src/HOL/NatDef.ML
 author nipkow Mon Apr 27 16:45:11 1998 +0200 (1998-04-27) changeset 4830 bd73675adbed parent 4821 bfbaea156f43 child 5069 3ea049f7979d permissions -rw-r--r--
Added a few lemmas.
Renamed expand_const -> split_const.
1 (*  Title:      HOL/NatDef.ML
2     ID:         \$Id\$
3     Author:     Tobias Nipkow, Cambridge University Computer Laboratory
4     Copyright   1991  University of Cambridge
6 Blast_tac proofs here can get PROOF FAILED of Ord theorems like order_refl
7 and order_less_irrefl.  We do not add the "nat" versions to the basic claset
8 because the type will be promoted to type class "order".
9 *)
11 goal 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";
15 val Nat_unfold = Nat_fun_mono RS (Nat_def RS def_lfp_Tarski);
17 (* Zero is a natural number -- this also justifies the type definition*)
18 goal thy "Zero_Rep: Nat";
19 by (stac Nat_unfold 1);
20 by (rtac (singletonI RS UnI1) 1);
21 qed "Zero_RepI";
23 val prems = goal thy "i: Nat ==> Suc_Rep(i) : Nat";
24 by (stac Nat_unfold 1);
25 by (rtac (imageI RS UnI2) 1);
26 by (resolve_tac prems 1);
27 qed "Suc_RepI";
29 (*** Induction ***)
31 val major::prems = goal 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 (blast_tac (claset() addIs prems) 1);
36 qed "Nat_induct";
38 val prems = goalw thy [Zero_def,Suc_def]
39     "[| P(0);   \
40 \       !!n. P(n) ==> P(Suc(n)) |]  ==> 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";
47 (*Perform induction on n. *)
48 local fun raw_nat_ind_tac a i =
49     res_inst_tac [("n",a)] nat_induct i  THEN  rename_last_tac a [""] (i+1)
50 in
51 val nat_ind_tac = Datatype.occs_in_prems raw_nat_ind_tac
52 end;
54 (*A special form of induction for reasoning about m<n and m-n*)
55 val prems = goal thy
56     "[| !!x. P x 0;  \
57 \       !!y. P 0 (Suc y);  \
58 \       !!x y. [| P x y |] ==> P (Suc x) (Suc y)  \
59 \    |] ==> P m n";
60 by (res_inst_tac [("x","m")] spec 1);
61 by (nat_ind_tac "n" 1);
62 by (rtac allI 2);
63 by (nat_ind_tac "x" 2);
64 by (REPEAT (ares_tac (prems@[allI]) 1 ORELSE etac spec 1));
65 qed "diff_induct";
67 (*Case analysis on the natural numbers*)
68 val prems = goal thy
69     "[| n=0 ==> P;  !!x. n = Suc(x) ==> P |] ==> P";
70 by (subgoal_tac "n=0 | (EX x. n = Suc(x))" 1);
71 by (fast_tac (claset() addSEs prems) 1);
72 by (nat_ind_tac "n" 1);
73 by (rtac (refl RS disjI1) 1);
74 by (Blast_tac 1);
75 qed "natE";
78 (*** Isomorphisms: Abs_Nat and Rep_Nat ***)
80 (*We can't take these properties as axioms, or take Abs_Nat==Inv(Rep_Nat),
81   since we assume the isomorphism equations will one day be given by Isabelle*)
83 goal thy "inj(Rep_Nat)";
84 by (rtac inj_inverseI 1);
85 by (rtac Rep_Nat_inverse 1);
86 qed "inj_Rep_Nat";
88 goal thy "inj_on Abs_Nat Nat";
89 by (rtac inj_on_inverseI 1);
