1 (* ======================================================================== *)
2 (* Infinite Ramsey's theorem. *)
4 (* Port to HOL Light of a HOL88 proof done on 9th May 1994 *)
5 (* ======================================================================== *)
7 (* ------------------------------------------------------------------------- *)
8 (* HOL88 compatibility. *)
9 (* ------------------------------------------------------------------------- *)
12 is_neg tm or is_imp tm;;
15 try dest_imp tm with Failure _ ->
16 try (dest_neg tm,mk_const("F",[]))
17 with Failure _ -> failwith "dest_neg_imp";;
19 (* ------------------------------------------------------------------------- *)
20 (* These get overwritten by the subgoal stuff. *)
21 (* ------------------------------------------------------------------------- *)
25 let prove_thm((s:string),g,t) = prove(g,t);;
27 (* ------------------------------------------------------------------------- *)
28 (* The quantifier movement conversions. *)
29 (* ------------------------------------------------------------------------- *)
31 let (CONV_OF_RCONV: conv -> conv) =
33 if is_abs tm then bndvar tm
34 else if is_comb tm then try get_bv (rand tm) with Failure _ -> get_bv (rator tm)
39 let th2 = ONCE_DEPTH_CONV (GEN_ALPHA_CONV v) (rhs(concl th1)) in
42 let (CONV_OF_THM: thm -> conv) =
43 CONV_OF_RCONV o REWR_CONV;;
45 let (X_FUN_EQ_CONV:term->conv) =
46 fun v -> (REWR_CONV FUN_EQ_THM) THENC GEN_ALPHA_CONV v;;
48 let (FUN_EQ_CONV:conv) =
50 let vars = frees tm in
51 let op,[ty1;ty2] = dest_type(type_of (lhs tm)) in
54 if (is_vartype ty1) then "x" else
55 hd(explode(fst(dest_type ty1))) in
56 let x = variant vars (mk_var(varnm,ty1)) in
58 else failwith "FUN_EQ_CONV";;
60 let (SINGLE_DEPTH_CONV:conv->conv) =
61 let rec SINGLE_DEPTH_CONV conv tm =
62 try conv tm with Failure _ ->
63 (SUB_CONV (SINGLE_DEPTH_CONV conv) THENC (TRY_CONV conv)) tm in
66 let (SKOLEM_CONV:conv) =
67 SINGLE_DEPTH_CONV (REWR_CONV SKOLEM_THM);;
69 let (X_SKOLEM_CONV:term->conv) =
70 fun v -> SKOLEM_CONV THENC GEN_ALPHA_CONV v;;
72 let EXISTS_UNIQUE_CONV tm =
73 let v = bndvar(rand tm) in
74 let th1 = REWR_CONV EXISTS_UNIQUE_THM tm in
75 let tm1 = rhs(concl th1) in
76 let vars = frees tm1 in
77 let v = variant vars v in
78 let v' = variant (v::vars) v in
80 (LAND_CONV(GEN_ALPHA_CONV v) THENC
81 RAND_CONV(BINDER_CONV(GEN_ALPHA_CONV v') THENC
82 GEN_ALPHA_CONV v)) tm1 in
85 let NOT_FORALL_CONV = CONV_OF_THM NOT_FORALL_THM;;
87 let NOT_EXISTS_CONV = CONV_OF_THM NOT_EXISTS_THM;;
89 let RIGHT_IMP_EXISTS_CONV = CONV_OF_THM RIGHT_IMP_EXISTS_THM;;
91 let FORALL_IMP_CONV = CONV_OF_RCONV
92 (REWR_CONV TRIV_FORALL_IMP_THM ORELSEC
93 REWR_CONV RIGHT_FORALL_IMP_THM ORELSEC
94 REWR_CONV LEFT_FORALL_IMP_THM);;
96 let EXISTS_AND_CONV = CONV_OF_RCONV
97 (REWR_CONV TRIV_EXISTS_AND_THM ORELSEC
98 REWR_CONV LEFT_EXISTS_AND_THM ORELSEC
99 REWR_CONV RIGHT_EXISTS_AND_THM);;
101 let LEFT_IMP_EXISTS_CONV = CONV_OF_THM LEFT_IMP_EXISTS_THM;;
103 let LEFT_AND_EXISTS_CONV tm =
104 let v = bndvar(rand(rand(rator tm))) in
105 (REWR_CONV LEFT_AND_EXISTS_THM THENC TRY_CONV (GEN_ALPHA_CONV v)) tm;;
107 let RIGHT_AND_EXISTS_CONV =
108 CONV_OF_THM RIGHT_AND_EXISTS_THM;;
110 let AND_FORALL_CONV = CONV_OF_THM AND_FORALL_THM;;
112 (* ------------------------------------------------------------------------- *)
113 (* The slew of named tautologies. *)
114 (* ------------------------------------------------------------------------- *)
116 let AND1_THM = TAUT `!t1 t2. t1 /\ t2 ==> t1`;;
118 let AND2_THM = TAUT `!t1 t2. t1 /\ t2 ==> t2`;;
120 let AND_IMP_INTRO = TAUT `!t1 t2 t3. t1 ==> t2 ==> t3 = t1 /\ t2 ==> t3`;;
122 let AND_INTRO_THM = TAUT `!t1 t2. t1 ==> t2 ==> t1 /\ t2`;;
124 let BOOL_EQ_DISTINCT = TAUT `~(T <=> F) /\ ~(F <=> T)`;;
126 let EQ_EXPAND = TAUT `!t1 t2. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2`;;
128 let EQ_IMP_THM = TAUT `!t1 t2. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)`;;
130 let FALSITY = TAUT `!t. F ==> t`;;
132 let F_IMP = TAUT `!t. ~t ==> t ==> F`;;
134 let IMP_DISJ_THM = TAUT `!t1 t2. t1 ==> t2 <=> ~t1 \/ t2`;;
136 let IMP_F = TAUT `!t. (t ==> F) ==> ~t`;;
138 let IMP_F_EQ_F = TAUT `!t. t ==> F <=> (t <=> F)`;;
140 let LEFT_AND_OVER_OR = TAUT
141 `!t1 t2 t3. t1 /\ (t2 \/ t3) <=> t1 /\ t2 \/ t1 /\ t3`;;
143 let LEFT_OR_OVER_AND = TAUT
144 `!t1 t2 t3. t1 \/ t2 /\ t3 <=> (t1 \/ t2) /\ (t1 \/ t3)`;;
146 let NOT_AND = TAUT `~(t /\ ~t)`;;
148 let NOT_F = TAUT `!t. ~t ==> (t <=> F)`;;
150 let OR_ELIM_THM = TAUT
151 `!t t1 t2. t1 \/ t2 ==> (t1 ==> t) ==> (t2 ==> t) ==> t`;;
153 let OR_IMP_THM = TAUT `!t1 t2. (t1 <=> t2 \/ t1) <=> t2 ==> t1`;;
155 let OR_INTRO_THM1 = TAUT `!t1 t2. t1 ==> t1 \/ t2`;;
157 let OR_INTRO_THM2 = TAUT `!t1 t2. t2 ==> t1 \/ t2`;;
159 let RIGHT_AND_OVER_OR = TAUT
160 `!t1 t2 t3. (t2 \/ t3) /\ t1 <=> t2 /\ t1 \/ t3 /\ t1`;;
162 let RIGHT_OR_OVER_AND = TAUT
163 `!t1 t2 t3. t2 /\ t3 \/ t1 <=> (t2 \/ t1) /\ (t3 \/ t1)`;;
165 (* ------------------------------------------------------------------------- *)
166 (* This is an overwrite -- is there any point in what I have? *)
167 (* ------------------------------------------------------------------------- *)
169 let is_type = can get_type_arity;;
171 (* ------------------------------------------------------------------------- *)
172 (* I suppose this is also useful. *)
173 (* ------------------------------------------------------------------------- *)
175 let is_constant = can get_const_type;;
177 (* ------------------------------------------------------------------------- *)
179 (* ------------------------------------------------------------------------- *)
181 let null l = l = [];;
183 (* ------------------------------------------------------------------------- *)
185 (* ------------------------------------------------------------------------- *)
187 let type_tyvars = type_vars_in_term o curry mk_var "x";;
191 try term_match [] u t with Failure _ ->
192 try find_mt(rator t) with Failure _ ->
193 try find_mt(rand t) with Failure _ ->
194 try find_mt(snd(dest_abs t))
195 with Failure _ -> failwith "find_match" in
196 fun t -> let _,tmin,tyin = find_mt t in
199 let rec mk_primed_var(name,ty) =
200 if can get_const_type name then mk_primed_var(name^"'",ty)
201 else mk_var(name,ty);;
204 let rec subst_occs slist tm =
205 let applic,noway = partition (fun (i,(t,x)) -> aconv tm x) slist in
206 let sposs = map (fun (l,z) -> let l1,l2 = partition ((=) 1) l in
207 (l1,z),(l2,z)) applic in
208 let racts,rrest = unzip sposs in
209 let acts = filter (fun t -> not (fst t = [])) racts in
210 let trest = map (fun (n,t) -> (map (C (-) 1) n,t)) rrest in
211 let urest = filter (fun t -> not (fst t = [])) trest in
