author  paulson 
Tue, 25 Nov 2003 10:37:03 +0100  
changeset 14267  b963e9cee2a0 
parent 14266  08b34c902618 
child 14302  6c24235e8d5d 
permissions  rwrr 
923  1 
(* 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 + Ring_and_Field: 
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subsection {* Type @{text ind} *} 

14 

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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 \<noteq> Zero_Rep" 
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26 

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subsection {* Type nat *} 

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text {* Type definition *} 

30 

31 
consts 

32 
Nat :: "ind set" 

33 

34 
inductive Nat 

35 
intros 

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Zero_RepI: "Zero_Rep : Nat" 

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

38 

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global 

40 

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typedef (open Nat) 

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nat = Nat by (rule exI, rule Nat.Zero_RepI) 
13449  43 

44 
instance nat :: ord .. 

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instance nat :: zero .. 

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instance nat :: one .. 

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48 

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text {* Abstract constants and syntax *} 

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consts 

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Suc :: "nat => nat" 

53 
pred_nat :: "(nat * nat) set" 

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local 

56 

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defs 

58 
Zero_nat_def: "0 == Abs_Nat Zero_Rep" 

59 
Suc_def: "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))" 

60 
One_nat_def [simp]: "1 == Suc 0" 

61 

62 
 {* nat operations *} 

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

64 

65 
less_def: "m < n == (m, n) : trancl pred_nat" 

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le_def: "m \<le> (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) 

74 
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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79 

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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_on_inverseI) 
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apply (rule Rep_Nat_inverse) 
85 
done 

86 

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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) 

90 
done 

91 

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text {* Distinctness of constructors *} 

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lemma Suc_not_Zero [iff]: "Suc m \<noteq> 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) 

98 
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 \<noteq> 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" 

105 
by (rule notE, rule Suc_not_Zero) 

106 

107 
lemma Zero_neq_Suc: "0 = Suc m ==> R" 

108 
by (rule Suc_neq_Zero, erule sym) 

109 

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

111 

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lemma inj_Suc: "inj Suc" 

113 
apply (unfold Suc_def) 

13585  114 
apply (rule inj_onI) 
13449  115 
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]) 

119 
done 

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

122 
by (rule inj_Suc [THEN injD]) 

123 

124 
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) 

128 
done 

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lemma nat_not_singleton: "(\<forall>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 

138 
induction nat_induct 

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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 t \<noteq> t" 
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by (rule not_sym, rule n_not_Suc_n) 
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146 
text {* A special form of induction for reasoning 

147 
about @{term "m < n"} and @{term "m  n"} *} 

148 

149 
theorem diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==> 

150 
(!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n" 

14208  151 
apply (rule_tac x = m in spec) 
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apply (induct_tac n) 
153 
prefer 2 

154 
apply (rule allI) 

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apply (induct_tac x, rules+) 
13449  156 
done 
157 

158 
subsection {* Basic properties of "less than" *} 

159 

160 
lemma wf_pred_nat: "wf pred_nat" 

14208  161 
apply (unfold wf_def pred_nat_def, clarify) 
162 
apply (induct_tac x, blast+) 

13449  163 
done 
164 

165 
lemma wf_less: "wf {(x, y::nat). x < y}" 

166 
apply (unfold less_def) 

14208  167 
apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_subset], blast) 
13449  168 
done 
169 

170 
lemma less_eq: "((m, n) : pred_nat^+) = (m < n)" 

171 
apply (unfold less_def) 

172 
apply (rule refl) 

173 
done 

174 

175 
subsubsection {* Introduction properties *} 

176 

177 
lemma less_trans: "i < j ==> j < k ==> i < (k::nat)" 

178 
apply (unfold less_def) 

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apply (rule trans_trancl [THEN transD], assumption+) 
13449  180 
done 
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182 
lemma lessI [iff]: "n < Suc n" 

183 
apply (unfold less_def pred_nat_def) 

184 
apply (simp add: r_into_trancl) 

185 
done 

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

14208  188 
apply (rule less_trans, assumption) 
13449  189 
apply (rule lessI) 
190 
done 

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

193 
apply (induct n) 

194 
apply (rule lessI) 

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apply (erule less_trans) 

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apply (rule lessI) 

197 
done 

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subsubsection {* Elimination properties *} 

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

202 
apply (unfold less_def) 

203 
apply (blast intro: wf_pred_nat wf_trancl [THEN wf_asym]) 

204 
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]) 

