author  berghofe 
Mon, 05 Aug 2002 14:27:42 +0200  
changeset 13449  43c9ec498291 
parent 12338  de0f4a63baa5 
child 13585  db4005b40cc6 
permissions  rwrr 
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(* Title: HOL/Nat.thy 
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ID: $Id$ 

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Author: Tobias Nipkow and Lawrence C Paulson 
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Type "nat" is a linear order, and a datatype; arithmetic operators +  
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and * (for div, mod and dvd, see theory Divides). 
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*) 
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header {* Natural numbers *} 
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theory Nat = Wellfounded_Recursion: 

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subsection {* Type @{text ind} *} 

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typedecl ind 

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consts 

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Zero_Rep :: ind 

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Suc_Rep :: "ind => ind" 

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axioms 

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 {* the axiom of infinity in 2 parts *} 

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inj_Suc_Rep: "inj Suc_Rep" 

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Suc_Rep_not_Zero_Rep: "Suc_Rep x ~= Zero_Rep" 

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subsection {* Type nat *} 

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text {* Type definition *} 

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consts 

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Nat :: "ind set" 

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inductive Nat 

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intros 

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Zero_RepI: "Zero_Rep : Nat" 

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Suc_RepI: "i : Nat ==> Suc_Rep i : Nat" 

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global 

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typedef (open Nat) 

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nat = "Nat" by (rule exI, rule Nat.Zero_RepI) 

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instance nat :: ord .. 

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instance nat :: zero .. 

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instance nat :: one .. 

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text {* Abstract constants and syntax *} 

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consts 

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Suc :: "nat => nat" 

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pred_nat :: "(nat * nat) set" 

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local 

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defs 

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Zero_nat_def: "0 == Abs_Nat Zero_Rep" 

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Suc_def: "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))" 

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One_nat_def [simp]: "1 == Suc 0" 

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 {* nat operations *} 

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pred_nat_def: "pred_nat == {(m, n). n = Suc m}" 

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less_def: "m < n == (m, n) : trancl pred_nat" 

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le_def: "m <= (n::nat) == ~ (n < m)" 

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text {* Induction *} 

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theorem nat_induct: "P 0 ==> (!!n. P n ==> P (Suc n)) ==> P n" 
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apply (unfold Zero_nat_def Suc_def) 

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apply (rule Rep_Nat_inverse [THEN subst])  {* types force good instantiation *} 

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apply (erule Rep_Nat [THEN Nat.induct]) 

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apply (rules elim: Abs_Nat_inverse [THEN subst]) 

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done 

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text {* Isomorphisms: @{text Abs_Nat} and @{text Rep_Nat} *} 

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lemma inj_Rep_Nat: "inj Rep_Nat" 

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apply (rule inj_inverseI) 

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apply (rule Rep_Nat_inverse) 

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done 

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lemma inj_on_Abs_Nat: "inj_on Abs_Nat Nat" 

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apply (rule inj_on_inverseI) 

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apply (erule Abs_Nat_inverse) 

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done 

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text {* Distinctness of constructors *} 

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lemma Suc_not_Zero [iff]: "Suc m ~= 0" 

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apply (unfold Zero_nat_def Suc_def) 

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apply (rule inj_on_Abs_Nat [THEN inj_on_contraD]) 

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apply (rule Suc_Rep_not_Zero_Rep) 

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apply (rule Rep_Nat Nat.Suc_RepI Nat.Zero_RepI)+ 

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done 

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lemma Zero_not_Suc [iff]: "0 ~= Suc m" 

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by (rule not_sym, rule Suc_not_Zero not_sym) 

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lemma Suc_neq_Zero: "Suc m = 0 ==> R" 

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by (rule notE, rule Suc_not_Zero) 

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lemma Zero_neq_Suc: "0 = Suc m ==> R" 

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by (rule Suc_neq_Zero, erule sym) 

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text {* Injectiveness of @{term Suc} *} 

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lemma inj_Suc: "inj Suc" 

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apply (unfold Suc_def) 

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apply (rule injI) 

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apply (drule inj_on_Abs_Nat [THEN inj_onD]) 

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apply (rule Rep_Nat Nat.Suc_RepI)+ 

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apply (drule inj_Suc_Rep [THEN injD]) 

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apply (erule inj_Rep_Nat [THEN injD]) 

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done 

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lemma Suc_inject: "Suc x = Suc y ==> x = y" 

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by (rule inj_Suc [THEN injD]) 

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lemma Suc_Suc_eq [iff]: "(Suc m = Suc n) = (m = n)" 

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apply (rule iffI) 

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apply (erule Suc_inject) 

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apply (erule arg_cong) 

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done 

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lemma nat_not_singleton: "(ALL x. x = (0::nat)) = False" 

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by auto 

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text {* @{typ nat} is a datatype *} 

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rep_datatype nat 
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distinct Suc_not_Zero Zero_not_Suc 
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inject Suc_Suc_eq 

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induction nat_induct 

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lemma n_not_Suc_n: "n ~= Suc n" 

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by (induct n) simp_all 

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lemma Suc_n_not_n: "Suc t ~= t" 

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by (rule not_sym, rule n_not_Suc_n) 

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text {* A special form of induction for reasoning 

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about @{term "m < n"} and @{term "m  n"} *} 

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theorem diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==> 

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(!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n" 

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apply (rule_tac x = "m" in spec) 

