(* ID: $Id: reflection.ML,v 1.21 2007/09/18 14:08:04 wenzelm Exp $ Author: Amine Chaieb, TU Muenchen A trial for automatical reification. *) signature REFLECTION = sig val genreify_tac: Proof.context -> thm list -> term option -> int -> tactic val reflection_tac: Proof.context -> thm list -> thm list -> term option -> int -> tactic val gen_reflection_tac: Proof.context -> (cterm -> thm) -> thm list -> thm list -> term option -> int -> tactic end; structure Reflection : REFLECTION = struct val ext2 = thm "ext2"; val nth_Cons_0 = thm "nth_Cons_0"; val nth_Cons_Suc = thm "nth_Cons_Suc"; (* Make a congruence rule out of a defining equation for the interpretation *) (* th is one defining equation of f, i.e. th is "f (Cp ?t1 ... ?tn) = P(f ?t1, .., f ?tn)" *) (* Cp is a constructor pattern and P is a pattern *) (* The result is: [|?A1 = f ?t1 ; .. ; ?An= f ?tn |] ==> P (?A1, .., ?An) = f (Cp ?t1 .. ?tn) *) (* + the a list of names of the A1 .. An, Those are fresh in the ctxt*) fun mk_congeq ctxt fs th = let val (f as Const(fN,fT)) = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> fst |> strip_comb |> fst val thy = ProofContext.theory_of ctxt val cert = Thm.cterm_of thy val (((_,_),[th']), ctxt') = Variable.import_thms true [th] ctxt val (lhs, rhs) = HOLogic.dest_eq (HOLogic.dest_Trueprop (Thm.prop_of th')) fun add_fterms (t as t1 $ t2) = if exists (fn f => could_unify (t |> strip_comb |> fst, f)) fs then insert (op aconv) t else add_fterms t1 #> add_fterms t2 | add_fterms (t as Abs(xn,xT,t')) = if (fN mem (term_consts t)) then (fn _ => [t]) else (fn _ => []) | add_fterms _ = I val fterms = add_fterms rhs [] val (xs, ctxt'') = Variable.variant_fixes (replicate (length fterms) "x") ctxt' val tys = map fastype_of fterms val vs = map Free (xs ~~ tys) val env = fterms ~~ vs (* FIXME!!!!*) fun replace_fterms (t as t1 $ t2) = (case AList.lookup (op aconv) env t of SOME v => v | NONE => replace_fterms t1 $ replace_fterms t2) | replace_fterms t = (case AList.lookup (op aconv) env t of SOME v => v | NONE => t) fun mk_def (Abs(x,xT,t),v) = HOLogic.mk_Trueprop ((HOLogic.all_const xT)$ Abs(x,xT,HOLogic.mk_eq(v$(Bound 0), t))) | mk_def (t, v) = HOLogic.mk_Trueprop (HOLogic.mk_eq (v, t)) fun tryext x = (x RS ext2 handle THM _ => x) val cong = (Goal.prove ctxt'' [] (map mk_def env) (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, replace_fterms rhs))) (fn x => LocalDefs.unfold_tac (#context x) (map tryext (#prems x)) THEN rtac th' 1)) RS sym val (cong' :: vars') = Variable.export ctxt'' ctxt (cong :: map (Drule.mk_term o cert) vs) val vs' = map (fst o fst o Term.dest_Var o Thm.term_of o Drule.dest_term) vars' in (vs', cong') end; (* congs is a list of pairs (P,th) where th is a theorem for *) (* [| f p1 = A1; ...; f pn = An|] ==> f (C p1 .. pn) = P *) val FWD = curry (op OF); (* da is the decomposition for atoms, ie. it returns ([],g) where g returns the right instance f (AtC n) = t , where AtC is the Atoms constructor and n is the number of the atom corresponding to t *) (* Generic decomp for reification : matches the actual term with the rhs of one cong rule. The result of the matching guides the proof synthesis: The matches of the introduced Variables A1 .. An are processed recursively The rest is instantiated in the cong rule,i.e. no reification is needed *) exception REIF of string; val bds = ref ([]: (typ * ((term list) * (term list))) list); fun index_of t = let val tt = HOLogic.listT (fastype_of t) in (case AList.lookup Type.could_unify (!bds) tt of NONE => error "index_of : type not found in environements!" | SOME (tbs,tats) => let val i = find_index_eq t tats val j = find_index_eq t tbs in (if j= ~1 then if i= ~1 then (bds := AList.update Type.could_unify (tt,(tbs,tats@[t])) (!bds) ; length tbs + length tats) else i else j) end) end; fun dest_listT (Type ("List.list", [T])) = T; fun