_Suc_add add_Suc_right [symmetric] ac_simps)
berghofe@13449
  1077
berghofe@13449
  1078
wenzelm@60758
  1079
subsubsection \<open>More results about difference\<close>
berghofe@13449
  1080
paulson@14267
  1081
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
nipkow@24438
  1082
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1083
berghofe@13449
  1084
lemma diff_less_Suc: "m - n < Suc m"
nipkow@24438
  1085
apply (induct m n rule: diff_induct)
nipkow@24438
  1086
apply (erule_tac [3] less_SucE)
nipkow@24438
  1087
apply (simp_all add: less_Suc_eq)
nipkow@24438
  1088
done
berghofe@13449
  1089
paulson@14267
  1090
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
nipkow@24438
  1091
by (induct m n rule: diff_induct) (simp_all add: le_SucI)
berghofe@13449
  1092
berghofe@13449
  1093
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
nipkow@24438
  1094
by (rule le_less_trans, rule diff_le_self)
berghofe@13449
  1095
berghofe@13449
  1096
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
nipkow@24438
  1097
by (cases n) (auto simp add: le_simps)
berghofe@13449
  1098
paulson@14267
  1099
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
nipkow@24438
  1100
by (induct j k rule: diff_induct) simp_all
berghofe@13449
  1101
paulson@14267
  1102
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
haftmann@57512
  1103
by (simp add: add.commute diff_add_assoc)
berghofe@13449
  1104
paulson@14267
  1105
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
haftmann@62365
  1106
by auto
berghofe@13449
  1107
paulson@14267
  1108
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
nipkow@24438
  1109
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1110
paulson@14267
  1111
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
nipkow@24438
  1112
by (rule iffD2, rule diff_is_0_eq)
berghofe@13449
  1113
berghofe@13449
  1114
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
nipkow@24438
  1115
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1116
wenzelm@22718
  1117
lemma less_imp_add_positive:
wenzelm@22718
  1118
  assumes "i < j"
wenzelm@22718
  1119
  shows "\<exists>k::nat. 0 < k & i + k = j"
wenzelm@22718
  1120
proof
wenzelm@22718
  1121
  from assms show "0 < j - i & i + (j - i) = j"
huffman@23476
  1122
    by (simp add: order_less_imp_le)
wenzelm@22718
  1123
qed
wenzelm@9436
  1124
wenzelm@60758
  1125
text \<open>a nice rewrite for bounded subtraction\<close>
haftmann@26072
  1126
lemma nat_minus_add_max:
haftmann@26072
  1127
  fixes n m :: nat
haftmann@26072
  1128
  shows "n - m + m = max n m"
haftmann@26072
  1129
    by (simp add: max_def not_le order_less_imp_le)
berghofe@13449
  1130
haftmann@26072
  1131
lemma nat_diff_split:
haftmann@62365
  1132
  fixes a b :: nat
haftmann@62365
  1133
  shows "P (a - b) \<longleftrightarrow> (a < b \<longrightarrow> P 0) \<and> (\<forall>d. a = b + d \<longrightarrow> P d)"
wenzelm@61799
  1134
    \<comment> \<open>elimination of \<open>-\<close> on \<open>nat\<close>\<close>
haftmann@62365
  1135
  by (cases "a < b")
haftmann@62365
  1136
    (auto simp add: not_less le_less dest!: add_eq_self_zero [OF sym])
berghofe@13449
  1137
haftmann@26072
  1138
lemma nat_diff_split_asm:
haftmann@62365
  1139
  fixes a b :: nat
haftmann@62365
  1140
  shows "P (a - b) \<longleftrightarrow> \<not> (a < b \<and> \<not> P 0 \<or> (\<exists>d. a = b + d \<and> \<not> P d))"
wenzelm@61799
  1141
    \<comment> \<open>elimination of \<open>-\<close> on \<open>nat\<close> in assumptions\<close>
haftmann@62365
  1142
  by (auto split: nat_diff_split)
berghofe@13449
  1143
huffman@47255
  1144
lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
huffman@47255
  1145
  by simp
huffman@47255
  1146
huffman@47255
  1147
lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
huffman@47255
  1148
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
  1149
huffman@47255
  1150
lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
huffman@47255
  1151
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
  1152
huffman@47255
  1153
lemma Suc_diff_eq_diff_pred: "0 < n ==> Suc m - n = m - (n - 1)"
huffman@47255
  1154
  unfolding One_nat_def by (cases n) simp_all
huffman@47255
  1155
huffman@47255
  1156
lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
huffman@47255
  1157
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
  1158
huffman@47255
  1159
lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
huffman@47255
  1160
  by (fact Let_def)
huffman@47255
  1161
berghofe@13449
  1162
wenzelm@60758
  1163
subsubsection \<open>Monotonicity of multiplication\<close>
berghofe@13449
  1164
paulson@14267
  1165
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
nipkow@24438
  1166
by (simp add: mult_right_mono)
berghofe@13449
  1167
paulson@14267
  1168
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
nipkow@24438
  1169
by (simp add: mult_left_mono)
berghofe@13449
  1170
wenzelm@61799
  1171
text \<open>\<open>\<le>\<close> monotonicity, BOTH arguments\<close>
paulson@14267
  1172
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
nipkow@24438
  1173
by (simp add: mult_mono)
berghofe@13449
  1174
berghofe@13449
  1175
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
nipkow@24438
  1176
by (simp add: mult_strict_right_mono)
berghofe@13449
  1177
wenzelm@61799
  1178
text\<open>Differs from the standard \<open>zero_less_mult_iff\<close> in that
wenzelm@60758
  1179
      there are no negative numbers.\<close>
paulson@14266
  1180
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
berghofe@13449
  1181
  apply (induct m)
wenzelm@22718
  1182
   apply simp
wenzelm@22718
  1183
  apply (case_tac n)
wenzelm@22718
  1184
   apply simp_all
berghofe@13449
  1185
  done
berghofe@13449
  1186
huffman@30079
  1187
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)"
berghofe@13449
  1188
  apply (induct m)
wenzelm@22718
  1189
   apply simp
wenzelm@22718
  1190
  apply (case_tac n)
wenzelm@22718
  1191
   apply simp_all
berghofe@13449
  1192
  done
berghofe@13449
  1193
paulson@14341
  1194
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
berghofe@13449
  1195
  apply (safe intro!: mult_less_mono1)
wenzelm@47988
  1196
  apply (cases k, auto)
berghofe@13449
  1197
