author  haftmann 
Wed, 17 Feb 2016 21:51:56 +0100  
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permissions  rwrr 
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(* Title: HOL/Nat.thy 
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Author: Tobias Nipkow and Lawrence C Paulson and Markus Wenzel 
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Type "nat" is a linear order, and a datatype; arithmetic operators +  
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and * (for div and mod, see theory Divides). 
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*) 
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section \<open>Natural numbers\<close> 
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theory Nat 
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imports Inductive Typedef Fun Fields 
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begin 
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ML_file "~~/src/Tools/rat.ML" 
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named_theorems arith "arith facts  only ground formulas" 

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ML_file "Tools/arith_data.ML" 
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ML_file "~~/src/Provers/Arith/fast_lin_arith.ML" 

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subsection \<open>Type \<open>ind\<close>\<close> 
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typedecl ind 

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axiomatization Zero_Rep :: ind and Suc_Rep :: "ind => ind" where 
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\<comment> \<open>the axiom of infinity in 2 parts\<close> 
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Suc_Rep_inject: "Suc_Rep x = Suc_Rep y ==> x = y" and 
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Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep" 
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subsection \<open>Type nat\<close> 
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text \<open>Type definition\<close> 

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inductive Nat :: "ind \<Rightarrow> bool" where 
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Zero_RepI: "Nat Zero_Rep" 

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 Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)" 

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typedef nat = "{n. Nat n}" 
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morphisms Rep_Nat Abs_Nat 
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using Nat.Zero_RepI by auto 
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lemma Nat_Rep_Nat: 
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"Nat (Rep_Nat n)" 
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using Rep_Nat by simp 
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lemma Nat_Abs_Nat_inverse: 
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"Nat n \<Longrightarrow> Rep_Nat (Abs_Nat n) = n" 
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using Abs_Nat_inverse by simp 
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lemma Nat_Abs_Nat_inject: 
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"Nat n \<Longrightarrow> Nat m \<Longrightarrow> Abs_Nat n = Abs_Nat m \<longleftrightarrow> n = m" 
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using Abs_Nat_inject by simp 
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instantiation nat :: zero 
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begin 

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definition Zero_nat_def: 
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"0 = Abs_Nat Zero_Rep" 
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instance .. 

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end 

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definition Suc :: "nat \<Rightarrow> nat" where 
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"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))" 
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lemma Suc_not_Zero: "Suc m \<noteq> 0" 
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by (simp add: Zero_nat_def Suc_def Suc_RepI Zero_RepI Nat_Abs_Nat_inject Suc_Rep_not_Zero_Rep Nat_Rep_Nat) 
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lemma Zero_not_Suc: "0 \<noteq> Suc m" 
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by (rule not_sym, rule Suc_not_Zero not_sym) 
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lemma Suc_Rep_inject': "Suc_Rep x = Suc_Rep y \<longleftrightarrow> x = y" 
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by (rule iffI, rule Suc_Rep_inject) simp_all 
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lemma nat_induct0: 
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fixes n 
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assumes "P 0" and "\<And>n. P n \<Longrightarrow> P (Suc n)" 
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shows "P n" 
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using assms 
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apply (unfold Zero_nat_def Suc_def) 
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apply (rule Rep_Nat_inverse [THEN subst]) \<comment> \<open>types force good instantiation\<close> 
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apply (erule Nat_Rep_Nat [THEN Nat.induct]) 
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apply (iprover elim: Nat_Abs_Nat_inverse [THEN subst]) 
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done 
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free_constructors case_nat for 
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"0 :: nat" 
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 Suc pred 
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where 
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"pred (0 :: nat) = (0 :: nat)" 
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apply atomize_elim 
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apply (rename_tac n, induct_tac n rule: nat_induct0, auto) 
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apply (simp add: Suc_def Nat_Abs_Nat_inject Nat_Rep_Nat Suc_RepI Suc_Rep_inject' 
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Rep_Nat_inject) 
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apply (simp only: Suc_not_Zero) 
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done 
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\<comment> \<open>Avoid name clashes by prefixing the output of \<open>old_rep_datatype\<close> with \<open>old\<close>.\<close> 
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setup \<open>Sign.mandatory_path "old"\<close> 
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old_rep_datatype "0 :: nat" Suc 
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apply (erule nat_induct0, assumption) 
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apply (rule nat.inject) 
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apply (rule nat.distinct(1)) 
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done 
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setup \<open>Sign.parent_path\<close> 
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\<comment> \<open>But erase the prefix for properties that are not generated by \<open>free_constructors\<close>.\<close> 
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setup \<open>Sign.mandatory_path "nat"\<close> 
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declare 
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old.nat.inject[iff del] 
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old.nat.distinct(1)[simp del, induct_simp del] 
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lemmas induct = old.nat.induct 
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lemmas inducts = old.nat.inducts 
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lemmas rec = old.nat.rec 
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lemmas simps = nat.inject nat.distinct nat.case nat.rec 
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setup \<open>Sign.parent_path\<close> 
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abbreviation rec_nat :: "'a \<Rightarrow> (nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a" where 
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"rec_nat \<equiv> old.rec_nat" 
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declare nat.sel[code del] 
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hide_const (open) Nat.pred \<comment> \<open>hide everything related to the selector\<close> 
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hide_fact 
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nat.case_eq_if 
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nat.collapse 
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nat.expand 
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nat.sel 
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nat.exhaust_sel 
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nat.split_sel 
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nat.split_sel_asm 
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lemma nat_exhaust [case_names 0 Suc, cases type: nat]: 
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\<comment> \<open>for backward compatibility  names of variables differ\<close> 
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"(y = 0 \<Longrightarrow> P) \<Longrightarrow> (\<And>nat. y = Suc nat \<Longrightarrow> P) \<Longrightarrow> P" 
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by (rule old.nat.exhaust) 
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lemma nat_induct [case_names 0 Suc, induct type: nat]: 
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\<comment> \<open>for backward compatibility  names of variables differ\<close> 
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fixes n 
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assumes "P 0" and "\<And>n. P n \<Longrightarrow> P (Suc n)" 
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shows "P n" 
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using assms by (rule nat.induct) 
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hide_fact 
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nat_exhaust 
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nat_induct0 
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ML \<open> 
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val nat_basic_lfp_sugar = 
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let 

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val ctr_sugar = the (Ctr_Sugar.ctr_sugar_of_global @{theory} @{type_name nat}); 

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val recx = Logic.varify_types_global @{term rec_nat}; 

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val C = body_type (fastype_of recx); 

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in 

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{T = HOLogic.natT, fp_res_index = 0, C = C, fun_arg_Tsss = [[], [[HOLogic.natT, C]]], 

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ctr_sugar = ctr_sugar, recx = recx, rec_thms = @{thms nat.rec}} 

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end; 

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\<close> 
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setup \<open> 

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let 
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fun basic_lfp_sugars_of _ [@{typ nat}] _ _ ctxt = 