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apply (induct_tac n) 

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prefer 2 

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apply (rule allI) 

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apply (induct_tac x) 

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apply rules+ 

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done 

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subsection {* Basic properties of "less than" *} 

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lemma wf_pred_nat: "wf pred_nat" 

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apply (unfold wf_def pred_nat_def) 

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apply clarify 

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apply (induct_tac x) 

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apply blast+ 

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done 

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lemma wf_less: "wf {(x, y::nat). x < y}" 

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apply (unfold less_def) 

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apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_subset]) 

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apply blast 

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done 

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lemma less_eq: "((m, n) : pred_nat^+) = (m < n)" 

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apply (unfold less_def) 

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apply (rule refl) 

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done 

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subsubsection {* Introduction properties *} 

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lemma less_trans: "i < j ==> j < k ==> i < (k::nat)" 

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apply (unfold less_def) 

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apply (rule trans_trancl [THEN transD]) 

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apply assumption+ 

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done 

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lemma lessI [iff]: "n < Suc n" 

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apply (unfold less_def pred_nat_def) 

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apply (simp add: r_into_trancl) 

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done 

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lemma less_SucI: "i < j ==> i < Suc j" 

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apply (rule less_trans) 

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apply assumption 

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apply (rule lessI) 

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done 

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lemma zero_less_Suc [iff]: "0 < Suc n" 

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apply (induct n) 

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apply (rule lessI) 

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apply (erule less_trans) 

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apply (rule lessI) 

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done 

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subsubsection {* Elimination properties *} 

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lemma less_not_sym: "n < m ==> ~ m < (n::nat)" 

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apply (unfold less_def) 

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apply (blast intro: wf_pred_nat wf_trancl [THEN wf_asym]) 

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done 

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lemma less_asym: 

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assumes h1: "(n::nat) < m" and h2: "~ P ==> m < n" shows P 

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apply (rule contrapos_np) 

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apply (rule less_not_sym) 

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apply (rule h1) 

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apply (erule h2) 

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done 

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

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apply (unfold less_def) 

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apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_not_refl]) 

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done 

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lemma less_irrefl [elim!]: "(n::nat) < n ==> R" 

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by (rule notE, rule less_not_refl) 

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lemma less_not_refl2: "n < m ==> m ~= (n::nat)" by blast 

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lemma less_not_refl3: "(s::nat) < t ==> s ~= t" 

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by (rule not_sym, rule less_not_refl2) 

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lemma lessE: 

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assumes major: "i < k" 

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and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P" 

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shows P 

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apply (rule major [unfolded less_def pred_nat_def, THEN tranclE]) 

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apply simp_all 

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apply (erule p1) 

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apply (rule p2) 

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apply (simp add: less_def pred_nat_def) 

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apply assumption 

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done 

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lemma not_less0 [iff]: "~ n < (0::nat)" 

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by (blast elim: lessE) 

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

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by (rule notE, rule not_less0) 

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lemma less_SucE: assumes major: "m < Suc n" 

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and less: "m < n ==> P" and eq: "m = n ==> P" shows P 

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apply (rule major [THEN lessE]) 

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apply (rule eq) 

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apply blast 

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apply (rule less) 

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apply blast 

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done 

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

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by (blast elim!: less_SucE intro: less_trans) 

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

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by (simp add: less_Suc_eq) 

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lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)" 

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by (simp add: less_Suc_eq) 

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

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by (induct n) (fast elim: less_trans lessE)+ 

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text {* "Less than" is a linear ordering *} 

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lemma less_linear: "m < n  m = n  n < (m::nat)" 

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apply (induct_tac m) 

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apply (induct_tac n) 

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apply (rule refl [THEN disjI1, THEN disjI2]) 

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apply (rule zero_less_Suc [THEN disjI1]) 

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apply (blast intro: Suc_mono less_SucI elim: lessE) 

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done 

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lemma nat_neq_iff: "((m::nat) ~= n) = (m < n  n < m)" 

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using less_linear by blast 

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

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

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shows "P n m" 

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apply (rule less_linear [THEN disjE]) 

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apply (erule_tac [2] disjE) 

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apply (erule lessCase) 

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apply (erule sym [THEN eqCase]) 

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apply (erule major) 

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done 

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subsubsection {* Inductive (?) properties *} 

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lemma Suc_lessI: "m < n ==> Suc m ~= n ==> Suc m < n" 

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apply (simp add: nat_neq_iff) 

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apply (blast elim!: less_irrefl less_SucE elim: less_asym) 

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done 

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lemma Suc_lessD: "Suc m < n ==> m < n" 

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apply (induct n) 

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apply (fast intro!: lessI [THEN less_SucI] elim: less_trans lessE)+ 

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done 

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

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

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apply (rule major [THEN lessE]) 

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apply (erule lessI [THEN minor]) 

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apply (erule Suc_lessD [THEN minor]) 

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apply assumption 

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done 

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

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by (blast elim: lessE dest: Suc_lessD) 

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lemma Suc_less_eq [iff]: "(Suc m < Suc n) = (m < n)" 
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apply (rule iffI) 

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apply (erule Suc_less_SucD) 

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apply (erule Suc_mono) 

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done 

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lemma less_trans_Suc: 

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

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apply (induct k) 

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apply simp_all 

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apply (insert le) 

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apply (simp add: less_Suc_eq) 

