src/HOL/Library/reflection.ML

+(*  Title:      HOL/Library/reflection.ML
+    Author:     Amine Chaieb, TU Muenchen
+
+A trial for automatical reification.
+*)
+
+signature REFLECTION =
+sig
+  val genreify_tac: Proof.context -> thm list -> term option -> int -> tactic
+  val reflection_tac: Proof.context -> thm list -> thm list -> term option -> int -> tactic
+  val gen_reflection_tac: Proof.context -> (cterm -> thm)
+    -> thm list -> thm list -> term option -> int -> tactic
+end;
+
+structure Reflection : REFLECTION =
+struct
+
+val ext2 = @{thm ext2};
+val nth_Cons_0 = @{thm nth_Cons_0};
+val nth_Cons_Suc = @{thm nth_Cons_Suc};
+
+  (* Make a congruence rule out of a defining equation for the interpretation *)
+  (* th is one defining equation of f, i.e.
+     th is "f (Cp ?t1 ... ?tn) = P(f ?t1, .., f ?tn)" *)
+  (* Cp is a constructor pattern and P is a pattern *)
+
+  (* The result is:
+      [|?A1 = f ?t1 ; .. ; ?An= f ?tn |] ==> P (?A1, .., ?An) = f (Cp ?t1 .. ?tn) *)
+  (*  + the a list of names of the A1 .. An, Those are fresh in the ctxt*)
+
+
+fun mk_congeq ctxt fs th = 
+  let 
+   val (f as Const(fN,fT)) = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq 
+     |> fst |> strip_comb |> fst
+   val thy = ProofContext.theory_of ctxt
+   val cert = Thm.cterm_of thy
+   val (((_,_),[th']), ctxt') = Variable.import_thms true [th] ctxt
+   val (lhs, rhs) = HOLogic.dest_eq (HOLogic.dest_Trueprop (Thm.prop_of th'))
+   fun add_fterms (t as t1 $ t2) = 
+       if exists (fn f => Term.could_unify (t |> strip_comb |> fst, f)) fs then insert (op aconv) t
+       else add_fterms t1 #> add_fterms t2
+     | add_fterms (t as Abs(xn,xT,t')) = 
+       if exists_Const (fn (c, _) => c = fN) t then (fn _ => [t]) else (fn _ => [])
+     | add_fterms _ = I
+   val fterms = add_fterms rhs []
+   val (xs, ctxt'') = Variable.variant_fixes (replicate (length fterms) "x") ctxt'
+   val tys = map fastype_of fterms
+   val vs = map Free (xs ~~ tys)
+   val env = fterms ~~ vs
+		    (* FIXME!!!!*)	
+   fun replace_fterms (t as t1 $ t2) =
+       (case AList.lookup (op aconv) env t of
+	    SOME v => v
+	  | NONE => replace_fterms t1 $ replace_fterms t2)
+     | replace_fterms t = (case AList.lookup (op aconv) env t of
+			       SOME v => v
+			     | NONE => t)
+      
+   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)))
+     | mk_def (t, v) = HOLogic.mk_Trueprop (HOLogic.mk_eq (v, t))
+   fun tryext x = (x RS ext2 handle THM _ =>  x)
+   val cong = (Goal.prove ctxt'' [] (map mk_def env)
+			  (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, replace_fterms rhs)))
+			  (fn x => LocalDefs.unfold_tac (#context x) (map tryext (#prems x)) 
+							THEN rtac th' 1)) RS sym
+	      
+   val (cong' :: vars') = 
+       Variable.export ctxt'' ctxt (cong :: map (Drule.mk_term o cert) vs)
+   val vs' = map (fst o fst o Term.dest_Var o Thm.term_of o Drule.dest_term) vars'
+					      
+  in  (vs', cong') end; 
+ (* congs is a list of pairs (P,th) where th is a theorem for *)
+        (* [| f p1 = A1; ...; f pn = An|] ==> f (C p1 .. pn) = P *)
+val FWD = curry (op OF);
+
+ (* da is the decomposition for atoms, ie. it returns ([],g) where g
+ returns the right instance f (AtC n) = t , where AtC is the Atoms
+ constructor and n is the number of the atom corresponding to t *)
+
+(* Generic decomp for reification : matches the actual term with the
+rhs of one cong rule. The result of the matching guides the
+proof synthesis: The matches of the introduced Variables A1 .. An are
+processed recursively
+ The rest is instantiated in the cong rule,i.e. no reification is needed *)
+
+exception REIF of string;
+
+val bds = ref ([]: (typ * ((term list) * (term list))) list);
+
+fun index_of t = 
+ let 
+  val tt = HOLogic.listT (fastype_of t)
+ in 
+  (case AList.lookup Type.could_unify (!bds) tt of
+    NONE => error "index_of : type not found in environements!"