212 let tlist = urest @ noway in
215 let l,r = dest_comb tm in
216 let l',s' = subst_occs tlist l in
217 let r',s'' = subst_occs s' r in
219 else if is_abs tm then
220 let bv,bod = dest_abs tm in
221 let gv = genvar(type_of bv) in
222 let nbod = vsubst[gv,bv] bod in
223 let tm',s' = subst_occs tlist nbod in
224 alpha bv (mk_abs(gv,tm')),s'
228 let tm' = (fun (n,(t,x)) -> subst[t,x] tm) (hd acts) in
230 fun ilist slist tm -> fst(subst_occs (zip ilist slist) tm);;
232 (* ------------------------------------------------------------------------- *)
233 (* Note that the all-instantiating INST and INST_TYPE are not overwritten. *)
234 (* ------------------------------------------------------------------------- *)
236 let INST_TY_TERM(substl,insttyl) th =
237 let th' = INST substl (INST_TYPE insttyl th) in
238 if hyp th' = hyp th then th'
239 else failwith "INST_TY_TERM: Free term and/or type variables in hypotheses";;
241 (* ------------------------------------------------------------------------- *)
242 (* Conversions stuff. *)
243 (* ------------------------------------------------------------------------- *)
245 let RIGHT_CONV_RULE (conv:conv) th =
246 TRANS th (conv(rhs(concl th)));;
248 (* ------------------------------------------------------------------------- *)
250 (* ------------------------------------------------------------------------- *)
253 let pth = GENL [`a:A`; `b:A`]
254 (GEN_REWRITE_RULE I [GSYM CONTRAPOS_THM] (DISCH_ALL(SYM(ASSUME`a:A = b`))))
256 fun th -> try let l,r = dest_eq(dest_neg(concl th)) in
257 MP (SPECL [r; l] (INST_TYPE [type_of l,aty] pth)) th
258 with Failure _ -> failwith "NOT_EQ_SYM";;
261 try MP thi th with Failure _ ->
262 try let t = dest_neg (concl thi) in
263 MP(MP (SPEC t F_IMP) thi) th
264 with Failure _ -> failwith "NOT_MP";;
267 let mkall = AP_TERM (mk_const("!",[type_of x,mk_vartype "A"])) in
268 fun th -> try mkall (ABS x th)
269 with Failure _ -> failwith "FORALL_EQ";;
272 let mkex = AP_TERM (mk_const("?",[type_of x,mk_vartype "A"])) in
273 fun th -> try mkex (ABS x th)
274 with Failure _ -> failwith "EXISTS_EQ";;
277 let mksel = AP_TERM (mk_const("@",[type_of x,mk_vartype "A"])) in
278 fun th -> try mksel (ABS x th)
279 with Failure _ -> failwith "SELECT_EQ";;
282 try TRANS th (BETA_CONV(rhs(concl th)))
283 with Failure _ -> failwith "RIGHT_BETA";;
285 let rec LIST_BETA_CONV tm =
286 try let rat,rnd = dest_comb tm in
287 RIGHT_BETA(AP_THM(LIST_BETA_CONV rat)rnd)
288 with Failure _ -> REFL tm;;
290 let RIGHT_LIST_BETA th = TRANS th (LIST_BETA_CONV(snd(dest_eq(concl th))));;
292 let LIST_CONJ = end_itlist CONJ ;;
294 let rec CONJ_LIST n th =
295 try if n=1 then [th] else (CONJUNCT1 th)::(CONJ_LIST (n-1) (CONJUNCT2 th))
296 with Failure _ -> failwith "CONJ_LIST";;
298 let rec BODY_CONJUNCTS th =
299 if is_forall(concl th) then
300 BODY_CONJUNCTS (SPEC_ALL th) else
301 if is_conj (concl th) then
302 BODY_CONJUNCTS (CONJUNCT1 th) @ BODY_CONJUNCTS (CONJUNCT2 th)
305 let rec IMP_CANON th =
307 if is_conj w then IMP_CANON (CONJUNCT1 th) @ IMP_CANON (CONJUNCT2 th)
308 else if is_imp w then
309 let ante,conc = dest_neg_imp w in
311 let a,b = dest_conj ante in
313 (DISCH a (DISCH b (NOT_MP th (CONJ (ASSUME a) (ASSUME b)))))
314 else if is_disj ante then
315 let a,b = dest_disj ante in
316 IMP_CANON (DISCH a (NOT_MP th (DISJ1 (ASSUME a) b))) @
317 IMP_CANON (DISCH b (NOT_MP th (DISJ2 a (ASSUME b))))
318 else if is_exists ante then
319 let x,body = dest_exists ante in
320 let x' = variant (thm_frees th) x in
321 let body' = subst [x',x] body in
323 (DISCH body' (NOT_MP th (EXISTS (ante, x') (ASSUME body'))))
325 map (DISCH ante) (IMP_CANON (UNDISCH th))
326 else if is_forall w then
327 IMP_CANON (SPEC_ALL th)
330 let LIST_MP = rev_itlist (fun x y -> MP y x);;
333 let pth = TAUT`!t1 t2. t1 \/ t2 ==> ~t1 ==> t2` in
335 try let a,b = dest_disj(concl th) in MP (SPECL [a;b] pth) th
336 with Failure _ -> failwith "DISJ_IMP";;
339 let pth = TAUT`!t1 t2. (t1 ==> t2) ==> ~t1 \/ t2` in
341 try let a,b = dest_imp(concl th) in MP (SPECL [a;b] pth) th
342 with Failure _ -> failwith "IMP_ELIM";;
344 let DISJ_CASES_UNION dth ath bth =
345 DISJ_CASES dth (DISJ1 ath (concl bth)) (DISJ2 (concl ath) bth);;
348 try let ov = bndvar(rand(concl qth)) in
349 let bv,rth = SPEC_VAR qth in
350 let sth = ABS bv rth in
351 let cnv = ALPHA_CONV ov in
352 CONV_RULE(BINOP_CONV cnv) sth
353 with Failure _ -> failwith "MK_ABS";;
356 try let th1 = MK_ABS th in
357 CONV_RULE(LAND_CONV ETA_CONV) th1
358 with Failure _ -> failwith "HALF_MK_ABS";;
361 try let ov = bndvar(rand(concl qth)) in
362 let bv,rth = SPEC_VAR qth in
363 let sth = EXISTS_EQ bv rth in
364 let cnv = GEN_ALPHA_CONV ov in
365 CONV_RULE(BINOP_CONV cnv) sth
366 with Failure _ -> failwith "MK_EXISTS";;
368 let LIST_MK_EXISTS l th = itlist (fun x th -> MK_EXISTS(GEN x th)) l th;;
370 let IMP_CONJ th1 th2 =
371 let A1,C1 = dest_imp (concl th1) and A2,C2 = dest_imp (concl th2) in
372 let a1,a2 = CONJ_PAIR (ASSUME (mk_conj(A1,A2))) in
373 DISCH (mk_conj(A1,A2)) (CONJ (MP th1 a1) (MP th2 a2));;
376 if not (is_var x) then failwith "EXISTS_IMP: first argument not a variable"
378 try let ante,cncl = dest_imp(concl th) in
379 let th1 = EXISTS (mk_exists(x,cncl),x) (UNDISCH th) in
380 let asm = mk_exists(x,ante) in
381 DISCH asm (CHOOSE (x,ASSUME asm) th1)
382 with Failure _ -> failwith "EXISTS_IMP: variable free in assumptions";;
385 let CONJUNCTS_CONV (t1,t2) =
386 let rec build_conj thl t =
387 try let l,r = dest_conj t in
388 CONJ (build_conj thl l) (build_conj thl r)
389 with Failure _ -> find (fun th -> concl th = t) thl in
391 (DISCH t1 (build_conj (CONJUNCTS (ASSUME t1)) t2))
392 (DISCH t2 (build_conj (CONJUNCTS (ASSUME t2)) t1))
393 with Failure _ -> failwith "CONJUNCTS_CONV";;
395 let CONJ_SET_CONV l1 l2 =
396 try CONJUNCTS_CONV (list_mk_conj l1, list_mk_conj l2)
397 with Failure _ -> failwith "CONJ_SET_CONV";;
399 let FRONT_CONJ_CONV tml t =
401 if hd l = x then tl l else (hd l)::(remove x (tl l)) in
402 try CONJ_SET_CONV tml (t::(remove t tml))
403 with Failure _ -> failwith "FRONT_CONJ_CONV";;
406 let pth = TAUT`!t t1 t2. (t ==> (t1 <=> t2)) ==> (t /\ t1 <=> t /\ t2)` in
408 try let t1,t2 = dest_eq(concl th) in
409 MP (SPECL [t; t1; t2] pth) (DISCH t th)
410 with Failure _ -> failwith "CONJ_DISCH";;
412 let rec CONJ_DISCHL l th =
413 if l = [] then th else CONJ_DISCH (hd l) (CONJ_DISCHL (tl l) th);;
416 let wl,w = dest_thm th in
418 GSPEC (SPEC (genvar (type_of (fst (dest_forall w)))) th)
421 let ANTE_CONJ_CONV tm =
422 try let (a1,a2),c = (dest_conj F_F I) (dest_imp tm) in
423 let imp1 = MP (ASSUME tm) (CONJ (ASSUME a1) (ASSUME a2)) and
424 imp2 = LIST_MP [CONJUNCT1 (ASSUME (mk_conj(a1,a2)));
425 CONJUNCT2 (ASSUME (mk_conj(a1,a2)))]
