author  haftmann 
Thu, 10 May 2007 10:21:50 +0200  
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parent 22845  5f9138bcb3d7 
child 23001  3608f0362a91 
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 and Markus Wenzel 
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Type "nat" is a linear order, and a datatype; arithmetic operators +  
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and * (for div, mod and dvd, see theory Divides). 
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*) 
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13449  9 
header {* Natural numbers *} 
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15131  11 
theory Nat 
15140  12 
imports Wellfounded_Recursion Ring_and_Field 
21243  13 
uses ("arith_data.ML") 
15131  14 
begin 
13449  15 

16 
subsection {* Type @{text ind} *} 

17 

18 
typedecl ind 

19 

19573  20 
axiomatization 
21 
Zero_Rep :: ind and 

22 
Suc_Rep :: "ind => ind" 

23 
where 

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 {* the axiom of infinity in 2 parts *} 
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inj_Suc_Rep: "inj Suc_Rep" and 
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Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep" 
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13449  28 

29 
subsection {* Type nat *} 

30 

31 
text {* Type definition *} 

32 

22262  33 
inductive2 Nat :: "ind \<Rightarrow> bool" 
34 
where 

35 
Zero_RepI: "Nat Zero_Rep" 

36 
 Suc_RepI: "Nat i ==> Nat (Suc_Rep i)" 

13449  37 

38 
global 

39 

40 
typedef (open Nat) 

22262  41 
nat = "Collect Nat" 
21243  42 
proof 
22262  43 
from Nat.Zero_RepI 
44 
show "Zero_Rep : Collect Nat" .. 

21243  45 
qed 
13449  46 

47 
text {* Abstract constants and syntax *} 

48 

49 
consts 

50 
Suc :: "nat => nat" 

51 

52 
local 

53 

54 
defs 

18648  55 
Suc_def: "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))" 
22718  56 

57 
definition 

58 
pred_nat :: "(nat * nat) set" where 

59 
"pred_nat = {(m, n). n = Suc m}" 

13449  60 

21456  61 
instance nat :: "{ord, zero, one}" 
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Zero_nat_def: "0 == Abs_Nat Zero_Rep" 

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

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less_def: "m < n == (m, n) : pred_nat^+" 
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le_def: "m \<le> (n::nat) == ~ (n < m)" .. 
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lemmas [code func del] = less_def le_def 
13449  68 

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text {* Induction *} 

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22718  71 
lemma Rep_Nat': "Nat (Rep_Nat x)" 
72 
by (rule Rep_Nat [simplified mem_Collect_eq]) 

73 

74 
lemma Abs_Nat_inverse': "Nat y \<Longrightarrow> Rep_Nat (Abs_Nat y) = y" 

75 
by (rule Abs_Nat_inverse [simplified mem_Collect_eq]) 

22262  76 

13449  77 
theorem nat_induct: "P 0 ==> (!!n. P n ==> P (Suc n)) ==> P n" 
78 
apply (unfold Zero_nat_def Suc_def) 

79 
apply (rule Rep_Nat_inverse [THEN subst])  {* types force good instantiation *} 

22262  80 
apply (erule Rep_Nat' [THEN Nat.induct]) 
81 
apply (iprover elim: Abs_Nat_inverse' [THEN subst]) 

13449  82 
done 
83 

84 
text {* Distinctness of constructors *} 

85 

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lemma Suc_not_Zero [iff]: "Suc m \<noteq> 0" 
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by (simp add: Zero_nat_def Suc_def Abs_Nat_inject Rep_Nat' Suc_RepI Zero_RepI 
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Suc_Rep_not_Zero_Rep) 
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lemma Zero_not_Suc [iff]: "0 \<noteq> Suc m" 
13449  91 
by (rule not_sym, rule Suc_not_Zero not_sym) 
92 

93 
lemma Suc_neq_Zero: "Suc m = 0 ==> R" 

94 
by (rule notE, rule Suc_not_Zero) 

95 

96 
lemma Zero_neq_Suc: "0 = Suc m ==> R" 

97 
by (rule Suc_neq_Zero, erule sym) 

98 

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

100 

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lemma inj_Suc[simp]: "inj_on Suc N" 
22718  102 
by (simp add: Suc_def inj_on_def Abs_Nat_inject Rep_Nat' Suc_RepI 
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inj_Suc_Rep [THEN inj_eq] Rep_Nat_inject) 

13449  104 

105 
lemma Suc_inject: "Suc x = Suc y ==> x = y" 

106 
by (rule inj_Suc [THEN injD]) 

107 

108 
lemma Suc_Suc_eq [iff]: "(Suc m = Suc n) = (m = n)" 

15413  109 
by (rule inj_Suc [THEN inj_eq]) 
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lemma nat_not_singleton: "(\<forall>x. x = (0::nat)) = False" 
13449  112 
by auto 
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21411  114 
text {* size of a datatype value *} 
21243  115 

22473  116 
class size = type + 
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fixes size :: "'a \<Rightarrow> nat" 
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13449  119 
text {* @{typ nat} is a datatype *} 
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rep_datatype nat 
13449  122 
distinct Suc_not_Zero Zero_not_Suc 
123 
inject Suc_Suc_eq 

21411  124 
induction nat_induct 
125 

126 
declare nat.induct [case_names 0 Suc, induct type: nat] 

127 
declare nat.exhaust [case_names 0 Suc, cases type: nat] 

13449  128 

21672  129 
lemmas nat_rec_0 = nat.recs(1) 
130 
and nat_rec_Suc = nat.recs(2) 

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lemmas nat_case_0 = nat.cases(1) 

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and nat_case_Suc = nat.cases(2) 

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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" 
13449  140 
by (rule not_sym, rule n_not_Suc_n) 
141 

142 
text {* A special form of induction for reasoning 

143 
about @{term "m < n"} and @{term "m  n"} *} 

144 

145 
theorem diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==> 

146 
(!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n" 

14208  147 
apply (rule_tac x = m in spec) 
15251  148 
apply (induct n) 
13449  149 
prefer 2 
150 
apply (rule allI) 

17589  151 
apply (induct_tac x, iprover+) 
13449  152 
done 
153 

154 
subsection {* Basic properties of "less than" *} 

155 

156 
lemma wf_pred_nat: "wf pred_nat" 

14208  157 
apply (unfold wf_def pred_nat_def, clarify) 
158 
apply (induct_tac x, blast+) 

13449  159 
done 
160 

161 
lemma wf_less: "wf {(x, y::nat). x < y}" 

162 
apply (unfold less_def) 

14208  163 
apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_subset], blast) 
13449  164 
done 
165 

166 
lemma less_eq: "((m, n) : pred_nat^+) = (m < n)" 

167 
apply (unfold less_def) 

168 
apply (rule refl) 

169 
done 

170 

171 
subsubsection {* Introduction properties *} 

172 

173 
lemma less_trans: "i < j ==> j < k ==> i < (k::nat)" 

174 
apply (unfold less_def) 

14208  175 
apply (rule trans_trancl [THEN transD], assumption+) 
13449  176 
done 
177 

