(*  Title:      HOL/Library/reflection.ML

     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

   val genreif : Proof.context -> thm list -> term -> thm

 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 = Proof_Context.theory_of ctxt

    val cert = Thm.cterm_of thy

    val (((_,_),[th']), ctxt') = Variable.import 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 => Term.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 exists_Const (fn (c, _) => c = fN) 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 => Local_Defs.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);

 exception REIF of string;

 fun dest_listT (Type (@{type_name "list"}, [T])) = T;

 fun rearrange congs =

   let

     fun P (_, th) =

       let val @{term "Trueprop"}$(Const (@{const_name HOL.eq},_) $l$_) = concl_of th

       in can dest_Var l end

     val (yes,no) = List.partition P congs

   in no @ yes end

 fun genreif ctxt raw_eqs t =

   let

     fun index_of t bds =

       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 (fn t' => t' = t) tats

             val j = find_index (fn t' => t' = t) tbs

           in (if j = ~1 then

               if i = ~1

               then (length tbs + length tats,

                     AList.update Type.could_unify (tt,(tbs,tats@[t])) bds)

               else (i, bds) else (j, bds))

           end)

       end;

     (* 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 *)

     (* 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 *)

     fun decomp_genreif da cgns (t,ctxt) bds =

       let

         val thy = Proof_Context.theory_of ctxt

         val cert = cterm_of thy

         fun tryabsdecomp (s,ctxt) bds =

           (case s of

              Abs(xn,xT,ta) => (

                let

                  val ([xn],ctxt') = Variable.variant_fixes ["x"] ctxt

                  val (xn,ta) = Syntax_Trans.variant_abs (xn,xT,ta)  (* FIXME !? *)

                  val x = Free(xn,xT)

                  val bds = (case AList.lookup Type.could_unify bds (HOLogic.listT xT)

                           of NONE => error "tryabsdecomp: Type not found in the Environement"

                            | SOME (bsT,atsT) =>

                              (AList.update Type.could_unify (HOLogic.listT xT, ((x::bsT), atsT)) bds))

                in (([(ta, ctxt')],

                     fn ([th], bds) =>

                       (hd (Variable.export ctxt' ctxt [(Thm.forall_intr (cert x) th) COMP allI]),

                        let val (bsT,asT) = the(AList.lookup Type.could_unify bds (HOLogic.listT xT))

                        in AList.update Type.could_unify (HOLogic.listT xT,(tl bsT,asT)) bds

                        end)),

                    bds)

                end)

            | _ => da (s,ctxt) bds)

       in

         (case cgns of

           [] => tryabsdecomp (t,ctxt) bds

         | ((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)

                   (Vartab.empty, Vartab.empty)

               val (fnvs,invs) = List.partition (fn ((vn,_),_) => member (op =) vns vn) (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),

                  Library.apfst (FWD (Drule.instantiate_normalize (ctyenv, its) cong))), bds)

             end handle Pattern.MATCH => decomp_genreif da congs (t,ctxt) bds))

       end;

  (* looks for the atoms equation and instantiates it with the right number *)

     fun mk_decompatom eqs (t,ctxt) bds = (([], fn (_, bds) =>

       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 = @{const_name "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(@{const_name "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 [] bds = error "Can not find the atoms equation"

          | tryeqs (eq::eqs) bds = ((

           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 = Proof_Context.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) (Vartab.empty, Vartab.empty)

             val sbst = Envir.subst_term (tyenv, tmenv)

             val sbsT = Envir.subst_type 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 (the (AList.lookup (op =) tml (xn,0)))) xns (* FIXME : Express with sbst*)

             val (subst_ns, bds) = fold_map

                 (fn (Const _ $ vs $ n, Var (xn0,T)) => fn bds =>

                   let

                     val name = snd (the (AList.lookup (op =) tml xn0))

                     val (idx, bds) = index_of name bds

                   in ((cert n, idx |> (HOLogic.mk_nat #> cert)), bds) end) subst bds

             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(@{const_name "List.Cons"}, T --> lT --> lT))

                     val lT' = sbsT lT

                     val (bsT,asT) = the (AList.lookup Type.could_unify bds lT)

                     val vsn = the (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 = (Drule.instantiate_normalize (subst_ty, substt)  eq) RS sym

           in (hd (Variable.export ctxt'' ctxt [th]), bds) end)

           handle Pattern.MATCH => tryeqs eqs bds)

       in tryeqs (filter isat eqs) bds end), bds);

   (* 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 => fold (insert (op =)) (f |> fastype_of |> binder_types |> tl)

                             ) fs []

         val (vs, ctxt') = Variable.variant_fixes (replicate (length tys) "vs") ctxt

         val thy = Proof_Context.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, (the o the) (AList.lookup (op =) vstys t)))

                         (filter is_Var vs)

           in Thm.instantiate ([],subst) eq

           end

         val bds = AList.make (fn _ => ([],[])) tys

         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), bds)

       end

     val (congs, bds) = mk_congs ctxt raw_eqs

     val congs = rearrange congs

     val (th, bds) = divide_and_conquer' (decomp_genreif (mk_decompatom raw_eqs) congs) (t,ctxt) bds

     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 (Proof_Context.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' = Drule.instantiate_normalize ([], 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 (simpset_of ctxt) 1)

   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 = SUBGOAL (fn (goal, i) =>

   let

     val t = (case to of NONE => HOLogic.dest_Trueprop goal | SOME x => x)

     val th = genreif ctxt eqs t RS ssubst

   in rtac th i end);

     (* Reflection calls reification and uses the correctness *)

         (* theorem assumed to be the head of the list *)

 fun gen_reflection_tac ctxt conv corr_thms raw_eqs to = SUBGOAL (fn (goal, i) =>

   let

     val t = (case to of NONE => HOLogic.dest_Trueprop goal | SOME x => x)

     val th = genreflect ctxt conv corr_thms raw_eqs t RS ssubst

   in rtac th i THEN TRY (rtac TrueI i) end);  (* FIXME THEN_ALL_NEW !? *)

 fun reflection_tac ctxt = gen_reflection_tac ctxt (Codegen.evaluation_conv ctxt);

   (*FIXME why Codegen.evaluation_conv?  very specific...*)

 end