src/Pure/proofterm.ML
author wenzelm
Tue Jul 11 12:16:54 2006 +0200 (2006-07-11 ago)
changeset 20071 8f3e1ddb50e6
parent 20000 2fa767ab91aa
child 20147 7aa076a45cb4
permissions -rw-r--r--
replaced Term.variant(list) by Name.variant(_list);
     1 (*  Title:      Pure/proofterm.ML
     2     ID:         $Id$
     3     Author:     Stefan Berghofer, TU Muenchen
     4 
     5 LF style proof terms.
     6 *)
     7 
     8 infix 8 % %% %>;
     9 
    10 signature BASIC_PROOFTERM =
    11 sig
    12   val proofs: int ref
    13 
    14   datatype proof =
    15      PBound of int
    16    | Abst of string * typ option * proof
    17    | AbsP of string * term option * proof
    18    | % of proof * term option
    19    | %% of proof * proof
    20    | Hyp of term
    21    | PThm of (string * (string * string list) list) * proof * term * typ list option
    22    | PAxm of string * term * typ list option
    23    | Oracle of string * term * typ list option
    24    | MinProof of ((string * term) * proof) list * (string * term) list * (string * term) list;
    25 
    26   val %> : proof * term -> proof
    27 end;
    28 
    29 signature PROOFTERM =
    30 sig
    31   include BASIC_PROOFTERM
    32 
    33   val infer_derivs : (proof -> proof -> proof) -> bool * proof -> bool * proof -> bool * proof
    34   val infer_derivs' : (proof -> proof) -> (bool * proof -> bool * proof)
    35 
    36   (** primitive operations **)
    37   val proof_combt : proof * term list -> proof
    38   val proof_combt' : proof * term option list -> proof
    39   val proof_combP : proof * proof list -> proof
    40   val strip_combt : proof -> proof * term option list
    41   val strip_combP : proof -> proof * proof list
    42   val strip_thm : proof -> proof
    43   val map_proof_terms_option : (term -> term option) -> (typ -> typ option) -> proof -> proof
    44   val map_proof_terms : (term -> term) -> (typ -> typ) -> proof -> proof
    45   val fold_proof_terms : (term * 'a -> 'a) -> (typ * 'a -> 'a) -> 'a * proof -> 'a
    46   val add_prf_names : string list * proof -> string list
    47   val add_prf_tfree_names : string list * proof -> string list
    48   val add_prf_tvar_ixns : indexname list * proof -> indexname list
    49   val maxidx_of_proof : proof -> int
    50   val size_of_proof : proof -> int
    51   val change_type : typ list option -> proof -> proof
    52   val prf_abstract_over : term -> proof -> proof
    53   val prf_incr_bv : int -> int -> int -> int -> proof -> proof
    54   val incr_pboundvars : int -> int -> proof -> proof
    55   val prf_loose_bvar1 : proof -> int -> bool
    56   val prf_loose_Pbvar1 : proof -> int -> bool
    57   val prf_add_loose_bnos : int -> int -> proof ->
    58     int list * int list -> int list * int list
    59   val norm_proof : Envir.env -> proof -> proof
    60   val norm_proof' : Envir.env -> proof -> proof
    61   val prf_subst_bounds : term list -> proof -> proof
    62   val prf_subst_pbounds : proof list -> proof -> proof
    63   val freeze_thaw_prf : proof -> proof * (proof -> proof)
    64   val proof_of_min_axm : string * term -> proof
    65   val proof_of_min_thm : (string * term) * proof -> proof
    66 
    67   val thms_of_proof : proof -> (term * proof) list Symtab.table ->
    68     (term * proof) list Symtab.table
    69   val thms_of_proof' : proof -> (term * proof) list Symtab.table ->
    70     (term * proof) list Symtab.table
    71   val axms_of_proof : proof -> proof Symtab.table -> proof Symtab.table
    72   val oracles_of_proof : (string * term) list -> proof -> (string * term) list
    73 
    74   (** proof terms for specific inference rules **)
    75   val implies_intr_proof : term -> proof -> proof
    76   val forall_intr_proof : term -> string -> proof -> proof
    77   val varify_proof : term -> (string * sort) list -> proof -> proof
    78   val freezeT : term -> proof -> proof
    79   val rotate_proof : term list -> term -> int -> proof -> proof
    80   val permute_prems_prf : term list -> int -> int -> proof -> proof
    81   val generalize: string list * string list -> int -> proof -> proof
    82   val instantiate : ((indexname * sort) * typ) list * ((indexname * typ) * term) list
    83     -> proof -> proof
    84   val lift_proof : term -> int -> term -> proof -> proof
    85   val assumption_proof : term list -> term -> int -> proof -> proof
    86   val bicompose_proof : bool -> term list -> term list -> term list -> term option ->
    87     int -> proof -> proof -> proof
    88   val equality_axms : (string * term) list
    89   val reflexive_axm : proof
    90   val symmetric_axm : proof
    91   val transitive_axm : proof
    92   val equal_intr_axm : proof
    93   val equal_elim_axm : proof
    94   val abstract_rule_axm : proof
    95   val combination_axm : proof
    96   val reflexive : proof
    97   val symmetric : proof -> proof
    98   val transitive : term -> typ -> proof -> proof -> proof
    99   val abstract_rule : term -> string -> proof -> proof
   100   val combination : term -> term -> term -> term -> typ -> proof -> proof -> proof
   101   val equal_intr : term -> term -> proof -> proof -> proof
   102   val equal_elim : term -> term -> proof -> proof -> proof
   103   val axm_proof : string -> term -> proof
   104   val oracle_proof : string -> term -> proof
   105   val thm_proof : theory -> string * (string * string list) list ->
   106     term list -> term -> proof -> proof
   107   val get_name_tags : term list -> term -> proof -> string * (string * string list) list
   108 
   109   (** rewriting on proof terms **)
   110   val add_prf_rrules : (proof * proof) list -> theory -> theory
   111   val add_prf_rprocs : (string * (Term.typ list -> proof -> proof option)) list ->
   112     theory -> theory
   113   val rewrite_proof : theory -> (proof * proof) list *
   114     (string * (typ list -> proof -> proof option)) list -> proof -> proof
   115   val rewrite_proof_notypes : (proof * proof) list *
   116     (string * (typ list -> proof -> proof option)) list -> proof -> proof
   117   val init_data: theory -> theory
   118   
   119 end
   120 
   121 structure Proofterm : PROOFTERM =
   122 struct
   123 
   124 open Envir;
   125 
   126 datatype proof =
   127    PBound of int
   128  | Abst of string * typ option * proof
   129  | AbsP of string * term option * proof
   130  | op % of proof * term option
   131  | op %% of proof * proof
   132  | Hyp of term
   133  | PThm of (string * (string * string list) list) * proof * term * typ list option
   134  | PAxm of string * term * typ list option
   135  | Oracle of string * term * typ list option
   136  | MinProof of ((string * term) * proof) list * (string * term) list * (string * term) list;
   137 
   138 fun proof_of_min_axm (s, prop) = PAxm (s, prop, NONE);
   139 fun proof_of_min_thm ((s, prop), prf) = PThm ((s, []), prf, prop, NONE);
   140 
   141 val string_term_ord = prod_ord fast_string_ord Term.fast_term_ord;
   142 
   143 fun oracles_of_proof oras prf =
   144   let
   145     fun oras_of (Abst (_, _, prf)) = oras_of prf
   146       | oras_of (AbsP (_, _, prf)) = oras_of prf
   147       | oras_of (prf % _) = oras_of prf
   148       | oras_of (prf1 %% prf2) = oras_of prf1 #> oras_of prf2
   149       | oras_of (PThm ((name, _), prf, prop, _)) = (fn tabs as (thms, oras) =>
   150           case Symtab.lookup thms name of
   151             NONE => oras_of prf (Symtab.update (name, [prop]) thms, oras)
   152           | SOME ps => if member (op =) ps prop then tabs else
   153               oras_of prf (Symtab.update (name, prop::ps) thms, oras))
   154       | oras_of (Oracle (s, prop, _)) =
   155           apsnd (OrdList.insert string_term_ord (s, prop))
   156       | oras_of (MinProof (thms, _, oras)) =
   157           apsnd (OrdList.union string_term_ord oras) #>
   158           fold (oras_of o proof_of_min_thm) thms
   159       | oras_of _ = I
   160   in
   161     snd (oras_of prf (Symtab.empty, oras))
   162   end;
   163 
   164 fun thms_of_proof (Abst (_, _, prf)) = thms_of_proof prf
   165   | thms_of_proof (AbsP (_, _, prf)) = thms_of_proof prf
   166   | thms_of_proof (prf1 %% prf2) = thms_of_proof prf1 #> thms_of_proof prf2
   167   | thms_of_proof (prf % _) = thms_of_proof prf
   168   | thms_of_proof (prf' as PThm ((s, _), prf, prop, _)) = (fn tab =>