178 
lemma lessI [iff]: "n < Suc n" 

179 
apply (unfold less_def pred_nat_def) 

180 
apply (simp add: r_into_trancl) 

181 
done 

182 

183 
lemma less_SucI: "i < j ==> i < Suc j" 

14208  184 
apply (rule less_trans, assumption) 
13449  185 
apply (rule lessI) 
186 
done 

187 

188 
lemma zero_less_Suc [iff]: "0 < Suc n" 

189 
apply (induct n) 

190 
apply (rule lessI) 

191 
apply (erule less_trans) 

192 
apply (rule lessI) 

193 
done 

194 

195 
subsubsection {* Elimination properties *} 

196 

197 
lemma less_not_sym: "n < m ==> ~ m < (n::nat)" 

198 
apply (unfold less_def) 

199 
apply (blast intro: wf_pred_nat wf_trancl [THEN wf_asym]) 

200 
done 

201 

202 
lemma less_asym: 

203 
assumes h1: "(n::nat) < m" and h2: "~ P ==> m < n" shows P 

204 
apply (rule contrapos_np) 

205 
apply (rule less_not_sym) 

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apply (rule h1) 

207 
apply (erule h2) 

208 
done 

209 

210 
lemma less_not_refl: "~ n < (n::nat)" 

211 
apply (unfold less_def) 

212 
apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_not_refl]) 

213 
done 

214 

215 
lemma less_irrefl [elim!]: "(n::nat) < n ==> R" 

216 
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" 
13449  221 
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 

14208  227 
apply (rule major [unfolded less_def pred_nat_def, THEN tranclE], simp_all) 
13449  228 
apply (erule p1) 
229 
apply (rule p2) 

14208  230 
apply (simp add: less_def pred_nat_def, assumption) 
13449  231 
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]) 

14208  242 
apply (rule eq, blast) 
243 
apply (rule less, blast) 

13449  244 
done 
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246 
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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249 
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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258 
text {* "Less than" is a linear ordering *} 

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

15251  260 
apply (induct m) 
261 
apply (induct n) 

13449  262 
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) 

265 
done 

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14302  267 
text {* "Less than" is antisymmetric, sort of *} 
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lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n" 

22718  269 
apply(simp only:less_Suc_eq) 
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apply blast 

271 
done 

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lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n  n < m)" 
13449  274 
using less_linear by blast 
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276 
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m" 

277 
and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m" 

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shows "P n m" 

279 
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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286 

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subsubsection {* Inductive (?) properties *} 

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lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n" 
13449  290 
apply (simp add: nat_neq_iff) 
291 
apply (blast elim!: less_irrefl less_SucE elim: less_asym) 

292 
done 

293 

294 
lemma Suc_lessD: "Suc m < n ==> m < n" 

295 
apply (induct n) 

296 
apply (fast intro!: lessI [THEN less_SucI] elim: less_trans lessE)+ 

297 
done 

298 

299 
lemma Suc_lessE: assumes major: "Suc i < k" 

300 
and minor: "!!j. i < j ==> k = Suc j ==> P" shows P 

301 
apply (rule major [THEN lessE]) 

302 
apply (erule lessI [THEN minor]) 

14208  303 
apply (erule Suc_lessD [THEN minor], assumption) 
13449  304 
done 
305 

306 
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n" 

307 
by (blast elim: lessE dest: Suc_lessD) 

4104  308 

16635  309 
lemma Suc_less_eq [iff, code]: "(Suc m < Suc n) = (m < n)" 
13449  310 
apply (rule iffI) 
311 
apply (erule Suc_less_SucD) 

312 
apply (erule Suc_mono) 

313 
done 

314 

315 
lemma less_trans_Suc: 

316 
assumes le: "i < j" shows "j < k ==> Suc i < k" 

14208  317 
apply (induct k, simp_all) 
13449  318 
apply (insert le) 
319 
apply (simp add: less_Suc_eq) 

320 
apply (blast dest: Suc_lessD) 

321 
done 

322 

16635  323 
lemma [code]: "((n::nat) < 0) = False" by simp 
324 
lemma [code]: "(0 < Suc n) = True" by simp 

325 

13449  326 
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m  n < m"} *} 
327 
lemma not_less_eq: "(~ m < n) = (n < Suc m)" 

22718  328 
by (induct m n rule: diff_induct) simp_all 
13449  329 

330 
text {* Complete induction, aka courseofvalues induction *} 

331 
lemma nat_less_induct: 

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assumes prem: "!!n. \<forall>m::nat. m < n > P m ==> P n" shows "P n" 
22718  333 
apply (induct n rule: wf_induct [OF wf_pred_nat [THEN wf_trancl]]) 
13449  334 
apply (rule prem) 
14208  335 
apply (unfold less_def, assumption) 
13449  336 
done 
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14131  338 
lemmas less_induct = nat_less_induct [rule_format, case_names less] 
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21243  340 

14131  341 
subsection {* Properties of "less than or equal" *} 
13449  342 

343 
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)" 
22718  345 
unfolding le_def by (rule not_less_eq [symmetric]) 
13449  346 

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lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n" 
13449  348 
by (rule less_Suc_eq_le [THEN iffD2]) 
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lemma le0 [iff]: "(0::nat) \<le> n" 
22718  351 
unfolding le_def by (rule not_less0) 
13449  352 

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lemma Suc_n_not_le_n: "~ Suc n \<le> n" 
13449  354 
by (simp add: le_def) 
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lemma le_0_eq [iff]: "((i::nat) \<le> 0) = (i = 0)" 
13449  357 
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  360 
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" 
17589  363 
by (drule le_Suc_eq [THEN iffD1], iprover+) 
13449  364 

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lemma Suc_leI: "m < n ==> Suc(m) \<le> n" 
13449  366 
apply (simp add: le_def less_Suc_eq) 
367 
apply (blast elim!: less_irrefl less_asym) 

368 
done  {* formerly called lessD *} 

369 

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lemma Suc_leD: "Suc(m) \<le> n ==> m \<le> n" 
13449  371 
by (simp add: le_def less_Suc_eq) 
372 

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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  375 
apply (simp add: le_def less_Suc_eq) 
376 
using less_linear 

377 
apply blast 

378 
done 

379 

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lemma Suc_le_eq: "(Suc m \<le> n) = (m < n)" 
13449  381 
by (blast intro: Suc_leI Suc_le_lessD) 
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lemma le_SucI: "m \<le> n ==> m \<le> Suc n" 
13449  384 
by (unfold le_def) (blast dest: Suc_lessD) 
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lemma less_imp_le: "m < n ==> m \<le> (n::nat)" 
13449  387 
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  390 
lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq 
391 

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text {* Equivalence of @{term "m \<le> n"} and @{term "m < n  m = n"} *} 
13449  394 

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lemma le_imp_less_or_eq: "m \<le> n ==> m < n  m = (n::nat)" 
22718  396 
unfolding le_def 
13449  397 
using less_linear 
22718  398 
by (blast elim: less_irrefl less_asym) 
13449  399 

