author  haftmann 
Thu, 26 Jun 2008 10:07:01 +0200  
changeset 27368  9f90ac19e32b 
parent 27110  194aa674c2a1 
child 27487  c8a6ce181805 
permissions  rwrr 
11355  1 
(* Title: HOL/Library/Nat_Infinity.thy 
2 
ID: $Id$ 

27110  3 
Author: David von Oheimb, TU Muenchen; Florian Haftmann, TU Muenchen 
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*) 
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14706  6 
header {* Natural numbers with infinity *} 
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15131  8 
theory Nat_Infinity 
27368  9 
imports Plain Presburger 
15131  10 
begin 
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27110  12 
subsection {* Type definition *} 
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text {* 
11355  15 
We extend the standard natural numbers by a special value indicating 
27110  16 
infinity. 
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*} 
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datatype inat = Fin nat  Infty 
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21210  21 
notation (xsymbols) 
19736  22 
Infty ("\<infinity>") 
23 

21210  24 
notation (HTML output) 
19736  25 
Infty ("\<infinity>") 
26 

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27110  28 
subsection {* Constructors and numbers *} 
29 

30 
instantiation inat :: "{zero, one, number}" 

25594  31 
begin 
32 

33 
definition 

27110  34 
"0 = Fin 0" 
25594  35 

36 
definition 

27110  37 
[code inline]: "1 = Fin 1" 
25594  38 

39 
definition 

27110  40 
[code inline, code func del]: "number_of k = Fin (number_of k)" 
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25594  42 
instance .. 
43 

44 
end 

45 

27110  46 
definition iSuc :: "inat \<Rightarrow> inat" where 
47 
"iSuc i = (case i of Fin n \<Rightarrow> Fin (Suc n)  \<infinity> \<Rightarrow> \<infinity>)" 

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lemma Fin_0: "Fin 0 = 0" 
27110  50 
by (simp add: zero_inat_def) 
51 

52 
lemma Fin_1: "Fin 1 = 1" 

53 
by (simp add: one_inat_def) 

54 

55 
lemma Fin_number: "Fin (number_of k) = number_of k" 

56 
by (simp add: number_of_inat_def) 

57 

58 
lemma one_iSuc: "1 = iSuc 0" 

59 
by (simp add: zero_inat_def one_inat_def iSuc_def) 

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lemma Infty_ne_i0 [simp]: "\<infinity> \<noteq> 0" 
27110  62 
by (simp add: zero_inat_def) 
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lemma i0_ne_Infty [simp]: "0 \<noteq> \<infinity>" 
27110  65 
by (simp add: zero_inat_def) 
66 

67 
lemma zero_inat_eq [simp]: 

68 
"number_of k = (0\<Colon>inat) \<longleftrightarrow> number_of k = (0\<Colon>nat)" 

69 
"(0\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (0\<Colon>nat)" 

70 
unfolding zero_inat_def number_of_inat_def by simp_all 

71 

72 
lemma one_inat_eq [simp]: 

73 
"number_of k = (1\<Colon>inat) \<longleftrightarrow> number_of k = (1\<Colon>nat)" 

74 
"(1\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (1\<Colon>nat)" 

75 
unfolding one_inat_def number_of_inat_def by simp_all 

76 

77 
lemma zero_one_inat_neq [simp]: 

78 
"\<not> 0 = (1\<Colon>inat)" 

79 
"\<not> 1 = (0\<Colon>inat)" 

80 
unfolding zero_inat_def one_inat_def by simp_all 

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27110  82 
lemma Infty_ne_i1 [simp]: "\<infinity> \<noteq> 1" 
83 
by (simp add: one_inat_def) 

84 

85 
lemma i1_ne_Infty [simp]: "1 \<noteq> \<infinity>" 

86 
by (simp add: one_inat_def) 

87 

88 
lemma Infty_ne_number [simp]: "\<infinity> \<noteq> number_of k" 

89 
by (simp add: number_of_inat_def) 

90 

91 
lemma number_ne_Infty [simp]: "number_of k \<noteq> \<infinity>" 

92 
by (simp add: number_of_inat_def) 

93 

94 
lemma iSuc_Fin: "iSuc (Fin n) = Fin (Suc n)" 