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
berghofe@13449
  1198
  apply (blast intro: mult_le_mono1)
berghofe@13449
  1199
  done
berghofe@13449
  1200
berghofe@13449
  1201
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
haftmann@57512
  1202
by (simp add: mult.commute [of k])
berghofe@13449
  1203
paulson@14267
  1204
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
nipkow@24438
  1205
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1206
paulson@14267
  1207
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
nipkow@24438
  1208
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1209
berghofe@13449
  1210
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
nipkow@24438
  1211
by (subst mult_less_cancel1) simp
berghofe@13449
  1212
paulson@14267
  1213
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
nipkow@24438
  1214
by (subst mult_le_cancel1) simp
berghofe@13449
  1215
haftmann@26072
  1216
lemma le_square: "m \<le> m * (m::nat)"
haftmann@26072
  1217
  by (cases m) (auto intro: le_add1)
haftmann@26072
  1218
haftmann@26072
  1219
lemma le_cube: "(m::nat) \<le> m * (m * m)"
haftmann@26072
  1220
  by (cases m) (auto intro: le_add1)
berghofe@13449
  1221
wenzelm@61799
  1222
text \<open>Lemma for \<open>gcd\<close>\<close>
huffman@30128
  1223
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
berghofe@13449
  1224
  apply (drule sym)
berghofe@13449
  1225
  apply (rule disjCI)
berghofe@13449
  1226
  apply (rule nat_less_cases, erule_tac [2] _)
paulson@25157
  1227
   apply (drule_tac [2] mult_less_mono2)
nipkow@25162
  1228
    apply (auto)
berghofe@13449
  1229
  done
wenzelm@9436
  1230
haftmann@51263
  1231
lemma mono_times_nat:
haftmann@51263
  1232
  fixes n :: nat
haftmann@51263
  1233
  assumes "n > 0"
haftmann@51263
  1234
  shows "mono (times n)"
haftmann@51263
  1235
proof
haftmann@51263
  1236
  fix m q :: nat
haftmann@51263
  1237
  assume "m \<le> q"
haftmann@51263
  1238
  with assms show "n * m \<le> n * q" by simp
haftmann@51263
  1239
qed
haftmann@51263
  1240
wenzelm@60758
  1241
text \<open>the lattice order on @{typ nat}\<close>
haftmann@24995
  1242
haftmann@26072
  1243
instantiation nat :: distrib_lattice
haftmann@26072
  1244
begin
haftmann@24995
  1245
haftmann@26072
  1246
definition
wenzelm@61076
  1247
  "(inf :: nat \<Rightarrow> nat \<Rightarrow> nat) = min"
haftmann@24995
  1248
haftmann@26072
  1249
definition
wenzelm@61076
  1250
  "(sup :: nat \<Rightarrow> nat \<Rightarrow> nat) = max"
haftmann@24995
  1251
haftmann@26072
  1252
instance by intro_classes
haftmann@26072
  1253
  (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def
haftmann@26072
  1254
    intro: order_less_imp_le antisym elim!: order_trans order_less_trans)
haftmann@24995
  1255
haftmann@26072
  1256
end
haftmann@24995
  1257
haftmann@24995
  1258
wenzelm@60758
  1259
subsection \<open>Natural operation of natural numbers on functions\<close>
wenzelm@60758
  1260
wenzelm@60758
  1261
text \<open>
haftmann@30971
  1262
  We use the same logical constant for the power operations on
haftmann@30971
  1263
  functions and relations, in order to share the same syntax.
wenzelm@60758
  1264
\<close>
haftmann@30971
  1265
haftmann@45965
  1266
consts compow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@30971
  1267
haftmann@45965
  1268
abbreviation compower :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^^" 80) where
haftmann@30971
  1269
  "f ^^ n \<equiv> compow n f"
haftmann@30971
  1270
haftmann@30971
  1271
notation (latex output)
haftmann@30971
  1272
  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30971
  1273
wenzelm@61799
  1274
text \<open>\<open>f ^^ n = f o ... o f\<close>, the n-fold composition of \<open>f\<close>\<close>
haftmann@30971
  1275
haftmann@30971
  1276
overloading
haftmann@30971
  1277
  funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
haftmann@30971
  1278
begin
haftmann@30954
  1279
blanchet@55575
  1280
primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@44325
  1281
  "funpow 0 f = id"
haftmann@44325
  1282
| "funpow (Suc n) f = f o funpow n f"
haftmann@30954
  1283
haftmann@30971
  1284
end
haftmann@30971
  1285
lp15@62217
  1286
lemma funpow_0 [simp]: "(f ^^ 0) x = x"
lp15@62217
  1287
  by simp
lp15@62217
  1288
haftmann@49723
  1289
lemma funpow_Suc_right:
haftmann@49723
  1290
  "f ^^ Suc n = f ^^ n \<circ> f"
haftmann@49723
  1291
proof (induct n)
haftmann@49723
  1292
  case 0 then show ?case by simp
haftmann@49723
  1293
next
haftmann@49723
  1294
  fix n
haftmann@49723
  1295
  assume "f ^^ Suc n = f ^^ n \<circ> f"
haftmann@49723
  1296
  then show "f ^^ Suc (Suc n) = f ^^ Suc n \<circ> f"
haftmann@49723
  1297
    by (simp add: o_assoc)
haftmann@49723
  1298
qed
haftmann@49723
  1299
haftmann@49723
  1300
lemmas funpow_simps_right = funpow.simps(1) funpow_Suc_right
haftmann@49723
  1301
wenzelm@60758
  1302
text \<open>for code generation\<close>
haftmann@30971
  1303
haftmann@30971
  1304
definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@46028
  1305
  funpow_code_def [code_abbrev]: "funpow = compow"
haftmann@30954
  1306
haftmann@30971
  1307
lemma [code]:
haftmann@37430
  1308
  "funpow (Suc n) f = f o funpow n f"
haftmann@30971
  1309
  "funpow 0 f = id"
haftmann@37430
  1310
  by (simp_all add: funpow_code_def)
haftmann@30971
  1311
wenzelm@36176
  1312
hide_const (open) funpow
haftmann@30954
  1313
haftmann@30954
  1314
lemma funpow_add:
haftmann@30971
  1315
  "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
haftmann@30954
  1316
  by (induct m) simp_all
haftmann@30954
  1317
haftmann@37430
  1318
lemma funpow_mult:
haftmann@37430
  1319
  fixes f :: "'a \<Rightarrow> 'a"
haftmann@37430
  1320
  shows "(f ^^ m) ^^ n = f ^^ (m * n)"
haftmann@37430
  1321
  by (induct n) (simp_all add: funpow_add)
haftmann@37430
  1322
haftmann@30954
  1323
lemma funpow_swap1:
haftmann@30971
  1324
  "f ((f ^^ n) x) = (f ^^ n) (f x)"
haftmann@30954
  1325
proof -
haftmann@30971
  1326
  have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp
haftmann@30971
  1327
  also have "\<dots>  = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)
haftmann@30971
  1328
  also have "\<dots> = (f ^^ n) (f x)" by simp
haftmann@30954
  1329
  finally show ?thesis .
