author | nipkow |
Fri, 08 May 2009 08:06:43 +0200 | |
changeset 31077 | 28dd6fd3d184 |
parent 30663 | 0b6aff7451b2 |
child 31084 | f4db921165ce |
permissions | -rw-r--r-- |
11355 | 1 |
(* Title: HOL/Library/Nat_Infinity.thy |
27110 | 2 |
Author: David von Oheimb, TU Muenchen; Florian Haftmann, TU Muenchen |
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
3 |
*) |
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
4 |
|
14706 | 5 |
header {* Natural numbers with infinity *} |
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
6 |
|
15131 | 7 |
theory Nat_Infinity |
30663
0b6aff7451b2
Main is (Complex_Main) base entry point in library theories
haftmann
parents:
29668
diff
changeset
|
8 |
imports Main |
15131 | 9 |
begin |
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
10 |
|
27110 | 11 |
subsection {* Type definition *} |
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
12 |
|
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
13 |
text {* |
11355 | 14 |
We extend the standard natural numbers by a special value indicating |
27110 | 15 |
infinity. |
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
16 |
*} |
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
17 |
|
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
18 |
datatype inat = Fin nat | Infty |
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
19 |
|
21210 | 20 |
notation (xsymbols) |
19736 | 21 |
Infty ("\<infinity>") |
22 |
||
21210 | 23 |
notation (HTML output) |
19736 | 24 |
Infty ("\<infinity>") |
25 |
||
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
26 |
|
31077 | 27 |
lemma not_Infty_eq[simp]: "(x ~= Infty) = (EX i. x = Fin i)" |
28 |
by (cases x) auto |
|
29 |
||
30 |
||
27110 | 31 |
subsection {* Constructors and numbers *} |
32 |
||
33 |
instantiation inat :: "{zero, one, number}" |
|
25594 | 34 |
begin |
35 |
||
36 |
definition |
|
27110 | 37 |
"0 = Fin 0" |
25594 | 38 |
|
39 |
definition |
|
27110 | 40 |
[code inline]: "1 = Fin 1" |
25594 | 41 |
|
42 |
definition |
|
28562 | 43 |
[code inline, code del]: "number_of k = Fin (number_of k)" |
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
44 |
|
25594 | 45 |
instance .. |
46 |
||
47 |
end |
|
48 |
||
27110 | 49 |
definition iSuc :: "inat \<Rightarrow> inat" where |
50 |
"iSuc i = (case i of Fin n \<Rightarrow> Fin (Suc n) | \<infinity> \<Rightarrow> \<infinity>)" |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
51 |
|
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
52 |
lemma Fin_0: "Fin 0 = 0" |
27110 | 53 |
by (simp add: zero_inat_def) |
54 |
||
55 |
lemma Fin_1: "Fin 1 = 1" |
|
56 |
by (simp add: one_inat_def) |
|
57 |
||
58 |
lemma Fin_number: "Fin (number_of k) = number_of k" |
|
59 |
by (simp add: number_of_inat_def) |
|
60 |
||
61 |
lemma one_iSuc: "1 = iSuc 0" |
|
62 |
by (simp add: zero_inat_def one_inat_def iSuc_def) |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
63 |
|
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
64 |
lemma Infty_ne_i0 [simp]: "\<infinity> \<noteq> 0" |
27110 | 65 |
by (simp add: zero_inat_def) |
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
66 |
|
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
67 |
lemma i0_ne_Infty [simp]: "0 \<noteq> \<infinity>" |
27110 | 68 |
by (simp add: zero_inat_def) |
69 |
||
