src/HOL/Library/reflection.ML
author haftmann
Fri Jul 10 07:59:27 2009 +0200 (2009-07-10)
changeset 31986 a68f88d264f7
parent 31810 a6b800855cdd
child 32032 a6a6e8031c14
permissions -rw-r--r--
dropped find_index_eq
     1 (*  Title:      HOL/Library/reflection.ML
     2     Author:     Amine Chaieb, TU Muenchen
     3 
     4 A trial for automatical reification.
     5 *)
     6 
     7 signature REFLECTION =
     8 sig
     9   val genreify_tac: Proof.context -> thm list -> term option -> int -> tactic
    10   val reflection_tac: Proof.context -> thm list -> thm list -> term option -> int -> tactic
    11   val gen_reflection_tac: Proof.context -> (cterm -> thm)
    12     -> thm list -> thm list -> term option -> int -> tactic
    13   val genreif : Proof.context -> thm list -> term -> thm
    14 end;
    15 
    16 structure Reflection : REFLECTION =
    17 struct
    18 
    19 val ext2 = @{thm ext2};
    20 val nth_Cons_0 = @{thm nth_Cons_0};
    21 val nth_Cons_Suc = @{thm nth_Cons_Suc};
    22 
    23   (* Make a congruence rule out of a defining equation for the interpretation *)
    24   (* th is one defining equation of f, i.e.
    25      th is "f (Cp ?t1 ... ?tn) = P(f ?t1, .., f ?tn)" *)
    26   (* Cp is a constructor pattern and P is a pattern *)
    27 
    28   (* The result is:
    29       [|?A1 = f ?t1 ; .. ; ?An= f ?tn |] ==> P (?A1, .., ?An) = f (Cp ?t1 .. ?tn) *)
    30   (*  + the a list of names of the A1 .. An, Those are fresh in the ctxt*)
    31 
    32 fun mk_congeq ctxt fs th =
    33   let
    34    val (f as Const(fN,fT)) = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq
    35      |> fst |> strip_comb |> fst
    36    val thy = ProofContext.theory_of ctxt
    37    val cert = Thm.cterm_of thy
    38    val (((_,_),[th']), ctxt') = Variable.import true [th] ctxt
    39    val (lhs, rhs) = HOLogic.dest_eq (HOLogic.dest_Trueprop (Thm.prop_of th'))
    40    fun add_fterms (t as t1 $ t2) =
    41        if exists (fn f => Term.could_unify (t |> strip_comb |> fst, f)) fs then insert (op aconv) t
    42        else add_fterms t1 #> add_fterms t2
    43      | add_fterms (t as Abs(xn,xT,t')) =
    44        if exists_Const (fn (c, _) => c = fN) t then (fn _ => [t]) else (fn _ => [])
    45      | add_fterms _ = I
    46    val fterms = add_fterms rhs []
    47    val (xs, ctxt'') = Variable.variant_fixes (replicate (length fterms) "x") ctxt'
    48    val tys = map fastype_of fterms
    49    val vs = map Free (xs ~~ tys)
    50    val env = fterms ~~ vs
    51 		    (* FIXME!!!!*)
    52    fun replace_fterms (t as t1 $ t2) =
    53        (case AList.lookup (op aconv) env t of
    54 	    SOME v => v
    55 	  | NONE => replace_fterms t1 $ replace_fterms t2)
    56      | replace_fterms t = (case AList.lookup (op aconv) env t of
    57 			       SOME v => v
    58 			     | NONE => t)
    59 
    60    fun mk_def (Abs(x,xT,t),v) = HOLogic.mk_Trueprop ((HOLogic.all_const xT)$ Abs(x,xT,HOLogic.mk_eq(v$(Bound 0), t)))
    61      | mk_def (t, v) = HOLogic.mk_Trueprop (HOLogic.mk_eq (v, t))
    62    fun tryext x = (x RS ext2 handle THM _ =>  x)
    63    val cong = (Goal.prove ctxt'' [] (map mk_def env)
    64 			  (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, replace_fterms rhs)))
    65 			  (fn x => LocalDefs.unfold_tac (#context x) (map tryext (#prems x))
    66 							THEN rtac th' 1)) RS sym
    67 
    68    val (cong' :: vars') =
    69        Variable.export ctxt'' ctxt (cong :: map (Drule.mk_term o cert) vs)
    70    val vs' = map (fst o fst o Term.dest_Var o Thm.term_of o Drule.dest_term) vars'
    71 
    72   in  (vs', cong') end;
    73  (* congs is a list of pairs (P,th) where th is a theorem for *)
    74         (* [| f p1 = A1; ...; f pn = An|] ==> f (C p1 .. pn) = P *)
    75 val FWD = curry (op OF);
    76 
    77 
    78 exception REIF of string;
    79 
    80 fun dest_listT (Type (@{type_name "list"}, [T])) = T;
    81 
    82 (* This modified version of divide_and_conquer allows the threading
    83    of a state variable *)
    84 fun divide_and_conquer' decomp s x =
    85   let val ((ys, recomb), s') = decomp s x
    86   in recomb (fold_map (divide_and_conquer' decomp) ys s') end;
    87 
    88 fun rearrange congs =
    89   let
    90     fun P (_, th) =
    91       let val @{term "Trueprop"}$(Const ("op =",_) $l$_) = concl_of th
    92       in can dest_Var l end
    93     val (yes,no) = List.partition P congs
    94   in no @ yes end
    95 
    96 fun genreif ctxt raw_eqs t =
    97   let
    98     fun index_of t bds =
    99       let
   100         val tt = HOLogic.listT (fastype_of t)
