--- /dev/null
+(* ======================================================================== *)
+(* Infinite Ramsey's theorem. *)
+(* *)
+(* Port to HOL Light of a HOL88 proof done on 9th May 1994 *)
+(* ======================================================================== *)
+
+(* ------------------------------------------------------------------------- *)
+(* HOL88 compatibility. *)
+(* ------------------------------------------------------------------------- *)
+
+let is_neg_imp tm =
+ is_neg tm or is_imp tm;;
+
+let dest_neg_imp tm =
+ try dest_imp tm with Failure _ ->
+ try (dest_neg tm,mk_const("F",[]))
+ with Failure _ -> failwith "dest_neg_imp";;
+
+(* ------------------------------------------------------------------------- *)
+(* These get overwritten by the subgoal stuff. *)
+(* ------------------------------------------------------------------------- *)
+
+let PROVE = prove;;
+
+let prove_thm((s:string),g,t) = prove(g,t);;
+
+(* ------------------------------------------------------------------------- *)
+(* The quantifier movement conversions. *)
+(* ------------------------------------------------------------------------- *)
+
+let (CONV_OF_RCONV: conv -> conv) =
+ let rec get_bv tm =
+ if is_abs tm then bndvar tm
+ else if is_comb tm then try get_bv (rand tm) with Failure _ -> get_bv (rator tm)
+ else failwith "" in
+ fun conv tm ->
+ let v = get_bv tm in
+ let th1 = conv tm in
+ let th2 = ONCE_DEPTH_CONV (GEN_ALPHA_CONV v) (rhs(concl th1)) in
+ TRANS th1 th2;;
+
+let (CONV_OF_THM: thm -> conv) =
+ CONV_OF_RCONV o REWR_CONV;;
+
+let (X_FUN_EQ_CONV:term->conv) =
+ fun v -> (REWR_CONV FUN_EQ_THM) THENC GEN_ALPHA_CONV v;;
+
+let (FUN_EQ_CONV:conv) =
+ fun tm ->
+ let vars = frees tm in
+ let op,[ty1;ty2] = dest_type(type_of (lhs tm)) in
+ if op = "fun"
+ then let varnm =
+ if (is_vartype ty1) then "x" else
+ hd(explode(fst(dest_type ty1))) in
+ let x = variant vars (mk_var(varnm,ty1)) in
+ X_FUN_EQ_CONV x tm
+ else failwith "FUN_EQ_CONV";;
+
+let (SINGLE_DEPTH_CONV:conv->conv) =
+ let rec SINGLE_DEPTH_CONV conv tm =
+ try conv tm with Failure _ ->
+ (SUB_CONV (SINGLE_DEPTH_CONV conv) THENC (TRY_CONV conv)) tm in
+ SINGLE_DEPTH_CONV;;
+
+let (SKOLEM_CONV:conv) =
+ SINGLE_DEPTH_CONV (REWR_CONV SKOLEM_THM);;
+
+let (X_SKOLEM_CONV:term->conv) =
+ fun v -> SKOLEM_CONV THENC GEN_ALPHA_CONV v;;
+
+let EXISTS_UNIQUE_CONV tm =
+ let v = bndvar(rand tm) in
+ let th1 = REWR_CONV EXISTS_UNIQUE_THM tm in
+ let tm1 = rhs(concl th1) in
+ let vars = frees tm1 in
+ let v = variant vars v in
+ let v' = variant (v::vars) v in
+ let th2 =
+ (LAND_CONV(GEN_ALPHA_CONV v) THENC
+ RAND_CONV(BINDER_CONV(GEN_ALPHA_CONV v') THENC
+ GEN_ALPHA_CONV v)) tm1 in
+ TRANS th1 th2;;
+
+let NOT_FORALL_CONV = CONV_OF_THM NOT_FORALL_THM;;
+
+let NOT_EXISTS_CONV = CONV_OF_THM NOT_EXISTS_THM;;
+
+let RIGHT_IMP_EXISTS_CONV = CONV_OF_THM RIGHT_IMP_EXISTS_THM;;
+
+let FORALL_IMP_CONV = CONV_OF_RCONV
+ (REWR_CONV TRIV_FORALL_IMP_THM ORELSEC
+ REWR_CONV RIGHT_FORALL_IMP_THM ORELSEC
+ REWR_CONV LEFT_FORALL_IMP_THM);;
+
+let EXISTS_AND_CONV = CONV_OF_RCONV
+ (REWR_CONV TRIV_EXISTS_AND_THM ORELSEC
+ REWR_CONV LEFT_EXISTS_AND_THM ORELSEC
+ REWR_CONV RIGHT_EXISTS_AND_THM);;
+
+let LEFT_IMP_EXISTS_CONV = CONV_OF_THM LEFT_IMP_EXISTS_THM;;
+
+let LEFT_AND_EXISTS_CONV tm =
+ let v = bndvar(rand(rand(rator tm))) in
+ (REWR_CONV LEFT_AND_EXISTS_THM THENC TRY_CONV (GEN_ALPHA_CONV v)) tm;;
+
+let RIGHT_AND_EXISTS_CONV =
+ CONV_OF_THM RIGHT_AND_EXISTS_THM;;
+
+let AND_FORALL_CONV = CONV_OF_THM AND_FORALL_THM;;
+
+(* ------------------------------------------------------------------------- *)
+(* The slew of named tautologies. *)
+(* ------------------------------------------------------------------------- *)
+
+let AND1_THM = TAUT `!t1 t2. t1 /\ t2 ==> t1`;;
+
+let AND2_THM = TAUT `!t1 t2. t1 /\ t2 ==> t2`;;
+
+let AND_IMP_INTRO = TAUT `!t1 t2 t3. t1 ==> t2 ==> t3 = t1 /\ t2 ==> t3`;;
+
+let AND_INTRO_THM = TAUT `!t1 t2. t1 ==> t2 ==> t1 /\ t2`;;
+
+let BOOL_EQ_DISTINCT = TAUT `~(T <=> F) /\ ~(F <=> T)`;;
+
+let EQ_EXPAND = TAUT `!t1 t2. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2`;;
+
+let EQ_IMP_THM = TAUT `!t1 t2. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)`;;
+
+let FALSITY = TAUT `!t. F ==> t`;;
+
+let F_IMP = TAUT `!t. ~t ==> t ==> F`;;
+
+let IMP_DISJ_THM = TAUT `!t1 t2. t1 ==> t2 <=> ~t1 \/ t2`;;
+
+let IMP_F = TAUT `!t. (t ==> F) ==> ~t`;;
+
+let IMP_F_EQ_F = TAUT `!t. t ==> F <=> (t <=> F)`;;
+
+let LEFT_AND_OVER_OR = TAUT
+ `!t1 t2 t3. t1 /\ (t2 \/ t3) <=> t1 /\ t2 \/ t1 /\ t3`;;
+
+let LEFT_OR_OVER_AND = TAUT
+ `!t1 t2 t3. t1 \/ t2 /\ t3 <=> (t1 \/ t2) /\ (t1 \/ t3)`;;
+
+let NOT_AND = TAUT `~(t /\ ~t)`;;
+
+let NOT_F = TAUT `!t. ~t ==> (t <=> F)`;;
+
+let OR_ELIM_THM = TAUT
+ `!t t1 t2. t1 \/ t2 ==> (t1 ==> t) ==> (t2 ==> t) ==> t`;;
+
+let OR_IMP_THM = TAUT `!t1 t2. (t1 <=> t2 \/ t1) <=> t2 ==> t1`;;
+
+let OR_INTRO_THM1 = TAUT `!t1 t2. t1 ==> t1 \/ t2`;;
+
+let OR_INTRO_THM2 = TAUT `!t1 t2. t2 ==> t1 \/ t2`;;
+
+let RIGHT_AND_OVER_OR = TAUT
+ `!t1 t2 t3. (t2 \/ t3) /\ t1 <=> t2 /\ t1 \/ t3 /\ t1`;;
+
+let RIGHT_OR_OVER_AND = TAUT
+ `!t1 t2 t3. t2 /\ t3 \/ t1 <=> (t2 \/ t1) /\ (t3 \/ t1)`;;
+
+(* ------------------------------------------------------------------------- *)
+(* This is an overwrite -- is there any point in what I have? *)
+(* ------------------------------------------------------------------------- *)
+
+let is_type = can get_type_arity;;
+
+(* ------------------------------------------------------------------------- *)
+(* I suppose this is also useful. *)
+(* ------------------------------------------------------------------------- *)
+
+let is_constant = can get_const_type;;
+
+(* ------------------------------------------------------------------------- *)
+(* Misc. *)
+(* ------------------------------------------------------------------------- *)
+
+let null l = l = [];;
+
+(* ------------------------------------------------------------------------- *)
+(* Syntax. *)
+(* ------------------------------------------------------------------------- *)
+
+let type_tyvars = type_vars_in_term o curry mk_var "x";;
