author | nipkow |
Mon, 10 Apr 1995 08:40:58 +0200 | |
changeset 1024 | b86042000035 |
parent 972 | e61b058d58d2 |
child 1264 | 3eb91524b938 |
permissions | -rw-r--r-- |
923 | 1 |
(* Title: HOL/nat |
2 |
ID: $Id$ |
|
3 |
Author: Tobias Nipkow, Cambridge University Computer Laboratory |
|
4 |
Copyright 1991 University of Cambridge |
|
5 |
||
6 |
For nat.thy. Type nat is defined as a set (Nat) over the type ind. |
|
7 |
*) |
|
8 |
||
9 |
open Nat; |
|
10 |
||
11 |
goal Nat.thy "mono(%X. {Zero_Rep} Un (Suc_Rep``X))"; |
|
12 |
by (REPEAT (ares_tac [monoI, subset_refl, image_mono, Un_mono] 1)); |
|
13 |
qed "Nat_fun_mono"; |
|
14 |
||
15 |
val Nat_unfold = Nat_fun_mono RS (Nat_def RS def_lfp_Tarski); |
|
16 |
||
17 |
(* Zero is a natural number -- this also justifies the type definition*) |
|
18 |
goal Nat.thy "Zero_Rep: Nat"; |
|
19 |
by (rtac (Nat_unfold RS ssubst) 1); |
|
20 |
by (rtac (singletonI RS UnI1) 1); |
|
21 |
qed "Zero_RepI"; |
|
22 |
||
23 |
val prems = goal Nat.thy "i: Nat ==> Suc_Rep(i) : Nat"; |
|
24 |
by (rtac (Nat_unfold RS ssubst) 1); |
|
25 |
by (rtac (imageI RS UnI2) 1); |
|
26 |
by (resolve_tac prems 1); |
|
27 |
qed "Suc_RepI"; |
|
28 |
||
29 |
(*** Induction ***) |
|
30 |
||
31 |
val major::prems = goal Nat.thy |
|
32 |
"[| i: Nat; P(Zero_Rep); \ |
|
33 |
\ !!j. [| j: Nat; P(j) |] ==> P(Suc_Rep(j)) |] ==> P(i)"; |
|
34 |
by (rtac ([Nat_def, Nat_fun_mono, major] MRS def_induct) 1); |
|
35 |
by (fast_tac (set_cs addIs prems) 1); |
|
36 |
qed "Nat_induct"; |
|
37 |
||
38 |
val prems = goalw Nat.thy [Zero_def,Suc_def] |
|
39 |
"[| P(0); \ |
|
40 |
\ !!k. P(k) ==> P(Suc(k)) |] ==> P(n)"; |
|
41 |
by (rtac (Rep_Nat_inverse RS subst) 1); (*types force good instantiation*) |
|
42 |
by (rtac (Rep_Nat RS Nat_induct) 1); |
|
43 |
by (REPEAT (ares_tac prems 1 |
|
44 |
ORELSE eresolve_tac [Abs_Nat_inverse RS subst] 1)); |
|
45 |
qed "nat_induct"; |
|
46 |
||
47 |
(*Perform induction on n. *) |
|
48 |
fun nat_ind_tac a i = |
|
49 |
EVERY [res_inst_tac [("n",a)] nat_induct i, |
|
50 |
rename_last_tac a ["1"] (i+1)]; |
|
51 |
||
52 |
(*A special form of induction for reasoning about m<n and m-n*) |
|
53 |
val prems = goal Nat.thy |
|
54 |
"[| !!x. P x 0; \ |
|
55 |
\ !!y. P 0 (Suc y); \ |
|
56 |
\ !!x y. [| P x y |] ==> P (Suc x) (Suc y) \ |
|
57 |
\ |] ==> P m n"; |
|
58 |
by (res_inst_tac [("x","m")] spec 1); |
|
59 |
by (nat_ind_tac "n" 1); |
|
60 |
by (rtac allI 2); |
|
61 |
by (nat_ind_tac "x" 2); |
|
62 |
by (REPEAT (ares_tac (prems@[allI]) 1 ORELSE etac spec 1)); |
|
63 |
qed "diff_induct"; |
|
64 |
||
65 |
(*Case analysis on the natural numbers*) |
|
66 |
val prems = goal Nat.thy |
|
67 |
"[| n=0 ==> P; !!x. n = Suc(x) ==> P |] ==> P"; |
|
68 |
by (subgoal_tac "n=0 | (EX x. n = Suc(x))" 1); |
|
69 |
by (fast_tac (HOL_cs addSEs prems) 1); |
|
70 |
by (nat_ind_tac "n" 1); |
|
71 |
