src/HOL/Tools/reification.ML
author wenzelm
Wed Jul 08 21:33:00 2015 +0200 (2015-07-08)
changeset 60696 8304fb4fb823
parent 60642 48dd1cefb4ae
child 60752 b48830b670a1
permissions -rw-r--r--
clarified context;
     1 (*  Title:      HOL/Tools/reification.ML
     2     Author:     Amine Chaieb, TU Muenchen
     3 
     4 A trial for automatical reification.
     5 *)
     6 
     7 signature REIFICATION =
     8 sig
     9   val conv: Proof.context -> thm list -> conv
    10   val tac: Proof.context -> thm list -> term option -> int -> tactic
    11   val lift_conv: Proof.context -> conv -> term option -> int -> tactic
    12   val dereify: Proof.context -> thm list -> conv
    13 end;
    14 
    15 structure Reification : REIFICATION =
    16 struct
    17 
    18 fun dest_listT (Type (@{type_name "list"}, [T])) = T;
    19 
    20 val FWD = curry (op OF);
    21 
    22 fun rewrite_with ctxt eqs = Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps eqs);
    23 
    24 val pure_subst = @{lemma "x == y ==> PROP P y ==> PROP P x" by simp}
    25 
    26 fun lift_conv ctxt conv some_t = Subgoal.FOCUS (fn {context = ctxt', concl, ...} =>
    27   let
    28     val ct =
    29       (case some_t of
    30         NONE => Thm.dest_arg concl
    31       | SOME t => Thm.cterm_of ctxt' t)
    32     val thm = conv ct;
    33   in
    34     if Thm.is_reflexive thm then no_tac
    35     else ALLGOALS (rtac (pure_subst OF [thm]))
    36   end) ctxt;
    37 
    38 (* Make a congruence rule out of a defining equation for the interpretation
    39 
    40    th is one defining equation of f,
    41      i.e. th is "f (Cp ?t1 ... ?tn) = P(f ?t1, .., f ?tn)" 
    42    Cp is a constructor pattern and P is a pattern 
    43 
    44    The result is:
    45      [|?A1 = f ?t1 ; .. ; ?An= f ?tn |] ==> P (?A1, .., ?An) = f (Cp ?t1 .. ?tn)
    46        + the a list of names of the A1 .. An, Those are fresh in the ctxt *)
    47 
    48 fun mk_congeq ctxt fs th =
    49   let
    50     val Const (fN, _) = th |> Thm.prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq
    51       |> fst |> strip_comb |> fst;
    52     val ((_, [th']), ctxt') = Variable.import true [th] ctxt;
    53     val (lhs, rhs) = HOLogic.dest_eq (HOLogic.dest_Trueprop (Thm.prop_of th'));
    54     fun add_fterms (t as t1 $ t2) =
    55           if exists (fn f => Term.could_unify (t |> strip_comb |> fst, f)) fs
    56           then insert (op aconv) t
    57           else add_fterms t1 #> add_fterms t2
    58       | add_fterms (t as Abs _) =
    59           if exists_Const (fn (c, _) => c = fN) t
    60           then K [t]
    61           else K []
    62       | add_fterms _ = I;
    63     val fterms = add_fterms rhs [];
    64     val (xs, ctxt'') = Variable.variant_fixes (replicate (length fterms) "x") ctxt';
    65     val tys = map fastype_of fterms;
    66     val vs = map Free (xs ~~ tys);
    67     val env = fterms ~~ vs; (*FIXME*)
    68     fun replace_fterms (t as t1 $ t2) =
    69         (case AList.lookup (op aconv) env t of
    70             SOME v => v
    71           | NONE => replace_fterms t1 $ replace_fterms t2)
    72       | replace_fterms t =
    73         (case AList.lookup (op aconv) env t of
    74             SOME v => v
    75           | NONE => t);
    76     fun mk_def (Abs (x, xT, t), v) =
    77           HOLogic.mk_Trueprop (HOLogic.all_const xT $ Abs (x, xT, HOLogic.mk_eq (v $ Bound 0, t)))
    78       | mk_def (t, v) = HOLogic.mk_Trueprop (HOLogic.mk_eq (v, t));
    79     fun tryext x =
    80       (x RS @{lemma "(\<forall>x. f x = g x) \<Longrightarrow> f = g" by blast} handle THM _ => x);