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400 
lemma less_or_eq_imp_le: "m < n  m = n ==> m \<le> (n::nat)" 
22718  401 
unfolding le_def 
13449  402 
using less_linear 
22718  403 
by (blast elim!: less_irrefl elim: less_asym) 
13449  404 

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405 
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n  m=n)" 
17589  406 
by (iprover intro: less_or_eq_imp_le le_imp_less_or_eq) 
13449  407 

22718  408 
text {* Useful with @{text blast}. *} 
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409 
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n" 
22718  410 
by (rule less_or_eq_imp_le) (rule disjI2) 
13449  411 

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412 
lemma le_refl: "n \<le> (n::nat)" 
13449  413 
by (simp add: le_eq_less_or_eq) 
414 

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415 
lemma le_less_trans: "[ i \<le> j; j < k ] ==> i < (k::nat)" 
13449  416 
by (blast dest!: le_imp_less_or_eq intro: less_trans) 
417 

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418 
lemma less_le_trans: "[ i < j; j \<le> k ] ==> i < (k::nat)" 
13449  419 
by (blast dest!: le_imp_less_or_eq intro: less_trans) 
420 

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421 
lemma le_trans: "[ i \<le> j; j \<le> k ] ==> i \<le> (k::nat)" 
13449  422 
by (blast dest!: le_imp_less_or_eq intro: less_or_eq_imp_le less_trans) 
423 

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424 
lemma le_anti_sym: "[ m \<le> n; n \<le> m ] ==> m = (n::nat)" 
13449  425 
by (blast dest!: le_imp_less_or_eq elim!: less_irrefl elim: less_asym) 
426 

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427 
lemma Suc_le_mono [iff]: "(Suc n \<le> Suc m) = (n \<le> m)" 
13449  428 
by (simp add: le_simps) 
429 

430 
text {* Axiom @{text order_less_le} of class @{text order}: *} 

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431 
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)" 
13449  432 
by (simp add: le_def nat_neq_iff) (blast elim!: less_asym) 
433 

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434 
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n" 
13449  435 
by (rule iffD2, rule nat_less_le, rule conjI) 
436 

437 
text {* Axiom @{text linorder_linear} of class @{text linorder}: *} 

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438 
lemma nat_le_linear: "(m::nat) \<le> n  n \<le> m" 
13449  439 
apply (simp add: le_eq_less_or_eq) 
22718  440 
using less_linear by blast 
13449  441 

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442 
text {* Type {@typ nat} is a wellfounded linear order *} 
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443 

22318  444 
instance nat :: wellorder 
14691  445 
by intro_classes 
446 
(assumption  

447 
rule le_refl le_trans le_anti_sym nat_less_le nat_le_linear wf_less)+ 

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448 

22718  449 
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat] 
15921  450 

13449  451 
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)" 
452 
by (blast elim!: less_SucE) 

453 

454 
text {* 

455 
Rewrite @{term "n < Suc m"} to @{term "n = m"} 

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456 
if @{term "~ n < m"} or @{term "m \<le> n"} hold. 
13449  457 
Not suitable as default simprules because they often lead to looping 
458 
*} 

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459 
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)" 
13449  460 
by (rule not_less_less_Suc_eq, rule leD) 
461 

462 
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq 

463 

464 

465 
text {* 

22718  466 
Reorientation of the equations @{text "0 = x"} and @{text "1 = x"}. 
467 
No longer added as simprules (they loop) 

13449  468 
but via @{text reorient_simproc} in Bin 
469 
*} 

470 

471 
text {* Polymorphic, not just for @{typ nat} *} 

472 
lemma zero_reorient: "(0 = x) = (x = 0)" 

473 
by auto 

474 

475 
lemma one_reorient: "(1 = x) = (x = 1)" 

476 
by auto 

477 

21243  478 

13449  479 
subsection {* Arithmetic operators *} 
1660  480 

22473  481 
class power = type + 
21411  482 
fixes power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "\<^loc>^" 80) 
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483 

13449  484 
text {* arithmetic operators @{text "+ "} and @{text "*"} *} 
485 

21456  486 
instance nat :: "{plus, minus, times}" .. 
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487 

13449  488 
primrec 
489 
add_0: "0 + n = n" 

490 
add_Suc: "Suc m + n = Suc (m + n)" 

491 

492 
primrec 

493 
diff_0: "m  0 = m" 

494 
diff_Suc: "m  Suc n = (case m  n of 0 => 0  Suc k => k)" 

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495 

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496 
primrec 
13449  497 
mult_0: "0 * n = 0" 
498 
mult_Suc: "Suc m * n = n + (m * n)" 

499 

22718  500 
text {* These two rules ease the use of primitive recursion. 
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501 
NOTE USE OF @{text "=="} *} 
13449  502 
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c" 
503 
by simp 

504 

505 
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)" 

506 
by simp 

507 

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508 
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m" 
22718  509 
by (cases n) simp_all 
13449  510 

22718  511 
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0" 
512 
by (cases n) simp_all 

13449  513 

22718  514 
lemma neq0_conv [iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)" 
515 
by (cases n) simp_all 

13449  516 

517 
text {* This theorem is useful with @{text blast} *} 

518 
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n" 

17589  519 
by (rule iffD1, rule neq0_conv, iprover) 
13449  520 

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521 
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)" 
13449  522 
by (fast intro: not0_implies_Suc) 
523 

524 
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)" 

525 
apply (rule iffI) 

22718  526 
apply (rule ccontr) 
527 
apply simp_all 

13449  528 
done 
529 

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530 
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)" 
13449  531 
by (induct m') simp_all 
532 

533 
text {* Useful in certain inductive arguments *} 

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534 
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0  (\<exists>j. m = Suc j & j < n))" 
22718  535 
by (cases m) simp_all 
13449  536 

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537 
lemma nat_induct2: "[P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k))] ==> P n" 
13449  538 
apply (rule nat_less_induct) 
539 
apply (case_tac n) 

540 
apply (case_tac [2] nat) 

541 
apply (blast intro: less_trans)+ 

542 
done 

543 

21243  544 

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545 
subsection {* @{text LEAST} theorems for type @{typ nat}*} 
13449  546 

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547 
lemma Least_Suc: 
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548 
"[ P n; ~ P 0 ] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))" 
14208  549 
apply (case_tac "n", auto) 
13449  550 
apply (frule LeastI) 
551 
apply (drule_tac P = "%x. P (Suc x) " in LeastI) 

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552 
apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))") 
13449  553 
apply (erule_tac [2] Least_le) 
14208  554 
apply (case_tac "LEAST x. P x", auto) 
13449  555 
apply (drule_tac P = "%x. P (Suc x) " in Least_le) 
556 
apply (blast intro: order_antisym) 

557 
done 

558 

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559 
lemma Least_Suc2: 
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560 
"[P n; Q m; ~P 0; !k. P (Suc k) = Q k] ==> Least P = Suc (Least Q)" 
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561 
by (erule (1) Least_Suc [THEN ssubst], simp) 
13449  562 

