author | wenzelm |
Fri, 14 Oct 2016 21:35:02 +0200 | |
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parent 63979 | 95c3ae4baba8 |
child 64447 | e44f5c123f26 |
permissions | -rw-r--r-- |
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(* Title: HOL/Nat.thy |
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Author: Tobias Nipkow |
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Author: Lawrence C Paulson |
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Author: Markus Wenzel |
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Type "nat" is a linear order, and a datatype; arithmetic operators + - |
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and * (for div and mod, see theory Divides). |
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*) |
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section \<open>Natural numbers\<close> |
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theory Nat |
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imports Inductive Typedef Fun Rings |
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begin |
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named_theorems arith "arith facts -- only ground formulas" |
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ML_file "Tools/arith_data.ML" |
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subsection \<open>Type \<open>ind\<close>\<close> |
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typedecl ind |
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axiomatization Zero_Rep :: ind and Suc_Rep :: "ind \<Rightarrow> ind" |
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\<comment> \<open>The axiom of infinity in 2 parts:\<close> |
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where Suc_Rep_inject: "Suc_Rep x = Suc_Rep y \<Longrightarrow> x = y" |
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and Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep" |
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subsection \<open>Type nat\<close> |
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text \<open>Type definition\<close> |
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inductive Nat :: "ind \<Rightarrow> bool" |
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where |
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Zero_RepI: "Nat Zero_Rep" |
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| Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)" |
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typedef nat = "{n. Nat n}" |
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morphisms Rep_Nat Abs_Nat |
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using Nat.Zero_RepI by auto |
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lemma Nat_Rep_Nat: "Nat (Rep_Nat n)" |
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using Rep_Nat by simp |
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lemma Nat_Abs_Nat_inverse: "Nat n \<Longrightarrow> Rep_Nat (Abs_Nat n) = n" |
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using Abs_Nat_inverse by simp |
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lemma Nat_Abs_Nat_inject: "Nat n \<Longrightarrow> Nat m \<Longrightarrow> Abs_Nat n = Abs_Nat m \<longleftrightarrow> n = m" |
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using Abs_Nat_inject by simp |
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instantiation nat :: zero |
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begin |
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definition Zero_nat_def: "0 = Abs_Nat Zero_Rep" |
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instance .. |
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end |
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definition Suc :: "nat \<Rightarrow> nat" |
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where "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))" |
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lemma Suc_not_Zero: "Suc m \<noteq> 0" |
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by (simp add: Zero_nat_def Suc_def Suc_RepI Zero_RepI |
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Nat_Abs_Nat_inject Suc_Rep_not_Zero_Rep Nat_Rep_Nat) |
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lemma Zero_not_Suc: "0 \<noteq> Suc m" |
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by (rule not_sym) (rule Suc_not_Zero) |
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lemma Suc_Rep_inject': "Suc_Rep x = Suc_Rep y \<longleftrightarrow> x = y" |
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by (rule iffI, rule Suc_Rep_inject) simp_all |
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lemma nat_induct0: |
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assumes "P 0" |
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and "\<And>n. P n \<Longrightarrow> P (Suc n)" |
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shows "P n" |
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using assms |
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apply (unfold Zero_nat_def Suc_def) |
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apply (rule Rep_Nat_inverse [THEN subst]) \<comment> \<open>types force good instantiation\<close> |
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apply (erule Nat_Rep_Nat [THEN Nat.induct]) |
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apply (iprover elim: Nat_Abs_Nat_inverse [THEN subst]) |
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done |
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free_constructors case_nat for "0 :: nat" | Suc pred |
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where "pred (0 :: nat) = (0 :: nat)" |
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apply atomize_elim |
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apply (rename_tac n, induct_tac n rule: nat_induct0, auto) |
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apply (simp add: Suc_def Nat_Abs_Nat_inject Nat_Rep_Nat Suc_RepI Suc_Rep_inject' Rep_Nat_inject) |
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apply (simp only: Suc_not_Zero) |
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done |
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\<comment> \<open>Avoid name clashes by prefixing the output of \<open>old_rep_datatype\<close> with \<open>old\<close>.\<close> |
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setup \<open>Sign.mandatory_path "old"\<close> |
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old_rep_datatype "0 :: nat" Suc |
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apply (erule nat_induct0) |
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apply assumption |
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apply (rule nat.inject) |
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apply (rule nat.distinct(1)) |
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done |
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setup \<open>Sign.parent_path\<close> |
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\<comment> \<open>But erase the prefix for properties that are not generated by \<open>free_constructors\<close>.\<close> |
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setup \<open>Sign.mandatory_path "nat"\<close> |
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declare old.nat.inject[iff del] |
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and old.nat.distinct(1)[simp del, induct_simp del] |
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lemmas induct = old.nat.induct |
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lemmas inducts = old.nat.inducts |
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lemmas rec = old.nat.rec |
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lemmas simps = nat.inject nat.distinct nat.case nat.rec |
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setup \<open>Sign.parent_path\<close> |
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abbreviation rec_nat :: "'a \<Rightarrow> (nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a" |
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where "rec_nat \<equiv> old.rec_nat" |
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declare nat.sel[code del] |
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hide_const (open) Nat.pred \<comment> \<open>hide everything related to the selector\<close> |
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hide_fact |
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nat.case_eq_if |
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nat.collapse |
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nat.expand |
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nat.sel |
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nat.exhaust_sel |
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nat.split_sel |
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nat.split_sel_asm |
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lemma nat_exhaust [case_names 0 Suc, cases type: nat]: |
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"(y = 0 \<Longrightarrow> P) \<Longrightarrow> (\<And>nat. y = Suc nat \<Longrightarrow> P) \<Longrightarrow> P" |
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\<comment> \<open>for backward compatibility -- names of variables differ\<close> |
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by (rule old.nat.exhaust) |
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lemma nat_induct [case_names 0 Suc, induct type: nat]: |
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fixes n |
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assumes "P 0" and "\<And>n. P n \<Longrightarrow> P (Suc n)" |
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shows "P n" |
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\<comment> \<open>for backward compatibility -- names of variables differ\<close> |
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using assms by (rule nat.induct) |
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hide_fact |
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nat_exhaust |
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nat_induct0 |
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ML \<open> |
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val nat_basic_lfp_sugar = |
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let |
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val ctr_sugar = the (Ctr_Sugar.ctr_sugar_of_global @{theory} @{type_name nat}); |
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val recx = Logic.varify_types_global @{term rec_nat}; |
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val C = body_type (fastype_of recx); |
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in |
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{T = HOLogic.natT, fp_res_index = 0, C = C, fun_arg_Tsss = [[], [[HOLogic.natT, C]]], |
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ctr_sugar = ctr_sugar, recx = recx, rec_thms = @{thms nat.rec}} |
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end; |
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\<close> |
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setup \<open> |
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let |
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fun basic_lfp_sugars_of _ [@{typ nat}] _ _ ctxt = |
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([], [0], [nat_basic_lfp_sugar], [], [], [], TrueI (*dummy*), [], false, ctxt) |
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| basic_lfp_sugars_of bs arg_Ts callers callssss ctxt = |
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BNF_LFP_Rec_Sugar.default_basic_lfp_sugars_of bs arg_Ts callers callssss ctxt; |
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in |
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BNF_LFP_Rec_Sugar.register_lfp_rec_extension |
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{nested_simps = [], is_new_datatype = K (K true), basic_lfp_sugars_of = basic_lfp_sugars_of, |
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rewrite_nested_rec_call = NONE} |
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end |
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\<close> |
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text \<open>Injectiveness and distinctness lemmas\<close> |
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lemma inj_Suc[simp]: "inj_on Suc N" |
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by (simp add: inj_on_def) |
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lemma Suc_neq_Zero: "Suc m = 0 \<Longrightarrow> R" |
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by (rule notE) (rule Suc_not_Zero) |
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lemma Zero_neq_Suc: "0 = Suc m \<Longrightarrow> R" |
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by (rule Suc_neq_Zero) (erule sym) |
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lemma Suc_inject: "Suc x = Suc y \<Longrightarrow> x = y" |
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by (rule inj_Suc [THEN injD]) |
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lemma n_not_Suc_n: "n \<noteq> Suc n" |
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by (induct n) simp_all |
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lemma Suc_n_not_n: "Suc n \<noteq> n" |
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by (rule not_sym) (rule n_not_Suc_n) |
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text \<open>A special form of induction for reasoning about @{term "m < n"} and @{term "m - n"}.\<close> |
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lemma diff_induct: |
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assumes "\<And>x. P x 0" |
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and "\<And>y. P 0 (Suc y)" |
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and "\<And>x y. P x y \<Longrightarrow> P (Suc x) (Suc y)" |
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shows "P m n" |
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proof (induct n arbitrary: m) |
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case 0 |
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show ?case by (rule assms(1)) |
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next |
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case (Suc n) |
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show ?case |
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proof (induct m) |
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case 0 |
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show ?case by (rule assms(2)) |
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next |
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case (Suc m) |
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from \<open>P m n\<close> show ?case by (rule assms(3)) |
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qed |
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qed |
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subsection \<open>Arithmetic operators\<close> |
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instantiation nat :: comm_monoid_diff |
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begin |
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primrec plus_nat |
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where |
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add_0: "0 + n = (n::nat)" |
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| add_Suc: "Suc m + n = Suc (m + n)" |
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lemma add_0_right [simp]: "m + 0 = m" |
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for m :: nat |
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by (induct m) simp_all |
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lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)" |
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by (induct m) simp_all |
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declare add_0 [code] |
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lemma add_Suc_shift [code]: "Suc m + n = m + Suc n" |
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|
236 |
by simp |
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|
237 |
|
63588 | 238 |
primrec minus_nat |
239 |
where |
|
240 |
diff_0 [code]: "m - 0 = (m::nat)" |
|
241 |
| diff_Suc: "m - Suc n = (case m - n of 0 \<Rightarrow> 0 | Suc k \<Rightarrow> k)" |
|
24995 | 242 |
|
28514 | 243 |
declare diff_Suc [simp del] |
26072
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|
244 |
|
63588 | 245 |
lemma diff_0_eq_0 [simp, code]: "0 - n = 0" |
246 |
for n :: nat |
|
26072
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|
247 |
by (induct n) (simp_all add: diff_Suc) |
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|
248 |
|
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changeset
|
249 |
lemma diff_Suc_Suc [simp, code]: "Suc m - Suc n = m - n" |
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parents:
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|
250 |
by (induct n) (simp_all add: diff_Suc) |
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|
251 |
|
63110 | 252 |
instance |
253 |
proof |
|
26072
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parents:
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diff
changeset
|
254 |
fix n m q :: nat |
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parents:
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|
255 |
show "(n + m) + q = n + (m + q)" by (induct n) simp_all |
f65a7fa2da6c
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parents:
