src/Pure/proofterm.ML
author berghofe
Thu Apr 21 19:12:03 2005 +0200 (2005-04-21)
changeset 15797 a63605582573
parent 15632 bb178a7a69c1
child 16458 4c6fd0c01d28
permissions -rw-r--r--
- Eliminated nodup_vars check.
- Unification and matching functions now check types of term variables / sorts
of type variables when applying a substitution.
- Thm.instantiate now takes (ctyp * ctyp) list instead of (indexname * ctyp) list
as argument, to allow for proper instantiation of theorems containing
type variables with same name but different sorts.
     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    | op % of proof * term option
    19    | op %% 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 proof 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 : (term -> term) -> (typ -> typ) -> proof -> proof
    44   val fold_proof_terms : (term * 'a -> 'a) -> (typ * 'a -> 'a) -> 'a * proof -> 'a
    45   val add_prf_names : string list * proof -> string list
    46   val add_prf_tfree_names : string list * proof -> string list
    47   val add_prf_tvar_ixns : indexname list * proof -> indexname list
    48   val maxidx_of_proof : proof -> int
    49   val size_of_proof : proof -> int
    50   val change_type : typ list option -> proof -> proof
    51   val prf_abstract_over : term -> proof -> proof
    52   val prf_incr_bv : int -> int -> int -> int -> proof -> proof
    53   val incr_pboundvars : int -> int -> proof -> proof
    54   val prf_loose_bvar1 : proof -> int -> bool
    55   val prf_loose_Pbvar1 : proof -> int -> bool
    56   val prf_add_loose_bnos : int -> int -> proof ->
    57     int list * int list -> int list * int list
    58   val norm_proof : Envir.env -> proof -> proof
    59   val norm_proof' : Envir.env -> proof -> proof
    60   val prf_subst_bounds : term list -> proof -> proof
    61   val prf_subst_pbounds : proof list -> proof -> proof
    62   val freeze_thaw_prf : proof -> proof * (proof -> proof)
    63 
    64   val thms_of_proof : (term * proof) list Symtab.table -> proof ->
    65     (term * proof) list Symtab.table
    66   val axms_of_proof : proof Symtab.table -> proof -> proof Symtab.table
    67   val oracles_of_proof : proof list -> proof -> proof list
    68 
    69   (** proof terms for specific inference rules **)
    70   val implies_intr_proof : term -> proof -> proof
    71   val forall_intr_proof : term -> string -> proof -> proof
    72   val varify_proof : term -> (string * sort) list -> proof -> proof
    73   val freezeT : term -> proof -> proof
    74   val rotate_proof : term list -> term -> int -> proof -> proof
    75   val permute_prems_prf : term list -> int -> int -> proof -> proof
    76   val instantiate : (typ * typ) list -> (term * term) list -> proof -> proof
    77   val lift_proof : term -> int -> term -> proof -> proof
    78   val assumption_proof : term list -> term -> int -> proof -> proof
    79   val bicompose_proof : term list -> term list -> term list -> term option ->
    80     int -> proof -> proof -> proof
    81   val equality_axms : (string * term) list
    82   val reflexive_axm : proof
    83   val symmetric_axm : proof
    84   val transitive_axm : proof
    85   val equal_intr_axm : proof
    86   val equal_elim_axm : proof
    87   val abstract_rule_axm : proof
    88   val combination_axm : proof
    89   val reflexive : proof
    90   val symmetric : proof -> proof
    91   val transitive : term -> typ -> proof -> proof -> proof
    92   val abstract_rule : term -> string -> proof -> proof
    93   val combination : term -> term -> term -> term -> typ -> proof -> proof -> proof
    94   val equal_intr : term -> term -> proof -> proof -> proof
    95   val equal_elim : term -> term -> proof -> proof -> proof
    96   val axm_proof : string -> term -> proof
    97   val oracle_proof : string -> term -> proof
    98   val thm_proof : Sign.sg -> string * (string * string list) list ->
    99     term list -> term -> proof -> proof
   100   val get_name_tags : term list -> term -> proof -> string * (string * string list) list
   101 
   102   (** rewriting on proof terms **)
   103   val add_prf_rrules : (proof * proof) list -> theory -> theory
   104   val add_prf_rprocs : (string * (Term.typ list -> proof -> proof option)) list ->
   105     theory -> theory
   106   val rewrite_proof : Type.tsig -> (proof * proof) list *
   107     (string * (typ list -> proof -> proof option)) list -> proof -> proof
   108   val rewrite_proof_notypes : (proof * proof) list *
   109     (string * (typ list -> proof -> proof option)) list -> proof -> proof
   110   val init : theory -> theory
   111   
   112 end
   113 
   114 structure Proofterm : PROOFTERM =
   115 struct
   116 
   117 open Envir;
   118 
   119 datatype proof =
   120    PBound of int
   121  | Abst of string * typ option * proof
   122  | AbsP of string * term option * proof
   123  | op % of proof * term option
   124  | op %% of proof * proof
   125  | Hyp of term
   126  | PThm of (string * (string * string list) list) * proof * term * typ list option
   127  | PAxm of string * term * typ list option
   128  | Oracle of string * term * typ list option
   129  | MinProof of proof list;
   130 
   131 fun oracles_of_proof prfs prf =
   132   let
   133     fun oras_of (tabs, Abst (_, _, prf)) = oras_of (tabs, prf)
   134       | oras_of (tabs, AbsP (_, _, prf)) = oras_of (tabs, prf)
   135       | oras_of (tabs, prf % _) = oras_of (tabs, prf)
   136       | oras_of (tabs, prf1 %% prf2) = oras_of (oras_of (tabs, prf1), prf2)
   137       | oras_of (tabs as (thms, oras), PThm ((name, _), prf, prop, _)) =
   138           (case Symtab.lookup (thms, name) of
   139              NONE => oras_of ((Symtab.update ((name, [prop]), thms), oras), prf)
   140            | SOME ps => if prop mem ps then tabs else
   141                oras_of ((Symtab.update ((name, prop::ps), thms), oras), prf))
   142       | oras_of ((thms, oras), prf as Oracle _) = (thms, prf ins oras)
   143       | oras_of (tabs, MinProof prfs) = Library.foldl oras_of (tabs, prfs)
   144       | oras_of (tabs, _) = tabs
   145   in
   146     snd (oras_of ((Symtab.empty, prfs), prf))
   147   end;
   148 
   149 fun thms_of_proof tab (Abst (_, _, prf)) = thms_of_proof tab prf
   150   | thms_of_proof tab (AbsP (_, _, prf)) = thms_of_proof tab prf
   151   | thms_of_proof tab (prf1 %% prf2) = thms_of_proof (thms_of_proof tab prf1) prf2
   152   | thms_of_proof tab (prf % _) = thms_of_proof tab prf
   153   | thms_of_proof tab (prf' as PThm ((s, _), prf, prop, _)) =