   101       in
   102        (case AList.lookup Type.could_unify bds tt of
   103           NONE => error "index_of : type not found in environements!"
   104         | SOME (tbs,tats) =>
   105           let
   106             val i = find_index (fn t' => t' = t) tats
   107             val j = find_index (fn t' => t' = t) tbs
   108           in (if j = ~1 then
   109               if i = ~1
   110               then (length tbs + length tats,
   111                     AList.update Type.could_unify (tt,(tbs,tats@[t])) bds)
   112               else (i, bds) else (j, bds))
   113           end)
   114       end;
   115 
   116     (* Generic decomp for reification : matches the actual term with the
   117        rhs of one cong rule. The result of the matching guides the
   118        proof synthesis: The matches of the introduced Variables A1 .. An are
   119        processed recursively
   120        The rest is instantiated in the cong rule,i.e. no reification is needed *)
   121 
   122     (* da is the decomposition for atoms, ie. it returns ([],g) where g
   123        returns the right instance f (AtC n) = t , where AtC is the Atoms
   124        constructor and n is the number of the atom corresponding to t *)
   125     fun decomp_genreif da cgns (t,ctxt) bds =
   126       let
   127         val thy = ProofContext.theory_of ctxt
   128         val cert = cterm_of thy
   129         fun tryabsdecomp (s,ctxt) bds =
   130           (case s of
   131              Abs(xn,xT,ta) => (
   132                let
   133                  val ([xn],ctxt') = Variable.variant_fixes ["x"] ctxt
   134                  val (xn,ta) = variant_abs (xn,xT,ta)
   135                  val x = Free(xn,xT)
   136                  val bds = (case AList.lookup Type.could_unify bds (HOLogic.listT xT)
   137 		          of NONE => error "tryabsdecomp: Type not found in the Environement"
   138                            | SOME (bsT,atsT) =>
   139                              (AList.update Type.could_unify (HOLogic.listT xT, ((x::bsT), atsT)) bds))
   140                in (([(ta, ctxt')],
   141                     fn ([th], bds) =>
   142                       (hd (Variable.export ctxt' ctxt [(forall_intr (cert x) th) COMP allI]),
   143                        let val (bsT,asT) = the(AList.lookup Type.could_unify bds (HOLogic.listT xT))
   144 		       in AList.update Type.could_unify (HOLogic.listT xT,(tl bsT,asT)) bds
   145 		       end)),
   146                    bds)
   147                end)
   148            | _ => da (s,ctxt) bds)
   149       in (case cgns of
   150           [] => tryabsdecomp (t,ctxt) bds
   151         | ((vns,cong)::congs) => ((let
   152             val cert = cterm_of thy
   153             val certy = ctyp_of thy
   154             val (tyenv, tmenv) =
   155               Pattern.match thy
   156               ((fst o HOLogic.dest_eq o HOLogic.dest_Trueprop) (concl_of cong), t)
   157               (Envir.type_env (Envir.empty 0), Vartab.empty)
   158             val (fnvs,invs) = List.partition (fn ((vn,_),_) => vn mem vns) (Vartab.dest tmenv)
   159             val (fts,its) =
   160 	      (map (snd o snd) fnvs,
   161                map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t)) invs)
   162 	    val ctyenv = map (fn ((vn,vi),(s,ty)) => (certy (TVar((vn,vi),s)), certy ty)) (Vartab.dest tyenv)
   163           in ((fts ~~ (replicate (length fts) ctxt),
   164                Library.apfst (FWD (instantiate (ctyenv, its) cong))), bds)
   165           end)
   166         handle MATCH => decomp_genreif da congs (t,ctxt) bds))
   167       end;
   168 
   169  (* looks for the atoms equation and instantiates it with the right number *)
   170     fun mk_decompatom eqs (t,ctxt) bds = (([], fn (_, bds) =>
   171       let
   172         val tT = fastype_of t
   173         fun isat eq =
   174           let
   175             val rhs = eq |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
   176           in exists_Const
   177 	    (fn (n,ty) => n = @{const_name "List.nth"}
   178                           andalso
