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--
added Determinants to Library
     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