+
+let find_match u =
+ let rec find_mt t =
+ try term_match [] u t with Failure _ ->
+ try find_mt(rator t) with Failure _ ->
+ try find_mt(rand t) with Failure _ ->
+ try find_mt(snd(dest_abs t))
+ with Failure _ -> failwith "find_match" in
+ fun t -> let _,tmin,tyin = find_mt t in
+ tmin,tyin;;
+
+let rec mk_primed_var(name,ty) =
+ if can get_const_type name then mk_primed_var(name^"'",ty)
+ else mk_var(name,ty);;
+
+let subst_occs =
+ let rec subst_occs slist tm =
+ let applic,noway = partition (fun (i,(t,x)) -> aconv tm x) slist in
+ let sposs = map (fun (l,z) -> let l1,l2 = partition ((=) 1) l in
+ (l1,z),(l2,z)) applic in
+ let racts,rrest = unzip sposs in
+ let acts = filter (fun t -> not (fst t = [])) racts in
+ let trest = map (fun (n,t) -> (map (C (-) 1) n,t)) rrest in
+ let urest = filter (fun t -> not (fst t = [])) trest in
+ let tlist = urest @ noway in
+ if acts = [] then
+ if is_comb tm then
+ let l,r = dest_comb tm in
+ let l',s' = subst_occs tlist l in
+ let r',s'' = subst_occs s' r in
+ mk_comb(l',r'),s''
+ else if is_abs tm then
+ let bv,bod = dest_abs tm in
+ let gv = genvar(type_of bv) in
+ let nbod = vsubst[gv,bv] bod in
+ let tm',s' = subst_occs tlist nbod in
+ alpha bv (mk_abs(gv,tm')),s'
+ else
+ tm,tlist
+ else
+ let tm' = (fun (n,(t,x)) -> subst[t,x] tm) (hd acts) in
+ tm',tlist in
+ fun ilist slist tm -> fst(subst_occs (zip ilist slist) tm);;
+
+(* ------------------------------------------------------------------------- *)
+(* Note that the all-instantiating INST and INST_TYPE are not overwritten. *)
+(* ------------------------------------------------------------------------- *)
+
+let INST_TY_TERM(substl,insttyl) th =
+ let th' = INST substl (INST_TYPE insttyl th) in
+ if hyp th' = hyp th then th'
+ else failwith "INST_TY_TERM: Free term and/or type variables in hypotheses";;
+
+(* ------------------------------------------------------------------------- *)
+(* Conversions stuff. *)
+(* ------------------------------------------------------------------------- *)
+
+let RIGHT_CONV_RULE (conv:conv) th =
+ TRANS th (conv(rhs(concl th)));;
+
+(* ------------------------------------------------------------------------- *)
+(* Derived rules. *)
+(* ------------------------------------------------------------------------- *)
+
+let NOT_EQ_SYM =
+ let pth = GENL [`a:A`; `b:A`]
+ (GEN_REWRITE_RULE I [GSYM CONTRAPOS_THM] (DISCH_ALL(SYM(ASSUME`a:A = b`))))
+ and aty = `:A` in
+ fun th -> try let l,r = dest_eq(dest_neg(concl th)) in
+ MP (SPECL [r; l] (INST_TYPE [type_of l,aty] pth)) th
+ with Failure _ -> failwith "NOT_EQ_SYM";;
+
+let NOT_MP thi th =
+ try MP thi th with Failure _ ->
+ try let t = dest_neg (concl thi) in
+ MP(MP (SPEC t F_IMP) thi) th
+ with Failure _ -> failwith "NOT_MP";;
+
+let FORALL_EQ x =
+ let mkall = AP_TERM (mk_const("!",[type_of x,mk_vartype "A"])) in
+ fun th -> try mkall (ABS x th)
+ with Failure _ -> failwith "FORALL_EQ";;
+
+let EXISTS_EQ x =
+ let mkex = AP_TERM (mk_const("?",[type_of x,mk_vartype "A"])) in
+ fun th -> try mkex (ABS x th)
+ with Failure _ -> failwith "EXISTS_EQ";;
+
+let SELECT_EQ x =
+ let mksel = AP_TERM (mk_const("@",[type_of x,mk_vartype "A"])) in
+ fun th -> try mksel (ABS x th)
+ with Failure _ -> failwith "SELECT_EQ";;
+
+let RIGHT_BETA th =
+ try TRANS th (BETA_CONV(rhs(concl th)))
+ with Failure _ -> failwith "RIGHT_BETA";;
+
+let rec LIST_BETA_CONV tm =
+ try let rat,rnd = dest_comb tm in
+ RIGHT_BETA(AP_THM(LIST_BETA_CONV rat)rnd)
+ with Failure _ -> REFL tm;;
+
+let RIGHT_LIST_BETA th = TRANS th (LIST_BETA_CONV(snd(dest_eq(concl th))));;
+
+let LIST_CONJ = end_itlist CONJ ;;
+
+let rec CONJ_LIST n th =
+ try if n=1 then [th] else (CONJUNCT1 th)::(CONJ_LIST (n-1) (CONJUNCT2 th))
+ with Failure _ -> failwith "CONJ_LIST";;
+
+let rec BODY_CONJUNCTS th =
+ if is_forall(concl th) then
+ BODY_CONJUNCTS (SPEC_ALL th) else
+ if is_conj (concl th) then
+ BODY_CONJUNCTS (CONJUNCT1 th) @ BODY_CONJUNCTS (CONJUNCT2 th)
+ else [th];;
+
+let rec IMP_CANON th =
+ let w = concl th in
+ if is_conj w then IMP_CANON (CONJUNCT1 th) @ IMP_CANON (CONJUNCT2 th)
+ else if is_imp w then
+ let ante,conc = dest_neg_imp w in
+ if is_conj ante then
+ let a,b = dest_conj ante in
+ IMP_CANON
+ (DISCH a (DISCH b (NOT_MP th (CONJ (ASSUME a) (ASSUME b)))))
+ else if is_disj ante then
+ let a,b = dest_disj ante in
+ IMP_CANON (DISCH a (NOT_MP th (DISJ1 (ASSUME a) b))) @
+ IMP_CANON (DISCH b (NOT_MP th (DISJ2 a (ASSUME b))))
+ else if is_exists ante then
+ let x,body = dest_exists ante in
+ let x' = variant (thm_frees th) x in
+ let body' = subst [x',x] body in
+ IMP_CANON
+ (DISCH body' (NOT_MP th (EXISTS (ante, x') (ASSUME body'))))
+ else
+ map (DISCH ante) (IMP_CANON (UNDISCH th))
+ else if is_forall w then
+ IMP_CANON (SPEC_ALL th)
+ else [th];;
+
+let LIST_MP = rev_itlist (fun x y -> MP y x);;
+
+let DISJ_IMP =
+ let pth = TAUT`!t1 t2. t1 \/ t2 ==> ~t1 ==> t2` in
+ fun th ->
+ try let a,b = dest_disj(concl th) in MP (SPECL [a;b] pth) th
+ with Failure _ -> failwith "DISJ_IMP";;
+
+let IMP_ELIM =
+ let pth = TAUT`!t1 t2. (t1 ==> t2) ==> ~t1 \/ t2` in
+ fun th ->
+ try let a,b = dest_imp(concl th) in MP (SPECL [a;b] pth) th
+ with Failure _ -> failwith "IMP_ELIM";;
+
+let DISJ_CASES_UNION dth ath bth =
+ DISJ_CASES dth (DISJ1 ath (concl bth)) (DISJ2 (concl ath) bth);;
+
+let MK_ABS qth =
+ try let ov = bndvar(rand(concl qth)) in
+ let bv,rth = SPEC_VAR qth in
+ let sth = ABS bv rth in
+ let cnv = ALPHA_CONV ov in
+ CONV_RULE(BINOP_CONV cnv) sth
+ with Failure _ -> failwith "MK_ABS";;
+
+let HALF_MK_ABS th =
+ try let th1 = MK_ABS th in
+ CONV_RULE(LAND_CONV ETA_CONV) th1
+ with Failure _ -> failwith "HALF_MK_ABS";;
+
+let MK_EXISTS qth =
+ try let ov = bndvar(rand(concl qth)) in
+ let bv,rth = SPEC_VAR qth in
+ let sth = EXISTS_EQ bv rth in
+ let cnv = GEN_ALPHA_CONV ov in
+ CONV_RULE(BINOP_CONV cnv) sth
+ with Failure _ -> failwith "MK_EXISTS";;
+
+let LIST_MK_EXISTS l th = itlist (fun x th -> MK_EXISTS(GEN x th)) l th;;
+
+let IMP_CONJ th1 th2 =
+ let A1,C1 = dest_imp (concl th1) and A2,C2 = dest_imp (concl th2) in
