haftmann@29650: (*  Title:      HOL/Library/reflection.ML
haftmann@29650:     Author:     Amine Chaieb, TU Muenchen
wenzelm@20319: 
wenzelm@20319: A trial for automatical reification.
wenzelm@20319: *)
wenzelm@20319: 
wenzelm@29269: signature REFLECTION =
wenzelm@29269: sig
wenzelm@20319:   val genreify_tac: Proof.context -> thm list -> term option -> int -> tactic
chaieb@23648:   val reflection_tac: Proof.context -> thm list -> thm list -> term option -> int -> tactic
haftmann@21878:   val gen_reflection_tac: Proof.context -> (cterm -> thm)
chaieb@23648:     -> thm list -> thm list -> term option -> int -> tactic
wenzelm@20319: end;
wenzelm@20319: 
wenzelm@29269: structure Reflection : REFLECTION =
wenzelm@29269: struct
wenzelm@20319: 
haftmann@29650: val ext2 = @{thm ext2};
haftmann@29650: val nth_Cons_0 = @{thm nth_Cons_0};
haftmann@29650: val nth_Cons_Suc = @{thm nth_Cons_Suc};
wenzelm@21669: 
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: 
hoelzl@31386: fun mk_congeq ctxt fs th =
hoelzl@31386:   let
hoelzl@31386:    val (f as Const(fN,fT)) = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq
haftmann@29650:      |> fst |> strip_comb |> fst
chaieb@20374:    val thy = ProofContext.theory_of ctxt
chaieb@20374:    val cert = Thm.cterm_of thy
wenzelm@22568:    val (((_,_),[th']), ctxt') = Variable.import_thms true [th] ctxt
chaieb@20374:    val (lhs, rhs) = HOLogic.dest_eq (HOLogic.dest_Trueprop (Thm.prop_of th'))
hoelzl@31386:    fun add_fterms (t as t1 $ t2) =
wenzelm@29269:        if exists (fn f => Term.could_unify (t |> strip_comb |> fst, f)) fs then insert (op aconv) t
chaieb@20564:        else add_fterms t1 #> add_fterms t2
hoelzl@31386:      | add_fterms (t as Abs(xn,xT,t')) =
wenzelm@29273:        if exists_Const (fn (c, _) => c = fN) t then (fn _ => [t]) else (fn _ => [])
chaieb@20374:      | add_fterms _ = I
chaieb@20374:    val fterms = add_fterms rhs []
wenzelm@20797:    val (xs, ctxt'') = Variable.variant_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
hoelzl@31386: 		    (* 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)
hoelzl@31386: 
chaieb@20374:    fun mk_def (Abs(x,xT,t),v) = HOLogic.mk_Trueprop ((HOLogic.all_const xT)$ Abs(x,xT,HOLogic.mk_eq(v$(Bound 0), t)))
chaieb@20374:      | mk_def (t, v) = HOLogic.mk_Trueprop (HOLogic.mk_eq (v, t))
chaieb@23605:    fun tryext x = (x RS ext2 handle THM _ =>  x)
chaieb@20374:    val cong = (Goal.prove ctxt'' [] (map mk_def env)
chaieb@20374: 			  (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, replace_fterms rhs)))
hoelzl@31386: 			  (fn x => LocalDefs.unfold_tac (#context x) (map tryext (#prems x))
chaieb@20374: 							THEN rtac th' 1)) RS sym
hoelzl@31386: 
hoelzl@31386:    val (cong' :: vars') =
chaieb@20374:        Variable.export ctxt'' ctxt (cong :: map (Drule.mk_term o cert) vs)
chaieb@20374:    val vs' = map (fst o fst o Term.dest_Var o Thm.term_of o Drule.dest_term) vars'
hoelzl@31386: 
hoelzl@31386:   in  (vs', cong') end;
wenzelm@20319:  (* congs is a list of pairs (P,th) where th is a theorem for *)
wenzelm@20319:         (* [| f p1 = A1; ...; f pn = An|] ==> f (C p1 .. pn) = P *)
wenzelm@20319: val FWD = curry (op OF);
wenzelm@20319: 
wenzelm@20319: 
chaieb@20374: exception REIF of string;
chaieb@20374: 
hoelzl@29834: fun dest_listT (Type ("List.list", [T])) = T;
hoelzl@29834: 
hoelzl@31387: (* This modified version of divide_and_conquer allows the threading
hoelzl@31387:    of a state variable *)
hoelzl@31387: fun divide_and_conquer' decomp (s, x) =
hoelzl@31387:   let val (ys, recomb) = decomp (s, x)
hoelzl@31387:   in recomb (Library.foldl_map (divide_and_conquer' decomp) ys) end;
hoelzl@31387: 
hoelzl@31386: fun rearrange congs =
hoelzl@31386:   let
hoelzl@31386:     fun P (_, th) =
hoelzl@31386:       let val @{term "Trueprop"}$(Const ("op =",_) $l$_) = concl_of th
hoelzl@31386:       in can dest_Var l end
hoelzl@31386:     val (yes,no) = List.partition P congs
hoelzl@31386:   in no @ yes end
hoelzl@29834: 
hoelzl@29834: fun genreif ctxt raw_eqs t =
hoelzl@31386:   let
hoelzl@31387:     fun index_of bds t =
hoelzl@31386:       let
hoelzl@31386:         val tt = HOLogic.listT (fastype_of t)
hoelzl@31386:       in
hoelzl@31387:        (case AList.lookup Type.could_unify bds tt of
hoelzl@31386:           NONE => error "index_of : type not found in environements!"
