src/HOL/Library/reflection.ML
 author chaieb Mon Feb 09 17:21:46 2009 +0000 (2009-02-09) changeset 29847 af32126ee729 parent 29834 3237cfd177f3 child 30148 5d04b67a866e permissions -rw-r--r--
```     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  (* da is the decomposition for atoms, ie. it returns ([],g) where g
```
```    78  returns the right instance f (AtC n) = t , where AtC is the Atoms
```
```    79  constructor and n is the number of the atom corresponding to t *)
```
```    80
```
```    81 (* Generic decomp for reification : matches the actual term with the
```
```    82 rhs of one cong rule. The result of the matching guides the
```
```    83 proof synthesis: The matches of the introduced Variables A1 .. An are
```
```    84 processed recursively
```
```    85  The rest is instantiated in the cong rule,i.e. no reification is needed *)
```
```    86
```
```    87 exception REIF of string;
```
```    88
```
```    89 fun dest_listT (Type ("List.list", [T])) = T;
```
```    90
```
```    91 fun partition P [] = ([],[])
```
```    92   | partition P (x::xs) =
```
```    93      let val (yes,no) = partition P xs
```
```    94      in if P x then (x::yes,no) else (yes, x::no) end
```
```    95
```
```    96 fun rearrange congs =
```
```    97 let
```
```    98  fun P (_, th) =
```
```    99   let val @{term "Trueprop"}\$(Const ("op =",_) \$l\$_) = concl_of th
```
```   100   in can dest_Var l end
```
```   101  val (yes,no) = partition P congs
```
```   102  in no @ yes end
```
```   103
```
```   104 fun genreif ctxt raw_eqs t =
```
```   105  let
```
```   106 val bds = ref ([]: (typ * ((term list) * (term list))) list);
```
```   107
```
```   108 fun index_of t =
```
```   109  let
```
```   110   val tt = HOLogic.listT (fastype_of t)
```
```   111  in
```
```   112   (case AList.lookup Type.could_unify (!bds) tt of
```
```   113     NONE => error "index_of : type not found in environements!"
```
```   114   | SOME (tbs,tats) =>
```
```   115     let
```
```   116      val i = find_index_eq t tats
```
```   117      val j = find_index_eq t tbs
```
```   118     in (if j= ~1 then
```
```   119 	    if i= ~1
```
```   120 	    then (bds := AList.update Type.could_unify (tt,(tbs,tats@[t])) (!bds) ;
```
```   121 		  length tbs + length tats)
```
```   122 	    else i else j)
```
```   123     end)
```
```   124  end;
```
```   125
```
```   126 fun decomp_genreif da cgns (t,ctxt) =
```
```   127  let
```
```   128   val thy = ProofContext.theory_of ctxt
```
```   129   val cert = cterm_of thy
```
```   130   fun tryabsdecomp (s,ctxt) =
```
```   131    (case s of
```
```   132      Abs(xn,xT,ta) =>
```
```   133      (let
```
```   134        val ([xn],ctxt') = Variable.variant_fixes ["x"] ctxt
```
```   135        val (xn,ta) = variant_abs (xn,xT,ta)
```
```   136        val x = Free(xn,xT)
```
```   137        val _ = (case AList.lookup Type.could_unify (!bds) (HOLogic.listT xT)
```
```   138 		 of NONE => error "tryabsdecomp: Type not found in the Environement"
```
```   139 		  | SOME (bsT,atsT) =>
```
```   140 		    (bds := AList.update Type.could_unify (HOLogic.listT xT, ((x::bsT), atsT)) (!bds)))
```
```   141       in ([(ta, ctxt')] ,
```
```   142 	  fn [th] => ((let val (bsT,asT) = the(AList.lookup Type.could_unify (!bds) (HOLogic.listT xT))
```
```   143 		       in (bds := AList.update Type.could_unify (HOLogic.listT xT,(tl bsT,asT)) (!bds))
```
```   144 		       end) ;
```
```   145 		      hd (Variable.export ctxt' ctxt [(forall_intr (cert x) th) COMP allI])))
```
```   146 	end)
```
```   147     | _ => da (s,ctxt))
```
```   148   in
```
```   149   (case cgns of
```
```   150     [] => tryabsdecomp (t,ctxt)
```
```   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), FWD (instantiate (ctyenv, its) cong))
```
```   164     end)
```
```   165       handle MATCH => decomp_genreif da congs (t,ctxt)))
```
```   166   end;
```
```   167
```
```   168  (* looks for the atoms equation and instantiates it with the right number *)
```
```   169 fun mk_decompatom eqs (t,ctxt) =
```
```   170 let
```
```   171  val tT = fastype_of t
```
```   172  fun isat eq =