70 |
lemma zero_inat_eq [simp]: |
|
71 |
"number_of k = (0\<Colon>inat) \<longleftrightarrow> number_of k = (0\<Colon>nat)" |
|
72 |
"(0\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (0\<Colon>nat)" |
|
73 |
unfolding zero_inat_def number_of_inat_def by simp_all |
|
74 |
||
75 |
lemma one_inat_eq [simp]: |
|
76 |
"number_of k = (1\<Colon>inat) \<longleftrightarrow> number_of k = (1\<Colon>nat)" |
|
77 |
"(1\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (1\<Colon>nat)" |
|
78 |
unfolding one_inat_def number_of_inat_def by simp_all |
|
79 |
||
80 |
lemma zero_one_inat_neq [simp]: |
|
81 |
"\<not> 0 = (1\<Colon>inat)" |
|
82 |
"\<not> 1 = (0\<Colon>inat)" |
|
83 |
unfolding zero_inat_def one_inat_def by simp_all |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
84 |
|
27110 | 85 |
lemma Infty_ne_i1 [simp]: "\<infinity> \<noteq> 1" |
86 |
by (simp add: one_inat_def) |
|
87 |
||
88 |
lemma i1_ne_Infty [simp]: "1 \<noteq> \<infinity>" |
|
89 |
by (simp add: one_inat_def) |
|
90 |
||
91 |
lemma Infty_ne_number [simp]: "\<infinity> \<noteq> number_of k" |
|
92 |
by (simp add: number_of_inat_def) |
|
93 |
||
94 |
lemma number_ne_Infty [simp]: "number_of k \<noteq> \<infinity>" |
|
95 |
by (simp add: number_of_inat_def) |
|
96 |
||
97 |
lemma iSuc_Fin: "iSuc (Fin n) = Fin (Suc n)" |
|
98 |
by (simp add: iSuc_def) |
|
99 |
||
100 |
lemma iSuc_number_of: "iSuc (number_of k) = Fin (Suc (number_of k))" |
|
101 |
by (simp add: iSuc_Fin number_of_inat_def) |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
102 |
|
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
103 |
lemma iSuc_Infty [simp]: "iSuc \<infinity> = \<infinity>" |
27110 | 104 |
by (simp add: iSuc_def) |
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
105 |
|
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
106 |
lemma iSuc_ne_0 [simp]: "iSuc n \<noteq> 0" |
27110 | 107 |
by (simp add: iSuc_def zero_inat_def split: inat.splits) |
108 |
||
109 |
lemma zero_ne_iSuc [simp]: "0 \<noteq> iSuc n" |
|
110 |
by (rule iSuc_ne_0 [symmetric]) |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
111 |
|
27110 | 112 |
lemma iSuc_inject [simp]: "iSuc m = iSuc n \<longleftrightarrow> m = n" |
113 |
by (simp add: iSuc_def split: inat.splits) |
|
114 |
||
115 |
lemma number_of_inat_inject [simp]: |
|
116 |
"(number_of k \<Colon> inat) = number_of l \<longleftrightarrow> (number_of k \<Colon> nat) = number_of l" |
|
117 |
by (simp add: number_of_inat_def) |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
118 |
|
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
119 |
|
27110 | 120 |
subsection {* Addition *} |
121 |
||
122 |
instantiation inat :: comm_monoid_add |
|
123 |
begin |
|
124 |
||
125 |
definition |
|
126 |
[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)))" |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
127 |
|
27110 | 128 |
lemma plus_inat_simps [simp, code]: |
129 |
"Fin m + Fin n = Fin (m + n)" |
|
130 |
"\<infinity> + q = \<infinity>" |
|
131 |
"q + \<infinity> = \<infinity>" |
|
132 |
by (simp_all add: plus_inat_def split: inat.splits) |
|
133 |
||
134 |
instance proof |
|
135 |
fix n m q :: inat |
|
136 |
show "n + m + q = n + (m + q)" |
|
137 |
by (cases n, auto, cases m, auto, cases q, auto) |
|
138 |
show "n + m = m + n" |
|
139 |