426 (ASSUME (mk_imp(a1,mk_imp(a2,c)))) in
427 IMP_ANTISYM_RULE (DISCH_ALL (DISCH a1 (DISCH a2 imp1)))
428 (DISCH_ALL (DISCH (mk_conj(a1,a2)) imp2))
429 with Failure _ -> failwith "ANTE_CONJ_CONV";;
432 let check = let boolty = `:bool` in check (fun tm -> type_of tm = boolty) in
433 let clist = map (GEN `b:bool`)
434 (CONJUNCTS(SPEC `b:bool` EQ_CLAUSES)) in
435 let tb = hd clist and bt = hd(tl clist) in
436 let T = `T` and F = `F` in
438 try let l,r = (I F_F check) (dest_eq tm) in
439 if l = r then EQT_INTRO (REFL l) else
440 if l = T then SPEC r tb else
441 if r = T then SPEC l bt else fail()
442 with Failure _ -> failwith "bool_EQ_CONV";;
445 let T = `T` and F = `F` and vt = genvar`:A` and vf = genvar `:A` in
446 let gen = GENL [vt;vf] in
447 let CT,CF = (gen F_F gen) (CONJ_PAIR (SPECL [vt;vf] COND_CLAUSES)) in
449 let P,(u,v) = try dest_cond tm
450 with Failure _ -> failwith "COND_CONV: not a conditional" in
451 let ty = type_of u in
452 if (P=T) then SPEC v (SPEC u (INST_TYPE [ty,`:A`] CT)) else
453 if (P=F) then SPEC v (SPEC u (INST_TYPE [ty,`:A`] CF)) else
454 if (u=v) then SPEC u (SPEC P (INST_TYPE [ty,`:A`] COND_ID)) else
456 let cnd = AP_TERM (rator tm) (ALPHA v u) in
457 let thm = SPEC u (SPEC P (INST_TYPE [ty,`:A`] COND_ID)) in
459 failwith "COND_CONV: can't simplify conditional";;
461 let SUBST_MATCH eqth th =
462 let tm_inst,ty_inst = find_match (lhs(concl eqth)) (concl th) in
463 SUBS [INST tm_inst (INST_TYPE ty_inst eqth)] th;;
465 let SUBST thl pat th =
466 let eqs,vs = unzip thl in
467 let gvs = map (genvar o type_of) vs in
468 let gpat = subst (zip gvs vs) pat in
469 let ls,rs = unzip (map (dest_eq o concl) eqs) in
470 let ths = map (ASSUME o mk_eq) (zip gvs rs) in
471 let th1 = ASSUME gpat in
472 let th2 = SUBS ths th1 in
473 let th3 = itlist DISCH (map concl ths) (DISCH gpat th2) in
474 let th4 = INST (zip ls gvs) th3 in
475 MP (rev_itlist (C MP) eqs th4) th;;
477 (* let GSUBS = ... *)
478 (* let SUBS_OCCS = ... *)
480 (* A poor thing but mine own. The old ones use mk_thm and the commented
481 out functions are bogus. *)
483 let SUBST_CONV thvars template tm =
484 let thms,vars = unzip thvars in
485 let gvs = map (genvar o type_of) vars in
486 let gtemplate = subst (zip gvs vars) template in
487 SUBST (zip thms gvs) (mk_eq(template,gtemplate)) (REFL tm);;
489 (* ------------------------------------------------------------------------- *)
490 (* Filtering rewrites. *)
491 (* ------------------------------------------------------------------------- *)
493 let FILTER_PURE_ASM_REWRITE_RULE f thl th =
494 PURE_REWRITE_RULE ((map ASSUME (filter f (hyp th))) @ thl) th
496 and FILTER_ASM_REWRITE_RULE f thl th =
497 REWRITE_RULE ((map ASSUME (filter f (hyp th))) @ thl) th
499 and FILTER_PURE_ONCE_ASM_REWRITE_RULE f thl th =
500 PURE_ONCE_REWRITE_RULE ((map ASSUME (filter f (hyp th))) @ thl) th
502 and FILTER_ONCE_ASM_REWRITE_RULE f thl th =
503 ONCE_REWRITE_RULE ((map ASSUME (filter f (hyp th))) @ thl) th;;
505 let (FILTER_PURE_ASM_REWRITE_TAC: (term->bool) -> thm list -> tactic) =
507 PURE_REWRITE_TAC (filter (f o concl) (map snd asl) @ thl) (asl,w)
509 and (FILTER_ASM_REWRITE_TAC: (term->bool) -> thm list -> tactic) =
511 REWRITE_TAC (filter (f o concl) (map snd asl) @ thl) (asl,w)
513 and (FILTER_PURE_ONCE_ASM_REWRITE_TAC: (term->bool) -> thm list -> tactic) =
515 PURE_ONCE_REWRITE_TAC (filter (f o concl) (map snd asl) @ thl) (asl,w)
517 and (FILTER_ONCE_ASM_REWRITE_TAC: (term->bool) -> thm list -> tactic) =
519 ONCE_REWRITE_TAC (filter (f o concl) (map snd asl) @ thl) (asl,w);;
521 (* ------------------------------------------------------------------------- *)
523 (* ------------------------------------------------------------------------- *)
525 let (X_CASES_THENL: term list list -> thm_tactic list -> thm_tactic) =
527 end_itlist DISJ_CASES_THEN2
528 (map (fun (vars,ttac) -> EVERY_TCL (map X_CHOOSE_THEN vars) ttac)
531 let (X_CASES_THEN: term list list -> thm_tactical) =
533 end_itlist DISJ_CASES_THEN2
534 (map (fun vars -> EVERY_TCL (map X_CHOOSE_THEN vars) ttac) varsl);;
536 let (CASES_THENL: thm_tactic list -> thm_tactic) =
537 fun ttacl -> end_itlist DISJ_CASES_THEN2 (map (REPEAT_TCL CHOOSE_THEN) ttacl);;
539 (* ------------------------------------------------------------------------- *)
541 (* ------------------------------------------------------------------------- *)
543 let (DISCARD_TAC: thm_tactic) =
546 if exists (aconv (concl th)) (truth::(map (concl o snd) asl))
548 else failwith "DISCARD_TAC";;
550 let (CHECK_ASSUME_TAC: thm_tactic) =
552 FIRST [CONTR_TAC gth; ACCEPT_TAC gth;
553 DISCARD_TAC gth; ASSUME_TAC gth];;
555 let (FILTER_GEN_TAC: term -> tactic) =
557 if is_forall w & not (tm = fst(dest_forall w)) then
559 else failwith "FILTER_GEN_TAC";;
561 let (FILTER_DISCH_THEN: thm_tactic -> term -> tactic) =
562 fun ttac tm (asl,w) ->
563 if is_neg_imp w & not (free_in tm (fst(dest_neg_imp w))) then
564 DISCH_THEN ttac (asl,w)
565 else failwith "FILTER_DISCH_THEN";;
567 let FILTER_STRIP_THEN ttac tm =
568 FIRST [FILTER_GEN_TAC tm; FILTER_DISCH_THEN ttac tm; CONJ_TAC];;
570 let FILTER_DISCH_TAC = FILTER_DISCH_THEN STRIP_ASSUME_TAC;;
572 let FILTER_STRIP_TAC = FILTER_STRIP_THEN STRIP_ASSUME_TAC;;
574 (* ------------------------------------------------------------------------- *)
575 (* Conversions for quantifier movement using proforma theorems. *)
576 (* ------------------------------------------------------------------------- *)
580 (* ------------------------------------------------------------------------- *)
581 (* Resolution stuff. *)
582 (* ------------------------------------------------------------------------- *)
586 if is_neg (concl th) then true,(NOT_ELIM th) else (false,th) in
587 let rec canon fl th =
590 let (th1,th2) = CONJ_PAIR th in (canon fl th1) @ (canon fl th2) else
591 if ((is_imp w) & not(is_neg w)) then
592 let ante,conc = dest_neg_imp w in
593 if (is_conj ante) then
594 let a,b = dest_conj ante in
595 let cth = NOT_MP th (CONJ (ASSUME a) (ASSUME b)) in
596 let th1 = DISCH b cth and th2 = DISCH a cth in
597 (canon true (DISCH a th1)) @ (canon true (DISCH b th2)) else
598 if (is_disj ante) then
599 let a,b = dest_disj ante in
600 let ath = DISJ1 (ASSUME a) b and bth = DISJ2 a (ASSUME b) in
601 let th1 = DISCH a (NOT_MP th ath) and
602 th2 = DISCH b (NOT_MP th bth) in
603 (canon true th1) @ (canon true th2) else
604 if (is_exists ante) then
605 let v,body = dest_exists ante in
606 let newv = variant (thm_frees th) v in
607 let newa = subst [newv,v] body in
608 let th1 = NOT_MP th (EXISTS (ante, newv) (ASSUME newa)) in
609 canon true (DISCH newa th1) else
610 map (GEN_ALL o (DISCH ante)) (canon true (UNDISCH th)) else