   169       case Symtab.lookup tab s of
   170         NONE => thms_of_proof prf (Symtab.update (s, [(prop, prf')]) tab)
   171       | SOME ps => if exists (fn (p, _) => p = prop) ps then tab else
   172           thms_of_proof prf (Symtab.update (s, (prop, prf')::ps) tab))
   173   | thms_of_proof (MinProof (prfs, _, _)) = fold (thms_of_proof o proof_of_min_thm) prfs
   174   | thms_of_proof _ = I;
   175 
   176 (* this version does not recursively descend into proofs of (named) theorems *)
   177 fun thms_of_proof' (Abst (_, _, prf)) = thms_of_proof' prf
   178   | thms_of_proof' (AbsP (_, _, prf)) = thms_of_proof' prf
   179   | thms_of_proof' (prf1 %% prf2) = thms_of_proof' prf1 #> thms_of_proof' prf2
   180   | thms_of_proof' (prf % _) = thms_of_proof' prf
   181   | thms_of_proof' (PThm (("", _), prf, prop, _)) = thms_of_proof' prf
   182   | thms_of_proof' (prf' as PThm ((s, _), _, prop, _)) = (fn tab =>
   183       case Symtab.lookup tab s of
   184         NONE => Symtab.update (s, [(prop, prf')]) tab
   185       | SOME ps => if exists (fn (p, _) => p = prop) ps then tab else
   186           Symtab.update (s, (prop, prf')::ps) tab)
   187   | thms_of_proof' (MinProof (prfs, _, _)) = fold (thms_of_proof' o proof_of_min_thm) prfs
   188   | thms_of_proof' _ = I;
   189 
   190 fun axms_of_proof (Abst (_, _, prf)) = axms_of_proof prf
   191   | axms_of_proof (AbsP (_, _, prf)) = axms_of_proof prf
   192   | axms_of_proof (prf1 %% prf2) = axms_of_proof prf1 #> axms_of_proof prf2
   193   | axms_of_proof (prf % _) = axms_of_proof prf
   194   | axms_of_proof (prf as PAxm (s, _, _)) = Symtab.update (s, prf)
   195   | axms_of_proof (MinProof (_, prfs, _)) = fold (axms_of_proof o proof_of_min_axm) prfs
   196   | axms_of_proof _ = I;
   197 
   198 (** collect all theorems, axioms and oracles **)
   199 
   200 fun map3 f g h (thms, axms, oras) = (f thms, g axms, h oras);
   201 
   202 fun mk_min_proof (Abst (_, _, prf)) = mk_min_proof prf
   203   | mk_min_proof (AbsP (_, _, prf)) = mk_min_proof prf
   204   | mk_min_proof (prf % _) = mk_min_proof prf
   205   | mk_min_proof (prf1 %% prf2) = mk_min_proof prf1 #> mk_min_proof prf2
   206   | mk_min_proof (PThm ((s, _), prf, prop, _)) =
   207       map3 (OrdList.insert (string_term_ord o pairself fst) ((s, prop), prf)) I I
   208   | mk_min_proof (PAxm (s, prop, _)) =
   209       map3 I (OrdList.insert string_term_ord (s, prop)) I
   210   | mk_min_proof (Oracle (s, prop, _)) =
   211       map3 I I (OrdList.insert string_term_ord (s, prop))
   212   | mk_min_proof (MinProof (thms, axms, oras)) =
   213       map3 (OrdList.union (string_term_ord o pairself fst) thms)
   214         (OrdList.union string_term_ord axms) (OrdList.union string_term_ord oras)
   215   | mk_min_proof _ = I;
   216 
   217 (** proof objects with different levels of detail **)
   218 
   219 val proofs = ref 2;
   220 
   221 fun err_illegal_level i =
   222   error ("Illegal level of detail for proof objects: " ^ string_of_int i);
   223 
   224 fun if_ora b = if b then oracles_of_proof else K;
   225 val min_proof = MinProof ([], [], []);
   226 
   227 fun infer_derivs f (ora1, prf1) (ora2, prf2) =
   228   let val ora = ora1 orelse ora2 in
   229     (ora,
   230       case !proofs of
   231         2 => f prf1 prf2
   232       | 1 => MinProof (([], [], []) |> mk_min_proof prf1 |> mk_min_proof prf2)
   233       | 0 => if ora then MinProof ([], [], if_ora ora2 (if_ora ora1 [] prf1) prf2) else min_proof
   234       | i => err_illegal_level i)
   235   end;
   236 
   237 fun infer_derivs' f = infer_derivs (K f) (false, min_proof);
   238 
   239 fun (prf %> t) = prf % SOME t;
   240 
   241 val proof_combt = Library.foldl (op %>);
   242 val proof_combt' = Library.foldl (op %);
   243 val proof_combP = Library.foldl (op %%);
   244 
   245 fun strip_combt prf = 
   246     let fun stripc (prf % t, ts) = stripc (prf, t::ts)
   247           | stripc  x =  x 
   248     in  stripc (prf, [])  end;
   249 
   250 fun strip_combP prf = 
   251     let fun stripc (prf %% prf', prfs) = stripc (prf, prf'::prfs)
   252           | stripc  x =  x
   253     in  stripc (prf, [])  end;
   254 
   255 fun strip_thm prf = (case strip_combt (fst (strip_combP prf)) of
   256       (PThm (_, prf', _, _), _) => prf'
   257     | _ => prf);
   258 
   259 val mk_Abst = foldr (fn ((s, T:typ), prf) => Abst (s, NONE, prf));
   260 fun mk_AbsP (i, prf) = funpow i (fn prf => AbsP ("H", NONE, prf)) prf;
   261 
   262 fun apsome f NONE = raise SAME
   263   | apsome f (SOME x) = (case f x of NONE => raise SAME | some => some);
   264 
   265 fun apsome' f NONE = raise SAME
   266   | apsome' f (SOME x) = SOME (f x);
   267 
   268 fun map_proof_terms_option f g =
   269   let
   270     fun map_typs (T :: Ts) =
   271           (case g T of
   272             NONE => T :: map_typs Ts
   273           | SOME T' => T' :: (map_typs Ts handle SAME => Ts))
   274       | map_typs [] = raise SAME;
   275 
   276     fun mapp (Abst (s, T, prf)) = (Abst (s, apsome g T, mapph prf)
   277           handle SAME => Abst (s, T, mapp prf))
   278       | mapp (AbsP (s, t, prf)) = (AbsP (s, apsome f t, mapph prf)
   279           handle SAME => AbsP (s, t, mapp prf))
   280       | mapp (prf % t) = (mapp prf % (apsome f t handle SAME => t)
   281           handle SAME => prf % apsome f t)
   282       | mapp (prf1 %% prf2) = (mapp prf1 %% mapph prf2
   283           handle SAME => prf1 %% mapp prf2)
   284       | mapp (PThm (a, prf, prop, SOME Ts)) =
   285           PThm (a, prf, prop, SOME (map_typs Ts))
   286       | mapp (PAxm (a, prop, SOME Ts)) =
   287           PAxm (a, prop, SOME (map_typs Ts))
   288       | mapp _ = raise SAME
   289     and mapph prf = (mapp prf handle SAME => prf)
   290 
   291   in mapph end;
   292 
   293 fun same f x =
   294   let val x' = f x
   295   in if x = x' then raise SAME else x' end;
   296 
   297 fun map_proof_terms f g =
   298   map_proof_terms_option
   299    (fn t => SOME (same f t) handle SAME => NONE)
   300    (fn T => SOME (same g T) handle SAME => NONE);
   301 
   302 fun fold_proof_terms f g (a, Abst (_, SOME T, prf)) = fold_proof_terms f g (g (T, a), prf)
   303   | fold_proof_terms f g (a, Abst (_, NONE, prf)) = fold_proof_terms f g (a, prf)
   304   | fold_proof_terms f g (a, AbsP (_, SOME t, prf)) = fold_proof_terms f g (f (t, a), prf)
   305   | fold_proof_terms f g (a, AbsP (_, NONE, prf)) = fold_proof_terms f g (a, prf)
   306   | fold_proof_terms f g (a, prf % SOME t) = f (t, fold_proof_terms f g (a, prf))
   307   | fold_proof_terms f g (a, prf % NONE) = fold_proof_terms f g (a, prf)
   308   | fold_proof_terms f g (a, prf1 %% prf2) = fold_proof_terms f g
   309       (fold_proof_terms f g (a, prf1), prf2)
   310   | fold_proof_terms _ g (a, PThm (_, _, _, SOME Ts)) = foldr g a Ts
   311   | fold_proof_terms _ g (a, PAxm (_, prop, SOME Ts)) = foldr g a Ts
   312   | fold_proof_terms _ _ (a, _) = a;
   313 
   314 val add_prf_names = fold_proof_terms add_term_names ((uncurry K) o swap);
   315 val add_prf_tfree_names = fold_proof_terms add_term_tfree_names add_typ_tfree_names;
   316 val add_prf_tvar_ixns = fold_proof_terms add_term_tvar_ixns (add_typ_ixns o swap);
   317 
   318 fun maxidx_of_proof prf = fold_proof_terms
   319   (Int.max o apfst maxidx_of_term) (Int.max o apfst maxidx_of_typ) (~1, prf); 
   320 
   321 fun size_of_proof (Abst (_, _, prf)) = 1 + size_of_proof prf
   322   | size_of_proof (AbsP (_, t, prf)) = 1 + size_of_proof prf
   323   | size_of_proof (prf1 %% prf2) = size_of_proof prf1 + size_of_proof prf2
   324   | size_of_proof (prf % _) = 1 + size_of_proof prf
   325   | size_of_proof _ = 1;
   326 
   327 fun change_type opTs (PThm (name, prf, prop, _)) = PThm (name, prf, prop, opTs)
   328   | change_type opTs (PAxm (name, prop, _)) = PAxm (name, prop, opTs)
   329   | change_type opTs (Oracle (name, prop, _)) = Oracle (name, prop, opTs)
   330   | change_type _ prf = prf;
   331 
   332 
   333 (***** utilities *****)
   334 
   335 fun strip_abs (_::Ts) (Abs (_, _, t)) = strip_abs Ts t
   336   | strip_abs _ t = t;
   337 
   338 fun mk_abs Ts t = Library.foldl (fn (t', T) => Abs ("", T, t')) (t, Ts);
   339 
   340 
   341 (*Abstraction of a proof term over its occurrences of v, 
   342     which must contain no loose bound variables.