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([], [0], [nat_basic_lfp_sugar], [], [], [], TrueI (*dummy*), [], false, ctxt) 
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 basic_lfp_sugars_of bs arg_Ts callers callssss ctxt = 
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BNF_LFP_Rec_Sugar.default_basic_lfp_sugars_of bs arg_Ts callers callssss ctxt; 

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in 

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BNF_LFP_Rec_Sugar.register_lfp_rec_extension 

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{nested_simps = [], is_new_datatype = K (K true), basic_lfp_sugars_of = basic_lfp_sugars_of, 

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rewrite_nested_rec_call = NONE} 

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end 

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\<close> 
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text \<open>Injectiveness and distinctness lemmas\<close> 

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lemma inj_Suc[simp]: "inj_on Suc N" 
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by (simp add: inj_on_def) 
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lemma Suc_neq_Zero: "Suc m = 0 \<Longrightarrow> R" 
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by (rule notE, rule Suc_not_Zero) 
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lemma Zero_neq_Suc: "0 = Suc m \<Longrightarrow> R" 
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by (rule Suc_neq_Zero, erule sym) 
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lemma Suc_inject: "Suc x = Suc y \<Longrightarrow> x = y" 
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by (rule inj_Suc [THEN injD]) 
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lemma n_not_Suc_n: "n \<noteq> Suc n" 
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by (induct n) simp_all 
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lemma Suc_n_not_n: "Suc n \<noteq> n" 
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by (rule not_sym, rule n_not_Suc_n) 
13449  199 

60758  200 
text \<open>A special form of induction for reasoning 
201 
about @{term "m < n"} and @{term "m  n"}\<close> 

13449  202 

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lemma diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==> 
13449  204 
(!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n" 
14208  205 
apply (rule_tac x = m in spec) 
15251  206 
apply (induct n) 
13449  207 
prefer 2 
208 
apply (rule allI) 

17589  209 
apply (induct_tac x, iprover+) 
13449  210 
done 
211 

24995  212 

60758  213 
subsection \<open>Arithmetic operators\<close> 
24995  214 

49388  215 
instantiation nat :: comm_monoid_diff 
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begin 
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primrec plus_nat where 
61076  219 
add_0: "0 + n = (n::nat)" 
44325  220 
 add_Suc: "Suc m + n = Suc (m + n)" 
24995  221 

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lemma add_0_right [simp]: "m + 0 = (m::nat)" 
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223 
by (induct m) simp_all 
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224 

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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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28514  228 
declare add_0 [code] 
229 

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lemma add_Suc_shift [code]: "Suc m + n = m + Suc n" 
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by simp 
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primrec minus_nat where 
61076  234 
diff_0 [code]: "m  0 = (m::nat)" 
39793  235 
 diff_Suc: "m  Suc n = (case m  n of 0 => 0  Suc k => k)" 
24995  236 

28514  237 
declare diff_Suc [simp del] 
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238 

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lemma diff_0_eq_0 [simp, code]: "0  n = (0::nat)" 
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by (induct n) (simp_all add: diff_Suc) 
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241 

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lemma diff_Suc_Suc [simp, code]: "Suc m  Suc n = m  n" 
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by (induct n) (simp_all add: diff_Suc) 
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244 

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instance proof 
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246 
fix n m q :: nat 
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show "(n + m) + q = n + (m + q)" by (induct n) simp_all 
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show "n + m = m + n" by (induct n) simp_all 
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show "m + n  m = n" by (induct m) simp_all 
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show "n  m  q = n  (m + q)" by (induct q) (simp_all add: diff_Suc) 
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show "0 + n = n" by simp 
49388  252 
show "0  n = 0" by simp 
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qed 
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254 

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255 
end 
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256 

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hide_fact (open) add_0 add_0_right diff_0 
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258 

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instantiation nat :: comm_semiring_1_cancel 
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begin 
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261 

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definition 
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One_nat_def [simp]: "1 = Suc 0" 
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primrec times_nat where 
61076  266 
mult_0: "0 * n = (0::nat)" 
44325  267 
 mult_Suc: "Suc m * n = n + (m * n)" 
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lemma mult_0_right [simp]: "(m::nat) * 0 = 0" 
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by (induct m) simp_all 
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271 

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lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)" 
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by (induct m) (simp_all add: add.left_commute) 
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274 

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lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)" 
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by (induct m) (simp_all add: add.assoc) 
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instance proof 
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fix n m q :: nat 
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show "0 \<noteq> (1::nat)" unfolding One_nat_def by simp 
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show "1 * n = n" unfolding One_nat_def by simp 
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show "n * m = m * n" by (induct n) simp_all 
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show "(n * m) * q = n * (m * q)" by (induct n) (simp_all add: add_mult_distrib) 
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show "(n + m) * q = n * q + m * q" by (rule add_mult_distrib) 
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next 
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fix k m n :: nat 
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show "k * ((m::nat)  n) = (k * m)  (k * n)" 
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288 
by (induct m n rule: diff_induct) simp_all 
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qed 
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290 

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291 
end 
24995  292 

60758  293 
text \<open>Difference distributes over multiplication\<close> 
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294 

24af00b010cf
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lemma diff_mult_distrib: 
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"((m::nat)  n) * k = (m * k)  (n * k)" 
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by (fact left_diff_distrib') 
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298 

24af00b010cf
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lemma diff_mult_distrib2: 
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"k * ((m::nat)  n) = (k * m)  (k * n)" 
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by (fact right_diff_distrib') 
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302 

24af00b010cf
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60758  304 
subsubsection \<open>Addition\<close> 
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lemma nat_add_left_cancel: 
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307 
fixes k m n :: nat 
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shows "k + m = k + n \<longleftrightarrow> m = n" 
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309 
by (fact add_left_cancel) 
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310 

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311 
lemma nat_add_right_cancel: 
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312 
fixes k m n :: nat 
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shows "m + k = n + k \<longleftrightarrow> m = n" 
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314 
by (fact add_right_cancel) 
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315 

61799  316 
text \<open>Reasoning about \<open>m + 0 = 0\<close>, etc.\<close> 
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317 

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lemma add_is_0 [iff]: 
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319 
fixes m n :: nat 
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shows "(m + n = 0) = (m = 0 & n = 0)" 
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321 
by (cases m) simp_all 
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322 

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323 
lemma add_is_1: 
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"(m+n= Suc 0) = (m= Suc 0 & n=0  m=0 & n= Suc 0)" 
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325 
by (cases m) simp_all 
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326 

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327 
lemma one_is_add: 
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"(Suc 0 = m + n) = (m = Suc 0 & n = 0  m = 0 & n = Suc 0)" 
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329 
by (rule trans, rule eq_commute, rule add_is_1) 
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330 

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331 
lemma add_eq_self_zero: 
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332 
fixes m n :: nat 
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333 
shows "m + n = m \<Longrightarrow> n = 0" 
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334 
by (induct m) simp_all 
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335 

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lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N" 
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337 
apply (induct k) 
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338 
apply simp 
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339 
apply(drule comp_inj_on[OF _ inj_Suc]) 
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340 
apply (simp add:o_def) 
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341 
done 
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342 