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|
256 |
show "n + m = m + n" by (induct n) simp_all |
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59582
diff
changeset
|
257 |
show "m + n - m = n" by (induct m) simp_all |
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59582
diff
changeset
|
258 |
show "n - m - q = n - (m + q)" by (induct q) (simp_all add: diff_Suc) |
26072
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parents:
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diff
changeset
|
259 |
show "0 + n = n" by simp |
49388 | 260 |
show "0 - n = 0" by simp |
26072
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|
261 |
qed |
f65a7fa2da6c
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parents:
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diff
changeset
|
262 |
|
f65a7fa2da6c
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parents:
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changeset
|
263 |
end |
f65a7fa2da6c
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parents:
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|
264 |
|
36176
3fe7e97ccca8
replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
wenzelm
parents:
35828
diff
changeset
|
265 |
hide_fact (open) add_0 add_0_right diff_0 |
35047
1b2bae06c796
hide fact Nat.add_0_right; make add_0_right from Groups priority
haftmann
parents:
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diff
changeset
|
266 |
|
26072
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|
267 |
instantiation nat :: comm_semiring_1_cancel |
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|
268 |
begin |
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|
269 |
|
63588 | 270 |
definition One_nat_def [simp]: "1 = Suc 0" |
271 |
||
272 |
primrec times_nat |
|
273 |
where |
|
274 |
mult_0: "0 * n = (0::nat)" |
|
275 |
| mult_Suc: "Suc m * n = n + (m * n)" |
|
276 |
||
277 |
lemma mult_0_right [simp]: "m * 0 = 0" |
|
278 |
for m :: nat |
|
26072
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|
279 |
by (induct m) simp_all |
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parents:
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changeset
|
280 |
|
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|
281 |
lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
282 |
by (induct m) (simp_all add: add.left_commute) |
26072
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parents:
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changeset
|
283 |
|
63588 | 284 |
lemma add_mult_distrib: "(m + n) * k = (m * k) + (n * k)" |
285 |
for m n k :: nat |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
286 |
by (induct m) (simp_all add: add.assoc) |
26072
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parents:
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changeset
|
287 |
|
63110 | 288 |
instance |
289 |
proof |
|
290 |
fix k n m q :: nat |
|
63588 | 291 |
show "0 \<noteq> (1::nat)" |
292 |
by simp |
|
293 |
show "1 * n = n" |
|
294 |
by simp |
|
295 |
show "n * m = m * n" |
|
296 |
by (induct n) simp_all |
|
297 |
show "(n * m) * q = n * (m * q)" |
|
298 |
by (induct n) (simp_all add: add_mult_distrib) |
|
299 |
show "(n + m) * q = n * q + m * q" |
|
300 |
by (rule add_mult_distrib) |
|
63110 | 301 |
show "k * (m - n) = (k * m) - (k * n)" |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60427
diff
changeset
|
302 |
by (induct m n rule: diff_induct) simp_all |
26072
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haftmann
parents:
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diff
changeset
|
303 |
qed |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25563
diff
changeset
|
304 |
|
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25563
diff
changeset
|
305 |
end |
24995 | 306 |
|
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60427
diff
changeset
|
307 |
|
60758 | 308 |
subsubsection \<open>Addition\<close> |
26072
f65a7fa2da6c
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haftmann
parents:
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|
309 |
|
61799 | 310 |
text \<open>Reasoning about \<open>m + 0 = 0\<close>, etc.\<close> |
26072
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haftmann
parents:
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diff
changeset
|
311 |
|
63588 | 312 |
lemma add_is_0 [iff]: "m + n = 0 \<longleftrightarrow> m = 0 \<and> n = 0" |
313 |
for m n :: nat |
|
26072
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parents:
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diff
changeset
|
314 |
by (cases m) simp_all |
f65a7fa2da6c
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parents:
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diff
changeset
|
315 |
|
63110 | 316 |
lemma add_is_1: "m + n = Suc 0 \<longleftrightarrow> m = Suc 0 \<and> n = 0 | m = 0 \<and> n = Suc 0" |
26072
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parents:
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diff
changeset
|
317 |
by (cases m) simp_all |
f65a7fa2da6c
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haftmann
parents:
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diff
changeset
|
318 |
|
63110 | 319 |
lemma one_is_add: "Suc 0 = m + n \<longleftrightarrow> m = Suc 0 \<and> n = 0 | m = 0 \<and> n = Suc 0" |
26072
f65a7fa2da6c
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haftmann
parents:
25928
diff
changeset
|
320 |
by (rule trans, rule eq_commute, rule add_is_1) |
f65a7fa2da6c
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haftmann
parents:
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diff
changeset
|
321 |
|
63588 | 322 |
lemma add_eq_self_zero: "m + n = m \<Longrightarrow> n = 0" |
323 |
for m n :: nat |
|
26072
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haftmann
parents:
25928
diff
changeset
|
324 |
by (induct m) simp_all |
f65a7fa2da6c
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haftmann
parents:
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diff
changeset
|
325 |
|
63588 | 326 |
lemma inj_on_add_nat [simp]: "inj_on (\<lambda>n. n + k) N" |
327 |
for k :: nat |
|
328 |
proof (induct k) |
|
329 |
case 0 |
|
330 |
then show ?case by simp |
|
331 |
next |
|
332 |
case (Suc k) |
|
333 |
show ?case |
|
334 |
using comp_inj_on [OF Suc inj_Suc] by (simp add: o_def) |
|
335 |
qed |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
336 |
|
47208 | 337 |
lemma Suc_eq_plus1: "Suc n = n + 1" |
63588 | 338 |
by simp |
47208 | 339 |
|
340 |
lemma Suc_eq_plus1_left: "Suc n = 1 + n" |
|
63588 | 341 |
by simp |
47208 | 342 |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
343 |
|
60758 | 344 |
subsubsection \<open>Difference\<close> |
26072
f65a7fa2da6c
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haftmann
parents:
25928
diff
changeset
|
345 |
|
f65a7fa2da6c
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haftmann
parents:
25928
diff
changeset
|
346 |
lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k" |
62365 | 347 |
by (simp add: diff_diff_add) |
26072
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haftmann
parents:
25928
diff
changeset
|
348 |
|
30093 | 349 |
lemma diff_Suc_1 [simp]: "Suc n - 1 = n" |
63588 | 350 |
by simp |
351 |
||
30093 | 352 |
|
60758 | 353 |
subsubsection \<open>Multiplication\<close> |
26072
f65a7fa2da6c
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haftmann
parents:
25928
diff
changeset
|
354 |
|
63110 | 355 |
lemma mult_is_0 [simp]: "m * n = 0 \<longleftrightarrow> m = 0 \<or> n = 0" for m n :: nat |
26072
f65a7fa2da6c
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haftmann
parents:
25928
diff
changeset
|
356 |
by (induct m) auto |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
357 |
|
63110 | 358 |
lemma mult_eq_1_iff [simp]: "m * n = Suc 0 \<longleftrightarrow> m = Suc 0 \<and> n = Suc 0" |
63588 | 359 |
proof (induct m) |
360 |
case 0 |
|
361 |
then show ?case by simp |
|
362 |
next |
|
363 |
case (Suc m) |
|
364 |
then show ?case by (induct n) auto |
|
365 |
qed |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
366 |
|
63110 | 367 |
lemma one_eq_mult_iff [simp]: "Suc 0 = m * n \<longleftrightarrow> m = Suc 0 \<and> n = Suc 0" |
26072
f65a7fa2da6c
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haftmann
parents:
25928
diff
changeset
|
368 |
apply (rule trans) |
63588 | 369 |
apply (rule_tac [2] mult_eq_1_iff) |
370 |
apply fastforce |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
371 |
done |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
372 |
|
63588 | 373 |
lemma nat_mult_eq_1_iff [simp]: "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1" |
374 |
for m n :: nat |
|
30079
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30056
diff
changeset
|
375 |
unfolding One_nat_def by (rule mult_eq_1_iff) |
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30056
diff
changeset
|
376 |
|
63588 | 377 |
lemma nat_1_eq_mult_iff [simp]: "1 = m * n \<longleftrightarrow> m = 1 \<and> n = 1" |
378 |
for m n :: nat |
|
30079
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30056
diff
changeset
|
379 |
unfolding One_nat_def by (rule one_eq_mult_iff) |
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30056
diff
changeset
|
380 |
|
63588 | 381 |
lemma mult_cancel1 [simp]: "k * m = k * n \<longleftrightarrow> m = n \<or> k = 0" |
382 |
for k m n :: nat |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
383 |
proof - |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
384 |
have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n" |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
385 |
proof (induct n arbitrary: m) |
63110 | 386 |
case 0 |
387 |
then show "m = 0" by simp |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
388 |
next |
63110 | 389 |
case (Suc n) |
390 |
then show "m = Suc n" |
|
391 |
by (cases m) (simp_all add: eq_commute [of 0]) |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
392 |
qed |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
393 |
then show ?thesis by auto |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
394 |
qed |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
395 |
|
63588 | 396 |
lemma mult_cancel2 [simp]: "m * k = n * k \<longleftrightarrow> m = n \<or> k = 0" |
397 |
for k m n :: nat |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
398 |
by (simp add: mult.commute) |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
399 |
|
63110 | 400 |
lemma Suc_mult_cancel1: "Suc k * m = Suc k * n \<longleftrightarrow> m = n" |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
401 |
by (subst mult_cancel1) simp |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
402 |
|
24995 | 403 |
|
60758 | 404 |
subsection \<open>Orders on @{typ nat}\<close> |
405 |
||
406 |
subsubsection \<open>Operation definition\<close> |
|
24995 | 407 |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
408 |
instantiation nat :: linorder |
25510 | 409 |
begin |
410 |
||
63588 | 411 |
primrec less_eq_nat |
412 |
where |
|
413 |
"(0::nat) \<le> n \<longleftrightarrow> True" |
|
414 |
| "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)" |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
415 |
|
28514 | 416 |
declare less_eq_nat.simps [simp del] |
63110 | 417 |
|
63588 | 418 |
lemma le0 [iff]: "0 \<le> n" for |
419 |
n :: nat |
|
63110 | 420 |
by (simp add: less_eq_nat.simps) |
421 |
||
63588 | 422 |
lemma [code]: "0 \<le> n \<longleftrightarrow> True" |
423 |
for n :: nat |
|
63110 | 424 |
by simp |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
425 |
|
63588 | 426 |
definition less_nat |
427 |
where less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m" |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
428 |
|
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
429 |
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m" |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
430 |
by (simp add: less_eq_nat.simps(2)) |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
431 |
|
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
432 |
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n" |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
433 |
unfolding less_eq_Suc_le .. |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
434 |
|
63588 | 435 |
lemma le_0_eq [iff]: "n \<le> 0 \<longleftrightarrow> n = 0" |
436 |
for n :: nat |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
437 |
by (induct n) (simp_all add: less_eq_nat.simps(2)) |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
438 |
|
63588 | 439 |
lemma not_less0 [iff]: "\<not> n < 0" |
440 |
for n :: nat |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
441 |
by (simp add: less_eq_Suc_le) |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
442 |
|
63588 | 443 |
lemma less_nat_zero_code [code]: "n < 0 \<longleftrightarrow> False" |
444 |
for n :: nat |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
445 |
by simp |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
446 |
|
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
447 |
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n" |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
448 |
by (simp add: less_eq_Suc_le) |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
449 |
|
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
450 |
lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n" |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
451 |
by (simp add: less_eq_Suc_le) |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
452 |
|
56194 | 453 |
lemma Suc_less_eq2: "Suc n < m \<longleftrightarrow> (\<exists>m'. m = Suc m' \<and> n < m')" |
454 |
by (cases m) auto |
|
455 |
||
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
456 |
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n" |
63110 | 457 |
by (induct m arbitrary: n) (simp_all add: less_eq_nat.simps(2) split: nat.splits) |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
458 |
|
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
459 |
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n" |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
460 |
by (cases n) (auto intro: le_SucI) |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
461 |
|
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
462 |
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n" |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
463 |
by (simp add: less_eq_Suc_le) (erule Suc_leD) |
24995 | 464 |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
465 |
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n" |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
466 |
by (simp add: less_eq_Suc_le) (erule Suc_leD) |
25510 | 467 |
|
26315
cb3badaa192e
removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents:
26300
diff
changeset
|
468 |
instance |
cb3badaa192e
removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents:
26300
diff
changeset
|
469 |
proof |
63110 | 470 |
fix n m q :: nat |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60427
diff
changeset
|
471 |
show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n" |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
472 |
proof (induct n arbitrary: m) |
63110 | 473 |
case 0 |
63588 | 474 |
then show ?case |
475 |
by (cases m) (simp_all add: less_eq_Suc_le) |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
476 |
next |
63110 | 477 |
case (Suc n) |
63588 | 478 |
then show ?case |
479 |
by (cases m) (simp_all add: less_eq_Suc_le) |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
480 |
qed |
63588 | 481 |
show "n \<le> n" |
482 |
by (induct n) simp_all |
|
63110 | 483 |
then show "n = m" if "n \<le> m" and "m \<le> n" |
484 |
using that by (induct n arbitrary: m) |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
485 |
(simp_all add: less_eq_nat.simps(2) split: nat.splits) |
63110 | 486 |
show "n \<le> q" if "n \<le> m" and "m \<le> q" |
487 |
using that |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
488 |
proof (induct n arbitrary: m q) |
63110 | 489 |
case 0 |
490 |
show ?case by simp |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
491 |
next |
63110 | 492 |
case (Suc n) |
493 |
then show ?case |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
494 |
by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify, |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
495 |
simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify, |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
496 |
simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits) |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
497 |
qed |
63110 | 498 |
show "n \<le> m \<or> m \<le> n" |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
499 |
by (induct n arbitrary: m) |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
500 |
(simp_all add: less_eq_nat.simps(2) split: nat.splits) |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
501 |
qed |
25510 | 502 |
|
503 |
end |
|
13449 | 504 |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52435
diff
changeset
|
505 |
instantiation nat :: order_bot |
29652 | 506 |
begin |
507 |
||
63588 | 508 |
definition bot_nat :: nat |
509 |
where "bot_nat = 0" |
|
510 |
||
511 |
instance |
|
512 |
by standard (simp add: bot_nat_def) |
|
29652 | 513 |
|
514 |
end |
|
515 |
||
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
516 |
instance nat :: no_top |
61169 | 517 |
by standard (auto intro: less_Suc_eq_le [THEN iffD2]) |
52289 | 518 |
|
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset
|
519 |
|
60758 | 520 |
subsubsection \<open>Introduction properties\<close> |
13449 | 521 |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
522 |
lemma lessI [iff]: "n < Suc n" |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
523 |
by (simp add: less_Suc_eq_le) |
