(* ======================================================================== *)
(* 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]);;