by (rtac (refl RS disjI1) 1); |
|
72 |
by (fast_tac HOL_cs 1); |
|
73 |
qed "natE"; |
|
74 |
||
75 |
(*** Isomorphisms: Abs_Nat and Rep_Nat ***) |
|
76 |
||
77 |
(*We can't take these properties as axioms, or take Abs_Nat==Inv(Rep_Nat), |
|
78 |
since we assume the isomorphism equations will one day be given by Isabelle*) |
|
79 |
||
80 |
goal Nat.thy "inj(Rep_Nat)"; |
|
81 |
by (rtac inj_inverseI 1); |
|
82 |
by (rtac Rep_Nat_inverse 1); |
|
83 |
qed "inj_Rep_Nat"; |
|
84 |
||
85 |
goal Nat.thy "inj_onto Abs_Nat Nat"; |
|
86 |
by (rtac inj_onto_inverseI 1); |
|
87 |
by (etac Abs_Nat_inverse 1); |
|
88 |
qed "inj_onto_Abs_Nat"; |
|
89 |
||
90 |
(*** Distinctness of constructors ***) |
|
91 |
||
92 |
goalw Nat.thy [Zero_def,Suc_def] "Suc(m) ~= 0"; |
|
93 |
by (rtac (inj_onto_Abs_Nat RS inj_onto_contraD) 1); |
|
94 |
by (rtac Suc_Rep_not_Zero_Rep 1); |
|
95 |
by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI, Zero_RepI] 1)); |
|
96 |
qed "Suc_not_Zero"; |
|
97 |
||
98 |
bind_thm ("Zero_not_Suc", (Suc_not_Zero RS not_sym)); |
|
99 |
||
100 |
bind_thm ("Suc_neq_Zero", (Suc_not_Zero RS notE)); |
|
101 |
val Zero_neq_Suc = sym RS Suc_neq_Zero; |
|
102 |
||
103 |
(** Injectiveness of Suc **) |
|
104 |
||
105 |
goalw Nat.thy [Suc_def] "inj(Suc)"; |
|
106 |
by (rtac injI 1); |
|
107 |
by (dtac (inj_onto_Abs_Nat RS inj_ontoD) 1); |
|
108 |
by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI] 1)); |
|
109 |
by (dtac (inj_Suc_Rep RS injD) 1); |
|
110 |
by (etac (inj_Rep_Nat RS injD) 1); |
|
111 |
qed "inj_Suc"; |
|
112 |
||
113 |
val Suc_inject = inj_Suc RS injD;; |
|
114 |
||
115 |
goal Nat.thy "(Suc(m)=Suc(n)) = (m=n)"; |
|
116 |
by (EVERY1 [rtac iffI, etac Suc_inject, etac arg_cong]); |
|
117 |
qed "Suc_Suc_eq"; |
|
118 |
||
119 |
goal Nat.thy "n ~= Suc(n)"; |
|
120 |
by (nat_ind_tac "n" 1); |
|
121 |
by (ALLGOALS(asm_simp_tac (HOL_ss addsimps [Zero_not_Suc,Suc_Suc_eq]))); |
|
122 |
qed "n_not_Suc_n"; |
|
123 |
||
124 |
val Suc_n_not_n = n_not_Suc_n RS not_sym; |
|
125 |
||
126 |
(*** nat_case -- the selection operator for nat ***) |
|
127 |
||
128 |
goalw Nat.thy [nat_case_def] "nat_case a f 0 = a"; |
|
129 |
by (fast_tac (set_cs addIs [select_equality] addEs [Zero_neq_Suc]) 1); |
|
130 |
qed "nat_case_0"; |
|
131 |
||
132 |
goalw Nat.thy [nat_case_def] "nat_case a f (Suc k) = f(k)"; |
|
133 |
by (fast_tac (set_cs addIs [select_equality] |
|
134 |
addEs [make_elim Suc_inject, Suc_neq_Zero]) 1); |
|
135 |
qed "nat_case_Suc"; |
|
136 |
||
137 |
(** Introduction rules for 'pred_nat' **) |
|
138 |
||
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
962
diff
changeset
|
139 |
goalw Nat.thy [pred_nat_def] "(n, Suc(n)) : pred_nat"; |
923 | 140 |
by (fast_tac set_cs 1); |
141 |
qed "pred_natI"; |
|
142 |
||
143 |
val major::prems = goalw Nat.thy [pred_nat_def] |
|
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