13449 | 524 |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
525 |
lemma zero_less_Suc [iff]: "0 < Suc n" |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
526 |
by (simp add: less_Suc_eq_le) |
13449 | 527 |
|
528 |
||
60758 | 529 |
subsubsection \<open>Elimination properties\<close> |
13449 | 530 |
|
63588 | 531 |
lemma less_not_refl: "\<not> n < n" |
532 |
for n :: nat |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
533 |
by (rule order_less_irrefl) |
13449 | 534 |
|
63588 | 535 |
lemma less_not_refl2: "n < m \<Longrightarrow> m \<noteq> n" |
536 |
for m n :: nat |
|
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60427
diff
changeset
|
537 |
by (rule not_sym) (rule less_imp_neq) |
13449 | 538 |
|
63588 | 539 |
lemma less_not_refl3: "s < t \<Longrightarrow> s \<noteq> t" |
540 |
for s t :: nat |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
541 |
by (rule less_imp_neq) |
13449 | 542 |
|
63588 | 543 |
lemma less_irrefl_nat: "n < n \<Longrightarrow> R" |
544 |
for n :: nat |
|
26335
961bbcc9d85b
removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents:
26315
diff
changeset
|
545 |
by (rule notE, rule less_not_refl) |
13449 | 546 |
|
63588 | 547 |
lemma less_zeroE: "n < 0 \<Longrightarrow> R" |
548 |
for n :: nat |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
549 |
by (rule notE) (rule not_less0) |
13449 | 550 |
|
63110 | 551 |
lemma less_Suc_eq: "m < Suc n \<longleftrightarrow> m < n \<or> m = n" |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
552 |
unfolding less_Suc_eq_le le_less .. |
13449 | 553 |
|
30079
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30056
diff
changeset
|
554 |
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)" |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
555 |
by (simp add: less_Suc_eq) |
13449 | 556 |
|
63588 | 557 |
lemma less_one [iff]: "n < 1 \<longleftrightarrow> n = 0" |
558 |
for n :: nat |
|
30079
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30056
diff
changeset
|
559 |
unfolding One_nat_def by (rule less_Suc0) |
13449 | 560 |
|
63110 | 561 |
lemma Suc_mono: "m < n \<Longrightarrow> Suc m < Suc n" |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
562 |
by simp |
13449 | 563 |
|
63588 | 564 |
text \<open>"Less than" is antisymmetric, sort of.\<close> |
565 |
lemma less_antisym: "\<not> n < m \<Longrightarrow> n < Suc m \<Longrightarrow> m = n" |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
566 |
unfolding not_less less_Suc_eq_le by (rule antisym) |
14302 | 567 |
|
63588 | 568 |
lemma nat_neq_iff: "m \<noteq> n \<longleftrightarrow> m < n \<or> n < m" |
569 |
for m n :: nat |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
570 |
by (rule linorder_neq_iff) |
13449 | 571 |
|
572 |
||
60758 | 573 |
subsubsection \<open>Inductive (?) properties\<close> |
13449 | 574 |
|
63110 | 575 |
lemma Suc_lessI: "m < n \<Longrightarrow> Suc m \<noteq> n \<Longrightarrow> Suc m < n" |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60427
diff
changeset
|
576 |
unfolding less_eq_Suc_le [of m] le_less by simp |
13449 | 577 |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
578 |
lemma lessE: |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
579 |
assumes major: "i < k" |
63110 | 580 |
and 1: "k = Suc i \<Longrightarrow> P" |
581 |
and 2: "\<And>j. i < j \<Longrightarrow> k = Suc j \<Longrightarrow> P" |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
582 |
shows P |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
583 |
proof - |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
584 |
from major have "\<exists>j. i \<le> j \<and> k = Suc j" |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
585 |
unfolding less_eq_Suc_le by (induct k) simp_all |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
586 |
then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i" |
63110 | 587 |
by (auto simp add: less_le) |
588 |
with 1 2 show P by auto |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
589 |
qed |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
590 |
|
63110 | 591 |
lemma less_SucE: |
592 |
assumes major: "m < Suc n" |
|
593 |
and less: "m < n \<Longrightarrow> P" |
|
594 |
and eq: "m = n \<Longrightarrow> P" |
|
595 |
shows P |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
596 |
apply (rule major [THEN lessE]) |
63588 | 597 |
apply (rule eq) |
598 |
apply blast |
|
599 |
apply (rule less) |
|
600 |
apply blast |
|
13449 | 601 |
done |
602 |
||
63110 | 603 |
lemma Suc_lessE: |
604 |
assumes major: "Suc i < k" |
|
605 |
and minor: "\<And>j. i < j \<Longrightarrow> k = Suc j \<Longrightarrow> P" |
|
606 |
shows P |
|
13449 | 607 |
apply (rule major [THEN lessE]) |
63588 | 608 |
apply (erule lessI [THEN minor]) |
609 |
apply (erule Suc_lessD [THEN minor]) |
|
610 |
apply assumption |
|
13449 | 611 |
done |
612 |
||
63110 | 613 |
lemma Suc_less_SucD: "Suc m < Suc n \<Longrightarrow> m < n" |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
614 |
by simp |
13449 | 615 |
|
616 |
lemma less_trans_Suc: |
|
63110 | 617 |
assumes le: "i < j" |
618 |
shows "j < k \<Longrightarrow> Suc i < k" |
|
63588 | 619 |
proof (induct k) |
620 |
case 0 |
|
621 |
then show ?case by simp |
|
622 |
next |
|
623 |
case (Suc k) |
|
624 |
with le show ?case |
|
625 |
by simp (auto simp add: less_Suc_eq dest: Suc_lessD) |
|
626 |
qed |
|
627 |
||
628 |
text \<open>Can be used with \<open>less_Suc_eq\<close> to get @{prop "n = m \<or> n < m"}.\<close> |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
629 |
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m" |
63588 | 630 |
by (simp only: not_less less_Suc_eq_le) |
13449 | 631 |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
632 |
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m" |
63588 | 633 |
by (simp only: not_le Suc_le_eq) |
634 |
||
635 |
text \<open>Properties of "less than or equal".\<close> |
|
13449 | 636 |
|
63110 | 637 |
lemma le_imp_less_Suc: "m \<le> n \<Longrightarrow> m < Suc n" |
63588 | 638 |
by (simp only: less_Suc_eq_le) |
13449 | 639 |
|
63110 | 640 |
lemma Suc_n_not_le_n: "\<not> Suc n \<le> n" |
63588 | 641 |
by (simp add: not_le less_Suc_eq_le) |
642 |
||
643 |
lemma le_Suc_eq: "m \<le> Suc n \<longleftrightarrow> m \<le> n \<or> m = Suc n" |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
644 |
by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq) |
13449 | 645 |
|
63110 | 646 |
lemma le_SucE: "m \<le> Suc n \<Longrightarrow> (m \<le> n \<Longrightarrow> R) \<Longrightarrow> (m = Suc n \<Longrightarrow> R) \<Longrightarrow> R" |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
647 |
by (drule le_Suc_eq [THEN iffD1], iprover+) |
13449 | 648 |
|
63588 | 649 |
lemma Suc_leI: "m < n \<Longrightarrow> Suc m \<le> n" |
650 |
by (simp only: Suc_le_eq) |
|
651 |
||
652 |
text \<open>Stronger version of \<open>Suc_leD\<close>.\<close> |
|
63110 | 653 |
lemma Suc_le_lessD: "Suc m \<le> n \<Longrightarrow> m < n" |
63588 | 654 |
by (simp only: Suc_le_eq) |
13449 | 655 |
|
63110 | 656 |
lemma less_imp_le_nat: "m < n \<Longrightarrow> m \<le> n" for m n :: nat |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
657 |
unfolding less_eq_Suc_le by (rule Suc_leD) |
13449 | 658 |
|
61799 | 659 |
text \<open>For instance, \<open>(Suc m < Suc n) = (Suc m \<le> n) = (m < n)\<close>\<close> |
26315
cb3badaa192e
removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents:
26300
diff
changeset
|
660 |
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq |
13449 | 661 |
|
662 |
||
63110 | 663 |
text \<open>Equivalence of \<open>m \<le> n\<close> and \<open>m < n \<or> m = n\<close>\<close> |
664 |
||
63588 | 665 |
lemma less_or_eq_imp_le: "m < n \<or> m = n \<Longrightarrow> m \<le> n" |
666 |
for m n :: nat |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
667 |
unfolding le_less . |
13449 | 668 |
|
63588 | 669 |
lemma le_eq_less_or_eq: "m \<le> n \<longleftrightarrow> m < n \<or> m = n" |
670 |
for m n :: nat |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
671 |
by (rule le_less) |
13449 | 672 |
|
61799 | 673 |
text \<open>Useful with \<open>blast\<close>.\<close> |
63588 | 674 |
lemma eq_imp_le: "m = n \<Longrightarrow> m \<le> n" |
675 |
for m n :: nat |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
676 |
by auto |
13449 | 677 |
|
63588 | 678 |
lemma le_refl: "n \<le> n" |
679 |
for n :: nat |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
680 |
by simp |
13449 | 681 |
|
63588 | 682 |
lemma le_trans: "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k" |
683 |
for i j k :: nat |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
684 |
by (rule order_trans) |
13449 | 685 |
|
63588 | 686 |
lemma le_antisym: "m \<le> n \<Longrightarrow> n \<le> m \<Longrightarrow> m = n" |
687 |
for m n :: nat |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
688 |
by (rule antisym) |
13449 | 689 |
|
63588 | 690 |
lemma nat_less_le: "m < n \<longleftrightarrow> m \<le> n \<and> m \<noteq> n" |
691 |
for m n :: nat |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
692 |
by (rule less_le) |
13449 | 693 |
|
63588 | 694 |
lemma le_neq_implies_less: "m \<le> n \<Longrightarrow> m \<noteq> n \<Longrightarrow> m < n" |
695 |
for m n :: nat |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
696 |
unfolding less_le .. |
13449 | 697 |
|
63588 | 698 |
lemma nat_le_linear: "m \<le> n | n \<le> m" |
699 |
for m n :: nat |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
700 |
by (rule linear) |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
701 |
|
22718 | 702 |
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat] |
15921 | 703 |
|
63110 | 704 |
lemma le_less_Suc_eq: "m \<le> n \<Longrightarrow> n < Suc m \<longleftrightarrow> n = m" |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
705 |
unfolding less_Suc_eq_le by auto |
13449 | 706 |
|
63110 | 707 |
lemma not_less_less_Suc_eq: "\<not> n < m \<Longrightarrow> n < Suc m \<longleftrightarrow> n = m" |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
708 |
unfolding not_less by (rule le_less_Suc_eq) |
13449 | 709 |
|
710 |
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq |
|
711 |
||
63110 | 712 |
lemma not0_implies_Suc: "n \<noteq> 0 \<Longrightarrow> \<exists>m. n = Suc m" |
713 |
by (cases n) simp_all |
|
714 |
||
715 |
lemma gr0_implies_Suc: "n > 0 \<Longrightarrow> \<exists>m. n = Suc m" |
|
716 |
by (cases n) simp_all |
|
717 |
||
63588 | 718 |
lemma gr_implies_not0: "m < n \<Longrightarrow> n \<noteq> 0" |
719 |
for m n :: nat |
|
63110 | 720 |
by (cases n) simp_all |
721 |
||
63588 | 722 |
lemma neq0_conv[iff]: "n \<noteq> 0 \<longleftrightarrow> 0 < n" |
723 |
for n :: nat |
|
63110 | 724 |
by (cases n) simp_all |
25140 | 725 |
|
61799 | 726 |
text \<open>This theorem is useful with \<open>blast\<close>\<close> |
63588 | 727 |
lemma gr0I: "(n = 0 \<Longrightarrow> False) \<Longrightarrow> 0 < n" |
728 |
for n :: nat |
|
729 |
by (rule neq0_conv[THEN iffD1]) iprover |
|
63110 | 730 |
|
731 |
lemma gr0_conv_Suc: "0 < n \<longleftrightarrow> (\<exists>m. n = Suc m)" |
|
732 |
by (fast intro: not0_implies_Suc) |
|
733 |
||
63588 | 734 |
lemma not_gr0 [iff]: "\<not> 0 < n \<longleftrightarrow> n = 0" |
735 |
for n :: nat |
|
63110 | 736 |
using neq0_conv by blast |
737 |
||
738 |
lemma Suc_le_D: "Suc n \<le> m' \<Longrightarrow> \<exists>m. m' = Suc m" |
|
739 |
by (induct m') simp_all |
|
13449 | 740 |
|
60758 | 741 |
text \<open>Useful in certain inductive arguments\<close> |
63110 | 742 |
lemma less_Suc_eq_0_disj: "m < Suc n \<longleftrightarrow> m = 0 \<or> (\<exists>j. m = Suc j \<and> j < n)" |
743 |
by (cases m) simp_all |
|
13449 | 744 |
|
745 |
||
60758 | 746 |
subsubsection \<open>Monotonicity of Addition\<close> |
13449 | 747 |
|
63110 | 748 |
lemma Suc_pred [simp]: "n > 0 \<Longrightarrow> Suc (n - Suc 0) = n" |
749 |
by (simp add: diff_Suc split: nat.split) |
|
750 |
||
751 |
lemma Suc_diff_1 [simp]: "0 < n \<Longrightarrow> Suc (n - 1) = n" |
|
752 |
unfolding One_nat_def by (rule Suc_pred) |
|
753 |
||
63588 | 754 |
lemma nat_add_left_cancel_le [simp]: "k + m \<le> k + n \<longleftrightarrow> m \<le> n" |
755 |
for k m n :: nat |
|
63110 | 756 |
by (induct k) simp_all |
757 |
||
63588 | 758 |
lemma nat_add_left_cancel_less [simp]: "k + m < k + n \<longleftrightarrow> m < n" |
759 |
for k m n :: nat |
|
63110 | 760 |
by (induct k) simp_all |
761 |
||
63588 | 762 |
lemma add_gr_0 [iff]: "m + n > 0 \<longleftrightarrow> m > 0 \<or> n > 0" |
763 |
for m n :: nat |
|
63110 | 764 |
by (auto dest: gr0_implies_Suc) |
13449 | 765 |
|
60758 | 766 |
text \<open>strict, in 1st argument\<close> |
63588 | 767 |
lemma add_less_mono1: "i < j \<Longrightarrow> i + k < j + k" |
768 |
for i j k :: nat |
|
63110 | 769 |
by (induct k) simp_all |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
770 |
|
60758 | 771 |
text \<open>strict, in both arguments\<close> |
63588 | 772 |
lemma add_less_mono: "i < j \<Longrightarrow> k < l \<Longrightarrow> i + k < j + l" |
773 |
for i j k l :: nat |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
774 |
apply (rule add_less_mono1 [THEN less_trans], assumption+) |
63588 | 775 |
apply (induct j) |
776 |
apply simp_all |
|
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
777 |
done |
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
778 |
|
61799 | 779 |
text \<open>Deleted \<open>less_natE\<close>; use \<open>less_imp_Suc_add RS exE\<close>\<close> |
63110 | 780 |
lemma less_imp_Suc_add: "m < n \<Longrightarrow> \<exists>k. n = Suc (m + k)" |
63588 | 781 |
proof (induct n) |
782 |
case 0 |
|
783 |
then show ?case by simp |
|
784 |
next |
|
785 |
case Suc |
|
786 |
then show ?case |
|
787 |
by (simp add: order_le_less) |
|
788 |
(blast elim!: less_SucE intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric]) |
|
789 |
qed |
|
790 |
||
791 |
lemma le_Suc_ex: "k \<le> l \<Longrightarrow> (\<exists>n. l = k + n)" |
|
792 |
for k l :: nat |
|
56194 | 793 |
by (auto simp: less_Suc_eq_le[symmetric] dest: less_imp_Suc_add) |
794 |
||
61799 | 795 |
text \<open>strict, in 1st argument; proof is by induction on \<open>k > 0\<close>\<close> |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
796 |
lemma mult_less_mono2: |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
797 |
fixes i j :: nat |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
798 |
assumes "i < j" and "0 < k" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
799 |
shows "k * i < k * j" |
63110 | 800 |
using \<open>0 < k\<close> |
801 |
proof (induct k) |
|
802 |
case 0 |
|
803 |
then show ?case by simp |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
804 |
next |
63110 | 805 |
case (Suc k) |
806 |
with \<open>i < j\<close> show ?case |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
807 |
by (cases k) (simp_all add: add_less_mono) |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
808 |
qed |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
809 |
|
60758 | 810 |
text \<open>Addition is the inverse of subtraction: |
811 |
if @{term "n \<le> m"} then @{term "n + (m - n) = m"}.\<close> |
|
63588 | 812 |
lemma add_diff_inverse_nat: "\<not> m < n \<Longrightarrow> n + (m - n) = m" |
813 |
for m n :: nat |
|
63110 | 814 |
by (induct m n rule: diff_induct) simp_all |
815 |
||
63588 | 816 |
lemma nat_le_iff_add: "m \<le> n \<longleftrightarrow> (\<exists>k. n = m + k)" |
817 |
for m n :: nat |
|
63110 | 818 |
using nat_add_left_cancel_le[of m 0] by (auto dest: le_Suc_ex) |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62365
diff
changeset
|
819 |
|
63588 | 820 |
text \<open>The naturals form an ordered \<open>semidom\<close> and a \<open>dioid\<close>.\<close> |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62365
diff
changeset
|
821 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34208
diff
changeset
|
822 |
instance nat :: linordered_semidom |
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset
|
823 |
proof |
63110 | 824 |
fix m n q :: nat |
63588 | 825 |
show "0 < (1::nat)" |
826 |
by simp |
|
827 |
show "m \<le> n \<Longrightarrow> q + m \<le> q + n" |
|
828 |
by simp |
|
829 |
show "m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n" |
|
830 |
by (simp add: mult_less_mono2) |
|
831 |
show "m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> m * n \<noteq> 0" |
|
832 |
by simp |
|
63110 | 833 |
show "n \<le> m \<Longrightarrow> (m - n) + n = m" |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60427
diff
changeset
|
834 |
by (simp add: add_diff_inverse_nat add.commute linorder_not_less) |
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62365
diff
changeset
|
835 |
qed |
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62365
diff
changeset
|
836 |
|
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62365
diff
changeset
|
837 |
instance nat :: dioid |
63110 | 838 |
by standard (rule nat_le_iff_add) |
63588 | 839 |
|
63145 | 840 |
declare le0[simp del] \<comment> \<open>This is now @{thm zero_le}\<close> |
841 |
declare le_0_eq[simp del] \<comment> \<open>This is now @{thm le_zero_eq}\<close> |
|
842 |
declare not_less0[simp del] \<comment> \<open>This is now @{thm not_less_zero}\<close> |
|
843 |
declare not_gr0[simp del] \<comment> \<open>This is now @{thm not_gr_zero}\<close> |
|
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62365
diff
changeset
|
844 |
|
63110 | 845 |
instance nat :: ordered_cancel_comm_monoid_add .. |
846 |
instance nat :: ordered_cancel_comm_monoid_diff .. |
|
847 |
||
44817 | 848 |
|
60758 | 849 |
subsubsection \<open>@{term min} and @{term max}\<close> |
44817 | 850 |
|
851 |
lemma mono_Suc: "mono Suc" |
|
63110 | 852 |
by (rule monoI) simp |
853 |
||
63588 | 854 |
lemma min_0L [simp]: "min 0 n = 0" |
855 |
for n :: nat |
|
63110 | 856 |
by (rule min_absorb1) simp |
857 |
||
63588 | 858 |
lemma min_0R [simp]: "min n 0 = 0" |
859 |
for n :: nat |
|
63110 | 860 |
by (rule min_absorb2) simp |