decomp_genreif da cgns (t,ctxt) = let val thy = ProofContext.theory_of ctxt val cert = cterm_of thy fun tryabsdecomp (s,ctxt) = (case s of Abs(xn,xT,ta) => (let val ([xn],ctxt') = Variable.variant_fixes ["x"] ctxt val (xn,ta) = variant_abs (xn,xT,ta) val x = Free(xn,xT) val _ = (case AList.lookup Type.could_unify (!bds) (HOLogic.listT xT) of NONE => error "tryabsdecomp: Type not found in the Environement" | SOME (bsT,atsT) => (bds := AList.update Type.could_unify (HOLogic.listT xT, ((x::bsT), atsT)) (!bds))) in ([(ta, ctxt')] , fn [th] => ((let val (bsT,asT) = the(AList.lookup Type.could_unify (!bds) (HOLogic.listT xT)) in (bds := AList.update Type.could_unify (HOLogic.listT xT,(tl bsT,asT)) (!bds)) end) ; hd (Variable.export ctxt' ctxt [(forall_intr (cert x) th) COMP allI]))) end) | _ => da (s,ctxt)) in (case cgns of [] => tryabsdecomp (t,ctxt) | ((vns,cong)::congs) => ((let val cert = cterm_of thy val certy = ctyp_of thy val (tyenv, tmenv) = Pattern.match thy ((fst o HOLogic.dest_eq o HOLogic.dest_Trueprop) (concl_of cong), t) (Envir.type_env (Envir.empty 0),Term.Vartab.empty) val (fnvs,invs) = List.partition (fn ((vn,_),_) => vn mem vns) (Vartab.dest tmenv) val (fts,its) = (map (snd o snd) fnvs, map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t)) invs) val ctyenv = map (fn ((vn,vi),(s,ty)) => (certy (TVar((vn,vi),s)), certy ty)) (Vartab.dest tyenv) in (fts ~~ (replicate (length fts) ctxt), FWD (instantiate (ctyenv, its) cong)) end) handle MATCH => decomp_genreif da congs (t,ctxt))) end; (* looks for the atoms equation and instantiates it with the right number *) fun mk_decompatom eqs (t,ctxt) = let val tT = fastype_of t fun isat eq = let val rhs = eq |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd in exists_Const (fn (n,ty) => n="List.nth" andalso AList.defined Type.could_unify (!bds) (domain_type ty)) rhs andalso Type.could_unify (fastype_of rhs, tT) end fun get_nths t acc = case t of Const("List.nth",_)$vs$n => insert (fn ((a,_),(b,_)) => a aconv b) (t,(vs,n)) acc | t1$t2 => get_nths t1 (get_nths t2 acc) | Abs(_,_,t') => get_nths t' acc | _ => acc fun tryeqs [] = error "Can not find the atoms equation" | tryeqs (eq::eqs) = (( let val rhs = eq |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd val nths = get_nths rhs [] val (vss,ns) = fold_rev (fn (_,(vs,n)) => fn (vss,ns) => (insert (op aconv) vs vss, insert (op aconv) n ns)) nths ([],[]) val (vsns, ctxt') = Variable.variant_fixes (replicate (length vss) "vs") ctxt val (xns, ctxt'') = Variable.variant_fixes (replicate (length nths) "x") ctxt' val thy = ProofContext.theory_of ctxt'' val cert = cterm_of thy val certT = ctyp_of thy val vsns_map = vss ~~ vsns val xns_map = (fst (split_list nths)) ~~ xns val subst = map (fn (nt, xn) => (nt, Var ((xn,0), fastype_of nt))) xns_map val rhs_P = subst_free subst rhs val (tyenv, tmenv) = Pattern.match thy (rhs_P, t) (Envir.type_env (Envir.empty 0),Term.Vartab.empty) val sbst = Envir.subst_vars (tyenv, tmenv) val sbsT = Envir.typ_subst_TVars tyenv val subst_ty = map (fn (n,(s,t)) => (certT (TVar (n, s)), certT t)) (Vartab.dest tyenv) val tml = Vartab.dest tmenv val t's = map (fn xn => snd (valOf (AList.lookup (op =) tml (xn,0)))) xns (* FIXME : Express with sbst*) val subst_ns = map (fn (Const _ $ vs $ n, Var (xn0,T)) => (cert n, snd (valOf (AList.lookup (op =) tml xn0)) |> (index_of #> HOLogic.mk_nat #> cert))) subst val subst_vs = let fun ty (Const _ $ (vs as Var (vsn,lT)) $ n, Var (xn0,T)) = (certT T, certT (sbsT T)) fun h (Const _ $ (vs as Var (vsn,lT)) $ n, Var (xn0,T)) = let val cns = sbst (Const("List.list.Cons", T --> lT --> lT)) val lT' = sbsT lT val (bsT,asT) = the (AList.lookup Type.could_unify (!bds) lT) val vsn = valOf (AList.lookup (op =) vsns_map vs) val cvs = cert (fold_rev (fn x => fn xs => cns$x$xs) bsT (Free (vsn, lT'))) in (cert