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apply (blast dest: Suc_lessD) 

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done 

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text {* Can be used with @{text less_Suc_eq} to get @{term "n = m  n < m"} *} 

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lemma not_less_eq: "(~ m < n) = (n < Suc m)" 

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apply (rule_tac m = "m" and n = "n" in diff_induct) 

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apply simp_all 

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done 

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text {* Complete induction, aka courseofvalues induction *} 

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lemma nat_less_induct: 

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assumes prem: "!!n. ALL m::nat. m < n > P m ==> P n" shows "P n" 

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apply (rule_tac a=n in wf_induct) 

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apply (rule wf_pred_nat [THEN wf_trancl]) 

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apply (rule prem) 

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apply (unfold less_def) 

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apply assumption 

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done 

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subsection {* Properties of "less or equal than" *} 

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text {* Was @{text le_eq_less_Suc}, but this orientation is more useful *} 

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lemma less_Suc_eq_le: "(m < Suc n) = (m <= n)" 

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by (unfold le_def, rule not_less_eq [symmetric]) 

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lemma le_imp_less_Suc: "m <= n ==> m < Suc n" 

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by (rule less_Suc_eq_le [THEN iffD2]) 

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lemma le0 [iff]: "(0::nat) <= n" 

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by (unfold le_def, rule not_less0) 

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lemma Suc_n_not_le_n: "~ Suc n <= n" 

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by (simp add: le_def) 

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lemma le_0_eq [iff]: "((i::nat) <= 0) = (i = 0)" 

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by (induct i) (simp_all add: le_def) 

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lemma le_Suc_eq: "(m <= Suc n) = (m <= n  m = Suc n)" 

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by (simp del: less_Suc_eq_le add: less_Suc_eq_le [symmetric] less_Suc_eq) 

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lemma le_SucE: "m <= Suc n ==> (m <= n ==> R) ==> (m = Suc n ==> R) ==> R" 

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by (drule le_Suc_eq [THEN iffD1], rules+) 

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lemma leI: "~ n < m ==> m <= (n::nat)" by (simp add: le_def) 

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lemma leD: "m <= n ==> ~ n < (m::nat)" 

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by (simp add: le_def) 

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lemmas leE = leD [elim_format] 

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lemma not_less_iff_le: "(~ n < m) = (m <= (n::nat))" 

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by (blast intro: leI elim: leE) 

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lemma not_leE: "~ m <= n ==> n<(m::nat)" 

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by (simp add: le_def) 

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lemma not_le_iff_less: "(~ n <= m) = (m < (n::nat))" 

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by (simp add: le_def) 

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lemma Suc_leI: "m < n ==> Suc(m) <= n" 

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apply (simp add: le_def less_Suc_eq) 

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apply (blast elim!: less_irrefl less_asym) 

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done  {* formerly called lessD *} 

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lemma Suc_leD: "Suc(m) <= n ==> m <= n" 

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by (simp add: le_def less_Suc_eq) 

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text {* Stronger version of @{text Suc_leD} *} 

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lemma Suc_le_lessD: "Suc m <= n ==> m < n" 

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apply (simp add: le_def less_Suc_eq) 

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using less_linear 

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apply blast 

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done 

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lemma Suc_le_eq: "(Suc m <= n) = (m < n)" 

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by (blast intro: Suc_leI Suc_le_lessD) 

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lemma le_SucI: "m <= n ==> m <= Suc n" 

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by (unfold le_def) (blast dest: Suc_lessD) 

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lemma less_imp_le: "m < n ==> m <= (n::nat)" 

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by (unfold le_def) (blast elim: less_asym) 

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text {* For instance, @{text "(Suc m < Suc n) = (Suc m <= n) = (m < n)"} *} 

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lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq 

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text {* Equivalence of @{term "m <= n"} and @{term "m < n  m = n"} *} 

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lemma le_imp_less_or_eq: "m <= n ==> m < n  m = (n::nat)" 

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apply (unfold le_def) 

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using less_linear 

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apply (blast elim: less_irrefl less_asym) 

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done 

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lemma less_or_eq_imp_le: "m < n  m = n ==> m <= (n::nat)" 

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apply (unfold le_def) 

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using less_linear 

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apply (blast elim!: less_irrefl elim: less_asym) 

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done 

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lemma le_eq_less_or_eq: "(m <= (n::nat)) = (m < n  m=n)" 

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by (rules intro: less_or_eq_imp_le le_imp_less_or_eq) 

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text {* Useful with @{text Blast}. *} 

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lemma eq_imp_le: "(m::nat) = n ==> m <= n" 

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by (rule less_or_eq_imp_le, rule disjI2) 

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lemma le_refl: "n <= (n::nat)" 

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by (simp add: le_eq_less_or_eq) 

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lemma le_less_trans: "[ i <= j; j < k ] ==> i < (k::nat)" 

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by (blast dest!: le_imp_less_or_eq intro: less_trans) 

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lemma less_le_trans: "[ i < j; j <= k ] ==> i < (k::nat)" 

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by (blast dest!: le_imp_less_or_eq intro: less_trans) 

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lemma le_trans: "[ i <= j; j <= k ] ==> i <= (k::nat)" 

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by (blast dest!: le_imp_less_or_eq intro: less_or_eq_imp_le less_trans) 

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lemma le_anti_sym: "[ m <= n; n <= m ] ==> m = (n::nat)" 