217 
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 \<noteq> (n::nat)" by blast 
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lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> 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], 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, 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, blast) 
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apply (rule less, 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) \<noteq> 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 \<noteq> 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]) 

300 
apply (erule lessI [THEN minor]) 

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apply (erule Suc_lessD [THEN minor], assumption) 
13449  302 
done 
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lemma Suc_less_SucD: "Suc m < Suc n ==> m < n" 

305 
by (blast elim: lessE dest: Suc_lessD) 

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13449  307 
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) 

311 
done 

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lemma less_trans_Suc: 

314 
assumes le: "i < j" shows "j < k ==> Suc i < k" 

14208  315 
apply (induct k, simp_all) 
13449  316 
apply (insert le) 
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apply (simp add: less_Suc_eq) 

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apply (blast dest: Suc_lessD) 

319 
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)" 

14208  323 
by (rule_tac m = m and n = n in diff_induct, simp_all) 
13449  324 

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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. \<forall>m::nat. m < n > P m ==> P n" shows "P n" 
13449  328 
apply (rule_tac a=n in wf_induct) 
329 
apply (rule wf_pred_nat [THEN wf_trancl]) 

330 
apply (rule prem) 

14208  331 
apply (unfold less_def, assumption) 
13449  332 
done 
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14131  334 
lemmas less_induct = nat_less_induct [rule_format, case_names less] 
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subsection {* Properties of "less than or equal" *} 

13449  337 

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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 \<le> n)" 
13449  340 
by (unfold le_def, rule not_less_eq [symmetric]) 
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lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n" 
13449  343 
by (rule less_Suc_eq_le [THEN iffD2]) 
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lemma le0 [iff]: "(0::nat) \<le> n" 
13449  346 
by (unfold le_def, rule not_less0) 
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lemma Suc_n_not_le_n: "~ Suc n \<le> n" 
13449  349 
by (simp add: le_def) 
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lemma le_0_eq [iff]: "((i::nat) \<le> 0) = (i = 0)" 
13449  352 
by (induct i) (simp_all add: le_def) 
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lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n  m = Suc n)" 
13449  355 
by (simp del: less_Suc_eq_le add: less_Suc_eq_le [symmetric] less_Suc_eq) 
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lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R" 
13449  358 
by (drule le_Suc_eq [THEN iffD1], rules+) 
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lemma leI: "~ n < m ==> m \<le> (n::nat)" by (simp add: le_def) 
13449  361 

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lemma leD: "m \<le> n ==> ~ n < (m::nat)" 
13449  363 
by (simp add: le_def) 
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365 
lemmas leE = leD [elim_format] 

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lemma not_less_iff_le: "(~ n < m) = (m \<le> (n::nat))" 
13449  368 
by (blast intro: leI elim: leE) 
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lemma not_leE: "~ m \<le> n ==> n<(m::nat)" 
13449  371 
by (simp add: le_def) 
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lemma not_le_iff_less: "(~ n \<le> m) = (m < (n::nat))" 
13449  374 
by (simp add: le_def) 
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lemma Suc_leI: "m < n ==> Suc(m) \<le> n" 
13449  377 
apply (simp add: le_def less_Suc_eq) 
378 
apply (blast elim!: less_irrefl less_asym) 

379 
done  {* formerly called lessD *} 

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lemma Suc_leD: "Suc(m) \<le> n ==> m \<le> n" 
13449  382 
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 \<le> n ==> m < n" 
13449  386 
apply (simp add: le_def less_Suc_eq) 
387 
using less_linear 

388 
apply blast 

389 
done 

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lemma Suc_le_eq: "(Suc m \<le> n) = (m < n)" 
13449  392 
by (blast intro: Suc_leI Suc_le_lessD) 
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lemma le_SucI: "m \<le> n ==> m \<le> Suc n" 
13449  395 
by (unfold le_def) (blast dest: Suc_lessD) 
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lemma less_imp_le: "m < n ==> m \<le> (n::nat)" 
13449  398 
by (unfold le_def) (blast elim: less_asym) 
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text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *} 
13449  401 
lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq 
402 

403 

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404 
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n  m = n"} *} 
13449  405 

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406 
lemma le_imp_less_or_eq: "m \<le> n ==> m < n  m = (n::nat)" 
13449  407 
apply (unfold le_def) 
408 
using less_linear 

409 
apply (blast elim: less_irrefl less_asym) 

410 
done 

411 

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412 
lemma less_or_eq_imp_le: "m < n  m = n ==> m \<le> (n::nat)" 
13449  413 
apply (unfold le_def) 
414 
using less_linear 