by (cases n, auto, cases m, auto) |
|
140 |
show "0 + n = n" |
|
141 |
by (cases n) (simp_all add: zero_inat_def) |
|
26089 | 142 |
qed |
143 |
||
27110 | 144 |
end |
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
145 |
|
27110 | 146 |
lemma plus_inat_0 [simp]: |
147 |
"0 + (q\<Colon>inat) = q" |
|
148 |
"(q\<Colon>inat) + 0 = q" |
|
149 |
by (simp_all add: plus_inat_def zero_inat_def split: inat.splits) |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
150 |
|
27110 | 151 |
lemma plus_inat_number [simp]: |
29012 | 152 |
"(number_of k \<Colon> inat) + number_of l = (if k < Int.Pls then number_of l |
153 |
else if l < Int.Pls then number_of k else number_of (k + l))" |
|
27110 | 154 |
unfolding number_of_inat_def plus_inat_simps nat_arith(1) if_distrib [symmetric, of _ Fin] .. |
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
155 |
|
27110 | 156 |
lemma iSuc_number [simp]: |
157 |
"iSuc (number_of k) = (if neg (number_of k \<Colon> int) then 1 else number_of (Int.succ k))" |
|
158 |
unfolding iSuc_number_of |
|
159 |
unfolding one_inat_def number_of_inat_def Suc_nat_number_of if_distrib [symmetric] .. |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
160 |
|
27110 | 161 |
lemma iSuc_plus_1: |
162 |
"iSuc n = n + 1" |
|
163 |
by (cases n) (simp_all add: iSuc_Fin one_inat_def) |
|
164 |
||
165 |
lemma plus_1_iSuc: |
|
166 |
"1 + q = iSuc q" |
|
167 |
"q + 1 = iSuc q" |
|
168 |
unfolding iSuc_plus_1 by (simp_all add: add_ac) |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
169 |
|
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
170 |
|
29014 | 171 |
subsection {* Multiplication *} |
172 |
||
173 |
instantiation inat :: comm_semiring_1 |
|
174 |
begin |
|
175 |
||
176 |
definition |
|
177 |
times_inat_def [code del]: |
|
178 |
"m * n = (case m of \<infinity> \<Rightarrow> if n = 0 then 0 else \<infinity> | Fin m \<Rightarrow> |
|
179 |
(case n of \<infinity> \<Rightarrow> if m = 0 then 0 else \<infinity> | Fin n \<Rightarrow> Fin (m * n)))" |
|
180 |
||
181 |
lemma times_inat_simps [simp, code]: |
|
182 |
"Fin m * Fin n = Fin (m * n)" |
|
183 |
"\<infinity> * \<infinity> = \<infinity>" |
|
184 |
"\<infinity> * Fin n = (if n = 0 then 0 else \<infinity>)" |
|
185 |
"Fin m * \<infinity> = (if m = 0 then 0 else \<infinity>)" |
|
186 |
unfolding times_inat_def zero_inat_def |
|
187 |
by (simp_all split: inat.split) |
|
188 |
||
189 |
instance proof |
|
190 |
fix a b c :: inat |
|
191 |
show "(a * b) * c = a * (b * c)" |
|
192 |
unfolding times_inat_def zero_inat_def |
|
193 |
by (simp split: inat.split) |
|
194 |
show "a * b = b * a" |
|
195 |
unfolding times_inat_def zero_inat_def |
|
196 |
by (simp split: inat.split) |
|
197 |
show "1 * a = a" |
|
198 |
unfolding times_inat_def zero_inat_def one_inat_def |
|
199 |
by (simp split: inat.split) |
|
200 |
show "(a + b) * c = a * c + b * c" |
|
201 |
unfolding times_inat_def zero_inat_def |
|
202 |
by (simp split: inat.split add: left_distrib) |
|
203 |
show "0 * a = 0" |
|
204 |
unfolding times_inat_def zero_inat_def |
|
205 |
by (simp split: inat.split) |
|
206 |
show "a * 0 = 0" |
|
207 |
unfolding times_inat_def zero_inat_def |
|
208 |
by (simp split: inat.split) |
|
209 |
show "(0::inat) \<noteq> 1" |
|
210 |
unfolding zero_inat_def one_inat_def |
|
211 |
by simp |
|
212 |