563 

564 
subsection {* @{term min} and @{term max} *} 

565 

566 
lemma min_0L [simp]: "min 0 n = (0::nat)" 

567 
by (rule min_leastL) simp 

568 

569 
lemma min_0R [simp]: "min n 0 = (0::nat)" 

570 
by (rule min_leastR) simp 

571 

572 
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)" 

573 
by (simp add: min_of_mono) 

574 

22191  575 
lemma min_Suc1: 
576 
"min (Suc n) m = (case m of 0 => 0  Suc m' => Suc(min n m'))" 

22718  577 
by (simp split: nat.split) 
22191  578 

579 
lemma min_Suc2: 

580 
"min m (Suc n) = (case m of 0 => 0  Suc m' => Suc(min m' n))" 

581 
by (simp split: nat.split) 

582 

13449  583 
lemma max_0L [simp]: "max 0 n = (n::nat)" 
584 
by (rule max_leastL) simp 

585 

586 
lemma max_0R [simp]: "max n 0 = (n::nat)" 

587 
by (rule max_leastR) simp 

588 

589 
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)" 

590 
by (simp add: max_of_mono) 

591 

22191  592 
lemma max_Suc1: 
593 
"max (Suc n) m = (case m of 0 => Suc n  Suc m' => Suc(max n m'))" 

22718  594 
by (simp split: nat.split) 
22191  595 

596 
lemma max_Suc2: 

597 
"max m (Suc n) = (case m of 0 => Suc n  Suc m' => Suc(max m' n))" 

598 
by (simp split: nat.split) 

599 

13449  600 

601 
subsection {* Basic rewrite rules for the arithmetic operators *} 

602 

603 
text {* Difference *} 

604 

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605 
lemma diff_0_eq_0 [simp, code]: "0  n = (0::nat)" 
15251  606 
by (induct n) simp_all 
13449  607 

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608 
lemma diff_Suc_Suc [simp, code]: "Suc(m)  Suc(n) = m  n" 
15251  609 
by (induct n) simp_all 
13449  610 

611 

612 
text {* 

613 
Could be (and is, below) generalized in various ways 

614 
However, none of the generalizations are currently in the simpset, 

615 
and I dread to think what happens if I put them in 

616 
*} 

617 
lemma Suc_pred [simp]: "0 < n ==> Suc (n  Suc 0) = n" 

618 
by (simp split add: nat.split) 

619 

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620 
declare diff_Suc [simp del, code del] 
13449  621 

622 

623 
subsection {* Addition *} 

624 

625 
lemma add_0_right [simp]: "m + 0 = (m::nat)" 

626 
by (induct m) simp_all 

627 

628 
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)" 

629 
by (induct m) simp_all 

630 

19890  631 
lemma add_Suc_shift [code]: "Suc m + n = m + Suc n" 
632 
by simp 

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633 

13449  634 

635 
text {* Associative law for addition *} 

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636 
lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)" 
13449  637 
by (induct m) simp_all 
638 

639 
text {* Commutative law for addition *} 

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640 
lemma nat_add_commute: "m + n = n + (m::nat)" 
13449  641 
by (induct m) simp_all 
642 

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643 
lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)" 
13449  644 
apply (rule mk_left_commute [of "op +"]) 
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645 
apply (rule nat_add_assoc) 
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646 
apply (rule nat_add_commute) 
13449  647 
done 
648 

14331  649 
lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))" 
13449  650 
by (induct k) simp_all 
651 

14331  652 
lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))" 
13449  653 
by (induct k) simp_all 
654 

14331  655 
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))" 
13449  656 
by (induct k) simp_all 
657 

14331  658 
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))" 
13449  659 
by (induct k) simp_all 
660 

661 
text {* Reasoning about @{text "m + 0 = 0"}, etc. *} 

662 

22718  663 
lemma add_is_0 [iff]: fixes m :: nat shows "(m + n = 0) = (m = 0 & n = 0)" 
664 
by (cases m) simp_all 

13449  665 

666 
lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0  m=0 & n= Suc 0)" 

22718  667 
by (cases m) simp_all 
13449  668 

669 
lemma one_is_add: "(Suc 0 = m + n) = (m = Suc 0 & n = 0  m = 0 & n = Suc 0)" 

670 
by (rule trans, rule eq_commute, rule add_is_1) 

671 

672 
lemma add_gr_0 [iff]: "!!m::nat. (0 < m + n) = (0 < m  0 < n)" 

673 
by (simp del: neq0_conv add: neq0_conv [symmetric]) 

674 

675 
lemma add_eq_self_zero: "!!m::nat. m + n = m ==> n = 0" 

676 
apply (drule add_0_right [THEN ssubst]) 

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677 
apply (simp add: nat_add_assoc del: add_0_right) 
13449  678 
done 
679 

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680 
lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N" 
22718  681 
apply (induct k) 
682 
apply simp 

683 
apply(drule comp_inj_on[OF _ inj_Suc]) 

684 
apply (simp add:o_def) 

685 
done 

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686 

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687 

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688 
subsection {* Multiplication *} 
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689 

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690 
text {* right annihilation in product *} 
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691 
lemma mult_0_right [simp]: "(m::nat) * 0 = 0" 
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692 
by (induct m) simp_all 
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693 

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694 
text {* right successor law for multiplication *} 
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695 
lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)" 
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696 
by (induct m) (simp_all add: nat_add_left_commute) 
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changeset

697 

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

698 
text {* Commutative law for multiplication *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

699 
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

700 
by (induct m) simp_all 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

701 

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

702 
text {* addition distributes over multiplication *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

703 
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

704 
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

705 

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

706 
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

707 
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

708 

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

709 
text {* Associative law for multiplication *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

710 
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

711 
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

712 

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

713 

14740  714 
text{*The naturals form a @{text comm_semiring_1_cancel}*} 
14738  715 
instance nat :: comm_semiring_1_cancel 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

716 
proof 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

717 
fix i j k :: nat 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

718 
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

719 
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

720 
show "0 + i = i" by simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

721 
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

722 
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

723 
show "1 * i = i" by simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

724 
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

725 
show "0 \<noteq> (1::nat)" by simp 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

726 
assume "k+i = k+j" thus "i=j" by simp 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

727 
qed 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

728 

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

729 
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0  n=0)" 
15251  730 
apply (induct m) 
22718  731 
apply (induct_tac [2] n) 
732 
apply simp_all 

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

733 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

734 

21243  735 

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

736 
subsection {* Monotonicity of Addition *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

737 

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

738 
text {* strict, in 1st argument *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

739 
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

740 
by (induct k) simp_all 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

741 

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

742 
text {* strict, in both arguments *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

743 
lemma add_less_mono: "[i < j; k < l] ==> i + k < j + (l::nat)" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

744 
apply (rule add_less_mono1 [THEN less_trans], assumption+) 
15251  745 
apply (induct j, simp_all) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

746 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

747 

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

748 
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

749 
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

750 
apply (induct n) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

751 
apply (simp_all add: order_le_less) 
22718  752 
apply (blast elim!: less_SucE 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