   343   The resulting proof term is ready to become the body of an Abst.*)
   344 
   345 fun prf_abstract_over v =
   346   let
   347     fun abst' lev u = if v aconv u then Bound lev else
   348       (case u of
   349          Abs (a, T, t) => Abs (a, T, abst' (lev + 1) t)
   350        | f $ t => (abst' lev f $ absth' lev t handle SAME => f $ abst' lev t)
   351        | _ => raise SAME)
   352     and absth' lev t = (abst' lev t handle SAME => t);
   353 
   354     fun abst lev (AbsP (a, t, prf)) =
   355           (AbsP (a, apsome' (abst' lev) t, absth lev prf)
   356            handle SAME => AbsP (a, t, abst lev prf))
   357       | abst lev (Abst (a, T, prf)) = Abst (a, T, abst (lev + 1) prf)
   358       | abst lev (prf1 %% prf2) = (abst lev prf1 %% absth lev prf2
   359           handle SAME => prf1 %% abst lev prf2)
   360       | abst lev (prf % t) = (abst lev prf % Option.map (absth' lev) t
   361           handle SAME => prf % apsome' (abst' lev) t)
   362       | abst _ _ = raise SAME
   363     and absth lev prf = (abst lev prf handle SAME => prf)
   364 
   365   in absth 0 end;
   366 
   367 
   368 (*increments a proof term's non-local bound variables
   369   required when moving a proof term within abstractions
   370      inc is  increment for bound variables
   371      lev is  level at which a bound variable is considered 'loose'*)
   372 
   373 fun incr_bv' inct tlev t = incr_bv (inct, tlev, t);
   374 
   375 fun prf_incr_bv' incP inct Plev tlev (PBound i) =
   376       if i >= Plev then PBound (i+incP) else raise SAME 
   377   | prf_incr_bv' incP inct Plev tlev (AbsP (a, t, body)) =
   378       (AbsP (a, apsome' (same (incr_bv' inct tlev)) t,
   379          prf_incr_bv incP inct (Plev+1) tlev body) handle SAME =>
   380            AbsP (a, t, prf_incr_bv' incP inct (Plev+1) tlev body))
   381   | prf_incr_bv' incP inct Plev tlev (Abst (a, T, body)) =
   382       Abst (a, T, prf_incr_bv' incP inct Plev (tlev+1) body)
   383   | prf_incr_bv' incP inct Plev tlev (prf %% prf') = 
   384       (prf_incr_bv' incP inct Plev tlev prf %% prf_incr_bv incP inct Plev tlev prf'
   385        handle SAME => prf %% prf_incr_bv' incP inct Plev tlev prf')
   386   | prf_incr_bv' incP inct Plev tlev (prf % t) = 
   387       (prf_incr_bv' incP inct Plev tlev prf % Option.map (incr_bv' inct tlev) t
   388        handle SAME => prf % apsome' (same (incr_bv' inct tlev)) t)
   389   | prf_incr_bv' _ _ _ _ _ = raise SAME
   390 and prf_incr_bv incP inct Plev tlev prf =
   391       (prf_incr_bv' incP inct Plev tlev prf handle SAME => prf);
   392 
   393 fun incr_pboundvars  0 0 prf = prf
   394   | incr_pboundvars incP inct prf = prf_incr_bv incP inct 0 0 prf;
   395 
   396 
   397 fun prf_loose_bvar1 (prf1 %% prf2) k = prf_loose_bvar1 prf1 k orelse prf_loose_bvar1 prf2 k
   398   | prf_loose_bvar1 (prf % SOME t) k = prf_loose_bvar1 prf k orelse loose_bvar1 (t, k)
   399   | prf_loose_bvar1 (_ % NONE) _ = true
   400   | prf_loose_bvar1 (AbsP (_, SOME t, prf)) k = loose_bvar1 (t, k) orelse prf_loose_bvar1 prf k
   401   | prf_loose_bvar1 (AbsP (_, NONE, _)) k = true
   402   | prf_loose_bvar1 (Abst (_, _, prf)) k = prf_loose_bvar1 prf (k+1)
   403   | prf_loose_bvar1 _ _ = false;
   404 
   405 fun prf_loose_Pbvar1 (PBound i) k = i = k
   406   | prf_loose_Pbvar1 (prf1 %% prf2) k = prf_loose_Pbvar1 prf1 k orelse prf_loose_Pbvar1 prf2 k
   407   | prf_loose_Pbvar1 (prf % _) k = prf_loose_Pbvar1 prf k
   408   | prf_loose_Pbvar1 (AbsP (_, _, prf)) k = prf_loose_Pbvar1 prf (k+1)
   409   | prf_loose_Pbvar1 (Abst (_, _, prf)) k = prf_loose_Pbvar1 prf k
   410   | prf_loose_Pbvar1 _ _ = false;
   411 
   412 fun prf_add_loose_bnos plev tlev (PBound i) (is, js) =
   413       if i < plev then (is, js) else (insert (op =) (i-plev) is, js)
   414   | prf_add_loose_bnos plev tlev (prf1 %% prf2) p =
   415       prf_add_loose_bnos plev tlev prf2
   416         (prf_add_loose_bnos plev tlev prf1 p)
   417   | prf_add_loose_bnos plev tlev (prf % opt) (is, js) =
   418       prf_add_loose_bnos plev tlev prf (case opt of
   419           NONE => (is, insert (op =) ~1 js)
   420         | SOME t => (is, add_loose_bnos (t, tlev, js)))
   421   | prf_add_loose_bnos plev tlev (AbsP (_, opt, prf)) (is, js) =
   422       prf_add_loose_bnos (plev+1) tlev prf (case opt of
   423           NONE => (is, insert (op =) ~1 js)
   424         | SOME t => (is, add_loose_bnos (t, tlev, js)))
   425   | prf_add_loose_bnos plev tlev (Abst (_, _, prf)) p =
   426       prf_add_loose_bnos plev (tlev+1) prf p
   427   | prf_add_loose_bnos _ _ _ _ = ([], []);
   428 
   429 
   430 (**** substitutions ****)
   431 
   432 fun del_conflicting_tvars envT T = Term.instantiateT
   433   (map_filter (fn ixnS as (_, S) =>
   434      (Type.lookup (envT, ixnS); NONE) handle TYPE _ =>
   435         SOME (ixnS, TFree ("'dummy", S))) (typ_tvars T)) T;
   436 
   437 fun del_conflicting_vars env t = Term.instantiate
   438   (map_filter (fn ixnS as (_, S) =>
   439      (Type.lookup (type_env env, ixnS); NONE) handle TYPE _ =>
   440         SOME (ixnS, TFree ("'dummy", S))) (term_tvars t),
   441    map_filter (fn Var (ixnT as (_, T)) =>
   442      (Envir.lookup (env, ixnT); NONE) handle TYPE _ =>
   443         SOME (ixnT, Free ("dummy", T))) (term_vars t)) t;
   444 
   445 fun norm_proof env =
   446   let
   447     val envT = type_env env;
   448     fun msg s = warning ("type conflict in norm_proof:\n" ^ s);
   449     fun htype f t = f env t handle TYPE (s, _, _) =>
   450       (msg s; f env (del_conflicting_vars env t));
   451     fun htypeT f T = f envT T handle TYPE (s, _, _) =>
   452       (msg s; f envT (del_conflicting_tvars envT T));
   453     fun htypeTs f Ts = f envT Ts handle TYPE (s, _, _) =>
   454       (msg s; f envT (map (del_conflicting_tvars envT) Ts));
   455     fun norm (Abst (s, T, prf)) = (Abst (s, apsome' (htypeT norm_type_same) T, normh prf)
   456           handle SAME => Abst (s, T, norm prf))
   457       | norm (AbsP (s, t, prf)) = (AbsP (s, apsome' (htype norm_term_same) t, normh prf)
   458           handle SAME => AbsP (s, t, norm prf))
   459       | norm (prf % t) = (norm prf % Option.map (htype norm_term) t
   460           handle SAME => prf % apsome' (htype norm_term_same) t)
   461       | norm (prf1 %% prf2) = (norm prf1 %% normh prf2
   462           handle SAME => prf1 %% norm prf2)
   463       | norm (PThm (s, prf, t, Ts)) = PThm (s, prf, t, apsome' (htypeTs norm_types_same) Ts)
   464       | norm (PAxm (s, prop, Ts)) = PAxm (s, prop, apsome' (htypeTs norm_types_same) Ts)
   465       | norm _ = raise SAME
   466     and normh prf = (norm prf handle SAME => prf);
   467   in normh end;
   468 
   469 (***** Remove some types in proof term (to save space) *****)
   470 
   471 fun remove_types (Abs (s, _, t)) = Abs (s, dummyT, remove_types t)
   472   | remove_types (t $ u) = remove_types t $ remove_types u
   473   | remove_types (Const (s, _)) = Const (s, dummyT)
   474   | remove_types t = t;