962
diff
changeset
|
144 |
"[| p : pred_nat; !!x n. [| p = (n, Suc(n)) |] ==> R \ |
923 | 145 |
\ |] ==> R"; |
146 |
by (rtac (major RS CollectE) 1); |
|
147 |
by (REPEAT (eresolve_tac ([asm_rl,exE]@prems) 1)); |
|
148 |
qed "pred_natE"; |
|
149 |
||
150 |
goalw Nat.thy [wf_def] "wf(pred_nat)"; |
|
151 |
by (strip_tac 1); |
|
152 |
by (nat_ind_tac "x" 1); |
|
153 |
by (fast_tac (HOL_cs addSEs [mp, pred_natE, Pair_inject, |
|
154 |
make_elim Suc_inject]) 2); |
|
155 |
by (fast_tac (HOL_cs addSEs [mp, pred_natE, Pair_inject, Zero_neq_Suc]) 1); |
|
156 |
qed "wf_pred_nat"; |
|
157 |
||
158 |
||
159 |
(*** nat_rec -- by wf recursion on pred_nat ***) |
|
160 |
||
161 |
bind_thm ("nat_rec_unfold", (wf_pred_nat RS (nat_rec_def RS def_wfrec))); |
|
162 |
||
163 |
(** conversion rules **) |
|
164 |
||
165 |
goal Nat.thy "nat_rec 0 c h = c"; |
|
166 |
by (rtac (nat_rec_unfold RS trans) 1); |
|
167 |
by (simp_tac (HOL_ss addsimps [nat_case_0]) 1); |
|
168 |
qed "nat_rec_0"; |
|
169 |
||
170 |
goal Nat.thy "nat_rec (Suc n) c h = h n (nat_rec n c h)"; |
|
171 |
by (rtac (nat_rec_unfold RS trans) 1); |
|
172 |
by (simp_tac (HOL_ss addsimps [nat_case_Suc, pred_natI, cut_apply]) 1); |
|
173 |
qed "nat_rec_Suc"; |
|
174 |
||
175 |
(*These 2 rules ease the use of primitive recursion. NOTE USE OF == *) |
|
176 |
val [rew] = goal Nat.thy |
|
177 |
"[| !!n. f(n) == nat_rec n c h |] ==> f(0) = c"; |
|
178 |
by (rewtac rew); |
|
179 |
by (rtac nat_rec_0 1); |
|
180 |
qed "def_nat_rec_0"; |
|
181 |
||
182 |
val [rew] = goal Nat.thy |
|
183 |
"[| !!n. f(n) == nat_rec n c h |] ==> f(Suc(n)) = h n (f n)"; |
|
184 |
by (rewtac rew); |
|
185 |
by (rtac nat_rec_Suc 1); |
|
186 |
qed "def_nat_rec_Suc"; |
|
187 |
||
188 |
fun nat_recs def = |
|
189 |
[standard (def RS def_nat_rec_0), |
|
190 |
standard (def RS def_nat_rec_Suc)]; |
|
191 |
||
192 |
||
193 |
(*** Basic properties of "less than" ***) |
|
194 |
||
195 |
(** Introduction properties **) |
|
196 |
||
197 |
val prems = goalw Nat.thy [less_def] "[| i<j; j<k |] ==> i<(k::nat)"; |
|
198 |
by (rtac (trans_trancl RS transD) 1); |
|
199 |
by (resolve_tac prems 1); |
|
200 |
by (resolve_tac prems 1); |
|
201 |
qed "less_trans"; |
|
202 |
||
203 |
goalw Nat.thy [less_def] "n < Suc(n)"; |
|
204 |
by (rtac (pred_natI RS r_into_trancl) 1); |
|
205 |
qed "lessI"; |
|
206 |
||
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
962
diff
changeset
|
207 |
(* i(j ==> i(Suc(j) *) |
923 | 208 |
val less_SucI = lessI RSN (2, less_trans); |
209 |
||
210 |
goal Nat.thy "0 < Suc(n)"; |
|
211 |
by (nat_ind_tac "n" 1); |
|
212 |
by (rtac lessI 1); |
|
213 |
by (etac less_trans 1); |
|
214 |
by (rtac lessI 1); |
|
215 |
qed "zero_less_Suc"; |
|
216 |
||
217 |
(** Elimination properties **) |
|
218 |
||
219 |
val prems = goalw Nat.thy [less_def] "n<m ==> ~ m<(n::nat)"; |
|
220 |
