author  noschinl 
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permissions  rwrr 
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(* Title: HOL/Library/Nat_Infinity.thy 
27110  2 
Author: David von Oheimb, TU Muenchen; Florian Haftmann, TU Muenchen 
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Contributions: David Trachtenherz, 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 
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Main is (Complex_Main) base entry point in library theories
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imports Main 
15131  10 
begin 
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27110  12 
subsection {* Type definition *} 
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text {* 
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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>") 
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21210  24 
notation (HTML output) 
19736  25 
Infty ("\<infinity>") 
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31084  28 
lemma not_Infty_eq[iff]: "(x ~= Infty) = (EX i. x = Fin i)" 
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by (cases x) auto 

30 

31 
lemma not_Fin_eq [iff]: "(ALL y. x ~= Fin y) = (x = Infty)" 

31077  32 
by (cases x) auto 
33 

34 

41855  35 
primrec the_Fin :: "inat \<Rightarrow> nat" 
36 
where "the_Fin (Fin n) = n" 

37 

38 

27110  39 
subsection {* Constructors and numbers *} 
40 

41 
instantiation inat :: "{zero, one, number}" 

25594  42 
begin 
43 

44 
definition 

27110  45 
"0 = Fin 0" 
25594  46 

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definition 

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[code_unfold]: "1 = Fin 1" 
25594  49 

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definition 

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[code_unfold, code del]: "number_of k = Fin (number_of k)" 
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25594  53 
instance .. 
54 

55 
end 

56 

27110  57 
definition iSuc :: "inat \<Rightarrow> inat" where 
58 
"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  61 
by (simp add: zero_inat_def) 
62 

63 
lemma Fin_1: "Fin 1 = 1" 

64 
by (simp add: one_inat_def) 

65 

66 
lemma Fin_number: "Fin (number_of k) = number_of k" 

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by (simp add: number_of_inat_def) 

68 

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lemma one_iSuc: "1 = iSuc 0" 

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by (simp add: zero_inat_def one_inat_def iSuc_def) 

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lemma Infty_ne_i0 [simp]: "\<infinity> \<noteq> 0" 
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by (simp add: zero_inat_def) 
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lemma i0_ne_Infty [simp]: "0 \<noteq> \<infinity>" 
27110  76 
by (simp add: zero_inat_def) 
77 

78 
lemma zero_inat_eq [simp]: 

79 
"number_of k = (0\<Colon>inat) \<longleftrightarrow> number_of k = (0\<Colon>nat)" 

80 
"(0\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (0\<Colon>nat)" 

81 
unfolding zero_inat_def number_of_inat_def by simp_all 

82 

83 
lemma one_inat_eq [simp]: 

84 
"number_of k = (1\<Colon>inat) \<longleftrightarrow> number_of k = (1\<Colon>nat)" 

85 
"(1\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (1\<Colon>nat)" 

86 
unfolding one_inat_def number_of_inat_def by simp_all 

87 

88 
lemma zero_one_inat_neq [simp]: 

89 
"\<not> 0 = (1\<Colon>inat)" 

90 
"\<not> 1 = (0\<Colon>inat)" 

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unfolding zero_inat_def one_inat_def by simp_all 

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27110  93 
lemma Infty_ne_i1 [simp]: "\<infinity> \<noteq> 1" 
94 
by (simp add: one_inat_def) 

95 

96 
lemma i1_ne_Infty [simp]: "1 \<noteq> \<infinity>" 

97 
by (simp add: one_inat_def) 

98 

99 
lemma Infty_ne_number [simp]: "\<infinity> \<noteq> number_of k" 

100 
by (simp add: number_of_inat_def) 

101 

102 
lemma number_ne_Infty [simp]: "number_of k \<noteq> \<infinity>" 

103 
by (simp add: number_of_inat_def) 

104 

105 
lemma iSuc_Fin: "iSuc (Fin n) = Fin (Suc n)" 

106 
by (simp add: iSuc_def) 

107 

108 
lemma iSuc_number_of: "iSuc (number_of k) = Fin (Suc (number_of k))" 

109 
by (simp add: iSuc_Fin number_of_inat_def) 

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lemma iSuc_Infty [simp]: "iSuc \<infinity> = \<infinity>" 
27110  112 
by (simp add: iSuc_def) 
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lemma iSuc_ne_0 [simp]: "iSuc n \<noteq> 0" 
27110  115 
by (simp add: iSuc_def zero_inat_def split: inat.splits) 
116 