90 by (etac Abs_Nat_inverse 1);
91 qed "inj_on_Abs_Nat";
93 (*** Distinctness of constructors ***)
95 goalw thy [Zero_def,Suc_def] "Suc(m) ~= 0";
96 by (rtac (inj_on_Abs_Nat RS inj_on_contraD) 1);
97 by (rtac Suc_Rep_not_Zero_Rep 1);
98 by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI, Zero_RepI] 1));
99 qed "Suc_not_Zero";
101 bind_thm ("Zero_not_Suc", Suc_not_Zero RS not_sym);
105 bind_thm ("Suc_neq_Zero", (Suc_not_Zero RS notE));
106 val Zero_neq_Suc = sym RS Suc_neq_Zero;
108 (** Injectiveness of Suc **)
110 goalw thy [Suc_def] "inj(Suc)";
111 by (rtac injI 1);
112 by (dtac (inj_on_Abs_Nat RS inj_onD) 1);
113 by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI] 1));
114 by (dtac (inj_Suc_Rep RS injD) 1);
115 by (etac (inj_Rep_Nat RS injD) 1);
116 qed "inj_Suc";
118 val Suc_inject = inj_Suc RS injD;
120 goal thy "(Suc(m)=Suc(n)) = (m=n)";
121 by (EVERY1 [rtac iffI, etac Suc_inject, etac arg_cong]);
122 qed "Suc_Suc_eq";
126 goal thy "n ~= Suc(n)";
127 by (nat_ind_tac "n" 1);
128 by (ALLGOALS Asm_simp_tac);
129 qed "n_not_Suc_n";
131 bind_thm ("Suc_n_not_n", n_not_Suc_n RS not_sym);
133 goal thy "!!n. n ~= 0 ==> EX m. n = Suc m";
134 by (rtac natE 1);
135 by (REPEAT (Blast_tac 1));
136 qed "not0_implies_Suc";
139 (*** nat_case -- the selection operator for nat ***)
141 goalw thy [nat_case_def] "nat_case a f 0 = a";
142 by (Blast_tac 1);
143 qed "nat_case_0";
145 goalw thy [nat_case_def] "nat_case a f (Suc k) = f(k)";
146 by (Blast_tac 1);
147 qed "nat_case_Suc";
149 goalw thy [wf_def, pred_nat_def] "wf(pred_nat)";
150 by (Clarify_tac 1);
151 by (nat_ind_tac "x" 1);
152 by (ALLGOALS Blast_tac);
153 qed "wf_pred_nat";
156 (*** nat_rec -- by wf recursion on pred_nat ***)
158 (* The unrolling rule for nat_rec *)
159 goal thy
160    "nat_rec c d = wfrec pred_nat (%f. nat_case c (%m. d m (f m)))";
161   by (simp_tac (HOL_ss addsimps [nat_rec_def]) 1);
162 bind_thm("nat_rec_unfold", wf_pred_nat RS
163                             ((result() RS eq_reflection) RS def_wfrec));
165 (*---------------------------------------------------------------------------
166  * Old:
167  * bind_thm ("nat_rec_unfold", (wf_pred_nat RS (nat_rec_def RS def_wfrec)));
168  *---------------------------------------------------------------------------*)
170 (** conversion rules **)
172 goal thy "nat_rec c h 0 = c";
173 by (rtac (nat_rec_unfold RS trans) 1);
174 by (simp_tac (simpset() addsimps [nat_case_0]) 1);
175 qed "nat_rec_0";
177 goal thy "nat_rec c h (Suc n) = h n (nat_rec c h n)";
178 by (rtac (nat_rec_unfold RS trans) 1);
179 by (simp_tac (simpset() addsimps [nat_case_Suc, pred_nat_def, cut_apply]) 1);
180 qed "nat_rec_Suc";
182 (*These 2 rules ease the use of primitive recursion.  NOTE USE OF == *)
183 val [rew] = goal thy
184     "[| !!n. f(n) == nat_rec c h n |] ==> f(0) = c";