   154       (case Symtab.lookup (tab, s) of
   155          NONE => thms_of_proof (Symtab.update ((s, [(prop, prf')]), tab)) prf
   156        | SOME ps => if exists (equal prop o fst) ps then tab else
   157            thms_of_proof (Symtab.update ((s, (prop, prf')::ps), tab)) prf)
   158   | thms_of_proof tab (MinProof prfs) = Library.foldl (uncurry thms_of_proof) (tab, prfs)
   159   | thms_of_proof tab _ = tab;
   160 
   161 fun axms_of_proof tab (Abst (_, _, prf)) = axms_of_proof tab prf
   162   | axms_of_proof tab (AbsP (_, _, prf)) = axms_of_proof tab prf
   163   | axms_of_proof tab (prf1 %% prf2) = axms_of_proof (axms_of_proof tab prf1) prf2
   164   | axms_of_proof tab (prf % _) = axms_of_proof tab prf
   165   | axms_of_proof tab (prf as PAxm (s, _, _)) = Symtab.update ((s, prf), tab)
   166   | axms_of_proof tab (MinProof prfs) = Library.foldl (uncurry axms_of_proof) (tab, prfs)
   167   | axms_of_proof tab _ = tab;
   168 
   169 (** collect all theorems, axioms and oracles **)
   170 
   171 fun mk_min_proof (prfs, Abst (_, _, prf)) = mk_min_proof (prfs, prf)
   172   | mk_min_proof (prfs, AbsP (_, _, prf)) = mk_min_proof (prfs, prf)
   173   | mk_min_proof (prfs, prf % _) = mk_min_proof (prfs, prf)
   174   | mk_min_proof (prfs, prf1 %% prf2) = mk_min_proof (mk_min_proof (prfs, prf1), prf2)
   175   | mk_min_proof (prfs, prf as PThm _) = prf ins prfs
   176   | mk_min_proof (prfs, prf as PAxm _) = prf ins prfs
   177   | mk_min_proof (prfs, prf as Oracle _) = prf ins prfs
   178   | mk_min_proof (prfs, MinProof prfs') = prfs union prfs'
   179   | mk_min_proof (prfs, _) = prfs;
   180 
   181 (** proof objects with different levels of detail **)
   182 
   183 val proofs = ref 2;
   184 
   185 fun err_illegal_level i =
   186   error ("Illegal level of detail for proof objects: " ^ string_of_int i);
   187 
   188 fun if_ora b = if b then oracles_of_proof else K;
   189 
   190 fun infer_derivs f (ora1, prf1) (ora2, prf2) =
   191   (ora1 orelse ora2, 
   192    case !proofs of
   193      2 => f prf1 prf2
   194    | 1 => MinProof (mk_min_proof (mk_min_proof ([], prf1), prf2))
   195    | 0 => MinProof (if_ora ora2 (if_ora ora1 [] prf1) prf2)
   196    | i => err_illegal_level i);
   197 
   198 fun infer_derivs' f = infer_derivs (K f) (false, MinProof []);
   199 
   200 fun (prf %> t) = prf % SOME t;
   201 
   202 val proof_combt = Library.foldl (op %>);
   203 val proof_combt' = Library.foldl (op %);
   204 val proof_combP = Library.foldl (op %%);
   205 
   206 fun strip_combt prf = 
   207     let fun stripc (prf % t, ts) = stripc (prf, t::ts)
   208           | stripc  x =  x 
   209     in  stripc (prf, [])  end;
   210 
   211 fun strip_combP prf = 
   212     let fun stripc (prf %% prf', prfs) = stripc (prf, prf'::prfs)
   213           | stripc  x =  x
   214     in  stripc (prf, [])  end;
   215 
   216 fun strip_thm prf = (case strip_combt (fst (strip_combP prf)) of
   217       (PThm (_, prf', _, _), _) => prf'
   218     | _ => prf);
   219 
   220 val mk_Abst = foldr (fn ((s, T:typ), prf) => Abst (s, NONE, prf));
   221 fun mk_AbsP (i, prf) = funpow i (fn prf => AbsP ("H", NONE, prf)) prf;
   222 
   223 fun apsome' f NONE = raise SAME
   224   | apsome' f (SOME x) = SOME (f x);
   225 
   226 fun same f x =
   227   let val x' = f x
   228   in if x = x' then raise SAME else x' end;
   229 
   230 fun map_proof_terms f g =
   231   let
   232     fun mapp (Abst (s, T, prf)) = (Abst (s, apsome' (same g) T, mapph prf)
   233           handle SAME => Abst (s, T, mapp prf))
   234       | mapp (AbsP (s, t, prf)) = (AbsP (s, apsome' (same f) t, mapph prf)
   235           handle SAME => AbsP (s, t, mapp prf))
   236       | mapp (prf % t) = (mapp prf % Option.map f t
   237           handle SAME => prf % apsome' (same f) t)
   238       | mapp (prf1 %% prf2) = (mapp prf1 %% mapph prf2
   239           handle SAME => prf1 %% mapp prf2)
   240       | mapp (PThm (a, prf, prop, SOME Ts)) =
   241           PThm (a, prf, prop, SOME (same (map g) Ts))
   242       | mapp (PAxm (a, prop, SOME Ts)) =
   243           PAxm (a, prop, SOME (same (map g) Ts))
   244       | mapp _ = raise SAME
   245     and mapph prf = (mapp prf handle SAME => prf)
   246 
   247   in mapph end;
   248 
   249 fun fold_proof_terms f g (a, Abst (_, SOME T, prf)) = fold_proof_terms f g (g (T, a), prf)
   250   | fold_proof_terms f g (a, Abst (_, NONE, prf)) = fold_proof_terms f g (a, prf)
   251   | fold_proof_terms f g (a, AbsP (_, SOME t, prf)) = fold_proof_terms f g (f (t, a), prf)
   252   | fold_proof_terms f g (a, AbsP (_, NONE, prf)) = fold_proof_terms f g (a, prf)
   253   | fold_proof_terms f g (a, prf % SOME t) = f (t, fold_proof_terms f g (a, prf))
   254   | fold_proof_terms f g (a, prf % NONE) = fold_proof_terms f g (a, prf)
   255   | fold_proof_terms f g (a, prf1 %% prf2) = fold_proof_terms f g
   256       (fold_proof_terms f g (a, prf1), prf2)
   257   | fold_proof_terms _ g (a, PThm (_, _, _, SOME Ts)) = foldr g a Ts
   258   | fold_proof_terms _ g (a, PAxm (_, prop, SOME Ts)) = foldr g a Ts
   259   | fold_proof_terms _ _ (a, _) = a;
   260 
   261 val add_prf_names = fold_proof_terms add_term_names ((uncurry K) o swap);
   262 val add_prf_tfree_names = fold_proof_terms add_term_tfree_names add_typ_tfree_names;
   263 val add_prf_tvar_ixns = fold_proof_terms add_term_tvar_ixns (add_typ_ixns o swap);
   264 
   265 fun maxidx_of_proof prf = fold_proof_terms
   266   (Int.max o apfst maxidx_of_term) (Int.max o apfst maxidx_of_typ) (~1, prf); 
   267 
   268 fun size_of_proof (Abst (_, _, prf)) = 1 + size_of_proof prf
   269   | size_of_proof (AbsP (_, t, prf)) = 1 + size_of_proof prf
   270   | size_of_proof (prf1 %% prf2) = size_of_proof prf1 + size_of_proof prf2
   271   | size_of_proof (prf % _) = 1 + size_of_proof prf
   272   | size_of_proof _ = 1;
   273 
   274 fun change_type opTs (PThm (name, prf, prop, _)) = PThm (name, prf, prop, opTs)
   275   | change_type opTs (PAxm (name, prop, _)) = PAxm (name, prop, opTs)
   276   | change_type opTs (Oracle (name, prop, _)) = Oracle (name, prop, opTs)
   277   | change_type _ prf = prf;
   278 
   279 
   280 (***** utilities *****)
   281 
   282 fun strip_abs (_::Ts) (Abs (_, _, t)) = strip_abs Ts t
   283   | strip_abs _ t = t;
   284 
   285 fun mk_abs Ts t = Library.foldl (fn (t', T) => Abs ("", T, t')) (t, Ts);
   286 
   287 
   288 (*Abstraction of a proof term over its occurrences of v, 
   289     which must contain no loose bound variables.