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

118 
by (rule iSuc_ne_0 [symmetric]) 

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27110  120 
lemma iSuc_inject [simp]: "iSuc m = iSuc n \<longleftrightarrow> m = n" 
121 
by (simp add: iSuc_def split: inat.splits) 

122 

123 
lemma number_of_inat_inject [simp]: 

124 
"(number_of k \<Colon> inat) = number_of l \<longleftrightarrow> (number_of k \<Colon> nat) = number_of l" 

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by (simp add: number_of_inat_def) 

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27110  128 
subsection {* Addition *} 
129 

130 
instantiation inat :: comm_monoid_add 

131 
begin 

132 

38167  133 
definition [nitpick_simp]: 
37765  134 
"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  136 
lemma plus_inat_simps [simp, code]: 
137 
"Fin m + Fin n = Fin (m + n)" 

138 
"\<infinity> + q = \<infinity>" 

139 
"q + \<infinity> = \<infinity>" 

140 
by (simp_all add: plus_inat_def split: inat.splits) 

141 

142 
instance proof 

143 
fix n m q :: inat 

144 
show "n + m + q = n + (m + q)" 

145 
by (cases n, auto, cases m, auto, cases q, auto) 

146 
show "n + m = m + n" 

147 
by (cases n, auto, cases m, auto) 

148 
show "0 + n = n" 

149 
by (cases n) (simp_all add: zero_inat_def) 

26089  150 
qed 
151 

27110  152 
end 
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27110  154 
lemma plus_inat_0 [simp]: 
155 
"0 + (q\<Colon>inat) = q" 

156 
"(q\<Colon>inat) + 0 = q" 

157 
by (simp_all add: plus_inat_def zero_inat_def split: inat.splits) 

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27110  159 
lemma plus_inat_number [simp]: 
29012  160 
"(number_of k \<Colon> inat) + number_of l = (if k < Int.Pls then number_of l 
161 
else if l < Int.Pls then number_of k else number_of (k + l))" 

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unfolding number_of_inat_def plus_inat_simps nat_arith(1) if_distrib [symmetric, of _ Fin] .. 
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27110  164 
lemma iSuc_number [simp]: 
165 
"iSuc (number_of k) = (if neg (number_of k \<Colon> int) then 1 else number_of (Int.succ k))" 

166 
unfolding iSuc_number_of 

167 
unfolding one_inat_def number_of_inat_def Suc_nat_number_of if_distrib [symmetric] .. 

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27110  169 
lemma iSuc_plus_1: 
170 
"iSuc n = n + 1" 

171 
by (cases n) (simp_all add: iSuc_Fin one_inat_def) 

172 

173 
lemma plus_1_iSuc: 

174 
"1 + q = iSuc q" 

175 
"q + 1 = iSuc q" 

41853  176 
by (simp_all add: iSuc_plus_1 add_ac) 
177 

178 
lemma iadd_Suc: "iSuc m + n = iSuc (m + n)" 

179 
by (simp_all add: iSuc_plus_1 add_ac) 

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41853  181 
lemma iadd_Suc_right: "m + iSuc n = iSuc (m + n)" 
182 
by (simp only: add_commute[of m] iadd_Suc) 

183 

184 
lemma iadd_is_0: "(m + n = (0::inat)) = (m = 0 \<and> n = 0)" 

185 
by (cases m, cases n, simp_all add: zero_inat_def) 

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29014  187 
subsection {* Multiplication *} 
188 

189 
instantiation inat :: comm_semiring_1 

190 
begin 

191 

38167  192 
definition times_inat_def [nitpick_simp]: 
29014  193 
"m * n = (case m of \<infinity> \<Rightarrow> if n = 0 then 0 else \<infinity>  Fin m \<Rightarrow> 
194 
(case n of \<infinity> \<Rightarrow> if m = 0 then 0 else \<infinity>  Fin n \<Rightarrow> Fin (m * n)))" 

195 

196 
lemma times_inat_simps [simp, code]: 

197 
"Fin m * Fin n = Fin (m * n)" 

198 
"\<infinity> * \<infinity> = \<infinity>" 

199 
"\<infinity> * Fin n = (if n = 0 then 0 else \<infinity>)" 

200 
"Fin m * \<infinity> = (if m = 0 then 0 else \<infinity>)" 