47208  343 
lemma Suc_eq_plus1: "Suc n = n + 1" 
344 
unfolding One_nat_def by simp 

345 

346 
lemma Suc_eq_plus1_left: "Suc n = 1 + n" 

347 
unfolding One_nat_def by simp 

348 

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349 

60758  350 
subsubsection \<open>Difference\<close> 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

351 

61076  352 
lemma diff_self_eq_0 [simp]: "(m::nat)  m = 0" 
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset

353 
by (fact diff_cancel) 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

354 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

355 
lemma diff_diff_left: "(i::nat)  j  k = i  (j + k)" 
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset

356 
by (fact diff_diff_add) 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

357 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

358 
lemma Suc_diff_diff [simp]: "(Suc m  n)  Suc k = m  n  k" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

359 
by (simp add: diff_diff_left) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

360 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

361 
lemma diff_commute: "(i::nat)  j  k = i  k  j" 
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset

362 
by (fact diff_right_commute) 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

363 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

364 
lemma diff_add_inverse: "(n + m)  n = (m::nat)" 
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset

365 
by (fact add_diff_cancel_left') 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

366 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

367 
lemma diff_add_inverse2: "(m + n)  n = (m::nat)" 
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset

368 
by (fact add_diff_cancel_right') 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

369 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

370 
lemma diff_cancel: "(k + m)  (k + n) = m  (n::nat)" 
59815
cce82e360c2f
explicit commutative additive inverse operation;
haftmann
parents:
59582
diff
changeset

371 
by (fact add_diff_cancel_left) 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

372 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

373 
lemma diff_cancel2: "(m + k)  (n + k) = m  (n::nat)" 
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset

374 
by (fact add_diff_cancel_right) 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

375 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

376 
lemma diff_add_0: "n  (n + m) = (0::nat)" 
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset

377 
by (fact diff_add_zero) 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

378 

30093  379 
lemma diff_Suc_1 [simp]: "Suc n  1 = n" 
380 
unfolding One_nat_def by simp 

381 

60758  382 
subsubsection \<open>Multiplication\<close> 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

383 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

384 
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)" 
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset

385 
by (fact distrib_left) 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

386 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

387 
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0  n=0)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

388 
by (induct m) auto 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

389 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

390 
lemmas nat_distrib = 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

391 
add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

392 

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

393 
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = Suc 0 & n = Suc 0)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

394 
apply (induct m) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

395 
apply simp 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

396 
apply (induct n) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

397 
apply auto 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

398 
done 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

399 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53986
diff
changeset

400 
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = Suc 0 & n = Suc 0)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

401 
apply (rule trans) 
44890
22f665a2e91c
new fastforce replacing fastsimp  less confusing name
nipkow
parents:
44848
diff
changeset

402 
apply (rule_tac [2] mult_eq_1_iff, fastforce) 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

403 
done 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

404 

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

405 
lemma nat_mult_eq_1_iff [simp]: "m * n = (1::nat) \<longleftrightarrow> m = 1 \<and> n = 1" 
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

406 
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

407 

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

408 
lemma nat_1_eq_mult_iff [simp]: "(1::nat) = m * n \<longleftrightarrow> m = 1 \<and> n = 1" 
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

409 
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

410 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

411 
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n  (k = (0::nat)))" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

412 
proof  
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

413 
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

414 
proof (induct n arbitrary: m) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

415 
case 0 then show "m = 0" by simp 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

416 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

417 
case (Suc n) then show "m = Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

418 
by (cases m) (simp_all add: eq_commute [of "0"]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

419 
qed 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

420 
then show ?thesis by auto 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

421 
qed 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

422 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

423 
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n  (k = (0::nat)))" 
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset

424 
by (simp add: mult.commute) 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

425 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

426 
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

427 
by (subst mult_cancel1) simp 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

428 

24995  429 

60758  430 
subsection \<open>Orders on @{typ nat}\<close> 
431 

432 
subsubsection \<open>Operation definition\<close> 

24995  433 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

434 
instantiation nat :: linorder 
25510  435 
begin 
436 

55575
a5e33e18fb5c
moved 'primrec' up (for real this time) and removed temporary 'old_primrec'
blanchet
parents:
55534
diff
changeset

437 
primrec less_eq_nat where 
61076  438 
"(0::nat) \<le> n \<longleftrightarrow> True" 
44325  439 
 "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

440 

28514  441 
declare less_eq_nat.simps [simp del] 
61076  442 
lemma le0 [iff]: "0 \<le> (n::nat)" by (simp add: less_eq_nat.simps) 
443 
lemma [code]: "(0::nat) \<le> n \<longleftrightarrow> True" by simp 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

444 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

445 
definition less_nat where 
28514  446 
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

447 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

448 
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

449 
by (simp add: less_eq_nat.simps(2)) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

450 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

451 
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

452 
unfolding less_eq_Suc_le .. 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

453 

61076  454 
lemma le_0_eq [iff]: "(n::nat) \<le> 0 \<longleftrightarrow> n = 0" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

455 
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

456 

61076  457 
lemma not_less0 [iff]: "\<not> n < (0::nat)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

458 
by (simp add: less_eq_Suc_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

459 

61076  460 
lemma less_nat_zero_code [code]: "n < (0::nat) \<longleftrightarrow> False" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

461 
by simp 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

462 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

463 
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

464 
by (simp add: less_eq_Suc_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

465 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

466 
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

467 
by (simp add: less_eq_Suc_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

468 

56194  469 
lemma Suc_less_eq2: "Suc n < m \<longleftrightarrow> (\<exists>m'. m = Suc m' \<and> n < m')" 
470 
by (cases m) auto 

471 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

472 
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

473 
by (induct m arbitrary: n) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

474 
(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

475 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

476 
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

477 
by (cases n) (auto intro: le_SucI) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

478 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

479 
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

480 
by (simp add: less_eq_Suc_le) (erule Suc_leD) 
24995  481 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

482 
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

483 
by (simp add: less_eq_Suc_le) (erule Suc_leD) 
25510  484 

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

485 
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

486 
proof 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

487 
fix n m :: 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

488 
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

489 
proof (induct n arbitrary: m) 
27679  490 
case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le) 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

491 
next 
27679  492 
case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le) 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

493 
qed 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

494 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

495 
fix n :: nat show "n \<le> n" by (induct n) simp_all 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

496 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

497 
fix n m :: nat assume "n \<le> m" and "m \<le> n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

498 
then show "n = m" 
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 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

502 
fix n m q :: nat assume "n \<le> m" and "m \<le> q" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

503 
then show "n \<le> q" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

504 
proof (induct n arbitrary: m q) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

505 
case 0 show ?case by simp 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

506 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

507 
case (Suc n) then show ?case 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

508 
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

509 
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

510 
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

511 
qed 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

512 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

513 
fix n m :: nat show "n \<le> m \<or> m \<le> n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

514 
by (induct n arbitrary: m) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

515 
(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

516 
qed 
25510  517 

518 
end 

13449  519 

52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
52435
diff
changeset

520 
instantiation nat :: order_bot 
29652  521 
begin 
522 

523 
definition bot_nat :: nat where 

524 
"bot_nat = 0" 