611 if (is_eq w & (type_of (rand w) = `:bool`)) then
612 let (th1,th2) = EQ_IMP_RULE th in
613 (if fl then [GEN_ALL th] else []) @ (canon true th1) @ (canon true th2) else
614 if (is_forall w) then
615 let vs,body = strip_forall w in
616 let fvs = thm_frees th in
617 let vfn = fun l -> variant (l @ fvs) in
618 let nvs = itlist (fun v nv -> let v' = vfn nv v in (v'::nv)) vs [] in
619 canon fl (SPECL nvs th) else
620 if fl then [GEN_ALL th] else [] in
621 fun th -> try let args = map (not_elim o SPEC_ALL) (CONJUNCTS (SPEC_ALL th)) in
622 let imps = flat (map (map GEN_ALL o (uncurry canon)) args) in
623 check (fun l -> l <> []) imps
625 failwith "RES_CANON: no implication is derivable from input thm.";;
627 let IMP_RES_THEN,RES_THEN =
629 let sth = SPEC_ALL impth in
630 let matchfn = (fun (a,b,c) -> b,c) o
631 term_match [] (fst(dest_neg_imp(concl sth))) in
632 fun th -> NOT_MP (INST_TY_TERM (matchfn (concl th)) sth) th in
633 let check st l = (if l = [] then failwith st else l) in
634 let IMP_RES_THEN ttac impth =
635 let ths = try RES_CANON impth with Failure _ -> failwith "IMP_RES_THEN: no implication" in
638 let l = itlist (fun th -> (@) (mapfilter (MATCH_MP th) asl)) ths [] in
639 let res = check "IMP_RES_THEN: no resolvents " l in
640 let tacs = check "IMP_RES_THEN: no tactics" (mapfilter ttac res) in
642 let RES_THEN ttac (asl,g) =
643 let asm = map snd asl in
644 let ths = itlist (@) (mapfilter RES_CANON asm) [] in
645 let imps = check "RES_THEN: no implication" ths in
646 let l = itlist (fun th -> (@) (mapfilter (MATCH_MP th) asm)) imps [] in
647 let res = check "RES_THEN: no resolvents " l in
648 let tacs = check "RES_THEN: no tactics" (mapfilter ttac res) in
649 EVERY tacs (asl,g) in
650 IMP_RES_THEN,RES_THEN;;
652 let IMP_RES_TAC th g =
653 try IMP_RES_THEN (REPEAT_GTCL IMP_RES_THEN STRIP_ASSUME_TAC) th g
654 with Failure _ -> ALL_TAC g;;
657 try RES_THEN (REPEAT_GTCL IMP_RES_THEN STRIP_ASSUME_TAC) g
658 with Failure _ -> ALL_TAC g;;
660 (* ------------------------------------------------------------------------- *)
661 (* Stuff for handling type definitions. *)
662 (* ------------------------------------------------------------------------- *)
664 let prove_rep_fn_one_one th =
665 try let thm = CONJUNCT1 th in
666 let A,R = (I F_F rator) (dest_comb(lhs(snd(dest_forall(concl thm))))) in
667 let _,[aty;rty] = dest_type (type_of R) in
668 let a = mk_primed_var("a",aty) in let a' = variant [a] a in
669 let a_eq_a' = mk_eq(a,a') and
670 Ra_eq_Ra' = mk_eq(mk_comb(R,a),mk_comb (R,a')) in
671 let th1 = AP_TERM A (ASSUME Ra_eq_Ra') in
672 let ga1 = genvar aty and ga2 = genvar aty in
673 let th2 = SUBST [SPEC a thm,ga1;SPEC a' thm,ga2] (mk_eq(ga1,ga2)) th1 in
674 let th3 = DISCH a_eq_a' (AP_TERM R (ASSUME a_eq_a')) in
675 GEN a (GEN a' (IMP_ANTISYM_RULE (DISCH Ra_eq_Ra' th2) th3))
676 with Failure _ -> failwith "prove_rep_fn_one_one";;
678 let prove_rep_fn_onto th =
679 try let [th1;th2] = CONJUNCTS th in
680 let r,eq = (I F_F rhs)(dest_forall(concl th2)) in
681 let RE,ar = dest_comb(lhs eq) and
682 sr = (mk_eq o (fun (x,y) -> y,x) o dest_eq) eq in
683 let a = mk_primed_var ("a",type_of ar) in
684 let sra = mk_eq(r,mk_comb(RE,a)) in
685 let ex = mk_exists(a,sra) in
686 let imp1 = EXISTS (ex,ar) (SYM(ASSUME eq)) in
687 let v = genvar (type_of r) and
689 s' = AP_TERM RE (SPEC a th1) in
690 let th = SUBST[SYM(ASSUME sra),v](mk_eq(mk_comb(RE,mk_comb(A,v)),v))s' in
691 let imp2 = CHOOSE (a,ASSUME ex) th in
692 let swap = IMP_ANTISYM_RULE (DISCH eq imp1) (DISCH ex imp2) in
693 GEN r (TRANS (SPEC r th2) swap)
694 with Failure _ -> failwith "prove_rep_fn_onto";;
696 let prove_abs_fn_onto th =
697 try let [th1;th2] = CONJUNCTS th in
698 let a,(A,R) = (I F_F ((I F_F rator)o dest_comb o lhs))
699 (dest_forall(concl th1)) in
700 let thm1 = EQT_ELIM(TRANS (SPEC (mk_comb (R,a)) th2)
701 (EQT_INTRO (AP_TERM R (SPEC a th1)))) in
702 let thm2 = SYM(SPEC a th1) in
703 let r,P = (I F_F (rator o lhs)) (dest_forall(concl th2)) in
704 let ex = mk_exists(r,mk_conj(mk_eq(a,mk_comb(A,r)),mk_comb(P,r))) in
705 GEN a (EXISTS(ex,mk_comb(R,a)) (CONJ thm2 thm1))
706 with Failure _ -> failwith "prove_abs_fn_onto";;
708 let prove_abs_fn_one_one th =
709 try let [th1;th2] = CONJUNCTS th in
710 let r,P = (I F_F (rator o lhs)) (dest_forall(concl th2)) and
711 A,R = (I F_F rator) (dest_comb(lhs(snd(dest_forall(concl th1))))) in
712 let r' = variant [r] r in
713 let as1 = ASSUME(mk_comb(P,r)) and as2 = ASSUME(mk_comb(P,r')) in
714 let t1 = EQ_MP (SPEC r th2) as1 and t2 = EQ_MP (SPEC r' th2) as2 in
715 let eq = (mk_eq(mk_comb(A,r),mk_comb(A,r'))) in
716 let v1 = genvar(type_of r) and v2 = genvar(type_of r) in
718 (SUBST [t1,v1;t2,v2] (mk_eq(v1,v2)) (AP_TERM R (ASSUME eq))) and
719 i2 = DISCH (mk_eq(r,r')) (AP_TERM A (ASSUME (mk_eq(r,r')))) in
720 let thm = IMP_ANTISYM_RULE i1 i2 in
721 let disch = DISCH (mk_comb(P,r)) (DISCH (mk_comb(P,r')) thm) in
723 with Failure _ -> failwith "prove_abs_fn_one_one";;
725 (* ------------------------------------------------------------------------- *)
726 (* AC rewriting needs to be wrapped up as a special conversion. *)
727 (* ------------------------------------------------------------------------- *)
729 let AC_CONV(assoc,sym) =
730 let th1 = SPEC_ALL assoc
731 and th2 = SPEC_ALL sym in
732 let th3 = GEN_REWRITE_RULE (RAND_CONV o LAND_CONV) [th2] th1 in
734 let th5 = GEN_REWRITE_RULE RAND_CONV [th4] th3 in
735 EQT_INTRO o AC(end_itlist CONJ [th2; th4; th5]);;
737 let AC_RULE ths = EQT_ELIM o AC_CONV ths;;
739 (* ------------------------------------------------------------------------- *)
740 (* The order of picking conditionals is different! *)
741 (* ------------------------------------------------------------------------- *)
743 let (COND_CASES_TAC :tactic) =
744 let is_good_cond tm =
745 try not(is_const(fst(dest_cond tm)))
746 with Failure _ -> false in
748 let cond = find_term (fun tm -> is_good_cond tm & free_in tm w) w in
749 let p,(t,u) = dest_cond cond in
750 let inst = INST_TYPE [type_of t, `:A`] COND_CLAUSES in
751 let (ct,cf) = CONJ_PAIR (SPEC u (SPEC t inst)) in
753 (fun th -> SUBST1_TAC (EQT_INTRO th) THEN
754 SUBST1_TAC ct THEN ASSUME_TAC th)
755 (fun th -> SUBST1_TAC (EQF_INTRO th) THEN
756 SUBST1_TAC cf THEN ASSUME_TAC th)
757 (SPEC p EXCLUDED_MIDDLE)
760 (* ------------------------------------------------------------------------- *)
761 (* MATCH_MP_TAC allows universals on the right of implication. *)
762 (* Here's a crude hack to allow it. *)
763 (* ------------------------------------------------------------------------- *)
765 let MATCH_MP_TAC th =
766 MATCH_MP_TAC th ORELSE
767 MATCH_MP_TAC(PURE_REWRITE_RULE[RIGHT_IMP_FORALL_THM] th);;