vs, cvs) end in map h subst end val cts = map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t)) (fold (AList.delete (fn (((a: string),_),(b,_)) => a = b)) (map (fn n => (n,0)) xns) tml) val substt = let val ih = Drule.cterm_rule (Thm.instantiate (subst_ty,[])) in map (fn (v,t) => (ih v, ih t)) (subst_ns@subst_vs@cts) end val th = (instantiate (subst_ty, substt) eq) RS sym in hd (Variable.export ctxt'' ctxt [th]) end) handle MATCH => tryeqs eqs) in ([], fn _ => tryeqs (filter isat eqs)) end; (* Generic reification procedure: *) (* creates all needed cong rules and then just uses the theorem synthesis *) fun mk_congs ctxt raw_eqs = let val fs = fold_rev (fn eq => insert (op =) (eq |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> fst |> strip_comb |> fst)) raw_eqs [] val tys = fold_rev (fn f => fn ts => (f |> fastype_of |> binder_types |> tl) union ts) fs [] val _ = bds := AList.make (fn _ => ([],[])) tys val (vs, ctxt') = Variable.variant_fixes (replicate (length tys) "vs") ctxt val thy = ProofContext.theory_of ctxt' val cert = cterm_of thy val vstys = map (fn (t,v) => (t,SOME (cert (Free(v,t))))) (tys ~~ vs) val is_Var = can dest_Var fun insteq eq vs = let val subst = map (fn (v as Var(n,t)) => (cert v, (valOf o valOf) (AList.lookup (op =) vstys t))) (filter is_Var vs) in Thm.instantiate ([],subst) eq end val eqs = map (fn eq => eq |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> fst |> strip_comb |> snd |> tl |> (insteq eq)) raw_eqs val (ps,congs) = split_list (map (mk_congeq ctxt' fs) eqs) in ps ~~ (Variable.export ctxt' ctxt congs) end fun partition P [] = ([],[]) | partition P (x::xs) = let val (yes,no) = partition P xs in if P x then (x::yes,no) else (yes, x::no) end fun rearrange congs = let fun P (_, th) = let val @{term "Trueprop"}$(Const ("op =",_) $l$_) = concl_of th in can dest_Var l end val (yes,no) = partition P congs in no @ yes end fun genreif ctxt raw_eqs t = let val congs = rearrange (mk_congs ctxt raw_eqs) val th = divide_and_conquer (decomp_genreif (mk_decompatom raw_eqs) congs) (t,ctxt) fun is_listVar (Var (_,t)) = can dest_listT t | is_listVar _ = false val vars = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd |> strip_comb |> snd |> filter is_listVar val cert = cterm_of (ProofContext.theory_of ctxt) val cvs = map (fn (v as Var(n,t)) => (cert v, the (AList.lookup Type.could_unify (!bds) t) |> snd |> HOLogic.mk_list (dest_listT t) |> cert)) vars val th' = instantiate ([], cvs) th val t' = (fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) th' val th'' = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, t'))) (fn _ => simp_tac (local_simpset_of ctxt) 1) val _ = bds := [] in FWD trans [th'',th'] end fun genreflect ctxt conv corr_thms raw_eqs t = let val reifth = genreif ctxt raw_eqs t fun trytrans [] = error "No suitable correctness theorem found" | trytrans (th::ths) = (FWD trans [reifth, th RS sym] handle THM _ => trytrans ths) val th = trytrans corr_thms val ft = (Thm.dest_arg1 o Thm.dest_arg o Thm.dest_arg o cprop_of) th val rth = conv ft in simplify (HOL_basic_ss addsimps raw_eqs addsimps [nth_Cons_0, nth_Cons_Suc]) (simplify (HOL_basic_ss addsimps [rth]) th) end fun genreify_tac ctxt eqs to i = (fn st => let val P = HOLogic.dest_Trueprop (List.nth (prems_of st, i - 1)) val t = (case to of NONE => P | SOME x => x) val th = (genreif ctxt eqs t) RS ssubst in rtac th i st end); (* Reflection calls reification and uses the correctness *) (* theorem assumed to be the dead of the list *) fun gen_reflection_tac ctxt conv corr_thms raw_eqs to i = (fn st => let val P = HOLogic.dest_Trueprop (nth (prems_of st) (i - 1)); val t = the_default P to; val th = genreflect ctxt conv corr_thms raw_eqs t RS ssubst; in (rtac th i THEN TRY(rtac TrueI i)) st end); fun reflection_tac ctxt = gen_reflection_tac ctxt Codegen.evaluation_conv; end