by(fast_tac (HOL_cs addIs ([wf_pred_nat, wf_trancl RS wf_asym]@prems))1); |
|
221 |
qed "less_not_sym"; |
|
222 |
||
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
962
diff
changeset
|
223 |
(* [| n(m; m(n |] ==> R *) |
923 | 224 |
bind_thm ("less_asym", (less_not_sym RS notE)); |
225 |
||
226 |
goalw Nat.thy [less_def] "~ n<(n::nat)"; |
|
227 |
by (rtac notI 1); |
|
228 |
by (etac (wf_pred_nat RS wf_trancl RS wf_anti_refl) 1); |
|
229 |
qed "less_not_refl"; |
|
230 |
||
972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
962
diff
changeset
|
231 |
(* n(n ==> R *) |
923 | 232 |
bind_thm ("less_anti_refl", (less_not_refl RS notE)); |
233 |
||
234 |
goal Nat.thy "!!m. n<m ==> m ~= (n::nat)"; |
|
235 |
by(fast_tac (HOL_cs addEs [less_anti_refl]) 1); |
|
236 |
qed "less_not_refl2"; |
|
237 |
||
238 |
||
239 |
val major::prems = goalw Nat.thy [less_def] |
|
240 |
"[| i<k; k=Suc(i) ==> P; !!j. [| i<j; k=Suc(j) |] ==> P \ |
|
241 |
\ |] ==> P"; |
|
242 |
by (rtac (major RS tranclE) 1); |
|
1024 | 243 |
by (REPEAT_FIRST (bound_hyp_subst_tac ORELSE' |
244 |
eresolve_tac (prems@[pred_natE, Pair_inject]))); |
|
245 |
by (rtac refl 1); |
|
923 | 246 |
qed "lessE"; |
247 |
||
248 |
goal Nat.thy "~ n<0"; |
|
249 |
by (rtac notI 1); |
|
250 |
by (etac lessE 1); |
|
251 |
by (etac Zero_neq_Suc 1); |
|
252 |
by (etac Zero_neq_Suc 1); |
|
253 |
qed "not_less0"; |
|
254 |
||
255 |
(* n<0 ==> R *) |
|
256 |
bind_thm ("less_zeroE", (not_less0 RS notE)); |
|
257 |
||
258 |
val [major,less,eq] = goal Nat.thy |
|
259 |
"[| m < Suc(n); m<n ==> P; m=n ==> P |] ==> P"; |
|
260 |
by (rtac (major RS lessE) 1); |
|
261 |
by (rtac eq 1); |
|
262 |
by (fast_tac (HOL_cs addSDs [Suc_inject]) 1); |
|
263 |
by (rtac less 1); |
|
264 |
by (fast_tac (HOL_cs addSDs [Suc_inject]) 1); |
|
265 |
qed "less_SucE"; |
|
266 |
||
267 |
goal Nat.thy "(m < Suc(n)) = (m < n | m = n)"; |
|
268 |
by (fast_tac (HOL_cs addSIs [lessI] |
|
269 |
addEs [less_trans, less_SucE]) 1); |
|
270 |
qed "less_Suc_eq"; |
|
271 |
||
272 |
||
273 |
(** Inductive (?) properties **) |
|
274 |
||
275 |
val [prem] = goal Nat.thy "Suc(m) < n ==> m<n"; |
|
276 |
by (rtac (prem RS rev_mp) 1); |
|
277 |
by (nat_ind_tac "n" 1); |
|
278 |
by (rtac impI 1); |
|
279 |
by (etac less_zeroE 1); |
|
280 |
by (fast_tac (HOL_cs addSIs [lessI RS less_SucI] |
|
281 |
addSDs [Suc_inject] |
|
282 |
addEs [less_trans, lessE]) 1); |
|
283 |
qed "Suc_lessD"; |
|
284 |
||
285 |
val [major,minor] = goal Nat.thy |
|
286 |
"[| Suc(i)<k; !!j. [| i<j; k=Suc(j) |] ==> P \ |
|
287 |
\ |] ==> P"; |
|
288 |
by (rtac (major RS lessE) 1); |
|
289 |
by (etac (lessI RS minor) 1); |
|
290 |
by (etac (Suc_lessD RS minor) 1); |
|
291 |
by (assume_tac 1); |
|
292 |
qed "Suc_lessE"; |
|
293 |
||
294 |
val [major] = goal Nat.thy "Suc(m) < Suc(n) ==> m<n"; |
|
295 |
by (rtac (major RS lessE) 1); |
|
296 |