525 

526 
instance proof 

527 
qed (simp add: bot_nat_def) 

528 

529 
end 

530 

51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

531 
instance nat :: no_top 
61169  532 
by standard (auto intro: less_Suc_eq_le [THEN iffD2]) 
52289  533 

51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51263
diff
changeset

534 

60758  535 
subsubsection \<open>Introduction properties\<close> 
13449  536 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

537 
lemma lessI [iff]: "n < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

538 
by (simp add: less_Suc_eq_le) 
13449  539 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

540 
lemma zero_less_Suc [iff]: "0 < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

541 
by (simp add: less_Suc_eq_le) 
13449  542 

543 

60758  544 
subsubsection \<open>Elimination properties\<close> 
13449  545 

546 
lemma less_not_refl: "~ n < (n::nat)" 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

547 
by (rule order_less_irrefl) 
13449  548 

26335
961bbcc9d85b
removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents:
26315
diff
changeset

549 
lemma less_not_refl2: "n < m ==> m \<noteq> (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

550 
by (rule not_sym) (rule less_imp_neq) 
13449  551 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

552 
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

553 
by (rule less_imp_neq) 
13449  554 

26335
961bbcc9d85b
removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents:
26315
diff
changeset

555 
lemma less_irrefl_nat: "(n::nat) < n ==> R" 
961bbcc9d85b
removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents:
26315
diff
changeset

556 
by (rule notE, rule less_not_refl) 
13449  557 

558 
lemma less_zeroE: "(n::nat) < 0 ==> R" 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

559 
by (rule notE) (rule not_less0) 
13449  560 

561 
lemma less_Suc_eq: "(m < Suc n) = (m < n  m = n)" 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

562 
unfolding less_Suc_eq_le le_less .. 
13449  563 

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

564 
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

565 
by (simp add: less_Suc_eq) 
13449  566 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53986
diff
changeset

567 
lemma less_one [iff]: "(n < (1::nat)) = (n = 0)" 
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

568 
unfolding One_nat_def by (rule less_Suc0) 
13449  569 

570 
lemma Suc_mono: "m < n ==> Suc m < Suc n" 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

571 
by simp 
13449  572 

60758  573 
text \<open>"Less than" is antisymmetric, sort of\<close> 
14302  574 
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

575 
unfolding not_less less_Suc_eq_le by (rule antisym) 
14302  576 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

577 
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n  n < m)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

578 
by (rule linorder_neq_iff) 
13449  579 

580 
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m" 

581 
and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m" 

582 
shows "P n m" 

583 
apply (rule less_linear [THEN disjE]) 

584 
apply (erule_tac [2] disjE) 

585 
apply (erule lessCase) 

586 
apply (erule sym [THEN eqCase]) 

587 
apply (erule major) 

588 
done 

589 

590 

60758  591 
subsubsection \<open>Inductive (?) properties\<close> 
13449  592 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

593 
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> 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

594 
unfolding less_eq_Suc_le [of m] le_less by simp 
13449  595 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

596 
lemma lessE: 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

597 
assumes major: "i < k" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

598 
and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

599 
shows P 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

600 
proof  
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

601 
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

602 
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

603 
then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

604 
by (clarsimp simp add: less_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

605 
with p1 p2 show P by auto 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

606 
qed 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

607 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

608 
lemma less_SucE: assumes major: "m < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

609 
and less: "m < n ==> P" and eq: "m = n ==> P" shows P 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

610 
apply (rule major [THEN lessE]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

611 
apply (rule eq, blast) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

612 
apply (rule less, blast) 
13449  613 
done 
614 

615 
lemma Suc_lessE: assumes major: "Suc i < k" 

616 
and minor: "!!j. i < j ==> k = Suc j ==> P" shows P 

617 
apply (rule major [THEN lessE]) 

618 
apply (erule lessI [THEN minor]) 

14208  619 
apply (erule Suc_lessD [THEN minor], assumption) 
13449  620 
done 
621 

622 
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n" 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

623 
by simp 
13449  624 

625 
lemma less_trans_Suc: 

626 
assumes le: "i < j" shows "j < k ==> Suc i < k" 

14208  627 
apply (induct k, simp_all) 
13449  628 
apply (insert le) 
629 
apply (simp add: less_Suc_eq) 

630 
apply (blast dest: Suc_lessD) 

631 
done 

632 

61799  633 
text \<open>Can be used with \<open>less_Suc_eq\<close> to get @{term "n = m  n < m"}\<close> 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

634 
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

635 
unfolding not_less less_Suc_eq_le .. 
13449  636 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

637 
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

638 
unfolding not_le Suc_le_eq .. 
21243  639 

60758  640 
text \<open>Properties of "less than or equal"\<close> 
13449  641 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

642 
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

643 
unfolding less_Suc_eq_le . 
13449  644 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

645 
lemma Suc_n_not_le_n: "~ Suc n \<le> n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

646 
unfolding not_le less_Suc_eq_le .. 
13449  647 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

648 
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n  m = Suc n)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

649 
by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq) 
13449  650 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

651 
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

652 
by (drule le_Suc_eq [THEN iffD1], iprover+) 
13449  653 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

654 
lemma Suc_leI: "m < n ==> Suc(m) \<le> n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

655 
unfolding Suc_le_eq . 
13449  656 

61799  657 
text \<open>Stronger version of \<open>Suc_leD\<close>\<close> 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

658 
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

659 
unfolding Suc_le_eq . 
13449  660 

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

661 
lemma less_imp_le_nat: "m < n ==> m \<le> (n::nat)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

662 
unfolding less_eq_Suc_le by (rule Suc_leD) 
13449  663 

61799  664 
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

665 
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq 
13449  666 

667 

60758  668 
text \<open>Equivalence of @{term "m \<le> n"} and @{term "m < n  m = n"}\<close> 
13449  669 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

670 
lemma less_or_eq_imp_le: "m < n  m = n ==> m \<le> (n::nat)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

671 
unfolding le_less . 
13449  672 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

673 
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n  m=n)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

674 
by (rule le_less) 
13449  675 

61799  676 
text \<open>Useful with \<open>blast\<close>.\<close> 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

677 
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

678 
by auto 
13449  679 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

680 
lemma le_refl: "n \<le> (n::nat)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

681 
by simp 
13449  682 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

683 
lemma le_trans: "[ i \<le> j; j \<le> k ] ==> i \<le> (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 

33657  686 
lemma le_antisym: "[ m \<le> n; n \<le> m ] ==> m = (n::nat)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

687 
by (rule antisym) 
13449  688 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

689 
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

690 
by (rule less_le) 
13449  691 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

692 
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

693 
unfolding less_le .. 
13449  694 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

695 
lemma nat_le_linear: "(m::nat) \<le> n  n \<le> m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

696 
by (rule linear) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

697 

22718  698 
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat] 
15921  699 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

700 
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

701 
unfolding less_Suc_eq_le by auto 
13449  702 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