hoelzl@31386:         | SOME (tbs,tats) =>
hoelzl@31386:           let
hoelzl@31386:             val i = find_index_eq t tats
hoelzl@31386:             val j = find_index_eq t tbs
hoelzl@31387:           in (if j = ~1 then
hoelzl@31387:               if i = ~1
hoelzl@31387:               then (AList.update Type.could_unify (tt,(tbs,tats@[t])) bds,
hoelzl@31386:                     length tbs + length tats)
hoelzl@31387:               else (bds, i) else (bds, j))
hoelzl@31386:           end)
hoelzl@31386:       end;
hoelzl@31386: 
hoelzl@31386:     (* Generic decomp for reification : matches the actual term with the
hoelzl@31386:        rhs of one cong rule. The result of the matching guides the
hoelzl@31386:        proof synthesis: The matches of the introduced Variables A1 .. An are
hoelzl@31386:        processed recursively
hoelzl@31386:        The rest is instantiated in the cong rule,i.e. no reification is needed *)
chaieb@20564: 
hoelzl@31386:     (* da is the decomposition for atoms, ie. it returns ([],g) where g
hoelzl@31386:        returns the right instance f (AtC n) = t , where AtC is the Atoms
hoelzl@31386:        constructor and n is the number of the atom corresponding to t *)
hoelzl@31387:     fun decomp_genreif da cgns (bds, (t,ctxt)) =
hoelzl@31386:       let
hoelzl@31386:         val thy = ProofContext.theory_of ctxt
chaieb@20374:         val cert = cterm_of thy
hoelzl@31387:         fun tryabsdecomp (bds, (s,ctxt)) =
hoelzl@31386:           (case s of
hoelzl@31386:              Abs(xn,xT,ta) => (
hoelzl@31386:                let
hoelzl@31386:                  val ([xn],ctxt') = Variable.variant_fixes ["x"] ctxt
hoelzl@31386:                  val (xn,ta) = variant_abs (xn,xT,ta)
hoelzl@31386:                  val x = Free(xn,xT)
hoelzl@31387:                  val bds = (case AList.lookup Type.could_unify bds (HOLogic.listT xT)
hoelzl@31386: 		          of NONE => error "tryabsdecomp: Type not found in the Environement"
hoelzl@31386:                            | SOME (bsT,atsT) =>
hoelzl@31387:                              (AList.update Type.could_unify (HOLogic.listT xT, ((x::bsT), atsT)) bds))
hoelzl@31387:                in ((bds, [(ta, ctxt')]),
hoelzl@31387:                   fn (bds, [th]) => (
hoelzl@31387:                     (let val (bsT,asT) = the(AList.lookup Type.could_unify bds (HOLogic.listT xT))
hoelzl@31387: 		     in AList.update Type.could_unify (HOLogic.listT xT,(tl bsT,asT)) bds
hoelzl@31387: 		     end),
hoelzl@31387:                     hd (Variable.export ctxt' ctxt [(forall_intr (cert x) th) COMP allI])))
hoelzl@31386:                end)
hoelzl@31387:            | _ => da (bds, (s,ctxt)))
hoelzl@31386:       in (case cgns of
hoelzl@31387:           [] => tryabsdecomp (bds, (t,ctxt))
hoelzl@31386:         | ((vns,cong)::congs) => ((let
hoelzl@31386:             val cert = cterm_of thy
hoelzl@31386:             val certy = ctyp_of thy
hoelzl@31386:             val (tyenv, tmenv) =
hoelzl@31386:               Pattern.match thy
hoelzl@31386:               ((fst o HOLogic.dest_eq o HOLogic.dest_Trueprop) (concl_of cong), t)
hoelzl@31386:               (Envir.type_env (Envir.empty 0), Vartab.empty)
hoelzl@31386:             val (fnvs,invs) = List.partition (fn ((vn,_),_) => vn mem vns) (Vartab.dest tmenv)
hoelzl@31386:             val (fts,its) =
hoelzl@31386: 	      (map (snd o snd) fnvs,