by (REPEAT (rtac lessI 1 |
|
297 |
ORELSE eresolve_tac [make_elim Suc_inject, ssubst, Suc_lessD] 1)); |
|
298 |
qed "Suc_less_SucD"; |
|
299 |
||
300 |
val prems = goal Nat.thy "m<n ==> Suc(m) < Suc(n)"; |
|
301 |
by (subgoal_tac "m<n --> Suc(m) < Suc(n)" 1); |
|
302 |
by (fast_tac (HOL_cs addIs prems) 1); |
|
303 |
by (nat_ind_tac "n" 1); |
|
304 |
by (rtac impI 1); |
|
305 |
by (etac less_zeroE 1); |
|
306 |
by (fast_tac (HOL_cs addSIs [lessI] |
|
307 |
addSDs [Suc_inject] |
|
308 |
addEs [less_trans, lessE]) 1); |
|
309 |
qed "Suc_mono"; |
|
310 |
||
311 |
goal Nat.thy "(Suc(m) < Suc(n)) = (m<n)"; |
|
312 |
by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]); |
|
313 |
qed "Suc_less_eq"; |
|
314 |
||
315 |
goal Nat.thy "~(Suc(n) < n)"; |
|
316 |
by(fast_tac (HOL_cs addEs [Suc_lessD RS less_anti_refl]) 1); |
|
317 |
qed "not_Suc_n_less_n"; |
|
318 |
||
319 |
(*"Less than" is a linear ordering*) |
|
320 |
goal Nat.thy "m<n | m=n | n<(m::nat)"; |
|
321 |
by (nat_ind_tac "m" 1); |
|
322 |
by (nat_ind_tac "n" 1); |
|
323 |
by (rtac (refl RS disjI1 RS disjI2) 1); |
|
324 |
by (rtac (zero_less_Suc RS disjI1) 1); |
|
325 |
by (fast_tac (HOL_cs addIs [lessI, Suc_mono, less_SucI] addEs [lessE]) 1); |
|
326 |
qed "less_linear"; |
|
327 |
||
328 |
(*Can be used with less_Suc_eq to get n=m | n<m *) |
|
329 |
goal Nat.thy "(~ m < n) = (n < Suc(m))"; |
|
330 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
|
331 |
by(ALLGOALS(asm_simp_tac (HOL_ss addsimps |
|
332 |
[not_less0,zero_less_Suc,Suc_less_eq]))); |
|
333 |
qed "not_less_eq"; |
|
334 |
||
335 |
(*Complete induction, aka course-of-values induction*) |
|
336 |
val prems = goalw Nat.thy [less_def] |
|
337 |
"[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |] ==> P(n)"; |
|
338 |
by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1); |
|
339 |
by (eresolve_tac prems 1); |
|
340 |
qed "less_induct"; |
|
341 |
||
342 |
||
343 |
(*** Properties of <= ***) |
|
344 |
||
345 |
goalw Nat.thy [le_def] "0 <= n"; |
|
346 |
by (rtac not_less0 1); |
|
347 |
qed "le0"; |
|
348 |
||
349 |
val nat_simps = [not_less0, less_not_refl, zero_less_Suc, lessI, |
|
350 |
Suc_less_eq, less_Suc_eq, le0, not_Suc_n_less_n, |
|
351 |
Suc_not_Zero, Zero_not_Suc, Suc_Suc_eq, |
|
352 |
n_not_Suc_n, Suc_n_not_n, |
|
353 |
nat_case_0, nat_case_Suc, nat_rec_0, nat_rec_Suc]; |
|
354 |
||
355 |
val nat_ss0 = sum_ss addsimps nat_simps; |
|
356 |
||
357 |
(*Prevents simplification of f and g: much faster*) |
|
358 |
qed_goal "nat_case_weak_cong" Nat.thy |
|
359 |
"m=n ==> nat_case a f m = nat_case a f n" |
|
360 |
(fn [prem] => [rtac (prem RS arg_cong) 1]); |
|
361 |
||
362 |
qed_goal "nat_rec_weak_cong" Nat.thy |
|
363 |
"m=n ==> nat_rec m a f = nat_rec n a f" |
|
364 |
(fn [prem] => [rtac (prem RS arg_cong) 1]); |
|
365 |
||
366 |
val prems = goalw Nat.thy [le_def] "~(n<m) ==> m<=(n::nat)"; |
