src/HOL/Library/reflection.ML
author wenzelm
Thu Feb 11 23:00:22 2010 +0100 (2010-02-11)
changeset 35115 446c5063e4fd
parent 32978 a473ba9a221d
child 35624 c4e29a0bb8c1
permissions -rw-r--r--
modernized translations;
formal markup of @{syntax_const} and @{const_syntax};
minor tuning;
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(*  Title:      HOL/Library/reflection.ML
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    Author:     Amine Chaieb, TU Muenchen
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A trial for automatical reification.
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*)
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signature REFLECTION =
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sig
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  val genreify_tac: Proof.context -> thm list -> term option -> int -> tactic
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  val reflection_tac: Proof.context -> thm list -> thm list -> term option -> int -> tactic
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  val gen_reflection_tac: Proof.context -> (cterm -> thm)
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    -> thm list -> thm list -> term option -> int -> tactic
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  val genreif : Proof.context -> thm list -> term -> thm
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end;
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structure Reflection : REFLECTION =
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struct
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val ext2 = @{thm ext2};
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val nth_Cons_0 = @{thm nth_Cons_0};
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val nth_Cons_Suc = @{thm nth_Cons_Suc};
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  (* Make a congruence rule out of a defining equation for the interpretation *)
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  (* th is one defining equation of f, i.e.
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     th is "f (Cp ?t1 ... ?tn) = P(f ?t1, .., f ?tn)" *)
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  (* Cp is a constructor pattern and P is a pattern *)
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  (* The result is:
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      [|?A1 = f ?t1 ; .. ; ?An= f ?tn |] ==> P (?A1, .., ?An) = f (Cp ?t1 .. ?tn) *)
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  (*  + the a list of names of the A1 .. An, Those are fresh in the ctxt*)
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fun mk_congeq ctxt fs th =
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  let
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   val (f as Const(fN,fT)) = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq
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     |> fst |> strip_comb |> fst
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   val thy = ProofContext.theory_of ctxt
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   val cert = Thm.cterm_of thy
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   val (((_,_),[th']), ctxt') = Variable.import true [th] ctxt
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   val (lhs, rhs) = HOLogic.dest_eq (HOLogic.dest_Trueprop (Thm.prop_of th'))
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   fun add_fterms (t as t1 $ t2) =
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       if exists (fn f => Term.could_unify (t |> strip_comb |> fst, f)) fs then insert (op aconv) t
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       else add_fterms t1 #> add_fterms t2
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     | add_fterms (t as Abs(xn,xT,t')) =
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       if exists_Const (fn (c, _) => c = fN) t then (fn _ => [t]) else (fn _ => [])
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     | add_fterms _ = I
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   val fterms = add_fterms rhs []
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   val (xs, ctxt'') = Variable.variant_fixes (replicate (length fterms) "x") ctxt'
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   val tys = map fastype_of fterms
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   val vs = map Free (xs ~~ tys)
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   val env = fterms ~~ vs
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                    (* FIXME!!!!*)
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   fun replace_fterms (t as t1 $ t2) =
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       (case AList.lookup (op aconv) env t of
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            SOME v => v
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          | NONE => replace_fterms t1 $ replace_fterms t2)
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     | replace_fterms t = (case AList.lookup (op aconv) env t of
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                               SOME v => v
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                             | NONE => t)
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   fun mk_def (Abs(x,xT,t),v) = HOLogic.mk_Trueprop ((HOLogic.all_const xT)$ Abs(x,xT,HOLogic.mk_eq(v$(Bound 0), t)))
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     | mk_def (t, v) = HOLogic.mk_Trueprop (HOLogic.mk_eq (v, t))
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   fun tryext x = (x RS ext2 handle THM _ =>  x)
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   val cong = (Goal.prove ctxt'' [] (map mk_def env)
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                          (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, replace_fterms rhs)))
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                          (fn x => LocalDefs.unfold_tac (#context x) (map tryext (#prems x))
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                                                        THEN rtac th' 1)) RS sym
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   val (cong' :: vars') =
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       Variable.export ctxt'' ctxt (cong :: map (Drule.mk_term o cert) vs)
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   val vs' = map (fst o fst o Term.dest_Var o Thm.term_of o Drule.dest_term) vars'
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  in  (vs', cong') end;
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 (* congs is a list of pairs (P,th) where th is a theorem for *)
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        (* [| f p1 = A1; ...; f pn = An|] ==> f (C p1 .. pn) = P *)
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val FWD = curry (op OF);
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exception REIF of string;
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fun dest_listT (Type (@{type_name "list"}, [T])) = T;
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fun rearrange congs =
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  let
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    fun P (_, th) =
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      let val @{term "Trueprop"}$(Const ("op =",_) $l$_) = concl_of th
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      in can dest_Var l end
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    val (yes,no) = List.partition P congs
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  in no @ yes end
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fun genreif ctxt raw_eqs t =
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  let
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    fun index_of t bds =
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      let
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        val tt = HOLogic.listT (fastype_of t)
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      in
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       (case AList.lookup Type.could_unify bds tt of
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          NONE => error "index_of : type not found in environements!"