415 
apply (blast elim!: less_irrefl elim: less_asym) 

416 
done 

417 

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418 
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n  m=n)" 
13449  419 
by (rules intro: less_or_eq_imp_le le_imp_less_or_eq) 
420 

421 
text {* Useful with @{text Blast}. *} 

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422 
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n" 
13449  423 
by (rule less_or_eq_imp_le, rule disjI2) 
424 

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425 
lemma le_refl: "n \<le> (n::nat)" 
13449  426 
by (simp add: le_eq_less_or_eq) 
427 

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428 
lemma le_less_trans: "[ i \<le> j; j < k ] ==> i < (k::nat)" 
13449  429 
by (blast dest!: le_imp_less_or_eq intro: less_trans) 
430 

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431 
lemma less_le_trans: "[ i < j; j \<le> k ] ==> i < (k::nat)" 
13449  432 
by (blast dest!: le_imp_less_or_eq intro: less_trans) 
433 

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434 
lemma le_trans: "[ i \<le> j; j \<le> k ] ==> i \<le> (k::nat)" 
13449  435 
by (blast dest!: le_imp_less_or_eq intro: less_or_eq_imp_le less_trans) 
436 

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437 
lemma le_anti_sym: "[ m \<le> n; n \<le> m ] ==> m = (n::nat)" 
13449  438 
by (blast dest!: le_imp_less_or_eq elim!: less_irrefl elim: less_asym) 
439 

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440 
lemma Suc_le_mono [iff]: "(Suc n \<le> Suc m) = (n \<le> m)" 
13449  441 
by (simp add: le_simps) 
442 

443 
text {* Axiom @{text order_less_le} of class @{text order}: *} 

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444 
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)" 
13449  445 
by (simp add: le_def nat_neq_iff) (blast elim!: less_asym) 
446 

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447 
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n" 
13449  448 
by (rule iffD2, rule nat_less_le, rule conjI) 
449 

450 
text {* Axiom @{text linorder_linear} of class @{text linorder}: *} 

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451 
lemma nat_le_linear: "(m::nat) \<le> n  n \<le> m" 
13449  452 
apply (simp add: le_eq_less_or_eq) 
453 
using less_linear 

454 
apply blast 

455 
done 

456 

457 
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)" 

458 
by (blast elim!: less_SucE) 

459 

460 

461 
text {* 

462 
Rewrite @{term "n < Suc m"} to @{term "n = m"} 

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463 
if @{term "~ n < m"} or @{term "m \<le> n"} hold. 
13449  464 
Not suitable as default simprules because they often lead to looping 
465 
*} 

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466 
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)" 
13449  467 
by (rule not_less_less_Suc_eq, rule leD) 
468 

469 
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq 

470 

471 

472 
text {* 

473 
Reorientation of the equations @{text "0 = x"} and @{text "1 = x"}. 

474 
No longer added as simprules (they loop) 

475 
but via @{text reorient_simproc} in Bin 

476 
*} 

477 

478 
text {* Polymorphic, not just for @{typ nat} *} 

479 
lemma zero_reorient: "(0 = x) = (x = 0)" 

480 
by auto 

481 

482 
lemma one_reorient: "(1 = x) = (x = 1)" 

483 
by auto 

484 

485 
text {* Type {@typ nat} is a wellfounded linear order *} 

486 

487 
instance nat :: order by (intro_classes, 

488 
(assumption  rule le_refl le_trans le_anti_sym nat_less_le)+) 

489 
instance nat :: linorder by (intro_classes, rule nat_le_linear) 

490 
instance nat :: wellorder by (intro_classes, rule wf_less) 

491 

492 
subsection {* Arithmetic operators *} 

1660  493 

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494 
axclass power < type 
10435  495 

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496 
consts 
13449  497 
power :: "('a::power) => nat => 'a" (infixr "^" 80) 
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498 

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499 

13449  500 
text {* arithmetic operators @{text "+ "} and @{text "*"} *} 
501 

502 
instance nat :: plus .. 

503 
instance nat :: minus .. 

504 
instance nat :: times .. 

505 
instance nat :: power .. 