185 by (rewtac rew);
186 by (rtac nat_rec_0 1);
187 qed "def_nat_rec_0";
189 val [rew] = goal thy
190     "[| !!n. f(n) == nat_rec c h n |] ==> f(Suc(n)) = h n (f n)";
191 by (rewtac rew);
192 by (rtac nat_rec_Suc 1);
193 qed "def_nat_rec_Suc";
195 fun nat_recs def =
196       [standard (def RS def_nat_rec_0),
197        standard (def RS def_nat_rec_Suc)];
200 (*** Basic properties of "less than" ***)
202 (*Used in TFL/post.sml*)
203 goalw thy [less_def] "(m,n) : pred_nat^+ = (m<n)";
204 by (rtac refl 1);
205 qed "less_eq";
207 (** Introduction properties **)
209 val prems = goalw thy [less_def] "[| i<j;  j<k |] ==> i<(k::nat)";
210 by (rtac (trans_trancl RS transD) 1);
211 by (resolve_tac prems 1);
212 by (resolve_tac prems 1);
213 qed "less_trans";
215 goalw thy [less_def, pred_nat_def] "n < Suc(n)";
216 by (simp_tac (simpset() addsimps [r_into_trancl]) 1);
217 qed "lessI";
220 (* i<j ==> i<Suc(j) *)
221 bind_thm("less_SucI", lessI RSN (2, less_trans));
224 goal thy "0 < Suc(n)";
225 by (nat_ind_tac "n" 1);
226 by (rtac lessI 1);
227 by (etac less_trans 1);
228 by (rtac lessI 1);
229 qed "zero_less_Suc";
232 (** Elimination properties **)
234 val prems = goalw thy [less_def] "n<m ==> ~ m<(n::nat)";
235 by (blast_tac (claset() addIs ([wf_pred_nat, wf_trancl RS wf_asym]@prems))1);
236 qed "less_not_sym";
238 (* [| n<m; m<n |] ==> R *)
239 bind_thm ("less_asym", (less_not_sym RS notE));
241 goalw thy [less_def] "~ n<(n::nat)";
242 by (rtac notI 1);
243 by (etac (wf_pred_nat RS wf_trancl RS wf_irrefl) 1);
244 qed "less_not_refl";
246 (* n<n ==> R *)
247 bind_thm ("less_irrefl", (less_not_refl RS notE));
249 goal thy "!!m. n<m ==> m ~= (n::nat)";
250 by (blast_tac (claset() addSEs [less_irrefl]) 1);
251 qed "less_not_refl2";
254 val major::prems = goalw thy [less_def, pred_nat_def]
255     "[| i<k;  k=Suc(i) ==> P;  !!j. [| i<j;  k=Suc(j) |] ==> P \
256 \    |] ==> P";
257 by (rtac (major RS tranclE) 1);
258 by (ALLGOALS Full_simp_tac);
259 by (REPEAT_FIRST (bound_hyp_subst_tac ORELSE'
260                   eresolve_tac (prems@[asm_rl, Pair_inject])));
261 qed "lessE";
263 goal thy "~ n<0";
264 by (rtac notI 1);
265 by (etac lessE 1);
266 by (etac Zero_neq_Suc 1);
267 by (etac Zero_neq_Suc 1);
268 qed "not_less0";
272 (* n<0 ==> R *)
273 bind_thm ("less_zeroE", not_less0 RS notE);
275 val [major,less,eq] = goal thy
276     "[| m < Suc(n);  m<n ==> P;  m=n ==> P |] ==> P";
277 by (rtac (major RS lessE) 1);
278 by (rtac eq 1);
279 by (Blast_tac 1);
280 by (rtac less 1);
281 by (Blast_tac 1);
282 qed "less_SucE";
284 goal thy "(m < Suc(n)) = (m < n | m = n)";
285 by (blast_tac (claset() addSEs [less_SucE] addIs [less_trans]) 1);
286 qed "less_Suc_eq";
288 goal thy "(n<1) = (n=0)";
289 by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);
290 qed "less_one";