703 
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

704 
unfolding not_less by (rule le_less_Suc_eq) 
13449  705 

706 
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq 

707 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

708 
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m" 
25162  709 
by (cases n) simp_all 
710 

711 
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m" 

712 
by (cases n) simp_all 

13449  713 

22718  714 
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0" 
25162  715 
by (cases n) simp_all 
13449  716 

25162  717 
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)" 
718 
by (cases n) simp_all 

25140  719 

61799  720 
text \<open>This theorem is useful with \<open>blast\<close>\<close> 
13449  721 
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n" 
25162  722 
by (rule neq0_conv[THEN iffD1], iprover) 
13449  723 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

724 
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)" 
25162  725 
by (fast intro: not0_implies_Suc) 
13449  726 

54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53986
diff
changeset

727 
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)" 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

728 
using neq0_conv by blast 
13449  729 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

730 
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)" 
25162  731 
by (induct m') simp_all 
13449  732 

60758  733 
text \<open>Useful in certain inductive arguments\<close> 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

734 
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0  (\<exists>j. m = Suc j & j < n))" 
25162  735 
by (cases m) simp_all 
13449  736 

737 

60758  738 
subsubsection \<open>Monotonicity of Addition\<close> 
13449  739 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

740 
lemma Suc_pred [simp]: "n>0 ==> Suc (n  Suc 0) = n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

741 
by (simp add: diff_Suc split: nat.split) 
13449  742 

30128
365ee7319b86
revert some Suc 0 lemmas back to their original forms; added some simp rules for (1::nat)
huffman
parents:
30093
diff
changeset

743 
lemma Suc_diff_1 [simp]: "0 < n ==> Suc (n  1) = n" 
365ee7319b86
revert some Suc 0 lemmas back to their original forms; added some simp rules for (1::nat)
huffman
parents:
30093
diff
changeset

744 
unfolding One_nat_def by (rule Suc_pred) 
365ee7319b86
revert some Suc 0 lemmas back to their original forms; added some simp rules for (1::nat)
huffman
parents:
30093
diff
changeset

745 

14331  746 
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))" 
25162  747 
by (induct k) simp_all 
13449  748 

14331  749 
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))" 
25162  750 
by (induct k) simp_all 
13449  751 

25162  752 
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0  n>0)" 
753 
by(auto dest:gr0_implies_Suc) 

13449  754 

60758  755 
text \<open>strict, in 1st argument\<close> 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

756 
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)" 
25162  757 
by (induct k) simp_all 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

758 

60758  759 
text \<open>strict, in both arguments\<close> 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

760 
lemma add_less_mono: "[i < j; k < l] ==> i + k < j + (l::nat)" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

761 
apply (rule add_less_mono1 [THEN less_trans], assumption+) 
15251  762 
apply (induct j, simp_all) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

763 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

764 

61799  765 
text \<open>Deleted \<open>less_natE\<close>; use \<open>less_imp_Suc_add RS exE\<close>\<close> 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

766 
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

767 
apply (induct n) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

768 
apply (simp_all add: order_le_less) 
22718  769 
apply (blast elim!: less_SucE 
35047
1b2bae06c796
hide fact Nat.add_0_right; make add_0_right from Groups priority
haftmann
parents:
35028
diff
changeset

770 
intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric]) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

771 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

772 

56194  773 
lemma le_Suc_ex: "(k::nat) \<le> l \<Longrightarrow> (\<exists>n. l = k + n)" 
774 
by (auto simp: less_Suc_eq_le[symmetric] dest: less_imp_Suc_add) 

775 

61799  776 
text \<open>strict, in 1st argument; proof is by induction on \<open>k > 0\<close>\<close> 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

777 
lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j" 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

778 
apply(auto simp: gr0_conv_Suc) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

779 
apply (induct_tac m) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

780 
apply (simp_all add: add_less_mono) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

781 
done 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

782 

60758  783 
text \<open>Addition is the inverse of subtraction: 
784 
if @{term "n \<le> m"} then @{term "n + (m  n) = m"}.\<close> 

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

785 
lemma add_diff_inverse_nat: "~ m < n ==> n + (m  n) = (m::nat)" 
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

786 
by (induct m n rule: diff_induct) simp_all 
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

787 

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

788 

61799  789 
text\<open>The naturals form an ordered \<open>semidom\<close>\<close> 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34208
diff
changeset

790 
instance nat :: linordered_semidom 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

791 
proof 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

792 
show "0 < (1::nat)" by simp 
52289  793 
show "\<And>m n q :: nat. m \<le> n \<Longrightarrow> q + m \<le> q + n" by simp 
794 
show "\<And>m n q :: nat. m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n" by (simp add: mult_less_mono2) 

59833
ab828c2c5d67
clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents:
59816
diff
changeset

795 
show "\<And>m n :: nat. m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> m * n \<noteq> 0" by simp 
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

796 
show "\<And>m n :: nat. n \<le> m ==> (m  n) + n = (m::nat)" 
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

797 
by (simp add: add_diff_inverse_nat add.commute linorder_not_less) 
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

798 
qed 
30056  799 

44817  800 

60758  801 
subsubsection \<open>@{term min} and @{term max}\<close> 
44817  802 

803 
lemma mono_Suc: "mono Suc" 

804 
by (rule monoI) simp 

805 

806 
lemma min_0L [simp]: "min 0 n = (0::nat)" 

45931  807 
by (rule min_absorb1) simp 
44817  808 

809 
lemma min_0R [simp]: "min n 0 = (0::nat)" 

45931  810 
by (rule min_absorb2) simp 
44817  811 

812 
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)" 

813 
by (simp add: mono_Suc min_of_mono) 

814 

815 
lemma min_Suc1: 

816 
"min (Suc n) m = (case m of 0 => 0  Suc m' => Suc(min n m'))" 

817 
by (simp split: nat.split) 

818 

819 
lemma min_Suc2: 

820 
"min m (Suc n) = (case m of 0 => 0  Suc m' => Suc(min m' n))" 

821 
by (simp split: nat.split) 

822 

823 
lemma max_0L [simp]: "max 0 n = (n::nat)" 

45931  824 
by (rule max_absorb2) simp 
44817  825 

826 
lemma max_0R [simp]: "max n 0 = (n::nat)" 

45931  827 
by (rule max_absorb1) simp 
44817  828 

829 
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)" 

830 
by (simp add: mono_Suc max_of_mono) 

831 

832 
lemma max_Suc1: 

833 
"max (Suc n) m = (case m of 0 => Suc n  Suc m' => Suc(max n m'))" 

834 
by (simp split: nat.split) 

835 

836 
lemma max_Suc2: 

837 
"max m (Suc n) = (case m of 0 => Suc n  Suc m' => Suc(max m' n))" 

838 
by (simp split: nat.split) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

839 

44817  840 
lemma nat_mult_min_left: 
841 
fixes m n q :: nat 

842 
shows "min m n * q = min (m * q) (n * q)" 

843 
by (simp add: min_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans) 

844 

845 
lemma nat_mult_min_right: 

