wenzelm@20319: (* wenzelm@20319: ID: $Id$ wenzelm@20319: Author: Amine Chaieb, TU Muenchen wenzelm@20319: wenzelm@20319: A trial for automatical reification. wenzelm@20319: *) wenzelm@20319: wenzelm@20319: signature REFLECTION = sig wenzelm@20319: val genreify_tac: Proof.context -> thm list -> term option -> int -> tactic wenzelm@20319: val reflection_tac: Proof.context -> thm list -> term option -> int -> tactic wenzelm@20319: end; wenzelm@20319: wenzelm@20319: structure Reflection : REFLECTION wenzelm@20319: = struct wenzelm@20319: chaieb@20374: val ext2 = thm "ext2"; chaieb@20374: (* Make a congruence rule out of a defining equation for the interpretation *) chaieb@20374: (* th is one defining equation of f, i.e. chaieb@20374: th is "f (Cp ?t1 ... ?tn) = P(f ?t1, .., f ?tn)" *) chaieb@20374: (* Cp is a constructor pattern and P is a pattern *) chaieb@20374: chaieb@20374: (* The result is: chaieb@20374: [|?A1 = f ?t1 ; .. ; ?An= f ?tn |] ==> P (?A1, .., ?An) = f (Cp ?t1 .. ?tn) *) chaieb@20374: (* + the a list of names of the A1 .. An, Those are fresh in the ctxt*) chaieb@20374: chaieb@20374: chaieb@20374: fun mk_congeq ctxt fs th = chaieb@20374: let chaieb@20564: val (f as Const(fN,fT)) = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq chaieb@20564: |> fst |> strip_comb |> fst chaieb@20374: val thy = ProofContext.theory_of ctxt chaieb@20374: val cert = Thm.cterm_of thy chaieb@20374: val (((_,_),[th']), ctxt') = Variable.import true [th] ctxt chaieb@20374: val (lhs, rhs) = HOLogic.dest_eq (HOLogic.dest_Trueprop (Thm.prop_of th')) chaieb@20564: fun add_fterms (t as t1 $ t2) = chaieb@20564: if exists (fn f => (t |> strip_comb |> fst) aconv f) fs then insert (op aconv) t chaieb@20564: else add_fterms t1 #> add_fterms t2 chaieb@20374: | add_fterms (t as Abs(xn,xT,t')) = chaieb@20564: if (fN mem (term_consts t)) then (fn _ => [t]) else (fn _ => []) chaieb@20374: | add_fterms _ = I chaieb@20374: val fterms = add_fterms rhs [] chaieb@20374: val (xs, ctxt'') = Variable.invent_fixes (replicate (length fterms) "x") ctxt' chaieb@20374: val tys = map fastype_of fterms chaieb@20374: val vs = map Free (xs ~~ tys) chaieb@20374: val env = fterms ~~ vs chaieb@20564: (* FIXME!!!!*) chaieb@20564: fun replace_fterms (t as t1 $ t2) = chaieb@20374: (case AList.lookup (op aconv) env t of chaieb@20374: SOME v => v chaieb@20564: | NONE => replace_fterms t1 $ replace_fterms t2) chaieb@20374: | replace_fterms t = (case AList.lookup (op aconv) env t of chaieb@20374: SOME v => v chaieb@20374: | NONE => t) chaieb@20374: chaieb@20374: 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))) chaieb@20374: | mk_def (t, v) = HOLogic.mk_Trueprop (HOLogic.mk_eq (v, t)) chaieb@20374: fun tryext x = (x RS ext2 handle _ => x) chaieb@20374: val cong = (Goal.prove ctxt'' [] (map mk_def env) chaieb@20374: (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, replace_fterms rhs))) chaieb@20374: (fn x => LocalDefs.unfold_tac (#context x) (map tryext (#prems x)) chaieb@20374: THEN rtac th' 1)) RS sym chaieb@20374: chaieb@20374: val (cong' :: vars') = chaieb@20374: Variable.export ctxt'' ctxt (cong :: map (Drule.mk_term o cert) vs) chaieb@20374: val vs' = map (fst o fst o Term.dest_Var o Thm.term_of o Drule.dest_term) vars' chaieb@20374: chaieb@20374: in (vs', cong') end; wenzelm@20319: (* congs is a list of pairs (P,th) where th is a theorem for *) wenzelm@20319: (* [| f p1 = A1; ...; f pn = An|] ==> f (C p1 .. pn) = P *) wenzelm@20319: val FWD = curry (op OF); wenzelm@20319: wenzelm@20319: (* da is the decomposition for atoms, ie. it returns ([],g) where g wenzelm@20319: returns the right instance f (AtC n) = t , where