   475 
   476 fun remove_types_env (Envir.Envir {iTs, asol, maxidx}) =
   477   Envir.Envir {iTs = iTs, asol = Vartab.map (apsnd remove_types) asol,
   478     maxidx = maxidx};
   479 
   480 fun norm_proof' env prf = norm_proof (remove_types_env env) prf;
   481 
   482 (**** substitution of bound variables ****)
   483 
   484 fun prf_subst_bounds args prf =
   485   let
   486     val n = length args;
   487     fun subst' lev (Bound i) =
   488          (if i<lev then raise SAME    (*var is locally bound*)
   489           else  incr_boundvars lev (List.nth (args, i-lev))
   490                   handle Subscript => Bound (i-n)  (*loose: change it*))
   491       | subst' lev (Abs (a, T, body)) = Abs (a, T,  subst' (lev+1) body)
   492       | subst' lev (f $ t) = (subst' lev f $ substh' lev t
   493           handle SAME => f $ subst' lev t)
   494       | subst' _ _ = raise SAME
   495     and substh' lev t = (subst' lev t handle SAME => t);
   496 
   497     fun subst lev (AbsP (a, t, body)) = (AbsP (a, apsome' (subst' lev) t, substh lev body)
   498           handle SAME => AbsP (a, t, subst lev body))
   499       | subst lev (Abst (a, T, body)) = Abst (a, T, subst (lev+1) body)
   500       | subst lev (prf %% prf') = (subst lev prf %% substh lev prf'
   501           handle SAME => prf %% subst lev prf')
   502       | subst lev (prf % t) = (subst lev prf % Option.map (substh' lev) t
   503           handle SAME => prf % apsome' (subst' lev) t)
   504       | subst _ _ = raise SAME
   505     and substh lev prf = (subst lev prf handle SAME => prf)
   506   in case args of [] => prf | _ => substh 0 prf end;
   507 
   508 fun prf_subst_pbounds args prf =
   509   let
   510     val n = length args;
   511     fun subst (PBound i) Plev tlev =
   512  	 (if i < Plev then raise SAME    (*var is locally bound*)
   513           else incr_pboundvars Plev tlev (List.nth (args, i-Plev))
   514                  handle Subscript => PBound (i-n)  (*loose: change it*))
   515       | subst (AbsP (a, t, body)) Plev tlev = AbsP (a, t, subst body (Plev+1) tlev)
   516       | subst (Abst (a, T, body)) Plev tlev = Abst (a, T, subst body Plev (tlev+1))
   517       | subst (prf %% prf') Plev tlev = (subst prf Plev tlev %% substh prf' Plev tlev
   518           handle SAME => prf %% subst prf' Plev tlev)
   519       | subst (prf % t) Plev tlev = subst prf Plev tlev % t
   520       | subst  prf _ _ = raise SAME
   521     and substh prf Plev tlev = (subst prf Plev tlev handle SAME => prf)
   522   in case args of [] => prf | _ => substh prf 0 0 end;
   523 
   524 
   525 (**** Freezing and thawing of variables in proof terms ****)
   526 
   527 fun frzT names =
   528   map_type_tvar (fn (ixn, xs) => TFree ((the o AList.lookup (op =) names) ixn, xs));
   529 
   530 fun thawT names =
   531   map_type_tfree (fn (s, xs) => case AList.lookup (op =) names s of
   532       NONE => TFree (s, xs)
   533     | SOME ixn => TVar (ixn, xs));
   534 
   535 fun freeze names names' (t $ u) =
   536       freeze names names' t $ freeze names names' u
   537   | freeze names names' (Abs (s, T, t)) =
   538       Abs (s, frzT names' T, freeze names names' t)
   539   | freeze names names' (Const (s, T)) = Const (s, frzT names' T)
   540   | freeze names names' (Free (s, T)) = Free (s, frzT names' T)
   541   | freeze names names' (Var (ixn, T)) =
   542       Free ((the o AList.lookup (op =) names) ixn, frzT names' T)
   543   | freeze names names' t = t;
   544 
   545 fun thaw names names' (t $ u) =
   546       thaw names names' t $ thaw names names' u
   547   | thaw names names' (Abs (s, T, t)) =
   548       Abs (s, thawT names' T, thaw names names' t)
   549   | thaw names names' (Const (s, T)) = Const (s, thawT names' T)
   550   | thaw names names' (Free (s, T)) = 
   551       let val T' = thawT names' T
   552       in case AList.lookup (op =) names s of
   553           NONE => Free (s, T')
   554         | SOME ixn => Var (ixn, T')
   555       end
   556   | thaw names names' (Var (ixn, T)) = Var (ixn, thawT names' T)
   557   | thaw names names' t = t;
   558 
   559 fun freeze_thaw_prf prf =
   560   let
   561     val (fs, Tfs, vs, Tvs) = fold_proof_terms
   562       (fn (t, (fs, Tfs, vs, Tvs)) =>
   563          (add_term_frees (t, fs), add_term_tfree_names (t, Tfs),
   564           add_term_vars (t, vs), add_term_tvar_ixns (t, Tvs)))
   565       (fn (T, (fs, Tfs, vs, Tvs)) =>
   566          (fs, add_typ_tfree_names (T, Tfs),
   567           vs, add_typ_ixns (Tvs, T)))
   568             (([], [], [], []), prf);
   569     val fs' = map (fst o dest_Free) fs;
   570     val vs' = map (fst o dest_Var) vs;
   571     val names = vs' ~~ Name.variant_list fs' (map fst vs');
   572     val names' = Tvs ~~ Name.variant_list Tfs (map fst Tvs);
   573     val rnames = map swap names;
   574     val rnames' = map swap names';
   575   in
   576     (map_proof_terms (freeze names names') (frzT names') prf,
   577      map_proof_terms (thaw rnames rnames') (thawT rnames'))
   578   end;
   579 
   580 
   581 (***** implication introduction *****)
   582 
   583 fun implies_intr_proof h prf =
   584   let
   585     fun abshyp i (Hyp t) = if h aconv t then PBound i else raise SAME
   586       | abshyp i (Abst (s, T, prf)) = Abst (s, T, abshyp i prf)
   587       | abshyp i (AbsP (s, t, prf)) = AbsP (s, t, abshyp (i+1) prf)
   588       | abshyp i (prf % t) = abshyp i prf % t
   589       | abshyp i (prf1 %% prf2) = (abshyp i prf1 %% abshyph i prf2
   590           handle SAME => prf1 %% abshyp i prf2)
   591       | abshyp _ _ = raise SAME
   592     and abshyph i prf = (abshyp i prf handle SAME => prf)
   593   in
   594     AbsP ("H", NONE (*h*), abshyph 0 prf)
   595   end;
   596 
   597 
   598 (***** forall introduction *****)
   599 
   600 fun forall_intr_proof x a prf = Abst (a, NONE, prf_abstract_over x prf);
   601 
   602 
   603 (***** varify *****)
   604 
   605 fun varify_proof t fixed prf =
   606   let
   607     val fs = Term.fold_types (Term.fold_atyps
   608       (fn TFree v => if member (op =) fixed v then I else insert (op =) v | _ => I)) t [];
   609     val ixns = add_term_tvar_ixns (t, []);
   610     val fmap = fs ~~ Name.variant_list (map #1 ixns) (map fst fs)
   611     fun thaw (f as (a, S)) =
   612       (case AList.lookup (op =) fmap f of
   613         NONE => TFree f
   614       | SOME b => TVar ((b, 0), S));
   615   in map_proof_terms (map_term_types (map_type_tfree thaw)) (map_type_tfree thaw) prf
   616   end;
   617 
   618 
   619 local
   620 
   621 fun new_name (ix, (pairs,used)) =
   622   let val v = Name.variant used (string_of_indexname ix)
   623   in  ((ix, v) :: pairs, v :: used)  end;
   624 
   625 fun freeze_one alist (ix, sort) = (case AList.lookup (op =) alist ix of
   626     NONE => TVar (ix, sort)
   627   | SOME name => TFree (name, sort));
   628 
   629 in
   630 
   631 fun freezeT t prf =
   632   let
   633     val used = it_term_types add_typ_tfree_names (t, [])