hoelzl@31386:                map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t)) invs)
hoelzl@31386: 	    val ctyenv = map (fn ((vn,vi),(s,ty)) => (certy (TVar((vn,vi),s)), certy ty)) (Vartab.dest tyenv)
hoelzl@31387:           in ((bds, fts ~~ (replicate (length fts) ctxt)),
hoelzl@31387:               Library.apsnd (FWD (instantiate (ctyenv, its) cong)))
hoelzl@31386:           end)
hoelzl@31387:         handle MATCH => decomp_genreif da congs (bds, (t,ctxt))))
hoelzl@31386:       end;
chaieb@23648: 
chaieb@23648:  (* looks for the atoms equation and instantiates it with the right number *)
hoelzl@31387:     fun mk_decompatom eqs (bds, (t,ctxt)) = ((bds, []), fn (bds, _) =>
hoelzl@31386:       let
hoelzl@31386:         val tT = fastype_of t
hoelzl@31386:         fun isat eq =
hoelzl@31386:           let
hoelzl@31386:             val rhs = eq |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
hoelzl@31386:           in exists_Const
hoelzl@31386: 	    (fn (n,ty) => n="List.nth"
hoelzl@31386:                           andalso
hoelzl@31387: 			  AList.defined Type.could_unify bds (domain_type ty)) rhs
hoelzl@31386:             andalso Type.could_unify (fastype_of rhs, tT)
hoelzl@31386:           end
hoelzl@31386: 
hoelzl@31386:         fun get_nths t acc =
hoelzl@31386:           case t of
hoelzl@31386:             Const("List.nth",_)$vs$n => insert (fn ((a,_),(b,_)) => a aconv b) (t,(vs,n)) acc
hoelzl@31386:           | t1$t2 => get_nths t1 (get_nths t2 acc)
hoelzl@31386:           | Abs(_,_,t') => get_nths t'  acc
hoelzl@31386:           | _ => acc
chaieb@23548: 
hoelzl@31386:         fun
hoelzl@31387:            tryeqs bds [] = error "Can not find the atoms equation"
hoelzl@31387:          | tryeqs bds (eq::eqs) = ((
hoelzl@31386:           let
hoelzl@31386:             val rhs = eq |> prop_of |> HOLogic.dest_Trueprop  |> HOLogic.dest_eq |> snd
hoelzl@31386:             val nths = get_nths rhs []
hoelzl@31386:             val (vss,ns) = fold_rev (fn (_,(vs,n)) => fn (vss,ns) =>
hoelzl@31386:                                       (insert (op aconv) vs vss, insert (op aconv) n ns)) nths ([],[])
hoelzl@31386:             val (vsns, ctxt') = Variable.variant_fixes (replicate (length vss) "vs") ctxt
hoelzl@31386:             val (xns, ctxt'') = Variable.variant_fixes (replicate (length nths) "x") ctxt'
hoelzl@31386:             val thy = ProofContext.theory_of ctxt''
hoelzl@31386:             val cert = cterm_of thy
hoelzl@31386:             val certT = ctyp_of thy
hoelzl@31386:             val vsns_map = vss ~~ vsns
hoelzl@31386:             val xns_map = (fst (split_list nths)) ~~ xns
hoelzl@31386:             val subst = map (fn (nt, xn) => (nt, Var ((xn,0), fastype_of nt))) xns_map
hoelzl@31386:             val rhs_P = subst_free subst rhs
hoelzl@31386:             val (tyenv, tmenv) = Pattern.match
hoelzl@31386:                               thy (rhs_P, t)
hoelzl@31386:                               (Envir.type_env (Envir.empty 0), Vartab.empty)
hoelzl@31386:             val sbst = Envir.subst_vars (tyenv, tmenv)
hoelzl@31386:             val sbsT = Envir.typ_subst_TVars tyenv
hoelzl@31386:             val subst_ty = map (fn (n,(s,t)) => (certT (TVar (n, s)), certT t))
hoelzl@31386:                                (Vartab.dest tyenv)
hoelzl@31386:             val tml = Vartab.dest tmenv