846 
fixes m n q :: nat 

847 
shows "m * min n q = min (m * n) (m * q)" 

848 
by (simp add: min_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans) 

849 

850 
lemma nat_add_max_left: 

851 
fixes m n q :: nat 

852 
shows "max m n + q = max (m + q) (n + q)" 

853 
by (simp add: max_def) 

854 

855 
lemma nat_add_max_right: 

856 
fixes m n q :: nat 

857 
shows "m + max n q = max (m + n) (m + q)" 

858 
by (simp add: max_def) 

859 

860 
lemma nat_mult_max_left: 

861 
fixes m n q :: nat 

862 
shows "max m n * q = max (m * q) (n * q)" 

863 
by (simp add: max_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans) 

864 

865 
lemma nat_mult_max_right: 

866 
fixes m n q :: nat 

867 
shows "m * max n q = max (m * n) (m * q)" 

868 
by (simp add: max_def not_le) (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

869 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

870 

60758  871 
subsubsection \<open>Additional theorems about @{term "op \<le>"}\<close> 
872 

873 
text \<open>Complete induction, aka courseofvalues induction\<close> 

26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

874 

27823  875 
instance nat :: wellorder proof 
876 
fix P and n :: nat 

877 
assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n" 

878 
have "\<And>q. q \<le> n \<Longrightarrow> P q" 

879 
proof (induct n) 

880 
case (0 n) 

26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

881 
have "P 0" by (rule step) auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

882 
thus ?case using 0 by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

883 
next 
27823  884 
case (Suc m n) 
885 
then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq) 

26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

886 
thus ?case 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

887 
proof 
27823  888 
assume "n \<le> m" thus "P n" by (rule Suc(1)) 
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

889 
next 
27823  890 
assume n: "n = Suc m" 
891 
show "P n" 

892 
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

893 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

894 
qed 
27823  895 
then show "P n" by auto 
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

896 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

897 

57015  898 

899 
lemma Least_eq_0[simp]: "P(0::nat) \<Longrightarrow> Least P = 0" 

900 
by (rule Least_equality[OF _ le0]) 

901 

27823  902 
lemma Least_Suc: 
903 
"[ P n; ~ P 0 ] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))" 

47988  904 
apply (cases n, auto) 
27823  905 
apply (frule LeastI) 
906 
apply (drule_tac P = "%x. P (Suc x) " in LeastI) 

907 
apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))") 

908 
apply (erule_tac [2] Least_le) 

47988  909 
apply (cases "LEAST x. P x", auto) 
27823  910 
apply (drule_tac P = "%x. P (Suc x) " in Least_le) 
911 
apply (blast intro: order_antisym) 

912 
done 

913 

914 
lemma Least_Suc2: 

915 
"[P n; Q m; ~P 0; !k. P (Suc k) = Q k] ==> Least P = Suc (Least Q)" 

916 
apply (erule (1) Least_Suc [THEN ssubst]) 

917 
apply simp 

918 
done 

919 

920 
lemma ex_least_nat_le: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k\<le>n. (\<forall>i<k. \<not>P i) & P(k)" 

921 
apply (cases n) 

922 
apply blast 

923 
apply (rule_tac x="LEAST k. P(k)" in exI) 

924 
apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex) 

925 
done 

926 

927 
lemma ex_least_nat_less: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k<n. (\<forall>i\<le>k. \<not>P i) & P(k+1)" 

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

928 
unfolding One_nat_def 
27823  929 
apply (cases n) 
930 
apply blast 

931 
apply (frule (1) ex_least_nat_le) 

932 
apply (erule exE) 

933 
apply (case_tac k) 

934 
apply simp 

935 
apply (rename_tac k1) 

936 
apply (rule_tac x=k1 in exI) 

937 
apply (auto simp add: less_eq_Suc_le) 

938 
done 

939 

26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

940 
lemma nat_less_induct: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

941 
assumes "!!n. \<forall>m::nat. m < n > P m ==> P n" shows "P n" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

942 
using assms less_induct by blast 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

943 

4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

944 
lemma measure_induct_rule [case_names less]: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

945 
fixes f :: "'a \<Rightarrow> nat" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

946 
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

947 
shows "P a" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

948 
by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step) 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

949 

60758  950 
text \<open>old style induction rules:\<close> 
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

951 
lemma measure_induct: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

952 
fixes f :: "'a \<Rightarrow> nat" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

953 
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

954 
by (rule measure_induct_rule [of f P a]) iprover 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

955 

4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

956 
lemma full_nat_induct: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

957 
assumes step: "(!!n. (ALL m. Suc m <= n > P m) ==> P n)" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

958 
shows "P n" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

959 
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

960 

60758  961 
text\<open>An induction rule for estabilishing binary relations\<close> 
22718  962 
lemma less_Suc_induct: 
19870  963 
assumes less: "i < j" 
964 
and step: "!!i. P i (Suc i)" 

31714  965 
and trans: "!!i j k. i < j ==> j < k ==> P i j ==> P j k ==> P i k" 
19870  966 
shows "P i j" 
967 
proof  

31714  968 
from less obtain k where j: "j = Suc (i + k)" by (auto dest: less_imp_Suc_add) 
22718  969 
have "P i (Suc (i + k))" 
19870  970 
proof (induct k) 
22718  971 
case 0 
972 
show ?case by (simp add: step) 

19870  973 
next 
974 
case (Suc k) 

31714  975 
have "0 + i < Suc k + i" by (rule add_less_mono1) simp 
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset

976 
hence "i < Suc (i + k)" by (simp add: add.commute) 
31714  977 
from trans[OF this lessI Suc step] 
978 
show ?case by simp 

19870  979 
qed 
22718  980 
thus "P i j" by (simp add: j) 
19870  981 
qed 
982 

60758  983 
text \<open>The method of infinite descent, frequently used in number theory. 
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

984 
Provided by Roelof Oosterhuis. 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

985 
$P(n)$ is true for all $n\in\mathbb{N}$ if 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

986 
\begin{itemize} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

987 
\item case ``0'': given $n=0$ prove $P(n)$, 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

988 
\item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

989 
a smaller integer $m$ such that $\neg P(m)$. 
60758  990 
\end{itemize}\<close> 
991 

992 
text\<open>A compact version without explicit base case:\<close> 

26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

993 
lemma infinite_descent: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

994 
"\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow> \<exists>m<n. \<not> P m \<rbrakk> \<Longrightarrow> P n" 
47988  995 
by (induct n rule: less_induct) auto 
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

996 

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

997 
lemma infinite_descent0[case_names 0 smaller]: 
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

998 
"\<lbrakk> P 0; !!n. n>0 \<Longrightarrow> \<not> P n \<Longrightarrow> (\<exists>m::nat. m < n \<and> \<not>P m) \<rbrakk> \<Longrightarrow> P n" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

999 
by (rule infinite_descent) (case_tac "n>0", auto) 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1000 