+  | SOME (tbs,tats) =>
+    let
+     val i = find_index_eq t tats
+     val j = find_index_eq t tbs 
+    in (if j= ~1 then 
+	    if i= ~1 
+	    then (bds := AList.update Type.could_unify (tt,(tbs,tats@[t])) (!bds) ; 
+		  length tbs + length tats) 
+	    else i else j)
+    end)
+ end;
+
+fun dest_listT (Type ("List.list", [T])) = T;
+
+fun decomp_genreif da cgns (t,ctxt) =
+ let 
+  val thy = ProofContext.theory_of ctxt 
+  val cert = cterm_of thy
+  fun tryabsdecomp (s,ctxt) = 
+   (case s of 
+     Abs(xn,xT,ta) => 
+     (let
+       val ([xn],ctxt') = Variable.variant_fixes ["x"] ctxt
+       val (xn,ta) = variant_abs (xn,xT,ta)
+       val x = Free(xn,xT)
+       val _ = (case AList.lookup Type.could_unify (!bds) (HOLogic.listT xT)
+		 of NONE => error "tryabsdecomp: Type not found in the Environement"
+		  | SOME (bsT,atsT) => 
+		    (bds := AList.update Type.could_unify (HOLogic.listT xT, ((x::bsT), atsT)) (!bds)))
+      in ([(ta, ctxt')] , 
+	  fn [th] => ((let val (bsT,asT) = the(AList.lookup Type.could_unify (!bds) (HOLogic.listT xT))
+		       in (bds := AList.update Type.could_unify (HOLogic.listT xT,(tl bsT,asT)) (!bds))
+		       end) ; 
+		      hd (Variable.export ctxt' ctxt [(forall_intr (cert x) th) COMP allI])))
+	end)
+    | _ => da (s,ctxt))
+  in 
+  (case cgns of 
+    [] => tryabsdecomp (t,ctxt)
+  | ((vns,cong)::congs) => ((let
+        val cert = cterm_of thy
+	val certy = ctyp_of thy
+        val (tyenv, tmenv) =
+        Pattern.match thy
+        ((fst o HOLogic.dest_eq o HOLogic.dest_Trueprop) (concl_of cong), t)
+        (Envir.type_env (Envir.empty 0), Vartab.empty)
+        val (fnvs,invs) = List.partition (fn ((vn,_),_) => vn mem vns) (Vartab.dest tmenv)
+        val (fts,its) = 
+	    (map (snd o snd) fnvs,
+             map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t)) invs)
+	val ctyenv = map (fn ((vn,vi),(s,ty)) => (certy (TVar((vn,vi),s)), certy ty)) (Vartab.dest tyenv)
+    in (fts ~~ (replicate (length fts) ctxt), FWD (instantiate (ctyenv, its) cong))
+    end)
+      handle MATCH => decomp_genreif da congs (t,ctxt)))
+  end;
+
+ (* looks for the atoms equation and instantiates it with the right number *)
+
+
+fun mk_decompatom eqs (t,ctxt) =
+let 
+ val tT = fastype_of t
+ fun isat eq = 
+  let 
+   val rhs = eq |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
+   in exists_Const 
+	  (fn (n,ty) => n="List.nth" 
+			andalso 
+			AList.defined Type.could_unify (!bds) (domain_type ty)) rhs 
+	  andalso Type.could_unify (fastype_of rhs, tT)
+   end
+ fun get_nths t acc = 
+  case t of
+    Const("List.nth",_)$vs$n => insert (fn ((a,_),(b,_)) => a aconv b) (t,(vs,n)) acc
+  | t1$t2 => get_nths t1 (get_nths t2 acc)
+  | Abs(_,_,t') => get_nths t'  acc
+  | _ => acc
+
+ fun 
+   tryeqs [] = error "Can not find the atoms equation"
+ | tryeqs (eq::eqs) = ((
+  let 
+   val rhs = eq |> prop_of |> HOLogic.dest_Trueprop  |> HOLogic.dest_eq |> snd
+   val nths = get_nths rhs []
+   val (vss,ns) = fold_rev (fn (_,(vs,n)) => fn (vss,ns) => 