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506 

13449  507 
text {* size of a datatype value; overloaded *} 
508 
consts size :: "'a => nat" 

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509 

13449  510 
primrec 
511 
add_0: "0 + n = n" 

512 
add_Suc: "Suc m + n = Suc (m + n)" 

513 

514 
primrec 

515 
diff_0: "m  0 = m" 

516 
diff_Suc: "m  Suc n = (case m  n of 0 => 0  Suc k => k)" 

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517 

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518 
primrec 
13449  519 
mult_0: "0 * n = 0" 
520 
mult_Suc: "Suc m * n = n + (m * n)" 

521 

522 
text {* These 2 rules ease the use of primitive recursion. NOTE USE OF @{text "=="} *} 

523 
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c" 

524 
by simp 

525 

526 
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)" 

527 
by simp 

528 

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529 
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m" 
13449  530 
by (case_tac n) simp_all 
531 

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532 
lemma gr_implies_not0: "!!n::nat. m<n ==> n \<noteq> 0" 
13449  533 
by (case_tac n) simp_all 
534 

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535 
lemma neq0_conv [iff]: "!!n::nat. (n \<noteq> 0) = (0 < n)" 
13449  536 
by (case_tac n) simp_all 
537 

538 
text {* This theorem is useful with @{text blast} *} 

539 
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n" 

540 
by (rule iffD1, rule neq0_conv, rules) 

541 

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542 
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)" 
13449  543 
by (fast intro: not0_implies_Suc) 
544 

545 
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)" 

546 
apply (rule iffI) 

14208  547 
apply (rule ccontr, simp_all) 
13449  548 
done 
549 

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550 
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)" 
13449  551 
by (induct m') simp_all 
552 

553 
text {* Useful in certain inductive arguments *} 

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554 
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0  (\<exists>j. m = Suc j & j < n))" 
13449  555 
by (case_tac m) simp_all 
556 

557 
lemma nat_induct2: "P 0 ==> P (Suc 0) ==> (!!k. P k ==> P (Suc (Suc k))) ==> P n" 

558 
apply (rule nat_less_induct) 

559 
apply (case_tac n) 

560 
apply (case_tac [2] nat) 

561 
apply (blast intro: less_trans)+ 

562 
done 

563 

564 
subsection {* @{text LEAST} theorems for type @{typ nat} by specialization *} 

565 

566 
lemmas LeastI = wellorder_LeastI 

567 
lemmas Least_le = wellorder_Least_le 

568 
lemmas not_less_Least = wellorder_not_less_Least 

569 

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570 
lemma Least_Suc: 
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571 
"[ P n; ~ P 0 ] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))" 
14208  572 
apply (case_tac "n", auto) 
13449  573 
apply (frule LeastI) 
574 
apply (drule_tac P = "%x. P (Suc x) " in LeastI) 

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575 
apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))") 
13449  576 
apply (erule_tac [2] Least_le) 
14208  577 
apply (case_tac "LEAST x. P x", auto) 
13449  578 
apply (drule_tac P = "%x. P (Suc x) " in Least_le) 
579 
apply (blast intro: order_antisym) 

580 
done 

581 

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582 
lemma Least_Suc2: 
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583 
"[P n; Q m; ~P 0; !k. P (Suc k) = Q k] ==> Least P = Suc (Least Q)" 
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584 
by (erule (1) Least_Suc [THEN ssubst], simp) 
13449  585 

586 

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587 

13449  588 
subsection {* @{term min} and @{term max} *} 
589 

590 
lemma min_0L [simp]: "min 0 n = (0::nat)" 

591 
by (rule min_leastL) simp 

592 

593 
lemma min_0R [simp]: "min n 0 = (0::nat)" 

594 
by (rule min_leastR) simp 

595 

596 
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)" 

597 
by (simp add: min_of_mono) 

598 

599 
lemma max_0L [simp]: "max 0 n = (n::nat)" 

600 
by (rule max_leastL) simp 

601 

602 
lemma max_0R [simp]: "max n 0 = (n::nat)" 

603 
by (rule max_leastR) simp 

604 

605 
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)" 

606 
by (simp add: max_of_mono) 

607 

608 

609 
subsection {* Basic rewrite rules for the arithmetic operators *} 

610 

611 
text {* Difference *} 

612 

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613 
lemma diff_0_eq_0 [simp, code]: "0  n = (0::nat)" 
13449  614 
by (induct_tac n) simp_all 
615 

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616 
lemma diff_Suc_Suc [simp, code]: "Suc(m)  Suc(n) = m  n" 
13449  617 
by (induct_tac n) simp_all 
618 

619 

620 
text {* 

621 
Could be (and is, below) generalized in various ways 

622 
However, none of the generalizations are currently in the simpset, 

623 
and I dread to think what happens if I put them in 

624 
*} 

625 
lemma Suc_pred [simp]: "0 < n ==> Suc (n  Suc 0) = n" 