haftmann@30954
  1330
qed
haftmann@30954
  1331
haftmann@38621
  1332
lemma comp_funpow:
haftmann@38621
  1333
  fixes f :: "'a \<Rightarrow> 'a"
haftmann@38621
  1334
  shows "comp f ^^ n = comp (f ^^ n)"
haftmann@38621
  1335
  by (induct n) simp_all
haftmann@30954
  1336
hoelzl@54496
  1337
lemma Suc_funpow[simp]: "Suc ^^ n = (op + n)"
hoelzl@54496
  1338
  by (induct n) simp_all
hoelzl@54496
  1339
hoelzl@54496
  1340
lemma id_funpow[simp]: "id ^^ n = id"
hoelzl@54496
  1341
  by (induct n) simp_all
haftmann@38621
  1342
hoelzl@59000
  1343
lemma funpow_mono:
hoelzl@59000
  1344
  fixes f :: "'a \<Rightarrow> ('a::lattice)"
hoelzl@59000
  1345
  shows "mono f \<Longrightarrow> A \<le> B \<Longrightarrow> (f ^^ n) A \<le> (f ^^ n) B"
hoelzl@59000
  1346
  by (induct n arbitrary: A B)
hoelzl@59000
  1347
     (auto simp del: funpow.simps(2) simp add: funpow_Suc_right mono_def)
hoelzl@59000
  1348
wenzelm@60758
  1349
subsection \<open>Kleene iteration\<close>
nipkow@45833
  1350
haftmann@52729
  1351
lemma Kleene_iter_lpfp:
haftmann@52729
  1352
assumes "mono f" and "f p \<le> p" shows "(f^^k) (bot::'a::order_bot) \<le> p"
nipkow@45833
  1353
proof(induction k)
nipkow@45833
  1354
  case 0 show ?case by simp
nipkow@45833
  1355
next
nipkow@45833
  1356
  case Suc
nipkow@45833
  1357
  from monoD[OF assms(1) Suc] assms(2)
nipkow@45833
  1358
  show ?case by simp
nipkow@45833
  1359
qed
nipkow@45833
  1360
nipkow@45833
  1361
lemma lfp_Kleene_iter: assumes "mono f" and "(f^^Suc k) bot = (f^^k) bot"
nipkow@45833
  1362
shows "lfp f = (f^^k) bot"
nipkow@45833
  1363
proof(rule antisym)
nipkow@45833
  1364
  show "lfp f \<le> (f^^k) bot"
nipkow@45833
  1365
  proof(rule lfp_lowerbound)
nipkow@45833
  1366
    show "f ((f^^k) bot) \<le> (f^^k) bot" using assms(2) by simp
nipkow@45833
  1367
  qed
nipkow@45833
  1368
next
nipkow@45833
  1369
  show "(f^^k) bot \<le> lfp f"
nipkow@45833
  1370
    using Kleene_iter_lpfp[OF assms(1)] lfp_unfold[OF assms(1)] by simp
nipkow@45833
  1371
qed
nipkow@45833
  1372
hoelzl@60636
  1373
lemma mono_pow:
hoelzl@60636
  1374
  fixes f :: "'a \<Rightarrow> 'a::complete_lattice"
hoelzl@60636
  1375
  shows "mono f \<Longrightarrow> mono (f ^^ n)"
hoelzl@60636
  1376
  by (induction n) (auto simp: mono_def)
hoelzl@60636
  1377
hoelzl@60636
  1378
lemma lfp_funpow:
hoelzl@60636
  1379
  assumes f: "mono f" shows "lfp (f ^^ Suc n) = lfp f"
hoelzl@60636
  1380
proof (rule antisym)
hoelzl@60636
  1381
  show "lfp f \<le> lfp (f ^^ Suc n)"
hoelzl@60636
  1382
  proof (rule lfp_lowerbound)
hoelzl@60636
  1383
    have "f (lfp (f ^^ Suc n)) = lfp (\<lambda>x. f ((f ^^ n) x))"
hoelzl@60636
  1384
      unfolding funpow_Suc_right by (simp add: lfp_rolling f mono_pow comp_def)
hoelzl@60636
  1385
    then show "f (lfp (f ^^ Suc n)) \<le> lfp (f ^^ Suc n)"
hoelzl@60636
  1386
      by (simp add: comp_def)
hoelzl@60636
  1387
  qed
hoelzl@60636
  1388
  have "(f^^n) (lfp f) = lfp f" for n
hoelzl@60636
  1389
    by (induction n) (auto intro: f lfp_unfold[symmetric])
hoelzl@60636
  1390
  then show "lfp (f^^Suc n) \<le> lfp f"
hoelzl@60636
  1391
    by (intro lfp_lowerbound) (simp del: funpow.simps)
hoelzl@60636
  1392
qed
hoelzl@60636
  1393
hoelzl@60636
  1394
lemma gfp_funpow:
hoelzl@60636
  1395
  assumes f: "mono f" shows "gfp (f ^^ Suc n) = gfp f"
hoelzl@60636
  1396
proof (rule antisym)
hoelzl@60636
  1397
  show "gfp f \<ge> gfp (f ^^ Suc n)"
hoelzl@60636
  1398
  proof (rule gfp_upperbound)
hoelzl@60636
  1399
    have "f (gfp (f ^^ Suc n)) = gfp (\<lambda>x. f ((f ^^ n) x))"
hoelzl@60636
  1400
      unfolding funpow_Suc_right by (simp add: gfp_rolling f mono_pow comp_def)
hoelzl@60636
  1401
    then show "f (gfp (f ^^ Suc n)) \<ge> gfp (f ^^ Suc n)"
hoelzl@60636
  1402
      by (simp add: comp_def)
hoelzl@60636
  1403
  qed
hoelzl@60636
  1404
  have "(f^^n) (gfp f) = gfp f" for n
hoelzl@60636
  1405
    by (induction n) (auto intro: f gfp_unfold[symmetric])
hoelzl@60636
  1406
  then show "gfp (f^^Suc n) \<ge> gfp f"
hoelzl@60636
  1407
    by (intro gfp_upperbound) (simp del: funpow.simps)
hoelzl@60636
  1408
qed
nipkow@45833
  1409
wenzelm@61799
  1410
subsection \<open>Embedding of the naturals into any \<open>semiring_1\<close>: @{term of_nat}\<close>
haftmann@24196
  1411
haftmann@24196
  1412
context semiring_1
haftmann@24196
  1413
begin
haftmann@24196
  1414
haftmann@38621
  1415
definition of_nat :: "nat \<Rightarrow> 'a" where
haftmann@38621
  1416
  "of_nat n = (plus 1 ^^ n) 0"
haftmann@38621
  1417
haftmann@38621
  1418
lemma of_nat_simps [simp]:
haftmann@38621
  1419
  shows of_nat_0: "of_nat 0 = 0"
haftmann@38621
  1420
    and of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"
haftmann@38621
  1421
  by (simp_all add: of_nat_def)
haftmann@25193
  1422
haftmann@25193
  1423
lemma of_nat_1 [simp]: "of_nat 1 = 1"
haftmann@38621
  1424
  by (simp add: of_nat_def)
haftmann@25193
  1425
haftmann@25193
  1426
lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"
haftmann@57514
  1427
  by (induct m) (simp_all add: ac_simps)
haftmann@25193
  1428
lp15@61649
  1429
lemma of_nat_mult [simp]: "of_nat (m * n) = of_nat m * of_nat n"
haftmann@57514
  1430
  by (induct m) (simp_all add: ac_simps distrib_right)
haftmann@25193
  1431
eberlm@61531
  1432
lemma mult_of_nat_commute: "of_nat x * y = y * of_nat x"
eberlm@61531
  1433
  by (induction x) (simp_all add: algebra_simps)
eberlm@61531
  1434
blanchet@55575
  1435
primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@28514
  1436
  "of_nat_aux inc 0 i = i"
wenzelm@61799
  1437
| "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" \<comment> \<open>tail recursive\<close>
haftmann@25928
  1438
haftmann@30966
  1439
lemma of_nat_code:
haftmann@28514
  1440
  "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"
haftmann@28514
  1441
proof (induct n)
haftmann@28514
  1442
  case 0 then show ?case by simp
haftmann@28514
  1443
next
haftmann@28514
  1444
  case (Suc n)
haftmann@28514
  1445
  have "\<And>i. of_nat_aux (\<lambda>i. i + 1) n (i + 1) = of_nat_aux (\<lambda>i. i + 1) n i + 1"
haftmann@28514
  1446
    by (induct n) simp_all
haftmann@28514
  1447
  from this [of 0] have "of_nat_aux (\<lambda>i. i + 1) n 1 = of_nat_aux (\<lambda>i. i + 1) n 0 + 1"
haftmann@28514
  1448
    by simp
haftmann@57512
  1449
  with Suc show ?case by (simp add: add.commute)
haftmann@28514
  1450
qed
haftmann@30966
  1451
haftmann@24196
  1452
end
haftmann@24196
  1453
bulwahn@45231
  1454
declare of_nat_code [code]
haftmann@30966
  1455
wenzelm@60758
  1456
text\<open>Class for unital semirings with characteristic zero.