AtC is the Atoms wenzelm@20319: constructor and n is the number of the atom corresponding to t *) wenzelm@20319: wenzelm@20319: (* Generic decomp for reification : matches the actual term with the wenzelm@20319: rhs of one cong rule. The result of the matching guides the wenzelm@20319: proof synthesis: The matches of the introduced Variables A1 .. An are wenzelm@20319: processed recursively wenzelm@20319: The rest is instantiated in the cong rule,i.e. no reification is needed *) wenzelm@20319: chaieb@20374: exception REIF of string; chaieb@20374: chaieb@20564: val bds = ref ([]: (typ * ((term list) * (term list))) list); chaieb@20564: chaieb@20564: fun index_of t = chaieb@20564: let chaieb@20564: val tt = HOLogic.listT (fastype_of t) chaieb@20564: in chaieb@20564: (case AList.lookup (op =) (!bds) tt of chaieb@20564: NONE => error "index_of : type not found in environements!" chaieb@20564: | SOME (tbs,tats) => chaieb@20564: let chaieb@20564: val i = find_index_eq t tats chaieb@20564: val j = find_index_eq t tbs chaieb@20564: in (if j= ~1 then chaieb@20564: if i= ~1 chaieb@20564: then (bds := AList.update (op =) (tt,(tbs,tats@[t])) (!bds) ; chaieb@20564: length tbs + length tats) chaieb@20564: else i else j) chaieb@20564: end) chaieb@20564: end; chaieb@20564: chaieb@20564: fun dest_listT (Type ("List.list", [T])) = T; chaieb@20374: chaieb@20374: fun decomp_genreif da cgns (t,ctxt) = chaieb@20564: let chaieb@20564: val thy = ProofContext.theory_of ctxt chaieb@20564: val cert = cterm_of thy chaieb@20564: fun tryabsdecomp (s,ctxt) = chaieb@20564: (case s of chaieb@20564: Abs(xn,xT,ta) => chaieb@20564: (let chaieb@20564: val ([xn],ctxt') = Variable.invent_fixes ["x"] ctxt chaieb@20564: val (xn,ta) = variant_abs (xn,xT,ta) chaieb@20564: val x = Free(xn,xT) chaieb@20564: val _ = (case AList.lookup (op =) (!bds) (HOLogic.listT xT) chaieb@20564: of NONE => error "tryabsdecomp: Type not found in the Environement" chaieb@20564: | SOME (bsT,atsT) => chaieb@20564: (bds := AList.update (op =) (HOLogic.listT xT, ((x::bsT), atsT)) (!bds))) chaieb@20564: in ([(ta, ctxt')] , chaieb@20564: fn [th] => ((let val (bsT,asT) = the(AList.lookup (op =) (!bds) (HOLogic.listT xT)) chaieb@20564: in (bds := AList.update (op =) (HOLogic.listT xT,(tl bsT,asT)) (!bds)) chaieb@20564: end) ; chaieb@20564: hd (Variable.export ctxt' ctxt [(forall_intr (cert x) th) COMP allI]))) chaieb@20374: end) chaieb@20374: | _ => da (s,ctxt)) chaieb@20374: in chaieb@20374: (case cgns of chaieb@20374: [] => tryabsdecomp (t,ctxt) chaieb@20374: | ((vns,cong)::congs) => ((let chaieb@20374: val cert = cterm_of thy chaieb@20564: val certy = ctyp_of thy chaieb@20564: val (tyenv, tmenv) = wenzelm@20319: Pattern.match thy wenzelm@20319: ((fst o HOLogic.dest_eq o HOLogic.dest_Trueprop) (concl_of cong), t) wenzelm@20319: (Envir.type_env (Envir.empty 0),Term.Vartab.empty) wenzelm@20319: val (fnvs,invs) = List.partition (fn ((vn,_),_) => vn mem vns) (Vartab.dest tmenv) chaieb@20564: val (fts,its) = chaieb@20564: (map (snd o snd) fnvs, chaieb@20564: map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t)) invs) chaieb@20564: val ctyenv = map (fn ((vn,vi),(s,ty)) => (certy (TVar((vn,vi),s)), certy ty)) (Vartab.dest tyenv) chaieb@20564: in (fts ~~ (replicate (length fts) ctxt), FWD (instantiate (ctyenv, its) cong)) wenzelm@20319: end) chaieb@20564: handle MATCH => decomp_genreif da congs (t,ctxt))) chaieb@20564: end; wenzelm@20319: (* looks for the atoms equation and instantiates it with the right number *) chaieb@20374: chaieb@20374: fun mk_decompatom eqs (t,ctxt) = chaieb@20564: let chaieb@20564: val tT = fastype_of t chaieb@20564: fun isat eq = chaieb@20564: let chaieb@20564: val rhs = eq |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd chaieb@20564: in exists_Const chaieb@20564: (fn (n,ty) => n="List.nth" chaieb@20564: andalso chaieb@20564: AList.defined (op =) (!bds) (domain_type ty)) rhs chaieb@20564: andalso fastype_of rhs = tT chaieb@20564: end chaieb@20564: fun get_nth t = chaieb@20564: case t of chaieb@20564: Const("List.nth",_)$vs$n => (t,vs,n) chaieb@20564: | t1$t2 => (get_nth t1 handle REIF "get_nth" => get_nth t2) chaieb@20564: | Abs(_,_,t') => get_nth t' chaieb@20564: | _ => raise REIF "get_nth" chaieb@20564: val ([xn,vsn],ctxt') = Variable.invent_fixes ["x","vs"] ctxt chaieb@20564: val thy = ProofContext.theory_of ctxt' chaieb@20564: val cert = cterm_of thy chaieb@20564: fun tryeqs [] = raise REIF "Can not find the atoms equation" chaieb@20564: | tryeqs (eq::eqs) = (( chaieb@20564: let chaieb@20564: val rhs = eq |> prop_of |> HOLogic.dest_Trueprop chaieb@20564: |> HOLogic.dest_eq |> snd chaieb@20564: val (nt,vs,n) = get_nth rhs chaieb@20564: val ntT = fastype_of nt chaieb@20564: val ntlT = HOLogic.listT ntT chaieb@20564: val (bsT,asT) = the (AList.lookup (op =) (!bds) ntlT) chaieb@20564: val x = Var ((xn,0),ntT) chaieb@20564: val rhs_P = subst_free [(nt,x)] rhs chaieb@20564: val (_, tmenv) = Pattern.match chaieb@20564: thy (rhs_P, t) chaieb@20564: (Envir.type_env (Envir.empty 0),Term.Vartab.empty) chaieb@20564: val tml = Vartab.dest tmenv chaieb@20564: val SOME (_,t') = AList.lookup (op =) tml (xn,0) chaieb@20564: val cvs = chaieb@20564: cert (foldr (fn (x,xs) => Const("List.list.Cons", ntT --> ntlT --> ntlT)$x$xs) chaieb@20564: (Free(vsn,ntlT)) bsT) chaieb@20564: val cts = map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t)) chaieb@20564: (AList.delete (op =) (xn,0) tml) chaieb@20564: val th = (instantiate chaieb@20564: ([], chaieb@20564: [(cert vs, cvs),(cert n, t' |> index_of |> HOLogic.mk_nat |> cert)] chaieb@20564: @cts) eq) RS sym chaieb@20564: in hd (Variable.export ctxt' ctxt [th]) chaieb@20564: end) chaieb@20564: handle MATCH => tryeqs eqs) chaieb@20564: in ([], fn _ => tryeqs (filter isat eqs)) chaieb@20564: end; chaieb@20374: chaieb@20564: (* chaieb@20564: fun mk_decompatom eqs (t,ctxt) = chaieb@20564: let chaieb@20564: val tT = fastype_of t chaieb@20564: val tlT = HOLogic.listT tT chaieb@20564: val (bsT,asT) = (the (AList.lookup (op =) (!bds) tlT) chaieb@20564: handle Option => error "mk_decompatom: Type not found in the env.") chaieb@20564: fun isateq (_$_$(Const("List.nth",_)$vs$_)) = (fastype_of vs = tlT) chaieb@20564: | isateq _ = false chaieb@20564: in case List.find (isateq o HOLogic.dest_Trueprop o prop_of) eqs of chaieb@20564: NONE => raise REIF "Can not find the atoms equation" chaieb@20564: | SOME th => chaieb@20564: ([], chaieb@20564: fn ths => chaieb@20564: let chaieb@20564: val ([x], ctxt') = Variable.invent_fixes ["vs"] ctxt chaieb@20564: val cert = cterm_of (ProofContext.theory_of ctxt') chaieb@20564: val (Const("List.nth",_)$(vs as Var((vsn,vsi),_))$n) = chaieb@20564: (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) th chaieb@20564: val cvs = chaieb@20564: cert (foldr (fn (x,xs) => Const("List.list.Cons", tT --> tlT --> tlT)$x$xs) chaieb@20564: (Free(x,tlT)) bsT) chaieb@20564: val th' = (instantiate ([], chaieb@20564: [(cert vs, cvs), chaieb@20564: (cert n, cert (HOLogic.mk_nat(index_of t)))]) th) chaieb@20564: RS sym chaieb@20564: in hd (Variable.export ctxt' ctxt [th']) end) chaieb@20564: end; chaieb@20374: *) wenzelm@20319: (* Generic reification procedure: *) wenzelm@20319: (* creates all needed cong rules and then just uses the theorem synthesis *) chaieb@20564: chaieb@20564: fun mk_congs ctxt raw_eqs = chaieb@20564: let chaieb@20564: val fs = foldr (fn (eq,fns) => chaieb@20564: (eq |> prop_of |> HOLogic.dest_Trueprop chaieb@20564: |> HOLogic.dest_eq |> fst |> strip_comb chaieb@20564: |> fst) ins fns) [] raw_eqs chaieb@20564: val tys = foldr (fn (f,ts) => (f |> fastype_of |> binder_types |> split_last |> fst) chaieb@20564: union ts) [] fs chaieb@20564: val _ = bds := AList.make (fn _ => ([],[])) tys chaieb@20564: val (vs, ctxt') = Variable.invent_fixes (replicate (length tys) "vs") ctxt chaieb@20564: val thy = ProofContext.theory_of ctxt' chaieb@20564: val cert = cterm_of thy chaieb@20564: val vstys = map (fn (t,v) => (t,SOME (cert (Free(v,t))))) chaieb@20564: (tys ~~ vs) chaieb@20564: fun insteq eq ts = chaieb@20564: let val itms = map (fn t => t|> (AList.lookup (op =) vstys) |> the) ts chaieb@20564: in instantiate' [] itms eq chaieb@20564: end chaieb@20564: val eqs = map (fn eq => eq |> prop_of |> HOLogic.dest_Trueprop chaieb@20564: |> HOLogic.dest_eq |> fst |> strip_comb |> fst |> fastype_of chaieb@20564: |> binder_types |> split_last |> fst chaieb@20564: |> (insteq eq)) raw_eqs chaieb@20564: val (ps,congs) = split_list (map (mk_congeq ctxt' fs) eqs) chaieb@20564: in ps ~~ (Variable.export ctxt' ctxt congs) chaieb@20564: end; chaieb@20564: wenzelm@20319: fun genreif ctxt raw_eqs t = chaieb@20564: let chaieb@20564: val _ = bds := [] chaieb@20564: val congs = mk_congs ctxt raw_eqs chaieb@20564: val th = divide_and_conquer (decomp_genreif (mk_decompatom raw_eqs) congs) (t,ctxt) chaieb@20564: val tys = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd chaieb@20564: |> strip_comb |> fst |> fastype_of |> strip_type |> fst chaieb@20564: |> split_last |> fst chaieb@20564: val cert = cterm_of (ProofContext.theory_of ctxt) chaieb@20564: val cvs = map (fn t => t |> (AList.lookup (op =) (!bds)) |> the |> snd chaieb@20564: |> HOLogic.mk_list I (dest_listT t) |> cert |> SOME) chaieb@20564: tys chaieb@20564: val th' = (instantiate' [] cvs (th RS sym)) RS sym chaieb@20564: val t' = (fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) th' chaieb@20564: val th'' = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, t'))) chaieb@20564: (fn _ => Simp_tac 1) chaieb@20564: val _ = bds := [] chaieb@20564: in FWD trans [th'',th'] chaieb@20564: end; wenzelm@20319: wenzelm@20319: fun genreflect ctxt corr_thm raw_eqs t = wenzelm@20319: let val th = FWD trans [genreif ctxt raw_eqs t, corr_thm RS sym] wenzelm@20319: val ft = (snd o Thm.dest_comb o snd o Thm.dest_comb o snd o Thm.dest_comb o cprop_of) th wenzelm@20319: val rth = normalization_conv ft wenzelm@20319: in simplify (HOL_basic_ss addsimps raw_eqs addsimps [nth_Cons_0, nth_Cons_Suc]) wenzelm@20319: (simplify (HOL_basic_ss addsimps [rth]) th) wenzelm@20319: end wenzelm@20319: wenzelm@20319: fun genreify_tac ctxt eqs to i = (fn st => wenzelm@20319: let wenzelm@20319: val P = HOLogic.dest_Trueprop (List.nth (prems_of st, i - 1)) wenzelm@20319: val t = (case to of NONE => P | SOME x => x) wenzelm@20319: val th = (genreif ctxt eqs t) RS ssubst wenzelm@20319: in rtac th i st wenzelm@20319: end); wenzelm@20319: wenzelm@20319: (* Reflection calls reification and uses the correctness *) wenzelm@20319: (* theorem assumed to be the dead of the list *) wenzelm@20319: fun reflection_tac ctxt (corr_thm::raw_eqs) to i = wenzelm@20319: (fn st => wenzelm@20319: let val P = (HOLogic.dest_Trueprop (List.nth (prems_of st, i - 1))) wenzelm@20319: val t = (case to of NONE => P | SOME x => x) wenzelm@20319: val th = (genreflect ctxt corr_thm raw_eqs t) RS ssubst wenzelm@20319: in rtac th i st end); wenzelm@20319: chaieb@20564: end