+ let a1,a2 = CONJ_PAIR (ASSUME (mk_conj(A1,A2))) in
+ DISCH (mk_conj(A1,A2)) (CONJ (MP th1 a1) (MP th2 a2));;
+
+let EXISTS_IMP x =
+ if not (is_var x) then failwith "EXISTS_IMP: first argument not a variable"
+ else fun th ->
+ try let ante,cncl = dest_imp(concl th) in
+ let th1 = EXISTS (mk_exists(x,cncl),x) (UNDISCH th) in
+ let asm = mk_exists(x,ante) in
+ DISCH asm (CHOOSE (x,ASSUME asm) th1)
+ with Failure _ -> failwith "EXISTS_IMP: variable free in assumptions";;
+
+
+let CONJUNCTS_CONV (t1,t2) =
+ let rec build_conj thl t =
+ try let l,r = dest_conj t in
+ CONJ (build_conj thl l) (build_conj thl r)
+ with Failure _ -> find (fun th -> concl th = t) thl in
+ try IMP_ANTISYM_RULE
+ (DISCH t1 (build_conj (CONJUNCTS (ASSUME t1)) t2))
+ (DISCH t2 (build_conj (CONJUNCTS (ASSUME t2)) t1))
+ with Failure _ -> failwith "CONJUNCTS_CONV";;
+
+let CONJ_SET_CONV l1 l2 =
+ try CONJUNCTS_CONV (list_mk_conj l1, list_mk_conj l2)
+ with Failure _ -> failwith "CONJ_SET_CONV";;
+
+let FRONT_CONJ_CONV tml t =
+ let rec remove x l =
+ if hd l = x then tl l else (hd l)::(remove x (tl l)) in
+ try CONJ_SET_CONV tml (t::(remove t tml))
+ with Failure _ -> failwith "FRONT_CONJ_CONV";;
+
+let CONJ_DISCH =
+ let pth = TAUT`!t t1 t2. (t ==> (t1 <=> t2)) ==> (t /\ t1 <=> t /\ t2)` in
+ fun t th ->
+ try let t1,t2 = dest_eq(concl th) in
+ MP (SPECL [t; t1; t2] pth) (DISCH t th)
+ with Failure _ -> failwith "CONJ_DISCH";;
+
+let rec CONJ_DISCHL l th =
+ if l = [] then th else CONJ_DISCH (hd l) (CONJ_DISCHL (tl l) th);;
+
+let rec GSPEC th =
+ let wl,w = dest_thm th in
+ if is_forall w then
+ GSPEC (SPEC (genvar (type_of (fst (dest_forall w)))) th)
+ else th;;
+
+let ANTE_CONJ_CONV tm =
+ try let (a1,a2),c = (dest_conj F_F I) (dest_imp tm) in
+ let imp1 = MP (ASSUME tm) (CONJ (ASSUME a1) (ASSUME a2)) and
+ imp2 = LIST_MP [CONJUNCT1 (ASSUME (mk_conj(a1,a2)));
+ CONJUNCT2 (ASSUME (mk_conj(a1,a2)))]
+ (ASSUME (mk_imp(a1,mk_imp(a2,c)))) in
+ IMP_ANTISYM_RULE (DISCH_ALL (DISCH a1 (DISCH a2 imp1)))
+ (DISCH_ALL (DISCH (mk_conj(a1,a2)) imp2))
+ with Failure _ -> failwith "ANTE_CONJ_CONV";;
+
+let bool_EQ_CONV =
+ let check = let boolty = `:bool` in check (fun tm -> type_of tm = boolty) in
+ let clist = map (GEN `b:bool`)
+ (CONJUNCTS(SPEC `b:bool` EQ_CLAUSES)) in
+ let tb = hd clist and bt = hd(tl clist) in
+ let T = `T` and F = `F` in
+ fun tm ->
+ try let l,r = (I F_F check) (dest_eq tm) in
+ if l = r then EQT_INTRO (REFL l) else
+ if l = T then SPEC r tb else
+ if r = T then SPEC l bt else fail()
+ with Failure _ -> failwith "bool_EQ_CONV";;
+
+let COND_CONV =
+ let T = `T` and F = `F` and vt = genvar`:A` and vf = genvar `:A` in
+ let gen = GENL [vt;vf] in
+ let CT,CF = (gen F_F gen) (CONJ_PAIR (SPECL [vt;vf] COND_CLAUSES)) in
+ fun tm ->
+ let P,(u,v) = try dest_cond tm
+ with Failure _ -> failwith "COND_CONV: not a conditional" in
+ let ty = type_of u in
+ if (P=T) then SPEC v (SPEC u (INST_TYPE [ty,`:A`] CT)) else
+ if (P=F) then SPEC v (SPEC u (INST_TYPE [ty,`:A`] CF)) else
+ if (u=v) then SPEC u (SPEC P (INST_TYPE [ty,`:A`] COND_ID)) else
+ if (aconv u v) then
+ let cnd = AP_TERM (rator tm) (ALPHA v u) in
+ let thm = SPEC u (SPEC P (INST_TYPE [ty,`:A`] COND_ID)) in
+ TRANS cnd thm else
+ failwith "COND_CONV: can't simplify conditional";;
+
+let SUBST_MATCH eqth th =
+ let tm_inst,ty_inst = find_match (lhs(concl eqth)) (concl th) in
+ SUBS [INST tm_inst (INST_TYPE ty_inst eqth)] th;;
+
+let SUBST thl pat th =
+ let eqs,vs = unzip thl in
+ let gvs = map (genvar o type_of) vs in
+ let gpat = subst (zip gvs vs) pat in
+ let ls,rs = unzip (map (dest_eq o concl) eqs) in
+ let ths = map (ASSUME o mk_eq) (zip gvs rs) in
+ let th1 = ASSUME gpat in
+ let th2 = SUBS ths th1 in
+ let th3 = itlist DISCH (map concl ths) (DISCH gpat th2) in
+ let th4 = INST (zip ls gvs) th3 in
+ MP (rev_itlist (C MP) eqs th4) th;;
+
+(* let GSUBS = ... *)
+(* let SUBS_OCCS = ... *)
+
+(* A poor thing but mine own. The old ones use mk_thm and the commented
+ out functions are bogus. *)
+
+let SUBST_CONV thvars template tm =
+ let thms,vars = unzip thvars in
+ let gvs = map (genvar o type_of) vars in
+ let gtemplate = subst (zip gvs vars) template in
+ SUBST (zip thms gvs) (mk_eq(template,gtemplate)) (REFL tm);;
+
+(* ------------------------------------------------------------------------- *)
+(* Filtering rewrites. *)
+(* ------------------------------------------------------------------------- *)
+
+let FILTER_PURE_ASM_REWRITE_RULE f thl th =
+ PURE_REWRITE_RULE ((map ASSUME (filter f (hyp th))) @ thl) th
+
+and FILTER_ASM_REWRITE_RULE f thl th =
+ REWRITE_RULE ((map ASSUME (filter f (hyp th))) @ thl) th
+
+and FILTER_PURE_ONCE_ASM_REWRITE_RULE f thl th =
+ PURE_ONCE_REWRITE_RULE ((map ASSUME (filter f (hyp th))) @ thl) th
+
+and FILTER_ONCE_ASM_REWRITE_RULE f thl th =
+ ONCE_REWRITE_RULE ((map ASSUME (filter f (hyp th))) @ thl) th;;
+
+let (FILTER_PURE_ASM_REWRITE_TAC: (term->bool) -> thm list -> tactic) =
+ fun f thl (asl,w) ->
+ PURE_REWRITE_TAC (filter (f o concl) (map snd asl) @ thl) (asl,w)
+
+and (FILTER_ASM_REWRITE_TAC: (term->bool) -> thm list -> tactic) =
+ fun f thl (asl,w) ->
+ REWRITE_TAC (filter (f o concl) (map snd asl) @ thl) (asl,w)
+
+and (FILTER_PURE_ONCE_ASM_REWRITE_TAC: (term->bool) -> thm list -> tactic) =
+ fun f thl (asl,w) ->
+ PURE_ONCE_REWRITE_TAC (filter (f o concl) (map snd asl) @ thl) (asl,w)
+
+and (FILTER_ONCE_ASM_REWRITE_TAC: (term->bool) -> thm list -> tactic) =
+ fun f thl (asl,w) ->
+ ONCE_REWRITE_TAC (filter (f o concl) (map snd asl) @ thl) (asl,w);;
+
+(* ------------------------------------------------------------------------- *)
+(* Tacticals. *)
+(* ------------------------------------------------------------------------- *)
+
+let (X_CASES_THENL: term list list -> thm_tactic list -> thm_tactic) =
+ fun varsl ttacl ->
+ end_itlist DISJ_CASES_THEN2
+ (map (fun (vars,ttac) -> EVERY_TCL (map X_CHOOSE_THEN vars) ttac)
+ (zip varsl ttacl));;
+
+let (X_CASES_THEN: term list list -> thm_tactical) =
+ fun varsl ttac ->
+ end_itlist DISJ_CASES_THEN2
+ (map (fun vars -> EVERY_TCL (map X_CHOOSE_THEN vars) ttac) varsl);;
+