   634     and tvars = map #1 (it_term_types add_typ_tvars (t, []));
   635     val (alist, _) = foldr new_name ([], used) tvars;
   636   in
   637     (case alist of
   638       [] => prf (*nothing to do!*)
   639     | _ =>
   640       let val frzT = map_type_tvar (freeze_one alist)
   641       in map_proof_terms (map_term_types frzT) frzT prf end)
   642   end;
   643 
   644 end;
   645 
   646 
   647 (***** rotate assumptions *****)
   648 
   649 fun rotate_proof Bs Bi m prf =
   650   let
   651     val params = Term.strip_all_vars Bi;
   652     val asms = Logic.strip_imp_prems (Term.strip_all_body Bi);
   653     val i = length asms;
   654     val j = length Bs;
   655   in
   656     mk_AbsP (j+1, proof_combP (prf, map PBound
   657       (j downto 1) @ [mk_Abst (mk_AbsP (i,
   658         proof_combP (proof_combt (PBound i, map Bound ((length params - 1) downto 0)),
   659           map PBound (((i-m-1) downto 0) @ ((i-1) downto (i-m)))))) params]))
   660   end;
   661 
   662 
   663 (***** permute premises *****)
   664 
   665 fun permute_prems_prf prems j k prf =
   666   let val n = length prems
   667   in mk_AbsP (n, proof_combP (prf,
   668     map PBound ((n-1 downto n-j) @ (k-1 downto 0) @ (n-j-1 downto k))))
   669   end;
   670 
   671 
   672 (***** generalization *****)
   673 
   674 fun generalize (tfrees, frees) idx =
   675   map_proof_terms_option
   676     (Term.generalize_option (tfrees, frees) idx)
   677     (Term.generalizeT_option tfrees idx);
   678 
   679 
   680 (***** instantiation *****)
   681 
   682 fun instantiate (instT, inst) =
   683   map_proof_terms_option
   684     (Term.instantiate_option (instT, map (apsnd remove_types) inst))
   685     (Term.instantiateT_option instT);
   686 
   687 
   688 (***** lifting *****)
   689 
   690 fun lift_proof Bi inc prop prf =
   691   let
   692     fun lift'' Us Ts t = strip_abs Ts (Logic.incr_indexes (Us, inc) (mk_abs Ts t));
   693 
   694     fun lift' Us Ts (Abst (s, T, prf)) =
   695           (Abst (s, apsome' (same (Logic.incr_tvar inc)) T, lifth' Us (dummyT::Ts) prf)
   696            handle SAME => Abst (s, T, lift' Us (dummyT::Ts) prf))
   697       | lift' Us Ts (AbsP (s, t, prf)) =
   698           (AbsP (s, apsome' (same (lift'' Us Ts)) t, lifth' Us Ts prf)
   699            handle SAME => AbsP (s, t, lift' Us Ts prf))
   700       | lift' Us Ts (prf % t) = (lift' Us Ts prf % Option.map (lift'' Us Ts) t
   701           handle SAME => prf % apsome' (same (lift'' Us Ts)) t)
   702       | lift' Us Ts (prf1 %% prf2) = (lift' Us Ts prf1 %% lifth' Us Ts prf2
   703           handle SAME => prf1 %% lift' Us Ts prf2)
   704       | lift' _ _ (PThm (s, prf, prop, Ts)) =
   705           PThm (s, prf, prop, apsome' (same (map (Logic.incr_tvar inc))) Ts)
   706       | lift' _ _ (PAxm (s, prop, Ts)) =
   707           PAxm (s, prop, apsome' (same (map (Logic.incr_tvar inc))) Ts)
   708       | lift' _ _ _ = raise SAME
   709     and lifth' Us Ts prf = (lift' Us Ts prf handle SAME => prf);
   710 
   711     val ps = map (Logic.lift_all inc Bi) (Logic.strip_imp_prems prop);
   712     val k = length ps;
   713 
   714     fun mk_app (b, (i, j, prf)) = 
   715           if b then (i-1, j, prf %% PBound i) else (i, j-1, prf %> Bound j);
   716 
   717     fun lift Us bs i j (Const ("==>", _) $ A $ B) =
   718 	    AbsP ("H", NONE (*A*), lift Us (true::bs) (i+1) j B)
   719       | lift Us bs i j (Const ("all", _) $ Abs (a, T, t)) = 
   720 	    Abst (a, NONE (*T*), lift (T::Us) (false::bs) i (j+1) t)
   721       | lift Us bs i j _ = proof_combP (lifth' (rev Us) [] prf,
   722             map (fn k => (#3 (foldr mk_app (i-1, j-1, PBound k) bs)))
   723               (i + k - 1 downto i));
   724   in
   725     mk_AbsP (k, lift [] [] 0 0 Bi)
   726   end;
   727 
   728 
   729 (***** proof by assumption *****)
   730 
   731 fun mk_asm_prf (Const ("==>", _) $ A $ B) i = AbsP ("H", NONE (*A*), mk_asm_prf B (i+1))
   732   | mk_asm_prf (Const ("all", _) $ Abs (a, T, t)) i = Abst (a, NONE (*T*), mk_asm_prf t i)
   733   | mk_asm_prf _ i = PBound i;
   734 
   735 fun assumption_proof Bs Bi n prf =
   736   mk_AbsP (length Bs, proof_combP (prf,
   737     map PBound (length Bs - 1 downto 0) @ [mk_asm_prf Bi (~n)]));
   738 
   739 
   740 (***** Composition of object rule with proof state *****)
   741 
   742 fun flatten_params_proof i j n (Const ("==>", _) $ A $ B, k) =
   743       AbsP ("H", NONE (*A*), flatten_params_proof (i+1) j n (B, k))
   744   | flatten_params_proof i j n (Const ("all", _) $ Abs (a, T, t), k) =
   745       Abst (a, NONE (*T*), flatten_params_proof i (j+1) n (t, k))
   746   | flatten_params_proof i j n (_, k) = proof_combP (proof_combt (PBound (k+i),
   747       map Bound (j-1 downto 0)), map PBound (remove (op =) (i-n) (i-1 downto 0)));
   748 
   749 fun bicompose_proof flatten Bs oldAs newAs A n rprf sprf =
   750   let
   751     val la = length newAs;
   752     val lb = length Bs;
   753   in
   754     mk_AbsP (lb+la, proof_combP (sprf,
   755       map PBound (lb + la - 1 downto la)) %%
   756         proof_combP (rprf, (if n>0 then [mk_asm_prf (the A) (~n)] else []) @
   757           map (if flatten then flatten_params_proof 0 0 n else PBound o snd)
   758             (oldAs ~~ (la - 1 downto 0))))
   759   end;
   760 
   761 
   762 (***** axioms for equality *****)
   763 
   764 val aT = TFree ("'a", []);
   765 val bT = TFree ("'b", []);
   766 val x = Free ("x", aT);
   767 val y = Free ("y", aT);
   768 val z = Free ("z", aT);
   769 val A = Free ("A", propT);
   770 val B = Free ("B", propT);
   771 val f = Free ("f", aT --> bT);
   772 val g = Free ("g", aT --> bT);
   773 
   774 local open Logic in
   775 
   776 val equality_axms =
   777   [("reflexive", mk_equals (x, x)),
   778    ("symmetric", mk_implies (mk_equals (x, y), mk_equals (y, x))),
   779    ("transitive", list_implies ([mk_equals (x, y), mk_equals (y, z)], mk_equals (x, z))),
   780    ("equal_intr", list_implies ([mk_implies (A, B), mk_implies (B, A)], mk_equals (A, B))),
   781    ("equal_elim", list_implies ([mk_equals (A, B), A], B)),
   782    ("abstract_rule", Logic.mk_implies
   783       (all aT $ Abs ("x", aT, equals bT $ (f $ Bound 0) $ (g $ Bound 0)),
   784        equals (aT --> bT) $
   785          Abs ("x", aT, f $ Bound 0) $ Abs ("x", aT, g $ Bound 0))),
   786    ("combination", Logic.list_implies
   787       ([Logic.mk_equals (f, g), Logic.mk_equals (x, y)],
   788        Logic.mk_equals (f $ x, g $ y)))];
   789 
   790 val [reflexive_axm, symmetric_axm, transitive_axm, equal_intr_axm,
   791   equal_elim_axm, abstract_rule_axm, combination_axm] =
   792     map (fn (s, t) => PAxm ("ProtoPure." ^ s, varify t, NONE)) equality_axms;
   793 
   794 end;
   795 
   796 val reflexive = reflexive_axm % NONE;
   797 
   798 fun symmetric (prf as PAxm ("ProtoPure.reflexive", _, _) % _) = prf
   799   | symmetric prf = symmetric_axm % NONE % NONE %% prf;
   800 