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        | SOME (tbs,tats) =>
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          let
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            val i = find_index (fn t' => t' = t) tats
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            val j = find_index (fn t' => t' = t) tbs
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          in (if j = ~1 then
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              if i = ~1
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              then (length tbs + length tats,
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                    AList.update Type.could_unify (tt,(tbs,tats@[t])) bds)
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              else (i, bds) else (j, bds))
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          end)
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      end;
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    (* Generic decomp for reification : matches the actual term with the
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       rhs of one cong rule. The result of the matching guides the
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       proof synthesis: The matches of the introduced Variables A1 .. An are
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       processed recursively
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       The rest is instantiated in the cong rule,i.e. no reification is needed *)
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    (* da is the decomposition for atoms, ie. it returns ([],g) where g
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       returns the right instance f (AtC n) = t , where AtC is the Atoms
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       constructor and n is the number of the atom corresponding to t *)
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    fun decomp_genreif da cgns (t,ctxt) bds =
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      let
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        val thy = ProofContext.theory_of ctxt
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        val cert = cterm_of thy
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        fun tryabsdecomp (s,ctxt) bds =
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          (case s of
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             Abs(xn,xT,ta) => (
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               let
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                 val ([xn],ctxt') = Variable.variant_fixes ["x"] ctxt
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                 val (xn,ta) = variant_abs (xn,xT,ta)
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                 val x = Free(xn,xT)
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                 val bds = (case AList.lookup Type.could_unify bds (HOLogic.listT xT)
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                          of NONE => error "tryabsdecomp: Type not found in the Environement"
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                           | SOME (bsT,atsT) =>
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                             (AList.update Type.could_unify (HOLogic.listT xT, ((x::bsT), atsT)) bds))
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               in (([(ta, ctxt')],
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                    fn ([th], bds) =>
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                      (hd (Variable.export ctxt' ctxt [(forall_intr (cert x) th) COMP allI]),
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                       let val (bsT,asT) = the(AList.lookup Type.could_unify bds (HOLogic.listT xT))
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                       in AList.update Type.could_unify (HOLogic.listT xT,(tl bsT,asT)) bds
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                       end)),
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                   bds)
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               end)
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           | _ => da (s,ctxt) bds)
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      in (case cgns of
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          [] => tryabsdecomp (t,ctxt) bds
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        | ((vns,cong)::congs) => ((let
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            val cert = cterm_of thy
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            val certy = ctyp_of thy
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            val (tyenv, tmenv) =
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              Pattern.match thy
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                ((fst o HOLogic.dest_eq o HOLogic.dest_Trueprop) (concl_of cong), t)
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                (Vartab.empty, Vartab.empty)
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            val (fnvs,invs) = List.partition (fn ((vn,_),_) => vn mem vns) (Vartab.dest tmenv)
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            val (fts,its) =
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              (map (snd o snd) fnvs,
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               map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t)) invs)
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            val ctyenv = map (fn ((vn,vi),(s,ty)) => (certy (TVar((vn,vi),s)), certy ty)) (Vartab.dest tyenv)
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          in ((fts ~~ (replicate (length fts) ctxt),
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               Library.apfst (FWD (instantiate (ctyenv, its) cong))), bds)
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          end)
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        handle MATCH => decomp_genreif da congs (t,ctxt) bds))
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      end;
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 (* looks for the atoms equation and instantiates it with the right number *)
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    fun mk_decompatom eqs (t,ctxt) bds = (([], fn (_, bds) =>
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      let
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        val tT = fastype_of t
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        fun isat eq =
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          let
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            val rhs = eq |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
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          in exists_Const
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            (fn (n,ty) => n = @{const_name "List.nth"}
