src/HOL/Library/reflection.ML
author hoelzl
Tue Jun 02 18:38:13 2009 +0200 (2009-06-02)
changeset 31386 8624b75a7784
parent 30969 fd9c89419358
child 31387 c4a3c3e9dc8e
permissions -rw-r--r--
corrected spacing in reflection
     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 end;
    14 
    15 structure Reflection : REFLECTION =
    16 struct
    17 
    18 val ext2 = @{thm ext2};
    19 val nth_Cons_0 = @{thm nth_Cons_0};
    20 val nth_Cons_Suc = @{thm nth_Cons_Suc};
    21 
    22   (* Make a congruence rule out of a defining equation for the interpretation *)
    23   (* th is one defining equation of f, i.e.
    24      th is "f (Cp ?t1 ... ?tn) = P(f ?t1, .., f ?tn)" *)
    25   (* Cp is a constructor pattern and P is a pattern *)
    26 
    27   (* The result is:
    28       [|?A1 = f ?t1 ; .. ; ?An= f ?tn |] ==> P (?A1, .., ?An) = f (Cp ?t1 .. ?tn) *)
    29   (*  + the a list of names of the A1 .. An, Those are fresh in the ctxt*)
    30 
    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_thms 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$(Bound 0), t)))
    61      | mk_def (t, v) = HOLogic.mk_Trueprop (HOLogic.mk_eq (v, t))
    62    fun tryext x = (x RS ext2 handle THM _ =>  x)
    63    val cong = (Goal.prove ctxt'' [] (map mk_def env)
    64 			  (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, replace_fterms rhs)))
    65 			  (fn x => LocalDefs.unfold_tac (#context x) (map tryext (#prems x))
    66 							THEN rtac th' 1)) RS sym
    67 
    68    val (cong' :: vars') =
    69        Variable.export ctxt'' ctxt (cong :: map (Drule.mk_term o cert) vs)
    70    val vs' = map (fst o fst o Term.dest_Var o Thm.term_of o Drule.dest_term) vars'
    71 
    72   in  (vs', cong') end;
    73  (* congs is a list of pairs (P,th) where th is a theorem for *)
    74         (* [| f p1 = A1; ...; f pn = An|] ==> f (C p1 .. pn) = P *)
    75 val FWD = curry (op OF);
    76 
    77 
    78 exception REIF of string;
    79 
    80 fun dest_listT (Type ("List.list", [T])) = T;
    81 
    82 fun rearrange congs =
    83   let
    84     fun P (_, th) =
    85       let val @{term "Trueprop"}$(Const ("op =",_) $l$_) = concl_of th
    86       in can dest_Var l end
    87     val (yes,no) = List.partition P congs
    88   in no @ yes end
    89 
    90 fun genreif ctxt raw_eqs t =
    91   let
    92     val bds = ref ([]: (typ * ((term list) * (term list))) list);
    93 
    94     fun index_of t =
    95       let
    96         val tt = HOLogic.listT (fastype_of t)