769 (* ------------------------------------------------------------------------- *)
770 (* Various theorems have different names. *)
771 (* ------------------------------------------------------------------------- *)
773 let ZERO_LESS_EQ = LE_0;;
775 let LESS_EQ_MONO = LE_SUC;;
777 let NOT_LESS = NOT_LT;;
781 let LESS_EQ_REFL = LE_REFL;;
783 let LESS_EQUAL_ANTISYM = GEN_ALL(fst(EQ_IMP_RULE(SPEC_ALL LE_ANTISYM)));;
785 let NOT_LESS_0 = GEN_ALL(EQF_ELIM(SPEC_ALL(CONJUNCT1 LT)));;
787 let LESS_TRANS = LT_TRANS;;
789 let LESS_LEMMA1 = GEN_ALL(fst(EQ_IMP_RULE(SPEC_ALL(CONJUNCT2 LT))));;
791 let LESS_SUC_REFL = prove(`!n. n < SUC n`,REWRITE_TAC[LT]);;
793 let FACT_LESS = FACT_LT;;
795 let LESS_EQ_SUC_REFL = prove(`!n. n <= SUC n`,REWRITE_TAC[LE; LE_REFL]);;
797 let LESS_EQ_ADD = LE_ADD;;
799 let GREATER_EQ = GE;;
801 let LESS_EQUAL_ADD = GEN_ALL(fst(EQ_IMP_RULE(SPEC_ALL LE_EXISTS)));;
803 let LESS_EQ_IMP_LESS_SUC = GEN_ALL(snd(EQ_IMP_RULE(SPEC_ALL LT_SUC_LE)));;
805 let LESS_IMP_LESS_OR_EQ = LT_IMP_LE;;
807 let LESS_MONO_ADD = GEN_ALL(snd(EQ_IMP_RULE(SPEC_ALL LT_ADD_RCANCEL)));;
809 let LESS_SUC = prove(`!m n. m < n ==> m < (SUC n)`,MESON_TAC[LT]);;
811 let LESS_CASES = LTE_CASES;;
813 let LESS_EQ = GSYM LE_SUC_LT;;
815 let LESS_OR_EQ = LE_LT;;
817 let LESS_ADD_1 = GEN_ALL(fst(EQ_IMP_RULE(SPEC_ALL
818 (REWRITE_RULE[ADD1] LT_EXISTS))));;
820 let SUC_SUB1 = prove(`!m. SUC m - 1 = m`,
821 REWRITE_TAC[num_CONV `1`; SUB_SUC; SUB_0]);;
823 let LESS_MONO_EQ = LT_SUC;;
825 let LESS_ADD_SUC = prove (`!m n. m < m + SUC n`,
826 REWRITE_TAC[ADD_CLAUSES; LT_SUC_LE; LE_ADD]);;
828 let LESS_REFL = LT_REFL;;
830 let INV_SUC_EQ = SUC_INJ;;
832 let LESS_EQ_CASES = LE_CASES;;
834 let LESS_EQ_TRANS = LE_TRANS;;
836 let LESS_THM = CONJUNCT2 LT;;
840 let LESS_EQ_0 = CONJUNCT1 LE;;
842 let OR_LESS = GEN_ALL(fst(EQ_IMP_RULE(SPEC_ALL LE_SUC_LT)));;
844 let SUB_EQUAL_0 = SUB_REFL;;
846 let SUB_MONO_EQ = SUB_SUC;;
848 let NOT_SUC_LESS_EQ = prove (`!n m. ~(SUC n <= m) <=> m <= n`,
849 REWRITE_TAC[NOT_LE; LT] THEN
852 let SUC_NOT = GSYM NOT_SUC;;
854 let LESS_LESS_CASES = prove(`!m n:num. (m = n) \/ m < n \/ n < m`,
855 MESON_TAC[LT_CASES]);;
857 let NOT_LESS_EQUAL = NOT_LE;;
859 let LESS_EQ_EXISTS = LE_EXISTS;;
861 let LESS_MONO_ADD_EQ = LT_ADD_RCANCEL;;
863 let LESS_LESS_EQ_TRANS = LTE_TRANS;;
865 let SUB_SUB = ARITH_RULE
866 `!b c. c <= b ==> (!a:num. a - (b - c) = (a + c) - b)`;;
868 let LESS_CASES_IMP = ARITH_RULE
869 `!m n:num. ~(m < n) /\ ~(m = n) ==> n < m`;;
871 let SUB_LESS_EQ = ARITH_RULE
872 `!n m:num. (n - m) <= n`;;
874 let SUB_EQ_EQ_0 = ARITH_RULE
875 `!m n:num. (m - n = m) <=> (m = 0) \/ (n = 0)`;;
877 let SUB_LEFT_LESS_EQ = ARITH_RULE
878 `!m n p:num. m <= (n - p) <=> (m + p) <= n \/ m <= 0`;;
880 let SUB_LEFT_GREATER_EQ =
881 ARITH_RULE `!m n p:num. m >= (n - p) <=> (m + p) >= n`;;
883 let LESS_EQ_LESS_TRANS = LET_TRANS;;
885 let LESS_0_CASES = ARITH_RULE `!m. (0 = m) \/ 0 < m`;;
887 let LESS_OR = ARITH_RULE `!m n. m < n ==> (SUC m) <= n`;;
891 (!m n. (SUC m) - n = (if m < n then 0 else SUC(m - n)))`;;
893 let LESS_MULT_MONO = prove
894 (`!m i n. ((SUC n) * m) < ((SUC n) * i) <=> m < i`,
895 REWRITE_TAC[LT_MULT_LCANCEL; NOT_SUC]);;
897 let LESS_MONO_MULT = prove
898 (`!m n p. m <= n ==> (m * p) <= (n * p)`,
899 SIMP_TAC[LE_MULT_RCANCEL]);;
901 let LESS_MULT2 = prove
902 (`!m n. 0 < m /\ 0 < n ==> 0 < (m * n)`,
903 REWRITE_TAC[LT_MULT]);;
905 let SUBSET_FINITE = prove
906 (`!s. FINITE s ==> (!t. t SUBSET s ==> FINITE t)`,
907 MESON_TAC[FINITE_SUBSET]);;
909 let LESS_EQ_SUC = prove
910 (`!n. m <= SUC n <=> (m = SUC n) \/ m <= n`,
913 let ANTE_RES_THEN ttac th = FIRST_ASSUM(fun t -> ttac (MATCH_MP t th));;
915 let IMP_RES_THEN ttac th = FIRST_ASSUM(fun t -> ttac (MATCH_MP th t));;
917 (* ------------------------------------------------------------------------ *)
918 (* Set theory lemmas. *)
919 (* ------------------------------------------------------------------------ *)
921 let INFINITE_MEMBER = prove(
922 `!s. INFINITE(s:A->bool) ==> ?x. x IN s`,
923 GEN_TAC THEN DISCH_TAC THEN
924 SUBGOAL_THEN `~(s:A->bool = {})` MP_TAC THENL
925 [UNDISCH_TAC `INFINITE (s:A->bool)` THEN
926 CONV_TAC CONTRAPOS_CONV THEN REWRITE_TAC[] THEN
927 DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[INFINITE; FINITE_EMPTY];
928 REWRITE_TAC[EXTENSION; NOT_IN_EMPTY] THEN
929 PURE_ONCE_REWRITE_TAC[NOT_FORALL_THM] THEN
932 let INFINITE_CHOOSE = prove(
933 `!s:A->bool. INFINITE(s) ==> ((@) s) IN s`,
934 GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP INFINITE_MEMBER) THEN
935 DISCH_THEN(MP_TAC o SELECT_RULE) THEN REWRITE_TAC[IN] THEN
936 CONV_TAC(ONCE_DEPTH_CONV ETA_CONV) THEN REWRITE_TAC[]);;
938 let INFINITE_DELETE = prove(
939 `!(t:A->bool) x. INFINITE (t DELETE x) = INFINITE(t)`,
940 REWRITE_TAC[INFINITE; FINITE_DELETE]);;
942 let INFINITE_INSERT = prove(
943 `!(x:A) t. INFINITE(x INSERT t) = INFINITE(t)`,
944 REWRITE_TAC[INFINITE; FINITE_INSERT]);;
946 let SIZE_INSERT = prove(
947 `!(x:A) t. ~(x IN t) /\ t HAS_SIZE n ==> (x INSERT t) HAS_SIZE (SUC n)`,
948 SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_RULES]);;
950 let SIZE_DELETE = prove(
951 `!(x:A) t. x IN t /\ t HAS_SIZE (SUC n) ==> (t DELETE x) HAS_SIZE n`,
952 SIMP_TAC[HAS_SIZE_SUC]);;
954 let SIZE_EXISTS = prove(
955 `!s N. s HAS_SIZE (SUC N) ==> ?x:A. x IN s`,
956 SIMP_TAC[HAS_SIZE_SUC; GSYM MEMBER_NOT_EMPTY]);;
958 let SUBSET_DELETE = prove(
959 `!s t (x:A). s SUBSET t ==> (s DELETE x) SUBSET t`,
960 REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_TRANS THEN
961 EXISTS_TAC `s:A->bool` THEN ASM_REWRITE_TAC[DELETE_SUBSET]);;
963 let INFINITE_FINITE_CHOICE = prove(
964 `!n (s:A->bool). INFINITE(s) ==> ?t. t SUBSET s /\ t HAS_SIZE n`,
965 INDUCT_TAC THEN GEN_TAC THEN DISCH_TAC THENL
966 [EXISTS_TAC `{}:A->bool` THEN
967 REWRITE_TAC[HAS_SIZE; EMPTY_SUBSET; HAS_SIZE_0];
968 FIRST_ASSUM(UNDISCH_TAC o check is_forall o concl) THEN
969 DISCH_THEN(MP_TAC o SPEC `s DELETE ((@) s :A)`) THEN
970 ASM_REWRITE_TAC[INFINITE_DELETE] THEN
971 DISCH_THEN(X_CHOOSE_THEN `t:A->bool` STRIP_ASSUME_TAC) THEN
972 EXISTS_TAC `((@) s :A) INSERT t` THEN CONJ_TAC THENL
973 [REWRITE_TAC[INSERT_SUBSET] THEN CONJ_TAC THENL
974 [MATCH_MP_TAC INFINITE_CHOOSE THEN ASM_REWRITE_TAC[];
975 REWRITE_TAC[SUBSET] THEN
976 RULE_ASSUM_TAC(REWRITE_RULE[SUBSET]) THEN
977 GEN_TAC THEN DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN
978 REWRITE_TAC[IN_DELETE] THEN CONV_TAC(EQT_INTRO o TAUT)];
979 MATCH_MP_TAC SIZE_INSERT THEN ASM_REWRITE_TAC[] THEN
980 DISCH_TAC THEN UNDISCH_TAC `t SUBSET (s DELETE ((@) s:A))` THEN
981 REWRITE_TAC[SUBSET; IN_DELETE] THEN
982 DISCH_THEN(IMP_RES_THEN MP_TAC) THEN REWRITE_TAC[]]]);;
984 let IMAGE_WOP_LEMMA = prove(
985 `!N (t:num->bool) (u:A->bool).