   290   The resulting proof term is ready to become the body of an Abst.*)
   291 
   292 fun prf_abstract_over v =
   293   let
   294     fun abst' lev u = if v aconv u then Bound lev else
   295       (case u of
   296          Abs (a, T, t) => Abs (a, T, abst' (lev + 1) t)
   297        | f $ t => (abst' lev f $ absth' lev t handle SAME => f $ abst' lev t)
   298        | _ => raise SAME)
   299     and absth' lev t = (abst' lev t handle SAME => t);
   300 
   301     fun abst lev (AbsP (a, t, prf)) =
   302           (AbsP (a, apsome' (abst' lev) t, absth lev prf)
   303            handle SAME => AbsP (a, t, abst lev prf))
   304       | abst lev (Abst (a, T, prf)) = Abst (a, T, abst (lev + 1) prf)
   305       | abst lev (prf1 %% prf2) = (abst lev prf1 %% absth lev prf2
   306           handle SAME => prf1 %% abst lev prf2)
   307       | abst lev (prf % t) = (abst lev prf % Option.map (absth' lev) t
   308           handle SAME => prf % apsome' (abst' lev) t)
   309       | abst _ _ = raise SAME
   310     and absth lev prf = (abst lev prf handle SAME => prf)
   311 
   312   in absth 0 end;
   313 
   314 
   315 (*increments a proof term's non-local bound variables
   316   required when moving a proof term within abstractions
   317      inc is  increment for bound variables
   318      lev is  level at which a bound variable is considered 'loose'*)
   319 
   320 fun incr_bv' inct tlev t = incr_bv (inct, tlev, t);
   321 
   322 fun prf_incr_bv' incP inct Plev tlev (PBound i) =
   323       if i >= Plev then PBound (i+incP) else raise SAME 
   324   | prf_incr_bv' incP inct Plev tlev (AbsP (a, t, body)) =
   325       (AbsP (a, apsome' (same (incr_bv' inct tlev)) t,
   326          prf_incr_bv incP inct (Plev+1) tlev body) handle SAME =>
   327            AbsP (a, t, prf_incr_bv' incP inct (Plev+1) tlev body))
   328   | prf_incr_bv' incP inct Plev tlev (Abst (a, T, body)) =
   329       Abst (a, T, prf_incr_bv' incP inct Plev (tlev+1) body)
   330   | prf_incr_bv' incP inct Plev tlev (prf %% prf') = 
   331       (prf_incr_bv' incP inct Plev tlev prf %% prf_incr_bv incP inct Plev tlev prf'
   332        handle SAME => prf %% prf_incr_bv' incP inct Plev tlev prf')
   333   | prf_incr_bv' incP inct Plev tlev (prf % t) = 
   334       (prf_incr_bv' incP inct Plev tlev prf % Option.map (incr_bv' inct tlev) t
   335        handle SAME => prf % apsome' (same (incr_bv' inct tlev)) t)
   336   | prf_incr_bv' _ _ _ _ _ = raise SAME
   337 and prf_incr_bv incP inct Plev tlev prf =
   338       (prf_incr_bv' incP inct Plev tlev prf handle SAME => prf);
   339 
   340 fun incr_pboundvars  0 0 prf = prf
   341   | incr_pboundvars incP inct prf = prf_incr_bv incP inct 0 0 prf;
   342 
   343 
   344 fun prf_loose_bvar1 (prf1 %% prf2) k = prf_loose_bvar1 prf1 k orelse prf_loose_bvar1 prf2 k
   345   | prf_loose_bvar1 (prf % SOME t) k = prf_loose_bvar1 prf k orelse loose_bvar1 (t, k)
   346   | prf_loose_bvar1 (_ % NONE) _ = true
   347   | prf_loose_bvar1 (AbsP (_, SOME t, prf)) k = loose_bvar1 (t, k) orelse prf_loose_bvar1 prf k
   348   | prf_loose_bvar1 (AbsP (_, NONE, _)) k = true
   349   | prf_loose_bvar1 (Abst (_, _, prf)) k = prf_loose_bvar1 prf (k+1)
   350   | prf_loose_bvar1 _ _ = false;
   351 
   352 fun prf_loose_Pbvar1 (PBound i) k = i = k
   353   | prf_loose_Pbvar1 (prf1 %% prf2) k = prf_loose_Pbvar1 prf1 k orelse prf_loose_Pbvar1 prf2 k
   354   | prf_loose_Pbvar1 (prf % _) k = prf_loose_Pbvar1 prf k
   355   | prf_loose_Pbvar1 (AbsP (_, _, prf)) k = prf_loose_Pbvar1 prf (k+1)
   356   | prf_loose_Pbvar1 (Abst (_, _, prf)) k = prf_loose_Pbvar1 prf k
   357   | prf_loose_Pbvar1 _ _ = false;
   358 
   359 fun prf_add_loose_bnos plev tlev (PBound i) (is, js) =
   360       if i < plev then (is, js) else ((i-plev) ins is, js)
   361   | prf_add_loose_bnos plev tlev (prf1 %% prf2) p =
   362       prf_add_loose_bnos plev tlev prf2
   363         (prf_add_loose_bnos plev tlev prf1 p)
   364   | prf_add_loose_bnos plev tlev (prf % opt) (is, js) =
   365       prf_add_loose_bnos plev tlev prf (case opt of
   366           NONE => (is, ~1 ins js)
   367         | SOME t => (is, add_loose_bnos (t, tlev, js)))
   368   | prf_add_loose_bnos plev tlev (AbsP (_, opt, prf)) (is, js) =
   369       prf_add_loose_bnos (plev+1) tlev prf (case opt of
   370           NONE => (is, ~1 ins js)
   371         | SOME t => (is, add_loose_bnos (t, tlev, js)))
   372   | prf_add_loose_bnos plev tlev (Abst (_, _, prf)) p =
   373       prf_add_loose_bnos plev (tlev+1) prf p
   374   | prf_add_loose_bnos _ _ _ _ = ([], []);
   375 
   376 
   377 (**** substitutions ****)
   378 
   379 fun norm_proof env =
   380   let
   381     val envT = type_env env;
   382     fun norm (Abst (s, T, prf)) = (Abst (s, apsome' (norm_type_same envT) T, normh prf)
   383           handle SAME => Abst (s, T, norm prf))
   384       | norm (AbsP (s, t, prf)) = (AbsP (s, apsome' (norm_term_same env) t, normh prf)
   385           handle SAME => AbsP (s, t, norm prf))
   386       | norm (prf % t) = (norm prf % Option.map (norm_term env) t
   387           handle SAME => prf % apsome' (norm_term_same env) t)
   388       | norm (prf1 %% prf2) = (norm prf1 %% normh prf2
   389           handle SAME => prf1 %% norm prf2)
   390       | norm (PThm (s, prf, t, Ts)) = PThm (s, prf, t, apsome' (norm_types_same envT) Ts)
   391       | norm (PAxm (s, prop, Ts)) = PAxm (s, prop, apsome' (norm_types_same envT) Ts)
   392       | norm _ = raise SAME
   393     and normh prf = (norm prf handle SAME => prf);
   394   in normh end;
   395 
   396 (***** Remove some types in proof term (to save space) *****)
   397 
   398 fun remove_types (Abs (s, _, t)) = Abs (s, dummyT, remove_types t)
   399   | remove_types (t $ u) = remove_types t $ remove_types u
   400   | remove_types (Const (s, _)) = Const (s, dummyT)
   401   | remove_types t = t;
   402 
   403 fun remove_types_env (Envir.Envir {iTs, asol, maxidx}) =
   404   Envir.Envir {iTs = iTs, asol = Vartab.map (apsnd remove_types) asol,
   405     maxidx = maxidx};
   406 
   407 fun norm_proof' env prf = norm_proof (remove_types_env env) prf;
   408 
   409 (**** substitution of bound variables ****)
   410 
   411 fun prf_subst_bounds args prf =
   412   let
   413     val n = length args;
   414     fun subst' lev (Bound i) =
   415          (if i<lev then raise SAME    (*var is locally bound*)
   416           else  incr_boundvars lev (List.nth (args, i-lev))
   417                   handle Subscript => Bound (i-n)  (*loose: change it*))