    97       in
    98        (case AList.lookup Type.could_unify (!bds) tt of
    99           NONE => error "index_of : type not found in environements!"
   100         | SOME (tbs,tats) =>
   101           let
   102             val i = find_index_eq t tats
   103             val j = find_index_eq t tbs
   104           in (if j= ~1 then
   105 	      if i= ~1
   106               then (bds := AList.update Type.could_unify (tt,(tbs,tats@[t])) (!bds) ;
   107                     length tbs + length tats)
   108               else i else j)
   109           end)
   110       end;
   111 
   112     (* Generic decomp for reification : matches the actual term with the
   113        rhs of one cong rule. The result of the matching guides the
   114        proof synthesis: The matches of the introduced Variables A1 .. An are
   115        processed recursively
   116        The rest is instantiated in the cong rule,i.e. no reification is needed *)
   117 
   118     (* da is the decomposition for atoms, ie. it returns ([],g) where g
   119        returns the right instance f (AtC n) = t , where AtC is the Atoms
   120        constructor and n is the number of the atom corresponding to t *)
   121     fun decomp_genreif da cgns (t,ctxt) =
   122       let
   123         val thy = ProofContext.theory_of ctxt
   124         val cert = cterm_of thy
   125         fun tryabsdecomp (s,ctxt) =
   126           (case s of
   127              Abs(xn,xT,ta) => (
   128                let
   129                  val ([xn],ctxt') = Variable.variant_fixes ["x"] ctxt
   130                  val (xn,ta) = variant_abs (xn,xT,ta)
   131                  val x = Free(xn,xT)
   132                  val _ = (case AList.lookup Type.could_unify (!bds) (HOLogic.listT xT)
   133 		          of NONE => error "tryabsdecomp: Type not found in the Environement"
   134                            | SOME (bsT,atsT) =>
   135                              (bds := AList.update Type.could_unify (HOLogic.listT xT, ((x::bsT), atsT)) (!bds)))
   136                in ([(ta, ctxt')] ,
   137                   fn [th] => ((let val (bsT,asT) = the(AList.lookup Type.could_unify (!bds) (HOLogic.listT xT))
   138 		               in (bds := AList.update Type.could_unify (HOLogic.listT xT,(tl bsT,asT)) (!bds))
   139 		               end) ;
   140                               hd (Variable.export ctxt' ctxt [(forall_intr (cert x) th) COMP allI])))
   141                end)
   142            | _ => da (s,ctxt))
   143       in (case cgns of
   144           [] => tryabsdecomp (t,ctxt)
   145         | ((vns,cong)::congs) => ((let
   146             val cert = cterm_of thy
   147             val certy = ctyp_of thy
   148             val (tyenv, tmenv) =
   149               Pattern.match thy
   150               ((fst o HOLogic.dest_eq o HOLogic.dest_Trueprop) (concl_of cong), t)
   151               (Envir.type_env (Envir.empty 0), Vartab.empty)
   152             val (fnvs,invs) = List.partition (fn ((vn,_),_) => vn mem vns) (Vartab.dest tmenv)
   153             val (fts,its) =
   154 	      (map (snd o snd) fnvs,
   155                map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t)) invs)
   156 	    val ctyenv = map (fn ((vn,vi),(s,ty)) => (certy (TVar((vn,vi),s)), certy ty)) (Vartab.dest tyenv)
   157           in (fts ~~ (replicate (length fts) ctxt), FWD (instantiate (ctyenv, its) cong))
   158           end)
   159         handle MATCH => decomp_genreif da congs (t,ctxt)))
   160       end;
   161 
   162  (* looks for the atoms equation and instantiates it with the right number *)
   163     fun mk_decompatom eqs (t,ctxt) =
   164       let
   165         val tT = fastype_of t
   166         fun isat eq =
   167           let
   168             val rhs = eq |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
   169           in exists_Const
   170 	    (fn (n,ty) => n="List.nth"
   171                           andalso
   172 			  AList.defined Type.could_unify (!bds) (domain_type ty)) rhs