|
367 |
by (resolve_tac prems 1); |
|
368 |
qed "leI"; |
|
369 |
||
370 |
val prems = goalw Nat.thy [le_def] "m<=n ==> ~(n<(m::nat))"; |
|
371 |
by (resolve_tac prems 1); |
|
372 |
qed "leD"; |
|
373 |
||
374 |
val leE = make_elim leD; |
|
375 |
||
376 |
goalw Nat.thy [le_def] "!!m. ~ m <= n ==> n<(m::nat)"; |
|
377 |
by (fast_tac HOL_cs 1); |
|
378 |
qed "not_leE"; |
|
379 |
||
380 |
goalw Nat.thy [le_def] "!!m. m < n ==> Suc(m) <= n"; |
|
381 |
by(simp_tac nat_ss0 1); |
|
382 |
by (fast_tac (HOL_cs addEs [less_anti_refl,less_asym]) 1); |
|
383 |
qed "lessD"; |
|
384 |
||
385 |
goalw Nat.thy [le_def] "!!m. Suc(m) <= n ==> m <= n"; |
|
386 |
by(asm_full_simp_tac nat_ss0 1); |
|
387 |
by(fast_tac HOL_cs 1); |
|
388 |
qed "Suc_leD"; |
|
389 |
||
390 |
goalw Nat.thy [le_def] "!!m. m < n ==> m <= (n::nat)"; |
|
391 |
by (fast_tac (HOL_cs addEs [less_asym]) 1); |
|
392 |
qed "less_imp_le"; |
|
393 |
||
394 |
goalw Nat.thy [le_def] "!!m. m <= n ==> m < n | m=(n::nat)"; |
|
395 |
by (cut_facts_tac [less_linear] 1); |
|
396 |
by (fast_tac (HOL_cs addEs [less_anti_refl,less_asym]) 1); |
|
397 |
qed "le_imp_less_or_eq"; |
|
398 |
||
399 |
goalw Nat.thy [le_def] "!!m. m<n | m=n ==> m <=(n::nat)"; |
|
400 |
by (cut_facts_tac [less_linear] 1); |
|
401 |
by (fast_tac (HOL_cs addEs [less_anti_refl,less_asym]) 1); |
|
402 |
by (flexflex_tac); |
|
403 |
qed "less_or_eq_imp_le"; |
|
404 |
||
405 |
goal Nat.thy "(x <= (y::nat)) = (x < y | x=y)"; |
|
406 |
by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1)); |
|
407 |
qed "le_eq_less_or_eq"; |
|
408 |
||
409 |
goal Nat.thy "n <= (n::nat)"; |
|
410 |
by(simp_tac (HOL_ss addsimps [le_eq_less_or_eq]) 1); |
|
411 |
qed "le_refl"; |
|
412 |
||
413 |
val prems = goal Nat.thy "!!i. [| i <= j; j < k |] ==> i < (k::nat)"; |
|
414 |
by (dtac le_imp_less_or_eq 1); |
|
415 |
by (fast_tac (HOL_cs addIs [less_trans]) 1); |
|
416 |
qed "le_less_trans"; |
|
417 |
||
418 |
goal Nat.thy "!!i. [| i < j; j <= k |] ==> i < (k::nat)"; |
|
419 |
by (dtac le_imp_less_or_eq 1); |
|
420 |
by (fast_tac (HOL_cs addIs [less_trans]) 1); |
|
421 |
qed "less_le_trans"; |
|
422 |
||
423 |
goal Nat.thy "!!i. [| i <= j; j <= k |] ==> i <= (k::nat)"; |
|
424 |
by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq, |
|
425 |
rtac less_or_eq_imp_le, fast_tac (HOL_cs addIs [less_trans])]); |
|
426 |
qed "le_trans"; |
|
427 |
||
428 |
val prems = goal Nat.thy "!!m. [| m <= n; n <= m |] ==> m = (n::nat)"; |
|
429 |
by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq, |
|
430 |
fast_tac (HOL_cs addEs [less_anti_refl,less_asym])]); |
|
431 |
qed "le_anti_sym"; |
|
432 |
||
433 |
goal Nat.thy "(Suc(n) <= Suc(m)) = (n <= m)"; |
|
434 |
by (simp_tac (nat_ss0 addsimps [le_eq_less_or_eq]) 1); |
|
435 |
qed "Suc_le_mono"; |
|
436 |
||
962 | 437 |
val nat_ss = nat_ss0 addsimps [le_refl,Suc_le_mono]; |