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 {* @{text order_less_irrefl} could make this proof fail *} 

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by (blast dest!: le_imp_less_or_eq elim!: less_irrefl elim: less_asym) 

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lemma Suc_le_mono [iff]: "(Suc n <= Suc m) = (n <= m)" 

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by (simp add: le_simps) 

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text {* Axiom @{text order_less_le} of class @{text order}: *} 

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lemma nat_less_le: "((m::nat) < n) = (m <= n & m ~= n)" 

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by (simp add: le_def nat_neq_iff) (blast elim!: less_asym) 

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lemma le_neq_implies_less: "(m::nat) <= n ==> m ~= n ==> m < n" 

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by (rule iffD2, rule nat_less_le, rule conjI) 

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text {* Axiom @{text linorder_linear} of class @{text linorder}: *} 

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lemma nat_le_linear: "(m::nat) <= n  n <= m" 

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apply (simp add: le_eq_less_or_eq) 

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using less_linear 

468 
apply blast 

469 
done 

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lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)" 

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by (blast elim!: less_SucE) 

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474 

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text {* 

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Rewrite @{term "n < Suc m"} to @{term "n = m"} 

477 
if @{term "~ n < m"} or @{term "m <= n"} hold. 

478 
Not suitable as default simprules because they often lead to looping 

479 
*} 

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lemma le_less_Suc_eq: "m <= n ==> (n < Suc m) = (n = m)" 

481 
by (rule not_less_less_Suc_eq, rule leD) 

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lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq 

484 

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text {* 

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Reorientation of the equations @{text "0 = x"} and @{text "1 = x"}. 

488 
No longer added as simprules (they loop) 

489 
but via @{text reorient_simproc} in Bin 

490 
*} 

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text {* Polymorphic, not just for @{typ nat} *} 

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lemma zero_reorient: "(0 = x) = (x = 0)" 

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by auto 

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lemma one_reorient: "(1 = x) = (x = 1)" 

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by auto 

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text {* Type {@typ nat} is a wellfounded linear order *} 

500 

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instance nat :: order by (intro_classes, 

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(assumption  rule le_refl le_trans le_anti_sym nat_less_le)+) 

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instance nat :: linorder by (intro_classes, rule nat_le_linear) 

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instance nat :: wellorder by (intro_classes, rule wf_less) 

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subsection {* Arithmetic operators *} 

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axclass power < type 
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consts 
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power :: "('a::power) => nat => 'a" (infixr "^" 80) 
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text {* arithmetic operators @{text "+ "} and @{text "*"} *} 
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instance nat :: plus .. 

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instance nat :: minus .. 

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instance nat :: times .. 

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instance nat :: power .. 

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text {* size of a datatype value; overloaded *} 
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consts size :: "'a => nat" 

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primrec 
525 
add_0: "0 + n = n" 

526 
add_Suc: "Suc m + n = Suc (m + n)" 

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primrec 

529 
diff_0: "m  0 = m" 

530 
diff_Suc: "m  Suc n = (case m  n of 0 => 0  Suc k => k)" 

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531 

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532 
primrec 
13449  533 
mult_0: "0 * n = 0" 
534 
mult_Suc: "Suc m * n = n + (m * n)" 

535 

536 
text {* These 2 rules ease the use of primitive recursion. NOTE USE OF @{text "=="} *} 

537 
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c" 

538 
by simp 

539 

540 
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)" 

541 
by simp 

542 

543 
lemma not0_implies_Suc: "n ~= 0 ==> EX m. n = Suc m" 

544 
by (case_tac n) simp_all 

545 

546 
lemma gr_implies_not0: "!!n::nat. m<n ==> n ~= 0" 

547 
by (case_tac n) simp_all 

548 

549 
lemma neq0_conv [iff]: "!!n::nat. (n ~= 0) = (0 < n)" 

550 
by (case_tac n) simp_all 

551 

552 
text {* This theorem is useful with @{text blast} *} 

553 
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n" 

554 
by (rule iffD1, rule neq0_conv, rules) 

555 

556 
lemma gr0_conv_Suc: "(0 < n) = (EX m. n = Suc m)" 

557 
by (fast intro: not0_implies_Suc) 

558 

559 
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)" 

560 
apply (rule iffI) 

561 
apply (rule ccontr) 

562 
apply simp_all 

563 
done 

564 

565 
lemma Suc_le_D: "(Suc n <= m') ==> (? m. m' = Suc m)" 

566 
by (induct m') simp_all 

567 

568 
text {* Useful in certain inductive arguments *} 

569 
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0  (EX j. m = Suc j & j < n))" 

570 
by (case_tac m) simp_all 

571 

572 
lemma nat_induct2: "P 0 ==> P (Suc 0) ==> (!!k. P k ==> P (Suc (Suc k))) ==> P n" 

573 
apply (rule nat_less_induct) 

574 
apply (case_tac n) 

575 
apply (case_tac [2] nat) 

576 
apply (blast intro: less_trans)+ 

577 
done 

578 

579 
subsection {* @{text LEAST} theorems for type @{typ nat} by specialization *} 

580 

581 
lemmas LeastI = wellorder_LeastI 

582 
lemmas Least_le = wellorder_Least_le 

583 
lemmas not_less_Least = wellorder_not_less_Least 

584 

585 
lemma Least_Suc: "[ P n; ~ P 0 ] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))" 