986 u SUBSET (IMAGE f t) /\ u HAS_SIZE (SUC N) ==>
987 ?n v. (u = (f n) INSERT v) /\
988 !y. y IN v ==> ?m. (y = f m) /\ n < m`,
989 REPEAT STRIP_TAC THEN
990 MP_TAC(SPEC `\n:num. ?y:A. y IN u /\ (y = f n)` num_WOP) THEN BETA_TAC THEN
991 DISCH_THEN(MP_TAC o fst o EQ_IMP_RULE) THEN
992 FIRST_ASSUM(X_CHOOSE_TAC `y:A` o MATCH_MP SIZE_EXISTS) THEN
993 FIRST_ASSUM(MP_TAC o SPEC `y:A` o REWRITE_RULE[SUBSET]) THEN
994 ASM_REWRITE_TAC[IN_IMAGE] THEN
995 DISCH_THEN(X_CHOOSE_THEN `n:num` STRIP_ASSUME_TAC) THEN
996 W(C SUBGOAL_THEN (fun t ->REWRITE_TAC[t]) o
997 funpow 2 (fst o dest_imp) o snd) THENL
998 [MAP_EVERY EXISTS_TAC [`n:num`; `y:A`] THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN
999 DISCH_THEN(X_CHOOSE_THEN `m:num` (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN
1000 DISCH_THEN(X_CHOOSE_THEN `x:A` STRIP_ASSUME_TAC) THEN
1001 MAP_EVERY EXISTS_TAC [`m:num`; `u DELETE (x:A)`] THEN CONJ_TAC THENL
1002 [ASM_REWRITE_TAC[] THEN CONV_TAC SYM_CONV THEN
1003 MATCH_MP_TAC INSERT_DELETE THEN FIRST_ASSUM(SUBST1_TAC o SYM) THEN
1004 FIRST_ASSUM MATCH_ACCEPT_TAC;
1005 X_GEN_TAC `z:A` THEN REWRITE_TAC[IN_DELETE] THEN STRIP_TAC THEN
1006 FIRST_ASSUM(MP_TAC o SPEC `z:A` o REWRITE_RULE[SUBSET]) THEN
1007 ASM_REWRITE_TAC[IN_IMAGE] THEN
1008 DISCH_THEN(X_CHOOSE_THEN `k:num` STRIP_ASSUME_TAC) THEN
1009 EXISTS_TAC `k:num` THEN ASM_REWRITE_TAC[GSYM NOT_LESS_EQUAL] THEN
1010 REWRITE_TAC[LESS_OR_EQ; DE_MORGAN_THM] THEN CONJ_TAC THENL
1011 [DISCH_THEN(ANTE_RES_THEN (MP_TAC o CONV_RULE NOT_EXISTS_CONV)) THEN
1012 DISCH_THEN(MP_TAC o SPEC `z:A`) THEN REWRITE_TAC[] THEN
1013 CONJ_TAC THEN FIRST_ASSUM MATCH_ACCEPT_TAC;
1014 DISCH_THEN SUBST_ALL_TAC THEN
1015 UNDISCH_TAC `~(z:A = x)` THEN ASM_REWRITE_TAC[]]]);;
1017 (* ------------------------------------------------------------------------ *)
1018 (* Lemma about finite colouring of natural numbers. *)
1019 (* ------------------------------------------------------------------------ *)
1021 let COLOURING_LEMMA = prove(
1022 `!M col s. (INFINITE(s) /\ !n:A. n IN s ==> col(n) <= M) ==>
1023 ?c t. t SUBSET s /\ INFINITE(t) /\ !n:A. n IN t ==> (col(n) = c)`,
1025 [REWRITE_TAC[LESS_EQ_0] THEN REPEAT STRIP_TAC THEN
1026 MAP_EVERY EXISTS_TAC [`0`; `s:A->bool`] THEN
1027 ASM_REWRITE_TAC[SUBSET_REFL];
1028 REPEAT STRIP_TAC THEN SUBGOAL_THEN
1029 `INFINITE { x:A | x IN s /\ (col x = SUC M) } \/
1030 INFINITE { x:A | x IN s /\ col x <= M}`
1031 DISJ_CASES_TAC THENL
1032 [UNDISCH_TAC `INFINITE(s:A->bool)` THEN
1033 REWRITE_TAC[INFINITE; GSYM DE_MORGAN_THM; GSYM FINITE_UNION] THEN
1034 CONV_TAC CONTRAPOS_CONV THEN REWRITE_TAC[] THEN
1035 DISCH_THEN(MATCH_MP_TAC o MATCH_MP SUBSET_FINITE) THEN
1036 REWRITE_TAC[SUBSET; IN_UNION] THEN
1037 REWRITE_TAC[IN_ELIM_THM] THEN
1038 GEN_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[GSYM LESS_EQ_SUC] THEN
1039 FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[];
1040 MAP_EVERY EXISTS_TAC [`SUC M`; `{ x:A | x IN s /\ (col x = SUC M)}`] THEN
1041 ASM_REWRITE_TAC[SUBSET] THEN REWRITE_TAC[IN_ELIM_THM] THEN
1042 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[];
1043 SUBGOAL_THEN `!n:A. n IN { x | x IN s /\ col x <= M } ==> col(n) <= M`
1045 [GEN_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN
1046 DISCH_THEN(MATCH_ACCEPT_TAC o CONJUNCT2);
1047 FIRST_X_ASSUM(MP_TAC o SPECL [`col:A->num`;
1048 `{ x:A | x IN s /\ col x <= M}`]) THEN
1050 MATCH_MP_TAC(TAUT `(c ==> d) ==> (b ==> c) ==> b ==> d`) THEN
1051 DISCH_THEN(X_CHOOSE_THEN `c:num` (X_CHOOSE_TAC `t:A->bool`)) THEN
1052 MAP_EVERY EXISTS_TAC [`c:num`; `t:A->bool`] THEN
1053 ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_TRANS THEN
1054 EXISTS_TAC `{ x:A | x IN s /\ col x <= M }` THEN ASM_REWRITE_TAC[] THEN
1055 REWRITE_TAC[SUBSET] THEN REWRITE_TAC[IN_ELIM_THM] THEN
1056 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]]]]);;
1058 let COLOURING_THM = prove(
1059 `!M col. (!n. col n <= M) ==>
1060 ?c s. INFINITE(s) /\ !n:num. n IN s ==> (col(n) = c)`,
1061 REPEAT STRIP_TAC THEN MP_TAC
1062 (ISPECL [`M:num`; `col:num->num`; `UNIV:num->bool`] COLOURING_LEMMA) THEN
1063 ASM_REWRITE_TAC[num_INFINITE] THEN
1064 DISCH_THEN(X_CHOOSE_THEN `c:num` (X_CHOOSE_TAC `t:num->bool`)) THEN
1065 MAP_EVERY EXISTS_TAC [`c:num`; `t:num->bool`] THEN ASM_REWRITE_TAC[]);;
1067 (* ------------------------------------------------------------------------ *)
1068 (* Simple approach via lemmas then induction over size of coloured sets. *)
1069 (* ------------------------------------------------------------------------ *)
1071 let RAMSEY_LEMMA1 = prove(
1072 `(!C s. INFINITE(s:A->bool) /\
1073 (!t. t SUBSET s /\ t HAS_SIZE N ==> C(t) <= M)
1074 ==> ?t c. INFINITE(t) /\ t SUBSET s /\
1075 (!u. u SUBSET t /\ u HAS_SIZE N ==> (C(u) = c)))
1076 ==> !C s. INFINITE(s:A->bool) /\
1077 (!t. t SUBSET s /\ t HAS_SIZE (SUC N) ==> C(t) <= M)
1078 ==> ?t c. INFINITE(t) /\ t SUBSET s /\ ~(((@) s) IN t) /\
1079 (!u. u SUBSET t /\ u HAS_SIZE N
1080 ==> (C(((@) s) INSERT u) = c))`,