qed |
|
213 |
||
214 |
end |
|
215 |
||
216 |
lemma mult_iSuc: "iSuc m * n = n + m * n" |
|
29667 | 217 |
unfolding iSuc_plus_1 by (simp add: algebra_simps) |
29014 | 218 |
|
219 |
lemma mult_iSuc_right: "m * iSuc n = m + m * n" |
|
29667 | 220 |
unfolding iSuc_plus_1 by (simp add: algebra_simps) |
29014 | 221 |
|
29023 | 222 |
lemma of_nat_eq_Fin: "of_nat n = Fin n" |
223 |
apply (induct n) |
|
224 |
apply (simp add: Fin_0) |
|
225 |
apply (simp add: plus_1_iSuc iSuc_Fin) |
|
226 |
done |
|
227 |
||
228 |
instance inat :: semiring_char_0 |
|
229 |
by default (simp add: of_nat_eq_Fin) |
|
230 |
||
29014 | 231 |
|
27110 | 232 |
subsection {* Ordering *} |
233 |
||
234 |
instantiation inat :: ordered_ab_semigroup_add |
|
235 |
begin |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
236 |
|
27110 | 237 |
definition |
238 |
[code del]: "m \<le> n = (case n of Fin n1 \<Rightarrow> (case m of Fin m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False) |
|
239 |
| \<infinity> \<Rightarrow> True)" |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
240 |
|
27110 | 241 |
definition |
242 |
[code del]: "m < n = (case m of Fin m1 \<Rightarrow> (case n of Fin n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True) |
|
243 |
| \<infinity> \<Rightarrow> False)" |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
244 |
|
27110 | 245 |
lemma inat_ord_simps [simp]: |
246 |
"Fin m \<le> Fin n \<longleftrightarrow> m \<le> n" |
|
247 |
"Fin m < Fin n \<longleftrightarrow> m < n" |
|
248 |
"q \<le> \<infinity>" |
|
249 |
"q < \<infinity> \<longleftrightarrow> q \<noteq> \<infinity>" |
|
250 |
"\<infinity> \<le> q \<longleftrightarrow> q = \<infinity>" |
|
251 |
"\<infinity> < q \<longleftrightarrow> False" |
|
252 |
by (simp_all add: less_eq_inat_def less_inat_def split: inat.splits) |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
253 |
|
27110 | 254 |
lemma inat_ord_code [code]: |
255 |
"Fin m \<le> Fin n \<longleftrightarrow> m \<le> n" |
|
256 |
"Fin m < Fin n \<longleftrightarrow> m < n" |
|
257 |
"q \<le> \<infinity> \<longleftrightarrow> True" |
|
258 |
"Fin m < \<infinity> \<longleftrightarrow> True" |
|
259 |
"\<infinity> \<le> Fin n \<longleftrightarrow> False" |
|
260 |
"\<infinity> < q \<longleftrightarrow> False" |
|
261 |
by simp_all |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
262 |
|
27110 | 263 |
instance by default |
264 |
(auto simp add: less_eq_inat_def less_inat_def plus_inat_def split: inat.splits) |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
265 |
|
27110 | 266 |
end |
267 |
||
31077 | 268 |
instance inat :: linorder |
269 |
by intro_classes (auto simp add: less_eq_inat_def split: inat.splits) |
|
270 |
||
29014 | 271 |
instance inat :: pordered_comm_semiring |
272 |
proof |
|
273 |
fix a b c :: inat |
|
274 |
assume "a \<le> b" and "0 \<le> c" |
|
275 |
thus "c * a \<le> c * b" |
|
276 |
unfolding times_inat_def less_eq_inat_def zero_inat_def |
|
277 |
by (simp split: inat.splits) |
|
278 |
qed |
|
279 |
||
27110 | 280 |
lemma inat_ord_number [simp]: |
281 |
"(number_of m \<Colon> inat) \<le> number_of n \<longleftrightarrow> (number_of m \<Colon> nat) \<le> number_of n" |
|
282 |
"(number_of m \<Colon> inat) < number_of n \<longleftrightarrow> (number_of m \<Colon> nat) < number_of n" |
|
283 |