   801 fun transitive _ _ (PAxm ("ProtoPure.reflexive", _, _) % _) prf2 = prf2
   802   | transitive _ _ prf1 (PAxm ("ProtoPure.reflexive", _, _) % _) = prf1
   803   | transitive u (Type ("prop", [])) prf1 prf2 =
   804       transitive_axm % NONE % SOME (remove_types u) % NONE %% prf1 %% prf2
   805   | transitive u T prf1 prf2 =
   806       transitive_axm % NONE % NONE % NONE %% prf1 %% prf2;
   807 
   808 fun abstract_rule x a prf =
   809   abstract_rule_axm % NONE % NONE %% forall_intr_proof x a prf;
   810 
   811 fun check_comb (PAxm ("ProtoPure.combination", _, _) % f % g % _ % _ %% prf %% _) =
   812       is_some f orelse check_comb prf
   813   | check_comb (PAxm ("ProtoPure.transitive", _, _) % _ % _ % _ %% prf1 %% prf2) =
   814       check_comb prf1 andalso check_comb prf2
   815   | check_comb (PAxm ("ProtoPure.symmetric", _, _) % _ % _ %% prf) = check_comb prf
   816   | check_comb _ = false;
   817 
   818 fun combination f g t u (Type (_, [T, U])) prf1 prf2 =
   819   let
   820     val f = Envir.beta_norm f;
   821     val g = Envir.beta_norm g;
   822     val prf =  if check_comb prf1 then
   823         combination_axm % NONE % NONE
   824       else (case prf1 of
   825           PAxm ("ProtoPure.reflexive", _, _) % _ =>
   826             combination_axm %> remove_types f % NONE
   827         | _ => combination_axm %> remove_types f %> remove_types g)
   828   in
   829     (case T of
   830        Type ("fun", _) => prf %
   831          (case head_of f of
   832             Abs _ => SOME (remove_types t)
   833           | Var _ => SOME (remove_types t)
   834           | _ => NONE) %
   835          (case head_of g of
   836             Abs _ => SOME (remove_types u)
   837           | Var _ => SOME (remove_types u)
   838           | _ => NONE) %% prf1 %% prf2
   839      | _ => prf % NONE % NONE %% prf1 %% prf2)
   840   end;
   841 
   842 fun equal_intr A B prf1 prf2 =
   843   equal_intr_axm %> remove_types A %> remove_types B %% prf1 %% prf2;
   844 
   845 fun equal_elim A B prf1 prf2 =
   846   equal_elim_axm %> remove_types A %> remove_types B %% prf1 %% prf2;
   847 
   848 
   849 (***** axioms and theorems *****)
   850 
   851 fun vars_of t = rev (fold_aterms (fn v as Var _ => insert (op =) v | _ => I) t []);
   852 
   853 fun test_args _ [] = true
   854   | test_args is (Bound i :: ts) =
   855       not (member (op =) is i) andalso test_args (i :: is) ts
   856   | test_args _ _ = false;
   857 
   858 fun is_fun (Type ("fun", _)) = true
   859   | is_fun (TVar _) = true
   860   | is_fun _ = false;
   861 
   862 fun add_funvars Ts (vs, t) =
   863   if is_fun (fastype_of1 (Ts, t)) then
   864     vs union map_filter (fn Var (ixn, T) =>
   865       if is_fun T then SOME ixn else NONE | _ => NONE) (vars_of t)
   866   else vs;
   867 
   868 fun add_npvars q p Ts (vs, Const ("==>", _) $ t $ u) =
   869       add_npvars q p Ts (add_npvars q (not p) Ts (vs, t), u)
   870   | add_npvars q p Ts (vs, Const ("all", Type (_, [Type (_, [T, _]), _])) $ t) =
   871       add_npvars q p Ts (vs, if p andalso q then betapply (t, Var (("",0), T)) else t)
   872   | add_npvars q p Ts (vs, Abs (_, T, t)) = add_npvars q p (T::Ts) (vs, t)
   873   | add_npvars _ _ Ts (vs, t) = add_npvars' Ts (vs, t)
   874 and add_npvars' Ts (vs, t) = (case strip_comb t of
   875     (Var (ixn, _), ts) => if test_args [] ts then vs
   876       else Library.foldl (add_npvars' Ts)
   877         (AList.update (op =) (ixn,
   878           Library.foldl (add_funvars Ts) ((these ooo AList.lookup) (op =) vs ixn, ts)) vs, ts)
   879   | (Abs (_, T, u), ts) => Library.foldl (add_npvars' (T::Ts)) (vs, u :: ts)
   880   | (_, ts) => Library.foldl (add_npvars' Ts) (vs, ts));
   881 
   882 fun prop_vars (Const ("==>", _) $ P $ Q) = prop_vars P union prop_vars Q
   883   | prop_vars (Const ("all", _) $ Abs (_, _, t)) = prop_vars t
   884   | prop_vars t = (case strip_comb t of
   885       (Var (ixn, _), _) => [ixn] | _ => []);
   886 
   887 fun is_proj t =
   888   let
   889     fun is_p i t = (case strip_comb t of
   890         (Bound j, []) => false
   891       | (Bound j, ts) => j >= i orelse exists (is_p i) ts
   892       | (Abs (_, _, u), _) => is_p (i+1) u
   893       | (_, ts) => exists (is_p i) ts)
   894   in (case strip_abs_body t of
   895         Bound _ => true
   896       | t' => is_p 0 t')
   897   end;
   898 
   899 fun needed_vars prop = 
   900   Library.foldl op union ([], map op ins (add_npvars true true [] ([], prop))) union
   901   prop_vars prop;
   902 
   903 fun gen_axm_proof c name prop =
   904   let
   905     val nvs = needed_vars prop;
   906     val args = map (fn (v as Var (ixn, _)) =>
   907         if member (op =) nvs ixn then SOME v else NONE) (vars_of prop) @
   908       map SOME (sort Term.term_ord (term_frees prop));
   909   in
   910     proof_combt' (c (name, prop, NONE), args)
   911   end;
   912 
   913 val axm_proof = gen_axm_proof PAxm;
   914 
   915 val dummy = Const (Term.dummy_patternN, dummyT);
   916 
   917 fun oracle_proof name prop =
   918   if !proofs = 0 then Oracle (name, dummy, NONE)
   919   else gen_axm_proof Oracle name prop;
   920 
   921 fun shrink_proof thy =
   922   let
   923     val compress_typ = Compress.typ thy;
   924     val compress_term = Compress.term thy;
   925   
   926     fun shrink ls lev (prf as Abst (a, T, body)) =
   927           let val (b, is, ch, body') = shrink ls (lev+1) body
   928           in (b, is, ch, if ch then Abst (a, Option.map compress_typ T, body') else prf) end
   929       | shrink ls lev (prf as AbsP (a, t, body)) =
   930           let val (b, is, ch, body') = shrink (lev::ls) lev body
   931           in (b orelse member (op =) is 0, map_filter (fn 0 => NONE | i => SOME (i-1)) is,
   932             ch, if ch then AbsP (a, Option.map compress_term t, body') else prf)
   933           end
   934       | shrink ls lev prf =
   935           let val (is, ch, _, prf') = shrink' ls lev [] [] prf
   936           in (false, is, ch, prf') end
   937     and shrink' ls lev ts prfs (prf as prf1 %% prf2) =
   938           let
   939             val p as (_, is', ch', prf') = shrink ls lev prf2;
   940             val (is, ch, ts', prf'') = shrink' ls lev ts (p::prfs) prf1
   941           in (is union is', ch orelse ch', ts',
   942               if ch orelse ch' then prf'' %% prf' else prf)
   943           end
   944       | shrink' ls lev ts prfs (prf as prf1 % t) =
   945           let val (is, ch, (ch', t')::ts', prf') = shrink' ls lev (t::ts) prfs prf1
   946           in (is, ch orelse ch', ts',
   947               if ch orelse ch' then prf' % Option.map compress_term t' else prf) end
   948       | shrink' ls lev ts prfs (prf as PBound i) =
   949           (if exists (fn SOME (Bound j) => lev-j <= List.nth (ls, i) | _ => true) ts
   950              orelse has_duplicates (op =)
   951                (Library.foldl (fn (js, SOME (Bound j)) => j :: js | (js, _) => js) ([], ts))
