src/HOL/ex/reflection.ML
author haftmann
Mon Dec 18 08:21:34 2006 +0100 (2006-12-18)
changeset 21878 cfc07819bb47
parent 21757 093ca3efb132
child 22199 b617ddd200eb
permissions -rw-r--r--
added gen_reflection_tac
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(*
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    ID:         $Id$
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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 = 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 -> term option -> int -> tactic
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  val gen_reflection_tac: Proof.context -> (cterm -> thm)
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    -> thm list -> term option -> int -> tactic
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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 => (t |> strip_comb |> fst) aconv 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 (fN mem (term_consts 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 _ =>  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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 (* 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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(* 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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exception REIF of string;
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val bds = ref ([]: (typ * ((term list) * (term list))) list);
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fun index_of t = 
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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 (op =) (!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_eq t tats
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     val j = find_index_eq t tbs 
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    in (if j= ~1 then 
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	    if i= ~1 
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	    then (bds := AList.update (op =) (tt,(tbs,tats@[t])) (!bds) ; 
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		  length tbs + length tats) 
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	    else i else j)
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    end)
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 end;
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fun dest_listT (Type ("List.list", [T])) = T;
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fun decomp_genreif da cgns (t,ctxt) =
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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) = 
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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 _ = (case AList.lookup (op =) (!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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		    (bds := AList.update (op =) (HOLogic.listT xT, ((x::bsT), atsT)) (!bds)))
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      in ([(ta, ctxt')] , 
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	  fn [th] => ((let val (bsT,asT) = the(AList.lookup (op =) (!bds) (HOLogic.listT xT))
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		       in (bds := AList.update (op =) (HOLogic.listT xT,(tl bsT,asT)) (!bds))
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		       end) ; 
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		      hd (Variable.export ctxt' ctxt [(forall_intr (cert x) th) COMP allI])))
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	end)
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    | _ => da (s,ctxt))
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  in 
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  (case cgns of 
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    [] => tryabsdecomp (t,ctxt)
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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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        (Envir.type_env (Envir.empty 0),Term.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), FWD (instantiate (ctyenv, its) cong))
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    end)
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      handle MATCH => decomp_genreif da congs (t,ctxt)))
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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) =
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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="List.nth" 
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			andalso 
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			AList.defined (op =) (!bds) (domain_type ty)) rhs 
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	  andalso fastype_of rhs = tT
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   end
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  fun get_nth t = 
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   case t of
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     Const("List.nth",_)$vs$n => (t,vs,n)
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   | t1$t2 => (get_nth t1 handle REIF "get_nth" => get_nth t2)
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   | Abs(_,_,t') => get_nth t'
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   | _ => raise REIF "get_nth"
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  val ([xn,vsn],ctxt') = Variable.variant_fixes ["x","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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  fun tryeqs [] = raise REIF "Can not find the atoms equation"
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    | tryeqs (eq::eqs) = ((
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      let 
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       val rhs = eq |> prop_of |> HOLogic.dest_Trueprop 
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			  |> HOLogic.dest_eq |> snd
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       val (nt,vs,n) = get_nth rhs
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       val ntT = fastype_of nt
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       val ntlT = HOLogic.listT ntT
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       val (bsT,asT) = the (AList.lookup (op =) (!bds) ntlT) 
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       val x = Var ((xn,0),ntT)
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       val rhs_P = subst_free [(nt,x)] rhs
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       val (_, tmenv) = Pattern.match 
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			    thy (rhs_P, t)
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			    (Envir.type_env (Envir.empty 0),Term.Vartab.empty)
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       val tml = Vartab.dest tmenv
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       val SOME (_,t') = AList.lookup (op =) tml (xn,0)
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       val cvs = 
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	   cert (fold_rev (fn x => fn xs => Const("List.list.Cons", ntT --> ntlT --> ntlT)$x$xs)
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		       bsT (Free (vsn, ntlT)))
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       val cts = map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t)) 
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		     (AList.delete (op =) (xn,0) tml)
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       val th = (instantiate 
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		     ([],
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		      [(cert vs, cvs),(cert n, t' |> index_of |> IntInf.fromInt |> HOLogic.mk_nat |> cert)]
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		      @cts) eq) RS sym
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      in hd (Variable.export ctxt' ctxt [th])
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      end)
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	  handle MATCH => tryeqs eqs)
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 in ([], fn _ => tryeqs (filter isat eqs))
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 end;
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(* 
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fun mk_decompatom eqs (t,ctxt) =
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 let 
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  val tT = fastype_of t
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  val tlT = HOLogic.listT tT
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  val (bsT,asT) = (the (AList.lookup (op =) (!bds) tlT) 
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		   handle Option => error "mk_decompatom: Type not found in the env.")