44817 | 861 |
|
862 |
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)" |
|
63110 | 863 |
by (simp add: mono_Suc min_of_mono) |
864 |
||
865 |
lemma min_Suc1: "min (Suc n) m = (case m of 0 \<Rightarrow> 0 | Suc m' \<Rightarrow> Suc(min n m'))" |
|
866 |
by (simp split: nat.split) |
|
867 |
||
868 |
lemma min_Suc2: "min m (Suc n) = (case m of 0 \<Rightarrow> 0 | Suc m' \<Rightarrow> Suc(min m' n))" |
|
869 |
by (simp split: nat.split) |
|
870 |
||
63588 | 871 |
lemma max_0L [simp]: "max 0 n = n" |
872 |
for n :: nat |
|
63110 | 873 |
by (rule max_absorb2) simp |
874 |
||
63588 | 875 |
lemma max_0R [simp]: "max n 0 = n" |
876 |
for n :: nat |
|
63110 | 877 |
by (rule max_absorb1) simp |
878 |
||
879 |
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc (max m n)" |
|
880 |
by (simp add: mono_Suc max_of_mono) |
|
881 |
||
882 |
lemma max_Suc1: "max (Suc n) m = (case m of 0 \<Rightarrow> Suc n | Suc m' \<Rightarrow> Suc (max n m'))" |
|
883 |
by (simp split: nat.split) |
|
884 |
||
885 |
lemma max_Suc2: "max m (Suc n) = (case m of 0 \<Rightarrow> Suc n | Suc m' \<Rightarrow> Suc (max m' n))" |
|
886 |
by (simp split: nat.split) |
|
887 |
||
63588 | 888 |
lemma nat_mult_min_left: "min m n * q = min (m * q) (n * q)" |
889 |
for m n q :: nat |
|
63110 | 890 |
by (simp add: min_def not_le) |
891 |
(auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans) |
|
892 |
||
63588 | 893 |
lemma nat_mult_min_right: "m * min n q = min (m * n) (m * q)" |
894 |
for m n q :: nat |
|
63110 | 895 |
by (simp add: min_def not_le) |
896 |
(auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans) |
|
897 |
||
63588 | 898 |
lemma nat_add_max_left: "max m n + q = max (m + q) (n + q)" |
899 |
for m n q :: nat |
|
44817 | 900 |
by (simp add: max_def) |
901 |
||
63588 | 902 |
lemma nat_add_max_right: "m + max n q = max (m + n) (m + q)" |
903 |
for m n q :: nat |
|
44817 | 904 |
by (simp add: max_def) |
905 |
||
63588 | 906 |
lemma nat_mult_max_left: "max m n * q = max (m * q) (n * q)" |
907 |
for m n q :: nat |
|
63110 | 908 |
by (simp add: max_def not_le) |
909 |
(auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans) |
|
910 |
||
63588 | 911 |
lemma nat_mult_max_right: "m * max n q = max (m * n) (m * q)" |
912 |
for m n q :: nat |
|
63110 | 913 |
by (simp add: max_def not_le) |
914 |
(auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans) |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
915 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
916 |
|
60758 | 917 |
subsubsection \<open>Additional theorems about @{term "op \<le>"}\<close> |
918 |
||
919 |
text \<open>Complete induction, aka course-of-values induction\<close> |
|
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
920 |
|
63110 | 921 |
instance nat :: wellorder |
922 |
proof |
|
27823 | 923 |
fix P and n :: nat |
63110 | 924 |
assume step: "(\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n" for n :: nat |
27823 | 925 |
have "\<And>q. q \<le> n \<Longrightarrow> P q" |
926 |
proof (induct n) |
|
927 |
case (0 n) |
|
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
928 |
have "P 0" by (rule step) auto |
63588 | 929 |
with 0 show ?case by auto |
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
930 |
next |
27823 | 931 |
case (Suc m n) |
63588 | 932 |
then have "n \<le> m \<or> n = Suc m" |
933 |
by (simp add: le_Suc_eq) |
|
63110 | 934 |
then show ?case |
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
935 |
proof |
63110 | 936 |
assume "n \<le> m" |
937 |
then show "P n" by (rule Suc(1)) |
|
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
938 |
next |
27823 | 939 |
assume n: "n = Suc m" |
63110 | 940 |
show "P n" by (rule step) (rule Suc(1), simp add: n le_simps) |
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
941 |
qed |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
942 |
qed |
27823 | 943 |
then show "P n" by auto |
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
944 |
qed |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
945 |
|
57015 | 946 |
|
63588 | 947 |
lemma Least_eq_0[simp]: "P 0 \<Longrightarrow> Least P = 0" |
948 |
for P :: "nat \<Rightarrow> bool" |
|
63110 | 949 |
by (rule Least_equality[OF _ le0]) |
950 |
||
951 |
lemma Least_Suc: "P n \<Longrightarrow> \<not> P 0 \<Longrightarrow> (LEAST n. P n) = Suc (LEAST m. P (Suc m))" |
|
63588 | 952 |
apply (cases n) |
953 |
apply auto |
|
27823 | 954 |
apply (frule LeastI) |
63588 | 955 |
apply (drule_tac P = "\<lambda>x. P (Suc x)" in LeastI) |
27823 | 956 |
apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))") |
63588 | 957 |
apply (erule_tac [2] Least_le) |
958 |
apply (cases "LEAST x. P x") |
|
959 |
apply auto |
|
960 |
apply (drule_tac P = "\<lambda>x. P (Suc x)" in Least_le) |
|
27823 | 961 |
apply (blast intro: order_antisym) |
962 |
done |
|
963 |
||
63110 | 964 |
lemma Least_Suc2: "P n \<Longrightarrow> Q m \<Longrightarrow> \<not> P 0 \<Longrightarrow> \<forall>k. P (Suc k) = Q k \<Longrightarrow> Least P = Suc (Least Q)" |
63588 | 965 |
by (erule (1) Least_Suc [THEN ssubst]) simp |
966 |
||
967 |
lemma ex_least_nat_le: "\<not> P 0 \<Longrightarrow> P n \<Longrightarrow> \<exists>k\<le>n. (\<forall>i<k. \<not> P i) \<and> P k" |
|
968 |
for P :: "nat \<Rightarrow> bool" |
|
27823 | 969 |
apply (cases n) |
970 |
apply blast |
|
63110 | 971 |
apply (rule_tac x="LEAST k. P k" in exI) |
27823 | 972 |
apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex) |
973 |
done |
|
974 |
||
63588 | 975 |
lemma ex_least_nat_less: "\<not> P 0 \<Longrightarrow> P n \<Longrightarrow> \<exists>k<n. (\<forall>i\<le>k. \<not> P i) \<and> P (k + 1)" |
976 |
for P :: "nat \<Rightarrow> bool" |
|
27823 | 977 |
apply (cases n) |
978 |
apply blast |
|
979 |
apply (frule (1) ex_least_nat_le) |
|
980 |
apply (erule exE) |
|
981 |
apply (case_tac k) |
|
982 |
apply simp |
|
983 |
apply (rename_tac k1) |
|
984 |
apply (rule_tac x=k1 in exI) |
|
985 |
apply (auto simp add: less_eq_Suc_le) |
|
986 |
done |
|
987 |
||
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
988 |
lemma nat_less_induct: |
63110 | 989 |
fixes P :: "nat \<Rightarrow> bool" |
990 |
assumes "\<And>n. \<forall>m. m < n \<longrightarrow> P m \<Longrightarrow> P n" |
|
991 |
shows "P n" |
|
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
992 |
using assms less_induct by blast |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
993 |
|
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
994 |
lemma measure_induct_rule [case_names less]: |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
995 |
fixes f :: "'a \<Rightarrow> nat" |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
996 |
assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x" |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
997 |
shows "P a" |
63110 | 998 |
by (induct m \<equiv> "f a" arbitrary: a rule: less_induct) (auto intro: step) |
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
999 |
|
60758 | 1000 |
text \<open>old style induction rules:\<close> |
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1001 |
lemma measure_induct: |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1002 |
fixes f :: "'a \<Rightarrow> nat" |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1003 |
shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a" |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1004 |
by (rule measure_induct_rule [of f P a]) iprover |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1005 |
|
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1006 |
lemma full_nat_induct: |
63110 | 1007 |
assumes step: "\<And>n. (\<forall>m. Suc m \<le> n \<longrightarrow> P m) \<Longrightarrow> P n" |
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1008 |
shows "P n" |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1009 |
by (rule less_induct) (auto intro: step simp:le_simps) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
1010 |
|
63110 | 1011 |
text\<open>An induction rule for establishing binary relations\<close> |
62683 | 1012 |
lemma less_Suc_induct [consumes 1]: |
63110 | 1013 |
assumes less: "i < j" |
1014 |
and step: "\<And>i. P i (Suc i)" |
|
1015 |
and trans: "\<And>i j k. i < j \<Longrightarrow> j < k \<Longrightarrow> P i j \<Longrightarrow> P j k \<Longrightarrow> P i k" |
|
19870 | 1016 |
shows "P i j" |
1017 |
proof - |
|
63110 | 1018 |
from less obtain k where j: "j = Suc (i + k)" |
1019 |
by (auto dest: less_imp_Suc_add) |
|
22718 | 1020 |
have "P i (Suc (i + k))" |
19870 | 1021 |
proof (induct k) |
22718 | 1022 |
case 0 |
1023 |
show ?case by (simp add: step) |
|
19870 | 1024 |
next |
1025 |
case (Suc k) |
|
31714 | 1026 |
have "0 + i < Suc k + i" by (rule add_less_mono1) simp |
63110 | 1027 |
then have "i < Suc (i + k)" by (simp add: add.commute) |
31714 | 1028 |
from trans[OF this lessI Suc step] |
1029 |
show ?case by simp |
|
19870 | 1030 |
qed |
63110 | 1031 |
then show "P i j" by (simp add: j) |
19870 | 1032 |
qed |
1033 |
||
63111 | 1034 |
text \<open> |
1035 |
The method of infinite descent, frequently used in number theory. |
|
1036 |
Provided by Roelof Oosterhuis. |
|
1037 |
\<open>P n\<close> is true for all natural numbers if |
|
1038 |
\<^item> case ``0'': given \<open>n = 0\<close> prove \<open>P n\<close> |
|
1039 |
\<^item> case ``smaller'': given \<open>n > 0\<close> and \<open>\<not> P n\<close> prove there exists |
|
1040 |
a smaller natural number \<open>m\<close> such that \<open>\<not> P m\<close>. |
|
1041 |
\<close> |
|
1042 |
||
63110 | 1043 |
lemma infinite_descent: "(\<And>n. \<not> P n \<Longrightarrow> \<exists>m<n. \<not> P m) \<Longrightarrow> P n" for P :: "nat \<Rightarrow> bool" |
63111 | 1044 |
\<comment> \<open>compact version without explicit base case\<close> |
63110 | 1045 |
by (induct n rule: less_induct) auto |
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1046 |
|
63111 | 1047 |
lemma infinite_descent0 [case_names 0 smaller]: |
63110 | 1048 |
fixes P :: "nat \<Rightarrow> bool" |
63111 | 1049 |
assumes "P 0" |
1050 |
and "\<And>n. n > 0 \<Longrightarrow> \<not> P n \<Longrightarrow> \<exists>m. m < n \<and> \<not> P m" |
|
63110 | 1051 |
shows "P n" |
1052 |
apply (rule infinite_descent) |
|
1053 |
using assms |
|
1054 |
apply (case_tac "n > 0") |
|
63588 | 1055 |
apply auto |
63110 | 1056 |
done |
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1057 |
|
60758 | 1058 |
text \<open> |
63111 | 1059 |
Infinite descent using a mapping to \<open>nat\<close>: |
1060 |
\<open>P x\<close> is true for all \<open>x \<in> D\<close> if there exists a \<open>V \<in> D \<Rightarrow> nat\<close> and |
|
1061 |
\<^item> case ``0'': given \<open>V x = 0\<close> prove \<open>P x\<close> |
|
1062 |
\<^item> ``smaller'': given \<open>V x > 0\<close> and \<open>\<not> P x\<close> prove |
|
1063 |
there exists a \<open>y \<in> D\<close> such that \<open>V y < V x\<close> and \<open>\<not> P y\<close>. |
|
1064 |
\<close> |
|
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1065 |
corollary infinite_descent0_measure [case_names 0 smaller]: |
63110 | 1066 |
fixes V :: "'a \<Rightarrow> nat" |
1067 |
assumes 1: "\<And>x. V x = 0 \<Longrightarrow> P x" |
|
1068 |
and 2: "\<And>x. V x > 0 \<Longrightarrow> \<not> P x \<Longrightarrow> \<exists>y. V y < V x \<and> \<not> P y" |
|
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1069 |
shows "P x" |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1070 |
proof - |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1071 |
obtain n where "n = V x" by auto |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1072 |
moreover have "\<And>x. V x = n \<Longrightarrow> P x" |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1073 |
proof (induct n rule: infinite_descent0) |
63110 | 1074 |
case 0 |
1075 |
with 1 show "P x" by auto |
|
1076 |
next |
|
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1077 |
case (smaller n) |
63110 | 1078 |
then obtain x where *: "V x = n " and "V x > 0 \<and> \<not> P x" by auto |
1079 |
with 2 obtain y where "V y < V x \<and> \<not> P y" by auto |
|
63111 | 1080 |
with * obtain m where "m = V y \<and> m < n \<and> \<not> P y" by auto |
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1081 |
then show ?case by auto |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1082 |
qed |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1083 |
ultimately show "P x" by auto |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1084 |
qed |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1085 |
|
63588 | 1086 |
text \<open>Again, without explicit base case:\<close> |
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1087 |
lemma infinite_descent_measure: |
63110 | 1088 |
fixes V :: "'a \<Rightarrow> nat" |
1089 |
assumes "\<And>x. \<not> P x \<Longrightarrow> \<exists>y. V y < V x \<and> \<not> P y" |
|
1090 |
shows "P x" |
|
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1091 |
proof - |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1092 |
from assms obtain n where "n = V x" by auto |
63110 | 1093 |
moreover have "\<And>x. V x = n \<Longrightarrow> P x" |
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1094 |
proof (induct n rule: infinite_descent, auto) |
63111 | 1095 |
show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" if "\<not> P x" for x |
1096 |
using assms and that by auto |
|
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1097 |
qed |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1098 |
ultimately show "P x" by auto |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1099 |
qed |
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset
|
1100 |
|
63111 | 1101 |
text \<open>A (clumsy) way of lifting \<open><\<close> monotonicity to \<open>\<le>\<close> monotonicity\<close> |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
1102 |
lemma less_mono_imp_le_mono: |
63110 | 1103 |
fixes f :: "nat \<Rightarrow> nat" |
1104 |
and i j :: nat |
|
1105 |
assumes "\<And>i j::nat. i < j \<Longrightarrow> f i < f j" |
|
1106 |
and "i \<le> j" |
|
1107 |
shows "f i \<le> f j" |
|
1108 |
using assms by (auto simp add: order_le_less) |
|
24438 | 1109 |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
1110 |
|
60758 | 1111 |
text \<open>non-strict, in 1st argument\<close> |
63588 | 1112 |
lemma add_le_mono1: "i \<le> j \<Longrightarrow> i + k \<le> j + k" |
1113 |
for i j k :: nat |
|
63110 | 1114 |
by (rule add_right_mono) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset
|
1115 |
|
60758 | 1116 |
text \<open>non-strict, in both arguments\<close> |
63588 | 1117 |
lemma add_le_mono: "i \<le> j \<Longrightarrow> k \<le> l \<Longrightarrow> i + k \<le> j + l" |
1118 |
for i j k l :: nat |
|
63110 | 1119 |
by (rule add_mono) |
1120 |
||
63588 | 1121 |
lemma le_add2: "n \<le> m + n" |
1122 |
for m n :: nat |
|
62608 | 1123 |
by simp |
13449 | 1124 |
|
63588 | 1125 |
lemma le_add1: "n \<le> n + m" |
1126 |
for m n :: nat |
|
62608 | 1127 |
by simp |
13449 | 1128 |
|
1129 |
lemma less_add_Suc1: "i < Suc (i + m)" |
|
63110 | 1130 |
by (rule le_less_trans, rule le_add1, rule lessI) |
13449 | 1131 |
|
1132 |
lemma less_add_Suc2: "i < Suc (m + i)" |
|
63110 | 1133 |
by (rule le_less_trans, rule le_add2, rule lessI) |
1134 |
||
1135 |
lemma less_iff_Suc_add: "m < n \<longleftrightarrow> (\<exists>k. n = Suc (m + k))" |
|
1136 |
by (iprover intro!: less_add_Suc1 less_imp_Suc_add) |
|
1137 |
||
63588 | 1138 |
lemma trans_le_add1: "i \<le> j \<Longrightarrow> i \<le> j + m" |
1139 |
for i j m :: nat |
|
63110 | 1140 |
by (rule le_trans, assumption, rule le_add1) |
1141 |
||
63588 | 1142 |
lemma trans_le_add2: "i \<le> j \<Longrightarrow> i \<le> m + j" |
1143 |
for i j m :: nat |
|
63110 | 1144 |
by (rule le_trans, assumption, rule le_add2) |
1145 |
||
63588 | 1146 |
lemma trans_less_add1: "i < j \<Longrightarrow> i < j + m" |
1147 |
for i j m :: nat |
|
63110 | 1148 |
by (rule less_le_trans, assumption, rule le_add1) |
1149 |
||
63588 | 1150 |
lemma trans_less_add2: "i < j \<Longrightarrow> i < m + j" |
1151 |
for i j m :: nat |
|
63110 | 1152 |
by (rule less_le_trans, assumption, rule le_add2) |
1153 |
||
63588 | 1154 |
lemma add_lessD1: "i + j < k \<Longrightarrow> i < k" |
1155 |
for i j k :: nat |
|
63110 | 1156 |
by (rule le_less_trans [of _ "i+j"]) (simp_all add: le_add1) |
1157 |
||
63588 | 1158 |
lemma not_add_less1 [iff]: "\<not> i + j < i" |
1159 |
for i j :: nat |
|
63110 | 1160 |
apply (rule notI) |
1161 |
apply (drule add_lessD1) |
|
1162 |
apply (erule less_irrefl [THEN notE]) |
|
1163 |
done |
|
1164 |
||
63588 | 1165 |
lemma not_add_less2 [iff]: "\<not> j + i < i" |
1166 |
for i j :: nat |
|
63110 | 1167 |
by (simp add: add.commute) |
1168 |
||
63588 | 1169 |
lemma add_leD1: "m + k \<le> n \<Longrightarrow> m \<le> n" |
1170 |
for k m n :: nat |
|
1171 |
by (rule order_trans [of _ "m + k"]) (simp_all add: le_add1) |
|
1172 |
||
1173 |
lemma add_leD2: "m + k \<le> n \<Longrightarrow> k \<le> n" |
|
1174 |
for k m n :: nat |
|
63110 | 1175 |
apply (simp add: add.commute) |
1176 |
apply (erule add_leD1) |
|
1177 |
done |
|
1178 |
||
63588 | 1179 |
lemma add_leE: "m + k \<le> n \<Longrightarrow> (m \<le> n \<Longrightarrow> k \<le> n \<Longrightarrow> R) \<Longrightarrow> R" |
1180 |
for k m n :: nat |
|
63110 | 1181 |
by (blast dest: add_leD1 add_leD2) |
1182 |
||
1183 |
text \<open>needs \<open>\<And>k\<close> for \<open>ac_simps\<close> to work\<close> |
|
63588 | 1184 |
lemma less_add_eq_less: "\<And>k. k < l \<Longrightarrow> m + l = k + n \<Longrightarrow> m < n" |
1185 |
for l m n :: nat |
|
63110 | 1186 |
by (force simp del: add_Suc_right simp add: less_iff_Suc_add add_Suc_right [symmetric] ac_simps) |
13449 | 1187 |
|
1188 |
||
60758 | 1189 |
subsubsection \<open>More results about difference\<close> |
13449 | 1190 |
|
63110 | 1191 |
lemma Suc_diff_le: "n \<le> m \<Longrightarrow> Suc m - n = Suc (m - n)" |
1192 |
by (induct m n rule: diff_induct) simp_all |