753 
intro!: add_0_right [symmetric] add_Suc_right [symmetric]) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

754 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

755 

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

756 
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

757 
lemma mult_less_mono2: "(i::nat) < j ==> 0 < k ==> k * i < k * j" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

758 
apply (erule_tac m1 = 0 in less_imp_Suc_add [THEN exE], simp) 
22718  759 
apply (induct_tac x) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

760 
apply (simp_all add: add_less_mono) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

761 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

762 

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

763 

14740  764 
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*} 
14738  765 
instance nat :: ordered_semidom 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

766 
proof 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

767 
fix i j k :: nat 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

768 
show "0 < (1::nat)" by simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

769 
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

770 
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

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

772 

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

773 
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

774 
by simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

775 

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

776 
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

777 
by simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

778 

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

779 

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

780 
subsection {* Additional theorems about "less than" *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

781 

19870  782 
text{*An induction rule for estabilishing binary relations*} 
22718  783 
lemma less_Suc_induct: 
19870  784 
assumes less: "i < j" 
785 
and step: "!!i. P i (Suc i)" 

786 
and trans: "!!i j k. P i j ==> P j k ==> P i k" 

787 
shows "P i j" 

788 
proof  

22718  789 
from less obtain k where j: "j = Suc(i+k)" by (auto dest: less_imp_Suc_add) 
790 
have "P i (Suc (i + k))" 

19870  791 
proof (induct k) 
22718  792 
case 0 
793 
show ?case by (simp add: step) 

19870  794 
next 
795 
case (Suc k) 

22718  796 
thus ?case by (auto intro: assms) 
19870  797 
qed 
22718  798 
thus "P i j" by (simp add: j) 
19870  799 
qed 
800 

801 

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

802 
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

803 
monotonicity to @{text "\<le>"} monotonicity *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

804 
lemma less_mono_imp_le_mono: 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

805 
assumes lt_mono: "!!i j::nat. i < j ==> f i < f j" 
22718  806 
and le: "i \<le> j" 
807 
shows "f i \<le> ((f j)::nat)" 

808 
using le 

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

809 
apply (simp add: order_le_less) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

810 
apply (blast intro!: lt_mono) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

811 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

812 

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

813 
text {* nonstrict, in 1st argument *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

814 
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

815 
by (rule add_right_mono) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

816 

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

817 
text {* nonstrict, in both arguments *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

818 
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

819 
by (rule add_mono) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

820 

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

821 
lemma le_add2: "n \<le> ((m + n)::nat)" 
22718  822 
by (insert add_right_mono [of 0 m n], simp) 
13449  823 

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

824 
lemma le_add1: "n \<le> ((n + m)::nat)" 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

825 
by (simp add: add_commute, rule le_add2) 
13449  826 

827 
lemma less_add_Suc1: "i < Suc (i + m)" 

828 
by (rule le_less_trans, rule le_add1, rule lessI) 

829 

830 
lemma less_add_Suc2: "i < Suc (m + i)" 

831 
by (rule le_less_trans, rule le_add2, rule lessI) 

832 

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

833 
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))" 
17589  834 
by (iprover intro!: less_add_Suc1 less_imp_Suc_add) 
13449  835 

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

836 
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m" 
13449  837 
by (rule le_trans, assumption, rule le_add1) 
838 

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

839 
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j" 
13449  840 
by (rule le_trans, assumption, rule le_add2) 
841 

842 
lemma trans_less_add1: "(i::nat) < j ==> i < j + m" 

843 
by (rule less_le_trans, assumption, rule le_add1) 

844 

845 
lemma trans_less_add2: "(i::nat) < j ==> i < m + j" 

846 
by (rule less_le_trans, assumption, rule le_add2) 

847 

848 
lemma add_lessD1: "i + j < (k::nat) ==> i < k" 

22718  849 
apply (rule le_less_trans [of _ "i+j"]) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

850 
apply (simp_all add: le_add1) 
13449  851 
done 
852 

853 
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))" 

854 
apply (rule notI) 

855 
apply (erule add_lessD1 [THEN less_irrefl]) 

856 
done 

857 

858 
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))" 

859 
by (simp add: add_commute not_add_less1) 

860 

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

861 
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)" 
22718  862 
apply (rule order_trans [of _ "m+k"]) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

863 
apply (simp_all add: le_add1) 
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paulson
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864 
done 
13449  865 

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866 
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)" 
13449  867 
apply (simp add: add_commute) 
868 
apply (erule add_leD1) 

869 
done 

870 

14267
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871 
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R" 
13449  872 
by (blast dest: add_leD1 add_leD2) 
873 

874 
text {* needs @{text "!!k"} for @{text add_ac} to work *} 

875 
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n" 

876 
by (force simp del: add_Suc_right 

877 
simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac) 

878 

879 

880 
subsection {* Difference *} 

881 

882 
lemma diff_self_eq_0 [simp]: "(m::nat)  m = 0" 

883 
by (induct m) simp_all 

884 

885 
text {* Addition is the inverse of subtraction: 

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886 
if @{term "n \<le> m"} then @{term "n + (m  n) = m"}. *} 
13449  887 
lemma add_diff_inverse: "~ m < n ==> n + (m  n) = (m::nat)" 
888 
by (induct m n rule: diff_induct) simp_all 

889 

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890 
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m  n) = (m::nat)" 
16796  891 
by (simp add: add_diff_inverse linorder_not_less) 
13449  892 

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893 
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m  n) + n = (m::nat)" 
13449  894 
by (simp add: le_add_diff_inverse add_commute) 
895 

896 

897 
subsection {* More results about difference *} 

898 

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899 
lemma Suc_diff_le: "n \<le> m ==> Suc m  n = Suc (m  n)" 
13449  900 
by (induct m n rule: diff_induct) simp_all 
901 

902 
lemma diff_less_Suc: "m  n < Suc m" 

903 
apply (induct m n rule: diff_induct) 

904 
apply (erule_tac [3] less_SucE) 

905 
apply (simp_all add: less_Suc_eq) 

906 
done 

907 

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908 
lemma diff_le_self [simp]: "m  n \<le> (m::nat)" 
13449  909 
by (induct m n rule: diff_induct) (simp_all add: le_SucI) 
910 

911 
lemma less_imp_diff_less: "(j::nat) < k ==> j  n < k" 

912 
by (rule le_less_trans, rule diff_le_self) 

913 

914 
lemma diff_diff_left: "(i::nat)  j  k = i  (j + k)" 

915 
by (induct i j rule: diff_induct) simp_all 

916 

917 
lemma Suc_diff_diff [simp]: "(Suc m  n)  Suc k = m  n  k" 

918 
by (simp add: diff_diff_left) 

919 

920 
lemma diff_Suc_less [simp]: "0<n ==> n  Suc i < n" 

22718  921 
by (cases n) (auto simp add: le_simps) 
13449  922 

923 
text {* This and the next few suggested by Florian Kammueller *} 

924 
lemma diff_commute: "(i::nat)  j  k = i  k  j" 

925 
by (simp add: diff_diff_left add_commute) 