hoelzl@31386:             val t's = map (fn xn => snd (valOf (AList.lookup (op =) tml (xn,0)))) xns (* FIXME : Express with sbst*)
hoelzl@31387:             val (bds, subst_ns) = Library.foldl_map
hoelzl@31387:                 (fn (bds, (Const _ $ vs $ n, Var (xn0,T))) =>
hoelzl@31387:                   let
hoelzl@31387:                     val name = snd (valOf (AList.lookup (op =) tml xn0))
hoelzl@31387:                     val (bds, idx) = index_of bds name
hoelzl@31387:                   in (bds, (cert n, idx |> (HOLogic.mk_nat #> cert))) end) (bds, subst)
hoelzl@31386:             val subst_vs =
hoelzl@31386:               let
hoelzl@31386:                 fun ty (Const _ $ (vs as Var (vsn,lT)) $ n, Var (xn0,T)) = (certT T, certT (sbsT T))
hoelzl@31386:                 fun h (Const _ $ (vs as Var (vsn,lT)) $ n, Var (xn0,T)) =
hoelzl@31386:                   let
hoelzl@31386:                     val cns = sbst (Const("List.list.Cons", T --> lT --> lT))
hoelzl@31386:                     val lT' = sbsT lT
hoelzl@31387:                     val (bsT,asT) = the (AList.lookup Type.could_unify bds lT)
hoelzl@31386:                     val vsn = valOf (AList.lookup (op =) vsns_map vs)
hoelzl@31386:                     val cvs = cert (fold_rev (fn x => fn xs => cns$x$xs) bsT (Free (vsn, lT')))
hoelzl@31386:                   in (cert vs, cvs) end
hoelzl@31386:               in map h subst end
hoelzl@31386:             val cts = map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t))
hoelzl@31386:                           (fold (AList.delete (fn (((a: string),_),(b,_)) => a = b))
hoelzl@31386:                                 (map (fn n => (n,0)) xns) tml)
hoelzl@31386:             val substt =
hoelzl@31386:               let val ih = Drule.cterm_rule (Thm.instantiate (subst_ty,[]))
hoelzl@31386:               in map (fn (v,t) => (ih v, ih t)) (subst_ns@subst_vs@cts)  end
hoelzl@31386:             val th = (instantiate (subst_ty, substt)  eq) RS sym
hoelzl@31387:           in (bds, hd (Variable.export ctxt'' ctxt [th])) end)
hoelzl@31387:           handle MATCH => tryeqs bds eqs)
hoelzl@31387:       in tryeqs bds (filter isat eqs) end);
chaieb@20374: 
wenzelm@20319:   (* Generic reification procedure: *)
wenzelm@20319:   (* creates all needed cong rules and then just uses the theorem synthesis *)
chaieb@20564: 
hoelzl@31386:     fun mk_congs ctxt raw_eqs =
hoelzl@31386:       let
hoelzl@31386:         val fs = fold_rev (fn eq =>
hoelzl@31386:                            insert (op =) (eq |> prop_of |> HOLogic.dest_Trueprop
hoelzl@31386:                            |> HOLogic.dest_eq |> fst |> strip_comb
hoelzl@31386:                            |> fst)) raw_eqs []
hoelzl@31386:         val tys = fold_rev (fn f => fold (insert (op =)) (f |> fastype_of |> binder_types |> tl)
hoelzl@31386:                             ) fs []
hoelzl@31386:         val (vs, ctxt') = Variable.variant_fixes (replicate (length tys) "vs") ctxt
hoelzl@31386:         val thy = ProofContext.theory_of ctxt'
hoelzl@31386:         val cert = cterm_of thy
hoelzl@31386:         val vstys = map (fn (t,v) => (t,SOME (cert (Free(v,t)))))
hoelzl@31386:                     (tys ~~ vs)
hoelzl@31386:         val is_Var = can dest_Var
hoelzl@31386:         fun insteq eq vs =
hoelzl@31386:           let
hoelzl@31386:             val subst = map (fn (v as Var(n,t)) => (cert v, (valOf o valOf) (AList.lookup (op =) vstys t)))