95 
by (simp add: iSuc_def) 

96 

97 
lemma iSuc_number_of: "iSuc (number_of k) = Fin (Suc (number_of k))" 

98 
by (simp add: iSuc_Fin number_of_inat_def) 

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lemma iSuc_Infty [simp]: "iSuc \<infinity> = \<infinity>" 
27110  101 
by (simp add: iSuc_def) 
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lemma iSuc_ne_0 [simp]: "iSuc n \<noteq> 0" 
27110  104 
by (simp add: iSuc_def zero_inat_def split: inat.splits) 
105 

106 
lemma zero_ne_iSuc [simp]: "0 \<noteq> iSuc n" 

107 
by (rule iSuc_ne_0 [symmetric]) 

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27110  109 
lemma iSuc_inject [simp]: "iSuc m = iSuc n \<longleftrightarrow> m = n" 
110 
by (simp add: iSuc_def split: inat.splits) 

111 

112 
lemma number_of_inat_inject [simp]: 

113 
"(number_of k \<Colon> inat) = number_of l \<longleftrightarrow> (number_of k \<Colon> nat) = number_of l" 

114 
by (simp add: number_of_inat_def) 

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27110  117 
subsection {* Addition *} 
118 

119 
instantiation inat :: comm_monoid_add 

120 
begin 

121 

122 
definition 

123 
[code del]: "m + n = (case m of \<infinity> \<Rightarrow> \<infinity>  Fin m \<Rightarrow> (case n of \<infinity> \<Rightarrow> \<infinity>  Fin n \<Rightarrow> Fin (m + n)))" 

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27110  125 
lemma plus_inat_simps [simp, code]: 
126 
"Fin m + Fin n = Fin (m + n)" 

127 
"\<infinity> + q = \<infinity>" 

128 
"q + \<infinity> = \<infinity>" 

129 
by (simp_all add: plus_inat_def split: inat.splits) 

130 

131 
instance proof 

132 
fix n m q :: inat 

133 
show "n + m + q = n + (m + q)" 

134 
by (cases n, auto, cases m, auto, cases q, auto) 

135 
show "n + m = m + n" 

136 
by (cases n, auto, cases m, auto) 

137 
show "0 + n = n" 

138 
by (cases n) (simp_all add: zero_inat_def) 

26089  139 
qed 
140 

27110  141 
end 
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27110  143 
lemma plus_inat_0 [simp]: 
144 
"0 + (q\<Colon>inat) = q" 

145 
"(q\<Colon>inat) + 0 = q" 

146 
by (simp_all add: plus_inat_def zero_inat_def split: inat.splits) 

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27110  148 
lemma plus_inat_number [simp]: 
149 
"(number_of k \<Colon> inat) + number_of l = (if neg (number_of k \<Colon> int) then number_of l 

150 
else if neg (number_of l \<Colon> int) then number_of k else number_of (k + l))" 

151 
unfolding number_of_inat_def plus_inat_simps nat_arith(1) if_distrib [symmetric, of _ Fin] .. 

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27110  153 
lemma iSuc_number [simp]: 
154 
"iSuc (number_of k) = (if neg (number_of k \<Colon> int) then 1 else number_of (Int.succ k))" 

155 
unfolding iSuc_number_of 

156 
unfolding one_inat_def number_of_inat_def Suc_nat_number_of if_distrib [symmetric] .. 

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27110  158 
lemma iSuc_plus_1: 
159 
"iSuc n = n + 1" 

160 
by (cases n) (simp_all add: iSuc_Fin one_inat_def) 

161 

162 
lemma plus_1_iSuc: 

163 
"1 + q = iSuc q" 

164 
"q + 1 = iSuc q" 

165 
unfolding iSuc_plus_1 by (simp_all add: add_ac) 

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27110  168 
subsection {* Ordering *} 
169 

170 
instantiation inat :: ordered_ab_semigroup_add 

171 
begin 

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27110  173 
definition 
174 
[code del]: "m \<le> n = (case n of Fin n1 \<Rightarrow> (case m of Fin m1 \<Rightarrow> m1 \<le> n1  \<infinity> \<Rightarrow> False) 