   952              orelse exists #1 prfs then [i] else [], false, map (pair false) ts, prf)
   953       | shrink' ls lev ts prfs (Hyp t) = ([], false, map (pair false) ts, Hyp (compress_term t))
   954       | shrink' ls lev ts prfs (prf as MinProof _) =
   955           ([], false, map (pair false) ts, prf)
   956       | shrink' ls lev ts prfs prf =
   957           let
   958             val prop = (case prf of PThm (_, _, prop, _) => prop | PAxm (_, prop, _) => prop
   959               | Oracle (_, prop, _) => prop | _ => error "shrink: proof not in normal form");
   960             val vs = vars_of prop;
   961             val (ts', ts'') = chop (length vs) ts;
   962             val insts = Library.take (length ts', map (fst o dest_Var) vs) ~~ ts';
   963             val nvs = Library.foldl (fn (ixns', (ixn, ixns)) =>
   964               insert (op =) ixn (case AList.lookup (op =) insts ixn of
   965                   SOME (SOME t) => if is_proj t then ixns union ixns' else ixns'
   966                 | _ => ixns union ixns'))
   967                   (needed prop ts'' prfs, add_npvars false true [] ([], prop));
   968             val insts' = map
   969               (fn (ixn, x as SOME _) => if member (op =) nvs ixn then (false, x) else (true, NONE)
   970                 | (_, x) => (false, x)) insts
   971           in ([], false, insts' @ map (pair false) ts'', prf) end
   972     and needed (Const ("==>", _) $ t $ u) ts ((b, _, _, _)::prfs) =
   973           (if b then map (fst o dest_Var) (vars_of t) else []) union needed u ts prfs
   974       | needed (Var (ixn, _)) (_::_) _ = [ixn]
   975       | needed _ _ _ = [];
   976   in shrink end;
   977 
   978 
   979 (**** Simple first order matching functions for terms and proofs ****)
   980 
   981 exception PMatch;
   982 
   983 (** see pattern.ML **)
   984 
   985 fun flt (i: int) = List.filter (fn n => n < i);
   986 
   987 fun fomatch Ts tymatch j =
   988   let
   989     fun mtch (instsp as (tyinsts, insts)) = fn
   990         (Var (ixn, T), t)  =>
   991           if j>0 andalso not (null (flt j (loose_bnos t)))
   992           then raise PMatch
   993           else (tymatch (tyinsts, fn () => (T, fastype_of1 (Ts, t))),
   994             (ixn, t) :: insts)
   995       | (Free (a, T), Free (b, U)) =>
   996 	  if a=b then (tymatch (tyinsts, K (T, U)), insts) else raise PMatch
   997       | (Const (a, T), Const (b, U))  =>
   998 	  if a=b then (tymatch (tyinsts, K (T, U)), insts) else raise PMatch
   999       | (f $ t, g $ u) => mtch (mtch instsp (f, g)) (t, u)
  1000       | (Bound i, Bound j) => if i=j then instsp else raise PMatch
  1001       | _ => raise PMatch
  1002   in mtch end;
  1003 
  1004 fun match_proof Ts tymatch =
  1005   let
  1006     fun optmatch _ inst (NONE, _) = inst
  1007       | optmatch _ _ (SOME _, NONE) = raise PMatch
  1008       | optmatch mtch inst (SOME x, SOME y) = mtch inst (x, y)
  1009 
  1010     fun matcht Ts j (pinst, tinst) (t, u) =
  1011       (pinst, fomatch Ts tymatch j tinst (t, Envir.beta_norm u));
  1012     fun matchT (pinst, (tyinsts, insts)) p =
  1013       (pinst, (tymatch (tyinsts, K p), insts));
  1014     fun matchTs inst (Ts, Us) = Library.foldl (uncurry matchT) (inst, Ts ~~ Us);
  1015 
  1016     fun mtch Ts i j (pinst, tinst) (Hyp (Var (ixn, _)), prf) =
  1017           if i = 0 andalso j = 0 then ((ixn, prf) :: pinst, tinst)
  1018           else (case apfst (flt i) (apsnd (flt j)
  1019                   (prf_add_loose_bnos 0 0 prf ([], []))) of
  1020               ([], []) => ((ixn, incr_pboundvars (~i) (~j) prf) :: pinst, tinst)
  1021             | ([], _) => if j = 0 then
  1022                    ((ixn, incr_pboundvars (~i) (~j) prf) :: pinst, tinst)
  1023                  else raise PMatch
  1024             | _ => raise PMatch)
  1025       | mtch Ts i j inst (prf1 % opt1, prf2 % opt2) =
  1026           optmatch (matcht Ts j) (mtch Ts i j inst (prf1, prf2)) (opt1, opt2)
  1027       | mtch Ts i j inst (prf1 %% prf2, prf1' %% prf2') =
  1028           mtch Ts i j (mtch Ts i j inst (prf1, prf1')) (prf2, prf2')
  1029       | mtch Ts i j inst (Abst (_, opT, prf1), Abst (_, opU, prf2)) =
  1030           mtch (the_default dummyT opU :: Ts) i (j+1)
  1031             (optmatch matchT inst (opT, opU)) (prf1, prf2)
  1032       | mtch Ts i j inst (prf1, Abst (_, opU, prf2)) =
  1033           mtch (the_default dummyT opU :: Ts) i (j+1) inst
  1034             (incr_pboundvars 0 1 prf1 %> Bound 0, prf2)
  1035       | mtch Ts i j inst (AbsP (_, opt, prf1), AbsP (_, opu, prf2)) =
  1036           mtch Ts (i+1) j (optmatch (matcht Ts j) inst (opt, opu)) (prf1, prf2)
  1037       | mtch Ts i j inst (prf1, AbsP (_, _, prf2)) =
  1038           mtch Ts (i+1) j inst (incr_pboundvars 1 0 prf1 %% PBound 0, prf2)
  1039       | mtch Ts i j inst (PThm ((name1, _), _, prop1, opTs),
  1040             PThm ((name2, _), _, prop2, opUs)) =
  1041           if name1=name2 andalso prop1=prop2 then
  1042             optmatch matchTs inst (opTs, opUs)
  1043           else raise PMatch
  1044       | mtch Ts i j inst (PAxm (s1, _, opTs), PAxm (s2, _, opUs)) =
  1045           if s1=s2 then optmatch matchTs inst (opTs, opUs)
  1046           else raise PMatch
  1047       | mtch _ _ _ inst (PBound i, PBound j) = if i = j then inst else raise PMatch
  1048       | mtch _ _ _ _ _ = raise PMatch
  1049   in mtch Ts 0 0 end;
  1050 
  1051 fun prf_subst (pinst, (tyinsts, insts)) =
  1052   let
  1053     val substT = Envir.typ_subst_TVars tyinsts;
  1054 
  1055     fun subst' lev (t as Var (ixn, _)) = (case AList.lookup (op =) insts ixn of
  1056           NONE => t
  1057         | SOME u => incr_boundvars lev u)
  1058       | subst' lev (Const (s, T)) = Const (s, substT T)
  1059       | subst' lev (Free (s, T)) = Free (s, substT T)
  1060       | subst' lev (Abs (a, T, body)) = Abs (a, substT T, subst' (lev+1) body)
  1061       | subst' lev (f $ t) = subst' lev f $ subst' lev t
  1062       | subst' _ t = t;
  1063 
  1064     fun subst plev tlev (AbsP (a, t, body)) =
  1065           AbsP (a, Option.map (subst' tlev) t, subst (plev+1) tlev body)
  1066       | subst plev tlev (Abst (a, T, body)) =
  1067           Abst (a, Option.map substT T, subst plev (tlev+1) body)
  1068       | subst plev tlev (prf %% prf') = subst plev tlev prf %% subst plev tlev prf'
  1069       | subst plev tlev (prf % t) = subst plev tlev prf % Option.map (subst' tlev) t
  1070       | subst plev tlev (prf as Hyp (Var (ixn, _))) = (case AList.lookup (op =) pinst ixn of
  1071           NONE => prf
  1072         | SOME prf' => incr_pboundvars plev tlev prf')
  1073       | subst _ _ (PThm (id, prf, prop, Ts)) =
  1074           PThm (id, prf, prop, Option.map (map substT) Ts)
  1075       | subst _ _ (PAxm (id, prop, Ts)) =
  1076           PAxm (id, prop, Option.map (map substT) Ts)
  1077       | subst _ _ t = t
  1078   in subst 0 0 end;
  1079 
  1080 (*A fast unification filter: true unless the two terms cannot be unified. 