   418       | subst' lev (Abs (a, T, body)) = Abs (a, T,  subst' (lev+1) body)
   419       | subst' lev (f $ t) = (subst' lev f $ substh' lev t
   420           handle SAME => f $ subst' lev t)
   421       | subst' _ _ = raise SAME
   422     and substh' lev t = (subst' lev t handle SAME => t);
   423 
   424     fun subst lev (AbsP (a, t, body)) = (AbsP (a, apsome' (subst' lev) t, substh lev body)
   425           handle SAME => AbsP (a, t, subst lev body))
   426       | subst lev (Abst (a, T, body)) = Abst (a, T, subst (lev+1) body)
   427       | subst lev (prf %% prf') = (subst lev prf %% substh lev prf'
   428           handle SAME => prf %% subst lev prf')
   429       | subst lev (prf % t) = (subst lev prf % Option.map (substh' lev) t
   430           handle SAME => prf % apsome' (subst' lev) t)
   431       | subst _ _ = raise SAME
   432     and substh lev prf = (subst lev prf handle SAME => prf)
   433   in case args of [] => prf | _ => substh 0 prf end;
   434 
   435 fun prf_subst_pbounds args prf =
   436   let
   437     val n = length args;
   438     fun subst (PBound i) Plev tlev =
   439  	 (if i < Plev then raise SAME    (*var is locally bound*)
   440           else incr_pboundvars Plev tlev (List.nth (args, i-Plev))
   441                  handle Subscript => PBound (i-n)  (*loose: change it*))
   442       | subst (AbsP (a, t, body)) Plev tlev = AbsP (a, t, subst body (Plev+1) tlev)
   443       | subst (Abst (a, T, body)) Plev tlev = Abst (a, T, subst body Plev (tlev+1))
   444       | subst (prf %% prf') Plev tlev = (subst prf Plev tlev %% substh prf' Plev tlev
   445           handle SAME => prf %% subst prf' Plev tlev)
   446       | subst (prf % t) Plev tlev = subst prf Plev tlev % t
   447       | subst  prf _ _ = raise SAME
   448     and substh prf Plev tlev = (subst prf Plev tlev handle SAME => prf)
   449   in case args of [] => prf | _ => substh prf 0 0 end;
   450 
   451 
   452 (**** Freezing and thawing of variables in proof terms ****)
   453 
   454 fun frzT names =
   455   map_type_tvar (fn (ixn, xs) => TFree (valOf (assoc (names, ixn)), xs));
   456 
   457 fun thawT names =
   458   map_type_tfree (fn (s, xs) => case assoc (names, s) of
   459       NONE => TFree (s, xs)
   460     | SOME ixn => TVar (ixn, xs));
   461 
   462 fun freeze names names' (t $ u) =
   463       freeze names names' t $ freeze names names' u
   464   | freeze names names' (Abs (s, T, t)) =
   465       Abs (s, frzT names' T, freeze names names' t)
   466   | freeze names names' (Const (s, T)) = Const (s, frzT names' T)
   467   | freeze names names' (Free (s, T)) = Free (s, frzT names' T)
   468   | freeze names names' (Var (ixn, T)) =
   469       Free (valOf (assoc (names, ixn)), frzT names' T)
   470   | freeze names names' t = t;
   471 
   472 fun thaw names names' (t $ u) =
   473       thaw names names' t $ thaw names names' u
   474   | thaw names names' (Abs (s, T, t)) =
   475       Abs (s, thawT names' T, thaw names names' t)
   476   | thaw names names' (Const (s, T)) = Const (s, thawT names' T)
   477   | thaw names names' (Free (s, T)) = 
   478       let val T' = thawT names' T
   479       in case assoc (names, s) of
   480           NONE => Free (s, T')
   481         | SOME ixn => Var (ixn, T')
   482       end
   483   | thaw names names' (Var (ixn, T)) = Var (ixn, thawT names' T)
   484   | thaw names names' t = t;
   485 
   486 fun freeze_thaw_prf prf =
   487   let
   488     val (fs, Tfs, vs, Tvs) = fold_proof_terms
   489       (fn (t, (fs, Tfs, vs, Tvs)) =>
   490          (add_term_frees (t, fs), add_term_tfree_names (t, Tfs),
   491           add_term_vars (t, vs), add_term_tvar_ixns (t, Tvs)))
   492       (fn (T, (fs, Tfs, vs, Tvs)) =>
   493          (fs, add_typ_tfree_names (T, Tfs),
   494           vs, add_typ_ixns (Tvs, T)))
   495             (([], [], [], []), prf);
   496     val fs' = map (fst o dest_Free) fs;
   497     val vs' = map (fst o dest_Var) vs;
   498     val names = vs' ~~ variantlist (map fst vs', fs');
   499     val names' = Tvs ~~ variantlist (map fst Tvs, Tfs);
   500     val rnames = map swap names;
   501     val rnames' = map swap names';
   502   in
   503     (map_proof_terms (freeze names names') (frzT names') prf,
   504      map_proof_terms (thaw rnames rnames') (thawT rnames'))
   505   end;
   506 
   507 
   508 (***** implication introduction *****)
   509 
   510 fun implies_intr_proof h prf =
   511   let
   512     fun abshyp i (Hyp t) = if h aconv t then PBound i else raise SAME
   513       | abshyp i (Abst (s, T, prf)) = Abst (s, T, abshyp i prf)
   514       | abshyp i (AbsP (s, t, prf)) = AbsP (s, t, abshyp (i+1) prf)
   515       | abshyp i (prf % t) = abshyp i prf % t
   516       | abshyp i (prf1 %% prf2) = (abshyp i prf1 %% abshyph i prf2
   517           handle SAME => prf1 %% abshyp i prf2)
   518       | abshyp _ _ = raise SAME
   519     and abshyph i prf = (abshyp i prf handle SAME => prf)
   520   in
   521     AbsP ("H", NONE (*h*), abshyph 0 prf)
   522   end;
   523 
   524 
   525 (***** forall introduction *****)
   526 
   527 fun forall_intr_proof x a prf = Abst (a, NONE, prf_abstract_over x prf);
   528 
   529 
   530 (***** varify *****)
   531 
   532 fun varify_proof t fixed prf =
   533   let
   534     val fs = add_term_tfrees (t, []) \\ fixed;
   535     val ixns = add_term_tvar_ixns (t, []);
   536     val fmap = fs ~~ variantlist (map fst fs, map #1 ixns)
   537     fun thaw (f as (a, S)) =
   538       (case assoc (fmap, f) of
   539         NONE => TFree f
   540       | SOME b => TVar ((b, 0), S));
   541   in map_proof_terms (map_term_types (map_type_tfree thaw)) (map_type_tfree thaw) prf
   542   end;
   543 
   544 
   545 local
   546 
   547 fun new_name (ix, (pairs,used)) =
   548   let val v = variant used (string_of_indexname ix)
   549   in  ((ix, v) :: pairs, v :: used)  end;
   550 
   551 fun freeze_one alist (ix, sort) = (case assoc (alist, ix) of
   552     NONE => TVar (ix, sort)
   553   | SOME name => TFree (name, sort));
   554 
   555 in
   556 
   557 fun freezeT t prf =
   558   let
   559     val used = it_term_types add_typ_tfree_names (t, [])
   560     and tvars = map #1 (it_term_types add_typ_tvars (t, []));
   561     val (alist, _) = foldr new_name ([], used) tvars;
   562   in
   563     (case alist of
   564       [] => prf (*nothing to do!*)
   565     | _ =>
   566       let val frzT = map_type_tvar (freeze_one alist)
   567       in map_proof_terms (map_term_types frzT) frzT prf end)
   568   end;
   569 