wenzelm@60758
  1457
 Includes non-ordered rings like the complex numbers.\<close>
haftmann@26072
  1458
haftmann@26072
  1459
class semiring_char_0 = semiring_1 +
haftmann@38621
  1460
  assumes inj_of_nat: "inj of_nat"
haftmann@26072
  1461
begin
haftmann@26072
  1462
haftmann@38621
  1463
lemma of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"
haftmann@38621
  1464
  by (auto intro: inj_of_nat injD)
haftmann@38621
  1465
wenzelm@60758
  1466
text\<open>Special cases where either operand is zero\<close>
haftmann@26072
  1467
blanchet@54147
  1468
lemma of_nat_0_eq_iff [simp]: "0 = of_nat n \<longleftrightarrow> 0 = n"
haftmann@38621
  1469
  by (fact of_nat_eq_iff [of 0 n, unfolded of_nat_0])
haftmann@26072
  1470
blanchet@54147
  1471
lemma of_nat_eq_0_iff [simp]: "of_nat m = 0 \<longleftrightarrow> m = 0"
haftmann@38621
  1472
  by (fact of_nat_eq_iff [of m 0, unfolded of_nat_0])
haftmann@26072
  1473
haftmann@60353
  1474
lemma of_nat_neq_0 [simp]:
haftmann@60353
  1475
  "of_nat (Suc n) \<noteq> 0"
haftmann@60353
  1476
  unfolding of_nat_eq_0_iff by simp
haftmann@60353
  1477
haftmann@60353
  1478
lemma of_nat_0_neq [simp]:
haftmann@60353
  1479
  "0 \<noteq> of_nat (Suc n)"
lp15@60562
  1480
  unfolding of_nat_0_eq_iff by simp
lp15@60562
  1481
haftmann@26072
  1482
end
haftmann@26072
  1483
haftmann@35028
  1484
context linordered_semidom
haftmann@25193
  1485
begin
haftmann@25193
  1486
huffman@47489
  1487
lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"
huffman@47489
  1488
  by (induct n) simp_all
haftmann@25193
  1489
huffman@47489
  1490
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"
huffman@47489
  1491
  by (simp add: not_less)
haftmann@25193
  1492
haftmann@25193
  1493
lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"
hoelzl@62376
  1494
  by (induct m n rule: diff_induct) (simp_all add: add_pos_nonneg)
haftmann@25193
  1495
haftmann@26072
  1496
lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"
haftmann@26072
  1497
  by (simp add: not_less [symmetric] linorder_not_less [symmetric])
haftmann@25193
  1498
huffman@47489
  1499
lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"
huffman@47489
  1500
  by simp
huffman@47489
  1501
huffman@47489
  1502
lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"
huffman@47489
  1503
  by simp
huffman@47489
  1504
wenzelm@61799
  1505
text\<open>Every \<open>linordered_semidom\<close> has characteristic zero.\<close>
haftmann@25193
  1506
haftmann@38621
  1507
subclass semiring_char_0 proof
haftmann@38621
  1508
qed (auto intro!: injI simp add: eq_iff)
haftmann@25193
  1509
wenzelm@60758
  1510
text\<open>Special cases where either operand is zero\<close>
haftmann@25193
  1511
blanchet@54147
  1512
lemma of_nat_le_0_iff [simp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"
haftmann@25193
  1513
  by (rule of_nat_le_iff [of _ 0, simplified])
haftmann@25193
  1514
haftmann@26072
  1515
lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"
haftmann@26072
  1516
  by (rule of_nat_less_iff [of 0, simplified])
haftmann@26072
  1517
haftmann@26072
  1518
end
haftmann@26072
  1519
haftmann@26072
  1520
context ring_1
haftmann@26072
  1521
begin
haftmann@26072
  1522
haftmann@26072
  1523
lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"
nipkow@29667
  1524
by (simp add: algebra_simps of_nat_add [symmetric])
haftmann@26072
  1525
haftmann@26072
  1526
end
haftmann@26072
  1527
haftmann@35028
  1528
context linordered_idom
haftmann@26072
  1529
begin
haftmann@26072
  1530
haftmann@26072
  1531
lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"
haftmann@26072
  1532
  unfolding abs_if by auto
haftmann@26072
  1533
haftmann@25193
  1534
end
haftmann@25193
  1535
haftmann@25193
  1536
lemma of_nat_id [simp]: "of_nat n = n"
huffman@35216
  1537
  by (induct n) simp_all
haftmann@25193
  1538
haftmann@25193
  1539
lemma of_nat_eq_id [simp]: "of_nat = id"
nipkow@39302
  1540
  by (auto simp add: fun_eq_iff)
haftmann@25193
  1541
haftmann@25193
  1542
wenzelm@60758
  1543
subsection \<open>The set of natural numbers\<close>
haftmann@25193
  1544
haftmann@26072
  1545
context semiring_1
haftmann@25193
  1546
begin
haftmann@25193
  1547
wenzelm@61070
  1548
definition Nats :: "'a set"  ("\<nat>")
wenzelm@61070
  1549
  where "\<nat> = range of_nat"
haftmann@25193
  1550
haftmann@26072
  1551
lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"
haftmann@26072
  1552
  by (simp add: Nats_def)
haftmann@26072
  1553
haftmann@26072
  1554
lemma Nats_0 [simp]: "0 \<in> \<nat>"
haftmann@26072
  1555
apply (simp add: Nats_def)
haftmann@26072
  1556
apply (rule range_eqI)
haftmann@26072
  1557
apply (rule of_nat_0 [symmetric])
haftmann@26072
  1558
done
haftmann@25193
  1559
haftmann@26072
  1560
lemma Nats_1 [simp]: "1 \<in> \<nat>"
haftmann@26072
  1561
apply (simp add: Nats_def)
haftmann@26072
  1562
apply (rule range_eqI)
haftmann@26072
  1563
apply (rule of_nat_1 [symmetric])
haftmann@26072
  1564
done
haftmann@25193
  1565
haftmann@26072
  1566
lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"
haftmann@26072
  1567
apply (auto simp add: Nats_def)
haftmann@26072
  1568
apply (rule range_eqI)
haftmann@26072
  1569
apply (rule of_nat_add [symmetric])
haftmann@26072
  1570
done
haftmann@26072
  1571
haftmann@26072
  1572
lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"
haftmann@26072
  1573
apply (auto simp add: Nats_def)
haftmann@26072
  1574
apply (rule range_eqI)
haftmann@26072
  1575
apply (rule of_nat_mult [symmetric])
haftmann@26072
  1576
done
haftmann@25193
  1577
huffman@35633
  1578
lemma Nats_cases [cases set: Nats]:
huffman@35633
  1579
  assumes "x \<in> \<nat>"
huffman@35633
  1580
  obtains (of_nat) n where "x = of_nat n"
huffman@35633
  1581
  unfolding Nats_def
huffman@35633
  1582
proof -
wenzelm@60758
  1583
  from \<open>x \<in> \<nat>\<close> have "x \<in> range of_nat" unfolding Nats_def .
huffman@35633
  1584
  then obtain n where "x = of_nat n" ..