|
13449 | 1193 |
|
1194 |
lemma diff_less_Suc: "m - n < Suc m" |
|
63588 | 1195 |
apply (induct m n rule: diff_induct) |
1196 |
apply (erule_tac [3] less_SucE) |
|
1197 |
apply (simp_all add: less_Suc_eq) |
|
1198 |
done |
|
1199 |
||
1200 |
lemma diff_le_self [simp]: "m - n \<le> m" |
|
1201 |
for m n :: nat |
|
63110 | 1202 |
by (induct m n rule: diff_induct) (simp_all add: le_SucI) |
1203 |
||
63588 | 1204 |
lemma less_imp_diff_less: "j < k \<Longrightarrow> j - n < k" |
1205 |
for j k n :: nat |
|
63110 | 1206 |
by (rule le_less_trans, rule diff_le_self) |
1207 |
||
1208 |
lemma diff_Suc_less [simp]: "0 < n \<Longrightarrow> n - Suc i < n" |
|
1209 |
by (cases n) (auto simp add: le_simps) |
|
1210 |
||
63588 | 1211 |
lemma diff_add_assoc: "k \<le> j \<Longrightarrow> (i + j) - k = i + (j - k)" |
1212 |
for i j k :: nat |
|
63110 | 1213 |
by (induct j k rule: diff_induct) simp_all |
1214 |
||
63588 | 1215 |
lemma add_diff_assoc [simp]: "k \<le> j \<Longrightarrow> i + (j - k) = i + j - k" |
1216 |
for i j k :: nat |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1217 |
by (fact diff_add_assoc [symmetric]) |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1218 |
|
63588 | 1219 |
lemma diff_add_assoc2: "k \<le> j \<Longrightarrow> (j + i) - k = (j - k) + i" |
1220 |
for i j k :: nat |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1221 |
by (simp add: ac_simps) |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1222 |
|
63588 | 1223 |
lemma add_diff_assoc2 [simp]: "k \<le> j \<Longrightarrow> j - k + i = j + i - k" |
1224 |
for i j k :: nat |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1225 |
by (fact diff_add_assoc2 [symmetric]) |
13449 | 1226 |
|
63588 | 1227 |
lemma le_imp_diff_is_add: "i \<le> j \<Longrightarrow> (j - i = k) = (j = k + i)" |
1228 |
for i j k :: nat |
|
63110 | 1229 |
by auto |
1230 |
||
63588 | 1231 |
lemma diff_is_0_eq [simp]: "m - n = 0 \<longleftrightarrow> m \<le> n" |
1232 |
for m n :: nat |
|
63110 | 1233 |
by (induct m n rule: diff_induct) simp_all |
1234 |
||
63588 | 1235 |
lemma diff_is_0_eq' [simp]: "m \<le> n \<Longrightarrow> m - n = 0" |
1236 |
for m n :: nat |
|
63110 | 1237 |
by (rule iffD2, rule diff_is_0_eq) |
1238 |
||
63588 | 1239 |
lemma zero_less_diff [simp]: "0 < n - m \<longleftrightarrow> m < n" |
1240 |
for m n :: nat |
|
63110 | 1241 |
by (induct m n rule: diff_induct) simp_all |
13449 | 1242 |
|
22718 | 1243 |
lemma less_imp_add_positive: |
1244 |
assumes "i < j" |
|
63110 | 1245 |
shows "\<exists>k::nat. 0 < k \<and> i + k = j" |
22718 | 1246 |
proof |
63110 | 1247 |
from assms show "0 < j - i \<and> i + (j - i) = j" |
23476 | 1248 |
by (simp add: order_less_imp_le) |
22718 | 1249 |
qed |
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
7702
diff
changeset
|
1250 |
|
60758 | 1251 |
text \<open>a nice rewrite for bounded subtraction\<close> |
63588 | 1252 |
lemma nat_minus_add_max: "n - m + m = max n m" |
1253 |
for m n :: nat |
|
1254 |
by (simp add: max_def not_le order_less_imp_le) |
|
13449 | 1255 |
|
63110 | 1256 |
lemma nat_diff_split: "P (a - b) \<longleftrightarrow> (a < b \<longrightarrow> P 0) \<and> (\<forall>d. a = b + d \<longrightarrow> P d)" |
1257 |
for a b :: nat |
|
63588 | 1258 |
\<comment> \<open>elimination of \<open>-\<close> on \<open>nat\<close>\<close> |
1259 |
by (cases "a < b") (auto simp add: not_less le_less dest!: add_eq_self_zero [OF sym]) |
|
13449 | 1260 |
|
63110 | 1261 |
lemma nat_diff_split_asm: "P (a - b) \<longleftrightarrow> \<not> (a < b \<and> \<not> P 0 \<or> (\<exists>d. a = b + d \<and> \<not> P d))" |
1262 |
for a b :: nat |
|
63588 | 1263 |
\<comment> \<open>elimination of \<open>-\<close> on \<open>nat\<close> in assumptions\<close> |
62365 | 1264 |
by (auto split: nat_diff_split) |
13449 | 1265 |
|
63110 | 1266 |
lemma Suc_pred': "0 < n \<Longrightarrow> n = Suc(n - 1)" |
47255
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset
|
1267 |
by simp |
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset
|
1268 |
|
63110 | 1269 |
lemma add_eq_if: "m + n = (if m = 0 then n else Suc ((m - 1) + n))" |
47255
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset
|
1270 |
unfolding One_nat_def by (cases m) simp_all |
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset
|
1271 |
|
63588 | 1272 |
lemma mult_eq_if: "m * n = (if m = 0 then 0 else n + ((m - 1) * n))" |
1273 |
for m n :: nat |
|
1274 |
by (cases m) simp_all |
|
47255
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset
|
1275 |
|
63110 | 1276 |
lemma Suc_diff_eq_diff_pred: "0 < n \<Longrightarrow> Suc m - n = m - (n - 1)" |
63588 | 1277 |
by (cases n) simp_all |
47255
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset
|
1278 |
|
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset
|
1279 |
lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n" |
63588 | 1280 |
by (cases m) simp_all |
1281 |
||
1282 |
lemma Let_Suc [simp]: "Let (Suc n) f \<equiv> f (Suc n)" |
|
47255
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset
|
1283 |
by (fact Let_def) |
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset
|
1284 |
|
13449 | 1285 |
|
60758 | 1286 |
subsubsection \<open>Monotonicity of multiplication\<close> |
13449 | 1287 |
|
63588 | 1288 |
lemma mult_le_mono1: "i \<le> j \<Longrightarrow> i * k \<le> j * k" |
1289 |
for i j k :: nat |
|
63110 | 1290 |
by (simp add: mult_right_mono) |
1291 |
||
63588 | 1292 |
lemma mult_le_mono2: "i \<le> j \<Longrightarrow> k * i \<le> k * j" |
1293 |
for i j k :: nat |
|
63110 | 1294 |
by (simp add: mult_left_mono) |
13449 | 1295 |
|
61799 | 1296 |
text \<open>\<open>\<le>\<close> monotonicity, BOTH arguments\<close> |
63588 | 1297 |
lemma mult_le_mono: "i \<le> j \<Longrightarrow> k \<le> l \<Longrightarrow> i * k \<le> j * l" |
1298 |
for i j k l :: nat |
|
63110 | 1299 |
by (simp add: mult_mono) |
1300 |
||
63588 | 1301 |
lemma mult_less_mono1: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> i * k < j * k" |
1302 |
for i j k :: nat |
|
63110 | 1303 |
by (simp add: mult_strict_right_mono) |
13449 | 1304 |
|
63588 | 1305 |
text \<open>Differs from the standard \<open>zero_less_mult_iff\<close> in that there are no negative numbers.\<close> |
1306 |
lemma nat_0_less_mult_iff [simp]: "0 < m * n \<longleftrightarrow> 0 < m \<and> 0 < n" |
|
1307 |
for m n :: nat |
|
1308 |
proof (induct m) |
|
1309 |
case 0 |
|
1310 |
then show ?case by simp |
|
1311 |
next |
|
1312 |
case (Suc m) |
|
1313 |
then show ?case by (cases n) simp_all |
|
1314 |
qed |
|
13449 | 1315 |
|
63110 | 1316 |
lemma one_le_mult_iff [simp]: "Suc 0 \<le> m * n \<longleftrightarrow> Suc 0 \<le> m \<and> Suc 0 \<le> n" |
63588 | 1317 |
proof (induct m) |
1318 |
case 0 |
|
1319 |
then show ?case by simp |
|
1320 |
next |
|
1321 |
case (Suc m) |
|
1322 |
then show ?case by (cases n) simp_all |
|
1323 |
qed |
|
1324 |
||
1325 |
lemma mult_less_cancel2 [simp]: "m * k < n * k \<longleftrightarrow> 0 < k \<and> m < n" |
|
1326 |
for k m n :: nat |
|
13449 | 1327 |
apply (safe intro!: mult_less_mono1) |
63588 | 1328 |
apply (cases k) |
1329 |
apply auto |
|
63110 | 1330 |
apply (simp add: linorder_not_le [symmetric]) |
13449 | 1331 |
apply (blast intro: mult_le_mono1) |
1332 |
done |
|
1333 |
||
63588 | 1334 |
lemma mult_less_cancel1 [simp]: "k * m < k * n \<longleftrightarrow> 0 < k \<and> m < n" |
1335 |
for k m n :: nat |
|
63110 | 1336 |
by (simp add: mult.commute [of k]) |
1337 |
||
63588 | 1338 |
lemma mult_le_cancel1 [simp]: "k * m \<le> k * n \<longleftrightarrow> (0 < k \<longrightarrow> m \<le> n)" |
1339 |
for k m n :: nat |
|
63110 | 1340 |
by (simp add: linorder_not_less [symmetric], auto) |
1341 |
||
63588 | 1342 |
lemma mult_le_cancel2 [simp]: "m * k \<le> n * k \<longleftrightarrow> (0 < k \<longrightarrow> m \<le> n)" |
1343 |
for k m n :: nat |
|
63110 | 1344 |
by (simp add: linorder_not_less [symmetric], auto) |
1345 |
||
1346 |
lemma Suc_mult_less_cancel1: "Suc k * m < Suc k * n \<longleftrightarrow> m < n" |
|
1347 |
by (subst mult_less_cancel1) simp |
|
1348 |
||
1349 |
lemma Suc_mult_le_cancel1: "Suc k * m \<le> Suc k * n \<longleftrightarrow> m \<le> n" |
|
1350 |
by (subst mult_le_cancel1) simp |
|
1351 |
||
63588 | 1352 |
lemma le_square: "m \<le> m * m" |
1353 |
for m :: nat |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1354 |
by (cases m) (auto intro: le_add1) |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1355 |
|
63588 | 1356 |
lemma le_cube: "m \<le> m * (m * m)" |
1357 |
for m :: nat |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1358 |
by (cases m) (auto intro: le_add1) |
13449 | 1359 |
|
61799 | 1360 |
text \<open>Lemma for \<open>gcd\<close>\<close> |
63588 | 1361 |
lemma mult_eq_self_implies_10: "m = m * n \<Longrightarrow> n = 1 \<or> m = 0" |
1362 |
for m n :: nat |
|
13449 | 1363 |
apply (drule sym) |
1364 |
apply (rule disjCI) |
|
63588 | 1365 |
apply (rule linorder_cases) |
1366 |
defer |
|
1367 |
apply assumption |
|
1368 |
apply (drule mult_less_mono2) |
|
1369 |
apply auto |
|
13449 | 1370 |
done |
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
7702
diff
changeset
|
1371 |
|
51263
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
51173
diff
changeset
|
1372 |
lemma mono_times_nat: |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
51173
diff
changeset
|
1373 |
fixes n :: nat |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
51173
diff
changeset
|
1374 |
assumes "n > 0" |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
51173
diff
changeset
|
1375 |
shows "mono (times n)" |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
51173
diff
changeset
|
1376 |
proof |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
51173
diff
changeset
|
1377 |
fix m q :: nat |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
51173
diff
changeset
|
1378 |
assume "m \<le> q" |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
51173
diff
changeset
|
1379 |
with assms show "n * m \<le> n * q" by simp |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
51173
diff
changeset
|
1380 |
qed |
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
51173
diff
changeset
|
1381 |
|
63588 | 1382 |
text \<open>The lattice order on @{typ nat}.\<close> |
24995 | 1383 |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1384 |
instantiation nat :: distrib_lattice |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1385 |
begin |
24995 | 1386 |
|
63110 | 1387 |
definition "(inf :: nat \<Rightarrow> nat \<Rightarrow> nat) = min" |
1388 |
||
1389 |
definition "(sup :: nat \<Rightarrow> nat \<Rightarrow> nat) = max" |
|
1390 |
||
1391 |
instance |
|
1392 |
by intro_classes |
|
1393 |
(auto simp add: inf_nat_def sup_nat_def max_def not_le min_def |
|
1394 |
intro: order_less_imp_le antisym elim!: order_trans order_less_trans) |
|
24995 | 1395 |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1396 |
end |
24995 | 1397 |
|
1398 |
||
60758 | 1399 |
subsection \<open>Natural operation of natural numbers on functions\<close> |
1400 |
||
1401 |
text \<open> |
|
30971 | 1402 |
We use the same logical constant for the power operations on |
1403 |
functions and relations, in order to share the same syntax. |
|
60758 | 1404 |
\<close> |
30971 | 1405 |
|
45965
2af982715e5c
generalized type signature to permit overloading on `set`
haftmann
parents:
45933
diff
changeset
|
1406 |
consts compow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" |
30971 | 1407 |
|
63110 | 1408 |
abbreviation compower :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^^" 80) |
1409 |
where "f ^^ n \<equiv> compow n f" |
|
30971 | 1410 |
|
1411 |
notation (latex output) |
|
1412 |
compower ("(_\<^bsup>_\<^esup>)" [1000] 1000) |
|
1413 |
||
63588 | 1414 |
text \<open>\<open>f ^^ n = f \<circ> \<dots> \<circ> f\<close>, the \<open>n\<close>-fold composition of \<open>f\<close>\<close> |
30971 | 1415 |
|
1416 |
overloading |
|
63110 | 1417 |
funpow \<equiv> "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)" |
30971 | 1418 |
begin |
30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset
|
1419 |
|
63588 | 1420 |
primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" |
1421 |
where |
|
1422 |
"funpow 0 f = id" |
|
1423 |
| "funpow (Suc n) f = f \<circ> funpow n f" |
|
30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset
|
1424 |
|
30971 | 1425 |
end |
1426 |
||
62217 | 1427 |
lemma funpow_0 [simp]: "(f ^^ 0) x = x" |
1428 |
by simp |
|
1429 |
||
63110 | 1430 |
lemma funpow_Suc_right: "f ^^ Suc n = f ^^ n \<circ> f" |
49723
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset
|
1431 |
proof (induct n) |
63110 | 1432 |
case 0 |
1433 |
then show ?case by simp |
|
49723
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset
|
1434 |
next |
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset
|
1435 |
fix n |
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset
|
1436 |
assume "f ^^ Suc n = f ^^ n \<circ> f" |
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset
|
1437 |
then show "f ^^ Suc (Suc n) = f ^^ Suc n \<circ> f" |
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset
|
1438 |
by (simp add: o_assoc) |
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset
|
1439 |
qed |
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset
|
1440 |
|
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset
|
1441 |
lemmas funpow_simps_right = funpow.simps(1) funpow_Suc_right |
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset
|
1442 |
|
63588 | 1443 |
text \<open>For code generation.\<close> |
30971 | 1444 |
|
63110 | 1445 |
definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" |
1446 |
where funpow_code_def [code_abbrev]: "funpow = compow" |
|
30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset
|
1447 |
|
30971 | 1448 |
lemma [code]: |
63110 | 1449 |
"funpow (Suc n) f = f \<circ> funpow n f" |
30971 | 1450 |
"funpow 0 f = id" |
37430 | 1451 |
by (simp_all add: funpow_code_def) |
30971 | 1452 |
|
36176
3fe7e97ccca8
replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
wenzelm
parents:
35828
diff
changeset
|
1453 |
hide_const (open) funpow |
30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset
|
1454 |
|
63110 | 1455 |
lemma funpow_add: "f ^^ (m + n) = f ^^ m \<circ> f ^^ n" |
30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset
|
1456 |
by (induct m) simp_all |
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset
|
1457 |
|
63588 | 1458 |
lemma funpow_mult: "(f ^^ m) ^^ n = f ^^ (m * n)" |
1459 |
for f :: "'a \<Rightarrow> 'a" |
|
37430 | 1460 |
by (induct n) (simp_all add: funpow_add) |
1461 |
||
63110 | 1462 |
lemma funpow_swap1: "f ((f ^^ n) x) = (f ^^ n) (f x)" |
30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset
|
1463 |
proof - |
30971 | 1464 |
have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp |
63588 | 1465 |
also have "\<dots> = (f ^^ n \<circ> f ^^ 1) x" by (simp only: funpow_add) |
30971 | 1466 |
also have "\<dots> = (f ^^ n) (f x)" by simp |
30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset
|
1467 |
finally show ?thesis . |
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset
|
1468 |
qed |
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset
|
1469 |
|
63588 | 1470 |
lemma comp_funpow: "comp f ^^ n = comp (f ^^ n)" |
1471 |
for f :: "'a \<Rightarrow> 'a" |
|
38621
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset
|
1472 |
by (induct n) simp_all |
30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset
|
1473 |
|
54496
178922b63b58
add lemmas Suc_funpow and id_funpow to simpset; add lemma map_add_upt
hoelzl
parents:
54411
diff
changeset
|
1474 |
lemma Suc_funpow[simp]: "Suc ^^ n = (op + n)" |
178922b63b58
add lemmas Suc_funpow and id_funpow to simpset; add lemma map_add_upt
hoelzl
parents:
54411
diff
changeset
|
1475 |
by (induct n) simp_all |
178922b63b58
add lemmas Suc_funpow and id_funpow to simpset; add lemma map_add_upt
hoelzl
parents:
54411
diff
changeset
|
1476 |
|
178922b63b58
add lemmas Suc_funpow and id_funpow to simpset; add lemma map_add_upt
hoelzl
parents:
54411
diff
changeset
|
1477 |
lemma id_funpow[simp]: "id ^^ n = id" |
178922b63b58
add lemmas Suc_funpow and id_funpow to simpset; add lemma map_add_upt
hoelzl
parents:
54411
diff
changeset
|
1478 |
by (induct n) simp_all |
38621
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset
|
1479 |
|
63110 | 1480 |
lemma funpow_mono: "mono f \<Longrightarrow> A \<le> B \<Longrightarrow> (f ^^ n) A \<le> (f ^^ n) B" |
63561
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1481 |
for f :: "'a \<Rightarrow> ('a::order)" |
59000 | 1482 |
by (induct n arbitrary: A B) |
1483 |
(auto simp del: funpow.simps(2) simp add: funpow_Suc_right mono_def) |
|
1484 |
||
63561
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1485 |
lemma funpow_mono2: |
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1486 |
assumes "mono f" |
63588 | 1487 |
and "i \<le> j" |
1488 |
and "x \<le> y" |
|
1489 |
and "x \<le> f x" |
|
63561
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1490 |
shows "(f ^^ i) x \<le> (f ^^ j) y" |
63588 | 1491 |
using assms(2,3) |
1492 |
proof (induct j arbitrary: y) |
|
1493 |
case 0 |
|
1494 |
then show ?case by simp |
|
1495 |
next |
|
63561
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1496 |
case (Suc j) |
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1497 |
show ?case |
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1498 |
proof(cases "i = Suc j") |
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1499 |
case True |
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1500 |
with assms(1) Suc show ?thesis |
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1501 |
by (simp del: funpow.simps add: funpow_simps_right monoD funpow_mono) |
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1502 |
next |
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1503 |
case False |
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1504 |