by (simp_all add: number_of_inat_def) |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
284 |
|
27110 | 285 |
lemma i0_lb [simp]: "(0\<Colon>inat) \<le> n" |
286 |
by (simp add: zero_inat_def less_eq_inat_def split: inat.splits) |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
287 |
|
27110 | 288 |
lemma i0_neq [simp]: "n \<le> (0\<Colon>inat) \<longleftrightarrow> n = 0" |
289 |
by (simp add: zero_inat_def less_eq_inat_def split: inat.splits) |
|
290 |
||
291 |
lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m \<Longrightarrow> R" |
|
292 |
by (simp add: zero_inat_def less_eq_inat_def split: inat.splits) |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
293 |
|
27110 | 294 |
lemma Infty_ilessE [elim!]: "\<infinity> < Fin m \<Longrightarrow> R" |
295 |
by simp |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
296 |
|
27110 | 297 |
lemma not_ilessi0 [simp]: "\<not> n < (0\<Colon>inat)" |
298 |
by (simp add: zero_inat_def less_inat_def split: inat.splits) |
|
299 |
||
300 |
lemma i0_eq [simp]: "(0\<Colon>inat) < n \<longleftrightarrow> n \<noteq> 0" |
|
301 |
by (simp add: zero_inat_def less_inat_def split: inat.splits) |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
302 |
|
27110 | 303 |
lemma iSuc_ile_mono [simp]: "iSuc n \<le> iSuc m \<longleftrightarrow> n \<le> m" |
304 |
by (simp add: iSuc_def less_eq_inat_def split: inat.splits) |
|
305 |
||
306 |
lemma iSuc_mono [simp]: "iSuc n < iSuc m \<longleftrightarrow> n < m" |
|
307 |
by (simp add: iSuc_def less_inat_def split: inat.splits) |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
308 |
|
27110 | 309 |
lemma ile_iSuc [simp]: "n \<le> iSuc n" |
310 |
by (simp add: iSuc_def less_eq_inat_def split: inat.splits) |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
311 |
|
11355 | 312 |
lemma not_iSuc_ilei0 [simp]: "\<not> iSuc n \<le> 0" |
27110 | 313 |
by (simp add: zero_inat_def iSuc_def less_eq_inat_def split: inat.splits) |
314 |
||
315 |
lemma i0_iless_iSuc [simp]: "0 < iSuc n" |
|
316 |
by (simp add: zero_inat_def iSuc_def less_inat_def split: inat.splits) |
|
317 |
||
318 |
lemma ileI1: "m < n \<Longrightarrow> iSuc m \<le> n" |
|
319 |
by (simp add: iSuc_def less_eq_inat_def less_inat_def split: inat.splits) |
|
320 |
||
321 |
lemma Suc_ile_eq: "Fin (Suc m) \<le> n \<longleftrightarrow> Fin m < n" |
|
322 |
by (cases n) auto |
|
323 |
||
324 |
lemma iless_Suc_eq [simp]: "Fin m < iSuc n \<longleftrightarrow> Fin m \<le> n" |
|
325 |
by (auto simp add: iSuc_def less_inat_def split: inat.splits) |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
326 |
|
27110 | 327 |
lemma min_inat_simps [simp]: |
328 |
"min (Fin m) (Fin n) = Fin (min m n)" |
|
329 |
"min q 0 = 0" |
|
330 |
"min 0 q = 0" |
|
331 |
"min q \<infinity> = q" |
|
332 |
"min \<infinity> q = q" |
|
333 |
by (auto simp add: min_def) |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
334 |
|
27110 | 335 |
lemma max_inat_simps [simp]: |
336 |
"max (Fin m) (Fin n) = Fin (max m n)" |
|
337 |
"max q 0 = q" |
|
338 |
"max 0 q = q" |
|
339 |
"max q \<infinity> = \<infinity>" |
|
340 |
"max \<infinity> q = \<infinity>" |
|
341 |
by (simp_all add: max_def) |
|
342 |
||
343 |
lemma Fin_ile: "n \<le> Fin m \<Longrightarrow> \<exists>k. n = Fin k" |
|
344 |
by (cases n) simp_all |
|
345 |
||
346 |