```
```   173   let
```
```   174    val rhs = eq |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
```
```   175    in exists_Const
```
```   176 	  (fn (n,ty) => n="List.nth"
```
```   177 			andalso
```
```   178 			AList.defined Type.could_unify (!bds) (domain_type ty)) rhs
```
```   179 	  andalso Type.could_unify (fastype_of rhs, tT)
```
```   180    end
```
```   181  fun get_nths t acc =
```
```   182   case t of
```
```   183     Const("List.nth",_)\$vs\$n => insert (fn ((a,_),(b,_)) => a aconv b) (t,(vs,n)) acc
```
```   184   | t1\$t2 => get_nths t1 (get_nths t2 acc)
```
```   185   | Abs(_,_,t') => get_nths t'  acc
```
```   186   | _ => acc
```
```   187
```
```   188  fun
```
```   189    tryeqs [] = error "Can not find the atoms equation"
```
```   190  | tryeqs (eq::eqs) = ((
```
```   191   let
```
```   192    val rhs = eq |> prop_of |> HOLogic.dest_Trueprop  |> HOLogic.dest_eq |> snd
```
```   193    val nths = get_nths rhs []
```
```   194    val (vss,ns) = fold_rev (fn (_,(vs,n)) => fn (vss,ns) =>
```
```   195                              (insert (op aconv) vs vss, insert (op aconv) n ns)) nths ([],[])
```
```   196    val (vsns, ctxt') = Variable.variant_fixes (replicate (length vss) "vs") ctxt
```
```   197    val (xns, ctxt'') = Variable.variant_fixes (replicate (length nths) "x") ctxt'
```
```   198    val thy = ProofContext.theory_of ctxt''
```
```   199    val cert = cterm_of thy
```
```   200    val certT = ctyp_of thy
```
```   201    val vsns_map = vss ~~ vsns
```
```   202    val xns_map = (fst (split_list nths)) ~~ xns
```
```   203    val subst = map (fn (nt, xn) => (nt, Var ((xn,0), fastype_of nt))) xns_map
```
```   204    val rhs_P = subst_free subst rhs
```
```   205    val (tyenv, tmenv) = Pattern.match
```
```   206 	                    thy (rhs_P, t)
```
```   207 	                    (Envir.type_env (Envir.empty 0), Vartab.empty)
```
```   208    val sbst = Envir.subst_vars (tyenv, tmenv)
```
```   209    val sbsT = Envir.typ_subst_TVars tyenv
```
```   210    val subst_ty = map (fn (n,(s,t)) => (certT (TVar (n, s)), certT t))
```
```   211                       (Vartab.dest tyenv)
```
```   212    val tml = Vartab.dest tmenv
```
```   213    val t's = map (fn xn => snd (valOf (AList.lookup (op =) tml (xn,0)))) xns (* FIXME : Express with sbst*)
```
```   214    val subst_ns = map (fn (Const _ \$ vs \$ n, Var (xn0,T)) =>
```
```   215                           (cert n, snd (valOf (AList.lookup (op =) tml xn0))
```
```   216                              |> (index_of #> HOLogic.mk_nat #> cert)))
```
```   217                       subst
```
```   218    val subst_vs =
```
```   219     let
```
```   220      fun ty (Const _ \$ (vs as Var (vsn,lT)) \$ n, Var (xn0,T)) = (certT T, certT (sbsT T))
```
```   221      fun h (Const _ \$ (vs as Var (vsn,lT)) \$ n, Var (xn0,T)) =
```
```   222       let
```
```   223        val cns = sbst (Const("List.list.Cons", T --> lT --> lT))
```
```   224        val lT' = sbsT lT
```
```   225        val (bsT,asT) = the (AList.lookup Type.could_unify (!bds) lT)
```
```   226        val vsn = valOf (AList.lookup (op =) vsns_map vs)
```
```   227        val cvs = cert (fold_rev (fn x => fn xs => cns\$x\$xs) bsT (Free (vsn, lT')))
```
```   228       in (cert vs, cvs) end
```
```   229     in map h subst end
```
```   230    val cts = map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t))
```
```   231                  (fold (AList.delete (fn (((a: string),_),(b,_)) => a = b))
```
```   232                        (map (fn n => (n,0)) xns) tml)
```
```   233    val substt =
```
```   234     let val ih = Drule.cterm_rule (Thm.instantiate (subst_ty,[]))
```
```   235     in map (fn (v,t) => (ih v, ih t)) (subst_ns@subst_vs@cts)  end
```
```   236    val th = (instantiate (subst_ty, substt)  eq) RS sym
```
```   237   in  hd (Variable.export ctxt'' ctxt [th]) end)
```
```   238  handle MATCH => tryeqs eqs)
```
```   239 in ([], fn _ => tryeqs (filter isat eqs))
```
```   240 end;
```
```   241
```
```   242   (* Generic reification procedure: *)
```
```   243   (* creates all needed cong rules and then just uses the theorem synthesis *)