    81     val cong =
    82       (Goal.prove ctxt'' [] (map mk_def env)
    83         (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, replace_fterms rhs)))
    84         (fn {context, prems, ...} =>
    85           Local_Defs.unfold_tac context (map tryext prems) THEN rtac th' 1)) RS sym;
    86     val (cong' :: vars') =
    87       Variable.export ctxt'' ctxt (cong :: map (Drule.mk_term o Thm.cterm_of ctxt'') vs);
    88     val vs' = map (fst o fst o Term.dest_Var o Thm.term_of o Drule.dest_term) vars';
    89 
    90   in (vs', cong') end;
    91 
    92 (* congs is a list of pairs (P,th) where th is a theorem for
    93      [| f p1 = A1; ...; f pn = An|] ==> f (C p1 .. pn) = P *)
    94 
    95 fun rearrange congs =
    96   let
    97     fun P (_, th) =
    98       let val @{term "Trueprop"} $ (Const (@{const_name HOL.eq}, _) $ l $ _) = Thm.concl_of th
    99       in can dest_Var l end;
   100     val (yes, no) = List.partition P congs;
   101   in no @ yes end;
   102 
   103 fun dereify ctxt eqs =
   104   rewrite_with ctxt (eqs @ @{thms nth_Cons_0 nth_Cons_Suc});
   105 
   106 fun index_of t bds =
   107   let
   108     val tt = HOLogic.listT (fastype_of t);
   109   in
   110     (case AList.lookup Type.could_unify bds tt of
   111         NONE => error "index_of: type not found in environements!"
   112       | SOME (tbs, tats) =>
   113           let
   114             val i = find_index (fn t' => t' = t) tats;
   115             val j = find_index (fn t' => t' = t) tbs;
   116           in
   117             if j = ~1 then
   118               if i = ~1
   119               then (length tbs + length tats, AList.update Type.could_unify (tt, (tbs, tats @ [t])) bds)
   120               else (i, bds)
   121             else (j, bds)
   122           end)
   123   end;
   124 
   125 (* Generic decomp for reification : matches the actual term with the
   126    rhs of one cong rule. The result of the matching guides the
   127    proof synthesis: The matches of the introduced Variables A1 .. An are
   128    processed recursively
   129    The rest is instantiated in the cong rule,i.e. no reification is needed *)
   130 
   131 (* da is the decomposition for atoms, ie. it returns ([],g) where g
   132    returns the right instance f (AtC n) = t , where AtC is the Atoms
   133    constructor and n is the number of the atom corresponding to t *)
   134 fun decomp_reify da cgns (ct, ctxt) bds =
   135   let
   136     val thy = Proof_Context.theory_of ctxt;
   137     fun tryabsdecomp (ct, ctxt) bds =
   138       (case Thm.term_of ct of
   139         Abs (_, xT, ta) =>
   140           let
   141             val ([raw_xn], ctxt') = Variable.variant_fixes ["x"] ctxt;
   142             val (xn, ta) = Syntax_Trans.variant_abs (raw_xn, xT, ta);  (* FIXME !? *)
   143             val x = Free (xn, xT);
   144             val cx = Thm.cterm_of ctxt' x;
   145             val cta = Thm.cterm_of ctxt' ta;
   146             val bds = (case AList.lookup Type.could_unify bds (HOLogic.listT xT) of
   147                 NONE => error "tryabsdecomp: Type not found in the Environement"
   148               | SOME (bsT, atsT) => AList.update Type.could_unify (HOLogic.listT xT,
   149                   (x :: bsT, atsT)) bds);
   150            in (([(cta, ctxt')],
   151                 fn ([th], bds) =>
   152                   (hd (Variable.export ctxt' ctxt [(Thm.forall_intr cx th) COMP allI]),
   153                    let
   154                      val (bsT, asT) = the (AList.lookup Type.could_unify bds (HOLogic.listT xT));