293 val prems = goal thy "m<n ==> n ~= 0";
294 by (res_inst_tac [("n","n")] natE 1);
295 by (cut_facts_tac prems 1);
296 by (ALLGOALS Asm_full_simp_tac);
297 qed "gr_implies_not0";
299 goal thy "(n ~= 0) = (0 < n)";
300 by (rtac natE 1);
301 by (Blast_tac 1);
302 by (Blast_tac 1);
303 qed "neq0_conv";
306 (*This theorem is useful with blast_tac: (n=0 ==> False) ==> 0<n *)
307 bind_thm ("gr0I", [neq0_conv, notI] MRS iffD1);
309 goal thy "(~(0 < n)) = (n=0)";
310 by (rtac iffI 1);
311  by (etac swap 1);
312  by (ALLGOALS Asm_full_simp_tac);
313 qed "not_gr0";
316 goal thy "!!m. m<n ==> 0 < n";
317 by (dtac gr_implies_not0 1);
318 by (Asm_full_simp_tac 1);
319 qed "gr_implies_gr0";
323 goal thy "!!m n. m<n ==> Suc(m) < Suc(n)";
324 by (etac rev_mp 1);
325 by (nat_ind_tac "n" 1);
326 by (ALLGOALS (fast_tac (claset() addEs  [less_trans, lessE])));
327 qed "Suc_mono";
329 (*"Less than" is a linear ordering*)
330 goal thy "m<n | m=n | n<(m::nat)";
331 by (nat_ind_tac "m" 1);
332 by (nat_ind_tac "n" 1);
333 by (rtac (refl RS disjI1 RS disjI2) 1);
334 by (rtac (zero_less_Suc RS disjI1) 1);
335 by (blast_tac (claset() addIs [Suc_mono, less_SucI] addEs [lessE]) 1);
336 qed "less_linear";
338 goal thy "!!m::nat. (m ~= n) = (m<n | n<m)";
339 by (cut_facts_tac [less_linear] 1);
340 by (blast_tac (claset() addSEs [less_irrefl]) 1);
341 qed "nat_neq_iff";
343 qed_goal "nat_less_cases" thy
344    "[| (m::nat)<n ==> P n m; m=n ==> P n m; n<m ==> P n m |] ==> P n m"
345 ( fn [major,eqCase,lessCase] =>
346         [
347         (rtac (less_linear RS disjE) 1),
348         (etac disjE 2),
349         (etac lessCase 1),
350         (etac (sym RS eqCase) 1),
351         (etac major 1)
352         ]);
355 (** Inductive (?) properties **)
357 goal thy "!!m. [| m<n; Suc m ~= n |] ==> Suc(m) < n";
358 by (full_simp_tac (simpset() addsimps [nat_neq_iff]) 1);
359 by (blast_tac (claset() addSEs [less_irrefl, less_SucE] addEs [less_asym]) 1);
360 qed "Suc_lessI";
362 val [prem] = goal thy "Suc(m) < n ==> m<n";
363 by (rtac (prem RS rev_mp) 1);
364 by (nat_ind_tac "n" 1);
365 by (ALLGOALS (fast_tac (claset() addSIs [lessI RS less_SucI]
366                                  addEs  [less_trans, lessE])));
367 qed "Suc_lessD";
369 val [major,minor] = goal thy
370     "[| Suc(i)<k;  !!j. [| i<j;  k=Suc(j) |] ==> P \
371 \    |] ==> P";
372 by (rtac (major RS lessE) 1);
373 by (etac (lessI RS minor) 1);
374 by (etac (Suc_lessD RS minor) 1);
375 by (assume_tac 1);
376 qed "Suc_lessE";
378 goal thy "!!m n. Suc(m) < Suc(n) ==> m<n";
379 by (blast_tac (claset() addEs [lessE, make_elim Suc_lessD]) 1);
380 qed "Suc_less_SucD";
383 goal thy "(Suc(m) < Suc(n)) = (m<n)";
384 by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]);
385 qed "Suc_less_eq";
388 goal thy "~(Suc(n) < n)";