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  fun isateq (_$_$(Const("List.nth",_)$vs$_)) = (fastype_of vs = tlT)
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    | isateq _ = false
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 in case List.find (isateq o HOLogic.dest_Trueprop o prop_of) eqs of
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     NONE => raise REIF "Can not find the atoms equation"
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   | SOME th =>
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     ([],
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      fn ths =>
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        let 
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         val ([x], ctxt') = Variable.variant_fixes ["vs"] ctxt
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	 val cert = cterm_of (ProofContext.theory_of ctxt')
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         val (Const("List.nth",_)$(vs as Var((vsn,vsi),_))$n) = 
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             (snd o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) th
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         val cvs = 
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	     cert (foldr (fn (x,xs) => Const("List.list.Cons", tT --> tlT --> tlT)$x$xs) 
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			 (Free(x,tlT)) bsT)
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	 val th' = (instantiate ([],
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				 [(cert vs, cvs), 
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				  (cert n, cert  (HOLogic.mk_nat(index_of t)))]) th)
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		       RS sym
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        in hd (Variable.export ctxt' ctxt [th']) end)
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 end;
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*)
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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 => fn ts => (f |> fastype_of |> binder_types |> split_last |> fst) 
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				    union ts) fs []
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  val _ = bds := AList.make (fn _ => ([],[])) tys
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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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  fun insteq eq ts = 
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   let val itms = map (fn t => t|> (AList.lookup (op =) vstys) |> the) ts
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   in instantiate' [] itms eq
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   end
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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 |> fst |> fastype_of 
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			     |> binder_types |> split_last |> fst 
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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)
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 end;
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fun genreif ctxt raw_eqs t =
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 let 
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  val _ = bds := []
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  val congs = mk_congs ctxt raw_eqs
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  val th = divide_and_conquer (decomp_genreif (mk_decompatom raw_eqs) congs) (t,ctxt)
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  val tys = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
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	       |> strip_comb |> fst |> fastype_of |> strip_type |> fst
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	       |> split_last |> fst
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  val cert = cterm_of (ProofContext.theory_of ctxt)
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  val cvs = map (fn t => t |> (AList.lookup (op =) (!bds)) |> the |> snd 
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			   |> HOLogic.mk_list (dest_listT t) |> cert |> SOME)
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		tys
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  val th' = (instantiate' [] cvs (th RS sym)) RS sym
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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 1)
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  val _ = bds := []
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 in FWD trans [th'',th']
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 end;
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fun genreflect ctxt conv corr_thm raw_eqs t =
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    let val th = FWD trans [genreif ctxt raw_eqs t, corr_thm RS sym]
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        val ft = (snd o Thm.dest_comb o snd o Thm.dest_comb o snd o Thm.dest_comb 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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    val P = HOLogic.dest_Trueprop (List.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_thm :: 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_thm raw_eqs t
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      RS ssubst;
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  in rtac th i st end);
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fun reflection_tac ctxt = gen_reflection_tac ctxt NBE.normalization_conv;
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end