   155                    in
   156                      AList.update Type.could_unify (HOLogic.listT xT, (tl bsT, asT)) bds
   157                    end)),
   158                bds)
   159            end
   160        | _ => da (ct, ctxt) bds)
   161   in
   162     (case cgns of
   163       [] => tryabsdecomp (ct, ctxt) bds
   164     | ((vns, cong) :: congs) =>
   165         (let
   166           val (tyenv, tmenv) =
   167             Pattern.match thy
   168               ((fst o HOLogic.dest_eq o HOLogic.dest_Trueprop) (Thm.concl_of cong), Thm.term_of ct)
   169               (Vartab.empty, Vartab.empty);
   170           val (fnvs, invs) = List.partition (fn ((vn, _),_) => member (op =) vns vn) (Vartab.dest tmenv);
   171           val (fts, its) =
   172             (map (snd o snd) fnvs,
   173              map (fn ((vn, vi), (tT, t)) => (((vn, vi), tT), Thm.cterm_of ctxt t)) invs);
   174           val ctyenv =
   175             map (fn ((vn, vi), (s, ty)) => (((vn, vi), s), Thm.ctyp_of ctxt ty))
   176               (Vartab.dest tyenv);
   177         in
   178           ((map (Thm.cterm_of ctxt) fts ~~ replicate (length fts) ctxt,
   179              apfst (FWD (Drule.instantiate_normalize (ctyenv, its) cong))), bds)
   180         end handle Pattern.MATCH => decomp_reify da congs (ct, ctxt) bds))
   181   end;
   182 
   183 fun get_nths (t as (Const (@{const_name "List.nth"}, _) $ vs $ n)) =
   184       AList.update (op aconv) (t, (vs, n))
   185   | get_nths (t1 $ t2) = get_nths t1 #> get_nths t2
   186   | get_nths (Abs (_, _, t')) = get_nths t'
   187   | get_nths _ = I;
   188 
   189 fun tryeqs [] (ct, ctxt) bds = error "Cannot find the atoms equation"
   190   | tryeqs (eq :: eqs) (ct, ctxt) bds = ((
   191       let
   192         val rhs = eq |> Thm.prop_of |> HOLogic.dest_Trueprop  |> HOLogic.dest_eq |> snd;
   193         val nths = get_nths rhs [];
   194         val (vss, _) = 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 = Proof_Context.theory_of ctxt'';
   199         val vsns_map = vss ~~ vsns;
   200         val xns_map = fst (split_list nths) ~~ xns;
   201         val subst = map (fn (nt, xn) => (nt, Var ((xn, 0), fastype_of nt))) xns_map;
   202         val rhs_P = subst_free subst rhs;
   203         val (tyenv, tmenv) = Pattern.match thy (rhs_P, Thm.term_of ct) (Vartab.empty, Vartab.empty);
   204         val sbst = Envir.subst_term (tyenv, tmenv);
   205         val sbsT = Envir.subst_type tyenv;
   206         val subst_ty =
   207           map (fn (n, (s, t)) => ((n, s), Thm.ctyp_of ctxt'' t)) (Vartab.dest tyenv)
   208         val tml = Vartab.dest tmenv;
   209         val (subst_ns, bds) = fold_map
   210           (fn (Const _ $ _ $ n, Var (xn0, _)) => fn bds =>
   211             let
   212               val name = snd (the (AList.lookup (op =) tml xn0));
   213               val (idx, bds) = index_of name bds;
   214             in (apply2 (Thm.cterm_of ctxt'') (n, idx |> HOLogic.mk_nat), bds) end) subst bds;
   215         val subst_vs =
   216           let
   217             fun h (Const _ $ (vs as Var (_, lT)) $ _, Var (_, T)) =
   218               let
   219                 val cns = sbst (Const (@{const_name "List.Cons"}, T --> lT --> lT));
   220                 val lT' = sbsT lT;
   221                 val (bsT, _) = the (AList.lookup Type.could_unify bds lT);
   222                 val vsn = the (AList.lookup (op =) vsns_map vs);
   223                 val vs' = fold_rev (fn x => fn xs => cns $ x $xs) bsT (Free (vsn, lT'));
   224               in apply2 (Thm.cterm_of ctxt'') (vs, vs') end;