389 by (blast_tac (claset() addEs [Suc_lessD RS less_irrefl]) 1);
390 qed "not_Suc_n_less_n";
393 goal thy "!!i. i<j ==> j<k --> Suc i < k";
394 by (nat_ind_tac "k" 1);
395 by (ALLGOALS (asm_simp_tac (simpset())));
396 by (asm_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
397 by (blast_tac (claset() addDs [Suc_lessD]) 1);
398 qed_spec_mp "less_trans_Suc";
400 (*Can be used with less_Suc_eq to get n=m | n<m *)
401 goal thy "(~ m < n) = (n < Suc(m))";
402 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
403 by (ALLGOALS Asm_simp_tac);
404 qed "not_less_eq";
406 (*Complete induction, aka course-of-values induction*)
407 val prems = goalw thy [less_def]
408     "[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |]  ==>  P(n)";
409 by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1);
410 by (eresolve_tac prems 1);
411 qed "less_induct";
413 qed_goal "nat_induct2" thy
414 "[| P 0; P 1; !!k. P k ==> P (Suc (Suc k)) |] ==> P n" (fn prems => [
415         cut_facts_tac prems 1,
416         rtac less_induct 1,
417         res_inst_tac [("n","n")] natE 1,
418          hyp_subst_tac 1,
419          atac 1,
420         hyp_subst_tac 1,
421         res_inst_tac [("n","x")] natE 1,
422          hyp_subst_tac 1,
423          atac 1,
424         hyp_subst_tac 1,
425         resolve_tac prems 1,
426         dtac spec 1,
427         etac mp 1,
428         rtac (lessI RS less_trans) 1,
429         rtac (lessI RS Suc_mono) 1]);
431 (*** Properties of <= ***)
433 goalw thy [le_def] "(m <= n) = (m < Suc n)";
434 by (rtac not_less_eq 1);
435 qed "le_eq_less_Suc";
437 (*  m<=n ==> m < Suc n  *)
438 bind_thm ("le_imp_less_Suc", le_eq_less_Suc RS iffD1);
440 goalw thy [le_def] "0 <= n";
441 by (rtac not_less0 1);
442 qed "le0";
444 goalw thy [le_def] "~ Suc n <= n";
445 by (Simp_tac 1);
446 qed "Suc_n_not_le_n";
448 goalw thy [le_def] "(i <= 0) = (i = 0)";
449 by (nat_ind_tac "i" 1);
450 by (ALLGOALS Asm_simp_tac);
451 qed "le_0_eq";
454 Addsimps [(*less_Suc_eq, makes simpset non-confluent*) le0, le_0_eq,
455           Suc_n_not_le_n,
456           n_not_Suc_n, Suc_n_not_n,
457           nat_case_0, nat_case_Suc, nat_rec_0, nat_rec_Suc];
459 goal thy "!!m. (m <= Suc(n)) = (m<=n | m = Suc n)";
460 by (simp_tac (simpset() addsimps [le_eq_less_Suc]) 1);
461 by (blast_tac (claset() addSEs [less_SucE] addIs [less_SucI]) 1);
462 qed "le_Suc_eq";
464 (* [| m <= Suc n;  m <= n ==> R;  m = Suc n ==> R |] ==> R *)
465 bind_thm ("le_SucE", le_Suc_eq RS iffD1 RS disjE);
467 (*
468 goal thy "(Suc m < n | Suc m = n) = (m < n)";
469 by (stac (less_Suc_eq RS sym) 1);
470 by (rtac Suc_less_eq 1);
471 qed "Suc_le_eq";
473 this could make the simpset (with less_Suc_eq added again) more confluent,
474 but less_Suc_eq makes additional problems with terms of the form 0 < Suc (...)