175 
 \<infinity> \<Rightarrow> True)" 

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27110  177 
definition 
178 
[code del]: "m < n = (case m of Fin m1 \<Rightarrow> (case n of Fin n1 \<Rightarrow> m1 < n1  \<infinity> \<Rightarrow> True) 

179 
 \<infinity> \<Rightarrow> False)" 

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27110  181 
lemma inat_ord_simps [simp]: 
182 
"Fin m \<le> Fin n \<longleftrightarrow> m \<le> n" 

183 
"Fin m < Fin n \<longleftrightarrow> m < n" 

184 
"q \<le> \<infinity>" 

185 
"q < \<infinity> \<longleftrightarrow> q \<noteq> \<infinity>" 

186 
"\<infinity> \<le> q \<longleftrightarrow> q = \<infinity>" 

187 
"\<infinity> < q \<longleftrightarrow> False" 

188 
by (simp_all add: less_eq_inat_def less_inat_def split: inat.splits) 

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27110  190 
lemma inat_ord_code [code]: 
191 
"Fin m \<le> Fin n \<longleftrightarrow> m \<le> n" 

192 
"Fin m < Fin n \<longleftrightarrow> m < n" 

193 
"q \<le> \<infinity> \<longleftrightarrow> True" 

194 
"Fin m < \<infinity> \<longleftrightarrow> True" 

195 
"\<infinity> \<le> Fin n \<longleftrightarrow> False" 

196 
"\<infinity> < q \<longleftrightarrow> False" 

197 
by simp_all 

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27110  199 
instance by default 
200 
(auto simp add: less_eq_inat_def less_inat_def plus_inat_def split: inat.splits) 

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27110  202 
end 
203 

204 
lemma inat_ord_number [simp]: 

205 
"(number_of m \<Colon> inat) \<le> number_of n \<longleftrightarrow> (number_of m \<Colon> nat) \<le> number_of n" 

206 
"(number_of m \<Colon> inat) < number_of n \<longleftrightarrow> (number_of m \<Colon> nat) < number_of n" 

207 
by (simp_all add: number_of_inat_def) 

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27110  209 
lemma i0_lb [simp]: "(0\<Colon>inat) \<le> n" 
210 
by (simp add: zero_inat_def less_eq_inat_def split: inat.splits) 

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27110  212 
lemma i0_neq [simp]: "n \<le> (0\<Colon>inat) \<longleftrightarrow> n = 0" 
213 
by (simp add: zero_inat_def less_eq_inat_def split: inat.splits) 

214 

215 
lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m \<Longrightarrow> R" 

216 
by (simp add: zero_inat_def less_eq_inat_def split: inat.splits) 

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27110  218 
lemma Infty_ilessE [elim!]: "\<infinity> < Fin m \<Longrightarrow> R" 
219 
by simp 

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27110  221 
lemma not_ilessi0 [simp]: "\<not> n < (0\<Colon>inat)" 
222 
by (simp add: zero_inat_def less_inat_def split: inat.splits) 

223 

224 
lemma i0_eq [simp]: "(0\<Colon>inat) < n \<longleftrightarrow> n \<noteq> 0" 

225 
by (simp add: zero_inat_def less_inat_def split: inat.splits) 

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27110  227 
lemma iSuc_ile_mono [simp]: "iSuc n \<le> iSuc m \<longleftrightarrow> n \<le> m" 
228 
by (simp add: iSuc_def less_eq_inat_def split: inat.splits) 

229 

230 
lemma iSuc_mono [simp]: "iSuc n < iSuc m \<longleftrightarrow> n < m" 

231 
by (simp add: iSuc_def less_inat_def split: inat.splits) 

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27110  233 
lemma ile_iSuc [simp]: "n \<le> iSuc n" 
234 
by (simp add: iSuc_def less_eq_inat_def split: inat.splits) 

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11355  236 
lemma not_iSuc_ilei0 [simp]: "\<not> iSuc n \<le> 0" 
27110  237 
by (simp add: zero_inat_def iSuc_def less_eq_inat_def split: inat.splits) 
238 

239 
lemma i0_iless_iSuc [simp]: "0 < iSuc n" 