   570 end;
   571 
   572 
   573 (***** rotate assumptions *****)
   574 
   575 fun rotate_proof Bs Bi m prf =
   576   let
   577     val params = Term.strip_all_vars Bi;
   578     val asms = Logic.strip_imp_prems (Term.strip_all_body Bi);
   579     val i = length asms;
   580     val j = length Bs;
   581   in
   582     mk_AbsP (j+1, proof_combP (prf, map PBound
   583       (j downto 1) @ [mk_Abst (mk_AbsP (i,
   584         proof_combP (proof_combt (PBound i, map Bound ((length params - 1) downto 0)),
   585           map PBound (((i-m-1) downto 0) @ ((i-1) downto (i-m)))))) params]))
   586   end;
   587 
   588 
   589 (***** permute premises *****)
   590 
   591 fun permute_prems_prf prems j k prf =
   592   let val n = length prems
   593   in mk_AbsP (n, proof_combP (prf,
   594     map PBound ((n-1 downto n-j) @ (k-1 downto 0) @ (n-j-1 downto k))))
   595   end;
   596 
   597 
   598 (***** instantiation *****)
   599 
   600 fun instantiate vTs tpairs prf =
   601   map_proof_terms (subst_atomic (map (apsnd remove_types) tpairs) o
   602     map_term_types (typ_subst_atomic vTs)) (typ_subst_atomic vTs) prf;
   603 
   604 
   605 (***** lifting *****)
   606 
   607 fun lift_proof Bi inc prop prf =
   608   let
   609     val (_, lift_all) = Logic.lift_fns (Bi, inc);
   610 
   611     fun lift'' Us Ts t = strip_abs Ts (Logic.incr_indexes (Us, inc) (mk_abs Ts t));
   612 
   613     fun lift' Us Ts (Abst (s, T, prf)) =
   614           (Abst (s, apsome' (same (incr_tvar inc)) T, lifth' Us (dummyT::Ts) prf)
   615            handle SAME => Abst (s, T, lift' Us (dummyT::Ts) prf))
   616       | lift' Us Ts (AbsP (s, t, prf)) =
   617           (AbsP (s, apsome' (same (lift'' Us Ts)) t, lifth' Us Ts prf)
   618            handle SAME => AbsP (s, t, lift' Us Ts prf))
   619       | lift' Us Ts (prf % t) = (lift' Us Ts prf % Option.map (lift'' Us Ts) t
   620           handle SAME => prf % apsome' (same (lift'' Us Ts)) t)
   621       | lift' Us Ts (prf1 %% prf2) = (lift' Us Ts prf1 %% lifth' Us Ts prf2
   622           handle SAME => prf1 %% lift' Us Ts prf2)
   623       | lift' _ _ (PThm (s, prf, prop, Ts)) =
   624           PThm (s, prf, prop, apsome' (same (map (incr_tvar inc))) Ts)
   625       | lift' _ _ (PAxm (s, prop, Ts)) =
   626           PAxm (s, prop, apsome' (same (map (incr_tvar inc))) Ts)
   627       | lift' _ _ _ = raise SAME
   628     and lifth' Us Ts prf = (lift' Us Ts prf handle SAME => prf);
   629 
   630     val ps = map lift_all (Logic.strip_imp_prems prop);
   631     val k = length ps;
   632 
   633     fun mk_app (b, (i, j, prf)) = 
   634           if b then (i-1, j, prf %% PBound i) else (i, j-1, prf %> Bound j);
   635 
   636     fun lift Us bs i j (Const ("==>", _) $ A $ B) =
   637 	    AbsP ("H", NONE (*A*), lift Us (true::bs) (i+1) j B)
   638       | lift Us bs i j (Const ("all", _) $ Abs (a, T, t)) = 
   639 	    Abst (a, NONE (*T*), lift (T::Us) (false::bs) i (j+1) t)
   640       | lift Us bs i j _ = proof_combP (lifth' (rev Us) [] prf,
   641             map (fn k => (#3 (foldr mk_app (i-1, j-1, PBound k) bs)))
   642               (i + k - 1 downto i));
   643   in
   644     mk_AbsP (k, lift [] [] 0 0 Bi)
   645   end;
   646 
   647 
   648 (***** proof by assumption *****)
   649 
   650 fun mk_asm_prf (Const ("==>", _) $ A $ B) i = AbsP ("H", NONE (*A*), mk_asm_prf B (i+1))
   651   | mk_asm_prf (Const ("all", _) $ Abs (a, T, t)) i = Abst (a, NONE (*T*), mk_asm_prf t i)
   652   | mk_asm_prf _ i = PBound i;
   653 
   654 fun assumption_proof Bs Bi n prf =
   655   mk_AbsP (length Bs, proof_combP (prf,
   656     map PBound (length Bs - 1 downto 0) @ [mk_asm_prf Bi (~n)]));
   657 
   658 
   659 (***** Composition of object rule with proof state *****)
   660 
   661 fun flatten_params_proof i j n (Const ("==>", _) $ A $ B, k) =
   662       AbsP ("H", NONE (*A*), flatten_params_proof (i+1) j n (B, k))
   663   | flatten_params_proof i j n (Const ("all", _) $ Abs (a, T, t), k) =
   664       Abst (a, NONE (*T*), flatten_params_proof i (j+1) n (t, k))
   665   | flatten_params_proof i j n (_, k) = proof_combP (proof_combt (PBound (k+i),
   666       map Bound (j-1 downto 0)), map PBound (i-1 downto 0 \ i-n));
   667 
   668 fun bicompose_proof Bs oldAs newAs A n rprf sprf =
   669   let
   670     val la = length newAs;
   671     val lb = length Bs;
   672   in
   673     mk_AbsP (lb+la, proof_combP (sprf,
   674       map PBound (lb + la - 1 downto la)) %%
   675         proof_combP (rprf, (if n>0 then [mk_asm_prf (valOf A) (~n)] else []) @
   676           map (flatten_params_proof 0 0 n) (oldAs ~~ (la - 1 downto 0))))
   677   end;
   678 
   679 
   680 (***** axioms for equality *****)
   681 
   682 val aT = TFree ("'a", []);
   683 val bT = TFree ("'b", []);
   684 val x = Free ("x", aT);
   685 val y = Free ("y", aT);
   686 val z = Free ("z", aT);
   687 val A = Free ("A", propT);
   688 val B = Free ("B", propT);
   689 val f = Free ("f", aT --> bT);
   690 val g = Free ("g", aT --> bT);
   691 
   692 local open Logic in
   693 
   694 val equality_axms =
   695   [("reflexive", mk_equals (x, x)),
   696    ("symmetric", mk_implies (mk_equals (x, y), mk_equals (y, x))),
   697    ("transitive", list_implies ([mk_equals (x, y), mk_equals (y, z)], mk_equals (x, z))),
   698    ("equal_intr", list_implies ([mk_implies (A, B), mk_implies (B, A)], mk_equals (A, B))),
   699    ("equal_elim", list_implies ([mk_equals (A, B), A], B)),
   700    ("abstract_rule", Logic.mk_implies
   701       (all aT $ Abs ("x", aT, equals bT $ (f $ Bound 0) $ (g $ Bound 0)),
   702        equals (aT --> bT) $
   703          Abs ("x", aT, f $ Bound 0) $ Abs ("x", aT, g $ Bound 0))),
   704    ("combination", Logic.list_implies
   705       ([Logic.mk_equals (f, g), Logic.mk_equals (x, y)],
   706        Logic.mk_equals (f $ x, g $ y)))];
   707 
   708 val [reflexive_axm, symmetric_axm, transitive_axm, equal_intr_axm,
   709   equal_elim_axm, abstract_rule_axm, combination_axm] =
   710     map (fn (s, t) => PAxm ("ProtoPure." ^ s, varify t, NONE)) equality_axms;
   711 
   712 end;
   713 
   714 val reflexive = reflexive_axm % NONE;
   715 
   716 fun symmetric (prf as PAxm ("ProtoPure.reflexive", _, _) % _) = prf
   717   | symmetric prf = symmetric_axm % NONE % NONE %% prf;
   718 
   719 fun transitive _ _ (PAxm ("ProtoPure.reflexive", _, _) % _) prf2 = prf2
   720   | transitive _ _ prf1 (PAxm ("ProtoPure.reflexive", _, _) % _) = prf1
   721   | transitive u (Type ("prop", [])) prf1 prf2 =