475 *)
477 (*Prevents simplification of f and g: much faster*)
478 qed_goal "nat_case_weak_cong" thy
479   "m=n ==> nat_case a f m = nat_case a f n"
480   (fn [prem] => [rtac (prem RS arg_cong) 1]);
482 qed_goal "nat_rec_weak_cong" thy
483   "m=n ==> nat_rec a f m = nat_rec a f n"
484   (fn [prem] => [rtac (prem RS arg_cong) 1]);
486 qed_goal "split_nat_case" thy
487   "P(nat_case z s n) = ((n=0 --> P z) & (!m. n = Suc m --> P(s m)))"
488   (fn _ => [nat_ind_tac "n" 1, ALLGOALS Asm_simp_tac]);
490 val prems = goalw thy [le_def] "~n<m ==> m<=(n::nat)";
491 by (resolve_tac prems 1);
492 qed "leI";
494 val prems = goalw thy [le_def] "m<=n ==> ~ n < (m::nat)";
495 by (resolve_tac prems 1);
496 qed "leD";
498 val leE = make_elim leD;
500 goal thy "(~n<m) = (m<=(n::nat))";
501 by (blast_tac (claset() addIs [leI] addEs [leE]) 1);
502 qed "not_less_iff_le";
504 goalw thy [le_def] "!!m. ~ m <= n ==> n<(m::nat)";
505 by (Blast_tac 1);
506 qed "not_leE";
508 goalw thy [le_def] "(~n<=m) = (m<(n::nat))";
509 by (Simp_tac 1);
510 qed "not_le_iff_less";
512 goalw thy [le_def] "!!m. m < n ==> Suc(m) <= n";
513 by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);
514 by (blast_tac (claset() addSEs [less_irrefl,less_asym]) 1);
515 qed "Suc_leI";  (*formerly called lessD*)
517 goalw thy [le_def] "!!m. Suc(m) <= n ==> m <= n";
518 by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
519 qed "Suc_leD";
521 (* stronger version of Suc_leD *)
522 goalw thy [le_def]
523         "!!m. Suc m <= n ==> m < n";
524 by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
525 by (cut_facts_tac [less_linear] 1);
526 by (Blast_tac 1);
527 qed "Suc_le_lessD";
529 goal thy "(Suc m <= n) = (m < n)";
530 by (blast_tac (claset() addIs [Suc_leI, Suc_le_lessD]) 1);
531 qed "Suc_le_eq";
533 goalw thy [le_def] "!!m. m <= n ==> m <= Suc n";
534 by (blast_tac (claset() addDs [Suc_lessD]) 1);
535 qed "le_SucI";
538 bind_thm ("le_Suc", not_Suc_n_less_n RS leI);
540 goalw thy [le_def] "!!m. m < n ==> m <= (n::nat)";
541 by (blast_tac (claset() addEs [less_asym]) 1);
542 qed "less_imp_le";
544 (** Equivalence of m<=n and  m<n | m=n **)
546 goalw thy [le_def] "!!m. m <= n ==> m < n | m=(n::nat)";
547 by (cut_facts_tac [less_linear] 1);
548 by (blast_tac (claset() addEs [less_irrefl,less_asym]) 1);
549 qed "le_imp_less_or_eq";
551 goalw thy [le_def] "!!m. m<n | m=n ==> m <=(n::nat)";
552 by (cut_facts_tac [less_linear] 1);
553 by (blast_tac (claset() addSEs [less_irrefl] addEs [less_asym]) 1);
554 qed "less_or_eq_imp_le";
556 goal thy "(m <= (n::nat)) = (m < n | m=n)";
557 by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1));
558 qed "le_eq_less_or_eq";
560 (*Useful with Blast_tac.   m=n ==> m<=n *)
561 bind_thm ("eq_imp_le", disjI2 RS less_or_eq_imp_le);
563 goal thy "n <= (n::nat)";
564 by (simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
565 qed "le_refl";
567 goal thy "!!i. [| i <= j; j < k |] ==> i < (k::nat)";
568 by (blast_tac (claset() addSDs [le_imp_less_or_eq]
569 	                addIs [less_trans]) 1);
570 qed "le_less_trans";
572 goal thy "!!i. [| i < j; j <= k |] ==> i < (k::nat)";
573 by (blast_tac (claset() addSDs [le_imp_less_or_eq]
574 	                addIs [less_trans]) 1);
575 qed "less_le_trans";
577 goal thy "!!i. [| i <= j; j <= k |] ==> i <= (k::nat)";
578 by (blast_tac (claset() addSDs [le_imp_less_or_eq]