1081 DISCH_THEN((THEN) (REPEAT STRIP_TAC) o MP_TAC) THEN
1082 DISCH_THEN(MP_TAC o SPEC `\u. C (((@) s :A) INSERT u):num`) THEN
1083 DISCH_THEN(MP_TAC o SPEC `s DELETE ((@)s:A)`) THEN BETA_TAC THEN
1084 ASM_REWRITE_TAC[INFINITE_DELETE] THEN
1085 W(C SUBGOAL_THEN (fun t ->REWRITE_TAC[t]) o
1086 funpow 2 (fst o dest_imp) o snd) THENL
1087 [REPEAT STRIP_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN CONJ_TAC THENL
1088 [UNDISCH_TAC `t SUBSET (s DELETE ((@) s :A))` THEN
1089 REWRITE_TAC[SUBSET; IN_INSERT; IN_DELETE; NOT_IN_EMPTY] THEN
1090 DISCH_TAC THEN GEN_TAC THEN DISCH_THEN DISJ_CASES_TAC THEN
1091 ASM_REWRITE_TAC[] THENL
1092 [MATCH_MP_TAC INFINITE_CHOOSE;
1093 FIRST_ASSUM(ANTE_RES_THEN ASSUME_TAC)] THEN
1095 MATCH_MP_TAC SIZE_INSERT THEN ASM_REWRITE_TAC[] THEN
1096 DISCH_TAC THEN UNDISCH_TAC `t SUBSET (s DELETE ((@) s :A))` THEN
1097 ASM_REWRITE_TAC[SUBSET; IN_DELETE] THEN
1098 DISCH_THEN(MP_TAC o SPEC `(@)s:A`) THEN ASM_REWRITE_TAC[]];
1099 DISCH_THEN(X_CHOOSE_THEN `t:A->bool` MP_TAC) THEN
1100 DISCH_THEN(X_CHOOSE_THEN `c:num` STRIP_ASSUME_TAC) THEN
1101 MAP_EVERY EXISTS_TAC [`t:A->bool`; `c:num`] THEN ASM_REWRITE_TAC[] THEN
1102 RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_DELETE]) THEN CONJ_TAC THENL
1103 [REWRITE_TAC[SUBSET] THEN
1104 GEN_TAC THEN DISCH_THEN(ANTE_RES_THEN(fun th -> REWRITE_TAC[th]));
1105 DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN REWRITE_TAC[]]]);;
1107 let RAMSEY_LEMMA2 = prove(
1108 `(!C s. INFINITE(s:A->bool) /\
1109 (!t. t SUBSET s /\ t HAS_SIZE (SUC N) ==> C(t) <= M)
1110 ==> ?t c. INFINITE(t) /\ t SUBSET s /\ ~(((@) s) IN t) /\
1111 (!u. u SUBSET t /\ u HAS_SIZE N
1112 ==> (C(((@) s) INSERT u) = c)))
1113 ==> !C s. INFINITE(s:A->bool) /\
1114 (!t. t SUBSET s /\ t HAS_SIZE (SUC N) ==> C(t) <= M)
1115 ==> ?t x col. (!n. col n <= M) /\
1116 (!n. (t n) SUBSET s) /\
1117 (!n. t(SUC n) SUBSET (t n)) /\
1118 (!n. ~((x n) IN (t n))) /\
1119 (!n. x(SUC n) IN (t n)) /\
1121 (!n u. u SUBSET (t n) /\ u HAS_SIZE N
1122 ==> (C((x n) INSERT u) = col n))`,
1123 REPEAT STRIP_TAC THEN
1124 MP_TAC(ISPECL [`s:A->bool`; `\s (n:num). @t:A->bool. ?c:num.
1128 !u. u SUBSET t /\ u HAS_SIZE N ==> (C(((@) s) INSERT u) = c)`]
1129 num_Axiom) THEN DISCH_THEN(MP_TAC o BETA_RULE o EXISTENCE) THEN
1130 DISCH_THEN(X_CHOOSE_THEN `f:num->(A->bool)` STRIP_ASSUME_TAC) THEN
1132 `!n:num. (f n) SUBSET (s:A->bool) /\
1133 ?c. INFINITE(f(SUC n)) /\ f(SUC n) SUBSET (f n) /\
1134 ~(((@)(f n)) IN (f(SUC n))) /\
1135 !u. u SUBSET (f(SUC n)) /\ u HAS_SIZE N ==>
1136 (C(((@)(f n)) INSERT u) = c:num)`
1138 [MATCH_MP_TAC num_INDUCTION THEN REPEAT STRIP_TAC THENL
1139 [ASM_REWRITE_TAC[SUBSET_REFL];
1140 FIRST_ASSUM(SUBST1_TAC o SPEC `0`) THEN CONV_TAC SELECT_CONV THEN
1141 FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[];
1142 MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `f(n:num):A->bool` THEN
1143 CONJ_TAC THEN FIRST_ASSUM MATCH_ACCEPT_TAC;
1144 FIRST_ASSUM(SUBST1_TAC o SPEC `SUC n`) THEN CONV_TAC SELECT_CONV THEN
1145 FIRST_ASSUM MATCH_MP_TAC THEN CONJ_TAC THEN
1146 TRY(FIRST_ASSUM MATCH_ACCEPT_TAC) THEN REPEAT STRIP_TAC THEN
1147 FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
1148 REPEAT(MATCH_MP_TAC SUBSET_TRANS THEN
1149 FIRST_ASSUM(fun th -> EXISTS_TAC(rand(concl th)) THEN
1150 CONJ_TAC THENL [FIRST_ASSUM MATCH_ACCEPT_TAC; ALL_TAC])) THEN
1151 MATCH_ACCEPT_TAC SUBSET_REFL];
1152 PURE_REWRITE_TAC[LEFT_EXISTS_AND_THM; RIGHT_EXISTS_AND_THM;
1153 FORALL_AND_THM] THEN
1154 DISCH_THEN(REPEAT_TCL (CONJUNCTS_THEN2 ASSUME_TAC) MP_TAC) THEN
1155 DISCH_THEN(X_CHOOSE_TAC `col:num->num` o CONV_RULE SKOLEM_CONV) THEN
1156 MAP_EVERY EXISTS_TAC
1157 [`\n:num. f(SUC n):A->bool`; `\n:num. (@)(f n):A`] THEN
1158 BETA_TAC THEN EXISTS_TAC `col:num->num` THEN CONJ_TAC THENL
1159 [X_GEN_TAC `n:num` THEN
1160 FIRST_ASSUM(MP_TAC o MATCH_MP INFINITE_FINITE_CHOICE o SPEC `n:num`) THEN
1161 DISCH_THEN(CHOOSE_THEN MP_TAC o SPEC `N:num`) THEN
1162 DISCH_THEN(fun th -> STRIP_ASSUME_TAC th THEN
1163 ANTE_RES_THEN MP_TAC th) THEN
1164 DISCH_THEN(SUBST1_TAC o SYM) THEN FIRST_ASSUM MATCH_MP_TAC THEN
1166 [REWRITE_TAC[INSERT_SUBSET] THEN CONJ_TAC THENL
1167 [FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET]) THEN
1168 EXISTS_TAC `n:num` THEN MATCH_MP_TAC INFINITE_CHOOSE THEN
1169 SPEC_TAC(`n:num`,`n:num`) THEN INDUCT_TAC THEN
1170 TRY(FIRST_ASSUM MATCH_ACCEPT_TAC) THEN ASM_REWRITE_TAC[];
1171 MATCH_MP_TAC SUBSET_TRANS THEN
1172 EXISTS_TAC `f(SUC n):A->bool` THEN ASM_REWRITE_TAC[]];
1173 MATCH_MP_TAC SIZE_INSERT THEN ASM_REWRITE_TAC[] THEN
1174 UNDISCH_TAC `!n:num. ~(((@)(f n):A) IN (f(SUC n)))` THEN
1175 DISCH_THEN(MP_TAC o SPEC `n:num`) THEN CONV_TAC CONTRAPOS_CONV THEN
1177 FIRST_ASSUM(MATCH_ACCEPT_TAC o REWRITE_RULE[SUBSET])];
1178 REPEAT CONJ_TAC THEN TRY (FIRST_ASSUM MATCH_ACCEPT_TAC) THENL
1179 [GEN_TAC; INDUCT_TAC THENL
1181 FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET]) THEN
1182 EXISTS_TAC `SUC n`]] THEN