+let (CASES_THENL: thm_tactic list -> thm_tactic) =
+ fun ttacl -> end_itlist DISJ_CASES_THEN2 (map (REPEAT_TCL CHOOSE_THEN) ttacl);;
+
+(* ------------------------------------------------------------------------- *)
+(* Tactics. *)
+(* ------------------------------------------------------------------------- *)
+
+let (DISCARD_TAC: thm_tactic) =
+ let truth = `T` in
+ fun th (asl,w) ->
+ if exists (aconv (concl th)) (truth::(map (concl o snd) asl))
+ then ALL_TAC (asl,w)
+ else failwith "DISCARD_TAC";;
+
+let (CHECK_ASSUME_TAC: thm_tactic) =
+ fun gth ->
+ FIRST [CONTR_TAC gth; ACCEPT_TAC gth;
+ DISCARD_TAC gth; ASSUME_TAC gth];;
+
+let (FILTER_GEN_TAC: term -> tactic) =
+ fun tm (asl,w) ->
+ if is_forall w & not (tm = fst(dest_forall w)) then
+ GEN_TAC (asl,w)
+ else failwith "FILTER_GEN_TAC";;
+
+let (FILTER_DISCH_THEN: thm_tactic -> term -> tactic) =
+ fun ttac tm (asl,w) ->
+ if is_neg_imp w & not (free_in tm (fst(dest_neg_imp w))) then
+ DISCH_THEN ttac (asl,w)
+ else failwith "FILTER_DISCH_THEN";;
+
+let FILTER_STRIP_THEN ttac tm =
+ FIRST [FILTER_GEN_TAC tm; FILTER_DISCH_THEN ttac tm; CONJ_TAC];;
+
+let FILTER_DISCH_TAC = FILTER_DISCH_THEN STRIP_ASSUME_TAC;;
+
+let FILTER_STRIP_TAC = FILTER_STRIP_THEN STRIP_ASSUME_TAC;;
+
+(* ------------------------------------------------------------------------- *)
+(* Conversions for quantifier movement using proforma theorems. *)
+(* ------------------------------------------------------------------------- *)
+
+(* let ....... *)
+
+(* ------------------------------------------------------------------------- *)
+(* Resolution stuff. *)
+(* ------------------------------------------------------------------------- *)
+
+let RES_CANON =
+ let not_elim th =
+ if is_neg (concl th) then true,(NOT_ELIM th) else (false,th) in
+ let rec canon fl th =
+ let w = concl th in
+ if (is_conj w) then
+ let (th1,th2) = CONJ_PAIR th in (canon fl th1) @ (canon fl th2) else
+ if ((is_imp w) & not(is_neg w)) then
+ let ante,conc = dest_neg_imp w in
+ if (is_conj ante) then
+ let a,b = dest_conj ante in
+ let cth = NOT_MP th (CONJ (ASSUME a) (ASSUME b)) in
+ let th1 = DISCH b cth and th2 = DISCH a cth in
+ (canon true (DISCH a th1)) @ (canon true (DISCH b th2)) else
+ if (is_disj ante) then
+ let a,b = dest_disj ante in
+ let ath = DISJ1 (ASSUME a) b and bth = DISJ2 a (ASSUME b) in
+ let th1 = DISCH a (NOT_MP th ath) and
+ th2 = DISCH b (NOT_MP th bth) in
+ (canon true th1) @ (canon true th2) else
+ if (is_exists ante) then
+ let v,body = dest_exists ante in
+ let newv = variant (thm_frees th) v in
+ let newa = subst [newv,v] body in
+ let th1 = NOT_MP th (EXISTS (ante, newv) (ASSUME newa)) in
+ canon true (DISCH newa th1) else
+ map (GEN_ALL o (DISCH ante)) (canon true (UNDISCH th)) else
+ if (is_eq w & (type_of (rand w) = `:bool`)) then
+ let (th1,th2) = EQ_IMP_RULE th in
+ (if fl then [GEN_ALL th] else []) @ (canon true th1) @ (canon true th2) else
+ if (is_forall w) then
+ let vs,body = strip_forall w in
+ let fvs = thm_frees th in
+ let vfn = fun l -> variant (l @ fvs) in
+ let nvs = itlist (fun v nv -> let v' = vfn nv v in (v'::nv)) vs [] in
+ canon fl (SPECL nvs th) else
+ if fl then [GEN_ALL th] else [] in
+ fun th -> try let args = map (not_elim o SPEC_ALL) (CONJUNCTS (SPEC_ALL th)) in
+ let imps = flat (map (map GEN_ALL o (uncurry canon)) args) in
+ check (fun l -> l <> []) imps
+ with Failure _ ->
+ failwith "RES_CANON: no implication is derivable from input thm.";;
+
+let IMP_RES_THEN,RES_THEN =
+ let MATCH_MP impth =
+ let sth = SPEC_ALL impth in
+ let matchfn = (fun (a,b,c) -> b,c) o
+ term_match [] (fst(dest_neg_imp(concl sth))) in
+ fun th -> NOT_MP (INST_TY_TERM (matchfn (concl th)) sth) th in
+ let check st l = (if l = [] then failwith st else l) in
+ let IMP_RES_THEN ttac impth =
+ let ths = try RES_CANON impth with Failure _ -> failwith "IMP_RES_THEN: no implication" in
+ ASSUM_LIST
+ (fun asl ->
+ let l = itlist (fun th -> (@) (mapfilter (MATCH_MP th) asl)) ths [] in
+ let res = check "IMP_RES_THEN: no resolvents " l in
+ let tacs = check "IMP_RES_THEN: no tactics" (mapfilter ttac res) in
+ EVERY tacs) in
+ let RES_THEN ttac (asl,g) =
+ let asm = map snd asl in
+ let ths = itlist (@) (mapfilter RES_CANON asm) [] in
+ let imps = check "RES_THEN: no implication" ths in
+ let l = itlist (fun th -> (@) (mapfilter (MATCH_MP th) asm)) imps [] in
+ let res = check "RES_THEN: no resolvents " l in
+ let tacs = check "RES_THEN: no tactics" (mapfilter ttac res) in
+ EVERY tacs (asl,g) in
+ IMP_RES_THEN,RES_THEN;;
+
+let IMP_RES_TAC th g =
+ try IMP_RES_THEN (REPEAT_GTCL IMP_RES_THEN STRIP_ASSUME_TAC) th g
+ with Failure _ -> ALL_TAC g;;
+
+let RES_TAC g =
+ try RES_THEN (REPEAT_GTCL IMP_RES_THEN STRIP_ASSUME_TAC) g
+ with Failure _ -> ALL_TAC g;;
+
+(* ------------------------------------------------------------------------- *)
+(* Stuff for handling type definitions. *)
+(* ------------------------------------------------------------------------- *)
+
+let prove_rep_fn_one_one th =
+ try let thm = CONJUNCT1 th in
+ let A,R = (I F_F rator) (dest_comb(lhs(snd(dest_forall(concl thm))))) in
+ let _,[aty;rty] = dest_type (type_of R) in
+ let a = mk_primed_var("a",aty) in let a' = variant [a] a in
+ let a_eq_a' = mk_eq(a,a') and
+ Ra_eq_Ra' = mk_eq(mk_comb(R,a),mk_comb (R,a')) in
+ let th1 = AP_TERM A (ASSUME Ra_eq_Ra') in
+ let ga1 = genvar aty and ga2 = genvar aty in
+ let th2 = SUBST [SPEC a thm,ga1;SPEC a' thm,ga2] (mk_eq(ga1,ga2)) th1 in
+ let th3 = DISCH a_eq_a' (AP_TERM R (ASSUME a_eq_a')) in
+ GEN a (GEN a' (IMP_ANTISYM_RULE (DISCH Ra_eq_Ra' th2) th3))
+ with Failure _ -> failwith "prove_rep_fn_one_one";;
+
+let prove_rep_fn_onto th =
+ try let [th1;th2] = CONJUNCTS th in
+ let r,eq = (I F_F rhs)(dest_forall(concl th2)) in
+ let RE,ar = dest_comb(lhs eq) and
+ sr = (mk_eq o (fun (x,y) -> y,x) o dest_eq) eq in
+ let a = mk_primed_var ("a",type_of ar) in
+ let sra = mk_eq(r,mk_comb(RE,a)) in
+ let ex = mk_exists(a,sra) in
+ let imp1 = EXISTS (ex,ar) (SYM(ASSUME eq)) in
+ let v = genvar (type_of r) and
+ A = rator ar and
+ s' = AP_TERM RE (SPEC a th1) in