with assms(1,4) Suc show ?thesis |
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1505 |
by (simp del: funpow.simps add: funpow_simps_right le_eq_less_or_eq less_Suc_eq_le) |
63588 | 1506 |
(simp add: Suc.hyps monoD order_subst1) |
63561
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1507 |
qed |
63588 | 1508 |
qed |
63561
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1509 |
|
63110 | 1510 |
|
60758 | 1511 |
subsection \<open>Kleene iteration\<close> |
45833 | 1512 |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52435
diff
changeset
|
1513 |
lemma Kleene_iter_lpfp: |
63588 | 1514 |
fixes f :: "'a::order_bot \<Rightarrow> 'a" |
63110 | 1515 |
assumes "mono f" |
1516 |
and "f p \<le> p" |
|
63588 | 1517 |
shows "(f ^^ k) bot \<le> p" |
1518 |
proof (induct k) |
|
63110 | 1519 |
case 0 |
1520 |
show ?case by simp |
|
45833 | 1521 |
next |
1522 |
case Suc |
|
63588 | 1523 |
show ?case |
1524 |
using monoD[OF assms(1) Suc] assms(2) by simp |
|
45833 | 1525 |
qed |
1526 |
||
63110 | 1527 |
lemma lfp_Kleene_iter: |
1528 |
assumes "mono f" |
|
63588 | 1529 |
and "(f ^^ Suc k) bot = (f ^^ k) bot" |
1530 |
shows "lfp f = (f ^^ k) bot" |
|
63110 | 1531 |
proof (rule antisym) |
63588 | 1532 |
show "lfp f \<le> (f ^^ k) bot" |
63110 | 1533 |
proof (rule lfp_lowerbound) |
63588 | 1534 |
show "f ((f ^^ k) bot) \<le> (f ^^ k) bot" |
63110 | 1535 |
using assms(2) by simp |
45833 | 1536 |
qed |
63588 | 1537 |
show "(f ^^ k) bot \<le> lfp f" |
45833 | 1538 |
using Kleene_iter_lpfp[OF assms(1)] lfp_unfold[OF assms(1)] by simp |
1539 |
qed |
|
1540 |
||
63588 | 1541 |
lemma mono_pow: "mono f \<Longrightarrow> mono (f ^^ n)" |
1542 |
for f :: "'a \<Rightarrow> 'a::complete_lattice" |
|
63110 | 1543 |
by (induct n) (auto simp: mono_def) |
60636
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1544 |
|
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1545 |
lemma lfp_funpow: |
63110 | 1546 |
assumes f: "mono f" |
1547 |
shows "lfp (f ^^ Suc n) = lfp f" |
|
60636
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1548 |
proof (rule antisym) |
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1549 |
show "lfp f \<le> lfp (f ^^ Suc n)" |
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1550 |
proof (rule lfp_lowerbound) |
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1551 |
have "f (lfp (f ^^ Suc n)) = lfp (\<lambda>x. f ((f ^^ n) x))" |
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1552 |
unfolding funpow_Suc_right by (simp add: lfp_rolling f mono_pow comp_def) |
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1553 |
then show "f (lfp (f ^^ Suc n)) \<le> lfp (f ^^ Suc n)" |
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1554 |
by (simp add: comp_def) |
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1555 |
qed |
63588 | 1556 |
have "(f ^^ n) (lfp f) = lfp f" for n |
63979 | 1557 |
by (induct n) (auto intro: f lfp_fixpoint) |
63588 | 1558 |
then show "lfp (f ^^ Suc n) \<le> lfp f" |
60636
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1559 |
by (intro lfp_lowerbound) (simp del: funpow.simps) |
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1560 |
qed |
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1561 |
|
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1562 |
lemma gfp_funpow: |
63110 | 1563 |
assumes f: "mono f" |
1564 |
shows "gfp (f ^^ Suc n) = gfp f" |
|
60636
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1565 |
proof (rule antisym) |
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1566 |
show "gfp f \<ge> gfp (f ^^ Suc n)" |
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1567 |
proof (rule gfp_upperbound) |
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1568 |
have "f (gfp (f ^^ Suc n)) = gfp (\<lambda>x. f ((f ^^ n) x))" |
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1569 |
unfolding funpow_Suc_right by (simp add: gfp_rolling f mono_pow comp_def) |
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1570 |
then show "f (gfp (f ^^ Suc n)) \<ge> gfp (f ^^ Suc n)" |
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1571 |
by (simp add: comp_def) |
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1572 |
qed |
63588 | 1573 |
have "(f ^^ n) (gfp f) = gfp f" for n |
63979 | 1574 |
by (induct n) (auto intro: f gfp_fixpoint) |
63588 | 1575 |
then show "gfp (f ^^ Suc n) \<ge> gfp f" |
60636
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1576 |
by (intro gfp_upperbound) (simp del: funpow.simps) |
ee18efe9b246
add named theorems order_continuous_intros; lfp/gfp_funpow; bounded variant for lfp/gfp transfer
hoelzl
parents:
60562
diff
changeset
|
1577 |
qed |
45833 | 1578 |
|
63561
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1579 |
lemma Kleene_iter_gpfp: |
63588 | 1580 |
fixes f :: "'a::order_top \<Rightarrow> 'a" |
63561
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1581 |
assumes "mono f" |
63588 | 1582 |
and "p \<le> f p" |
1583 |
shows "p \<le> (f ^^ k) top" |
|
1584 |
proof (induct k) |
|
1585 |
case 0 |
|
1586 |
show ?case by simp |
|
63561
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1587 |
next |
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1588 |
case Suc |
63588 | 1589 |
show ?case |
1590 |
using monoD[OF assms(1) Suc] assms(2) by simp |
|
63561
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1591 |
qed |
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1592 |
|
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1593 |
lemma gfp_Kleene_iter: |
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1594 |
assumes "mono f" |
63588 | 1595 |
and "(f ^^ Suc k) top = (f ^^ k) top" |
1596 |
shows "gfp f = (f ^^ k) top" |
|
1597 |
(is "?lhs = ?rhs") |
|
1598 |
proof (rule antisym) |
|
1599 |
have "?rhs \<le> f ?rhs" |
|
1600 |
using assms(2) by simp |
|
1601 |
then show "?rhs \<le> ?lhs" |
|
1602 |
by (rule gfp_upperbound) |
|
63561
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1603 |
show "?lhs \<le> ?rhs" |
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1604 |
using Kleene_iter_gpfp[OF assms(1)] gfp_unfold[OF assms(1)] by simp |
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1605 |
qed |
fba08009ff3e
add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents:
63197
diff
changeset
|
1606 |
|
63110 | 1607 |
|
61799 | 1608 |
subsection \<open>Embedding of the naturals into any \<open>semiring_1\<close>: @{term of_nat}\<close> |
24196 | 1609 |
|
1610 |
context semiring_1 |
|
1611 |
begin |
|
1612 |
||
63110 | 1613 |
definition of_nat :: "nat \<Rightarrow> 'a" |
1614 |
where "of_nat n = (plus 1 ^^ n) 0" |
|
38621
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset
|
1615 |
|
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset
|
1616 |
lemma of_nat_simps [simp]: |
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset
|
1617 |
shows of_nat_0: "of_nat 0 = 0" |
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset
|
1618 |
and of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m" |
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset
|
1619 |
by (simp_all add: of_nat_def) |
25193 | 1620 |
|
1621 |
lemma of_nat_1 [simp]: "of_nat 1 = 1" |
|
38621
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset
|
1622 |
by (simp add: of_nat_def) |
25193 | 1623 |
|
1624 |
lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n" |
|
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
1625 |
by (induct m) (simp_all add: ac_simps) |
25193 | 1626 |
|
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
1627 |
lemma of_nat_mult [simp]: "of_nat (m * n) = of_nat m * of_nat n" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
1628 |
by (induct m) (simp_all add: ac_simps distrib_right) |
25193 | 1629 |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
1630 |
lemma mult_of_nat_commute: "of_nat x * y = y * of_nat x" |
63110 | 1631 |
by (induct x) (simp_all add: algebra_simps) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
1632 |
|
63588 | 1633 |
primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" |
1634 |
where |
|
1635 |
"of_nat_aux inc 0 i = i" |
|
1636 |
| "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" \<comment> \<open>tail recursive\<close> |
|
25928 | 1637 |
|
63110 | 1638 |
lemma of_nat_code: "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0" |
28514 | 1639 |
proof (induct n) |
63110 | 1640 |
case 0 |
1641 |
then show ?case by simp |
|
28514 | 1642 |
next |
1643 |
case (Suc n) |
|
1644 |
have "\<And>i. of_nat_aux (\<lambda>i. i + 1) n (i + 1) = of_nat_aux (\<lambda>i. i + 1) n i + 1" |
|
1645 |
by (induct n) simp_all |
|
1646 |
from this [of 0] have "of_nat_aux (\<lambda>i. i + 1) n 1 = of_nat_aux (\<lambda>i. i + 1) n 0 + 1" |
|
1647 |
by simp |
|
63588 | 1648 |
with Suc show ?case |
1649 |
by (simp add: add.commute) |
|
28514 | 1650 |
qed |
30966 | 1651 |
|
24196 | 1652 |
end |
1653 |
||
45231
d85a2fdc586c
replacing code_inline by code_unfold, removing obsolete code_unfold, code_inline del now that the ancient code generator is removed
bulwahn
parents:
44890
diff
changeset
|
1654 |
declare of_nat_code [code] |
30966 | 1655 |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1656 |
context ring_1 |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1657 |
begin |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1658 |
|
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1659 |
lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n" |
63110 | 1660 |
by (simp add: algebra_simps of_nat_add [symmetric]) |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1661 |
|
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1662 |
end |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1663 |
|
63110 | 1664 |
text \<open>Class for unital semirings with characteristic zero. |
60758 | 1665 |
Includes non-ordered rings like the complex numbers.\<close> |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1666 |
|
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1667 |
class semiring_char_0 = semiring_1 + |
38621
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset
|
1668 |
assumes inj_of_nat: "inj of_nat" |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1669 |
begin |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1670 |
|
38621
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset
|
1671 |
lemma of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n" |
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset
|
1672 |
by (auto intro: inj_of_nat injD) |
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset
|
1673 |
|
63110 | 1674 |
text \<open>Special cases where either operand is zero\<close> |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1675 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53986
diff
changeset
|
1676 |
lemma of_nat_0_eq_iff [simp]: "0 = of_nat n \<longleftrightarrow> 0 = n" |
38621
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset
|
1677 |
by (fact of_nat_eq_iff [of 0 n, unfolded of_nat_0]) |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1678 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53986
diff
changeset
|
1679 |
lemma of_nat_eq_0_iff [simp]: "of_nat m = 0 \<longleftrightarrow> m = 0" |
38621
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset
|
1680 |
by (fact of_nat_eq_iff [of m 0, unfolded of_nat_0]) |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1681 |
|
63588 | 1682 |
lemma of_nat_neq_0 [simp]: "of_nat (Suc n) \<noteq> 0" |
60353
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60175
diff
changeset
|
1683 |
unfolding of_nat_eq_0_iff by simp |
838025c6e278
implicit partial divison operation in integral domains
haftmann
parents:
60175
diff
changeset
|
1684 |
|
63588 | 1685 |
lemma of_nat_0_neq [simp]: "0 \<noteq> of_nat (Suc n)" |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60427
diff
changeset
|
1686 |
unfolding of_nat_0_eq_iff by simp |
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60427
diff
changeset
|
1687 |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1688 |
end |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1689 |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1690 |
class ring_char_0 = ring_1 + semiring_char_0 |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1691 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34208
diff
changeset
|
1692 |
context linordered_semidom |
25193 | 1693 |
begin |
1694 |
||
47489 | 1695 |
lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n" |
1696 |
by (induct n) simp_all |
|
25193 | 1697 |
|
47489 | 1698 |
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0" |
1699 |
by (simp add: not_less) |
|
25193 | 1700 |
|
1701 |
lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n" |
|
62376
85f38d5f8807
Rename ordered_comm_monoid_add to ordered_cancel_comm_monoid_add. Introduce ordreed_comm_monoid_add, canonically_ordered_comm_monoid and dioid. Setup nat, entat and ennreal as dioids.
hoelzl
parents:
62365
diff
changeset
|
1702 |
by (induct m n rule: diff_induct) (simp_all add: add_pos_nonneg) |
25193 | 1703 |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1704 |
lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n" |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1705 |
by (simp add: not_less [symmetric] linorder_not_less [symmetric]) |
25193 | 1706 |
|
47489 | 1707 |
lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n" |
1708 |
by simp |
|
1709 |
||
1710 |
lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n" |
|
1711 |
by simp |
|
1712 |
||
63110 | 1713 |
text \<open>Every \<open>linordered_semidom\<close> has characteristic zero.\<close> |
1714 |
||
1715 |
subclass semiring_char_0 |
|
1716 |
by standard (auto intro!: injI simp add: eq_iff) |
|
1717 |
||
1718 |
text \<open>Special cases where either operand is zero\<close> |
|
25193 | 1719 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53986
diff
changeset
|
1720 |
lemma of_nat_le_0_iff [simp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0" |
25193 | 1721 |
by (rule of_nat_le_iff [of _ 0, simplified]) |
1722 |
||
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1723 |
lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n" |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1724 |
by (rule of_nat_less_iff [of 0, simplified]) |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1725 |
|
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1726 |
end |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1727 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34208
diff
changeset
|
1728 |
context linordered_idom |
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1729 |
begin |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1730 |
|
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1731 |
lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n" |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1732 |
unfolding abs_if by auto |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1733 |
|
25193 | 1734 |
end |
1735 |
||
1736 |
lemma of_nat_id [simp]: "of_nat n = n" |
|
35216 | 1737 |
by (induct n) simp_all |
25193 | 1738 |
|
1739 |
lemma of_nat_eq_id [simp]: "of_nat = id" |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
1740 |
by (auto simp add: fun_eq_iff) |
25193 | 1741 |
|
1742 |
||
60758 | 1743 |
subsection \<open>The set of natural numbers\<close> |
25193 | 1744 |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1745 |
context semiring_1 |
25193 | 1746 |
begin |
1747 |
||
61070 | 1748 |
definition Nats :: "'a set" ("\<nat>") |
1749 |
where "\<nat> = range of_nat" |
|
25193 | 1750 |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1751 |
lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>" |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1752 |
by (simp add: Nats_def) |
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1753 |
|
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1754 |
lemma Nats_0 [simp]: "0 \<in> \<nat>" |
63588 | 1755 |
apply (simp add: Nats_def) |
1756 |
apply (rule range_eqI) |
|
1757 |
apply (rule of_nat_0 [symmetric]) |
|
1758 |
done |
|
25193 | 1759 |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1760 |
lemma Nats_1 [simp]: "1 \<in> \<nat>" |
63588 | 1761 |
apply (simp add: Nats_def) |
1762 |
apply (rule range_eqI) |
|
1763 |
apply (rule of_nat_1 [symmetric]) |
|
1764 |
done |
|
25193 | 1765 |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1766 |
lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>" |
63588 | 1767 |
apply (auto simp add: Nats_def) |
1768 |
apply (rule range_eqI) |
|
1769 |
apply (rule of_nat_add [symmetric]) |
|
1770 |
done |
|
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1771 |
|
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset
|
1772 |
lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>" |
63588 | 1773 |
apply (auto simp add: Nats_def) |
1774 |
apply (rule range_eqI) |
|
1775 |
apply (rule of_nat_mult [symmetric]) |
|
1776 |
done |
|
25193 | 1777 |
|
35633 | 1778 |
lemma Nats_cases [cases set: Nats]: |
1779 |
assumes "x \<in> \<nat>" |
|
1780 |
obtains (of_nat) n where "x = of_nat n" |
|
1781 |
unfolding Nats_def |
|
1782 |
proof - |
|
60758 | 1783 |
from \<open>x \<in> \<nat>\<close> have "x \<in> range of_nat" unfolding Nats_def . |
35633 | 1784 |
then obtain n where "x = of_nat n" .. |
1785 |
then show thesis .. |
|
1786 |
qed |
|
1787 |
||
63588 | 1788 |
lemma Nats_induct [case_names of_nat, induct set: Nats]: "x \<in> \<nat> \<Longrightarrow> (\<And>n. P (of_nat n)) \<Longrightarrow> P x" |
35633 | 1789 |
by (rule Nats_cases) auto |
1790 |
||
25193 | 1791 |
end |
1792 |
||
1793 |
||
60758 | 1794 |
subsection \<open>Further arithmetic facts concerning the natural numbers\<close> |
21243 | 1795 |
|
22845 | 1796 |
lemma subst_equals: |
63110 | 1797 |
assumes "t = s" and "u = t" |
22845 | 1798 |
shows "u = s" |
63110 | 1799 |
using assms(2,1) by (rule trans) |
22845 | 1800 |
|
48891 | 1801 |