201 
unfolding times_inat_def zero_inat_def 

202 
by (simp_all split: inat.split) 

203 

204 
instance proof 

205 
fix a b c :: inat 

206 
show "(a * b) * c = a * (b * c)" 

207 
unfolding times_inat_def zero_inat_def 

208 
by (simp split: inat.split) 

209 
show "a * b = b * a" 

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unfolding times_inat_def zero_inat_def 

211 
by (simp split: inat.split) 

212 
show "1 * a = a" 

213 
unfolding times_inat_def zero_inat_def one_inat_def 

214 
by (simp split: inat.split) 

215 
show "(a + b) * c = a * c + b * c" 

216 
unfolding times_inat_def zero_inat_def 

217 
by (simp split: inat.split add: left_distrib) 

218 
show "0 * a = 0" 

219 
unfolding times_inat_def zero_inat_def 

220 
by (simp split: inat.split) 

221 
show "a * 0 = 0" 

222 
unfolding times_inat_def zero_inat_def 

223 
by (simp split: inat.split) 

224 
show "(0::inat) \<noteq> 1" 

225 
unfolding zero_inat_def one_inat_def 

226 
by simp 

227 
qed 

228 

229 
end 

230 

231 
lemma mult_iSuc: "iSuc m * n = n + m * n" 

29667  232 
unfolding iSuc_plus_1 by (simp add: algebra_simps) 
29014  233 

234 
lemma mult_iSuc_right: "m * iSuc n = m + m * n" 

29667  235 
unfolding iSuc_plus_1 by (simp add: algebra_simps) 
29014  236 

29023  237 
lemma of_nat_eq_Fin: "of_nat n = Fin n" 
238 
apply (induct n) 

239 
apply (simp add: Fin_0) 

240 
apply (simp add: plus_1_iSuc iSuc_Fin) 

241 
done 

242 

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instance inat :: semiring_char_0 proof 
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have "inj Fin" by (rule injI) simp 
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then show "inj (\<lambda>n. of_nat n :: inat)" by (simp add: of_nat_eq_Fin) 
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qed 
29023  247 

41853  248 
lemma imult_is_0[simp]: "((m::inat) * n = 0) = (m = 0 \<or> n = 0)" 
249 
by(auto simp add: times_inat_def zero_inat_def split: inat.split) 

250 

251 
lemma imult_is_Infty: "((a::inat) * b = \<infinity>) = (a = \<infinity> \<and> b \<noteq> 0 \<or> b = \<infinity> \<and> a \<noteq> 0)" 

252 
by(auto simp add: times_inat_def zero_inat_def split: inat.split) 

253 

254 

255 
subsection {* Subtraction *} 

256 

257 
instantiation inat :: minus 

258 
begin 

259 

260 
definition diff_inat_def: 

261 
"a  b = (case a of (Fin x) \<Rightarrow> (case b of (Fin y) \<Rightarrow> Fin (x  y)  \<infinity> \<Rightarrow> 0) 

262 
 \<infinity> \<Rightarrow> \<infinity>)" 

263 

264 
instance .. 

265 

266 
end 

267 

268 
lemma idiff_Fin_Fin[simp,code]: "Fin a  Fin b = Fin (a  b)" 

269 
by(simp add: diff_inat_def) 

270 

271 
lemma idiff_Infty[simp,code]: "\<infinity>  n = \<infinity>" 

272 
by(simp add: diff_inat_def) 

273 

274 
lemma idiff_Infty_right[simp,code]: "Fin a  \<infinity> = 0" 

275 
by(simp add: diff_inat_def) 

276 

277 
lemma idiff_0[simp]: "(0::inat)  n = 0" 

278 
by (cases n, simp_all add: zero_inat_def) 

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280 
lemmas idiff_Fin_0[simp] = idiff_0[unfolded zero_inat_def] 

281 

282 
lemma idiff_0_right[simp]: "(n::inat)  0 = n" 

283 
by (cases n) (simp_all add: zero_inat_def) 

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285 
lemmas idiff_Fin_0_right[simp] = idiff_0_right[unfolded zero_inat_def] 

286 

287 
lemma idiff_self[simp]: "n \<noteq> \<infinity> \<Longrightarrow> (n::inat)  n = 0" 

288 
by(auto simp: zero_inat_def) 

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41855  290 
lemma iSuc_minus_iSuc [simp]: "iSuc n  iSuc m = n  m" 
291 
by(simp add: iSuc_def split: inat.split) 