586 
apply (case_tac "n") 

587 
apply auto 

588 
apply (frule LeastI) 

589 
apply (drule_tac P = "%x. P (Suc x) " in LeastI) 

590 
apply (subgoal_tac " (LEAST x. P x) <= Suc (LEAST x. P (Suc x))") 

591 
apply (erule_tac [2] Least_le) 

592 
apply (case_tac "LEAST x. P x") 

593 
apply auto 

594 
apply (drule_tac P = "%x. P (Suc x) " in Least_le) 

595 
apply (blast intro: order_antisym) 

596 
done 

597 

598 
lemma Least_Suc2: "[P n; Q m; ~P 0; !k. P (Suc k) = Q k] ==> Least P = Suc (Least Q)" 

599 
apply (erule (1) Least_Suc [THEN ssubst]) 

600 
apply simp 

601 
done 

602 

603 

604 
subsection {* @{term min} and @{term max} *} 

605 

606 
lemma min_0L [simp]: "min 0 n = (0::nat)" 

607 
by (rule min_leastL) simp 

608 

609 
lemma min_0R [simp]: "min n 0 = (0::nat)" 

610 
by (rule min_leastR) simp 

611 

612 
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)" 

613 
by (simp add: min_of_mono) 

614 

615 
lemma max_0L [simp]: "max 0 n = (n::nat)" 

616 
by (rule max_leastL) simp 

617 

618 
lemma max_0R [simp]: "max n 0 = (n::nat)" 

619 
by (rule max_leastR) simp 

620 

621 
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)" 

622 
by (simp add: max_of_mono) 

623 

624 

625 
subsection {* Basic rewrite rules for the arithmetic operators *} 

626 

627 
text {* Difference *} 

628 

629 
lemma diff_0_eq_0 [simp]: "0  n = (0::nat)" 

630 
by (induct_tac n) simp_all 

631 

632 
lemma diff_Suc_Suc [simp]: "Suc(m)  Suc(n) = m  n" 

633 
by (induct_tac n) simp_all 

634 

635 

636 
text {* 

637 
Could be (and is, below) generalized in various ways 

638 
However, none of the generalizations are currently in the simpset, 

639 
and I dread to think what happens if I put them in 

640 
*} 

641 
lemma Suc_pred [simp]: "0 < n ==> Suc (n  Suc 0) = n" 

642 
by (simp split add: nat.split) 

643 

644 
declare diff_Suc [simp del] 

645 

646 

647 
subsection {* Addition *} 

648 

649 
lemma add_0_right [simp]: "m + 0 = (m::nat)" 

650 
by (induct m) simp_all 

651 

652 
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)" 

653 
by (induct m) simp_all 

654 

655 

656 
text {* Associative law for addition *} 

657 
lemma add_assoc: "(m + n) + k = m + ((n + k)::nat)" 

658 
by (induct m) simp_all 

659 

660 
text {* Commutative law for addition *} 

661 
lemma add_commute: "m + n = n + (m::nat)" 

662 
by (induct m) simp_all 

663 

664 
lemma add_left_commute: "x + (y + z) = y + ((x + z)::nat)" 

665 
apply (rule mk_left_commute [of "op +"]) 

666 
apply (rule add_assoc) 

667 
apply (rule add_commute) 

668 
done 

669 

670 
text {* Addition is an ACoperator *} 

671 
lemmas add_ac = add_assoc add_commute add_left_commute 

672 

673 
lemma add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))" 

674 
by (induct k) simp_all 

675 

676 
lemma add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))" 

677 
by (induct k) simp_all 

678 

679 
lemma add_left_cancel_le [simp]: "(k + m <= k + n) = (m<=(n::nat))" 

680 
by (induct k) simp_all 

681 

682 
lemma add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))" 

683 
by (induct k) simp_all 

684 

685 
text {* Reasoning about @{text "m + 0 = 0"}, etc. *} 

686 

687 
lemma add_is_0 [iff]: "!!m::nat. (m + n = 0) = (m = 0 & n = 0)" 

688 
by (case_tac m) simp_all 

689 

690 
lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0  m=0 & n= Suc 0)" 

691 
by (case_tac m) simp_all 

692 

693 
lemma one_is_add: "(Suc 0 = m + n) = (m = Suc 0 & n = 0  m = 0 & n = Suc 0)" 

694 
by (rule trans, rule eq_commute, rule add_is_1) 

695 

696 
lemma add_gr_0 [iff]: "!!m::nat. (0 < m + n) = (0 < m  0 < n)" 

697 
by (simp del: neq0_conv add: neq0_conv [symmetric]) 

698 

699 
lemma add_eq_self_zero: "!!m::nat. m + n = m ==> n = 0" 

700 
apply (drule add_0_right [THEN ssubst]) 

701 
apply (simp add: add_assoc del: add_0_right) 

702 
done 

703 

704 
subsection {* Additional theorems about "less than" *} 

705 

706 
text {* Deleted @{text less_natE}; instead use @{text "less_imp_Suc_add RS exE"} *} 

707 
lemma less_imp_Suc_add: "m < n ==> (EX k. n = Suc (m + k))" 

708 
apply (induct n) 

709 
apply (simp_all add: order_le_less) 

710 
apply (blast elim!: less_SucE intro!: add_0_right [symmetric] add_Suc_right [symmetric]) 