579 	                addIs [less_or_eq_imp_le, less_trans]) 1);
580 qed "le_trans";
582 goal thy "!!m. [| m <= n; n <= m |] ==> m = (n::nat)";
583 (*order_less_irrefl could make this proof fail*)
584 by (blast_tac (claset() addSDs [le_imp_less_or_eq]
586 qed "le_anti_sym";
588 goal thy "(Suc(n) <= Suc(m)) = (n <= m)";
589 by (simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
590 qed "Suc_le_mono";
594 (* Axiom 'order_le_less' of class 'order': *)
595 goal thy "(m::nat) < n = (m <= n & m ~= n)";
596 by (simp_tac (simpset() addsimps [le_def, nat_neq_iff]) 1);
597 by (blast_tac (claset() addSEs [less_asym]) 1);
598 qed "nat_less_le";
600 (* Axiom 'linorder_linear' of class 'linorder': *)
601 goal thy "(m::nat) <= n | n <= m";
602 by (simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
603 by (cut_facts_tac [less_linear] 1);
604 by(Blast_tac 1);
605 qed "nat_le_linear";
608 (** max
610 goalw thy [max_def] "!!z::nat. (z <= max x y) = (z <= x | z <= y)";
611 by (simp_tac (simpset() addsimps [not_le_iff_less]) 1);
612 by (blast_tac (claset() addIs [less_imp_le, le_trans]) 1);
613 qed "le_max_iff_disj";
615 goalw thy [max_def] "!!z::nat. (max x y <= z) = (x <= z & y <= z)";
616 by (simp_tac (simpset() addsimps [not_le_iff_less]) 1);
617 by (blast_tac (claset() addIs [less_imp_le, le_trans]) 1);
618 qed "max_le_iff_conj";
621 (** min **)
623 goalw thy [min_def] "!!z::nat. (z <= min x y) = (z <= x & z <= y)";
624 by (simp_tac (simpset() addsimps [not_le_iff_less]) 1);
625 by (blast_tac (claset() addIs [less_imp_le, le_trans]) 1);
626 qed "le_min_iff_conj";
628 goalw thy [min_def] "!!z::nat. (min x y <= z) = (x <= z | y <= z)";
629 by (simp_tac (simpset() addsimps [not_le_iff_less] addsplits) 1);
630 by (blast_tac (claset() addIs [less_imp_le, le_trans]) 1);
631 qed "min_le_iff_disj";
632  **)
634 (** LEAST -- the least number operator **)
636 goal thy "(! m::nat. P m --> n <= m) = (! m. m < n --> ~ P m)";
637 by (blast_tac (claset() addIs [leI] addEs [leE]) 1);
638 val lemma = result();
640 (* This is an old def of Least for nat, which is derived for compatibility *)
641 goalw thy [Least_def]
642   "(LEAST n::nat. P n) == (@n. P(n) & (ALL m. m < n --> ~P(m)))";
643 by (simp_tac (simpset() addsimps [lemma]) 1);
644 qed "Least_nat_def";
646 val [prem1,prem2] = goalw thy [Least_nat_def]
647     "[| P(k::nat);  !!x. x<k ==> ~P(x) |] ==> (LEAST x. P(x)) = k";
648 by (rtac select_equality 1);
649 by (blast_tac (claset() addSIs [prem1,prem2]) 1);
650 by (cut_facts_tac [less_linear] 1);
651 by (blast_tac (claset() addSIs [prem1] addSDs [prem2]) 1);
652 qed "Least_equality";
654 val [prem] = goal thy "P(k::nat) ==> P(LEAST x. P(x))";
655 by (rtac (prem RS rev_mp) 1);
656 by (res_inst_tac [("n","k")] less_induct 1);
657 by (rtac impI 1);
658 by (rtac classical 1);
659 by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
660 by (assume_tac 1);
661 by (assume_tac 2);
662 by (Blast_tac 1);
663 qed "LeastI";
665 (*Proof is almost identical to the one above!*)
666 val [prem] = goal thy "P(k::nat) ==> (LEAST x. P(x)) <= k";
667 by (rtac (prem RS rev_mp) 1);
668 by (res_inst_tac [("n","k")] less_induct 1);
669 by (rtac impI 1);
670 by (rtac classical 1);
671 by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
672 by (assume_tac 1);
673 by (rtac le_refl 2);
674 by (blast_tac (claset() addIs [less_imp_le,le_trans]) 1);
675 qed "Least_le";