626 
by (simp split add: nat.split) 

627 

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628 
declare diff_Suc [simp del, code del] 
13449  629 

630 

631 
subsection {* Addition *} 

632 

633 
lemma add_0_right [simp]: "m + 0 = (m::nat)" 

634 
by (induct m) simp_all 

635 

636 
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)" 

637 
by (induct m) simp_all 

638 

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639 
lemma [code]: "Suc m + n = m + Suc n" by simp 
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640 

13449  641 

642 
text {* Associative law for addition *} 

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643 
lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)" 
13449  644 
by (induct m) simp_all 
645 

646 
text {* Commutative law for addition *} 

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647 
lemma nat_add_commute: "m + n = n + (m::nat)" 
13449  648 
by (induct m) simp_all 
649 

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650 
lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)" 
13449  651 
apply (rule mk_left_commute [of "op +"]) 
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652 
apply (rule nat_add_assoc) 
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653 
apply (rule nat_add_commute) 
13449  654 
done 
655 

656 
lemma add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))" 

657 
by (induct k) simp_all 

658 

659 
lemma add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))" 

660 
by (induct k) simp_all 

661 

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662 
lemma add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))" 
13449  663 
by (induct k) simp_all 
664 

665 
lemma add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))" 

666 
by (induct k) simp_all 

667 

668 
text {* Reasoning about @{text "m + 0 = 0"}, etc. *} 

669 

670 
lemma add_is_0 [iff]: "!!m::nat. (m + n = 0) = (m = 0 & n = 0)" 

671 
by (case_tac m) simp_all 

672 

673 
lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0  m=0 & n= Suc 0)" 

674 
by (case_tac m) simp_all 

675 

676 
lemma one_is_add: "(Suc 0 = m + n) = (m = Suc 0 & n = 0  m = 0 & n = Suc 0)" 

677 
by (rule trans, rule eq_commute, rule add_is_1) 

678 

679 
lemma add_gr_0 [iff]: "!!m::nat. (0 < m + n) = (0 < m  0 < n)" 

680 
by (simp del: neq0_conv add: neq0_conv [symmetric]) 

681 

682 
lemma add_eq_self_zero: "!!m::nat. m + n = m ==> n = 0" 

683 
apply (drule add_0_right [THEN ssubst]) 

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684 
apply (simp add: nat_add_assoc del: add_0_right) 
13449  685 
done 
686 

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687 
subsection {* Monotonicity of Addition *} 
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688 

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689 
text {* strict, in 1st argument *} 
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690 
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)" 
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691 
by (induct k) simp_all 
13449  692 

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693 
text {* strict, in both arguments *} 
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694 
lemma add_less_mono: "[i < j; k < l] ==> i + k < j + (l::nat)" 
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695 
apply (rule add_less_mono1 [THEN less_trans], assumption+) 
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696 
apply (induct_tac j, simp_all) 
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697 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

698 

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

699 
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

700 
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))" 
13449  701 
apply (induct n) 
702 
apply (simp_all add: order_le_less) 

703 
apply (blast elim!: less_SucE intro!: add_0_right [symmetric] add_Suc_right [symmetric]) 

704 
done 

705 

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

706 

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

707 
subsection {* Multiplication *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

708 

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

709 
text {* right annihilation in product *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

710 
lemma mult_0_right [simp]: "(m::nat) * 0 = 0" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

711 
by (induct m) simp_all 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

712 

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

713 
text {* right successor law for multiplication *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

714 
lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

715 
by (induct m) (simp_all add: nat_add_left_commute) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

716 

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

717 
text {* Commutative law for multiplication *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

718 
lemma nat_mult_commute: "m * n = n * (m::nat)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

719 
by (induct m) simp_all 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

720 

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

721 
text {* addition distributes over multiplication *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

722 
lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

723 
by (induct m) (simp_all add: nat_add_assoc nat_add_left_commute) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

724 

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

725 
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

726 
by (induct m) (simp_all add: nat_add_assoc) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

727 

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

728 
text {* Associative law for multiplication *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

729 
lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

730 
by (induct m) (simp_all add: add_mult_distrib) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

731 

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

732 
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0  n=0)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

733 
apply (induct_tac m) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

734 
apply (induct_tac [2] n, simp_all) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

735 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

736 

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

737 
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

738 
lemma mult_less_mono2: "(i::nat) < j ==> 0 < k ==> k * i < k * j" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