60758  1001 
text \<open> 
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1002 
Infinite descent using a mapping to $\mathbb{N}$: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1003 
$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1004 
\begin{itemize} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1005 
\item case ``0'': given $V(x)=0$ prove $P(x)$, 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1006 
\item case ``smaller'': given $V(x)>0$ and $\neg P(x)$ prove there exists a $y \in D$ such that $V(y)<V(x)$ and $~\neg P(y)$. 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1007 
\end{itemize} 
60758  1008 
NB: the proof also shows how to use the previous lemma.\<close> 
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1009 

4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1010 
corollary infinite_descent0_measure [case_names 0 smaller]: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1011 
assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1012 
and A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1013 
shows "P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1014 
proof  
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1015 
obtain n where "n = V x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1016 
moreover have "\<And>x. V x = n \<Longrightarrow> P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1017 
proof (induct n rule: infinite_descent0) 
61799  1018 
case 0 \<comment> "i.e. $V(x) = 0$" 
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1019 
with A0 show "P x" by auto 
61799  1020 
next \<comment> "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$" 
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1021 
case (smaller n) 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1022 
then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1023 
with A1 obtain y where "V y < V x \<and> \<not> P y" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1024 
with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1025 
then show ?case by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1026 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1027 
ultimately show "P x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1028 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1029 

60758  1030 
text\<open>Again, without explicit base case:\<close> 
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1031 
lemma infinite_descent_measure: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1032 
assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1033 
proof  
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1034 
from assms obtain n where "n = V x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1035 
moreover have "!!x. V x = n \<Longrightarrow> P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1036 
proof (induct n rule: infinite_descent, auto) 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1037 
fix x assume "\<not> P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1038 
with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1039 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1040 
ultimately show "P x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1041 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1042 

61799  1043 
text \<open>A [clumsy] way of lifting \<open><\<close> 
1044 
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

1045 
lemma less_mono_imp_le_mono: 
24438  1046 
"\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)" 
1047 
by (simp add: order_le_less) (blast) 

1048 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1049 

60758  1050 
text \<open>nonstrict, in 1st argument\<close> 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1051 
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)" 
24438  1052 
by (rule add_right_mono) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1053 

60758  1054 
text \<open>nonstrict, in both arguments\<close> 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1055 
lemma add_le_mono: "[ i \<le> j; k \<le> l ] ==> i + k \<le> j + (l::nat)" 
24438  1056 
by (rule add_mono) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1057 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1058 
lemma le_add2: "n \<le> ((m + n)::nat)" 
24438  1059 
by (insert add_right_mono [of 0 m n], simp) 
13449  1060 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1061 
lemma le_add1: "n \<le> ((n + m)::nat)" 
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset

1062 
by (simp add: add.commute, rule le_add2) 
13449  1063 

1064 
lemma less_add_Suc1: "i < Suc (i + m)" 

24438  1065 
by (rule le_less_trans, rule le_add1, rule lessI) 
13449  1066 

1067 
lemma less_add_Suc2: "i < Suc (m + i)" 

24438  1068 
by (rule le_less_trans, rule le_add2, rule lessI) 
13449  1069 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1070 
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))" 
24438  1071 
by (iprover intro!: less_add_Suc1 less_imp_Suc_add) 
13449  1072 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1073 
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m" 
24438  1074 
by (rule le_trans, assumption, rule le_add1) 
13449  1075 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1076 
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j" 
24438  1077 
by (rule le_trans, assumption, rule le_add2) 
13449  1078 

1079 
lemma trans_less_add1: "(i::nat) < j ==> i < j + m" 

24438  1080 
by (rule less_le_trans, assumption, rule le_add1) 
13449  1081 

1082 
lemma trans_less_add2: "(i::nat) < j ==> i < m + j" 

24438  1083 
by (rule less_le_trans, assumption, rule le_add2) 
13449  1084 

1085 
lemma add_lessD1: "i + j < (k::nat) ==> i < k" 

24438  1086 
apply (rule le_less_trans [of _ "i+j"]) 
1087 
apply (simp_all add: le_add1) 

1088 
done 

13449  1089 

1090 
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))" 

24438  1091 
apply (rule notI) 
26335
961bbcc9d85b
removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents:
26315
diff
changeset

1092 
apply (drule add_lessD1) 
961bbcc9d85b
removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents:
26315
diff
changeset

1093 
apply (erule less_irrefl [THEN notE]) 
24438  1094 
done 
13449  1095 

1096 
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))" 

57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset

1097 
by (simp add: add.commute) 
13449  1098 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1099 
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)" 
24438  1100 
apply (rule order_trans [of _ "m+k"]) 
1101 
apply (simp_all add: le_add1) 

1102 
done 

13449  1103 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1104 
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)" 
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset

1105 
apply (simp add: add.commute) 
24438  1106 
apply (erule add_leD1) 
1107 
done 

13449  1108 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1109 
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R" 
24438  1110 
by (blast dest: add_leD1 add_leD2) 
13449  1111 

61799  1112 
text \<open>needs \<open>!!k\<close> for \<open>ac_simps\<close> to work\<close> 
13449  1113 
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n" 
24438  1114 
by (force simp del: add_Suc_right 
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset

1115 
simp add: less_iff_Suc_add add_Suc_right [symmetric] ac_simps) 
13449  1116 

1117 

60758  1118 
subsubsection \<open>More results about difference\<close> 
13449  1119 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1120 
lemma Suc_diff_le: "n \<le> m ==> Suc m  n = Suc (m  n)" 
24438  1121 
by (induct m n rule: diff_induct) simp_all 
13449  1122 

1123 
lemma diff_less_Suc: "m  n < Suc m" 

24438  1124 
apply (induct m n rule: diff_induct) 
1125 
apply (erule_tac [3] less_SucE) 

1126 
apply (simp_all add: less_Suc_eq) 

1127 
done 

13449  1128 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1129 
lemma diff_le_self [simp]: "m  n \<le> (m::nat)" 
24438  1130 
by (induct m n rule: diff_induct) (simp_all add: le_SucI) 
13449  1131 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1132 
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1133 
by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1134 

52289  1135 
instance nat :: ordered_cancel_comm_monoid_diff 
1136 
proof 

1137 
show "\<And>m n :: nat. m \<le> n \<longleftrightarrow> (\<exists>q. n = m + q)" by (fact le_iff_add) 

1138 
qed 

1139 

13449  1140 
lemma less_imp_diff_less: "(j::nat) < k ==> j  n < k" 
24438  1141 
by (rule le_less_trans, rule diff_le_self) 
13449  1142 

1143 
lemma diff_Suc_less [simp]: "0<n ==> n  Suc i < n" 

24438  1144 
by (cases n) (auto simp add: le_simps) 
13449  1145 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1146 
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j)  k = i + (j  k)" 
24438  1147 
by (induct j k rule: diff_induct) simp_all 
13449  1148 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1149 
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i)  k = (j  k) + i" 
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset

1150 
by (simp add: add.commute diff_add_assoc) 
13449  1151 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1152 
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j  i = k) = (j = k + i)" 
24438  1153 
by (auto simp add: diff_add_inverse2) 
13449  1154 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1155 
lemma diff_is_0_eq [simp]: "((m::nat)  n = 0) = (m \<le> n)" 
24438  1156 
by (induct m n rule: diff_induct) simp_all 
13449  1157 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1158 
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat)  n = 0" 
24438  1159 
by (rule iffD2, rule diff_is_0_eq) 
13449  1160 

1161 
lemma zero_less_diff [simp]: "(0 < n  (m::nat)) = (m < n)" 

24438  1162 
by (induct m n rule: diff_induct) simp_all 
13449  1163 

22718  1164 
lemma less_imp_add_positive: 
1165 
assumes "i < j" 

1166 
shows "\<exists>k::nat. 0 < k & i + k = j" 

1167 
proof 

1168 
from assms show "0 < j  i & i + (j  i) = j" 

23476  1169 
by (simp add: order_less_imp_le) 
22718  1170 
qed 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
7702
diff
changeset

1171 

60758  1172 
text \<open>a nice rewrite for bounded subtraction\<close> 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1173 
lemma nat_minus_add_max: 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1174 
fixes n m :: nat 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1175 
shows "n  m + m = max n m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1176 
by (simp add: max_def not_le order_less_imp_le) 
13449  1177 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1178 
lemma nat_diff_split: 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1179 
"P(a  b::nat) = ((a<b > P 0) & (ALL d. a = b + d > P d))" 
61799  1180 
\<comment> \<open>elimination of \<open>\<close> on \<open>nat\<close>\<close> 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1181 
by (cases "a < b") 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1182 
(auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse 
57492
74bf65a1910a
Hypsubst preserves equality hypotheses
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
57200
diff
changeset

1183 
not_less le_less dest!: add_eq_self_zero add_eq_self_zero[OF sym]) 
13449  1184 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1185 
lemma nat_diff_split_asm: 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1186 
"P(a  b::nat) = (~ (a < b & ~ P 0  (EX d. a = b + d & ~ P d)))" 
61799  1187 
\<comment> \<open>elimination of \<open>\<close> on \<open>nat\<close> in assumptions\<close> 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1188 
by (auto split: nat_diff_split) 
13449  1189 

47255
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1190 
lemma Suc_pred': "0 < n ==> n = Suc(n  1)" 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1191 
by simp 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1192 

30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1193 
lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m  1) + n))" 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1194 
unfolding One_nat_def by (cases m) simp_all 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1195 

30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1196 
lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m  1) * n))" 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1197 
unfolding One_nat_def by (cases m) simp_all 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1198 

30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1199 
lemma Suc_diff_eq_diff_pred: "0 < n ==> Suc m  n = m  (n  1)" 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1200 
unfolding One_nat_def by (cases n) simp_all 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1201 

30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1202 
lemma diff_Suc_eq_diff_pred: "m  Suc n = (m  1)  n" 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1203 
unfolding One_nat_def by (cases m) simp_all 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1204 

30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1205 
lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)" 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1206 
by (fact Let_def) 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1207 

13449  1208 

60758  1209 
subsubsection \<open>Monotonicity of multiplication\<close> 
13449  1210 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1211 
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k" 
24438  1212 
by (simp add: mult_right_mono) 
13449  1213 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1214 
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j" 
24438  1215 
by (simp add: mult_left_mono) 
13449  1216 

61799  1217 
text \<open>\<open>\<le>\<close> monotonicity, BOTH arguments\<close> 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1218 
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l" 
24438  1219 
by (simp add: mult_mono) 
13449  1220 

1221 
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k" 

24438  1222 
by (simp add: mult_strict_right_mono) 
13449  1223 

61799  1224 
text\<open>Differs from the standard \<open>zero_less_mult_iff\<close> in that 
60758  1225 
there are no negative numbers.\<close> 
14266  1226 
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)" 
13449  1227 
apply (induct m) 
22718  1228 
apply simp 
1229 
apply (case_tac n) 

1230 
apply simp_all 

13449  1231 
done 
1232 

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

1233 
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)" 
13449  1234 
apply (induct m) 
22718  1235 
apply simp 
1236 
apply (case_tac n) 

1237 
apply simp_all 

13449  1238 
done 
1239 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

1240 
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)" 
13449  1241 
apply (safe intro!: mult_less_mono1) 
47988  1242 
apply (cases k, auto) 
13449  1243 
apply (simp del: le_0_eq add: linorder_not_le [symmetric]) 
1244 
apply (blast intro: mult_le_mono1) 

1245 
done 

1246 

1247 
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)" 

57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset

1248 
by (simp add: mult.commute [of k]) 
13449  1249 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1250 
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k > m \<le> n)" 
24438  1251 
by (simp add: linorder_not_less [symmetric], auto) 
13449  1252 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1253 
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k > m \<le> n)" 
24438  1254 
by (simp add: linorder_not_less [symmetric], auto) 
13449  1255 

1256 
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)" 

24438  1257 
by (subst mult_less_cancel1) simp 
13449  1258 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1259 
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)" 
24438  1260 
by (subst mult_le_cancel1) simp 
13449  1261 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1262 
lemma le_square: "m \<le> m * (m::nat)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1263 
by (cases m) (auto intro: le_add1) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1264 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1265 
lemma le_cube: "(m::nat) \<le> m * (m * m)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1266 
by (cases m) (auto intro: le_add1) 
13449  1267 

61799  1268 
text \<open>Lemma for \<open>gcd\<close>\<close> 
30128
365ee7319b86
revert some Suc 0 lemmas back to their original forms; added some simp rules for (1::nat)
huffman
parents:
30093
diff
changeset

1269 
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1  m = 0" 
13449  1270 
apply (drule sym) 
1271 
apply (rule disjCI) 

1272 
apply (rule nat_less_cases, erule_tac [2] _) 

25157  1273 
apply (drule_tac [2] mult_less_mono2) 
25162  1274 
apply (auto) 
13449  1275 
done 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
7702
diff
changeset

1276 

51263
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
51173
diff
changeset

1277 
lemma mono_times_nat: 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
51173
diff
changeset

1278 
fixes n :: nat 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
51173
diff
changeset

1279 
assumes "n > 0" 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
51173
diff
changeset

1280 
shows "mono (times n)" 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
51173
diff
changeset

1281 
proof 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
51173
diff
changeset

1282 
fix m q :: nat 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
51173
diff
changeset

1283 
assume "m \<le> q" 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
51173
diff
changeset

1284 
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

1285 
qed 
31e786e0e6a7
turned example into library for comparing growth of functions
haftmann
parents:
51173
diff
changeset

1286 

60758  1287 
text \<open>the lattice order on @{typ nat}\<close> 
24995  1288 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1289 
instantiation nat :: distrib_lattice 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1290 
begin 
24995  1291 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1292 
definition 
61076  1293 
"(inf :: nat \<Rightarrow> nat \<Rightarrow> nat) = min" 
24995  1294 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1295 
definition 
61076  1296 
"(sup :: nat \<Rightarrow> nat \<Rightarrow> nat) = max" 
24995  1297 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1298 
instance by intro_classes 