hoelzl@31386:                         (filter is_Var vs)
hoelzl@31386:           in Thm.instantiate ([],subst) eq
hoelzl@31386:           end
chaieb@20564: 
hoelzl@31387:         val bds = AList.make (fn _ => ([],[])) tys
hoelzl@31386:         val eqs = map (fn eq => eq |> prop_of |> HOLogic.dest_Trueprop
hoelzl@31386:   	                           |> HOLogic.dest_eq |> fst |> strip_comb |> snd |> tl
hoelzl@31386:                                    |> (insteq eq)) raw_eqs
hoelzl@31386:         val (ps,congs) = split_list (map (mk_congeq ctxt' fs) eqs)
hoelzl@31387:       in (bds, ps ~~ (Variable.export ctxt' ctxt congs))
hoelzl@31386:       end
hoelzl@31386: 
hoelzl@31387:     val (bds, congs) = mk_congs ctxt raw_eqs
hoelzl@31387:     val congs = rearrange congs
hoelzl@31387:     val (bds, th) = divide_and_conquer' (decomp_genreif (mk_decompatom raw_eqs) congs) (bds, (t,ctxt))
hoelzl@31386:     fun is_listVar (Var (_,t)) = can dest_listT t
hoelzl@31386:          | is_listVar _ = false
hoelzl@31386:     val vars = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
hoelzl@31386: 	          |> strip_comb |> snd |> filter is_listVar
hoelzl@31386:     val cert = cterm_of (ProofContext.theory_of ctxt)
hoelzl@31386:     val cvs = map (fn (v as Var(n,t)) => (cert v,
hoelzl@31387:                   the (AList.lookup Type.could_unify bds t) |> snd |> HOLogic.mk_list (dest_listT t) |> cert)) vars
hoelzl@31386:     val th' = instantiate ([], cvs) th
hoelzl@31386:     val t' = (fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) th'
hoelzl@31386:     val th'' = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, t')))
hoelzl@31386: 	       (fn _ => simp_tac (local_simpset_of ctxt) 1)
hoelzl@31386:   in FWD trans [th'',th']
hoelzl@31386:   end
wenzelm@20319: 
chaieb@23648: 
chaieb@23648: fun genreflect ctxt conv corr_thms raw_eqs t =
hoelzl@31386:   let
hoelzl@31386:     val reifth = genreif ctxt raw_eqs t
hoelzl@31386:     fun trytrans [] = error "No suitable correctness theorem found"
hoelzl@31386:       | trytrans (th::ths) =
hoelzl@31386:            (FWD trans [reifth, th RS sym] handle THM _ => trytrans ths)
hoelzl@31386:     val th = trytrans corr_thms
hoelzl@31386:     val ft = (Thm.dest_arg1 o Thm.dest_arg o Thm.dest_arg o cprop_of) th
hoelzl@31386:     val rth = conv ft
hoelzl@31386:   in simplify (HOL_basic_ss addsimps raw_eqs addsimps [nth_Cons_0, nth_Cons_Suc])
hoelzl@31386:              (simplify (HOL_basic_ss addsimps [rth]) th)
hoelzl@31386:   end
wenzelm@20319: 
wenzelm@20319: fun genreify_tac ctxt eqs to i = (fn st =>
wenzelm@20319:   let
hoelzl@29805:     fun P () = HOLogic.dest_Trueprop (List.nth (prems_of st, i - 1))
hoelzl@29805:     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 *)
chaieb@23648: fun gen_reflection_tac ctxt conv corr_thms raw_eqs to i = (fn st =>
haftmann@21878:   let
haftmann@21878:     val P = HOLogic.dest_Trueprop (nth (prems_of st) (i - 1));
haftmann@21878:     val t = the_default P to;
chaieb@23648:     val th = genreflect ctxt conv corr_thms raw_eqs t
haftmann@21878:       RS ssubst;
nipkow@23791:   in (rtac th i THEN TRY(rtac TrueI i)) st end);
haftmann@21878: 
nipkow@23791: fun reflection_tac ctxt = gen_reflection_tac ctxt Codegen.evaluation_conv;
haftmann@30969:   (*FIXME why Codegen.evaluation_conv?  very specific...*)
haftmann@29650: 
wenzelm@20797: end