739 
apply (erule_tac m1 = 0 in less_imp_Suc_add [THEN exE], simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

740 
apply (induct_tac x) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

741 
apply (simp_all add: add_less_mono) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

742 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

743 

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

744 
text{*The Naturals Form an Ordered Semiring*} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

745 
instance nat :: ordered_semiring 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

746 
proof 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

747 
fix i j k :: nat 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

748 
show "(i + j) + k = i + (j + k)" by (rule nat_add_assoc) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

749 
show "i + j = j + i" by (rule nat_add_commute) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

750 
show "0 + i = i" by simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

751 
show "(i * j) * k = i * (j * k)" by (rule nat_mult_assoc) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

752 
show "i * j = j * i" by (rule nat_mult_commute) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

753 
show "1 * i = i" by simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

754 
show "(i + j) * k = i * k + j * k" by (simp add: add_mult_distrib) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

755 
show "0 \<noteq> (1::nat)" by simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

756 
show "i \<le> j ==> k + i \<le> k + j" by simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

757 
show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

758 
qed 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

759 

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

760 
lemma nat_mult_1: "(1::nat) * n = n" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

761 
by simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

762 

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

763 
lemma nat_mult_1_right: "n * (1::nat) = n" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

764 
by simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

765 

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

766 

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

767 
subsection {* Additional theorems about "less than" *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

768 

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

769 
text {* A [clumsy] way of lifting @{text "<"} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

770 
monotonicity to @{text "\<le>"} monotonicity *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

771 
lemma less_mono_imp_le_mono: 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

772 
assumes lt_mono: "!!i j::nat. i < j ==> f i < f j" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

773 
and le: "i \<le> j" shows "f i \<le> ((f j)::nat)" using le 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

774 
apply (simp add: order_le_less) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

775 
apply (blast intro!: lt_mono) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

776 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

777 

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

778 
text {* nonstrict, in 1st argument *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

779 
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

780 
by (rule add_right_mono) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

781 

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

782 
text {* nonstrict, in both arguments *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

783 
lemma add_le_mono: "[ i \<le> j; k \<le> l ] ==> i + k \<le> j + (l::nat)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

784 
by (rule add_mono) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

785 

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

786 
lemma le_add2: "n \<le> ((m + n)::nat)" 
14208  787 
apply (induct m, simp_all) 
13449  788 
apply (erule le_SucI) 
789 
done 

790 

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

791 
lemma le_add1: "n \<le> ((n + m)::nat)" 
13449  792 
apply (simp add: add_ac) 
793 
apply (rule le_add2) 

794 
done 

795 

796 
lemma less_add_Suc1: "i < Suc (i + m)" 

797 
by (rule le_less_trans, rule le_add1, rule lessI) 

798 

799 
lemma less_add_Suc2: "i < Suc (m + i)" 

800 
by (rule le_less_trans, rule le_add2, rule lessI) 

801 

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

802 
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))" 
13449  803 
by (rules intro!: less_add_Suc1 less_imp_Suc_add) 
804 

805 

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

806 
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m" 
13449  807 
by (rule le_trans, assumption, rule le_add1) 
808 

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

809 
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j" 
13449  810 
by (rule le_trans, assumption, rule le_add2) 
811 

812 
lemma trans_less_add1: "(i::nat) < j ==> i < j + m" 

813 
by (rule less_le_trans, assumption, rule le_add1) 

814 

815 
lemma trans_less_add2: "(i::nat) < j ==> i < m + j" 

816 
by (rule less_le_trans, assumption, rule le_add2) 

817 

818 
lemma add_lessD1: "i + j < (k::nat) ==> i < k" 

14208  819 
apply (induct j, simp_all) 
13449  820 
apply (blast dest: Suc_lessD) 
821 
done 

822 

823 
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))" 

824 
apply (rule notI) 

825 
apply (erule add_lessD1 [THEN less_irrefl]) 

826 
done 

827 

828 
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))" 

829 
by (simp add: add_commute not_add_less1) 

830 

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

831 
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)" 
13449  832 
by (induct k) (simp_all add: le_simps) 
833 

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

834 
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)" 
13449  835 
apply (simp add: add_commute) 
836 
apply (erule add_leD1) 

837 
done 

838 

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

839 
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R" 
13449  840 
by (blast dest: add_leD1 add_leD2) 
841 

842 
text {* needs @{text "!!k"} for @{text add_ac} to work *} 

843 
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n" 