1183 MATCH_MP_TAC INFINITE_CHOOSE THEN ASM_REWRITE_TAC[]]]);;
1185 let RAMSEY_LEMMA3 = prove(
1186 `(!C s. INFINITE(s:A->bool) /\
1187 (!t. t SUBSET s /\ t HAS_SIZE (SUC N) ==> C(t) <= M)
1188 ==> ?t x col. (!n. col n <= M) /\
1189 (!n. (t n) SUBSET s) /\
1190 (!n. t(SUC n) SUBSET (t n)) /\
1191 (!n. ~((x n) IN (t n))) /\
1192 (!n. x(SUC n) IN (t n)) /\
1194 (!n u. u SUBSET (t n) /\ u HAS_SIZE N
1195 ==> (C((x n) INSERT u) = col n)))
1196 ==> !C s. INFINITE(s:A->bool) /\
1197 (!t. t SUBSET s /\ t HAS_SIZE (SUC N) ==> C(t) <= M)
1198 ==> ?t c. INFINITE(t) /\ t SUBSET s /\
1199 (!u. u SUBSET t /\ u HAS_SIZE (SUC N) ==> (C(u) = c))`,
1200 DISCH_THEN((THEN) (REPEAT STRIP_TAC) o MP_TAC) THEN
1201 DISCH_THEN(MP_TAC o SPECL [`C:(A->bool)->num`; `s:A->bool`]) THEN
1202 ASM_REWRITE_TAC[] THEN
1203 DISCH_THEN(X_CHOOSE_THEN `t:num->(A->bool)` MP_TAC) THEN
1204 DISCH_THEN(X_CHOOSE_THEN `x:num->A` MP_TAC) THEN
1205 DISCH_THEN(X_CHOOSE_THEN `col:num->num` STRIP_ASSUME_TAC) THEN
1206 MP_TAC(ISPECL [`M:num`; `col:num->num`; `UNIV:num->bool`]
1207 COLOURING_LEMMA) THEN ASM_REWRITE_TAC[num_INFINITE] THEN
1208 DISCH_THEN(X_CHOOSE_THEN `c:num` MP_TAC) THEN
1209 DISCH_THEN(X_CHOOSE_THEN `t:num->bool` STRIP_ASSUME_TAC) THEN
1210 MAP_EVERY EXISTS_TAC [`IMAGE (x:num->A) t`; `c:num`] THEN
1211 SUBGOAL_THEN `!m n. m <= n ==> (t n:A->bool) SUBSET (t m)` ASSUME_TAC THENL
1212 [REPEAT GEN_TAC THEN REWRITE_TAC[LESS_EQ_EXISTS] THEN
1213 DISCH_THEN(X_CHOOSE_THEN `d:num` SUBST1_TAC) THEN
1214 SPEC_TAC(`d:num`,`d:num`) THEN INDUCT_TAC THEN
1215 ASM_REWRITE_TAC[ADD_CLAUSES; SUBSET_REFL] THEN
1216 MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `t(m + d):A->bool` THEN
1217 ASM_REWRITE_TAC[]; ALL_TAC] THEN
1218 SUBGOAL_THEN `!m n. m < n ==> (x n:A) IN (t m)` ASSUME_TAC THENL
1219 [REPEAT GEN_TAC THEN
1220 DISCH_THEN(X_CHOOSE_THEN `d:num` SUBST1_TAC o MATCH_MP LESS_ADD_1) THEN
1221 FIRST_ASSUM(MP_TAC o SPECL [`m:num`; `m + d`]) THEN
1222 REWRITE_TAC[LESS_EQ_ADD; SUBSET] THEN DISCH_THEN MATCH_MP_TAC THEN
1223 ASM_REWRITE_TAC[GSYM ADD1; ADD_CLAUSES]; ALL_TAC] THEN
1224 SUBGOAL_THEN `!m n. ((x:num->A) m = x n) <=> (m = n)` ASSUME_TAC THENL
1225 [REPEAT GEN_TAC THEN EQ_TAC THENL
1226 [REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC
1227 (SPECL [`m:num`; `n:num`] LESS_LESS_CASES) THEN
1228 ASM_REWRITE_TAC[] THEN REPEAT DISCH_TAC THEN
1229 FIRST_ASSUM(ANTE_RES_THEN MP_TAC) THEN ASM_REWRITE_TAC[] THEN
1230 FIRST_ASSUM(UNDISCH_TAC o check is_eq o concl) THEN
1231 DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_REWRITE_TAC[];
1232 DISCH_THEN SUBST1_TAC THEN REFL_TAC]; ALL_TAC] THEN
1233 REPEAT CONJ_TAC THENL
1234 [UNDISCH_TAC `INFINITE(t:num->bool)` THEN
1235 MATCH_MP_TAC INFINITE_IMAGE_INJ THEN ASM_REWRITE_TAC[];
1236 REWRITE_TAC[SUBSET; IN_IMAGE] THEN GEN_TAC THEN
1237 DISCH_THEN(CHOOSE_THEN (SUBST1_TAC o CONJUNCT1)) THEN ASM_REWRITE_TAC[];
1238 GEN_TAC THEN DISCH_THEN(fun th -> STRIP_ASSUME_TAC th THEN MP_TAC th) THEN
1239 DISCH_THEN(MP_TAC o MATCH_MP IMAGE_WOP_LEMMA) THEN
1240 DISCH_THEN(X_CHOOSE_THEN `n:num` (X_CHOOSE_THEN `v:A->bool` MP_TAC)) THEN
1241 DISCH_THEN STRIP_ASSUME_TAC THEN ASM_REWRITE_TAC[] THEN
1242 SUBGOAL_THEN `c = (col:num->num) n` SUBST1_TAC THENL
1243 [CONV_TAC SYM_CONV THEN FIRST_ASSUM MATCH_MP_TAC THEN
1244 UNDISCH_TAC `u SUBSET (IMAGE (x:num->A) t)` THEN
1245 REWRITE_TAC[SUBSET; IN_IMAGE] THEN
1246 DISCH_THEN(MP_TAC o SPEC `(x:num->A) n`) THEN
1247 ASM_REWRITE_TAC[IN_INSERT] THEN
1248 DISCH_THEN(CHOOSE_THEN STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[];
1249 FIRST_ASSUM MATCH_MP_TAC THEN CONJ_TAC THENL
1250 [REWRITE_TAC[SUBSET] THEN GEN_TAC THEN
1251 DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN
1252 DISCH_THEN(X_CHOOSE_THEN `m:num` STRIP_ASSUME_TAC) THEN
1253 ASM_REWRITE_TAC[] THEN FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[];
1254 SUBGOAL_THEN `v = u DELETE ((x:num->A) n)` SUBST1_TAC THENL
1255 [ASM_REWRITE_TAC[] THEN REWRITE_TAC[DELETE_INSERT] THEN
1256 REWRITE_TAC[EXTENSION; IN_DELETE;
1257 TAUT `(a <=> a /\ b) <=> a ==> b`] THEN
1258 GEN_TAC THEN CONV_TAC CONTRAPOS_CONV THEN REWRITE_TAC[] THEN
1259 DISCH_THEN SUBST1_TAC THEN DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN
1260 ASM_REWRITE_TAC[] THEN
1261 DISCH_THEN(CHOOSE_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
1262 ASM_REWRITE_TAC[LESS_REFL];
1263 MATCH_MP_TAC SIZE_DELETE THEN CONJ_TAC THENL
1264 [ASM_REWRITE_TAC[IN_INSERT]; FIRST_ASSUM MATCH_ACCEPT_TAC]]]]]);;
1268 INFINITE(s:A->bool) /\
1269 (!t. t SUBSET s /\ t HAS_SIZE N ==> C(t) <= M)
1270 ==> ?t c. INFINITE(t) /\ t SUBSET s /\
1271 (!u. u SUBSET t /\ u HAS_SIZE N ==> (C(u) = c))`,
1272 GEN_TAC THEN INDUCT_TAC THENL
1273 [REPEAT STRIP_TAC THEN
1274 MAP_EVERY EXISTS_TAC [`s:A->bool`; `(C:(A->bool)->num) {}`] THEN
1275 ASM_REWRITE_TAC[HAS_SIZE_0] THEN
1276 REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[SUBSET_REFL];
1277 MAP_EVERY MATCH_MP_TAC [RAMSEY_LEMMA3; RAMSEY_LEMMA2; RAMSEY_LEMMA1] THEN
1278 POP_ASSUM MATCH_ACCEPT_TAC]);;