huffman@35633
  1585
  then show thesis ..
huffman@35633
  1586
qed
huffman@35633
  1587
huffman@35633
  1588
lemma Nats_induct [case_names of_nat, induct set: Nats]:
huffman@35633
  1589
  "x \<in> \<nat> \<Longrightarrow> (\<And>n. P (of_nat n)) \<Longrightarrow> P x"
huffman@35633
  1590
  by (rule Nats_cases) auto
huffman@35633
  1591
haftmann@25193
  1592
end
haftmann@25193
  1593
haftmann@25193
  1594
wenzelm@60758
  1595
subsection \<open>Further arithmetic facts concerning the natural numbers\<close>
wenzelm@21243
  1596
haftmann@22845
  1597
lemma subst_equals:
haftmann@22845
  1598
  assumes 1: "t = s" and 2: "u = t"
haftmann@22845
  1599
  shows "u = s"
haftmann@22845
  1600
  using 2 1 by (rule trans)
haftmann@22845
  1601
wenzelm@48891
  1602
ML_file "Tools/nat_arith.ML"
huffman@48559
  1603
huffman@48559
  1604
simproc_setup nateq_cancel_sums
huffman@48559
  1605
  ("(l::nat) + m = n" | "(l::nat) = m + n" | "Suc m = n" | "m = Suc n") =
wenzelm@60758
  1606
  \<open>fn phi => try o Nat_Arith.cancel_eq_conv\<close>
huffman@48559
  1607
huffman@48559
  1608
simproc_setup natless_cancel_sums
huffman@48559
  1609
  ("(l::nat) + m < n" | "(l::nat) < m + n" | "Suc m < n" | "m < Suc n") =
wenzelm@60758
  1610
  \<open>fn phi => try o Nat_Arith.cancel_less_conv\<close>
huffman@48559
  1611
huffman@48559
  1612
simproc_setup natle_cancel_sums
huffman@48559
  1613
  ("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" | "Suc m \<le> n" | "m \<le> Suc n") =
wenzelm@60758
  1614
  \<open>fn phi => try o Nat_Arith.cancel_le_conv\<close>
huffman@48559
  1615
huffman@48559
  1616
simproc_setup natdiff_cancel_sums
huffman@48559
  1617
  ("(l::nat) + m - n" | "(l::nat) - (m + n)" | "Suc m - n" | "m - Suc n") =
wenzelm@60758
  1618
  \<open>fn phi => try o Nat_Arith.cancel_diff_conv\<close>
wenzelm@24091
  1619
wenzelm@48891
  1620
ML_file "Tools/lin_arith.ML"
wenzelm@60758
  1621
setup \<open>Lin_Arith.global_setup\<close>
wenzelm@60758
  1622
declaration \<open>K Lin_Arith.setup\<close>
wenzelm@24091
  1623
wenzelm@61144
  1624
simproc_setup fast_arith_nat ("(m::nat) < n" | "(m::nat) \<le> n" | "(m::nat) = n") =
wenzelm@61144
  1625
  \<open>K Lin_Arith.simproc\<close>
wenzelm@43595
  1626
(* Because of this simproc, the arithmetic solver is really only
wenzelm@43595
  1627
useful to detect inconsistencies among the premises for subgoals which are
wenzelm@43595
  1628
*not* themselves (in)equalities, because the latter activate
wenzelm@43595
  1629
fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
wenzelm@43595
  1630
solver all the time rather than add the additional check. *)
wenzelm@43595
  1631
wenzelm@43595
  1632
wenzelm@21243
  1633
lemmas [arith_split] = nat_diff_split split_min split_max
wenzelm@21243
  1634
nipkow@27625
  1635
context order
nipkow@27625
  1636
begin
nipkow@27625
  1637
nipkow@27625
  1638
lemma lift_Suc_mono_le:
haftmann@53986
  1639
  assumes mono: "\<And>n. f n \<le> f (Suc n)" and "n \<le> n'"
krauss@27627
  1640
  shows "f n \<le> f n'"
krauss@27627
  1641
proof (cases "n < n'")
krauss@27627
  1642
  case True
haftmann@53986
  1643
  then show ?thesis
haftmann@53986
  1644
    by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
wenzelm@61799
  1645
qed (insert \<open>n \<le> n'\<close>, auto) \<comment> \<open>trivial for @{prop "n = n'"}\<close>
nipkow@27625
  1646
hoelzl@56020
  1647
lemma lift_Suc_antimono_le:
hoelzl@56020
  1648
  assumes mono: "\<And>n. f n \<ge> f (Suc n)" and "n \<le> n'"
hoelzl@56020
  1649
  shows "f n \<ge> f n'"
hoelzl@56020
  1650
proof (cases "n < n'")
hoelzl@56020
  1651
  case True
hoelzl@56020
  1652
  then show ?thesis
hoelzl@56020
  1653
    by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
wenzelm@61799
  1654
qed (insert \<open>n \<le> n'\<close>, auto) \<comment> \<open>trivial for @{prop "n = n'"}\<close>
hoelzl@56020
  1655
nipkow@27625
  1656
lemma lift_Suc_mono_less:
haftmann@53986
  1657
  assumes mono: "\<And>n. f n < f (Suc n)" and "n < n'"
krauss@27627
  1658
  shows "f n < f n'"
wenzelm@60758
  1659
using \<open>n < n'\<close>
haftmann@53986
  1660
by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
nipkow@27625
  1661
nipkow@27789
  1662
lemma lift_Suc_mono_less_iff:
haftmann@53986
  1663
  "(\<And>n. f n < f (Suc n)) \<Longrightarrow> f n < f m \<longleftrightarrow> n < m"
haftmann@53986
  1664
  by (blast intro: less_asym' lift_Suc_mono_less [of f]
haftmann@53986
  1665
    dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq [THEN iffD1])
nipkow@27789
  1666
nipkow@27625
  1667
end
nipkow@27625
  1668
haftmann@53986
  1669
lemma mono_iff_le_Suc:
haftmann@53986
  1670
  "mono f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
haftmann@37387
  1671
  unfolding mono_def by (auto intro: lift_Suc_mono_le [of f])
nipkow@27625
  1672
hoelzl@56020
  1673
lemma antimono_iff_le_Suc:
hoelzl@56020
  1674
  "antimono f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
hoelzl@56020
  1675
  unfolding antimono_def by (auto intro: lift_Suc_antimono_le [of f])
hoelzl@56020
  1676
nipkow@27789
  1677
lemma mono_nat_linear_lb:
haftmann@53986
  1678
  fixes f :: "nat \<Rightarrow> nat"
haftmann@53986
  1679
  assumes "\<And>m n. m < n \<Longrightarrow> f m < f n"
haftmann@53986
  1680
  shows "f m + k \<le> f (m + k)"
haftmann@53986
  1681
proof (induct k)
haftmann@53986
  1682
  case 0 then show ?case by simp
haftmann@53986
  1683
next
haftmann@53986
  1684
  case (Suc k)
haftmann@53986
  1685
  then have "Suc (f m + k) \<le> Suc (f (m + k))" by simp
haftmann@53986
  1686
  also from assms [of "m + k" "Suc (m + k)"] have "Suc (f (m + k)) \<le> f (Suc (m + k))"
haftmann@53986
  1687
    by (simp add: Suc_le_eq)
haftmann@53986
  1688
  finally show ?case by simp
haftmann@53986
  1689
qed
nipkow@27789
  1690
nipkow@27789
  1691
wenzelm@60758
  1692
text\<open>Subtraction laws, mostly by Clemens Ballarin\<close>
wenzelm@21243
  1693
wenzelm@21243
  1694
lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
nipkow@24438
  1695
by arith
wenzelm@21243
  1696
wenzelm@21243
  1697
lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
nipkow@24438
  1698
by arith
wenzelm@21243
  1699
haftmann@51173
  1700
lemma less_diff_conv2:
haftmann@51173
  1701
  fixes j k i :: nat
haftmann@51173
  1702
  assumes "k \<le> j"