844 
by (force simp del: add_Suc_right 

845 
simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac) 

846 

847 

848 

849 
subsection {* Difference *} 

850 

851 
lemma diff_self_eq_0 [simp]: "(m::nat)  m = 0" 

852 
by (induct m) simp_all 

853 

854 
text {* Addition is the inverse of subtraction: 

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

855 
if @{term "n \<le> m"} then @{term "n + (m  n) = m"}. *} 
13449  856 
lemma add_diff_inverse: "~ m < n ==> n + (m  n) = (m::nat)" 
857 
by (induct m n rule: diff_induct) simp_all 

858 

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

859 
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m  n) = (m::nat)" 
13449  860 
by (simp add: add_diff_inverse not_less_iff_le) 
861 

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

862 
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m  n) + n = (m::nat)" 
13449  863 
by (simp add: le_add_diff_inverse add_commute) 
864 

865 

866 
subsection {* More results about difference *} 

867 

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

868 
lemma Suc_diff_le: "n \<le> m ==> Suc m  n = Suc (m  n)" 
13449  869 
by (induct m n rule: diff_induct) simp_all 
870 

871 
lemma diff_less_Suc: "m  n < Suc m" 

872 
apply (induct m n rule: diff_induct) 

873 
apply (erule_tac [3] less_SucE) 

874 
apply (simp_all add: less_Suc_eq) 

875 
done 

876 

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877 
lemma diff_le_self [simp]: "m  n \<le> (m::nat)" 
13449  878 
by (induct m n rule: diff_induct) (simp_all add: le_SucI) 
879 

880 
lemma less_imp_diff_less: "(j::nat) < k ==> j  n < k" 

881 
by (rule le_less_trans, rule diff_le_self) 

882 

883 
lemma diff_diff_left: "(i::nat)  j  k = i  (j + k)" 

884 
by (induct i j rule: diff_induct) simp_all 

885 

886 
lemma Suc_diff_diff [simp]: "(Suc m  n)  Suc k = m  n  k" 

887 
by (simp add: diff_diff_left) 

888 

889 
lemma diff_Suc_less [simp]: "0<n ==> n  Suc i < n" 

14208  890 
apply (case_tac "n", safe) 
13449  891 
apply (simp add: le_simps) 
892 
done 

893 

894 
text {* This and the next few suggested by Florian Kammueller *} 

895 
lemma diff_commute: "(i::nat)  j  k = i  k  j" 

896 
by (simp add: diff_diff_left add_commute) 

897 

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898 
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j)  k = i + (j  k)" 
13449  899 
by (induct j k rule: diff_induct) simp_all 
900 

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901 
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i)  k = (j  k) + i" 
13449  902 
by (simp add: add_commute diff_add_assoc) 
903 

904 
lemma diff_add_inverse: "(n + m)  n = (m::nat)" 

905 
by (induct n) simp_all 

906 

907 
lemma diff_add_inverse2: "(m + n)  n = (m::nat)" 

908 
by (simp add: diff_add_assoc) 

909 

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910 
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j  i = k) = (j = k + i)" 
13449  911 
apply safe 
912 
apply (simp_all add: diff_add_inverse2) 

913 
done 

914 

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915 
lemma diff_is_0_eq [simp]: "((m::nat)  n = 0) = (m \<le> n)" 
13449  916 
by (induct m n rule: diff_induct) simp_all 
917 

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918 
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat)  n = 0" 
13449  919 
by (rule iffD2, rule diff_is_0_eq) 
920 

921 
lemma zero_less_diff [simp]: "(0 < n  (m::nat)) = (m < n)" 

922 
by (induct m n rule: diff_induct) simp_all 

923 

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924 
lemma less_imp_add_positive: "i < j ==> \<exists>k::nat. 0 < k & i + k = j" 
13449  925 
apply (rule_tac x = "j  i" in exI) 
926 
apply (simp (no_asm_simp) add: add_diff_inverse less_not_sym) 

927 
done 

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928 

13449  929 
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k  i)" 
930 
apply (induct k i rule: diff_induct) 

931 
apply (simp_all (no_asm)) 

932 
apply rules 

933 
done 

934 

935 
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0" 

936 
apply (rule diff_self_eq_0 [THEN subst]) 

14208  937 
apply (rule zero_induct_lemma, rules+) 
13449  938 
done 
939 

940 
lemma diff_cancel: "(k + m)  (k + n) = m  (n::nat)" 