   179 			  AList.defined Type.could_unify bds (domain_type ty)) rhs
   180             andalso Type.could_unify (fastype_of rhs, tT)
   181           end
   182 
   183         fun get_nths t acc =
   184           case t of
   185             Const(@{const_name "List.nth"},_)$vs$n => insert (fn ((a,_),(b,_)) => a aconv b) (t,(vs,n)) acc
   186           | t1$t2 => get_nths t1 (get_nths t2 acc)
   187           | Abs(_,_,t') => get_nths t'  acc
   188           | _ => acc
   189 
   190         fun
   191            tryeqs [] bds = error "Can not find the atoms equation"
   192          | tryeqs (eq::eqs) bds = ((
   193           let
   194             val rhs = eq |> prop_of |> HOLogic.dest_Trueprop  |> HOLogic.dest_eq |> snd
   195             val nths = get_nths rhs []
   196             val (vss,ns) = fold_rev (fn (_,(vs,n)) => fn (vss,ns) =>
   197                                       (insert (op aconv) vs vss, insert (op aconv) n ns)) nths ([],[])
   198             val (vsns, ctxt') = Variable.variant_fixes (replicate (length vss) "vs") ctxt
   199             val (xns, ctxt'') = Variable.variant_fixes (replicate (length nths) "x") ctxt'
   200             val thy = ProofContext.theory_of ctxt''
   201             val cert = cterm_of thy
   202             val certT = ctyp_of thy
   203             val vsns_map = vss ~~ vsns
   204             val xns_map = (fst (split_list nths)) ~~ xns
   205             val subst = map (fn (nt, xn) => (nt, Var ((xn,0), fastype_of nt))) xns_map
   206             val rhs_P = subst_free subst rhs
   207             val (tyenv, tmenv) = Pattern.match
   208                               thy (rhs_P, t)
   209                               (Envir.type_env (Envir.empty 0), Vartab.empty)
   210             val sbst = Envir.subst_vars (tyenv, tmenv)
   211             val sbsT = Envir.typ_subst_TVars tyenv
   212             val subst_ty = map (fn (n,(s,t)) => (certT (TVar (n, s)), certT t))
   213                                (Vartab.dest tyenv)
   214             val tml = Vartab.dest tmenv
   215             val t's = map (fn xn => snd (the (AList.lookup (op =) tml (xn,0)))) xns (* FIXME : Express with sbst*)
   216             val (subst_ns, bds) = fold_map
   217                 (fn (Const _ $ vs $ n, Var (xn0,T)) => fn bds =>
   218                   let
   219                     val name = snd (the (AList.lookup (op =) tml xn0))
   220                     val (idx, bds) = index_of name bds
   221                   in ((cert n, idx |> (HOLogic.mk_nat #> cert)), bds) end) subst bds
   222             val subst_vs =
   223               let
   224                 fun ty (Const _ $ (vs as Var (vsn,lT)) $ n, Var (xn0,T)) = (certT T, certT (sbsT T))
   225                 fun h (Const _ $ (vs as Var (vsn,lT)) $ n, Var (xn0,T)) =
   226                   let
   227                     val cns = sbst (Const(@{const_name "List.Cons"}, T --> lT --> lT))
   228                     val lT' = sbsT lT
   229                     val (bsT,asT) = the (AList.lookup Type.could_unify bds lT)
   230                     val vsn = the (AList.lookup (op =) vsns_map vs)
   231                     val cvs = cert (fold_rev (fn x => fn xs => cns$x$xs) bsT (Free (vsn, lT')))
   232                   in (cert vs, cvs) end
   233               in map h subst end
   234             val cts = map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t))
   235                           (fold (AList.delete (fn (((a: string),_),(b,_)) => a = b))
   236                                 (map (fn n => (n,0)) xns) tml)
   237             val substt =
   238               let val ih = Drule.cterm_rule (Thm.instantiate (subst_ty,[]))
   239               in map (fn (v,t) => (ih v, ih t)) (subst_ns@subst_vs@cts)  end
   240             val th = (instantiate (subst_ty, substt)  eq) RS sym
   241           in (hd (Variable.export ctxt'' ctxt [th]), bds) end)
   242           handle MATCH => tryeqs eqs bds)
   243       in tryeqs (filter isat eqs) bds end), bds);
   244 
   245   (* Generic reification procedure: *)
   246   (* creates all needed cong rules and then just uses the theorem synthesis *)