ML_file "Tools/nat_arith.ML" |
48559
686cc7c47589
give Nat_Arith simprocs proper name bindings by using simproc_setup
huffman
parents:
47988
diff
changeset
|
1802 |
|
686cc7c47589
give Nat_Arith simprocs proper name bindings by using simproc_setup
huffman
parents:
47988
diff
changeset
|
1803 |
simproc_setup nateq_cancel_sums |
686cc7c47589
give Nat_Arith simprocs proper name bindings by using simproc_setup
huffman
parents:
47988
diff
changeset
|
1804 |
("(l::nat) + m = n" | "(l::nat) = m + n" | "Suc m = n" | "m = Suc n") = |
60758 | 1805 |
\<open>fn phi => try o Nat_Arith.cancel_eq_conv\<close> |
48559
686cc7c47589
give Nat_Arith simprocs proper name bindings by using simproc_setup
huffman
parents:
47988
diff
changeset
|
1806 |
|
686cc7c47589
give Nat_Arith simprocs proper name bindings by using simproc_setup
huffman
parents:
47988
diff
changeset
|
1807 |
simproc_setup natless_cancel_sums |
686cc7c47589
give Nat_Arith simprocs proper name bindings by using simproc_setup
huffman
parents:
47988
diff
changeset
|
1808 |
("(l::nat) + m < n" | "(l::nat) < m + n" | "Suc m < n" | "m < Suc n") = |
60758 | 1809 |
\<open>fn phi => try o Nat_Arith.cancel_less_conv\<close> |
48559
686cc7c47589
give Nat_Arith simprocs proper name bindings by using simproc_setup
huffman
parents:
47988
diff
changeset
|
1810 |
|
686cc7c47589
give Nat_Arith simprocs proper name bindings by using simproc_setup
huffman
parents:
47988
diff
changeset
|
1811 |
simproc_setup natle_cancel_sums |
686cc7c47589
give Nat_Arith simprocs proper name bindings by using simproc_setup
huffman
parents:
47988
diff
changeset
|
1812 |
("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" | "Suc m \<le> n" | "m \<le> Suc n") = |
60758 | 1813 |
\<open>fn phi => try o Nat_Arith.cancel_le_conv\<close> |
48559
686cc7c47589
give Nat_Arith simprocs proper name bindings by using simproc_setup
huffman
parents:
47988
diff
changeset
|
1814 |
|
686cc7c47589
give Nat_Arith simprocs proper name bindings by using simproc_setup
huffman
parents:
47988
diff
changeset
|
1815 |
simproc_setup natdiff_cancel_sums |
686cc7c47589
give Nat_Arith simprocs proper name bindings by using simproc_setup
huffman
parents:
47988
diff
changeset
|
1816 |
("(l::nat) + m - n" | "(l::nat) - (m + n)" | "Suc m - n" | "m - Suc n") = |
60758 | 1817 |
\<open>fn phi => try o Nat_Arith.cancel_diff_conv\<close> |
24091 | 1818 |
|
27625 | 1819 |
context order |
1820 |
begin |
|
1821 |
||
1822 |
lemma lift_Suc_mono_le: |
|
63588 | 1823 |
assumes mono: "\<And>n. f n \<le> f (Suc n)" |
1824 |
and "n \<le> n'" |
|
27627 | 1825 |
shows "f n \<le> f n'" |
1826 |
proof (cases "n < n'") |
|
1827 |
case True |
|
53986 | 1828 |
then show ?thesis |
62683 | 1829 |
by (induct n n' rule: less_Suc_induct) (auto intro: mono) |
63110 | 1830 |
next |
1831 |
case False |
|
1832 |
with \<open>n \<le> n'\<close> show ?thesis by auto |
|
1833 |
qed |
|
27625 | 1834 |
|
56020
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55642
diff
changeset
|
1835 |
lemma lift_Suc_antimono_le: |
63588 | 1836 |
assumes mono: "\<And>n. f n \<ge> f (Suc n)" |
1837 |
and "n \<le> n'" |
|
56020
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55642
diff
changeset
|
1838 |
shows "f n \<ge> f n'" |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55642
diff
changeset
|
1839 |
proof (cases "n < n'") |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55642
diff
changeset
|
1840 |
case True |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55642
diff
changeset
|
1841 |
then show ?thesis |
62683 | 1842 |
by (induct n n' rule: less_Suc_induct) (auto intro: mono) |
63110 | 1843 |
next |
1844 |
case False |
|
1845 |
with \<open>n \<le> n'\<close> show ?thesis by auto |
|
1846 |
qed |
|
56020
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55642
diff
changeset
|
1847 |
|
27625 | 1848 |
lemma lift_Suc_mono_less: |
63588 | 1849 |
assumes mono: "\<And>n. f n < f (Suc n)" |
1850 |
and "n < n'" |
|
27627 | 1851 |
shows "f n < f n'" |
63110 | 1852 |
using \<open>n < n'\<close> by (induct n n' rule: less_Suc_induct) (auto intro: mono) |
1853 |
||
1854 |
lemma lift_Suc_mono_less_iff: "(\<And>n. f n < f (Suc n)) \<Longrightarrow> f n < f m \<longleftrightarrow> n < m" |
|
53986 | 1855 |
by (blast intro: less_asym' lift_Suc_mono_less [of f] |
1856 |
dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq [THEN iffD1]) |
|
27789 | 1857 |
|
27625 | 1858 |
end |
1859 |
||
63110 | 1860 |
lemma mono_iff_le_Suc: "mono f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))" |
37387
3581483cca6c
qualified types "+" and nat; qualified constants Ball, Bex, Suc, curry; modernized some specifications
haftmann
parents:
36977
diff
changeset
|
1861 |
unfolding mono_def by (auto intro: lift_Suc_mono_le [of f]) |
27625 | 1862 |
|
63110 | 1863 |
lemma antimono_iff_le_Suc: "antimono f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)" |
56020
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55642
diff
changeset
|
1864 |
unfolding antimono_def by (auto intro: lift_Suc_antimono_le [of f]) |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55642
diff
changeset
|
1865 |
|
27789 | 1866 |
lemma mono_nat_linear_lb: |
53986 | 1867 |
fixes f :: "nat \<Rightarrow> nat" |
1868 |
assumes "\<And>m n. m < n \<Longrightarrow> f m < f n" |
|
1869 |
shows "f m + k \<le> f (m + k)" |
|
1870 |
proof (induct k) |
|
63110 | 1871 |
case 0 |
1872 |
then show ?case by simp |
|
53986 | 1873 |
next |
1874 |
case (Suc k) |
|
1875 |
then have "Suc (f m + k) \<le> Suc (f (m + k))" by simp |
|
1876 |
also from assms [of "m + k" "Suc (m + k)"] have "Suc (f (m + k)) \<le> f (Suc (m + k))" |
|
1877 |
by (simp add: Suc_le_eq) |
|
1878 |
finally show ?case by simp |
|
1879 |
qed |
|
27789 | 1880 |
|
1881 |
||
63110 | 1882 |
text \<open>Subtraction laws, mostly by Clemens Ballarin\<close> |
21243 | 1883 |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1884 |
lemma diff_less_mono: |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1885 |
fixes a b c :: nat |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1886 |
assumes "a < b" and "c \<le> a" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1887 |
shows "a - c < b - c" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1888 |
proof - |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1889 |
from assms obtain d e where "b = c + (d + e)" and "a = c + e" and "d > 0" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1890 |
by (auto dest!: le_Suc_ex less_imp_Suc_add simp add: ac_simps) |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1891 |
then show ?thesis by simp |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1892 |
qed |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1893 |
|
63588 | 1894 |
lemma less_diff_conv: "i < j - k \<longleftrightarrow> i + k < j" |
1895 |
for i j k :: nat |
|
63110 | 1896 |
by (cases "k \<le> j") (auto simp add: not_le dest: less_imp_Suc_add le_Suc_ex) |
1897 |
||
63588 | 1898 |
lemma less_diff_conv2: "k \<le> j \<Longrightarrow> j - k < i \<longleftrightarrow> j < i + k" |
1899 |
for j k i :: nat |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1900 |
by (auto dest: le_Suc_ex) |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1901 |
|
63588 | 1902 |
lemma le_diff_conv: "j - k \<le> i \<longleftrightarrow> j \<le> i + k" |
1903 |
for j k i :: nat |
|
63110 | 1904 |
by (cases "k \<le> j") (auto simp add: not_le dest!: less_imp_Suc_add le_Suc_ex) |
1905 |
||
63588 | 1906 |
lemma diff_diff_cancel [simp]: "i \<le> n \<Longrightarrow> n - (n - i) = i" |
1907 |
for i n :: nat |
|
63110 | 1908 |
by (auto dest: le_Suc_ex) |
1909 |
||
63588 | 1910 |
lemma diff_less [simp]: "0 < n \<Longrightarrow> 0 < m \<Longrightarrow> m - n < m" |
1911 |
for i n :: nat |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1912 |
by (auto dest: less_imp_Suc_add) |
21243 | 1913 |
|
60758 | 1914 |
text \<open>Simplification of relational expressions involving subtraction\<close> |
21243 | 1915 |
|
63588 | 1916 |
lemma diff_diff_eq: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k - (n - k) = m - n" |
1917 |
for m n k :: nat |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1918 |
by (auto dest!: le_Suc_ex) |
21243 | 1919 |
|
36176
3fe7e97ccca8
replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
wenzelm
parents:
35828
diff
changeset
|
1920 |
hide_fact (open) diff_diff_eq |
35064
1bdef0c013d3
hide fact names clashing with fact names from Group.thy
haftmann
parents:
35047
diff
changeset
|
1921 |
|
63588 | 1922 |
lemma eq_diff_iff: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k = n - k \<longleftrightarrow> m = n" |
1923 |
for m n k :: nat |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1924 |
by (auto dest: le_Suc_ex) |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1925 |
|
63588 | 1926 |
lemma less_diff_iff: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k < n - k \<longleftrightarrow> m < n" |
1927 |
for m n k :: nat |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1928 |
by (auto dest!: le_Suc_ex) |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1929 |
|
63588 | 1930 |
lemma le_diff_iff: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k \<le> n - k \<longleftrightarrow> m \<le> n" |
1931 |
for m n k :: nat |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1932 |
by (auto dest!: le_Suc_ex) |
21243 | 1933 |
|
63588 | 1934 |
lemma le_diff_iff': "a \<le> c \<Longrightarrow> b \<le> c \<Longrightarrow> c - a \<le> c - b \<longleftrightarrow> b \<le> a" |
1935 |
for a b c :: nat |
|
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
63040
diff
changeset
|
1936 |
by (force dest: le_Suc_ex) |
63110 | 1937 |
|
1938 |
||
1939 |
text \<open>(Anti)Monotonicity of subtraction -- by Stephan Merz\<close> |
|
1940 |
||
63588 | 1941 |
lemma diff_le_mono: "m \<le> n \<Longrightarrow> m - l \<le> n - l" |
1942 |
for m n l :: nat |
|
63648 | 1943 |
by (auto dest: less_imp_le less_imp_Suc_add split: nat_diff_split) |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1944 |
|
63588 | 1945 |
lemma diff_le_mono2: "m \<le> n \<Longrightarrow> l - n \<le> l - m" |
1946 |
for m n l :: nat |
|
63648 | 1947 |
by (auto dest: less_imp_le le_Suc_ex less_imp_Suc_add less_le_trans split: nat_diff_split) |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1948 |
|
63588 | 1949 |
lemma diff_less_mono2: "m < n \<Longrightarrow> m < l \<Longrightarrow> l - n < l - m" |
1950 |
for m n l :: nat |
|
63648 | 1951 |
by (auto dest: less_imp_Suc_add split: nat_diff_split) |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1952 |
|
63588 | 1953 |
lemma diffs0_imp_equal: "m - n = 0 \<Longrightarrow> n - m = 0 \<Longrightarrow> m = n" |
1954 |
for m n :: nat |
|
63648 | 1955 |
by (simp split: nat_diff_split) |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1956 |
|
63588 | 1957 |
lemma min_diff: "min (m - i) (n - i) = min m n - i" |
1958 |
for m n i :: nat |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1959 |
by (cases m n rule: le_cases) |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1960 |
(auto simp add: not_le min.absorb1 min.absorb2 min.absorb_iff1 [symmetric] diff_le_mono) |
26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset
|
1961 |
|
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60427
diff
changeset
|
1962 |
lemma inj_on_diff_nat: |
63110 | 1963 |
fixes k :: nat |
1964 |
assumes "\<forall>n \<in> N. k \<le> n" |
|
26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset
|
1965 |
shows "inj_on (\<lambda>n. n - k) N" |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset
|
1966 |
proof (rule inj_onI) |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset
|
1967 |
fix x y |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset
|
1968 |
assume a: "x \<in> N" "y \<in> N" "x - k = y - k" |
63110 | 1969 |
with assms have "x - k + k = y - k + k" by auto |
1970 |
with a assms show "x = y" by (auto simp add: eq_diff_iff) |
|
26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset
|
1971 |
qed |
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset
|
1972 |
|
63110 | 1973 |
text \<open>Rewriting to pull differences out\<close> |
1974 |
||
63588 | 1975 |
lemma diff_diff_right [simp]: "k \<le> j \<Longrightarrow> i - (j - k) = i + k - j" |
1976 |
for i j k :: nat |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1977 |
by (fact diff_diff_right) |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1978 |
|
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1979 |
lemma diff_Suc_diff_eq1 [simp]: |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1980 |
assumes "k \<le> j" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1981 |
shows "i - Suc (j - k) = i + k - Suc j" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1982 |
proof - |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1983 |
from assms have *: "Suc (j - k) = Suc j - k" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1984 |
by (simp add: Suc_diff_le) |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1985 |
from assms have "k \<le> Suc j" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1986 |
by (rule order_trans) simp |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1987 |
with diff_diff_right [of k "Suc j" i] * show ?thesis |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1988 |
by simp |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1989 |
qed |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1990 |
|
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1991 |
lemma diff_Suc_diff_eq2 [simp]: |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1992 |
assumes "k \<le> j" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1993 |
shows "Suc (j - k) - i = Suc j - (k + i)" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1994 |
proof - |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1995 |
from assms obtain n where "j = k + n" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1996 |
by (auto dest: le_Suc_ex) |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1997 |
moreover have "Suc n - i = (k + Suc n) - (k + i)" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1998 |
using add_diff_cancel_left [of k "Suc n" i] by simp |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
1999 |
ultimately show ?thesis by simp |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2000 |
qed |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2001 |
|
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2002 |
lemma Suc_diff_Suc: |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2003 |
assumes "n < m" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2004 |
shows "Suc (m - Suc n) = m - n" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2005 |
proof - |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2006 |
from assms obtain q where "m = n + Suc q" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2007 |
by (auto dest: less_imp_Suc_add) |
63040 | 2008 |
moreover define r where "r = Suc q" |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2009 |
ultimately have "Suc (m - Suc n) = r" and "m = n + r" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2010 |
by simp_all |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2011 |
then show ?thesis by simp |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2012 |
qed |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2013 |
|
63110 | 2014 |
lemma one_less_mult: "Suc 0 < n \<Longrightarrow> Suc 0 < m \<Longrightarrow> Suc 0 < m * n" |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2015 |
using less_1_mult [of n m] by (simp add: ac_simps) |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2016 |
|
63110 | 2017 |
lemma n_less_m_mult_n: "0 < n \<Longrightarrow> Suc 0 < m \<Longrightarrow> n < m * n" |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2018 |
using mult_strict_right_mono [of 1 m n] by simp |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2019 |
|
63110 | 2020 |
lemma n_less_n_mult_m: "0 < n \<Longrightarrow> Suc 0 < m \<Longrightarrow> n < n * m" |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2021 |
using mult_strict_left_mono [of 1 m n] by simp |
21243 | 2022 |
|
63110 | 2023 |
|
60758 | 2024 |
text \<open>Specialized induction principles that work "backwards":\<close> |
23001
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset
|
2025 |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2026 |
lemma inc_induct [consumes 1, case_names base step]: |
54411 | 2027 |
assumes less: "i \<le> j" |
63110 | 2028 |
and base: "P j" |
2029 |
and step: "\<And>n. i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P (Suc n) \<Longrightarrow> P n" |
|
23001
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset
|
2030 |
shows "P i" |
54411 | 2031 |
using less step |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2032 |
proof (induct "j - i" arbitrary: i) |
23001
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset
|
2033 |
case (0 i) |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2034 |
then have "i = j" by simp |
23001
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset
|
2035 |
with base show ?case by simp |
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset
|
2036 |
next |
54411 | 2037 |
case (Suc d n) |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2038 |
from Suc.hyps have "n \<noteq> j" by auto |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2039 |
with Suc have "n < j" by (simp add: less_le) |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2040 |
from \<open>Suc d = j - n\<close> have "d + 1 = j - n" by simp |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2041 |
then have "d + 1 - 1 = j - n - 1" by simp |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2042 |
then have "d = j - n - 1" by simp |
63588 | 2043 |
then have "d = j - (n + 1)" by (simp add: diff_diff_eq) |
2044 |
then have "d = j - Suc n" by simp |
|
2045 |