941 
by (induct k) simp_all 

942 

943 
lemma diff_cancel2: "(m + k)  (n + k) = m  (n::nat)" 

944 
by (simp add: diff_cancel add_commute) 

945 

946 
lemma diff_add_0: "n  (n + m) = (0::nat)" 

947 
by (induct n) simp_all 

948 

949 

950 
text {* Difference distributes over multiplication *} 

951 

952 
lemma diff_mult_distrib: "((m::nat)  n) * k = (m * k)  (n * k)" 

953 
by (induct m n rule: diff_induct) (simp_all add: diff_cancel) 

954 

955 
lemma diff_mult_distrib2: "k * ((m::nat)  n) = (k * m)  (k * n)" 

956 
by (simp add: diff_mult_distrib mult_commute [of k]) 

957 
 {* NOT added as rewrites, since sometimes they are used from righttoleft *} 

958 

959 
lemmas nat_distrib = 

960 
add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2 

961 

962 

963 
subsection {* Monotonicity of Multiplication *} 

964 

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965 
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k" 
13449  966 
by (induct k) (simp_all add: add_le_mono) 
967 

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968 
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j" 
13449  969 
apply (drule mult_le_mono1) 
970 
apply (simp add: mult_commute) 

971 
done 

972 

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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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973 
text {* @{text "\<le>"} monotonicity, BOTH arguments *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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974 
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l" 
13449  975 
apply (erule mult_le_mono1 [THEN le_trans]) 
976 
apply (erule mult_le_mono2) 

977 
done 

978 

979 
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k" 

980 
by (drule mult_less_mono2) (simp_all add: mult_commute) 

981 

14266  982 
text{*Differs from the standard @{text zero_less_mult_iff} in that 
983 
there are no negative numbers.*} 

984 
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)" 

13449  985 
apply (induct m) 
14208  986 
apply (case_tac [2] n, simp_all) 
13449  987 
done 
988 

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989 
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)" 
13449  990 
apply (induct m) 
14208  991 
apply (case_tac [2] n, simp_all) 
13449  992 
done 
993 

994 
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)" 

14208  995 
apply (induct_tac m, simp) 
996 
apply (induct_tac n, simp, fastsimp) 

13449  997 
done 
998 

999 
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = 1 & n = 1)" 

1000 
apply (rule trans) 

14208  1001 
apply (rule_tac [2] mult_eq_1_iff, fastsimp) 
13449  1002 
done 
1003 

1004 
lemma mult_less_cancel2: "((m::nat) * k < n * k) = (0 < k & m < n)" 

1005 
apply (safe intro!: mult_less_mono1) 

14208  1006 
apply (case_tac k, auto) 
13449  1007 
apply (simp del: le_0_eq add: linorder_not_le [symmetric]) 
1008 
apply (blast intro: mult_le_mono1) 

1009 
done 

1010 

1011 
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)" 

1012 
by (simp add: mult_less_cancel2 mult_commute [of k]) 

1013 

1014 
declare mult_less_cancel2 [simp] 

1015 

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1016 
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k > m \<le> n)" 
14208  1017 
by (simp add: linorder_not_less [symmetric], auto) 
13449  1018 

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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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1019 
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k > m \<le> n)" 
14208  1020 
by (simp add: linorder_not_less [symmetric], auto) 
13449  1021 

1022 
lemma mult_cancel2: "(m * k = n * k) = (m = n  (k = (0::nat)))" 

14208  1023 
apply (cut_tac less_linear, safe, auto) 
13449  1024 
apply (drule mult_less_mono1, assumption, simp)+ 
1025 
done 

1026 

1027 
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n  (k = (0::nat)))" 

1028 
by (simp add: mult_cancel2 mult_commute [of k]) 

1029 

1030 
declare mult_cancel2 [simp] 

1031 

1032 
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)" 

1033 
by (subst mult_less_cancel1) simp 

1034 

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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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1035 
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)" 
13449  1036 
by (subst mult_le_cancel1) simp 
1037 

1038 
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)" 

1039 
by (subst mult_cancel1) simp 

1040 

1041 

1042 
text {* Lemma for @{text gcd} *} 

1043 
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1  m = 0" 

1044 
apply (drule sym) 

1045 
apply (rule disjCI) 

1046 
apply (rule nat_less_cases, erule_tac [2] _) 

1047 
apply (fastsimp elim!: less_SucE) 

1048 
apply (fastsimp dest: mult_less_mono2) 

1049 
done 

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rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
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1050 

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1051 

923  1052 
end 