+                             (insert (op aconv) vs vss, insert (op aconv) n ns)) nths ([],[]) 
+   val (vsns, ctxt') = Variable.variant_fixes (replicate (length vss) "vs") ctxt
+   val (xns, ctxt'') = Variable.variant_fixes (replicate (length nths) "x") ctxt' 
+   val thy = ProofContext.theory_of ctxt''
+   val cert = cterm_of thy
+   val certT = ctyp_of thy
+   val vsns_map = vss ~~ vsns
+   val xns_map = (fst (split_list nths)) ~~ xns
+   val subst = map (fn (nt, xn) => (nt, Var ((xn,0), fastype_of nt))) xns_map
+   val rhs_P = subst_free subst rhs
+   val (tyenv, tmenv) = Pattern.match 
+	                    thy (rhs_P, t)
+	                    (Envir.type_env (Envir.empty 0), Vartab.empty)
+   val sbst = Envir.subst_vars (tyenv, tmenv)
+   val sbsT = Envir.typ_subst_TVars tyenv
+   val subst_ty = map (fn (n,(s,t)) => (certT (TVar (n, s)), certT t)) 
+                      (Vartab.dest tyenv)
+   val tml = Vartab.dest tmenv
+   val t's = map (fn xn => snd (valOf (AList.lookup (op =) tml (xn,0)))) xns (* FIXME : Express with sbst*)
+   val subst_ns = map (fn (Const _ $ vs $ n, Var (xn0,T)) => 
+                          (cert n, snd (valOf (AList.lookup (op =) tml xn0)) 
+                             |> (index_of #> HOLogic.mk_nat #> cert))) 
+                      subst
+   val subst_vs = 
+    let 
+     fun ty (Const _ $ (vs as Var (vsn,lT)) $ n, Var (xn0,T)) = (certT T, certT (sbsT T))
+     fun h (Const _ $ (vs as Var (vsn,lT)) $ n, Var (xn0,T)) = 
+      let 
+       val cns = sbst (Const("List.list.Cons", T --> lT --> lT))
+       val lT' = sbsT lT
+       val (bsT,asT) = the (AList.lookup Type.could_unify (!bds) lT)
+       val vsn = valOf (AList.lookup (op =) vsns_map vs)
+       val cvs = cert (fold_rev (fn x => fn xs => cns$x$xs) bsT (Free (vsn, lT')))
+      in (cert vs, cvs) end
+    in map h subst end
+   val cts = map (fn ((vn,vi),(tT,t)) => (cert(Var ((vn,vi),tT)), cert t)) 
+                 (fold (AList.delete (fn (((a: string),_),(b,_)) => a = b)) 
+                       (map (fn n => (n,0)) xns) tml)
+   val substt = 
+    let val ih = Drule.cterm_rule (Thm.instantiate (subst_ty,[]))
+    in map (fn (v,t) => (ih v, ih t)) (subst_ns@subst_vs@cts)  end
+   val th = (instantiate (subst_ty, substt)  eq) RS sym
+  in  hd (Variable.export ctxt'' ctxt [th]) end)
+ handle MATCH => tryeqs eqs)
+in ([], fn _ => tryeqs (filter isat eqs))
+end;
+
+  (* Generic reification procedure: *)
+  (* creates all needed cong rules and then just uses the theorem synthesis *)
+
+  fun mk_congs ctxt raw_eqs = 
+ let
+  val fs = fold_rev (fn eq =>
+		     insert (op =) (eq |> prop_of |> HOLogic.dest_Trueprop 
+			 |> HOLogic.dest_eq |> fst |> strip_comb 
+			 |> fst)) raw_eqs []
+  val tys = fold_rev (fn f => fn ts => (f |> fastype_of |> binder_types |> tl) 
+				    union ts) fs []
+  val _ = bds := AList.make (fn _ => ([],[])) tys
+  val (vs, ctxt') = Variable.variant_fixes (replicate (length tys) "vs") ctxt
+  val thy = ProofContext.theory_of ctxt'