   722       transitive_axm % NONE % SOME (remove_types u) % NONE %% prf1 %% prf2
   723   | transitive u T prf1 prf2 =
   724       transitive_axm % NONE % NONE % NONE %% prf1 %% prf2;
   725 
   726 fun abstract_rule x a prf =
   727   abstract_rule_axm % NONE % NONE %% forall_intr_proof x a prf;
   728 
   729 fun check_comb (PAxm ("ProtoPure.combination", _, _) % f % g % _ % _ %% prf %% _) =
   730       isSome f orelse check_comb prf
   731   | check_comb (PAxm ("ProtoPure.transitive", _, _) % _ % _ % _ %% prf1 %% prf2) =
   732       check_comb prf1 andalso check_comb prf2
   733   | check_comb (PAxm ("ProtoPure.symmetric", _, _) % _ % _ %% prf) = check_comb prf
   734   | check_comb _ = false;
   735 
   736 fun combination f g t u (Type (_, [T, U])) prf1 prf2 =
   737   let
   738     val f = Envir.beta_norm f;
   739     val g = Envir.beta_norm g;
   740     val prf =  if check_comb prf1 then
   741         combination_axm % NONE % NONE
   742       else (case prf1 of
   743           PAxm ("ProtoPure.reflexive", _, _) % _ =>
   744             combination_axm %> remove_types f % NONE
   745         | _ => combination_axm %> remove_types f %> remove_types g)
   746   in
   747     (case T of
   748        Type ("fun", _) => prf %
   749          (case head_of f of
   750             Abs _ => SOME (remove_types t)
   751           | Var _ => SOME (remove_types t)
   752           | _ => NONE) %
   753          (case head_of g of
   754             Abs _ => SOME (remove_types u)
   755           | Var _ => SOME (remove_types u)
   756           | _ => NONE) %% prf1 %% prf2
   757      | _ => prf % NONE % NONE %% prf1 %% prf2)
   758   end;
   759 
   760 fun equal_intr A B prf1 prf2 =
   761   equal_intr_axm %> remove_types A %> remove_types B %% prf1 %% prf2;
   762 
   763 fun equal_elim A B prf1 prf2 =
   764   equal_elim_axm %> remove_types A %> remove_types B %% prf1 %% prf2;
   765 
   766 
   767 (***** axioms and theorems *****)
   768 
   769 fun vars_of t = rev (foldl_aterms
   770   (fn (vs, v as Var _) => v ins vs | (vs, _) => vs) ([], t));
   771 
   772 fun test_args _ [] = true
   773   | test_args is (Bound i :: ts) =
   774       not (i mem is) andalso test_args (i :: is) ts
   775   | test_args _ _ = false;
   776 
   777 fun is_fun (Type ("fun", _)) = true
   778   | is_fun (TVar _) = true
   779   | is_fun _ = false;
   780 
   781 fun add_funvars Ts (vs, t) =
   782   if is_fun (fastype_of1 (Ts, t)) then
   783     vs union List.mapPartial (fn Var (ixn, T) =>
   784       if is_fun T then SOME ixn else NONE | _ => NONE) (vars_of t)
   785   else vs;
   786 
   787 fun add_npvars q p Ts (vs, Const ("==>", _) $ t $ u) =
   788       add_npvars q p Ts (add_npvars q (not p) Ts (vs, t), u)
   789   | add_npvars q p Ts (vs, Const ("all", Type (_, [Type (_, [T, _]), _])) $ t) =
   790       add_npvars q p Ts (vs, if p andalso q then betapply (t, Var (("",0), T)) else t)
   791   | add_npvars q p Ts (vs, Abs (_, T, t)) = add_npvars q p (T::Ts) (vs, t)
   792   | add_npvars _ _ Ts (vs, t) = add_npvars' Ts (vs, t)
   793 and add_npvars' Ts (vs, t) = (case strip_comb t of
   794     (Var (ixn, _), ts) => if test_args [] ts then vs
   795       else Library.foldl (add_npvars' Ts) (overwrite (vs,
   796         (ixn, Library.foldl (add_funvars Ts) (getOpt (assoc (vs, ixn), []), ts))), ts)
   797   | (Abs (_, T, u), ts) => Library.foldl (add_npvars' (T::Ts)) (vs, u :: ts)
   798   | (_, ts) => Library.foldl (add_npvars' Ts) (vs, ts));
   799 
   800 fun prop_vars (Const ("==>", _) $ P $ Q) = prop_vars P union prop_vars Q
   801   | prop_vars (Const ("all", _) $ Abs (_, _, t)) = prop_vars t
   802   | prop_vars t = (case strip_comb t of
   803       (Var (ixn, _), _) => [ixn] | _ => []);
   804 
   805 fun is_proj t =
   806   let
   807     fun is_p i t = (case strip_comb t of
   808         (Bound j, []) => false
   809       | (Bound j, ts) => j >= i orelse exists (is_p i) ts
   810       | (Abs (_, _, u), _) => is_p (i+1) u
   811       | (_, ts) => exists (is_p i) ts)
   812   in (case strip_abs_body t of
   813         Bound _ => true
   814       | t' => is_p 0 t')
   815   end;
   816 
   817 fun needed_vars prop = 
   818   Library.foldl op union ([], map op ins (add_npvars true true [] ([], prop))) union
   819   prop_vars prop;
   820 
   821 fun gen_axm_proof c name prop =
   822   let
   823     val nvs = needed_vars prop;
   824     val args = map (fn (v as Var (ixn, _)) =>
   825         if ixn mem nvs then SOME v else NONE) (vars_of prop) @
   826       map SOME (sort (make_ord atless) (term_frees prop));
   827   in
   828     proof_combt' (c (name, prop, NONE), args)
   829   end;
   830 
   831 val axm_proof = gen_axm_proof PAxm;
   832 val oracle_proof = gen_axm_proof Oracle;
   833 
   834 fun shrink ls lev (prf as Abst (a, T, body)) =
   835       let val (b, is, ch, body') = shrink ls (lev+1) body
   836       in (b, is, ch, if ch then Abst (a, T, body') else prf) end
   837   | shrink ls lev (prf as AbsP (a, t, body)) =
   838       let val (b, is, ch, body') = shrink (lev::ls) lev body
   839       in (b orelse 0 mem is, List.mapPartial (fn 0 => NONE | i => SOME (i-1)) is,
   840         ch, if ch then AbsP (a, t, body') else prf)
   841       end
   842   | shrink ls lev prf =
   843       let val (is, ch, _, prf') = shrink' ls lev [] [] prf
   844       in (false, is, ch, prf') end
   845 and shrink' ls lev ts prfs (prf as prf1 %% prf2) =
   846       let
   847         val p as (_, is', ch', prf') = shrink ls lev prf2;
   848         val (is, ch, ts', prf'') = shrink' ls lev ts (p::prfs) prf1
   849       in (is union is', ch orelse ch', ts',
   850           if ch orelse ch' then prf'' %% prf' else prf)
   851       end
   852   | shrink' ls lev ts prfs (prf as prf1 % t) =
   853       let val (is, ch, (ch', t')::ts', prf') = shrink' ls lev (t::ts) prfs prf1
   854       in (is, ch orelse ch', ts', if ch orelse ch' then prf' % t' else prf) end
   855   | shrink' ls lev ts prfs (prf as PBound i) =
   856       (if exists (fn SOME (Bound j) => lev-j <= List.nth (ls, i) | _ => true) ts
   857          orelse not (null (duplicates
   858            (Library.foldl (fn (js, SOME (Bound j)) => j :: js | (js, _) => js) ([], ts))))
   859          orelse exists #1 prfs then [i] else [], false, map (pair false) ts, prf)
   860   | shrink' ls lev ts prfs (prf as Hyp _) = ([], false, map (pair false) ts, prf)
   861   | shrink' ls lev ts prfs (prf as MinProof _) =
   862       ([], false, map (pair false) ts, prf)
   863   | shrink' ls lev ts prfs prf =