haftmann@51173
  1703
  shows "j - k < i \<longleftrightarrow> j < i + k"
haftmann@51173
  1704
  using assms by arith
haftmann@51173
  1705
wenzelm@21243
  1706
lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
nipkow@24438
  1707
by arith
wenzelm@21243
  1708
wenzelm@21243
  1709
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
nipkow@24438
  1710
by arith
wenzelm@21243
  1711
wenzelm@21243
  1712
(*Replaces the previous diff_less and le_diff_less, which had the stronger
wenzelm@21243
  1713
  second premise n\<le>m*)
wenzelm@21243
  1714
lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
nipkow@24438
  1715
by arith
wenzelm@21243
  1716
wenzelm@60758
  1717
text \<open>Simplification of relational expressions involving subtraction\<close>
wenzelm@21243
  1718
wenzelm@21243
  1719
lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
nipkow@24438
  1720
by (simp split add: nat_diff_split)
wenzelm@21243
  1721
wenzelm@36176
  1722
hide_fact (open) diff_diff_eq
haftmann@35064
  1723
wenzelm@21243
  1724
lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
nipkow@24438
  1725
by (auto split add: nat_diff_split)
wenzelm@21243
  1726
wenzelm@21243
  1727
lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
nipkow@24438
  1728
by (auto split add: nat_diff_split)
wenzelm@21243
  1729
wenzelm@21243
  1730
lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
nipkow@24438
  1731
by (auto split add: nat_diff_split)
wenzelm@21243
  1732
wenzelm@60758
  1733
text\<open>(Anti)Monotonicity of subtraction -- by Stephan Merz\<close>
wenzelm@21243
  1734
wenzelm@21243
  1735
(* Monotonicity of subtraction in first argument *)
wenzelm@21243
  1736
lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
nipkow@24438
  1737
by (simp split add: nat_diff_split)
wenzelm@21243
  1738
wenzelm@21243
  1739
lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
nipkow@24438
  1740
by (simp split add: nat_diff_split)
wenzelm@21243
  1741
wenzelm@21243
  1742
lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
nipkow@24438
  1743
by (simp split add: nat_diff_split)
wenzelm@21243
  1744
wenzelm@21243
  1745
lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
nipkow@24438
  1746
by (simp split add: nat_diff_split)
wenzelm@21243
  1747
bulwahn@26143
  1748
lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i"
nipkow@32437
  1749
by auto
bulwahn@26143
  1750
lp15@60562
  1751
lemma inj_on_diff_nat:
bulwahn@26143
  1752
  assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)"
bulwahn@26143
  1753
  shows "inj_on (\<lambda>n. n - k) N"
bulwahn@26143
  1754
proof (rule inj_onI)
bulwahn@26143
  1755
  fix x y
bulwahn@26143
  1756
  assume a: "x \<in> N" "y \<in> N" "x - k = y - k"
bulwahn@26143
  1757
  with k_le_n have "x - k + k = y - k + k" by auto
bulwahn@26143
  1758
  with a k_le_n show "x = y" by auto
bulwahn@26143
  1759
qed
bulwahn@26143
  1760
wenzelm@60758
  1761
text\<open>Rewriting to pull differences out\<close>
haftmann@26072
  1762
haftmann@26072
  1763
lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
haftmann@26072
  1764
by arith
haftmann@26072
  1765
haftmann@26072
  1766
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
haftmann@26072
  1767
by arith
haftmann@26072
  1768
haftmann@26072
  1769
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
haftmann@26072
  1770
by arith
haftmann@26072
  1771
noschinl@45933
  1772
lemma Suc_diff_Suc: "n < m \<Longrightarrow> Suc (m - Suc n) = m - n"
noschinl@45933
  1773
by simp
noschinl@45933
  1774
bulwahn@46350
  1775
(*The others are
bulwahn@46350
  1776
      i - j - k = i - (j + k),
bulwahn@46350
  1777
      k \<le> j ==> j - k + i = j + i - k,
bulwahn@46350
  1778
      k \<le> j ==> i + (j - k) = i + j - k *)
bulwahn@46350
  1779
lemmas add_diff_assoc = diff_add_assoc [symmetric]
bulwahn@46350
  1780
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
haftmann@62365
  1781
declare add_diff_assoc [simp] add_diff_assoc2[simp]
bulwahn@46350
  1782
wenzelm@61799
  1783
text\<open>At present we prove no analogue of \<open>not_less_Least\<close> or \<open>Least_Suc\<close>, since there appears to be no need.\<close>
wenzelm@60758
  1784
wenzelm@60758
  1785
text\<open>Lemmas for ex/Factorization\<close>
wenzelm@21243
  1786
wenzelm@21243
  1787
lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
nipkow@24438
  1788
by (cases m) auto
wenzelm@21243
  1789
wenzelm@21243
  1790
lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
nipkow@24438
  1791
by (cases m) auto
wenzelm@21243
  1792
wenzelm@21243
  1793
lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
nipkow@24438
  1794
by (cases m) auto
wenzelm@21243
  1795
wenzelm@60758
  1796
text \<open>Specialized induction principles that work "backwards":\<close>
krauss@23001
  1797
krauss@23001
  1798
lemma inc_induct[consumes 1, case_names base step]:
hoelzl@54411
  1799
  assumes less: "i \<le> j"
krauss@23001
  1800
  assumes base: "P j"
hoelzl@54411
  1801
  assumes step: "\<And>n. i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P (Suc n) \<Longrightarrow> P n"
krauss@23001
  1802
  shows "P i"
hoelzl@54411
  1803
  using less step
hoelzl@54411
  1804
proof (induct d\<equiv>"j - i" arbitrary: i)
krauss@23001
  1805
  case (0 i)
krauss@23001
  1806
  hence "i = j" by simp
krauss@23001
  1807
  with base show ?case by simp
krauss@23001
  1808
next
hoelzl@54411
  1809
  case (Suc d n)
hoelzl@54411
  1810
  hence "n \<le> n" "n < j" "P (Suc n)"
krauss@23001
  1811
    by simp_all
hoelzl@54411
  1812
  then show "P n" by fact
krauss@23001
  1813
qed
krauss@23001
  1814
krauss@23001
  1815
lemma strict_inc_induct[consumes 1, case_names base step]:
krauss@23001
  1816
  assumes less: "i < j"
krauss@23001
  1817
  assumes base: "!!i. j = Suc i ==> P i"
krauss@23001
  1818
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1819
  shows "P i"
krauss@23001
  1820
  using less
krauss@23001
  1821
proof (induct d=="j - i - 1" arbitrary: i)
krauss@23001
  1822
  case (0 i)
wenzelm@60758
  1823
  with \<open>i < j\<close> have "j = Suc i" by simp
krauss@23001
  1824
  with base show ?case by simp
krauss@23001
  1825
next
krauss@23001
  1826
  case (Suc d i)
krauss@23001
  1827
  hence "i < j" "P (Suc i)"
krauss@23001
  1828
    by simp_all
krauss@23001
  1829