  1081   Terms must be NORMAL.  Treats all Vars as distinct. *)
  1082 fun could_unify prf1 prf2 =
  1083   let
  1084     fun matchrands (prf1 %% prf2) (prf1' %% prf2') =
  1085           could_unify prf2 prf2' andalso matchrands prf1 prf1'
  1086       | matchrands (prf % SOME t) (prf' % SOME t') =
  1087           Term.could_unify (t, t') andalso matchrands prf prf'
  1088       | matchrands (prf % _) (prf' % _) = matchrands prf prf'
  1089       | matchrands _ _ = true
  1090 
  1091     fun head_of (prf %% _) = head_of prf
  1092       | head_of (prf % _) = head_of prf
  1093       | head_of prf = prf
  1094 
  1095   in case (head_of prf1, head_of prf2) of
  1096         (_, Hyp (Var _)) => true
  1097       | (Hyp (Var _), _) => true
  1098       | (PThm ((a, _), _, propa, _), PThm ((b, _), _, propb, _)) =>
  1099           a = b andalso propa = propb andalso matchrands prf1 prf2
  1100       | (PAxm (a, _, _), PAxm (b, _, _)) => a = b andalso matchrands prf1 prf2
  1101       | (PBound i, PBound j) =>  i = j andalso matchrands prf1 prf2
  1102       | (AbsP _, _) =>  true   (*because of possible eta equality*)
  1103       | (Abst _, _) =>  true
  1104       | (_, AbsP _) =>  true
  1105       | (_, Abst _) =>  true
  1106       | _ => false
  1107   end;
  1108 
  1109 (**** rewriting on proof terms ****)
  1110 
  1111 val skel0 = PBound 0;
  1112 
  1113 fun rewrite_prf tymatch (rules, procs) prf =
  1114   let
  1115     fun rew _ (Abst (_, _, body) % SOME t) = SOME (prf_subst_bounds [t] body, skel0)
  1116       | rew _ (AbsP (_, _, body) %% prf) = SOME (prf_subst_pbounds [prf] body, skel0)
  1117       | rew Ts prf = (case get_first (fn (_, r) => r Ts prf) procs of
  1118           SOME prf' => SOME (prf', skel0)
  1119         | NONE => get_first (fn (prf1, prf2) => SOME (prf_subst
  1120             (match_proof Ts tymatch ([], (Vartab.empty, [])) (prf1, prf)) prf2, prf2)
  1121                handle PMatch => NONE) (List.filter (could_unify prf o fst) rules));
  1122 
  1123     fun rew0 Ts (prf as AbsP (_, _, prf' %% PBound 0)) =
  1124           if prf_loose_Pbvar1 prf' 0 then rew Ts prf
  1125           else
  1126             let val prf'' = incr_pboundvars (~1) 0 prf'
  1127             in SOME (the_default (prf'', skel0) (rew Ts prf'')) end
  1128       | rew0 Ts (prf as Abst (_, _, prf' % SOME (Bound 0))) =
  1129           if prf_loose_bvar1 prf' 0 then rew Ts prf
  1130           else
  1131             let val prf'' = incr_pboundvars 0 (~1) prf'
  1132             in SOME (the_default (prf'', skel0) (rew Ts prf'')) end
  1133       | rew0 Ts prf = rew Ts prf;
  1134 
  1135     fun rew1 _ (Hyp (Var _)) _ = NONE
  1136       | rew1 Ts skel prf = (case rew2 Ts skel prf of
  1137           SOME prf1 => (case rew0 Ts prf1 of
  1138               SOME (prf2, skel') => SOME (the_default prf2 (rew1 Ts skel' prf2))
  1139             | NONE => SOME prf1)
  1140         | NONE => (case rew0 Ts prf of
  1141               SOME (prf1, skel') => SOME (the_default prf1 (rew1 Ts skel' prf1))
  1142             | NONE => NONE))
  1143 
  1144     and rew2 Ts skel (prf % SOME t) = (case prf of
  1145             Abst (_, _, body) =>
  1146               let val prf' = prf_subst_bounds [t] body
  1147               in SOME (the_default prf' (rew2 Ts skel0 prf')) end
  1148           | _ => (case rew1 Ts (case skel of skel' % _ => skel' | _ => skel0) prf of
  1149               SOME prf' => SOME (prf' % SOME t)
  1150             | NONE => NONE))
  1151       | rew2 Ts skel (prf % NONE) = Option.map (fn prf' => prf' % NONE)
  1152           (rew1 Ts (case skel of skel' % _ => skel' | _ => skel0) prf)
  1153       | rew2 Ts skel (prf1 %% prf2) = (case prf1 of
  1154             AbsP (_, _, body) =>
  1155               let val prf' = prf_subst_pbounds [prf2] body
  1156               in SOME (the_default prf' (rew2 Ts skel0 prf')) end
  1157           | _ =>
  1158             let val (skel1, skel2) = (case skel of
  1159                 skel1 %% skel2 => (skel1, skel2)
  1160               | _ => (skel0, skel0))
  1161             in case rew1 Ts skel1 prf1 of
  1162                 SOME prf1' => (case rew1 Ts skel2 prf2 of
  1163                     SOME prf2' => SOME (prf1' %% prf2')
  1164                   | NONE => SOME (prf1' %% prf2))
  1165               | NONE => (case rew1 Ts skel2 prf2 of
  1166                     SOME prf2' => SOME (prf1 %% prf2')
  1167                   | NONE => NONE)
  1168             end)
  1169       | rew2 Ts skel (Abst (s, T, prf)) = (case rew1 (the_default dummyT T :: Ts)
  1170               (case skel of Abst (_, _, skel') => skel' | _ => skel0) prf of
  1171             SOME prf' => SOME (Abst (s, T, prf'))
  1172           | NONE => NONE)
  1173       | rew2 Ts skel (AbsP (s, t, prf)) = (case rew1 Ts
  1174               (case skel of AbsP (_, _, skel') => skel' | _ => skel0) prf of
  1175             SOME prf' => SOME (AbsP (s, t, prf'))
  1176           | NONE => NONE)
  1177       | rew2 _ _ _ = NONE
  1178 
  1179   in the_default prf (rew1 [] skel0 prf) end;
  1180 
  1181 fun rewrite_proof thy = rewrite_prf (fn (tyenv, f) =>
  1182   Sign.typ_match thy (f ()) tyenv handle Type.TYPE_MATCH => raise PMatch);
  1183 
  1184 fun rewrite_proof_notypes rews = rewrite_prf fst rews;
  1185 
  1186 
  1187 (**** theory data ****)
  1188 
  1189 structure ProofData = TheoryDataFun
  1190 (struct
  1191   val name = "Pure/proof";
  1192   type T = ((proof * proof) list *
  1193     (string * (typ list -> proof -> proof option)) list);
  1194 
  1195   val empty = ([], []);
  1196   val copy = I;
  1197   val extend = I;
  1198   fun merge _ ((rules1, procs1), (rules2, procs2)) =
  1199     (merge_lists rules1 rules2, merge_alists procs1 procs2);
  1200   fun print _ _ = ();
  1201 end);
  1202 
  1203 val init_data = ProofData.init;
  1204 
  1205 fun add_prf_rrules rs thy =
  1206   let val r = ProofData.get thy
  1207   in ProofData.put (rs @ fst r, snd r) thy end;
  1208 
  1209 fun add_prf_rprocs ps thy =
  1210   let val r = ProofData.get thy
  1211   in ProofData.put (fst r, ps @ snd r) thy end;
  1212 
  1213 fun thm_proof thy (name, tags) hyps prop prf =
  1214   let
  1215     val prop = Logic.list_implies (hyps, prop);
  1216     val nvs = needed_vars prop;
  1217     val args = map (fn (v as Var (ixn, _)) =>
  1218         if member (op =) nvs ixn then SOME v else NONE) (vars_of prop) @
  1219       map SOME (sort Term.term_ord (term_frees prop));
  1220     val opt_prf = if ! proofs = 2 then
  1221         #4 (shrink_proof thy [] 0 (rewrite_prf fst (ProofData.get thy)
  1222           (foldr (uncurry implies_intr_proof) prf hyps)))
  1223       else MinProof (mk_min_proof prf ([], [], []));
  1224     val head = (case strip_combt (fst (strip_combP prf)) of
  1225         (PThm ((old_name, _), prf', prop', NONE), args') =>
  1226           if (old_name="" orelse old_name=name) andalso
  1227              prop = prop' andalso args = args' then
  1228             PThm ((name, tags), prf', prop, NONE)
  1229           else
  1230             PThm ((name, tags), opt_prf, prop, NONE)
  1231       | _ => PThm ((name, tags), opt_prf, prop, NONE))
  1232   in
  1233     proof_combP (proof_combt' (head, args), map Hyp hyps)
  1234   end;
  1235 
  1236 fun get_name_tags hyps prop prf =
  1237   let val prop = Logic.list_implies (hyps, prop) in
  1238     (case strip_combt (fst (strip_combP prf)) of
  1239       (PThm ((name, tags), _, prop', _), _) =>
  1240         if prop=prop' then (name, tags) else ("", [])
  1241     | (PAxm (name, prop', _), _) =>
  1242         if prop=prop' then (name, []) else ("", [])
  1243     | _ => ("", []))
  1244   end;
  1245 
  1246 end;
  1247 
  1248 structure BasicProofterm : BASIC_PROOFTERM = Proofterm;
  1249 open BasicProofterm;