   173             andalso Type.could_unify (fastype_of rhs, tT)
   174           end
   175 
   176         fun get_nths t acc =
   177           case t of
   178             Const("List.nth",_)$vs$n => insert (fn ((a,_),(b,_)) => a aconv b) (t,(vs,n)) acc
   179           | t1$t2 => get_nths t1 (get_nths t2 acc)
   180           | Abs(_,_,t') => get_nths t'  acc
   181           | _ => acc
   182 
   183         fun
   184            tryeqs [] = error "Can not find the atoms equation"
   185          | tryeqs (eq::eqs) = ((
   186           let
   187             val rhs = eq |> prop_of |> HOLogic.dest_Trueprop  |> HOLogic.dest_eq |> snd
   188             val nths = get_nths rhs []
   189             val (vss,ns) = fold_rev (fn (_,(vs,n)) => fn (vss,ns) =>
   190                                       (insert (op aconv) vs vss, insert (op aconv) n ns)) nths ([],[])
   191             val (vsns, ctxt') = Variable.variant_fixes (replicate (length vss) "vs") ctxt
   192             val (xns, ctxt'') = Variable.variant_fixes (replicate (length nths) "x") ctxt'
   193             val thy = ProofContext.theory_of ctxt''
   194             val cert = cterm_of thy
   195             val certT = ctyp_of thy
   196             val vsns_map = vss ~~ vsns
   197             val xns_map = (fst (split_list nths)) ~~ xns
   198             val subst = map (fn (nt, xn) => (nt, Var ((xn,0), fastype_of nt))) xns_map
   199             val rhs_P = subst_free subst rhs
   200             val (tyenv, tmenv) = Pattern.match
   201                               thy (rhs_P, t)
   202                               (Envir.type_env (Envir.empty 0), Vartab.empty)
   203             val sbst = Envir.subst_vars (tyenv, tmenv)
   204             val sbsT = Envir.typ_subst_TVars tyenv
   205             val subst_ty = map (fn (n,(s,t)) => (certT (TVar (n, s)), certT t))
   206                                (Vartab.dest tyenv)
   207             val tml = Vartab.dest tmenv
   208             val t's = map (fn xn => snd (valOf (AList.lookup (op =) tml (xn,0)))) xns (* FIXME : Express with sbst*)
   209             val subst_ns = map (fn (Const _ $ vs $ n, Var (xn0,T)) =>
   210                                    (cert n, snd (valOf (AList.lookup (op =) tml xn0))
   211                                       |> (index_of #> HOLogic.mk_nat #> cert)))
   212                                subst
   213             val subst_vs =
   214               let
   215                 fun ty (Const _ $ (vs as Var (vsn,lT)) $ n, Var (xn0,T)) = (certT T, certT (sbsT T))
   216                 fun h (Const _ $ (vs as Var (vsn,lT)) $ n, Var (xn0,T)) =
   217                   let
   218                     val cns = sbst (Const("List.list.Cons", T --> lT --> lT))
   219                     val lT' = sbsT lT
   220                     val (bsT,asT) = the (AList.lookup Type.could_unify (!bds) lT)
   221                     val vsn = valOf (AList.lookup (op =) vsns_map vs)
   222                     val cvs = cert (fold_rev (fn x => fn xs => cns$x$xs) bsT (Free (vsn, lT')))
   223                   in (cert vs, cvs) end
   224               in map h subst end
   225             val cts = map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t))
   226                           (fold (AList.delete (fn (((a: string),_),(b,_)) => a = b))
   227                                 (map (fn n => (n,0)) xns) tml)
   228             val substt =
   229               let val ih = Drule.cterm_rule (Thm.instantiate (subst_ty,[]))
   230               in map (fn (v,t) => (ih v, ih t)) (subst_ns@subst_vs@cts)  end
   231             val th = (instantiate (subst_ty, substt)  eq) RS sym
   232           in hd (Variable.export ctxt'' ctxt [th]) end)
   233           handle MATCH => tryeqs eqs)
   234       in ([], fn _ => tryeqs (filter isat eqs))
   235       end;
   236 
   237   (* Generic reification procedure: *)
   238   (* creates all needed cong rules and then just uses the theorem synthesis *)
   239 
   240     fun mk_congs ctxt raw_eqs =
   241       let