292 

293 
lemma iSuc_minus_1 [simp]: "iSuc n  1 = n" 

294 
by(simp add: one_inat_def iSuc_Fin[symmetric] zero_inat_def[symmetric]) 

295 

41853  296 
(*lemmas idiff_self_eq_0_Fin = idiff_self_eq_0[unfolded zero_inat_def]*) 
297 

29014  298 

27110  299 
subsection {* Ordering *} 
300 

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instantiation inat :: linordered_ab_semigroup_add 
27110  302 
begin 
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38167  304 
definition [nitpick_simp]: 
37765  305 
"m \<le> n = (case n of Fin n1 \<Rightarrow> (case m of Fin m1 \<Rightarrow> m1 \<le> n1  \<infinity> \<Rightarrow> False) 
27110  306 
 \<infinity> \<Rightarrow> True)" 
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38167  308 
definition [nitpick_simp]: 
37765  309 
"m < n = (case m of Fin m1 \<Rightarrow> (case n of Fin n1 \<Rightarrow> m1 < n1  \<infinity> \<Rightarrow> True) 
27110  310 
 \<infinity> \<Rightarrow> False)" 
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27110  312 
lemma inat_ord_simps [simp]: 
313 
"Fin m \<le> Fin n \<longleftrightarrow> m \<le> n" 

314 
"Fin m < Fin n \<longleftrightarrow> m < n" 

315 
"q \<le> \<infinity>" 

316 
"q < \<infinity> \<longleftrightarrow> q \<noteq> \<infinity>" 

317 
"\<infinity> \<le> q \<longleftrightarrow> q = \<infinity>" 

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

319 
by (simp_all add: less_eq_inat_def less_inat_def split: inat.splits) 

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27110  321 
lemma inat_ord_code [code]: 
322 
"Fin m \<le> Fin n \<longleftrightarrow> m \<le> n" 

323 
"Fin m < Fin n \<longleftrightarrow> m < n" 

324 
"q \<le> \<infinity> \<longleftrightarrow> True" 

325 
"Fin m < \<infinity> \<longleftrightarrow> True" 

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

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

328 
by simp_all 

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27110  330 
instance by default 
331 
(auto simp add: less_eq_inat_def less_inat_def plus_inat_def split: inat.splits) 

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27110  333 
end 
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instance inat :: ordered_comm_semiring 
29014  336 
proof 
337 
fix a b c :: inat 

338 
assume "a \<le> b" and "0 \<le> c" 

339 
thus "c * a \<le> c * b" 

340 
unfolding times_inat_def less_eq_inat_def zero_inat_def 

341 
by (simp split: inat.splits) 

342 
qed 

343 

27110  344 
lemma inat_ord_number [simp]: 
345 
"(number_of m \<Colon> inat) \<le> number_of n \<longleftrightarrow> (number_of m \<Colon> nat) \<le> number_of n" 

346 
"(number_of m \<Colon> inat) < number_of n \<longleftrightarrow> (number_of m \<Colon> nat) < number_of n" 

347 
by (simp_all add: number_of_inat_def) 

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27110  349 
lemma i0_lb [simp]: "(0\<Colon>inat) \<le> n" 
350 
by (simp add: zero_inat_def less_eq_inat_def split: inat.splits) 

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351 

41853  352 
lemma ile0_eq [simp]: "n \<le> (0\<Colon>inat) \<longleftrightarrow> n = 0" 
353 
by (simp add: zero_inat_def less_eq_inat_def split: inat.splits) 

27110  354 

355 
lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m \<Longrightarrow> R" 

356 
by (simp add: zero_inat_def less_eq_inat_def split: inat.splits) 

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27110  358 
lemma Infty_ilessE [elim!]: "\<infinity> < Fin m \<Longrightarrow> R" 
359 
by simp 

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41853  361 
lemma not_iless0 [simp]: "\<not> n < (0\<Colon>inat)" 
27110  362 
by (simp add: zero_inat_def less_inat_def split: inat.splits) 
363 

41853  364 
lemma i0_less [simp]: "(0\<Colon>inat) < n \<longleftrightarrow> n \<noteq> 0" 
365 
by (simp add: zero_inat_def less_inat_def split: inat.splits) 