   864       let
   865         val prop = (case prf of PThm (_, _, prop, _) => prop | PAxm (_, prop, _) => prop
   866           | Oracle (_, prop, _) => prop | _ => error "shrink: proof not in normal form");
   867         val vs = vars_of prop;
   868         val (ts', ts'') = splitAt (length vs, ts)
   869         val insts = Library.take (length ts', map (fst o dest_Var) vs) ~~ ts';
   870         val nvs = Library.foldl (fn (ixns', (ixn, ixns)) =>
   871           ixn ins (case assoc (insts, ixn) of
   872               SOME (SOME t) => if is_proj t then ixns union ixns' else ixns'
   873             | _ => ixns union ixns'))
   874               (needed prop ts'' prfs, add_npvars false true [] ([], prop));
   875         val insts' = map
   876           (fn (ixn, x as SOME _) => if ixn mem nvs then (false, x) else (true, NONE)
   877             | (_, x) => (false, x)) insts
   878       in ([], false, insts' @ map (pair false) ts'', prf) end
   879 and needed (Const ("==>", _) $ t $ u) ts ((b, _, _, _)::prfs) =
   880       (if b then map (fst o dest_Var) (vars_of t) else []) union needed u ts prfs
   881   | needed (Var (ixn, _)) (_::_) _ = [ixn]
   882   | needed _ _ _ = [];
   883 
   884 
   885 (**** Simple first order matching functions for terms and proofs ****)
   886 
   887 exception PMatch;
   888 
   889 (** see pattern.ML **)
   890 
   891 fun flt (i: int) = List.filter (fn n => n < i);
   892 
   893 fun fomatch Ts tymatch j =
   894   let
   895     fun mtch (instsp as (tyinsts, insts)) = fn
   896         (Var (ixn, T), t)  =>
   897           if j>0 andalso not (null (flt j (loose_bnos t)))
   898           then raise PMatch
   899           else (tymatch (tyinsts, fn () => (T, fastype_of1 (Ts, t))),
   900             (ixn, t) :: insts)
   901       | (Free (a, T), Free (b, U)) =>
   902 	  if a=b then (tymatch (tyinsts, K (T, U)), insts) else raise PMatch
   903       | (Const (a, T), Const (b, U))  =>
   904 	  if a=b then (tymatch (tyinsts, K (T, U)), insts) else raise PMatch
   905       | (f $ t, g $ u) => mtch (mtch instsp (f, g)) (t, u)
   906       | (Bound i, Bound j) => if i=j then instsp else raise PMatch
   907       | _ => raise PMatch
   908   in mtch end;
   909 
   910 fun match_proof Ts tymatch =
   911   let
   912     fun optmatch _ inst (NONE, _) = inst
   913       | optmatch _ _ (SOME _, NONE) = raise PMatch
   914       | optmatch mtch inst (SOME x, SOME y) = mtch inst (x, y)
   915 
   916     fun matcht Ts j (pinst, tinst) (t, u) =
   917       (pinst, fomatch Ts tymatch j tinst (t, Envir.beta_norm u));
   918     fun matchT (pinst, (tyinsts, insts)) p =
   919       (pinst, (tymatch (tyinsts, K p), insts));
   920     fun matchTs inst (Ts, Us) = Library.foldl (uncurry matchT) (inst, Ts ~~ Us);
   921 
   922     fun mtch Ts i j (pinst, tinst) (Hyp (Var (ixn, _)), prf) =
   923           if i = 0 andalso j = 0 then ((ixn, prf) :: pinst, tinst)
   924           else (case apfst (flt i) (apsnd (flt j)
   925                   (prf_add_loose_bnos 0 0 prf ([], []))) of
   926               ([], []) => ((ixn, incr_pboundvars (~i) (~j) prf) :: pinst, tinst)
   927             | ([], _) => if j = 0 then
   928                    ((ixn, incr_pboundvars (~i) (~j) prf) :: pinst, tinst)
   929                  else raise PMatch
   930             | _ => raise PMatch)
   931       | mtch Ts i j inst (prf1 % opt1, prf2 % opt2) =
   932           optmatch (matcht Ts j) (mtch Ts i j inst (prf1, prf2)) (opt1, opt2)
   933       | mtch Ts i j inst (prf1 %% prf2, prf1' %% prf2') =
   934           mtch Ts i j (mtch Ts i j inst (prf1, prf1')) (prf2, prf2')
   935       | mtch Ts i j inst (Abst (_, opT, prf1), Abst (_, opU, prf2)) =
   936           mtch (getOpt (opU,dummyT) :: Ts) i (j+1)
   937             (optmatch matchT inst (opT, opU)) (prf1, prf2)
   938       | mtch Ts i j inst (prf1, Abst (_, opU, prf2)) =
   939           mtch (getOpt (opU,dummyT) :: Ts) i (j+1) inst
   940             (incr_pboundvars 0 1 prf1 %> Bound 0, prf2)
   941       | mtch Ts i j inst (AbsP (_, opt, prf1), AbsP (_, opu, prf2)) =
   942           mtch Ts (i+1) j (optmatch (matcht Ts j) inst (opt, opu)) (prf1, prf2)
   943       | mtch Ts i j inst (prf1, AbsP (_, _, prf2)) =
   944           mtch Ts (i+1) j inst (incr_pboundvars 1 0 prf1 %% PBound 0, prf2)
   945       | mtch Ts i j inst (PThm ((name1, _), _, prop1, opTs),
   946             PThm ((name2, _), _, prop2, opUs)) =
   947           if name1=name2 andalso prop1=prop2 then
   948             optmatch matchTs inst (opTs, opUs)
   949           else raise PMatch
   950       | mtch Ts i j inst (PAxm (s1, _, opTs), PAxm (s2, _, opUs)) =
   951           if s1=s2 then optmatch matchTs inst (opTs, opUs)
   952           else raise PMatch
   953       | mtch _ _ _ inst (PBound i, PBound j) = if i = j then inst else raise PMatch
   954       | mtch _ _ _ _ _ = raise PMatch
   955   in mtch Ts 0 0 end;
   956 
   957 fun prf_subst (pinst, (tyinsts, insts)) =
   958   let
   959     val substT = Envir.typ_subst_TVars tyinsts;
   960 
   961     fun subst' lev (t as Var (ixn, _)) = (case assoc (insts, ixn) of
   962           NONE => t
   963         | SOME u => incr_boundvars lev u)
   964       | subst' lev (Const (s, T)) = Const (s, substT T)
   965       | subst' lev (Free (s, T)) = Free (s, substT T)
   966       | subst' lev (Abs (a, T, body)) = Abs (a, substT T, subst' (lev+1) body)
   967       | subst' lev (f $ t) = subst' lev f $ subst' lev t
   968       | subst' _ t = t;
   969 
   970     fun subst plev tlev (AbsP (a, t, body)) =
   971           AbsP (a, Option.map (subst' tlev) t, subst (plev+1) tlev body)
   972       | subst plev tlev (Abst (a, T, body)) =
   973           Abst (a, Option.map substT T, subst plev (tlev+1) body)
   974       | subst plev tlev (prf %% prf') = subst plev tlev prf %% subst plev tlev prf'
   975       | subst plev tlev (prf % t) = subst plev tlev prf % Option.map (subst' tlev) t
   976       | subst plev tlev (prf as Hyp (Var (ixn, _))) = (case assoc (pinst, ixn) of
   977           NONE => prf
   978         | SOME prf' => incr_pboundvars plev tlev prf')
   979       | subst _ _ (PThm (id, prf, prop, Ts)) =
   980           PThm (id, prf, prop, Option.map (map substT) Ts)
   981       | subst _ _ (PAxm (id, prop, Ts)) =
   982           PAxm (id, prop, Option.map (map substT) Ts)
   983       | subst _ _ t = t
   984   in subst 0 0 end;
   985 
   986 (*A fast unification filter: true unless the two terms cannot be unified. 