677 val [prem] = goal thy "k < (LEAST x. P(x)) ==> ~P(k::nat)";
678 by (rtac notI 1);
679 by (etac (rewrite_rule [le_def] Least_le RS notE) 1);
680 by (rtac prem 1);
681 qed "not_less_Least";
683 qed_goalw "Least_Suc" thy [Least_nat_def]
684  "!!P. [| ? n. P(Suc n); ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
685  (fn _ => [
686         rtac select_equality 1,
687         fold_goals_tac [Least_nat_def],
688         safe_tac (claset() addSEs [LeastI]),
689         rename_tac "j" 1,
690         res_inst_tac [("n","j")] natE 1,
691         Blast_tac 1,
692         blast_tac (claset() addDs [Suc_less_SucD, not_less_Least]) 1,
693         rename_tac "k n" 1,
694         res_inst_tac [("n","k")] natE 1,
695         Blast_tac 1,
696         hyp_subst_tac 1,
697         rewtac Least_nat_def,
698         rtac (select_equality RS arg_cong RS sym) 1,
699         Safe_tac,
700         dtac Suc_mono 1,
701         Blast_tac 1,
702         cut_facts_tac [less_linear] 1,
703         Safe_tac,
704         atac 2,
705         Blast_tac 2,
706         dtac Suc_mono 1,
707         Blast_tac 1]);
710 (*** Instantiation of transitivity prover ***)
712 structure Less_Arith =
713 struct
714 val nat_leI = leI;
715 val nat_leD = leD;
716 val lessI = lessI;
717 val zero_less_Suc = zero_less_Suc;
718 val less_reflE = less_irrefl;
719 val less_zeroE = less_zeroE;
720 val less_incr = Suc_mono;
721 val less_decr = Suc_less_SucD;
722 val less_incr_rhs = Suc_mono RS Suc_lessD;
723 val less_decr_lhs = Suc_lessD;
724 val less_trans_Suc = less_trans_Suc;
725 val leI = Suc_leI RS (Suc_le_mono RS iffD1);
726 val not_lessI = leI RS leD
727 val not_leI = prove_goal thy "!!m::nat. n < m ==> ~ m <= n"
728   (fn _ => [etac swap2 1, etac leD 1]);
729 val eqI = prove_goal thy "!!m. [| m < Suc n; n < Suc m |] ==> m=n"
730   (fn _ => [etac less_SucE 1,
731             blast_tac (claset() addSDs [Suc_less_SucD] addSEs [less_irrefl]
732                               addDs [less_trans_Suc]) 1,
733             assume_tac 1]);
734 val leD = le_eq_less_Suc RS iffD1;
735 val not_lessD = nat_leI RS leD;
736 val not_leD = not_leE
737 val eqD1 = prove_goal thy  "!!n. m = n ==> m < Suc n"
738  (fn _ => [etac subst 1, rtac lessI 1]);
739 val eqD2 = sym RS eqD1;
741 fun is_zero(t) =  t = Const("0",Type("nat",[]));
743 fun nnb T = T = Type("fun",[Type("nat",[]),
744                             Type("fun",[Type("nat",[]),
745                                         Type("bool",[])])])
747 fun decomp_Suc(Const("Suc",_)\$t) = let val (a,i) = decomp_Suc t in (a,i+1) end
748   | decomp_Suc t = (t,0);
750 fun decomp2(rel,T,lhs,rhs) =
751   if not(nnb T) then None else
752   let val (x,i) = decomp_Suc lhs
753       val (y,j) = decomp_Suc rhs
754   in case rel of
755        "op <"  => Some(x,i,"<",y,j)
756      | "op <=" => Some(x,i,"<=",y,j)
757      | "op ="  => Some(x,i,"=",y,j)
758      | _       => None
759   end;
761 fun negate(Some(x,i,rel,y,j)) = Some(x,i,"~"^rel,y,j)
762   | negate None = None;
764 fun decomp(_\$(Const(rel,T)\$lhs\$rhs)) = decomp2(rel,T,lhs,rhs)
765   | decomp(_\$(Const("Not",_)\$(Const(rel,T)\$lhs\$rhs))) =
766       negate(decomp2(rel,T,lhs,rhs))
767   | decomp _ = None
769 end;
771 structure Trans_Tac = Trans_Tac_Fun(Less_Arith);
773 open Trans_Tac;
775 (*** eliminates ~= in premises, which trans_tac cannot deal with ***)
776 bind_thm("nat_neqE", nat_neq_iff RS iffD1 RS disjE);