240 
by (simp add: zero_inat_def iSuc_def less_inat_def split: inat.splits) 

241 

242 
lemma ileI1: "m < n \<Longrightarrow> iSuc m \<le> n" 

243 
by (simp add: iSuc_def less_eq_inat_def less_inat_def split: inat.splits) 

244 

245 
lemma Suc_ile_eq: "Fin (Suc m) \<le> n \<longleftrightarrow> Fin m < n" 

246 
by (cases n) auto 

247 

248 
lemma iless_Suc_eq [simp]: "Fin m < iSuc n \<longleftrightarrow> Fin m \<le> n" 

249 
by (auto simp add: iSuc_def less_inat_def split: inat.splits) 

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27110  251 
lemma min_inat_simps [simp]: 
252 
"min (Fin m) (Fin n) = Fin (min m n)" 

253 
"min q 0 = 0" 

254 
"min 0 q = 0" 

255 
"min q \<infinity> = q" 

256 
"min \<infinity> q = q" 

257 
by (auto simp add: min_def) 

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27110  259 
lemma max_inat_simps [simp]: 
260 
"max (Fin m) (Fin n) = Fin (max m n)" 

261 
"max q 0 = q" 

262 
"max 0 q = q" 

263 
"max q \<infinity> = \<infinity>" 

264 
"max \<infinity> q = \<infinity>" 

265 
by (simp_all add: max_def) 

266 

267 
lemma Fin_ile: "n \<le> Fin m \<Longrightarrow> \<exists>k. n = Fin k" 

268 
by (cases n) simp_all 

269 

270 
lemma Fin_iless: "n < Fin m \<Longrightarrow> \<exists>k. n = Fin k" 

271 
by (cases n) simp_all 

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lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. Fin k < Y j" 
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apply (induct_tac k) 
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apply (simp (no_asm) only: Fin_0) 
27110  276 
apply (fast intro: le_less_trans [OF i0_lb]) 
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apply (erule exE) 
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apply (drule spec) 
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apply (erule exE) 
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apply (drule ileI1) 
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apply (rule iSuc_Fin [THEN subst]) 
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apply (rule exI) 
27110  283 
apply (erule (1) le_less_trans) 
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done 
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26089  286 

27110  287 
subsection {* Wellordering *} 
26089  288 

289 
lemma less_FinE: 

290 
"[ n < Fin m; !!k. n = Fin k ==> k < m ==> P ] ==> P" 

291 
by (induct n) auto 

292 

293 
lemma less_InftyE: 

294 
"[ n < Infty; !!k. n = Fin k ==> P ] ==> P" 

295 
by (induct n) auto 

296 

297 
lemma inat_less_induct: 

298 
assumes prem: "!!n. \<forall>m::inat. m < n > P m ==> P n" shows "P n" 

299 
proof  

300 
have P_Fin: "!!k. P (Fin k)" 

301 
apply (rule nat_less_induct) 

302 
apply (rule prem, clarify) 

303 
apply (erule less_FinE, simp) 

304 
done 

305 
show ?thesis 

306 
proof (induct n) 

307 
fix nat 

308 
show "P (Fin nat)" by (rule P_Fin) 

309 
next 

310 
show "P Infty" 

311 
apply (rule prem, clarify) 

312 
apply (erule less_InftyE) 

313 
apply (simp add: P_Fin) 

314 
done 

315 
qed 

316 
qed 

317 

318 
instance inat :: wellorder 

319 
proof 

320 
show "wf {(x::inat, y::inat). x < y}" 

321 
proof (rule wfUNIVI) 

322 
fix P and x :: inat 

323 
assume "\<forall>x::inat. (\<forall>y. (y, x) \<in> {(x, y). x < y} \<longrightarrow> P y) \<longrightarrow> P x" 

324 
hence 1: "!!x::inat. ALL y. y < x > P y ==> P x" by fast 

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thus "P x" by (rule inat_less_induct) 

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qed 

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qed 

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subsection {* Traditional theorem names *} 

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lemmas inat_defs = zero_inat_def one_inat_def number_of_inat_def iSuc_def 

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plus_inat_def less_eq_inat_def less_inat_def 

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lemmas inat_splits = inat.splits 

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end 