+ let th = SUBST[SYM(ASSUME sra),v](mk_eq(mk_comb(RE,mk_comb(A,v)),v))s' in
+ let imp2 = CHOOSE (a,ASSUME ex) th in
+ let swap = IMP_ANTISYM_RULE (DISCH eq imp1) (DISCH ex imp2) in
+ GEN r (TRANS (SPEC r th2) swap)
+ with Failure _ -> failwith "prove_rep_fn_onto";;
+
+let prove_abs_fn_onto th =
+ try let [th1;th2] = CONJUNCTS th in
+ let a,(A,R) = (I F_F ((I F_F rator)o dest_comb o lhs))
+ (dest_forall(concl th1)) in
+ let thm1 = EQT_ELIM(TRANS (SPEC (mk_comb (R,a)) th2)
+ (EQT_INTRO (AP_TERM R (SPEC a th1)))) in
+ let thm2 = SYM(SPEC a th1) in
+ let r,P = (I F_F (rator o lhs)) (dest_forall(concl th2)) in
+ let ex = mk_exists(r,mk_conj(mk_eq(a,mk_comb(A,r)),mk_comb(P,r))) in
+ GEN a (EXISTS(ex,mk_comb(R,a)) (CONJ thm2 thm1))
+ with Failure _ -> failwith "prove_abs_fn_onto";;
+
+let prove_abs_fn_one_one th =
+ try let [th1;th2] = CONJUNCTS th in
+ let r,P = (I F_F (rator o lhs)) (dest_forall(concl th2)) and
+ A,R = (I F_F rator) (dest_comb(lhs(snd(dest_forall(concl th1))))) in
+ let r' = variant [r] r in
+ let as1 = ASSUME(mk_comb(P,r)) and as2 = ASSUME(mk_comb(P,r')) in
+ let t1 = EQ_MP (SPEC r th2) as1 and t2 = EQ_MP (SPEC r' th2) as2 in
+ let eq = (mk_eq(mk_comb(A,r),mk_comb(A,r'))) in
+ let v1 = genvar(type_of r) and v2 = genvar(type_of r) in
+ let i1 = DISCH eq
+ (SUBST [t1,v1;t2,v2] (mk_eq(v1,v2)) (AP_TERM R (ASSUME eq))) and
+ i2 = DISCH (mk_eq(r,r')) (AP_TERM A (ASSUME (mk_eq(r,r')))) in
+ let thm = IMP_ANTISYM_RULE i1 i2 in
+ let disch = DISCH (mk_comb(P,r)) (DISCH (mk_comb(P,r')) thm) in
+ GEN r (GEN r' disch)
+ with Failure _ -> failwith "prove_abs_fn_one_one";;
+
+(* ------------------------------------------------------------------------- *)
+(* AC rewriting needs to be wrapped up as a special conversion. *)
+(* ------------------------------------------------------------------------- *)
+
+let AC_CONV(assoc,sym) =
+ let th1 = SPEC_ALL assoc
+ and th2 = SPEC_ALL sym in
+ let th3 = GEN_REWRITE_RULE (RAND_CONV o LAND_CONV) [th2] th1 in
+ let th4 = SYM th1 in
+ let th5 = GEN_REWRITE_RULE RAND_CONV [th4] th3 in
+ EQT_INTRO o AC(end_itlist CONJ [th2; th4; th5]);;
+
+let AC_RULE ths = EQT_ELIM o AC_CONV ths;;
+
+(* ------------------------------------------------------------------------- *)
+(* The order of picking conditionals is different! *)
+(* ------------------------------------------------------------------------- *)
+
+let (COND_CASES_TAC :tactic) =
+ let is_good_cond tm =
+ try not(is_const(fst(dest_cond tm)))
+ with Failure _ -> false in
+ fun (asl,w) ->
+ let cond = find_term (fun tm -> is_good_cond tm & free_in tm w) w in
+ let p,(t,u) = dest_cond cond in
+ let inst = INST_TYPE [type_of t, `:A`] COND_CLAUSES in
+ let (ct,cf) = CONJ_PAIR (SPEC u (SPEC t inst)) in
+ DISJ_CASES_THEN2
+ (fun th -> SUBST1_TAC (EQT_INTRO th) THEN
+ SUBST1_TAC ct THEN ASSUME_TAC th)
+ (fun th -> SUBST1_TAC (EQF_INTRO th) THEN
+ SUBST1_TAC cf THEN ASSUME_TAC th)
+ (SPEC p EXCLUDED_MIDDLE)
+ (asl,w) ;;
+
+(* ------------------------------------------------------------------------- *)
+(* MATCH_MP_TAC allows universals on the right of implication. *)
+(* Here's a crude hack to allow it. *)
+(* ------------------------------------------------------------------------- *)
+
+let MATCH_MP_TAC th =
+ MATCH_MP_TAC th ORELSE
+ MATCH_MP_TAC(PURE_REWRITE_RULE[RIGHT_IMP_FORALL_THM] th);;
+
+(* ------------------------------------------------------------------------- *)
+(* Various theorems have different names. *)
+(* ------------------------------------------------------------------------- *)
+
+let ZERO_LESS_EQ = LE_0;;
+
+let LESS_EQ_MONO = LE_SUC;;
+
+let NOT_LESS = NOT_LT;;
+
+let LESS_0 = LT_0;;
+
+let LESS_EQ_REFL = LE_REFL;;
+
+let LESS_EQUAL_ANTISYM = GEN_ALL(fst(EQ_IMP_RULE(SPEC_ALL LE_ANTISYM)));;
+
+let NOT_LESS_0 = GEN_ALL(EQF_ELIM(SPEC_ALL(CONJUNCT1 LT)));;
+
+let LESS_TRANS = LT_TRANS;;
+
+let LESS_LEMMA1 = GEN_ALL(fst(EQ_IMP_RULE(SPEC_ALL(CONJUNCT2 LT))));;
+
+let LESS_SUC_REFL = prove(`!n. n < SUC n`,REWRITE_TAC[LT]);;
+
+let FACT_LESS = FACT_LT;;
+
+let LESS_EQ_SUC_REFL = prove(`!n. n <= SUC n`,REWRITE_TAC[LE; LE_REFL]);;
+
+let LESS_EQ_ADD = LE_ADD;;
+
+let GREATER_EQ = GE;;
+
+let LESS_EQUAL_ADD = GEN_ALL(fst(EQ_IMP_RULE(SPEC_ALL LE_EXISTS)));;
+
+let LESS_EQ_IMP_LESS_SUC = GEN_ALL(snd(EQ_IMP_RULE(SPEC_ALL LT_SUC_LE)));;
+
+let LESS_IMP_LESS_OR_EQ = LT_IMP_LE;;
+
+let LESS_MONO_ADD = GEN_ALL(snd(EQ_IMP_RULE(SPEC_ALL LT_ADD_RCANCEL)));;
+
+let LESS_SUC = prove(`!m n. m < n ==> m < (SUC n)`,MESON_TAC[LT]);;
+
+let LESS_CASES = LTE_CASES;;
+
+let LESS_EQ = GSYM LE_SUC_LT;;
+
+let LESS_OR_EQ = LE_LT;;
+
+let LESS_ADD_1 = GEN_ALL(fst(EQ_IMP_RULE(SPEC_ALL
+ (REWRITE_RULE[ADD1] LT_EXISTS))));;
+
+let SUC_SUB1 = prove(`!m. SUC m - 1 = m`,
+ REWRITE_TAC[num_CONV `1`; SUB_SUC; SUB_0]);;
+
+let LESS_MONO_EQ = LT_SUC;;
+
+let LESS_ADD_SUC = prove (`!m n. m < m + SUC n`,
+ REWRITE_TAC[ADD_CLAUSES; LT_SUC_LE; LE_ADD]);;
+
+let LESS_REFL = LT_REFL;;
+
+let INV_SUC_EQ = SUC_INJ;;
+
+let LESS_EQ_CASES = LE_CASES;;
+
+let LESS_EQ_TRANS = LE_TRANS;;
+
+let LESS_THM = CONJUNCT2 LT;;
+
+let GREATER = GT;;
+
+let LESS_EQ_0 = CONJUNCT1 LE;;
+
+let OR_LESS = GEN_ALL(fst(EQ_IMP_RULE(SPEC_ALL LE_SUC_LT)));;
+
+let SUB_EQUAL_0 = SUB_REFL;;
+
+let SUB_MONO_EQ = SUB_SUC;;
+
+let NOT_SUC_LESS_EQ = prove (`!n m. ~(SUC n <= m) <=> m <= n`,
+ REWRITE_TAC[NOT_LE; LT] THEN
+ MESON_TAC[LE_LT]);;
+
+let SUC_NOT = GSYM NOT_SUC;;
+
+let LESS_LESS_CASES = prove(`!m n:num. (m = n) \/ m < n \/ n < m`,
+ MESON_TAC[LT_CASES]);;
+
+let NOT_LESS_EQUAL = NOT_LE;;
+
+let LESS_EQ_EXISTS = LE_EXISTS;;
+
+let LESS_MONO_ADD_EQ = LT_ADD_RCANCEL;;
+
+let LESS_LESS_EQ_TRANS = LTE_TRANS;;
+
+let SUB_SUB = ARITH_RULE
+ `!b c. c <= b ==> (!a:num. a - (b - c) = (a + c) - b)`;;
+
+let LESS_CASES_IMP = ARITH_RULE
+ `!m n:num. ~(m < n) /\ ~(m = n) ==> n < m`;;
+
+let SUB_LESS_EQ = ARITH_RULE
+ `!n m:num. (n - m) <= n`;;
+
+let SUB_EQ_EQ_0 = ARITH_RULE
+ `!m n:num. (m - n = m) <=> (m = 0) \/ (n = 0)`;;
+
+let SUB_LEFT_LESS_EQ = ARITH_RULE
+ `!m n p:num. m <= (n - p) <=> (m + p) <= n \/ m <= 0`;;
+
+let SUB_LEFT_GREATER_EQ =
+ ARITH_RULE `!m n p:num. m >= (n - p) <=> (m + p) >= n`;;