711 
done 

712 

713 
lemma le_add2: "n <= ((m + n)::nat)" 

714 
apply (induct m) 

715 
apply simp_all 

716 
apply (erule le_SucI) 

717 
done 

718 

719 
lemma le_add1: "n <= ((n + m)::nat)" 

720 
apply (simp add: add_ac) 

721 
apply (rule le_add2) 

722 
done 

723 

724 
lemma less_add_Suc1: "i < Suc (i + m)" 

725 
by (rule le_less_trans, rule le_add1, rule lessI) 

726 

727 
lemma less_add_Suc2: "i < Suc (m + i)" 

728 
by (rule le_less_trans, rule le_add2, rule lessI) 

729 

730 
lemma less_iff_Suc_add: "(m < n) = (EX k. n = Suc (m + k))" 

731 
by (rules intro!: less_add_Suc1 less_imp_Suc_add) 

732 

733 

734 
lemma trans_le_add1: "(i::nat) <= j ==> i <= j + m" 

735 
by (rule le_trans, assumption, rule le_add1) 

736 

737 
lemma trans_le_add2: "(i::nat) <= j ==> i <= m + j" 

738 
by (rule le_trans, assumption, rule le_add2) 

739 

740 
lemma trans_less_add1: "(i::nat) < j ==> i < j + m" 

741 
by (rule less_le_trans, assumption, rule le_add1) 

742 

743 
lemma trans_less_add2: "(i::nat) < j ==> i < m + j" 

744 
by (rule less_le_trans, assumption, rule le_add2) 

745 

746 
lemma add_lessD1: "i + j < (k::nat) ==> i < k" 

747 
apply (induct j) 

748 
apply simp_all 

749 
apply (blast dest: Suc_lessD) 

750 
done 

751 

752 
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))" 

753 
apply (rule notI) 

754 
apply (erule add_lessD1 [THEN less_irrefl]) 

755 
done 

756 

757 
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))" 

758 
by (simp add: add_commute not_add_less1) 

759 

760 
lemma add_leD1: "m + k <= n ==> m <= (n::nat)" 

761 
by (induct k) (simp_all add: le_simps) 

762 

763 
lemma add_leD2: "m + k <= n ==> k <= (n::nat)" 

764 
apply (simp add: add_commute) 

765 
apply (erule add_leD1) 

766 
done 

767 

768 
lemma add_leE: "(m::nat) + k <= n ==> (m <= n ==> k <= n ==> R) ==> R" 

769 
by (blast dest: add_leD1 add_leD2) 

770 

771 
text {* needs @{text "!!k"} for @{text add_ac} to work *} 

772 
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n" 

773 
by (force simp del: add_Suc_right 

774 
simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac) 

775 

776 

777 
subsection {* Monotonicity of Addition *} 

778 

779 
text {* strict, in 1st argument *} 

780 
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)" 

781 
by (induct k) simp_all 

782 

783 
text {* strict, in both arguments *} 

784 
lemma add_less_mono: "[i < j; k < l] ==> i + k < j + (l::nat)" 

785 
apply (rule add_less_mono1 [THEN less_trans]) 

786 
apply assumption+ 

787 
apply (induct_tac j) 

788 
apply simp_all 

789 
done 

790 

791 
text {* A [clumsy] way of lifting @{text "<"} 

792 
monotonicity to @{text "<="} monotonicity *} 

793 
lemma less_mono_imp_le_mono: 

794 
assumes lt_mono: "!!i j::nat. i < j ==> f i < f j" 

795 
and le: "i <= j" shows "f i <= ((f j)::nat)" using le 

796 
apply (simp add: order_le_less) 

797 
apply (blast intro!: lt_mono) 

798 
done 

799 

800 
text {* nonstrict, in 1st argument *} 

801 
lemma add_le_mono1: "i <= j ==> i + k <= j + (k::nat)" 

802 
apply (rule_tac f = "%j. j + k" in less_mono_imp_le_mono) 

803 
apply (erule add_less_mono1) 

804 
apply assumption 

805 
done 

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806 

13449  807 
text {* nonstrict, in both arguments *} 
808 
lemma add_le_mono: "[ i <= j; k <= l ] ==> i + k <= j + (l::nat)" 

809 
apply (erule add_le_mono1 [THEN le_trans]) 

810 
apply (simp add: add_commute) 

811 
done 

812 

813 

814 
subsection {* Multiplication *} 

815 

816 
text {* right annihilation in product *} 

817 
lemma mult_0_right [simp]: "(m::nat) * 0 = 0" 

818 
by (induct m) simp_all 

819 

820 
text {* right successor law for multiplication *} 

821 
lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)" 

822 
by (induct m) (simp_all add: add_ac) 

823 

824 
lemma mult_1: "(1::nat) * n = n" by simp 

825 

826 
lemma mult_1_right: "n * (1::nat) = n" by simp 

827 

828 
text {* Commutative law for multiplication *} 

829 
lemma mult_commute: "m * n = n * (m::nat)" 

830 
by (induct m) simp_all 

831 

832 
text {* addition distributes over multiplication *} 

833 
lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)" 

834 
by (induct m) (simp_all add: add_ac) 

835 

836 
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)" 

837 
by (induct m) (simp_all add: add_ac) 

838 

839 
text {* Associative law for multiplication *} 

840 
lemma mult_assoc: "(m * n) * k = m * ((n * k)::nat)" 