+  val cert = cterm_of thy
+  val vstys = map (fn (t,v) => (t,SOME (cert (Free(v,t))))) 
+		  (tys ~~ vs)
+  val is_Var = can dest_Var
+  fun insteq eq vs = 
+   let
+     val subst = map (fn (v as Var(n,t)) => (cert v, (valOf o valOf) (AList.lookup (op =) vstys t)))  
+  (filter is_Var vs)
+   in Thm.instantiate ([],subst) eq
+   end
+  val eqs = map (fn eq => eq |> prop_of |> HOLogic.dest_Trueprop 
+			     |> HOLogic.dest_eq |> fst |> strip_comb |> snd |> tl
+			     |> (insteq eq)) raw_eqs
+  val (ps,congs) = split_list (map (mk_congeq ctxt' fs) eqs)
+in ps ~~ (Variable.export ctxt' ctxt congs)
+end
+
+fun partition P [] = ([],[])
+  | partition P (x::xs) = 
+     let val (yes,no) = partition P xs
+     in if P x then (x::yes,no) else (yes, x::no) end
+
+fun rearrange congs = 
+let 
+ fun P (_, th) = 
+  let val @{term "Trueprop"}$(Const ("op =",_) $l$_) = concl_of th
+  in can dest_Var l end
+ val (yes,no) = partition P congs 
+ in no @ yes end
+
+
+
+fun genreif ctxt raw_eqs t =
+ let 
+  val congs = rearrange (mk_congs ctxt raw_eqs)
+  val th = divide_and_conquer (decomp_genreif (mk_decompatom raw_eqs) congs) (t,ctxt)
+  fun is_listVar (Var (_,t)) = can dest_listT t
+       | is_listVar _ = false
+  val vars = th |> prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd
+	       |> strip_comb |> snd |> filter is_listVar
+  val cert = cterm_of (ProofContext.theory_of ctxt)
+  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
+  val th' = instantiate ([], cvs) th
+  val t' = (fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) th'
+  val th'' = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, t')))
+			(fn _ => simp_tac (local_simpset_of ctxt) 1)
+  val _ = bds := []
+in FWD trans [th'',th']
+end
+
+
+fun genreflect ctxt conv corr_thms raw_eqs t =
+let 
+  val reifth = genreif ctxt raw_eqs t
+  fun trytrans [] = error "No suitable correctness theorem found"
+    | trytrans (th::ths) = 
+         (FWD trans [reifth, th RS sym] handle THM _ => trytrans ths)
+  val th = trytrans corr_thms
+  val ft = (Thm.dest_arg1 o Thm.dest_arg o Thm.dest_arg o cprop_of) th
+  val rth = conv ft
+in simplify (HOL_basic_ss addsimps raw_eqs addsimps [nth_Cons_0, nth_Cons_Suc])
+           (simplify (HOL_basic_ss addsimps [rth]) th)
+end
+
+fun genreify_tac ctxt eqs to i = (fn st =>
+  let
+    val P = HOLogic.dest_Trueprop (List.nth (prems_of st, i - 1))
+    val t = (case to of NONE => P | SOME x => x)
+    val th = (genreif ctxt eqs t) RS ssubst
+  in rtac th i st
+  end);
+
+    (* Reflection calls reification and uses the correctness *)
+        (* theorem assumed to be the dead of the list *)
+fun gen_reflection_tac ctxt conv corr_thms raw_eqs to i = (fn st =>
+  let
+    val P = HOLogic.dest_Trueprop (nth (prems_of st) (i - 1));
+    val t = the_default P to;
+    val th = genreflect ctxt conv corr_thms raw_eqs t
+      RS ssubst;
+  in (rtac th i THEN TRY(rtac TrueI i)) st end);
+
+fun reflection_tac ctxt = gen_reflection_tac ctxt Codegen.evaluation_conv;
+
+end