   247 
   248     fun mk_congs ctxt raw_eqs =
   249       let
   250         val fs = fold_rev (fn eq =>
   251                            insert (op =) (eq |> prop_of |> HOLogic.dest_Trueprop
   252                            |> HOLogic.dest_eq |> fst |> strip_comb
   253                            |> fst)) raw_eqs []
   254         val tys = fold_rev (fn f => fold (insert (op =)) (f |> fastype_of |> binder_types |> tl)
   255                             ) fs []
   256         val (vs, ctxt') = Variable.variant_fixes (replicate (length tys) "vs") ctxt
   257         val thy = ProofContext.theory_of ctxt'
   258         val cert = cterm_of thy
   259         val vstys = map (fn (t,v) => (t,SOME (cert (Free(v,t)))))
   260                     (tys ~~ vs)
   261         val is_Var = can dest_Var
   262         fun insteq eq vs =
   263           let
   264             val subst = map (fn (v as Var(n,t)) => (cert v, (the o the) (AList.lookup (op =) vstys t)))
   265                         (filter is_Var vs)
   266           in Thm.instantiate ([],subst) eq
   267           end
   268 
   269         val bds = AList.make (fn _ => ([],[])) tys
   270         val eqs = map (fn eq => eq |> prop_of |> HOLogic.dest_Trueprop
   271   	                           |> HOLogic.dest_eq |> fst |> strip_comb |> snd |> tl
   272                                    |> (insteq eq)) raw_eqs
   273         val (ps,congs) = split_list (map (mk_congeq ctxt' fs) eqs)
   274       in (ps ~~ (Variable.export ctxt' ctxt congs), bds)
   275       end
   276 
   277     val (congs, bds) = mk_congs ctxt raw_eqs
   278     val congs = rearrange congs
   279     val (th, bds) = divide_and_conquer' (decomp_genreif (mk_decompatom raw_eqs) congs) (t,ctxt) bds
   280     fun is_listVar (Var (_,t)) = can dest_listT t
   281          | is_listVar _ = false
   282     val vars = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
   283 	          |> strip_comb |> snd |> filter is_listVar
   284     val cert = cterm_of (ProofContext.theory_of ctxt)
   285     val cvs = map (fn (v as Var(n,t)) => (cert v,
   286                   the (AList.lookup Type.could_unify bds t) |> snd |> HOLogic.mk_list (dest_listT t) |> cert)) vars
   287     val th' = instantiate ([], cvs) th
   288     val t' = (fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) th'
   289     val th'' = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, t')))
   290 	       (fn _ => simp_tac (local_simpset_of ctxt) 1)
   291   in FWD trans [th'',th']
   292   end
   293 
   294 
   295 fun genreflect ctxt conv corr_thms raw_eqs t =
   296   let
   297     val reifth = genreif ctxt raw_eqs t
   298     fun trytrans [] = error "No suitable correctness theorem found"
   299       | trytrans (th::ths) =
   300            (FWD trans [reifth, th RS sym] handle THM _ => trytrans ths)
   301     val th = trytrans corr_thms
   302     val ft = (Thm.dest_arg1 o Thm.dest_arg o Thm.dest_arg o cprop_of) th
   303     val rth = conv ft
   304   in simplify (HOL_basic_ss addsimps raw_eqs addsimps [nth_Cons_0, nth_Cons_Suc])
   305              (simplify (HOL_basic_ss addsimps [rth]) th)
   306   end
   307 
   308 fun genreify_tac ctxt eqs to i = (fn st =>
   309   let
   310     fun P () = HOLogic.dest_Trueprop (nth (prems_of st) (i - 1))
   311     val t = (case to of NONE => P () | SOME x => x)
   312     val th = (genreif ctxt eqs t) RS ssubst
   313   in rtac th i st
   314   end);
   315 
   316     (* Reflection calls reification and uses the correctness *)
   317         (* theorem assumed to be the dead of the list *)
   318 fun gen_reflection_tac ctxt conv corr_thms raw_eqs to i = (fn st =>
   319   let
   320     val P = HOLogic.dest_Trueprop (nth (prems_of st) (i - 1));
   321     val t = the_default P to;
   322     val th = genreflect ctxt conv corr_thms raw_eqs t
   323       RS ssubst;
   324   in (rtac th i THEN TRY(rtac TrueI i)) st end);
   325 
   326 fun reflection_tac ctxt = gen_reflection_tac ctxt Codegen.evaluation_conv;
   327   (*FIXME why Codegen.evaluation_conv?  very specific...*)
   328 
   329 end