926 

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927 
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j)  k = i + (j  k)" 
13449  928 
by (induct j k rule: diff_induct) simp_all 
929 

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930 
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i)  k = (j  k) + i" 
13449  931 
by (simp add: add_commute diff_add_assoc) 
932 

933 
lemma diff_add_inverse: "(n + m)  n = (m::nat)" 

934 
by (induct n) simp_all 

935 

936 
lemma diff_add_inverse2: "(m + n)  n = (m::nat)" 

937 
by (simp add: diff_add_assoc) 

938 

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939 
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j  i = k) = (j = k + i)" 
22718  940 
by (auto simp add: diff_add_inverse2) 
13449  941 

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942 
lemma diff_is_0_eq [simp]: "((m::nat)  n = 0) = (m \<le> n)" 
13449  943 
by (induct m n rule: diff_induct) simp_all 
944 

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changeset

945 
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat)  n = 0" 
13449  946 
by (rule iffD2, rule diff_is_0_eq) 
947 

948 
lemma zero_less_diff [simp]: "(0 < n  (m::nat)) = (m < n)" 

949 
by (induct m n rule: diff_induct) simp_all 

950 

22718  951 
lemma less_imp_add_positive: 
952 
assumes "i < j" 

953 
shows "\<exists>k::nat. 0 < k & i + k = j" 

954 
proof 

955 
from assms show "0 < j  i & i + (j  i) = j" 

956 
by (simp add: add_diff_inverse less_not_sym) 

957 
qed 

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958 

13449  959 
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k  i)" 
960 
apply (induct k i rule: diff_induct) 

961 
apply (simp_all (no_asm)) 

17589  962 
apply iprover 
13449  963 
done 
964 

965 
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0" 

966 
apply (rule diff_self_eq_0 [THEN subst]) 

17589  967 
apply (rule zero_induct_lemma, iprover+) 
13449  968 
done 
969 

970 
lemma diff_cancel: "(k + m)  (k + n) = m  (n::nat)" 

971 
by (induct k) simp_all 

972 

973 
lemma diff_cancel2: "(m + k)  (n + k) = m  (n::nat)" 

974 
by (simp add: diff_cancel add_commute) 

975 

976 
lemma diff_add_0: "n  (n + m) = (0::nat)" 

977 
by (induct n) simp_all 

978 

979 

980 
text {* Difference distributes over multiplication *} 

981 

982 
lemma diff_mult_distrib: "((m::nat)  n) * k = (m * k)  (n * k)" 

983 
by (induct m n rule: diff_induct) (simp_all add: diff_cancel) 

984 

985 
lemma diff_mult_distrib2: "k * ((m::nat)  n) = (k * m)  (k * n)" 

986 
by (simp add: diff_mult_distrib mult_commute [of k]) 

987 
 {* NOT added as rewrites, since sometimes they are used from righttoleft *} 

988 

989 
lemmas nat_distrib = 

990 
add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2 

991 

992 

993 
subsection {* Monotonicity of Multiplication *} 

994 

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995 
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k" 
22718  996 
by (simp add: mult_right_mono) 
13449  997 

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998 
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j" 
22718  999 
by (simp add: mult_left_mono) 
13449  1000 

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1001 
text {* @{text "\<le>"} monotonicity, BOTH arguments *} 
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1002 
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l" 
22718  1003 
by (simp add: mult_mono) 
13449  1004 

1005 
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k" 

22718  1006 
by (simp add: mult_strict_right_mono) 
13449  1007 

14266  1008 
text{*Differs from the standard @{text zero_less_mult_iff} in that 
1009 
there are no negative numbers.*} 

1010 
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)" 

13449  1011 
apply (induct m) 
22718  1012 
apply simp 
1013 
apply (case_tac n) 

1014 
apply simp_all 

13449  1015 
done 
1016 

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1017 
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)" 
13449  1018 
apply (induct m) 
22718  1019 
apply simp 
1020 
apply (case_tac n) 

1021 
apply simp_all 

13449  1022 
done 
1023 

1024 
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)" 

22718  1025 
apply (induct m) 
1026 
apply simp 

1027 
apply (induct n) 

1028 
apply auto 

13449  1029 
done 
1030 

1031 
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = 1 & n = 1)" 

1032 
apply (rule trans) 

14208  1033 
apply (rule_tac [2] mult_eq_1_iff, fastsimp) 
13449  1034 
done 
1035 

14341
a09441bd4f1e
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paulson
parents:
14331
diff
changeset

1036 
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)" 
13449  1037 
apply (safe intro!: mult_less_mono1) 
14208  1038 
apply (case_tac k, auto) 
13449  1039 
apply (simp del: le_0_eq add: linorder_not_le [symmetric]) 
1040 
apply (blast intro: mult_le_mono1) 

1041 
done 

1042 

1043 
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)" 

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

1044 
by (simp add: mult_commute [of k]) 
13449  1045 

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changeset

1046 
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k > m \<le> n)" 
22718  1047 
by (simp add: linorder_not_less [symmetric], auto) 
13449  1048 

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changeset

1049 
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k > m \<le> n)" 
22718  1050 
by (simp add: linorder_not_less [symmetric], auto) 
13449  1051 

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

1052 
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n  (k = (0::nat)))" 
14208  1053 
apply (cut_tac less_linear, safe, auto) 
13449  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)))" 

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

1058 
by (simp add: mult_commute [of k]) 
13449  1059 

1060 
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)" 

1061 
by (subst mult_less_cancel1) simp 

1062 

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1063 
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)" 
13449  1064 
by (subst mult_le_cancel1) simp 
1065 

1066 
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)" 

1067 
by (subst mult_cancel1) simp 

1068 

1069 
text {* Lemma for @{text gcd} *} 

1070 
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1  m = 0" 

1071 
apply (drule sym) 

1072 
apply (rule disjCI) 

1073 
apply (rule nat_less_cases, erule_tac [2] _) 

1074 
apply (fastsimp elim!: less_SucE) 

1075 
apply (fastsimp dest: mult_less_mono2) 

1076 
done 

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

1077 

20588  1078 

18702  1079 
subsection {* Code generator setup *} 
1080 

22718  1081 
lemma one_is_Suc_zero [code inline]: "1 = Suc 0" 
20355  1082 
by simp 
1083 

20588  1084 
instance nat :: eq .. 
1085 

1086 
lemma [code func]: 

22718  1087 
"(0\<Colon>nat) = 0 \<longleftrightarrow> True" 
1088 
"Suc n = Suc m \<longleftrightarrow> n = m" 

1089 
"Suc n = 0 \<longleftrightarrow> False" 

1090 
"0 = Suc m \<longleftrightarrow> False" 

22348  1091 
by auto 
20588  1092 

1093 
lemma [code func]: 

22718  1094 
"(0\<Colon>nat) \<le> m \<longleftrightarrow> True" 
1095 
"Suc (n\<Colon>nat) \<le> m \<longleftrightarrow> n < m" 

1096 
"(n\<Colon>nat) < 0 \<longleftrightarrow> False" 