841 
by (induct m) (simp_all add: add_mult_distrib) 

842 

843 
lemma mult_left_commute: "x * (y * z) = y * ((x * z)::nat)" 

844 
apply (rule mk_left_commute [of "op *"]) 

845 
apply (rule mult_assoc) 

846 
apply (rule mult_commute) 

847 
done 

848 

849 
lemmas mult_ac = mult_assoc mult_commute mult_left_commute 

850 

851 
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0  n=0)" 

852 
apply (induct_tac m) 

853 
apply (induct_tac [2] n) 

854 
apply simp_all 

855 
done 

856 

857 

858 
subsection {* Difference *} 

859 

860 
lemma diff_self_eq_0 [simp]: "(m::nat)  m = 0" 

861 
by (induct m) simp_all 

862 

863 
text {* Addition is the inverse of subtraction: 

864 
if @{term "n <= m"} then @{term "n + (m  n) = m"}. *} 

865 
lemma add_diff_inverse: "~ m < n ==> n + (m  n) = (m::nat)" 

866 
by (induct m n rule: diff_induct) simp_all 

867 

868 
lemma le_add_diff_inverse [simp]: "n <= m ==> n + (m  n) = (m::nat)" 

869 
by (simp add: add_diff_inverse not_less_iff_le) 

870 

871 
lemma le_add_diff_inverse2 [simp]: "n <= m ==> (m  n) + n = (m::nat)" 

872 
by (simp add: le_add_diff_inverse add_commute) 

873 

874 

875 
subsection {* More results about difference *} 

876 

877 
lemma Suc_diff_le: "n <= m ==> Suc m  n = Suc (m  n)" 

878 
by (induct m n rule: diff_induct) simp_all 

879 

880 
lemma diff_less_Suc: "m  n < Suc m" 

881 
apply (induct m n rule: diff_induct) 

882 
apply (erule_tac [3] less_SucE) 

883 
apply (simp_all add: less_Suc_eq) 

884 
done 

885 

886 
lemma diff_le_self [simp]: "m  n <= (m::nat)" 

887 
by (induct m n rule: diff_induct) (simp_all add: le_SucI) 

888 

889 
lemma less_imp_diff_less: "(j::nat) < k ==> j  n < k" 

890 
by (rule le_less_trans, rule diff_le_self) 

891 

892 
lemma diff_diff_left: "(i::nat)  j  k = i  (j + k)" 

893 
by (induct i j rule: diff_induct) simp_all 

894 

895 
lemma Suc_diff_diff [simp]: "(Suc m  n)  Suc k = m  n  k" 

896 
by (simp add: diff_diff_left) 

897 

898 
lemma diff_Suc_less [simp]: "0<n ==> n  Suc i < n" 

899 
apply (case_tac "n") 

900 
apply safe 

901 
apply (simp add: le_simps) 

902 
done 

903 

904 
text {* This and the next few suggested by Florian Kammueller *} 

905 
lemma diff_commute: "(i::nat)  j  k = i  k  j" 

906 
by (simp add: diff_diff_left add_commute) 

907 

908 
lemma diff_add_assoc: "k <= (j::nat) ==> (i + j)  k = i + (j  k)" 

909 
by (induct j k rule: diff_induct) simp_all 

910 

911 
lemma diff_add_assoc2: "k <= (j::nat) ==> (j + i)  k = (j  k) + i" 

912 
by (simp add: add_commute diff_add_assoc) 

913 

914 
lemma diff_add_inverse: "(n + m)  n = (m::nat)" 

915 
by (induct n) simp_all 

916 

917 
lemma diff_add_inverse2: "(m + n)  n = (m::nat)" 

918 
by (simp add: diff_add_assoc) 

919 

920 
lemma le_imp_diff_is_add: "i <= (j::nat) ==> (j  i = k) = (j = k + i)" 

921 
apply safe 

922 
apply (simp_all add: diff_add_inverse2) 

923 
done 

924 

925 
lemma diff_is_0_eq [simp]: "((m::nat)  n = 0) = (m <= n)" 

926 
by (induct m n rule: diff_induct) simp_all 

927 

928 
lemma diff_is_0_eq' [simp]: "m <= n ==> (m::nat)  n = 0" 

929 
by (rule iffD2, rule diff_is_0_eq) 

930 

931 
lemma zero_less_diff [simp]: "(0 < n  (m::nat)) = (m < n)" 

932 
by (induct m n rule: diff_induct) simp_all 

933 

934 
lemma less_imp_add_positive: "i < j ==> EX k::nat. 0 < k & i + k = j" 

935 
apply (rule_tac x = "j  i" in exI) 

936 
apply (simp (no_asm_simp) add: add_diff_inverse less_not_sym) 

937 
done 

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938 

13449  939 
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k  i)" 
940 
apply (induct k i rule: diff_induct) 

941 
apply (simp_all (no_asm)) 

942 
apply rules 

943 
done 

944 

945 
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0" 

946 
apply (rule diff_self_eq_0 [THEN subst]) 

947 
apply (rule zero_induct_lemma) 

948 
apply rules+ 

949 
done 

950 

951 
lemma diff_cancel: "(k + m)  (k + n) = m  (n::nat)" 

952 
by (induct k) simp_all 

953 

954 
lemma diff_cancel2: "(m + k)  (n + k) = m  (n::nat)" 