+
+let LESS_EQ_LESS_TRANS = LET_TRANS;;
+
+let LESS_0_CASES = ARITH_RULE `!m. (0 = m) \/ 0 < m`;;
+
+let LESS_OR = ARITH_RULE `!m n. m < n ==> (SUC m) <= n`;;
+
+let SUB = ARITH_RULE
+ `(!m. 0 - m = 0) /\
+ (!m n. (SUC m) - n = (if m < n then 0 else SUC(m - n)))`;;
+
+let LESS_MULT_MONO = prove
+ (`!m i n. ((SUC n) * m) < ((SUC n) * i) <=> m < i`,
+ REWRITE_TAC[LT_MULT_LCANCEL; NOT_SUC]);;
+
+let LESS_MONO_MULT = prove
+ (`!m n p. m <= n ==> (m * p) <= (n * p)`,
+ SIMP_TAC[LE_MULT_RCANCEL]);;
+
+let LESS_MULT2 = prove
+ (`!m n. 0 < m /\ 0 < n ==> 0 < (m * n)`,
+ REWRITE_TAC[LT_MULT]);;
+
+let SUBSET_FINITE = prove
+ (`!s. FINITE s ==> (!t. t SUBSET s ==> FINITE t)`,
+ MESON_TAC[FINITE_SUBSET]);;
+
+let LESS_EQ_SUC = prove
+ (`!n. m <= SUC n <=> (m = SUC n) \/ m <= n`,
+ REWRITE_TAC[LE]);;
+
+let ANTE_RES_THEN ttac th = FIRST_ASSUM(fun t -> ttac (MATCH_MP t th));;
+
+let IMP_RES_THEN ttac th = FIRST_ASSUM(fun t -> ttac (MATCH_MP th t));;
+
+(* ------------------------------------------------------------------------ *)
+(* Set theory lemmas. *)
+(* ------------------------------------------------------------------------ *)
+
+let INFINITE_MEMBER = prove(
+ `!s. INFINITE(s:A->bool) ==> ?x. x IN s`,
+ GEN_TAC THEN DISCH_TAC THEN
+ SUBGOAL_THEN `~(s:A->bool = {})` MP_TAC THENL
+ [UNDISCH_TAC `INFINITE (s:A->bool)` THEN
+ CONV_TAC CONTRAPOS_CONV THEN REWRITE_TAC[] THEN
+ DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[INFINITE; FINITE_EMPTY];
+ REWRITE_TAC[EXTENSION; NOT_IN_EMPTY] THEN
+ PURE_ONCE_REWRITE_TAC[NOT_FORALL_THM] THEN
+ REWRITE_TAC[]]);;
+
+let INFINITE_CHOOSE = prove(
+ `!s:A->bool. INFINITE(s) ==> ((@) s) IN s`,
+ GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP INFINITE_MEMBER) THEN
+ DISCH_THEN(MP_TAC o SELECT_RULE) THEN REWRITE_TAC[IN] THEN
+ CONV_TAC(ONCE_DEPTH_CONV ETA_CONV) THEN REWRITE_TAC[]);;
+
+let INFINITE_DELETE = prove(
+ `!(t:A->bool) x. INFINITE (t DELETE x) = INFINITE(t)`,
+ REWRITE_TAC[INFINITE; FINITE_DELETE]);;
+
+let INFINITE_INSERT = prove(
+ `!(x:A) t. INFINITE(x INSERT t) = INFINITE(t)`,
+ REWRITE_TAC[INFINITE; FINITE_INSERT]);;
+
+let SIZE_INSERT = prove(
+ `!(x:A) t. ~(x IN t) /\ t HAS_SIZE n ==> (x INSERT t) HAS_SIZE (SUC n)`,
+ SIMP_TAC[HAS_SIZE; CARD_CLAUSES; FINITE_RULES]);;
+
+let SIZE_DELETE = prove(
+ `!(x:A) t. x IN t /\ t HAS_SIZE (SUC n) ==> (t DELETE x) HAS_SIZE n`,
+ SIMP_TAC[HAS_SIZE_SUC]);;
+
+let SIZE_EXISTS = prove(
+ `!s N. s HAS_SIZE (SUC N) ==> ?x:A. x IN s`,
+ SIMP_TAC[HAS_SIZE_SUC; GSYM MEMBER_NOT_EMPTY]);;
+
+let SUBSET_DELETE = prove(
+ `!s t (x:A). s SUBSET t ==> (s DELETE x) SUBSET t`,
+ REPEAT STRIP_TAC THEN MATCH_MP_TAC SUBSET_TRANS THEN
+ EXISTS_TAC `s:A->bool` THEN ASM_REWRITE_TAC[DELETE_SUBSET]);;
+
+let INFINITE_FINITE_CHOICE = prove(
+ `!n (s:A->bool). INFINITE(s) ==> ?t. t SUBSET s /\ t HAS_SIZE n`,
+ INDUCT_TAC THEN GEN_TAC THEN DISCH_TAC THENL
+ [EXISTS_TAC `{}:A->bool` THEN
+ REWRITE_TAC[HAS_SIZE; EMPTY_SUBSET; HAS_SIZE_0];
+ FIRST_ASSUM(UNDISCH_TAC o check is_forall o concl) THEN
+ DISCH_THEN(MP_TAC o SPEC `s DELETE ((@) s :A)`) THEN
+ ASM_REWRITE_TAC[INFINITE_DELETE] THEN
+ DISCH_THEN(X_CHOOSE_THEN `t:A->bool` STRIP_ASSUME_TAC) THEN
+ EXISTS_TAC `((@) s :A) INSERT t` THEN CONJ_TAC THENL
+ [REWRITE_TAC[INSERT_SUBSET] THEN CONJ_TAC THENL
+ [MATCH_MP_TAC INFINITE_CHOOSE THEN ASM_REWRITE_TAC[];
+ REWRITE_TAC[SUBSET] THEN
+ RULE_ASSUM_TAC(REWRITE_RULE[SUBSET]) THEN
+ GEN_TAC THEN DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN
+ REWRITE_TAC[IN_DELETE] THEN CONV_TAC(EQT_INTRO o TAUT)];
+ MATCH_MP_TAC SIZE_INSERT THEN ASM_REWRITE_TAC[] THEN
+ DISCH_TAC THEN UNDISCH_TAC `t SUBSET (s DELETE ((@) s:A))` THEN
+ REWRITE_TAC[SUBSET; IN_DELETE] THEN
+ DISCH_THEN(IMP_RES_THEN MP_TAC) THEN REWRITE_TAC[]]]);;
+
+let IMAGE_WOP_LEMMA = prove(
+ `!N (t:num->bool) (u:A->bool).
+ u SUBSET (IMAGE f t) /\ u HAS_SIZE (SUC N) ==>
+ ?n v. (u = (f n) INSERT v) /\
+ !y. y IN v ==> ?m. (y = f m) /\ n < m`,
+ REPEAT STRIP_TAC THEN
+ MP_TAC(SPEC `\n:num. ?y:A. y IN u /\ (y = f n)` num_WOP) THEN BETA_TAC THEN
+ DISCH_THEN(MP_TAC o fst o EQ_IMP_RULE) THEN
+ FIRST_ASSUM(X_CHOOSE_TAC `y:A` o MATCH_MP SIZE_EXISTS) THEN
+ FIRST_ASSUM(MP_TAC o SPEC `y:A` o REWRITE_RULE[SUBSET]) THEN
+ ASM_REWRITE_TAC[IN_IMAGE] THEN
+ DISCH_THEN(X_CHOOSE_THEN `n:num` STRIP_ASSUME_TAC) THEN
+ W(C SUBGOAL_THEN (fun t ->REWRITE_TAC[t]) o
+ funpow 2 (fst o dest_imp) o snd) THENL
+ [MAP_EVERY EXISTS_TAC [`n:num`; `y:A`] THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN
+ DISCH_THEN(X_CHOOSE_THEN `m:num` (CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN
+ DISCH_THEN(X_CHOOSE_THEN `x:A` STRIP_ASSUME_TAC) THEN
+ MAP_EVERY EXISTS_TAC [`m:num`; `u DELETE (x:A)`] THEN CONJ_TAC THENL
+ [ASM_REWRITE_TAC[] THEN CONV_TAC SYM_CONV THEN
+ MATCH_MP_TAC INSERT_DELETE THEN FIRST_ASSUM(SUBST1_TAC o SYM) THEN
+ FIRST_ASSUM MATCH_ACCEPT_TAC;
+ X_GEN_TAC `z:A` THEN REWRITE_TAC[IN_DELETE] THEN STRIP_TAC THEN
+ FIRST_ASSUM(MP_TAC o SPEC `z:A` o REWRITE_RULE[SUBSET]) THEN
+ ASM_REWRITE_TAC[IN_IMAGE] THEN
+ DISCH_THEN(X_CHOOSE_THEN `k:num` STRIP_ASSUME_TAC) THEN
+ EXISTS_TAC `k:num` THEN ASM_REWRITE_TAC[GSYM NOT_LESS_EQUAL] THEN
+ REWRITE_TAC[LESS_OR_EQ; DE_MORGAN_THM] THEN CONJ_TAC THENL
+ [DISCH_THEN(ANTE_RES_THEN (MP_TAC o CONV_RULE NOT_EXISTS_CONV)) THEN
+ DISCH_THEN(MP_TAC o SPEC `z:A`) THEN REWRITE_TAC[] THEN
+ CONJ_TAC THEN FIRST_ASSUM MATCH_ACCEPT_TAC;
+ DISCH_THEN SUBST_ALL_TAC THEN
+ UNDISCH_TAC `~(z:A = x)` THEN ASM_REWRITE_TAC[]]]);;
+
+(* ------------------------------------------------------------------------ *)
+(* Lemma about finite colouring of natural numbers. *)
+(* ------------------------------------------------------------------------ *)
+
+let COLOURING_LEMMA = prove(
+ `!M col s. (INFINITE(s) /\ !n:A. n IN s ==> col(n) <= M) ==>
+ ?c t. t SUBSET s /\ INFINITE(t) /\ !n:A. n IN t ==> (col(n) = c)`,
+ INDUCT_TAC THENL
+ [REWRITE_TAC[LESS_EQ_0] THEN REPEAT STRIP_TAC THEN
+ MAP_EVERY EXISTS_TAC [`0`; `s:A->bool`] THEN
+ ASM_REWRITE_TAC[SUBSET_REFL];
+ REPEAT STRIP_TAC THEN SUBGOAL_THEN
+ `INFINITE { x:A | x IN s /\ (col x = SUC M) } \/