```
```   244
```
```   245 fun mk_congs ctxt raw_eqs =
```
```   246 let
```
```   247   val fs = fold_rev (fn eq =>
```
```   248 		     insert (op =) (eq |> prop_of |> HOLogic.dest_Trueprop
```
```   249 			 |> HOLogic.dest_eq |> fst |> strip_comb
```
```   250 			 |> fst)) raw_eqs []
```
```   251   val tys = fold_rev (fn f => fn ts => (f |> fastype_of |> binder_types |> tl)
```
```   252 				    union ts) fs []
```
```   253   val _ = bds := AList.make (fn _ => ([],[])) tys
```
```   254   val (vs, ctxt') = Variable.variant_fixes (replicate (length tys) "vs") ctxt
```
```   255   val thy = ProofContext.theory_of ctxt'
```
```   256   val cert = cterm_of thy
```
```   257   val vstys = map (fn (t,v) => (t,SOME (cert (Free(v,t)))))
```
```   258 		  (tys ~~ vs)
```
```   259   val is_Var = can dest_Var
```
```   260   fun insteq eq vs =
```
```   261    let
```
```   262      val subst = map (fn (v as Var(n,t)) => (cert v, (valOf o valOf) (AList.lookup (op =) vstys t)))
```
```   263   (filter is_Var vs)
```
```   264    in Thm.instantiate ([],subst) eq
```
```   265    end
```
```   266   val eqs = map (fn eq => eq |> prop_of |> HOLogic.dest_Trueprop
```
```   267 			     |> HOLogic.dest_eq |> fst |> strip_comb |> snd |> tl
```
```   268 			     |> (insteq eq)) raw_eqs
```
```   269   val (ps,congs) = split_list (map (mk_congeq ctxt' fs) eqs)
```
```   270 in ps ~~ (Variable.export ctxt' ctxt congs)
```
```   271 end
```
```   272
```
```   273   val congs = rearrange (mk_congs ctxt raw_eqs)
```
```   274   val th = divide_and_conquer (decomp_genreif (mk_decompatom raw_eqs) congs) (t,ctxt)
```
```   275   fun is_listVar (Var (_,t)) = can dest_listT t
```
```   276        | is_listVar _ = false
```
```   277   val vars = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
```
```   278 	       |> strip_comb |> snd |> filter is_listVar
```
```   279   val cert = cterm_of (ProofContext.theory_of ctxt)
```
```   280   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
```
```   281   val th' = instantiate ([], cvs) th
```
```   282   val t' = (fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) th'
```
```   283   val th'' = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, t')))
```
```   284 			(fn _ => simp_tac (local_simpset_of ctxt) 1)
```
```   285   val _ = bds := []
```
```   286 in FWD trans [th'',th']
```
```   287 end
```
```   288
```
```   289
```
```   290 fun genreflect ctxt conv corr_thms raw_eqs t =
```
```   291 let
```
```   292   val reifth = genreif ctxt raw_eqs t
```
```   293   fun trytrans [] = error "No suitable correctness theorem found"
```
```   294     | trytrans (th::ths) =
```
```   295          (FWD trans [reifth, th RS sym] handle THM _ => trytrans ths)
```
```   296   val th = trytrans corr_thms
```
```   297   val ft = (Thm.dest_arg1 o Thm.dest_arg o Thm.dest_arg o cprop_of) th
```
```   298   val rth = conv ft
```
```   299 in simplify (HOL_basic_ss addsimps raw_eqs addsimps [nth_Cons_0, nth_Cons_Suc])
```
```   300            (simplify (HOL_basic_ss addsimps [rth]) th)
```
```   301 end
```
```   302
```
```   303 fun genreify_tac ctxt eqs to i = (fn st =>
```
```   304   let
```
```   305     fun P () = HOLogic.dest_Trueprop (List.nth (prems_of st, i - 1))
```
```   306     val t = (case to of NONE => P () | SOME x => x)
```
```   307     val th = (genreif ctxt eqs t) RS ssubst
```
```   308   in rtac th i st
```
```   309   end);
```
```   310
```
```   311     (* Reflection calls reification and uses the correctness *)
```
```   312         (* theorem assumed to be the dead of the list *)
```
```   313 fun gen_reflection_tac ctxt conv corr_thms raw_eqs to i = (fn st =>
```
```   314   let
```
```   315     val P = HOLogic.dest_Trueprop (nth (prems_of st) (i - 1));
```
```   316     val t = the_default P to;
```
```   317     val th = genreflect ctxt conv corr_thms raw_eqs t
```
```   318       RS ssubst;
```
```   319   in (rtac th i THEN TRY(rtac TrueI i)) st end);
```
```   320
```
```   321 fun reflection_tac ctxt = gen_reflection_tac ctxt Codegen.evaluation_conv;
```
```   322
```
```   323 end
```