955 
by (simp add: diff_cancel add_commute) 

956 

957 
lemma diff_add_0: "n  (n + m) = (0::nat)" 

958 
by (induct n) simp_all 

959 

960 

961 
text {* Difference distributes over multiplication *} 

962 

963 
lemma diff_mult_distrib: "((m::nat)  n) * k = (m * k)  (n * k)" 

964 
by (induct m n rule: diff_induct) (simp_all add: diff_cancel) 

965 

966 
lemma diff_mult_distrib2: "k * ((m::nat)  n) = (k * m)  (k * n)" 

967 
by (simp add: diff_mult_distrib mult_commute [of k]) 

968 
 {* NOT added as rewrites, since sometimes they are used from righttoleft *} 

969 

970 
lemmas nat_distrib = 

971 
add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2 

972 

973 

974 
subsection {* Monotonicity of Multiplication *} 

975 

976 
lemma mult_le_mono1: "i <= (j::nat) ==> i * k <= j * k" 

977 
by (induct k) (simp_all add: add_le_mono) 

978 

979 
lemma mult_le_mono2: "i <= (j::nat) ==> k * i <= k * j" 

980 
apply (drule mult_le_mono1) 

981 
apply (simp add: mult_commute) 

982 
done 

983 

984 
text {* @{text "<="} monotonicity, BOTH arguments *} 

985 
lemma mult_le_mono: "i <= (j::nat) ==> k <= l ==> i * k <= j * l" 

986 
apply (erule mult_le_mono1 [THEN le_trans]) 

987 
apply (erule mult_le_mono2) 

988 
done 

989 

990 
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *} 

991 
lemma mult_less_mono2: "(i::nat) < j ==> 0 < k ==> k * i < k * j" 

992 
apply (erule_tac m1 = "0" in less_imp_Suc_add [THEN exE]) 

993 
apply simp 

994 
apply (induct_tac x) 

995 
apply (simp_all add: add_less_mono) 

996 
done 

997 

998 
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k" 

999 
by (drule mult_less_mono2) (simp_all add: mult_commute) 

1000 

1001 
lemma zero_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)" 

1002 
apply (induct m) 

1003 
apply (case_tac [2] n) 

1004 
apply simp_all 

1005 
done 

1006 

1007 
lemma one_le_mult_iff [simp]: "(Suc 0 <= m * n) = (1 <= m & 1 <= n)" 

1008 
apply (induct m) 

1009 
apply (case_tac [2] n) 

1010 
apply simp_all 

1011 
done 

1012 

1013 
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)" 

1014 
apply (induct_tac m) 

1015 
apply simp 

1016 
apply (induct_tac n) 

1017 
apply simp 

1018 
apply fastsimp 

1019 
done 

1020 

1021 
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = 1 & n = 1)" 

1022 
apply (rule trans) 

1023 
apply (rule_tac [2] mult_eq_1_iff) 

1024 
apply fastsimp 

1025 
done 

1026 

1027 
lemma mult_less_cancel2: "((m::nat) * k < n * k) = (0 < k & m < n)" 

1028 
apply (safe intro!: mult_less_mono1) 

1029 
apply (case_tac k) 

1030 
apply auto 

1031 
apply (simp del: le_0_eq add: linorder_not_le [symmetric]) 

1032 
apply (blast intro: mult_le_mono1) 

1033 
done 

1034 

1035 
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)" 

1036 
by (simp add: mult_less_cancel2 mult_commute [of k]) 

1037 

1038 
declare mult_less_cancel2 [simp] 

1039 

1040 
lemma mult_le_cancel1 [simp]: "(k * (m::nat) <= k * n) = (0 < k > m <= n)" 

1041 
apply (simp add: linorder_not_less [symmetric]) 

1042 
apply auto 

1043 
done 

1044 

1045 
lemma mult_le_cancel2 [simp]: "((m::nat) * k <= n * k) = (0 < k > m <= n)" 

1046 
apply (simp add: linorder_not_less [symmetric]) 

1047 
apply auto 

1048 
done 

1049 

1050 
lemma mult_cancel2: "(m * k = n * k) = (m = n  (k = (0::nat)))" 

1051 
apply (cut_tac less_linear) 

1052 
apply safe 

1053 
apply auto 

1054 
apply (drule mult_less_mono1, assumption, simp)+ 

1055 
done 

1056 

1057 
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n  (k = (0::nat)))" 

1058 
by (simp add: mult_cancel2 mult_commute [of k]) 

1059 

1060 
declare mult_cancel2 [simp] 

1061 

1062 
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)" 

1063 
by (subst mult_less_cancel1) simp 

1064 

1065 
lemma Suc_mult_le_cancel1: "(Suc k * m <= Suc k * n) = (m <= n)" 

1066 
by (subst mult_le_cancel1) simp 

1067 

1068 
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)" 

1069 
by (subst mult_cancel1) simp 

1070 

1071 

1072 
text {* Lemma for @{text gcd} *} 

1073 
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1  m = 0" 

1074 
apply (drule sym) 

1075 
apply (rule disjCI) 

1076 
apply (rule nat_less_cases, erule_tac [2] _) 

1077 
apply (fastsimp elim!: less_SucE) 

1078 
apply (fastsimp dest: mult_less_mono2) 

1079 
done 

9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
7702
diff
changeset

1080 

923  1081 
end 