  thus "P i" by (rule step)
krauss@23001
  1830
qed
krauss@23001
  1831
krauss@23001
  1832
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
krauss@23001
  1833
  using inc_induct[of "k - i" k P, simplified] by blast
krauss@23001
  1834
krauss@23001
  1835
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
krauss@23001
  1836
  using inc_induct[of 0 k P] by blast
wenzelm@21243
  1837
wenzelm@60758
  1838
text \<open>Further induction rule similar to @{thm inc_induct}\<close>
nipkow@27625
  1839
bulwahn@46351
  1840
lemma dec_induct[consumes 1, case_names base step]:
hoelzl@54411
  1841
  "i \<le> j \<Longrightarrow> P i \<Longrightarrow> (\<And>n. i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P n \<Longrightarrow> P (Suc n)) \<Longrightarrow> P j"
bulwahn@46351
  1842
  by (induct j arbitrary: i) (auto simp: le_Suc_eq)
hoelzl@59000
  1843
hoelzl@59000
  1844
subsection \<open> Monotonicity of funpow \<close>
hoelzl@59000
  1845
hoelzl@59000
  1846
lemma funpow_increasing:
hoelzl@59000
  1847
  fixes f :: "'a \<Rightarrow> ('a::{lattice, order_top})"
hoelzl@59000
  1848
  shows "m \<le> n \<Longrightarrow> mono f \<Longrightarrow> (f ^^ n) \<top> \<le> (f ^^ m) \<top>"
hoelzl@59000
  1849
  by (induct rule: inc_induct)
hoelzl@59000
  1850
     (auto simp del: funpow.simps(2) simp add: funpow_Suc_right
hoelzl@59000
  1851
           intro: order_trans[OF _ funpow_mono])
hoelzl@59000
  1852
hoelzl@59000
  1853
lemma funpow_decreasing:
hoelzl@59000
  1854
  fixes f :: "'a \<Rightarrow> ('a::{lattice, order_bot})"
hoelzl@59000
  1855
  shows "m \<le> n \<Longrightarrow> mono f \<Longrightarrow> (f ^^ m) \<bottom> \<le> (f ^^ n) \<bottom>"
hoelzl@59000
  1856
  by (induct rule: dec_induct)
hoelzl@59000
  1857
     (auto simp del: funpow.simps(2) simp add: funpow_Suc_right
hoelzl@59000
  1858
           intro: order_trans[OF _ funpow_mono])
hoelzl@59000
  1859
hoelzl@59000
  1860
lemma mono_funpow:
hoelzl@60427
  1861
  fixes Q :: "'a::{lattice, order_bot} \<Rightarrow> 'a"
hoelzl@59000
  1862
  shows "mono Q \<Longrightarrow> mono (\<lambda>i. (Q ^^ i) \<bottom>)"
hoelzl@59000
  1863
  by (auto intro!: funpow_decreasing simp: mono_def)
blanchet@58377
  1864
hoelzl@60175
  1865
lemma antimono_funpow:
hoelzl@60427
  1866
  fixes Q :: "'a::{lattice, order_top} \<Rightarrow> 'a"
hoelzl@60175
  1867
  shows "mono Q \<Longrightarrow> antimono (\<lambda>i. (Q ^^ i) \<top>)"
hoelzl@60175
  1868
  by (auto intro!: funpow_increasing simp: antimono_def)
hoelzl@60175
  1869
wenzelm@60758
  1870
subsection \<open>The divides relation on @{typ nat}\<close>
haftmann@33274
  1871
haftmann@62365
  1872
lemma dvd_1_left [iff]:
haftmann@62365
  1873
  "Suc 0 dvd k"
haftmann@62365
  1874
  by (simp add: dvd_def)
haftmann@62365
  1875
haftmann@62365
  1876
lemma dvd_1_iff_1 [simp]:
haftmann@62365
  1877
  "m dvd Suc 0 \<longleftrightarrow> m = Suc 0"
haftmann@62365
  1878
  by (simp add: dvd_def)
haftmann@62365
  1879
haftmann@62365
  1880
lemma nat_dvd_1_iff_1 [simp]:
haftmann@62365
  1881
  "m dvd (1::nat) \<longleftrightarrow> m = 1"
haftmann@62365
  1882
  by (simp add: dvd_def)
haftmann@62365
  1883
haftmann@62365
  1884
lemma dvd_antisym:
haftmann@62365
  1885
  "m dvd n \<Longrightarrow> n dvd m \<Longrightarrow> m = (n::nat)"
haftmann@33274
  1886
  unfolding dvd_def
haftmann@57512
  1887
  by (force dest: mult_eq_self_implies_10 simp add: mult.assoc)
haftmann@33274
  1888
haftmann@62365
  1889
lemma dvd_diff_nat [simp]:
haftmann@62365
  1890
  "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m - n :: nat)"
haftmann@62365
  1891
  unfolding dvd_def
haftmann@62365
  1892
  by (blast intro: right_diff_distrib' [symmetric])
haftmann@62365
  1893
haftmann@62365
  1894
lemma dvd_diffD:
haftmann@62365
  1895
  "k dvd m - n \<Longrightarrow> k dvd n \<Longrightarrow> n \<le> m \<Longrightarrow> k dvd (m::nat)"
haftmann@33274
  1896
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
haftmann@33274
  1897
  apply (blast intro: dvd_add)
haftmann@33274
  1898
  done
haftmann@33274
  1899
haftmann@62365
  1900
lemma dvd_diffD1:
haftmann@62365
  1901
  "k dvd m - n \<Longrightarrow> k dvd m \<Longrightarrow> n \<le> m \<Longrightarrow> k dvd (n::nat)"
haftmann@62365
  1902
  by (drule_tac m = m in dvd_diff_nat) auto
haftmann@62365
  1903
haftmann@62365
  1904
lemma dvd_mult_cancel:
haftmann@62365
  1905
  fixes m n k :: nat
haftmann@62365
  1906
  assumes "k * m dvd k * n" and "0 < k"
haftmann@62365
  1907
  shows "m dvd n"
haftmann@62365
  1908
proof -
haftmann@62365
  1909
  from assms(1) obtain q where "k * n = (k * m) * q" ..
haftmann@62365
  1910
  then have "k * n = k * (m * q)" by (simp add: ac_simps)
haftmann@62365
  1911
  with \<open>0 < k\<close> have "n = m * q" by simp
haftmann@62365
  1912
  then show ?thesis ..
haftmann@62365
  1913
qed
haftmann@62365
  1914
  
haftmann@62365
  1915
lemma dvd_mult_cancel1:
haftmann@62365
  1916
  "0 < m \<Longrightarrow> m * n dvd m \<longleftrightarrow> n = (1::nat)"
haftmann@33274
  1917
  apply auto
haftmann@33274
  1918
   apply (subgoal_tac "m*n dvd m*1")
haftmann@33274
  1919
   apply (drule dvd_mult_cancel, auto)
haftmann@33274
  1920
  done
haftmann@33274
  1921
haftmann@62365
  1922
lemma dvd_mult_cancel2:
haftmann@62365
  1923
  "0 < m \<Longrightarrow> n * m dvd m \<longleftrightarrow> n = (1::nat)"
haftmann@62365
  1924
  using dvd_mult_cancel1 [of m n] by (simp add: ac_simps)
haftmann@62365
  1925
haftmann@62365
  1926
lemma dvd_imp_le:
haftmann@62365
  1927
  "k dvd n \<Longrightarrow> 0 < n \<Longrightarrow> k \<le> (n::nat)"
haftmann@62365
  1928
  by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
haftmann@33274
  1929
haftmann@33274
  1930
lemma nat_dvd_not_less:
haftmann@33274
  1931
  fixes m n :: nat
haftmann@33274
  1932
  shows "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"
haftmann@62365
  1933
  by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
haftmann@33274
  1934
haftmann@54222
  1935
lemma less_eq_dvd_minus:
haftmann@51173
  1936
  fixes m n :: nat
haftmann@54222
  1937
  assumes "m \<le> n"
haftmann@54222
  1938
  shows "m dvd n \<longleftrightarrow> m dvd n - m"
haftmann@51173
  1939
proof -
haftmann@54222
  1940
  from assms have "n = m + (n - m)" by simp
haftmann@51173
  1941
  then obtain q where "n = m + q" ..