1097 
"(n\<Colon>nat) < Suc m \<longleftrightarrow> n \<le> m" 

22348  1098 
using Suc_le_eq less_Suc_eq_le by simp_all 
20588  1099 

21243  1100 

1101 
subsection {* Further Arithmetic Facts Concerning the Natural Numbers *} 

1102 

22845  1103 
lemma subst_equals: 
1104 
assumes 1: "t = s" and 2: "u = t" 

1105 
shows "u = s" 

1106 
using 2 1 by (rule trans) 

1107 

21243  1108 
use "arith_data.ML" 
1109 
setup arith_setup 

1110 

1111 
text{*The following proofs may rely on the arithmetic proof procedures.*} 

1112 

1113 
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)" 

1114 
by (auto simp: le_eq_less_or_eq dest: less_imp_Suc_add) 

1115 

1116 
lemma pred_nat_trancl_eq_le: "((m, n) : pred_nat^*) = (m \<le> n)" 

22718  1117 
by (simp add: less_eq reflcl_trancl [symmetric] del: reflcl_trancl, arith) 
21243  1118 

1119 
lemma nat_diff_split: 

22718  1120 
"P(a  b::nat) = ((a<b > P 0) & (ALL d. a = b + d > P d))" 
21243  1121 
 {* elimination of @{text } on @{text nat} *} 
22718  1122 
by (cases "a<b" rule: case_split) (auto simp add: diff_is_0_eq [THEN iffD2]) 
21243  1123 

1124 
lemma nat_diff_split_asm: 

1125 
"P(a  b::nat) = (~ (a < b & ~ P 0  (EX d. a = b + d & ~ P d)))" 

1126 
 {* elimination of @{text } on @{text nat} in assumptions *} 

1127 
by (simp split: nat_diff_split) 

1128 

1129 
lemmas [arith_split] = nat_diff_split split_min split_max 

1130 

1131 

1132 
lemma le_square: "m \<le> m * (m::nat)" 

1133 
by (induct m) auto 

1134 

1135 
lemma le_cube: "(m::nat) \<le> m * (m * m)" 

1136 
by (induct m) auto 

1137 

1138 

1139 
text{*Subtraction laws, mostly by Clemens Ballarin*} 

1140 

1141 
lemma diff_less_mono: "[ a < (b::nat); c \<le> a ] ==> ac < bc" 

22718  1142 
by arith 
21243  1143 

1144 
lemma less_diff_conv: "(i < jk) = (i+k < (j::nat))" 

22718  1145 
by arith 
21243  1146 

1147 
lemma le_diff_conv: "(jk \<le> (i::nat)) = (j \<le> i+k)" 

22718  1148 
by arith 
21243  1149 

1150 
lemma le_diff_conv2: "k \<le> j ==> (i \<le> jk) = (i+k \<le> (j::nat))" 

22718  1151 
by arith 
21243  1152 

1153 
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n  (n  i) = i" 

22718  1154 
by arith 
21243  1155 

1156 
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m  k" 

22718  1157 
by arith 
21243  1158 

1159 
(*Replaces the previous diff_less and le_diff_less, which had the stronger 

1160 
second premise n\<le>m*) 

1161 
lemma diff_less[simp]: "!!m::nat. [ 0<n; 0<m ] ==> m  n < m" 

22718  1162 
by arith 
21243  1163 

1164 

1165 
(** Simplification of relational expressions involving subtraction **) 

1166 

1167 
lemma diff_diff_eq: "[ k \<le> m; k \<le> (n::nat) ] ==> ((mk)  (nk)) = (mn)" 

22718  1168 
by (simp split add: nat_diff_split) 
21243  1169 

1170 
lemma eq_diff_iff: "[ k \<le> m; k \<le> (n::nat) ] ==> (mk = nk) = (m=n)" 

22718  1171 
by (auto split add: nat_diff_split) 
21243  1172 

1173 
lemma less_diff_iff: "[ k \<le> m; k \<le> (n::nat) ] ==> (mk < nk) = (m<n)" 

22718  1174 
by (auto split add: nat_diff_split) 
21243  1175 

1176 
lemma le_diff_iff: "[ k \<le> m; k \<le> (n::nat) ] ==> (mk \<le> nk) = (m\<le>n)" 

22718  1177 
by (auto split add: nat_diff_split) 
21243  1178 

1179 

1180 
text{*(Anti)Monotonicity of subtraction  by Stephan Merz*} 

1181 

1182 
(* Monotonicity of subtraction in first argument *) 

1183 
lemma diff_le_mono: "m \<le> (n::nat) ==> (ml) \<le> (nl)" 

22718  1184 
by (simp split add: nat_diff_split) 
21243  1185 

1186 
lemma diff_le_mono2: "m \<le> (n::nat) ==> (ln) \<le> (lm)" 

22718  1187 
by (simp split add: nat_diff_split) 
21243  1188 

1189 
lemma diff_less_mono2: "[ m < (n::nat); m<l ] ==> (ln) < (lm)" 

22718  1190 
by (simp split add: nat_diff_split) 
21243  1191 

1192 
lemma diffs0_imp_equal: "!!m::nat. [ mn = 0; nm = 0 ] ==> m=n" 

22718  1193 
by (simp split add: nat_diff_split) 
21243  1194 

1195 
text{*Lemmas for ex/Factorization*} 

1196 

1197 
lemma one_less_mult: "[ Suc 0 < n; Suc 0 < m ] ==> Suc 0 < m*n" 

22718  1198 
by (cases m) auto 
21243  1199 

1200 
lemma n_less_m_mult_n: "[ Suc 0 < n; Suc 0 < m ] ==> n<m*n" 

22718  1201 
by (cases m) auto 
21243  1202 

1203 
lemma n_less_n_mult_m: "[ Suc 0 < n; Suc 0 < m ] ==> n<n*m" 

22718  1204 
by (cases m) auto 
21243  1205 

1206 

1207 
text{*Rewriting to pull differences out*} 

1208 

1209 
lemma diff_diff_right [simp]: "k\<le>j > i  (j  k) = i + (k::nat)  j" 

22718  1210 
by arith 
21243  1211 

1212 
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m  Suc (j  k) = m + k  Suc j" 

22718  1213 
by arith 
21243  1214 

1215 
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j  k)  m = Suc j  (k + m)" 

22718  1216 
by arith 
21243  1217 

1218 
(*The others are 

1219 
i  j  k = i  (j + k), 

1220 
k \<le> j ==> j  k + i = j + i  k, 

1221 
k \<le> j ==> i + (j  k) = i + j  k *) 

1222 
lemmas add_diff_assoc = diff_add_assoc [symmetric] 

1223 
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric] 

1224 
declare diff_diff_left [simp] add_diff_assoc [simp] add_diff_assoc2[simp] 

1225 

1226 
text{*At present we prove no analogue of @{text not_less_Least} or @{text 

1227 
Least_Suc}, since there appears to be no need.*} 

1228 

1229 
ML 

1230 
{* 

1231 
val pred_nat_trancl_eq_le = thm "pred_nat_trancl_eq_le"; 