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                          andalso
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                          AList.defined Type.could_unify bds (domain_type ty)) rhs
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            andalso Type.could_unify (fastype_of rhs, tT)
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          end
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        fun get_nths t acc =
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          case t of
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            Const(@{const_name "List.nth"},_)$vs$n => insert (fn ((a,_),(b,_)) => a aconv b) (t,(vs,n)) acc
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          | t1$t2 => get_nths t1 (get_nths t2 acc)
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          | Abs(_,_,t') => get_nths t'  acc
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          | _ => acc
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        fun
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           tryeqs [] bds = error "Can not find the atoms equation"
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         | tryeqs (eq::eqs) bds = ((
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          let
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            val rhs = eq |> prop_of |> HOLogic.dest_Trueprop  |> HOLogic.dest_eq |> snd
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            val nths = get_nths rhs []
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            val (vss,ns) = fold_rev (fn (_,(vs,n)) => fn (vss,ns) =>
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                                      (insert (op aconv) vs vss, insert (op aconv) n ns)) nths ([],[])
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            val (vsns, ctxt') = Variable.variant_fixes (replicate (length vss) "vs") ctxt
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            val (xns, ctxt'') = Variable.variant_fixes (replicate (length nths) "x") ctxt'
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            val thy = ProofContext.theory_of ctxt''
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            val cert = cterm_of thy
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            val certT = ctyp_of thy
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            val vsns_map = vss ~~ vsns
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            val xns_map = (fst (split_list nths)) ~~ xns
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            val subst = map (fn (nt, xn) => (nt, Var ((xn,0), fastype_of nt))) xns_map
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            val rhs_P = subst_free subst rhs
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            val (tyenv, tmenv) = Pattern.match thy (rhs_P, t) (Vartab.empty, Vartab.empty)
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            val sbst = Envir.subst_term (tyenv, tmenv)
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            val sbsT = Envir.subst_type tyenv
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            val subst_ty = map (fn (n,(s,t)) => (certT (TVar (n, s)), certT t))
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                               (Vartab.dest tyenv)
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            val tml = Vartab.dest tmenv
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            val t's = map (fn xn => snd (the (AList.lookup (op =) tml (xn,0)))) xns (* FIXME : Express with sbst*)
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            val (subst_ns, bds) = fold_map
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                (fn (Const _ $ vs $ n, Var (xn0,T)) => fn bds =>
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                  let
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                    val name = snd (the (AList.lookup (op =) tml xn0))
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                    val (idx, bds) = index_of name bds
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                  in ((cert n, idx |> (HOLogic.mk_nat #> cert)), bds) end) subst bds
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            val subst_vs =
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              let
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                fun ty (Const _ $ (vs as Var (vsn,lT)) $ n, Var (xn0,T)) = (certT T, certT (sbsT T))
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                fun h (Const _ $ (vs as Var (vsn,lT)) $ n, Var (xn0,T)) =
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                  let
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                    val cns = sbst (Const(@{const_name "List.Cons"}, T --> lT --> lT))
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                    val lT' = sbsT lT
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                    val (bsT,asT) = the (AList.lookup Type.could_unify bds lT)
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                    val vsn = the (AList.lookup (op =) vsns_map vs)
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                    val cvs = cert (fold_rev (fn x => fn xs => cns$x$xs) bsT (Free (vsn, lT')))
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                  in (cert vs, cvs) end
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              in map h subst end
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            val cts = map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t))
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                          (fold (AList.delete (fn (((a: string),_),(b,_)) => a = b))
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                                (map (fn n => (n,0)) xns) tml)
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            val substt =
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              let val ih = Drule.cterm_rule (Thm.instantiate (subst_ty,[]))
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              in map (fn (v,t) => (ih v, ih t)) (subst_ns@subst_vs@cts)  end
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            val th = (instantiate (subst_ty, substt)  eq) RS sym
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          in (hd (Variable.export ctxt'' ctxt [th]), bds) end)
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          handle MATCH => tryeqs eqs bds)
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      in tryeqs (filter isat eqs) bds end), bds);
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  (* Generic reification procedure: *)
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  (* creates all needed cong rules and then just uses the theorem synthesis *)
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    fun mk_congs ctxt raw_eqs =
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      let
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        val fs = fold_rev (fn eq =>