haftmann@58647
  1942
  then show ?thesis by (simp add: add.commute [of m])
haftmann@51173
  1943
qed
haftmann@51173
  1944
haftmann@51173
  1945
lemma dvd_minus_self:
haftmann@51173
  1946
  fixes m n :: nat
haftmann@51173
  1947
  shows "m dvd n - m \<longleftrightarrow> n < m \<or> m dvd n"
haftmann@51173
  1948
  by (cases "n < m") (auto elim!: dvdE simp add: not_less le_imp_diff_is_add)
haftmann@51173
  1949
haftmann@51173
  1950
lemma dvd_minus_add:
haftmann@51173
  1951
  fixes m n q r :: nat
haftmann@51173
  1952
  assumes "q \<le> n" "q \<le> r * m"
haftmann@51173
  1953
  shows "m dvd n - q \<longleftrightarrow> m dvd n + (r * m - q)"
haftmann@51173
  1954
proof -
haftmann@51173
  1955
  have "m dvd n - q \<longleftrightarrow> m dvd r * m + (n - q)"
haftmann@58649
  1956
    using dvd_add_times_triv_left_iff [of m r] by simp
wenzelm@53374
  1957
  also from assms have "\<dots> \<longleftrightarrow> m dvd r * m + n - q" by simp
wenzelm@53374
  1958
  also from assms have "\<dots> \<longleftrightarrow> m dvd (r * m - q) + n" by simp
haftmann@57512
  1959
  also have "\<dots> \<longleftrightarrow> m dvd n + (r * m - q)" by (simp add: add.commute)
haftmann@51173
  1960
  finally show ?thesis .
haftmann@51173
  1961
qed
haftmann@51173
  1962
haftmann@33274
  1963
haftmann@62365
  1964
subsection \<open>Aliasses\<close>
haftmann@44817
  1965
haftmann@44817
  1966
lemma nat_mult_1: "(1::nat) * n = n"
haftmann@58647
  1967
  by (fact mult_1_left)
lp15@60562
  1968
haftmann@44817
  1969
lemma nat_mult_1_right: "n * (1::nat) = n"
haftmann@58647
  1970
  by (fact mult_1_right)
haftmann@58647
  1971
haftmann@62365
  1972
lemma nat_add_left_cancel:
haftmann@62365
  1973
  fixes k m n :: nat
haftmann@62365
  1974
  shows "k + m = k + n \<longleftrightarrow> m = n"
haftmann@62365
  1975
  by (fact add_left_cancel)
haftmann@62365
  1976
haftmann@62365
  1977
lemma nat_add_right_cancel:
haftmann@62365
  1978
  fixes k m n :: nat
haftmann@62365
  1979
  shows "m + k = n + k \<longleftrightarrow> m = n"
haftmann@62365
  1980
  by (fact add_right_cancel)
haftmann@62365
  1981
haftmann@62365
  1982
lemma diff_mult_distrib:
haftmann@62365
  1983
  "((m::nat) - n) * k = (m * k) - (n * k)"
haftmann@62365
  1984
  by (fact left_diff_distrib')
haftmann@62365
  1985
haftmann@62365
  1986
lemma diff_mult_distrib2:
haftmann@62365
  1987
  "k * ((m::nat) - n) = (k * m) - (k * n)"
haftmann@62365
  1988
  by (fact right_diff_distrib')
haftmann@62365
  1989
haftmann@62365
  1990
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
haftmann@62365
  1991
  by (fact le_add_diff) \<comment> \<open>FIXME delete\<close>
haftmann@62365
  1992
haftmann@62365
  1993
lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
haftmann@62365
  1994
  by (fact le_diff_conv2) \<comment> \<open>FIXME delete\<close>
haftmann@62365
  1995
haftmann@62365
  1996
lemma diff_self_eq_0 [simp]: "(m::nat) - m = 0"
haftmann@62365
  1997
  by (fact diff_cancel)
haftmann@62365
  1998
haftmann@62365
  1999
lemma diff_diff_left [simp]: "(i::nat) - j - k = i - (j + k)"
haftmann@62365
  2000
  by (fact diff_diff_add)
haftmann@62365
  2001
haftmann@62365
  2002
lemma diff_commute: "(i::nat) - j - k = i - k - j"
haftmann@62365
  2003
  by (fact diff_right_commute)
haftmann@62365
  2004
haftmann@62365
  2005
lemma diff_add_inverse: "(n + m) - n = (m::nat)"
haftmann@62365
  2006
  by (fact add_diff_cancel_left')
haftmann@62365
  2007
haftmann@62365
  2008
lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
haftmann@62365
  2009
  by (fact add_diff_cancel_right')
haftmann@62365
  2010
haftmann@62365
  2011
lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
haftmann@62365
  2012
  by (fact add_diff_cancel_left)
haftmann@62365
  2013
haftmann@62365
  2014
lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
haftmann@62365
  2015
  by (fact add_diff_cancel_right)
haftmann@62365
  2016
haftmann@62365
  2017
lemma diff_add_0: "n - (n + m) = (0::nat)"
haftmann@62365
  2018
  by (fact diff_add_zero)
haftmann@62365
  2019
haftmann@62365
  2020
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
haftmann@62365
  2021
  by (fact distrib_left)
haftmann@62365
  2022
haftmann@62365
  2023
lemmas nat_distrib =
haftmann@62365
  2024
  add_mult_distrib distrib_left diff_mult_distrib diff_mult_distrib2
haftmann@62365
  2025
haftmann@44817
  2026
wenzelm@60758
  2027
subsection \<open>Size of a datatype value\<close>
haftmann@25193
  2028
haftmann@29608
  2029
class size =
wenzelm@61799
  2030
  fixes size :: "'a \<Rightarrow> nat" \<comment> \<open>see further theory \<open>Wellfounded\<close>\<close>
haftmann@23852
  2031
blanchet@58377
  2032
instantiation nat :: size
blanchet@58377
  2033
begin
blanchet@58377
  2034
blanchet@58377
  2035
definition size_nat where
wenzelm@61076
  2036
  [simp, code]: "size (n::nat) = n"
blanchet@58377
  2037
blanchet@58377
  2038
instance ..
blanchet@58377
  2039
blanchet@58377
  2040
end
blanchet@58377
  2041
blanchet@58377
  2042
wenzelm@60758
  2043
subsection \<open>Code module namespace\<close>
haftmann@33364
  2044
haftmann@52435
  2045
code_identifier
haftmann@52435
  2046
  code_module Nat \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  2047
huffman@47108
  2048
hide_const (open) of_nat_aux
huffman@47108
  2049
haftmann@25193
  2050
end