   242         val fs = fold_rev (fn eq =>
   243                            insert (op =) (eq |> prop_of |> HOLogic.dest_Trueprop
   244                            |> HOLogic.dest_eq |> fst |> strip_comb
   245                            |> fst)) raw_eqs []
   246         val tys = fold_rev (fn f => fold (insert (op =)) (f |> fastype_of |> binder_types |> tl)
   247                             ) fs []
   248         val _ = bds := AList.make (fn _ => ([],[])) tys
   249         val (vs, ctxt') = Variable.variant_fixes (replicate (length tys) "vs") ctxt
   250         val thy = ProofContext.theory_of ctxt'
   251         val cert = cterm_of thy
   252         val vstys = map (fn (t,v) => (t,SOME (cert (Free(v,t)))))
   253                     (tys ~~ vs)
   254         val is_Var = can dest_Var
   255         fun insteq eq vs =
   256           let
   257             val subst = map (fn (v as Var(n,t)) => (cert v, (valOf o valOf) (AList.lookup (op =) vstys t)))
   258                         (filter is_Var vs)
   259           in Thm.instantiate ([],subst) eq
   260           end
   261 
   262         val eqs = map (fn eq => eq |> prop_of |> HOLogic.dest_Trueprop
   263   	                           |> HOLogic.dest_eq |> fst |> strip_comb |> snd |> tl
   264                                    |> (insteq eq)) raw_eqs
   265         val (ps,congs) = split_list (map (mk_congeq ctxt' fs) eqs)
   266       in ps ~~ (Variable.export ctxt' ctxt congs)
   267       end
   268 
   269     val congs = rearrange (mk_congs ctxt raw_eqs)
   270     val th = divide_and_conquer (decomp_genreif (mk_decompatom raw_eqs) congs) (t,ctxt)
   271     fun is_listVar (Var (_,t)) = can dest_listT t
   272          | is_listVar _ = false
   273     val vars = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
   274 	          |> strip_comb |> snd |> filter is_listVar
   275     val cert = cterm_of (ProofContext.theory_of ctxt)
   276     val cvs = map (fn (v as Var(n,t)) => (cert v,
   277                   the (AList.lookup Type.could_unify (!bds) t) |> snd |> HOLogic.mk_list (dest_listT t) |> cert)) vars
   278     val th' = instantiate ([], cvs) th
   279     val t' = (fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) th'
   280     val th'' = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, t')))
   281 	       (fn _ => simp_tac (local_simpset_of ctxt) 1)
   282     val _ = bds := []
   283   in FWD trans [th'',th']
   284   end
   285 
   286 
   287 fun genreflect ctxt conv corr_thms raw_eqs t =
   288   let
   289     val reifth = genreif ctxt raw_eqs t
   290     fun trytrans [] = error "No suitable correctness theorem found"
   291       | trytrans (th::ths) =
   292            (FWD trans [reifth, th RS sym] handle THM _ => trytrans ths)
   293     val th = trytrans corr_thms
   294     val ft = (Thm.dest_arg1 o Thm.dest_arg o Thm.dest_arg o cprop_of) th
   295     val rth = conv ft
   296   in simplify (HOL_basic_ss addsimps raw_eqs addsimps [nth_Cons_0, nth_Cons_Suc])
   297              (simplify (HOL_basic_ss addsimps [rth]) th)
   298   end
   299 
   300 fun genreify_tac ctxt eqs to i = (fn st =>
   301   let
   302     fun P () = HOLogic.dest_Trueprop (List.nth (prems_of st, i - 1))
   303     val t = (case to of NONE => P () | SOME x => x)
   304     val th = (genreif ctxt eqs t) RS ssubst
   305   in rtac th i st
   306   end);
   307 
   308     (* Reflection calls reification and uses the correctness *)
   309         (* theorem assumed to be the dead of the list *)
   310 fun gen_reflection_tac ctxt conv corr_thms raw_eqs to i = (fn st =>
   311   let
   312     val P = HOLogic.dest_Trueprop (nth (prems_of st) (i - 1));
   313     val t = the_default P to;
   314     val th = genreflect ctxt conv corr_thms raw_eqs t
   315       RS ssubst;
   316   in (rtac th i THEN TRY(rtac TrueI i)) st end);
   317 
   318 fun reflection_tac ctxt = gen_reflection_tac ctxt Codegen.evaluation_conv;
   319   (*FIXME why Codegen.evaluation_conv?  very specific...*)
   320 
   321 end