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366 

27110  367 
lemma iSuc_ile_mono [simp]: "iSuc n \<le> iSuc m \<longleftrightarrow> n \<le> m" 
368 
by (simp add: iSuc_def less_eq_inat_def split: inat.splits) 

369 

370 
lemma iSuc_mono [simp]: "iSuc n < iSuc m \<longleftrightarrow> n < m" 

371 
by (simp add: iSuc_def less_inat_def split: inat.splits) 

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27110  373 
lemma ile_iSuc [simp]: "n \<le> iSuc n" 
374 
by (simp add: iSuc_def less_eq_inat_def split: inat.splits) 

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375 

11355  376 
lemma not_iSuc_ilei0 [simp]: "\<not> iSuc n \<le> 0" 
27110  377 
by (simp add: zero_inat_def iSuc_def less_eq_inat_def split: inat.splits) 
378 

379 
lemma i0_iless_iSuc [simp]: "0 < iSuc n" 

380 
by (simp add: zero_inat_def iSuc_def less_inat_def split: inat.splits) 

381 

41853  382 
lemma iless_iSuc0[simp]: "(n < iSuc 0) = (n = 0)" 
383 
by (simp add: zero_inat_def iSuc_def less_inat_def split: inat.split) 

384 

27110  385 
lemma ileI1: "m < n \<Longrightarrow> iSuc m \<le> n" 
386 
by (simp add: iSuc_def less_eq_inat_def less_inat_def split: inat.splits) 

387 

388 
lemma Suc_ile_eq: "Fin (Suc m) \<le> n \<longleftrightarrow> Fin m < n" 

389 
by (cases n) auto 

390 

391 
lemma iless_Suc_eq [simp]: "Fin m < iSuc n \<longleftrightarrow> Fin m \<le> n" 

392 
by (auto simp add: iSuc_def less_inat_def split: inat.splits) 

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393 

41853  394 
lemma imult_Infty: "(0::inat) < n \<Longrightarrow> \<infinity> * n = \<infinity>" 
395 
by (simp add: zero_inat_def less_inat_def split: inat.splits) 

396 

397 
lemma imult_Infty_right: "(0::inat) < n \<Longrightarrow> n * \<infinity> = \<infinity>" 

398 
by (simp add: zero_inat_def less_inat_def split: inat.splits) 

399 

400 
lemma inat_0_less_mult_iff: "(0 < (m::inat) * n) = (0 < m \<and> 0 < n)" 

401 
by (simp only: i0_less imult_is_0, simp) 

402 

403 
lemma mono_iSuc: "mono iSuc" 

404 
by(simp add: mono_def) 

405 

406 

27110  407 
lemma min_inat_simps [simp]: 
408 
"min (Fin m) (Fin n) = Fin (min m n)" 

409 
"min q 0 = 0" 

410 
"min 0 q = 0" 

411 
"min q \<infinity> = q" 

412 
"min \<infinity> q = q" 

413 
by (auto simp add: min_def) 

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414 

27110  415 
lemma max_inat_simps [simp]: 
416 
"max (Fin m) (Fin n) = Fin (max m n)" 

417 
"max q 0 = q" 

418 
"max 0 q = q" 

419 
"max q \<infinity> = \<infinity>" 

420 
"max \<infinity> q = \<infinity>" 

421 
by (simp_all add: max_def) 

422 

423 
lemma Fin_ile: "n \<le> Fin m \<Longrightarrow> \<exists>k. n = Fin k" 

424 
by (cases n) simp_all 

425 

426 
lemma Fin_iless: "n < Fin m \<Longrightarrow> \<exists>k. n = Fin k" 

427 
by (cases n) simp_all 

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428 

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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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430 
apply (induct_tac k) 
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apply (simp (no_asm) only: Fin_0) 
27110  432 
apply (fast intro: le_less_trans [OF i0_lb]) 
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433 
apply (erule exE) 
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434 
apply (drule spec) 
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435 
apply (erule exE) 
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436 
apply (drule ileI1) 
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437 
apply (rule iSuc_Fin [THEN subst]) 
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438 
apply (rule exI) 
27110  439 
apply (erule (1) le_less_trans) 
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440 
done 
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441 

29337  442 
instantiation inat :: "{bot, top}" 
443 
begin 

444 

445 
definition bot_inat :: inat where 

446 
"bot_inat = 0" 

447 

448 
definition top_inat :: inat where 

449 
"top_inat = \<infinity>" 