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                           insert (op =) (eq |> prop_of |> HOLogic.dest_Trueprop
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                           |> HOLogic.dest_eq |> fst |> strip_comb
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                           |> fst)) raw_eqs []
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        val tys = fold_rev (fn f => fold (insert (op =)) (f |> fastype_of |> binder_types |> tl)
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                            ) fs []
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        val (vs, ctxt') = Variable.variant_fixes (replicate (length tys) "vs") ctxt
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        val thy = ProofContext.theory_of ctxt'
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        val cert = cterm_of thy
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        val vstys = map (fn (t,v) => (t,SOME (cert (Free(v,t)))))
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                    (tys ~~ vs)
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        val is_Var = can dest_Var
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        fun insteq eq vs =
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          let
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            val subst = map (fn (v as Var(n,t)) => (cert v, (the o the) (AList.lookup (op =) vstys t)))
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                        (filter is_Var vs)
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          in Thm.instantiate ([],subst) eq
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          end
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        val bds = AList.make (fn _ => ([],[])) tys
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        val eqs = map (fn eq => eq |> prop_of |> HOLogic.dest_Trueprop
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                                   |> HOLogic.dest_eq |> fst |> strip_comb |> snd |> tl
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                                   |> (insteq eq)) raw_eqs
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        val (ps,congs) = split_list (map (mk_congeq ctxt' fs) eqs)
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      in (ps ~~ (Variable.export ctxt' ctxt congs), bds)
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      end
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    val (congs, bds) = mk_congs ctxt raw_eqs
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    val congs = rearrange congs
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    val (th, bds) = divide_and_conquer' (decomp_genreif (mk_decompatom raw_eqs) congs) (t,ctxt) bds
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    fun is_listVar (Var (_,t)) = can dest_listT t
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         | is_listVar _ = false
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    val vars = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
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                  |> strip_comb |> snd |> filter is_listVar
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    val cert = cterm_of (ProofContext.theory_of ctxt)
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    val cvs = map (fn (v as Var(n,t)) => (cert v,
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                  the (AList.lookup Type.could_unify bds t) |> snd |> HOLogic.mk_list (dest_listT t) |> cert)) vars
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    val th' = instantiate ([], cvs) th
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    val t' = (fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) th'
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    val th'' = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, t')))
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               (fn _ => simp_tac (simpset_of ctxt) 1)
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  in FWD trans [th'',th']
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  end
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fun genreflect ctxt conv corr_thms raw_eqs t =
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  let
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    val reifth = genreif ctxt raw_eqs t
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    fun trytrans [] = error "No suitable correctness theorem found"
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      | trytrans (th::ths) =
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           (FWD trans [reifth, th RS sym] handle THM _ => trytrans ths)
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    val th = trytrans corr_thms
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    val ft = (Thm.dest_arg1 o Thm.dest_arg o Thm.dest_arg o cprop_of) th
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    val rth = conv ft
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  in simplify (HOL_basic_ss addsimps raw_eqs addsimps [nth_Cons_0, nth_Cons_Suc])
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             (simplify (HOL_basic_ss addsimps [rth]) th)
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  end
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fun genreify_tac ctxt eqs to i = (fn st =>
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  let
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    fun P () = HOLogic.dest_Trueprop (nth (prems_of st) (i - 1))
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    val t = (case to of NONE => P () | SOME x => x)
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    val th = (genreif ctxt eqs t) RS ssubst
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  in rtac th i st
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  end);
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    (* Reflection calls reification and uses the correctness *)
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        (* theorem assumed to be the dead of the list *)
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fun gen_reflection_tac ctxt conv corr_thms raw_eqs to i = (fn st =>
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  let
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    val P = HOLogic.dest_Trueprop (nth (prems_of st) (i - 1));
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    val t = the_default P to;
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    val th = genreflect ctxt conv corr_thms raw_eqs t
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      RS ssubst;
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  in (rtac th i THEN TRY(rtac TrueI i)) st end);
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fun reflection_tac ctxt = gen_reflection_tac ctxt Codegen.evaluation_conv;
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  (*FIXME why Codegen.evaluation_conv?  very specific...*)
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end