1232 
val nat_diff_split = thm "nat_diff_split"; 

1233 
val nat_diff_split_asm = thm "nat_diff_split_asm"; 

1234 
val le_square = thm "le_square"; 

1235 
val le_cube = thm "le_cube"; 

1236 
val diff_less_mono = thm "diff_less_mono"; 

1237 
val less_diff_conv = thm "less_diff_conv"; 

1238 
val le_diff_conv = thm "le_diff_conv"; 

1239 
val le_diff_conv2 = thm "le_diff_conv2"; 

1240 
val diff_diff_cancel = thm "diff_diff_cancel"; 

1241 
val le_add_diff = thm "le_add_diff"; 

1242 
val diff_less = thm "diff_less"; 

1243 
val diff_diff_eq = thm "diff_diff_eq"; 

1244 
val eq_diff_iff = thm "eq_diff_iff"; 

1245 
val less_diff_iff = thm "less_diff_iff"; 

1246 
val le_diff_iff = thm "le_diff_iff"; 

1247 
val diff_le_mono = thm "diff_le_mono"; 

1248 
val diff_le_mono2 = thm "diff_le_mono2"; 

1249 
val diff_less_mono2 = thm "diff_less_mono2"; 

1250 
val diffs0_imp_equal = thm "diffs0_imp_equal"; 

1251 
val one_less_mult = thm "one_less_mult"; 

1252 
val n_less_m_mult_n = thm "n_less_m_mult_n"; 

1253 
val n_less_n_mult_m = thm "n_less_n_mult_m"; 

1254 
val diff_diff_right = thm "diff_diff_right"; 

1255 
val diff_Suc_diff_eq1 = thm "diff_Suc_diff_eq1"; 

1256 
val diff_Suc_diff_eq2 = thm "diff_Suc_diff_eq2"; 

1257 
*} 

1258 

22718  1259 

1260 
subsection{*Embedding of the Naturals into any 

1261 
@{text semiring_1_cancel}: @{term of_nat}*} 

21243  1262 

1263 
consts of_nat :: "nat => 'a::semiring_1_cancel" 

1264 

1265 
primrec 

1266 
of_nat_0: "of_nat 0 = 0" 

1267 
of_nat_Suc: "of_nat (Suc m) = of_nat m + 1" 

1268 

22920  1269 
lemma of_nat_id [simp]: "(of_nat n \<Colon> nat) = n" 
1270 
by (induct n) auto 

1271 

21243  1272 
lemma of_nat_1 [simp]: "of_nat 1 = 1" 
22718  1273 
by simp 
21243  1274 

1275 
lemma of_nat_add [simp]: "of_nat (m+n) = of_nat m + of_nat n" 

22718  1276 
by (induct m) (simp_all add: add_ac) 
21243  1277 

1278 
lemma of_nat_mult [simp]: "of_nat (m*n) = of_nat m * of_nat n" 

22718  1279 
by (induct m) (simp_all add: add_ac left_distrib) 
21243  1280 

1281 
lemma zero_le_imp_of_nat: "0 \<le> (of_nat m::'a::ordered_semidom)" 

22718  1282 
apply (induct m, simp_all) 
1283 
apply (erule order_trans) 

1284 
apply (rule less_add_one [THEN order_less_imp_le]) 

1285 
done 

21243  1286 

1287 
lemma less_imp_of_nat_less: 

22718  1288 
"m < n ==> of_nat m < (of_nat n::'a::ordered_semidom)" 
1289 
apply (induct m n rule: diff_induct, simp_all) 

1290 
apply (insert add_le_less_mono [OF zero_le_imp_of_nat zero_less_one], force) 

1291 
done 

21243  1292 

1293 
lemma of_nat_less_imp_less: 

22718  1294 
"of_nat m < (of_nat n::'a::ordered_semidom) ==> m < n" 
1295 
apply (induct m n rule: diff_induct, simp_all) 

1296 
apply (insert zero_le_imp_of_nat) 

1297 
apply (force simp add: linorder_not_less [symmetric]) 

1298 
done 

21243  1299 

1300 
lemma of_nat_less_iff [simp]: 

22718  1301 
"(of_nat m < (of_nat n::'a::ordered_semidom)) = (m<n)" 
1302 
by (blast intro: of_nat_less_imp_less less_imp_of_nat_less) 

21243  1303 

1304 
text{*Special cases where either operand is zero*} 

22718  1305 

1306 
lemma of_nat_0_less_iff [simp]: "((0::'a::ordered_semidom) < of_nat n) = (0 < n)" 

1307 
by (rule of_nat_less_iff [of 0, simplified]) 

1308 

1309 
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < (0::'a::ordered_semidom)" 

1310 
by (rule of_nat_less_iff [of _ 0, simplified]) 

21243  1311 

1312 
lemma of_nat_le_iff [simp]: 

22718  1313 
"(of_nat m \<le> (of_nat n::'a::ordered_semidom)) = (m \<le> n)" 
1314 
by (simp add: linorder_not_less [symmetric]) 

21243  1315 

1316 
text{*Special cases where either operand is zero*} 

22718  1317 
lemma of_nat_0_le_iff [simp]: "(0::'a::ordered_semidom) \<le> of_nat n" 
1318 
by (rule of_nat_le_iff [of 0, simplified]) 

1319 
lemma of_nat_le_0_iff [simp]: "(of_nat m \<le> (0::'a::ordered_semidom)) = (m = 0)" 

1320 
by (rule of_nat_le_iff [of _ 0, simplified]) 

21243  1321 

1322 
text{*The ordering on the @{text semiring_1_cancel} is necessary 

1323 
to exclude the possibility of a finite field, which indeed wraps back to 

1324 
zero.*} 

1325 
lemma of_nat_eq_iff [simp]: 

22718  1326 
"(of_nat m = (of_nat n::'a::ordered_semidom)) = (m = n)" 
1327 
by (simp add: order_eq_iff) 

21243  1328 

1329 
text{*Special cases where either operand is zero*} 

22718  1330 
lemma of_nat_0_eq_iff [simp]: "((0::'a::ordered_semidom) = of_nat n) = (0 = n)" 
1331 
by (rule of_nat_eq_iff [of 0, simplified]) 

1332 
lemma of_nat_eq_0_iff [simp]: "(of_nat m = (0::'a::ordered_semidom)) = (m = 0)" 

1333 
by (rule of_nat_eq_iff [of _ 0, simplified]) 

21243  1334 

1335 
lemma of_nat_diff [simp]: 

22718  1336 
"n \<le> m ==> of_nat (m  n) = of_nat m  (of_nat n :: 'a::ring_1)" 
1337 
by (simp del: of_nat_add 

1338 
add: compare_rls of_nat_add [symmetric] split add: nat_diff_split) 

21243  1339 

22483  1340 
instance nat :: distrib_lattice 
1341 
"inf \<equiv> min" 

1342 
"sup \<equiv> max" 

1343 
by intro_classes (auto simp add: inf_nat_def sup_nat_def) 

1344 

22157 