450 

451 
instance proof 

452 
qed (simp_all add: bot_inat_def top_inat_def) 

453 

454 
end 

455 

42993  456 
lemma finite_Fin_bounded: 
457 
assumes le_fin: "\<And>y. y \<in> A \<Longrightarrow> y \<le> Fin n" 

458 
shows "finite A" 

459 
proof (rule finite_subset) 

460 
show "finite (Fin ` {..n})" by blast 

461 

462 
have "A \<subseteq> {..Fin n}" using le_fin by fastsimp 

463 
also have "\<dots> \<subseteq> Fin ` {..n}" 

464 
by (rule subsetI) (case_tac x, auto) 

465 
finally show "A \<subseteq> Fin ` {..n}" . 

466 
qed 

467 

26089  468 

27110  469 
subsection {* Wellordering *} 
26089  470 

471 
lemma less_FinE: 

472 
"[ n < Fin m; !!k. n = Fin k ==> k < m ==> P ] ==> P" 

473 
by (induct n) auto 

474 

475 
lemma less_InftyE: 

476 
"[ n < Infty; !!k. n = Fin k ==> P ] ==> P" 

477 
by (induct n) auto 

478 

479 
lemma inat_less_induct: 

480 
assumes prem: "!!n. \<forall>m::inat. m < n > P m ==> P n" shows "P n" 

481 
proof  

482 
have P_Fin: "!!k. P (Fin k)" 

483 
apply (rule nat_less_induct) 

484 
apply (rule prem, clarify) 

485 
apply (erule less_FinE, simp) 

486 
done 

487 
show ?thesis 

488 
proof (induct n) 

489 
fix nat 

490 
show "P (Fin nat)" by (rule P_Fin) 

491 
next 

492 
show "P Infty" 

493 
apply (rule prem, clarify) 

494 
apply (erule less_InftyE) 

495 
apply (simp add: P_Fin) 

496 
done 

497 
qed 

498 
qed 

499 

500 
instance inat :: wellorder 

501 
proof 

27823  502 
fix P and n 
503 
assume hyp: "(\<And>n\<Colon>inat. (\<And>m\<Colon>inat. m < n \<Longrightarrow> P m) \<Longrightarrow> P n)" 

504 
show "P n" by (blast intro: inat_less_induct hyp) 

26089  505 
qed 
506 

42993  507 
subsection {* Complete Lattice *} 
508 

509 
instantiation inat :: complete_lattice 

510 
begin 

511 

512 
definition inf_inat :: "inat \<Rightarrow> inat \<Rightarrow> inat" where 

513 
"inf_inat \<equiv> min" 

514 

515 
definition sup_inat :: "inat \<Rightarrow> inat \<Rightarrow> inat" where 

516 
"sup_inat \<equiv> max" 

517 

518 
definition Inf_inat :: "inat set \<Rightarrow> inat" where 

519 
"Inf_inat A \<equiv> if A = {} then \<infinity> else (LEAST x. x \<in> A)" 

520 

521 
definition Sup_inat :: "inat set \<Rightarrow> inat" where 

522 
"Sup_inat A \<equiv> if A = {} then 0 

523 
else if finite A then Max A 

524 
else \<infinity>" 

525 
instance proof 

526 
fix x :: "inat" and A :: "inat set" 

527 
{ assume "x \<in> A" then show "Inf A \<le> x" 

528 
unfolding Inf_inat_def by (auto intro: Least_le) } 

529 
{ assume "\<And>y. y \<in> A \<Longrightarrow> x \<le> y" then show "x \<le> Inf A" 

530 
unfolding Inf_inat_def 

531 
by (cases "A = {}") (auto intro: LeastI2_ex) } 

532 
{ assume "x \<in> A" then show "x \<le> Sup A" 

533 
unfolding Sup_inat_def by (cases "finite A") auto } 

534 
{ assume "\<And>y. y \<in> A \<Longrightarrow> y \<le> x" then show "Sup A \<le> x" 

535 
unfolding Sup_inat_def using finite_Fin_bounded by auto } 

536 
qed (simp_all add: inf_inat_def sup_inat_def) 

537 
end 

538 

27110  539 

540 
subsection {* Traditional theorem names *} 

541 

542 
lemmas inat_defs = zero_inat_def one_inat_def number_of_inat_def iSuc_def 

543 
plus_inat_def less_eq_inat_def less_inat_def 

544 

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545 
end 