   225           in map h subst end;
   226         val cts =
   227           map (fn ((vn, vi), (tT, t)) => apply2 (Thm.cterm_of ctxt'') (Var ((vn, vi), tT), t))
   228             (fold (AList.delete (fn (((a : string), _), (b, _)) => a = b))
   229               (map (fn n => (n, 0)) xns) tml);
   230         val substt =
   231           let
   232             val ih = Drule.cterm_rule (Thm.instantiate (subst_ty, []));
   233           in map (apply2 ih) (subst_ns @ subst_vs @ cts) end;
   234         val th =
   235           (Drule.instantiate_normalize (subst_ty, map (apfst (dest_Var o Thm.term_of)) substt) eq)
   236             RS sym;
   237       in (hd (Variable.export ctxt'' ctxt [th]), bds) end)
   238       handle Pattern.MATCH => tryeqs eqs (ct, ctxt) bds);
   239 
   240 (* looks for the atoms equation and instantiates it with the right number *)
   241 
   242 fun mk_decompatom eqs (ct, ctxt) bds = (([], fn (_, bds) =>
   243   let
   244     val tT = fastype_of (Thm.term_of ct);
   245     fun isat eq =
   246       let
   247         val rhs = eq |> Thm.prop_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd;
   248       in exists_Const
   249         (fn (n, ty) => n = @{const_name "List.nth"}
   250           andalso AList.defined Type.could_unify bds (domain_type ty)) rhs
   251           andalso Type.could_unify (fastype_of rhs, tT)
   252       end;
   253   in tryeqs (filter isat eqs) (ct, ctxt) bds end), bds);
   254 
   255 (* Generic reification procedure: *)
   256 (* creates all needed cong rules and then just uses the theorem synthesis *)
   257 
   258 fun mk_congs ctxt eqs =
   259   let
   260     val fs = fold_rev (fn eq => insert (op =) (eq |> Thm.prop_of |> HOLogic.dest_Trueprop
   261       |> HOLogic.dest_eq |> fst |> strip_comb
   262       |> fst)) eqs [];
   263     val tys = fold_rev (fn f => fold (insert (op =)) (f |> fastype_of |> binder_types |> tl)) fs [];
   264     val (vs, ctxt') = Variable.variant_fixes (replicate (length tys) "vs") ctxt;
   265     val subst =
   266       the o AList.lookup (op =)
   267         (map2 (fn T => fn v => (T, Thm.cterm_of ctxt' (Free (v, T)))) tys vs);
   268     fun prep_eq eq =
   269       let
   270         val (_, _ :: vs) = eq |> Thm.prop_of |> HOLogic.dest_Trueprop
   271           |> HOLogic.dest_eq |> fst |> strip_comb;
   272         val subst = map_filter (fn Var v => SOME (v, subst (#2 v)) | _ => NONE) vs;
   273       in Thm.instantiate ([], subst) eq end;
   274     val (ps, congs) = map_split (mk_congeq ctxt' fs o prep_eq) eqs;
   275     val bds = AList.make (K ([], [])) tys;
   276   in (ps ~~ Variable.export ctxt' ctxt congs, bds) end
   277 
   278 fun conv ctxt eqs ct =
   279   let
   280     val (congs, bds) = mk_congs ctxt eqs;
   281     val congs = rearrange congs;
   282     val (th, bds') =
   283       apfst mk_eq (divide_and_conquer' (decomp_reify (mk_decompatom eqs) congs) (ct, ctxt) bds);
   284     fun is_list_var (Var (_, t)) = can dest_listT t
   285       | is_list_var _ = false;
   286     val vars = th |> Thm.prop_of |> Logic.dest_equals |> snd
   287       |> strip_comb |> snd |> filter is_list_var;
   288     val vs = map (fn Var (v as (_, T)) =>
   289       (v, the (AList.lookup Type.could_unify bds' T) |> snd |> HOLogic.mk_list (dest_listT T))) vars;
   290     val th' =
   291       Drule.instantiate_normalize ([], map (apsnd (Thm.cterm_of ctxt)) vs) th;
   292     val th'' = Thm.symmetric (dereify ctxt [] (Thm.lhs_of th'));
   293   in Thm.transitive th'' th' end;
   294 
   295 fun tac ctxt eqs =
   296   lift_conv ctxt (conv ctxt eqs);
   297 
   298 end;