+ INFINITE { x:A | x IN s /\ col x <= M}`
+ DISJ_CASES_TAC THENL
+ [UNDISCH_TAC `INFINITE(s:A->bool)` THEN
+ REWRITE_TAC[INFINITE; GSYM DE_MORGAN_THM; GSYM FINITE_UNION] THEN
+ CONV_TAC CONTRAPOS_CONV THEN REWRITE_TAC[] THEN
+ DISCH_THEN(MATCH_MP_TAC o MATCH_MP SUBSET_FINITE) THEN
+ REWRITE_TAC[SUBSET; IN_UNION] THEN
+ REWRITE_TAC[IN_ELIM_THM] THEN
+ GEN_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[GSYM LESS_EQ_SUC] THEN
+ FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[];
+ MAP_EVERY EXISTS_TAC [`SUC M`; `{ x:A | x IN s /\ (col x = SUC M)}`] THEN
+ ASM_REWRITE_TAC[SUBSET] THEN REWRITE_TAC[IN_ELIM_THM] THEN
+ REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[];
+ SUBGOAL_THEN `!n:A. n IN { x | x IN s /\ col x <= M } ==> col(n) <= M`
+ MP_TAC THENL
+ [GEN_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN
+ DISCH_THEN(MATCH_ACCEPT_TAC o CONJUNCT2);
+ FIRST_X_ASSUM(MP_TAC o SPECL [`col:A->num`;
+ `{ x:A | x IN s /\ col x <= M}`]) THEN
+ ASM_SIMP_TAC[] THEN
+ MATCH_MP_TAC(TAUT `(c ==> d) ==> (b ==> c) ==> b ==> d`) THEN
+ DISCH_THEN(X_CHOOSE_THEN `c:num` (X_CHOOSE_TAC `t:A->bool`)) THEN
+ MAP_EVERY EXISTS_TAC [`c:num`; `t:A->bool`] THEN
+ ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_TRANS THEN
+ EXISTS_TAC `{ x:A | x IN s /\ col x <= M }` THEN ASM_REWRITE_TAC[] THEN
+ REWRITE_TAC[SUBSET] THEN REWRITE_TAC[IN_ELIM_THM] THEN
+ REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]]]]);;
+
+let COLOURING_THM = prove(
+ `!M col. (!n. col n <= M) ==>
+ ?c s. INFINITE(s) /\ !n:num. n IN s ==> (col(n) = c)`,
+ REPEAT STRIP_TAC THEN MP_TAC
+ (ISPECL [`M:num`; `col:num->num`; `UNIV:num->bool`] COLOURING_LEMMA) THEN
+ ASM_REWRITE_TAC[num_INFINITE] THEN
+ DISCH_THEN(X_CHOOSE_THEN `c:num` (X_CHOOSE_TAC `t:num->bool`)) THEN
+ MAP_EVERY EXISTS_TAC [`c:num`; `t:num->bool`] THEN ASM_REWRITE_TAC[]);;
+
+(* ------------------------------------------------------------------------ *)
+(* Simple approach via lemmas then induction over size of coloured sets. *)
+(* ------------------------------------------------------------------------ *)
+
+let RAMSEY_LEMMA1 = prove(
+ `(!C s. INFINITE(s:A->bool) /\
+ (!t. t SUBSET s /\ t HAS_SIZE N ==> C(t) <= M)
+ ==> ?t c. INFINITE(t) /\ t SUBSET s /\
+ (!u. u SUBSET t /\ u HAS_SIZE N ==> (C(u) = c)))
+ ==> !C s. INFINITE(s:A->bool) /\
+ (!t. t SUBSET s /\ t HAS_SIZE (SUC N) ==> C(t) <= M)
+ ==> ?t c. INFINITE(t) /\ t SUBSET s /\ ~(((@) s) IN t) /\
+ (!u. u SUBSET t /\ u HAS_SIZE N
+ ==> (C(((@) s) INSERT u) = c))`,
+ DISCH_THEN((THEN) (REPEAT STRIP_TAC) o MP_TAC) THEN
+ DISCH_THEN(MP_TAC o SPEC `\u. C (((@) s :A) INSERT u):num`) THEN
+ DISCH_THEN(MP_TAC o SPEC `s DELETE ((@)s:A)`) THEN BETA_TAC THEN
+ ASM_REWRITE_TAC[INFINITE_DELETE] THEN
+ W(C SUBGOAL_THEN (fun t ->REWRITE_TAC[t]) o
+ funpow 2 (fst o dest_imp) o snd) THENL
+ [REPEAT STRIP_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN CONJ_TAC THENL
+ [UNDISCH_TAC `t SUBSET (s DELETE ((@) s :A))` THEN
+ REWRITE_TAC[SUBSET; IN_INSERT; IN_DELETE; NOT_IN_EMPTY] THEN
+ DISCH_TAC THEN GEN_TAC THEN DISCH_THEN DISJ_CASES_TAC THEN
+ ASM_REWRITE_TAC[] THENL
+ [MATCH_MP_TAC INFINITE_CHOOSE;
+ FIRST_ASSUM(ANTE_RES_THEN ASSUME_TAC)] THEN
+ ASM_REWRITE_TAC[];
+ MATCH_MP_TAC SIZE_INSERT THEN ASM_REWRITE_TAC[] THEN
+ DISCH_TAC THEN UNDISCH_TAC `t SUBSET (s DELETE ((@) s :A))` THEN
+ ASM_REWRITE_TAC[SUBSET; IN_DELETE] THEN
+ DISCH_THEN(MP_TAC o SPEC `(@)s:A`) THEN ASM_REWRITE_TAC[]];
+ DISCH_THEN(X_CHOOSE_THEN `t:A->bool` MP_TAC) THEN
+ DISCH_THEN(X_CHOOSE_THEN `c:num` STRIP_ASSUME_TAC) THEN
+ MAP_EVERY EXISTS_TAC [`t:A->bool`; `c:num`] THEN ASM_REWRITE_TAC[] THEN
+ RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; IN_DELETE]) THEN CONJ_TAC THENL
+ [REWRITE_TAC[SUBSET] THEN
+ GEN_TAC THEN DISCH_THEN(ANTE_RES_THEN(fun th -> REWRITE_TAC[th]));
+ DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN REWRITE_TAC[]]]);;
+
+let RAMSEY_LEMMA2 = prove(
+ `(!C s. INFINITE(s:A->bool) /\
+ (!t. t SUBSET s /\ t HAS_SIZE (SUC N) ==> C(t) <= M)
+ ==> ?t c. INFINITE(t) /\ t SUBSET s /\ ~(((@) s) IN t) /\
+ (!u. u SUBSET t /\ u HAS_SIZE N
+ ==> (C(((@) s) INSERT u) = c)))
+ ==> !C s. INFINITE(s:A->bool) /\
+ (!t. t SUBSET s /\ t HAS_SIZE (SUC N) ==> C(t) <= M)
+ ==> ?t x col. (!n. col n <= M) /\
+ (!n. (t n) SUBSET s) /\
+ (!n. t(SUC n) SUBSET (t n)) /\
+ (!n. ~((x n) IN (t n))) /\
+ (!n. x(SUC n) IN (t n)) /\
+ (!n. (x n) IN s) /\
+ (!n u. u SUBSET (t n) /\ u HAS_SIZE N
+ ==> (C((x n) INSERT u) = col n))`,
+ REPEAT STRIP_TAC THEN
+ MP_TAC(ISPECL [`s:A->bool`; `\s (n:num). @t:A->bool. ?c:num.
+ INFINITE(t) /\
+ t SUBSET s /\
+ ~(((@) s) IN t) /\
+ !u. u SUBSET t /\ u HAS_SIZE N ==> (C(((@) s) INSERT u) = c)`]
+ num_Axiom) THEN DISCH_THEN(MP_TAC o BETA_RULE o EXISTENCE) THEN
+ DISCH_THEN(X_CHOOSE_THEN `f:num->(A->bool)` STRIP_ASSUME_TAC) THEN
+ SUBGOAL_THEN
+ `!n:num. (f n) SUBSET (s:A->bool) /\
+ ?c. INFINITE(f(SUC n)) /\ f(SUC n) SUBSET (f n) /\
+ ~(((@)(f n)) IN (f(SUC n))) /\
+ !u. u SUBSET (f(SUC n)) /\ u HAS_SIZE N ==>
+ (C(((@)(f n)) INSERT u) = c:num)`
+ MP_TAC THENL
+ [MATCH_MP_TAC num_INDUCTION THEN REPEAT STRIP_TAC THENL
+ [ASM_REWRITE_TAC[SUBSET_REFL];
+ FIRST_ASSUM(SUBST1_TAC o SPEC `0`) THEN CONV_TAC SELECT_CONV THEN
+ FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[];
+ MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `f(n:num):A->bool` THEN
+ CONJ_TAC THEN FIRST_ASSUM MATCH_ACCEPT_TAC;
+ FIRST_ASSUM(SUBST1_TAC o SPEC `SUC n`) THEN CONV_TAC SELECT_CONV THEN
+ FIRST_ASSUM MATCH_MP_TAC THEN CONJ_TAC THEN
+ TRY(FIRST_ASSUM MATCH_ACCEPT_TAC) THEN REPEAT STRIP_TAC THEN
+ FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
+ REPEAT(MATCH_MP_TAC SUBSET_TRANS THEN
+ FIRST_ASSUM(fun th -> EXISTS_TAC(rand(concl th)) THEN
+ CONJ_TAC THENL [FIRST_ASSUM MATCH_ACCEPT_TAC; ALL_TAC])) THEN
+ MATCH_ACCEPT_TAC SUBSET_REFL];
+ PURE_REWRITE_TAC[LEFT_EXISTS_AND_THM; RIGHT_EXISTS_AND_THM;
+ FORALL_AND_THM] THEN
+ DISCH_THEN(REPEAT_TCL (CONJUNCTS_THEN2 ASSUME_TAC) MP_TAC) THEN
+ DISCH_THEN(X_CHOOSE_TAC `col:num->num` o CONV_RULE SKOLEM_CONV) THEN
+ MAP_EVERY EXISTS_TAC
+ [`\n:num. f(SUC n):A->bool`; `\n:num. (@)(f n):A`] THEN