moreover from \<open>n < j\<close> have "Suc n \<le> j" by (simp add: Suc_le_eq) |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2046 |
ultimately have "P (Suc n)" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2047 |
proof (rule Suc.hyps) |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2048 |
fix q |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2049 |
assume "Suc n \<le> q" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2050 |
then have "n \<le> q" by (simp add: Suc_le_eq less_imp_le) |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2051 |
moreover assume "q < j" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2052 |
moreover assume "P (Suc q)" |
63588 | 2053 |
ultimately show "P q" by (rule Suc.prems) |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2054 |
qed |
63588 | 2055 |
with order_refl \<open>n < j\<close> show "P n" by (rule Suc.prems) |
23001
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset
|
2056 |
qed |
63110 | 2057 |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2058 |
lemma strict_inc_induct [consumes 1, case_names base step]: |
23001
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset
|
2059 |
assumes less: "i < j" |
63110 | 2060 |
and base: "\<And>i. j = Suc i \<Longrightarrow> P i" |
2061 |
and step: "\<And>i. i < j \<Longrightarrow> P (Suc i) \<Longrightarrow> P i" |
|
23001
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset
|
2062 |
shows "P i" |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2063 |
using less proof (induct "j - i - 1" arbitrary: i) |
23001
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset
|
2064 |
case (0 i) |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2065 |
from \<open>i < j\<close> obtain n where "j = i + n" and "n > 0" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2066 |
by (auto dest!: less_imp_Suc_add) |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2067 |
with 0 have "j = Suc i" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2068 |
by (auto intro: order_antisym simp add: Suc_le_eq) |
23001
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset
|
2069 |
with base show ?case by simp |
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset
|
2070 |
next |
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset
|
2071 |
case (Suc d i) |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2072 |
from \<open>Suc d = j - i - 1\<close> have *: "Suc d = j - Suc i" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2073 |
by (simp add: diff_diff_add) |
63588 | 2074 |
then have "Suc d - 1 = j - Suc i - 1" by simp |
2075 |
then have "d = j - Suc i - 1" by simp |
|
2076 |
moreover from * have "j - Suc i \<noteq> 0" by auto |
|
2077 |
then have "Suc i < j" by (simp add: not_le) |
|
2078 |
ultimately have "P (Suc i)" by (rule Suc.hyps) |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2079 |
with \<open>i < j\<close> show "P i" by (rule step) |
23001
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset
|
2080 |
qed |
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset
|
2081 |
|
63110 | 2082 |
lemma zero_induct_lemma: "P k \<Longrightarrow> (\<And>n. P (Suc n) \<Longrightarrow> P n) \<Longrightarrow> P (k - i)" |
23001
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset
|
2083 |
using inc_induct[of "k - i" k P, simplified] by blast |
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset
|
2084 |
|
63110 | 2085 |
lemma zero_induct: "P k \<Longrightarrow> (\<And>n. P (Suc n) \<Longrightarrow> P n) \<Longrightarrow> P 0" |
23001
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset
|
2086 |
using inc_induct[of 0 k P] by blast |
21243 | 2087 |
|
63588 | 2088 |
text \<open>Further induction rule similar to @{thm inc_induct}.\<close> |
27625 | 2089 |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2090 |
lemma dec_induct [consumes 1, case_names base step]: |
54411 | 2091 |
"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" |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2092 |
proof (induct j arbitrary: i) |
63110 | 2093 |
case 0 |
2094 |
then show ?case by simp |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2095 |
next |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2096 |
case (Suc j) |
63110 | 2097 |
from Suc.prems consider "i \<le> j" | "i = Suc j" |
2098 |
by (auto simp add: le_Suc_eq) |
|
2099 |
then show ?case |
|
2100 |
proof cases |
|
2101 |
case 1 |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2102 |
moreover have "j < Suc j" by simp |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2103 |
moreover have "P j" using \<open>i \<le> j\<close> \<open>P i\<close> |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2104 |
proof (rule Suc.hyps) |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2105 |
fix q |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2106 |
assume "i \<le> q" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2107 |
moreover assume "q < j" then have "q < Suc j" |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2108 |
by (simp add: less_Suc_eq) |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2109 |
moreover assume "P q" |
63588 | 2110 |
ultimately show "P (Suc q)" by (rule Suc.prems) |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2111 |
qed |
63588 | 2112 |
ultimately show "P (Suc j)" by (rule Suc.prems) |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2113 |
next |
63110 | 2114 |
case 2 |
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2115 |
with \<open>P i\<close> show "P (Suc j)" by simp |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2116 |
qed |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2117 |
qed |
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2118 |
|
59000 | 2119 |
|
63110 | 2120 |
subsection \<open>Monotonicity of \<open>funpow\<close>\<close> |
59000 | 2121 |
|
63588 | 2122 |
lemma funpow_increasing: "m \<le> n \<Longrightarrow> mono f \<Longrightarrow> (f ^^ n) \<top> \<le> (f ^^ m) \<top>" |
2123 |
for f :: "'a::{lattice,order_top} \<Rightarrow> 'a" |
|
59000 | 2124 |
by (induct rule: inc_induct) |
63588 | 2125 |
(auto simp del: funpow.simps(2) simp add: funpow_Suc_right |
2126 |
intro: order_trans[OF _ funpow_mono]) |
|
2127 |
||
2128 |
lemma funpow_decreasing: "m \<le> n \<Longrightarrow> mono f \<Longrightarrow> (f ^^ m) \<bottom> \<le> (f ^^ n) \<bottom>" |
|
2129 |
for f :: "'a::{lattice,order_bot} \<Rightarrow> 'a" |
|
59000 | 2130 |
by (induct rule: dec_induct) |
63588 | 2131 |
(auto simp del: funpow.simps(2) simp add: funpow_Suc_right |
2132 |
intro: order_trans[OF _ funpow_mono]) |
|
2133 |
||
2134 |
lemma mono_funpow: "mono Q \<Longrightarrow> mono (\<lambda>i. (Q ^^ i) \<bottom>)" |
|
2135 |
for Q :: "'a::{lattice,order_bot} \<Rightarrow> 'a" |
|
59000 | 2136 |
by (auto intro!: funpow_decreasing simp: mono_def) |
58377
c6f93b8d2d8e
moved old 'size' generator together with 'old_datatype'
blanchet
parents:
58306
diff
changeset
|
2137 |
|
63588 | 2138 |
lemma antimono_funpow: "mono Q \<Longrightarrow> antimono (\<lambda>i. (Q ^^ i) \<top>)" |
2139 |
for Q :: "'a::{lattice,order_top} \<Rightarrow> 'a" |
|
60175 | 2140 |
by (auto intro!: funpow_increasing simp: antimono_def) |
2141 |
||
63110 | 2142 |
|
60758 | 2143 |
subsection \<open>The divides relation on @{typ nat}\<close> |
33274
b6ff7db522b5
moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents:
32772
diff
changeset
|
2144 |
|
63110 | 2145 |
lemma dvd_1_left [iff]: "Suc 0 dvd k" |
62365 | 2146 |
by (simp add: dvd_def) |
2147 |
||
63110 | 2148 |
lemma dvd_1_iff_1 [simp]: "m dvd Suc 0 \<longleftrightarrow> m = Suc 0" |
62365 | 2149 |
by (simp add: dvd_def) |
2150 |
||
63588 | 2151 |
lemma nat_dvd_1_iff_1 [simp]: "m dvd 1 \<longleftrightarrow> m = 1" |
2152 |
for m :: nat |
|
62365 | 2153 |
by (simp add: dvd_def) |
2154 |
||
63588 | 2155 |
lemma dvd_antisym: "m dvd n \<Longrightarrow> n dvd m \<Longrightarrow> m = n" |
2156 |
for m n :: nat |
|
63110 | 2157 |
unfolding dvd_def by (force dest: mult_eq_self_implies_10 simp add: mult.assoc) |
2158 |
||
63588 | 2159 |
lemma dvd_diff_nat [simp]: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m - n)" |
2160 |
for k m n :: nat |
|
63110 | 2161 |
unfolding dvd_def by (blast intro: right_diff_distrib' [symmetric]) |
2162 |
||
63588 | 2163 |
lemma dvd_diffD: "k dvd m - n \<Longrightarrow> k dvd n \<Longrightarrow> n \<le> m \<Longrightarrow> k dvd m" |
2164 |
for k m n :: nat |
|
33274
b6ff7db522b5
moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents:
32772
diff
changeset
|
2165 |
apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst]) |
b6ff7db522b5
moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents:
32772
diff
changeset
|
2166 |
apply (blast intro: dvd_add) |
b6ff7db522b5
moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents:
32772
diff
changeset
|
2167 |
done |
b6ff7db522b5
moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents:
32772
diff
changeset
|
2168 |
|
63588 | 2169 |
lemma dvd_diffD1: "k dvd m - n \<Longrightarrow> k dvd m \<Longrightarrow> n \<le> m \<Longrightarrow> k dvd n" |
2170 |
for k m n :: nat |
|
62365 | 2171 |
by (drule_tac m = m in dvd_diff_nat) auto |
2172 |
||
2173 |
lemma dvd_mult_cancel: |
|
2174 |
fixes m n k :: nat |
|
2175 |
assumes "k * m dvd k * n" and "0 < k" |
|
2176 |
shows "m dvd n" |
|
2177 |
proof - |
|
2178 |
from assms(1) obtain q where "k * n = (k * m) * q" .. |
|
2179 |
then have "k * n = k * (m * q)" by (simp add: ac_simps) |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2180 |
with \<open>0 < k\<close> have "n = m * q" by (auto simp add: mult_left_cancel) |
62365 | 2181 |
then show ?thesis .. |
2182 |
qed |
|
63110 | 2183 |
|
63588 | 2184 |
lemma dvd_mult_cancel1: "0 < m \<Longrightarrow> m * n dvd m \<longleftrightarrow> n = 1" |
2185 |
for m n :: nat |
|
33274
b6ff7db522b5
moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents:
32772
diff
changeset
|
2186 |
apply auto |
63588 | 2187 |
apply (subgoal_tac "m * n dvd m * 1") |
2188 |
apply (drule dvd_mult_cancel) |
|
2189 |
apply auto |
|
33274
b6ff7db522b5
moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents:
32772
diff
changeset
|
2190 |
done |
b6ff7db522b5
moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents:
32772
diff
changeset
|
2191 |
|
63588 | 2192 |
lemma dvd_mult_cancel2: "0 < m \<Longrightarrow> n * m dvd m \<longleftrightarrow> n = 1" |
2193 |
for m n :: nat |
|
62365 | 2194 |
using dvd_mult_cancel1 [of m n] by (simp add: ac_simps) |
2195 |
||
63588 | 2196 |
lemma dvd_imp_le: "k dvd n \<Longrightarrow> 0 < n \<Longrightarrow> k \<le> n" |
2197 |
for k n :: nat |
|
62365 | 2198 |
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc) |
33274
b6ff7db522b5
moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents:
32772
diff
changeset
|
2199 |
|
63588 | 2200 |
lemma nat_dvd_not_less: "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m" |
2201 |
for m n :: nat |
|
62365 | 2202 |
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc) |
33274
b6ff7db522b5
moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents:
32772
diff
changeset
|
2203 |
|
54222 | 2204 |
lemma less_eq_dvd_minus: |
51173 | 2205 |
fixes m n :: nat |
54222 | 2206 |
assumes "m \<le> n" |
2207 |
shows "m dvd n \<longleftrightarrow> m dvd n - m" |
|
51173 | 2208 |
proof - |
54222 | 2209 |
from assms have "n = m + (n - m)" by simp |
51173 | 2210 |
then obtain q where "n = m + q" .. |
58647 | 2211 |
then show ?thesis by (simp add: add.commute [of m]) |
51173 | 2212 |
qed |
2213 |
||
63588 | 2214 |
lemma dvd_minus_self: "m dvd n - m \<longleftrightarrow> n < m \<or> m dvd n" |
2215 |
for m n :: nat |
|
62481
b5d8e57826df
tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents:
62378
diff
changeset
|
2216 |
by (cases "n < m") (auto elim!: dvdE simp add: not_less le_imp_diff_is_add dest: less_imp_le) |
51173 | 2217 |
|
2218 |
lemma dvd_minus_add: |
|
2219 |
fixes m n q r :: nat |
|
2220 |
assumes "q \<le> n" "q \<le> r * m" |
|
2221 |
shows "m dvd n - q \<longleftrightarrow> m dvd n + (r * m - q)" |
|
2222 |
proof - |
|
2223 |
have "m dvd n - q \<longleftrightarrow> m dvd r * m + (n - q)" |
|
58649
a62065b5e1e2
generalized and consolidated some theorems concerning divisibility
haftmann
parents:
58647
diff
changeset
|
2224 |
using dvd_add_times_triv_left_iff [of m r] by simp |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
52729
diff
changeset
|
2225 |
also from assms have "\<dots> \<longleftrightarrow> m dvd r * m + n - q" by simp |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
52729
diff
changeset
|
2226 |
also from assms have "\<dots> \<longleftrightarrow> m dvd (r * m - q) + n" by simp |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
2227 |
also have "\<dots> \<longleftrightarrow> m dvd n + (r * m - q)" by (simp add: add.commute) |
51173 | 2228 |
finally show ?thesis . |
2229 |
qed |
|
2230 |
||
33274
b6ff7db522b5
moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents:
32772
diff
changeset
|
2231 |
|
62365 | 2232 |
subsection \<open>Aliasses\<close> |
44817 | 2233 |
|
63588 | 2234 |
lemma nat_mult_1: "1 * n = n" |
2235 |
for n :: nat |
|
58647 | 2236 |
by (fact mult_1_left) |
60562
24af00b010cf
Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents:
60427
diff
changeset
|
2237 |
|
63588 | 2238 |
lemma nat_mult_1_right: "n * 1 = n" |
2239 |
for n :: nat |
|
58647 | 2240 |
by (fact mult_1_right) |
2241 |
||
63588 | 2242 |
lemma nat_add_left_cancel: "k + m = k + n \<longleftrightarrow> m = n" |
2243 |
for k m n :: nat |
|
62365 | 2244 |
by (fact add_left_cancel) |
2245 |
||
63588 | 2246 |
lemma nat_add_right_cancel: "m + k = n + k \<longleftrightarrow> m = n" |
2247 |
for k m n :: nat |
|
62365 | 2248 |
by (fact add_right_cancel) |
2249 |
||
63588 | 2250 |
lemma diff_mult_distrib: "(m - n) * k = (m * k) - (n * k)" |
2251 |
for k m n :: nat |
|
62365 | 2252 |
by (fact left_diff_distrib') |
2253 |
||
63588 | 2254 |
lemma diff_mult_distrib2: "k * (m - n) = (k * m) - (k * n)" |
2255 |
for k m n :: nat |
|
62365 | 2256 |
by (fact right_diff_distrib') |
2257 |
||
63588 | 2258 |
lemma le_add_diff: "k \<le> n \<Longrightarrow> m \<le> n + m - k" |
2259 |
for k m n :: nat |
|
63110 | 2260 |
by (fact le_add_diff) (* FIXME delete *) |
2261 |
||
63588 | 2262 |
lemma le_diff_conv2: "k \<le> j \<Longrightarrow> (i \<le> j - k) = (i + k \<le> j)" |
2263 |
for i j k :: nat |
|
63110 | 2264 |
by (fact le_diff_conv2) (* FIXME delete *) |
2265 |
||
63588 | 2266 |
lemma diff_self_eq_0 [simp]: "m - m = 0" |
2267 |
for m :: nat |
|
62365 | 2268 |
by (fact diff_cancel) |
2269 |
||
63588 | 2270 |
lemma diff_diff_left [simp]: "i - j - k = i - (j + k)" |
2271 |
for i j k :: nat |
|
62365 | 2272 |
by (fact diff_diff_add) |
2273 |
||
63588 | 2274 |
lemma diff_commute: "i - j - k = i - k - j" |
2275 |
for i j k :: nat |
|
62365 | 2276 |
by (fact diff_right_commute) |
2277 |
||
63588 | 2278 |
lemma diff_add_inverse: "(n + m) - n = m" |
2279 |
for m n :: nat |
|
62365 | 2280 |
by (fact add_diff_cancel_left') |
2281 |
||
63588 | 2282 |
lemma diff_add_inverse2: "(m + n) - n = m" |
2283 |
for m n :: nat |
|
62365 | 2284 |
by (fact add_diff_cancel_right') |
2285 |
||
63588 | 2286 |
lemma diff_cancel: "(k + m) - (k + n) = m - n" |
2287 |
for k m n :: nat |
|
62365 | 2288 |
by (fact add_diff_cancel_left) |
2289 |
||
63588 | 2290 |
lemma diff_cancel2: "(m + k) - (n + k) = m - n" |
2291 |
for k m n :: nat |
|
62365 | 2292 |
by (fact add_diff_cancel_right) |
2293 |
||
63588 | 2294 |
lemma diff_add_0: "n - (n + m) = 0" |
2295 |
for m n :: nat |
|
62365 | 2296 |
by (fact diff_add_zero) |
2297 |
||
63588 | 2298 |
lemma add_mult_distrib2: "k * (m + n) = (k * m) + (k * n)" |
2299 |
for k m n :: nat |
|
62365 | 2300 |
by (fact distrib_left) |
2301 |
||
2302 |
lemmas nat_distrib = |
|
2303 |
add_mult_distrib distrib_left diff_mult_distrib diff_mult_distrib2 |
|
2304 |
||
44817 | 2305 |
|
60758 | 2306 |
subsection \<open>Size of a datatype value\<close> |
25193 | 2307 |
|
29608 | 2308 |
class size = |
61799 | 2309 |
fixes size :: "'a \<Rightarrow> nat" \<comment> \<open>see further theory \<open>Wellfounded\<close>\<close> |
23852 | 2310 |
|
58377
c6f93b8d2d8e
moved old 'size' generator together with 'old_datatype'
blanchet
parents:
58306
diff
changeset
|
2311 |
instantiation nat :: size |
c6f93b8d2d8e
moved old 'size' generator together with 'old_datatype'
blanchet
parents:
58306
diff
changeset
|
2312 |
begin |
c6f93b8d2d8e
moved old 'size' generator together with 'old_datatype'
blanchet
parents:
58306
diff
changeset
|
2313 |
|
63110 | 2314 |
definition size_nat where [simp, code]: "size (n::nat) = n" |
58377
c6f93b8d2d8e
moved old 'size' generator together with 'old_datatype'
blanchet
parents:
58306
diff
changeset
|
2315 |
|
c6f93b8d2d8e
moved old 'size' generator together with 'old_datatype'
blanchet
parents:
58306
diff
changeset
|
2316 |
instance .. |
c6f93b8d2d8e
moved old 'size' generator together with 'old_datatype'
blanchet
parents:
58306
diff
changeset
|
2317 |
|
c6f93b8d2d8e
moved old 'size' generator together with 'old_datatype'
blanchet
parents:
58306
diff
changeset
|
2318 |
end |
c6f93b8d2d8e
moved old 'size' generator together with 'old_datatype'
blanchet
parents:
58306
diff
changeset
|
2319 |
|
c6f93b8d2d8e
moved old 'size' generator together with 'old_datatype'
blanchet
parents:
58306
diff
changeset
|
2320 |
|
60758 | 2321 |
subsection \<open>Code module namespace\<close> |
33364 | 2322 |
|
52435
6646bb548c6b
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
52289
diff
changeset
|
2323 |
code_identifier |
6646bb548c6b
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents:
52289
diff
changeset
|
2324 |
code_module Nat \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith |
33364 | 2325 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46351
diff
changeset
|
2326 |
hide_const (open) of_nat_aux |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46351
diff
changeset
|
2327 |
|
25193 | 2328 |
end |