lemma Fin_iless: "n < Fin m \<Longrightarrow> \<exists>k. n = Fin k" |
|
347 |
by (cases n) simp_all |
|
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
348 |
|
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
349 |
lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. Fin k < Y j" |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
350 |
apply (induct_tac k) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
351 |
apply (simp (no_asm) only: Fin_0) |
27110 | 352 |
apply (fast intro: le_less_trans [OF i0_lb]) |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
353 |
apply (erule exE) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
354 |
apply (drule spec) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
355 |
apply (erule exE) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
356 |
apply (drule ileI1) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
357 |
apply (rule iSuc_Fin [THEN subst]) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
358 |
apply (rule exI) |
27110 | 359 |
apply (erule (1) le_less_trans) |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
360 |
done |
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
361 |
|
29337 | 362 |
instantiation inat :: "{bot, top}" |
363 |
begin |
|
364 |
||
365 |
definition bot_inat :: inat where |
|
366 |
"bot_inat = 0" |
|
367 |
||
368 |
definition top_inat :: inat where |
|
369 |
"top_inat = \<infinity>" |
|
370 |
||
371 |
instance proof |
|
372 |
qed (simp_all add: bot_inat_def top_inat_def) |
|
373 |
||
374 |
end |
|
375 |
||
26089 | 376 |
|
27110 | 377 |
subsection {* Well-ordering *} |
26089 | 378 |
|
379 |
lemma less_FinE: |
|
380 |
"[| n < Fin m; !!k. n = Fin k ==> k < m ==> P |] ==> P" |
|
381 |
by (induct n) auto |
|
382 |
||
383 |
lemma less_InftyE: |
|
384 |
"[| n < Infty; !!k. n = Fin k ==> P |] ==> P" |
|
385 |
by (induct n) auto |
|
386 |
||
387 |
lemma inat_less_induct: |
|
388 |
assumes prem: "!!n. \<forall>m::inat. m < n --> P m ==> P n" shows "P n" |
|
389 |
proof - |
|
390 |
have P_Fin: "!!k. P (Fin k)" |
|
391 |
apply (rule nat_less_induct) |
|
392 |
apply (rule prem, clarify) |
|
393 |
apply (erule less_FinE, simp) |
|
394 |
done |
|
395 |
show ?thesis |
|
396 |
proof (induct n) |
|
397 |
fix nat |
|
398 |
show "P (Fin nat)" by (rule P_Fin) |
|
399 |
next |
|
400 |
show "P Infty" |
|
401 |
apply (rule prem, clarify) |
|
402 |
apply (erule less_InftyE) |
|
403 |
apply (simp add: P_Fin) |
|
404 |
done |
|
405 |
qed |
|
406 |
qed |
|
407 |
||
408 |
instance inat :: wellorder |
|
409 |
proof |
|
27823 | 410 |
fix P and n |
411 |
assume hyp: "(\<And>n\<Colon>inat. (\<And>m\<Colon>inat. m < n \<Longrightarrow> P m) \<Longrightarrow> P n)" |
|
412 |
show "P n" by (blast intro: inat_less_induct hyp) |
|
26089 | 413 |
qed |
414 |
||
27110 | 415 |
|
416 |
subsection {* Traditional theorem names *} |
|
417 |
||
418 |
lemmas inat_defs = zero_inat_def one_inat_def number_of_inat_def iSuc_def |
|
419 |
plus_inat_def less_eq_inat_def less_inat_def |
|
420 |
||
421 |
lemmas inat_splits = inat.splits |
|
422 |
||
31077 | 423 |
|
424 |
instance inat :: linorder |
|
425 |
by intro_classes (auto simp add: inat_defs split: inat.splits) |
|
426 |
||
11351
c5c403d30c77
added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff
changeset
|
427 |
end |