+ BETA_TAC THEN EXISTS_TAC `col:num->num` THEN CONJ_TAC THENL
+ [X_GEN_TAC `n:num` THEN
+ FIRST_ASSUM(MP_TAC o MATCH_MP INFINITE_FINITE_CHOICE o SPEC `n:num`) THEN
+ DISCH_THEN(CHOOSE_THEN MP_TAC o SPEC `N:num`) THEN
+ DISCH_THEN(fun th -> STRIP_ASSUME_TAC th THEN
+ ANTE_RES_THEN MP_TAC th) THEN
+ DISCH_THEN(SUBST1_TAC o SYM) THEN FIRST_ASSUM MATCH_MP_TAC THEN
+ CONJ_TAC THENL
+ [REWRITE_TAC[INSERT_SUBSET] THEN CONJ_TAC THENL
+ [FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET]) THEN
+ EXISTS_TAC `n:num` THEN MATCH_MP_TAC INFINITE_CHOOSE THEN
+ SPEC_TAC(`n:num`,`n:num`) THEN INDUCT_TAC THEN
+ TRY(FIRST_ASSUM MATCH_ACCEPT_TAC) THEN ASM_REWRITE_TAC[];
+ MATCH_MP_TAC SUBSET_TRANS THEN
+ EXISTS_TAC `f(SUC n):A->bool` THEN ASM_REWRITE_TAC[]];
+ MATCH_MP_TAC SIZE_INSERT THEN ASM_REWRITE_TAC[] THEN
+ UNDISCH_TAC `!n:num. ~(((@)(f n):A) IN (f(SUC n)))` THEN
+ DISCH_THEN(MP_TAC o SPEC `n:num`) THEN CONV_TAC CONTRAPOS_CONV THEN
+ REWRITE_TAC[] THEN
+ FIRST_ASSUM(MATCH_ACCEPT_TAC o REWRITE_RULE[SUBSET])];
+ REPEAT CONJ_TAC THEN TRY (FIRST_ASSUM MATCH_ACCEPT_TAC) THENL
+ [GEN_TAC; INDUCT_TAC THENL
+ [ASM_REWRITE_TAC[];
+ FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET]) THEN
+ EXISTS_TAC `SUC n`]] THEN
+ MATCH_MP_TAC INFINITE_CHOOSE THEN ASM_REWRITE_TAC[]]]);;
+
+let RAMSEY_LEMMA3 = prove(
+ `(!C s. INFINITE(s:A->bool) /\
+ (!t. t SUBSET s /\ t HAS_SIZE (SUC N) ==> C(t) <= M)
+ ==> ?t x col. (!n. col n <= M) /\
+ (!n. (t n) SUBSET s) /\
+ (!n. t(SUC n) SUBSET (t n)) /\
+ (!n. ~((x n) IN (t n))) /\
+ (!n. x(SUC n) IN (t n)) /\
+ (!n. (x n) IN s) /\
+ (!n u. u SUBSET (t n) /\ u HAS_SIZE N
+ ==> (C((x n) INSERT u) = col n)))
+ ==> !C s. INFINITE(s:A->bool) /\
+ (!t. t SUBSET s /\ t HAS_SIZE (SUC N) ==> C(t) <= M)
+ ==> ?t c. INFINITE(t) /\ t SUBSET s /\
+ (!u. u SUBSET t /\ u HAS_SIZE (SUC N) ==> (C(u) = c))`,
+ DISCH_THEN((THEN) (REPEAT STRIP_TAC) o MP_TAC) THEN
+ DISCH_THEN(MP_TAC o SPECL [`C:(A->bool)->num`; `s:A->bool`]) THEN
+ ASM_REWRITE_TAC[] THEN
+ DISCH_THEN(X_CHOOSE_THEN `t:num->(A->bool)` MP_TAC) THEN
+ DISCH_THEN(X_CHOOSE_THEN `x:num->A` MP_TAC) THEN
+ DISCH_THEN(X_CHOOSE_THEN `col:num->num` STRIP_ASSUME_TAC) THEN
+ MP_TAC(ISPECL [`M:num`; `col:num->num`; `UNIV:num->bool`]
+ COLOURING_LEMMA) THEN ASM_REWRITE_TAC[num_INFINITE] THEN
+ DISCH_THEN(X_CHOOSE_THEN `c:num` MP_TAC) THEN
+ DISCH_THEN(X_CHOOSE_THEN `t:num->bool` STRIP_ASSUME_TAC) THEN
+ MAP_EVERY EXISTS_TAC [`IMAGE (x:num->A) t`; `c:num`] THEN
+ SUBGOAL_THEN `!m n. m <= n ==> (t n:A->bool) SUBSET (t m)` ASSUME_TAC THENL
+ [REPEAT GEN_TAC THEN REWRITE_TAC[LESS_EQ_EXISTS] THEN
+ DISCH_THEN(X_CHOOSE_THEN `d:num` SUBST1_TAC) THEN
+ SPEC_TAC(`d:num`,`d:num`) THEN INDUCT_TAC THEN
+ ASM_REWRITE_TAC[ADD_CLAUSES; SUBSET_REFL] THEN
+ MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `t(m + d):A->bool` THEN
+ ASM_REWRITE_TAC[]; ALL_TAC] THEN
+ SUBGOAL_THEN `!m n. m < n ==> (x n:A) IN (t m)` ASSUME_TAC THENL
+ [REPEAT GEN_TAC THEN
+ DISCH_THEN(X_CHOOSE_THEN `d:num` SUBST1_TAC o MATCH_MP LESS_ADD_1) THEN
+ FIRST_ASSUM(MP_TAC o SPECL [`m:num`; `m + d`]) THEN
+ REWRITE_TAC[LESS_EQ_ADD; SUBSET] THEN DISCH_THEN MATCH_MP_TAC THEN
+ ASM_REWRITE_TAC[GSYM ADD1; ADD_CLAUSES]; ALL_TAC] THEN
+ SUBGOAL_THEN `!m n. ((x:num->A) m = x n) <=> (m = n)` ASSUME_TAC THENL
+ [REPEAT GEN_TAC THEN EQ_TAC THENL
+ [REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC
+ (SPECL [`m:num`; `n:num`] LESS_LESS_CASES) THEN
+ ASM_REWRITE_TAC[] THEN REPEAT DISCH_TAC THEN
+ FIRST_ASSUM(ANTE_RES_THEN MP_TAC) THEN ASM_REWRITE_TAC[] THEN
+ FIRST_ASSUM(UNDISCH_TAC o check is_eq o concl) THEN
+ DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_REWRITE_TAC[];
+ DISCH_THEN SUBST1_TAC THEN REFL_TAC]; ALL_TAC] THEN
+ REPEAT CONJ_TAC THENL
+ [UNDISCH_TAC `INFINITE(t:num->bool)` THEN
+ MATCH_MP_TAC INFINITE_IMAGE_INJ THEN ASM_REWRITE_TAC[];
+ REWRITE_TAC[SUBSET; IN_IMAGE] THEN GEN_TAC THEN
+ DISCH_THEN(CHOOSE_THEN (SUBST1_TAC o CONJUNCT1)) THEN ASM_REWRITE_TAC[];
+ GEN_TAC THEN DISCH_THEN(fun th -> STRIP_ASSUME_TAC th THEN MP_TAC th) THEN
+ DISCH_THEN(MP_TAC o MATCH_MP IMAGE_WOP_LEMMA) THEN
+ DISCH_THEN(X_CHOOSE_THEN `n:num` (X_CHOOSE_THEN `v:A->bool` MP_TAC)) THEN
+ DISCH_THEN STRIP_ASSUME_TAC THEN ASM_REWRITE_TAC[] THEN
+ SUBGOAL_THEN `c = (col:num->num) n` SUBST1_TAC THENL
+ [CONV_TAC SYM_CONV THEN FIRST_ASSUM MATCH_MP_TAC THEN
+ UNDISCH_TAC `u SUBSET (IMAGE (x:num->A) t)` THEN
+ REWRITE_TAC[SUBSET; IN_IMAGE] THEN
+ DISCH_THEN(MP_TAC o SPEC `(x:num->A) n`) THEN
+ ASM_REWRITE_TAC[IN_INSERT] THEN
+ DISCH_THEN(CHOOSE_THEN STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[];
+ FIRST_ASSUM MATCH_MP_TAC THEN CONJ_TAC THENL
+ [REWRITE_TAC[SUBSET] THEN GEN_TAC THEN
+ DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN
+ DISCH_THEN(X_CHOOSE_THEN `m:num` STRIP_ASSUME_TAC) THEN
+ ASM_REWRITE_TAC[] THEN FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[];
+ SUBGOAL_THEN `v = u DELETE ((x:num->A) n)` SUBST1_TAC THENL
+ [ASM_REWRITE_TAC[] THEN REWRITE_TAC[DELETE_INSERT] THEN
+ REWRITE_TAC[EXTENSION; IN_DELETE;
+ TAUT `(a <=> a /\ b) <=> a ==> b`] THEN
+ GEN_TAC THEN CONV_TAC CONTRAPOS_CONV THEN REWRITE_TAC[] THEN
+ DISCH_THEN SUBST1_TAC THEN DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN
+ ASM_REWRITE_TAC[] THEN
+ DISCH_THEN(CHOOSE_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
+ ASM_REWRITE_TAC[LESS_REFL];
+ MATCH_MP_TAC SIZE_DELETE THEN CONJ_TAC THENL
+ [ASM_REWRITE_TAC[IN_INSERT]; FIRST_ASSUM MATCH_ACCEPT_TAC]]]]]);;
+
+let RAMSEY = prove(
+ `!M N C s.
+ INFINITE(s:A->bool) /\
+ (!t. t SUBSET s /\ t HAS_SIZE N ==> C(t) <= M)
+ ==> ?t c. INFINITE(t) /\ t SUBSET s /\
+ (!u. u SUBSET t /\ u HAS_SIZE N ==> (C(u) = c))`,
+ GEN_TAC THEN INDUCT_TAC THENL
+ [REPEAT STRIP_TAC THEN
+ MAP_EVERY EXISTS_TAC [`s:A->bool`; `(C:(A->bool)->num) {}`] THEN
+ ASM_REWRITE_TAC[HAS_SIZE_0] THEN
+ REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[SUBSET_REFL];
+ MAP_EVERY MATCH_MP_TAC [RAMSEY_LEMMA3; RAMSEY_LEMMA2; RAMSEY_LEMMA1] THEN
+ POP_ASSUM MATCH_ACCEPT_TAC]);;
git://colo12-c703.uibk.ac.at / Infinite Ramsey's theorem/.git / commitdiff