   987   Terms must be NORMAL.  Treats all Vars as distinct. *)
   988 fun could_unify prf1 prf2 =
   989   let
   990     fun matchrands (prf1 %% prf2) (prf1' %% prf2') =
   991           could_unify prf2 prf2' andalso matchrands prf1 prf1'
   992       | matchrands (prf % SOME t) (prf' % SOME t') =
   993           Term.could_unify (t, t') andalso matchrands prf prf'
   994       | matchrands (prf % _) (prf' % _) = matchrands prf prf'
   995       | matchrands _ _ = true
   996 
   997     fun head_of (prf %% _) = head_of prf
   998       | head_of (prf % _) = head_of prf
   999       | head_of prf = prf
  1000 
  1001   in case (head_of prf1, head_of prf2) of
  1002         (_, Hyp (Var _)) => true
  1003       | (Hyp (Var _), _) => true
  1004       | (PThm ((a, _), _, propa, _), PThm ((b, _), _, propb, _)) =>
  1005           a = b andalso propa = propb andalso matchrands prf1 prf2
  1006       | (PAxm (a, _, _), PAxm (b, _, _)) => a = b andalso matchrands prf1 prf2
  1007       | (PBound i, PBound j) =>  i = j andalso matchrands prf1 prf2
  1008       | (AbsP _, _) =>  true   (*because of possible eta equality*)
  1009       | (Abst _, _) =>  true
  1010       | (_, AbsP _) =>  true
  1011       | (_, Abst _) =>  true
  1012       | _ => false
  1013   end;
  1014 
  1015 (**** rewriting on proof terms ****)
  1016 
  1017 val skel0 = PBound 0;
  1018 
  1019 fun rewrite_prf tymatch (rules, procs) prf =
  1020   let
  1021     fun rew _ (Abst (_, _, body) % SOME t) = SOME (prf_subst_bounds [t] body, skel0)
  1022       | rew _ (AbsP (_, _, body) %% prf) = SOME (prf_subst_pbounds [prf] body, skel0)
  1023       | rew Ts prf = (case get_first (fn (_, r) => r Ts prf) procs of
  1024           SOME prf' => SOME (prf', skel0)
  1025         | NONE => get_first (fn (prf1, prf2) => SOME (prf_subst
  1026             (match_proof Ts tymatch ([], (Vartab.empty, [])) (prf1, prf)) prf2, prf2)
  1027                handle PMatch => NONE) (List.filter (could_unify prf o fst) rules));
  1028 
  1029     fun rew0 Ts (prf as AbsP (_, _, prf' %% PBound 0)) =
  1030           if prf_loose_Pbvar1 prf' 0 then rew Ts prf
  1031           else
  1032             let val prf'' = incr_pboundvars (~1) 0 prf'
  1033             in SOME (getOpt (rew Ts prf'', (prf'', skel0))) end
  1034       | rew0 Ts (prf as Abst (_, _, prf' % SOME (Bound 0))) =
  1035           if prf_loose_bvar1 prf' 0 then rew Ts prf
  1036           else
  1037             let val prf'' = incr_pboundvars 0 (~1) prf'
  1038             in SOME (getOpt (rew Ts prf'', (prf'', skel0))) end
  1039       | rew0 Ts prf = rew Ts prf;
  1040 
  1041     fun rew1 _ (Hyp (Var _)) _ = NONE
  1042       | rew1 Ts skel prf = (case rew2 Ts skel prf of
  1043           SOME prf1 => (case rew0 Ts prf1 of
  1044               SOME (prf2, skel') => SOME (getOpt (rew1 Ts skel' prf2, prf2))
  1045             | NONE => SOME prf1)
  1046         | NONE => (case rew0 Ts prf of
  1047               SOME (prf1, skel') => SOME (getOpt (rew1 Ts skel' prf1, prf1))
  1048             | NONE => NONE))
  1049 
  1050     and rew2 Ts skel (prf % SOME t) = (case prf of
  1051             Abst (_, _, body) =>
  1052               let val prf' = prf_subst_bounds [t] body
  1053               in SOME (getOpt (rew2 Ts skel0 prf', prf')) end
  1054           | _ => (case rew1 Ts (case skel of skel' % _ => skel' | _ => skel0) prf of
  1055               SOME prf' => SOME (prf' % SOME t)
  1056             | NONE => NONE))
  1057       | rew2 Ts skel (prf % NONE) = Option.map (fn prf' => prf' % NONE)
  1058           (rew1 Ts (case skel of skel' % _ => skel' | _ => skel0) prf)
  1059       | rew2 Ts skel (prf1 %% prf2) = (case prf1 of
  1060             AbsP (_, _, body) =>
  1061               let val prf' = prf_subst_pbounds [prf2] body
  1062               in SOME (getOpt (rew2 Ts skel0 prf', prf')) end
  1063           | _ =>
  1064             let val (skel1, skel2) = (case skel of
  1065                 skel1 %% skel2 => (skel1, skel2)
  1066               | _ => (skel0, skel0))
  1067             in case rew1 Ts skel1 prf1 of
  1068                 SOME prf1' => (case rew1 Ts skel2 prf2 of
  1069                     SOME prf2' => SOME (prf1' %% prf2')
  1070                   | NONE => SOME (prf1' %% prf2))
  1071               | NONE => (case rew1 Ts skel2 prf2 of
  1072                     SOME prf2' => SOME (prf1 %% prf2')
  1073                   | NONE => NONE)
  1074             end)
  1075       | rew2 Ts skel (Abst (s, T, prf)) = (case rew1 (getOpt (T,dummyT) :: Ts)
  1076               (case skel of Abst (_, _, skel') => skel' | _ => skel0) prf of
  1077             SOME prf' => SOME (Abst (s, T, prf'))
  1078           | NONE => NONE)
  1079       | rew2 Ts skel (AbsP (s, t, prf)) = (case rew1 Ts
  1080               (case skel of AbsP (_, _, skel') => skel' | _ => skel0) prf of
  1081             SOME prf' => SOME (AbsP (s, t, prf'))
  1082           | NONE => NONE)
  1083       | rew2 _ _ _ = NONE
  1084 
  1085   in getOpt (rew1 [] skel0 prf, prf) end;
  1086 
  1087 fun rewrite_proof tsig = rewrite_prf (fn (tab, f) =>
  1088   Type.typ_match tsig (tab, f ()) handle Type.TYPE_MATCH => raise PMatch);
  1089 
  1090 fun rewrite_proof_notypes rews = rewrite_prf fst rews;
  1091 
  1092 (**** theory data ****)
  1093 
  1094 (* data kind 'Pure/proof' *)
  1095 
  1096 structure ProofArgs =
  1097 struct
  1098   val name = "Pure/proof";
  1099   type T = ((proof * proof) list *
  1100     (string * (typ list -> proof -> proof option)) list);
  1101 
  1102   val empty = ([], []);
  1103   val copy = I;
  1104   val prep_ext = I;
  1105   fun merge ((rules1, procs1), (rules2, procs2)) =
  1106     (merge_lists rules1 rules2, merge_alists procs1 procs2);
  1107   fun print _ _ = ();
  1108 end;
  1109 
  1110 structure ProofData = TheoryDataFun(ProofArgs);
  1111 
  1112 val init = ProofData.init;
  1113 
  1114 fun add_prf_rrules rs thy =
  1115   let val r = ProofData.get thy
  1116   in ProofData.put (rs @ fst r, snd r) thy end;
  1117 
  1118 fun add_prf_rprocs ps thy =
  1119   let val r = ProofData.get thy
  1120   in ProofData.put (fst r, ps @ snd r) thy end;
  1121 
  1122 fun thm_proof sign (name, tags) hyps prop prf =
  1123   let
  1124     val prop = Logic.list_implies (hyps, prop);
  1125     val nvs = needed_vars prop;
  1126     val args = map (fn (v as Var (ixn, _)) =>
  1127         if ixn mem nvs then SOME v else NONE) (vars_of prop) @
  1128       map SOME (sort (make_ord atless) (term_frees prop));
  1129     val opt_prf = if ! proofs = 2 then
  1130         #4 (shrink [] 0 (rewrite_prf fst (ProofData.get_sg sign)
  1131           (foldr (uncurry implies_intr_proof) prf hyps)))
  1132       else MinProof (mk_min_proof ([], prf));
  1133     val head = (case strip_combt (fst (strip_combP prf)) of
  1134         (PThm ((old_name, _), prf', prop', NONE), args') =>
  1135           if (old_name="" orelse old_name=name) andalso
  1136              prop = prop' andalso args = args' then
  1137             PThm ((name, tags), prf', prop, NONE)
  1138           else
  1139             PThm ((name, tags), opt_prf, prop, NONE)
  1140       | _ => PThm ((name, tags), opt_prf, prop, NONE))
  1141   in
  1142     proof_combP (proof_combt' (head, args), map Hyp hyps)
  1143   end;
  1144 
  1145 fun get_name_tags hyps prop prf =
  1146   let val prop = Logic.list_implies (hyps, prop) in
  1147     (case strip_combt (fst (strip_combP prf)) of
  1148       (PThm ((name, tags), _, prop', _), _) =>
  1149         if prop=prop' then (name, tags) else ("", [])
  1150     | (PAxm (name, prop', _), _) =>
  1151         if prop=prop' then (name, []) else ("", [])
  1152     | _ => ("", []))
  1153   end;
  1154 
  1155 end;
  1156 
  1157 structure BasicProofterm : BASIC_PROOFTERM = Proofterm;
  1158 open BasicProofterm;