src/Pure/proofterm.ML
author wenzelm
Sat Dec 13 15:00:39 2008 +0100 (2008-12-13)
changeset 29091 b81fe045e799
parent 28971 300ec36a19af
child 29261 7ee78cc8ef2c
permissions -rw-r--r--
Context.display_names;
     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      MinProof
    16    | PBound of int
    17    | Abst of string * typ option * proof
    18    | AbsP of string * term option * proof
    19    | op % of proof * term option
    20    | op %% of proof * proof
    21    | Hyp of term
    22    | PAxm of string * term * typ list option
    23    | Oracle of string * term * typ list option
    24    | Promise of serial * term * typ list
    25    | PThm of serial * ((string * term * typ list option) * proof_body lazy)
    26   and proof_body = PBody of
    27     {oracles: (string * term) OrdList.T,
    28      thms: (serial * (string * term * proof_body lazy)) OrdList.T,
    29      proof: proof}
    30 
    31   val %> : proof * term -> proof
    32 end;
    33 
    34 signature PROOFTERM =
    35 sig
    36   include BASIC_PROOFTERM
    37 
    38   type oracle = string * term
    39   type pthm = serial * (string * term * proof_body lazy)
    40   val force_body: proof_body lazy ->
    41     {oracles: oracle OrdList.T, thms: pthm OrdList.T, proof: proof}
    42   val force_proof: proof_body lazy -> proof
    43   val proof_of: proof_body -> proof
    44   val fold_body_thms: (string * term * proof_body -> 'a -> 'a) -> proof_body list -> 'a -> 'a
    45   val fold_proof_atoms: bool -> (proof -> 'a -> 'a) -> proof list -> 'a -> 'a
    46 
    47   val oracle_ord: oracle * oracle -> order
    48   val thm_ord: pthm * pthm -> order
    49   val make_proof_body: proof -> proof_body
    50   val merge_oracles: oracle OrdList.T -> oracle OrdList.T -> oracle OrdList.T
    51   val make_oracles: proof -> oracle OrdList.T
    52   val merge_thms: pthm OrdList.T -> pthm OrdList.T -> pthm OrdList.T
    53   val make_thms: proof -> pthm OrdList.T
    54 
    55   (** primitive operations **)
    56   val proof_combt: proof * term list -> proof
    57   val proof_combt': proof * term option list -> proof
    58   val proof_combP: proof * proof list -> proof
    59   val strip_combt: proof -> proof * term option list
    60   val strip_combP: proof -> proof * proof list
    61   val strip_thm: proof_body -> proof_body
    62   val map_proof_terms_option: (term -> term option) -> (typ -> typ option) -> proof -> proof
    63   val map_proof_terms: (term -> term) -> (typ -> typ) -> proof -> proof
    64   val fold_proof_terms: (term -> 'a -> 'a) -> (typ -> 'a -> 'a) -> proof -> 'a -> 'a
    65   val maxidx_proof: proof -> int -> int
    66   val size_of_proof: proof -> int
    67   val change_type: typ list option -> proof -> proof
    68   val prf_abstract_over: term -> proof -> proof
    69   val prf_incr_bv: int -> int -> int -> int -> proof -> proof
    70   val incr_pboundvars: int -> int -> proof -> proof
    71   val prf_loose_bvar1: proof -> int -> bool
    72   val prf_loose_Pbvar1: proof -> int -> bool
    73   val prf_add_loose_bnos: int -> int -> proof -> int list * int list -> int list * int list
    74   val norm_proof: Envir.env -> proof -> proof
    75   val norm_proof': Envir.env -> proof -> proof
    76   val prf_subst_bounds: term list -> proof -> proof
    77   val prf_subst_pbounds: proof list -> proof -> proof
    78   val freeze_thaw_prf: proof -> proof * (proof -> proof)
    79 
    80   (** proof terms for specific inference rules **)
    81   val implies_intr_proof: term -> proof -> proof
    82   val forall_intr_proof: term -> string -> proof -> proof
    83   val varify_proof: term -> (string * sort) list -> proof -> proof
    84   val freezeT: term -> proof -> proof
    85   val rotate_proof: term list -> term -> int -> proof -> proof
    86   val permute_prems_prf: term list -> int -> int -> proof -> proof
    87   val generalize: string list * string list -> int -> proof -> proof
    88   val instantiate: ((indexname * sort) * typ) list * ((indexname * typ) * term) list
    89     -> proof -> proof
    90   val lift_proof: term -> int -> term -> proof -> proof
    91   val assumption_proof: term list -> term -> int -> proof -> proof
    92   val bicompose_proof: bool -> term list -> term list -> term list -> term option ->
    93     int -> int -> proof -> proof -> proof
    94   val equality_axms: (string * term) list
    95   val reflexive_axm: proof
    96   val symmetric_axm: proof
    97   val transitive_axm: proof
    98   val equal_intr_axm: proof
    99   val equal_elim_axm: proof
   100   val abstract_rule_axm: proof
   101   val combination_axm: proof
   102   val reflexive: proof
   103   val symmetric: proof -> proof
   104   val transitive: term -> typ -> proof -> proof -> proof
   105   val abstract_rule: term -> string -> proof -> proof
   106   val combination: term -> term -> term -> term -> typ -> proof -> proof -> proof
   107   val equal_intr: term -> term -> proof -> proof -> proof
   108   val equal_elim: term -> term -> proof -> proof -> proof
   109   val axm_proof: string -> term -> proof
   110   val oracle_proof: string -> term -> proof
   111   val promise_proof: theory -> serial -> term -> proof
   112   val fulfill_proof: theory -> (serial * proof) list -> proof_body -> proof_body
   113   val thm_proof: theory -> string -> term list -> term ->
   114     (serial * proof) list lazy -> proof_body -> pthm * proof
   115   val get_name: term list -> term -> proof -> string
   116 
   117   (** rewriting on proof terms **)
   118   val add_prf_rrule: proof * proof -> theory -> theory
   119   val add_prf_rproc: (typ list -> proof -> proof option) -> theory -> theory
   120   val rewrite_proof: theory -> (proof * proof) list *
   121     (typ list -> proof -> proof option) list -> proof -> proof
   122   val rewrite_proof_notypes: (proof * proof) list *
   123     (typ list -> proof -> proof option) list -> proof -> proof
   124   val rew_proof: theory -> proof -> proof
   125 end
   126 
   127 structure Proofterm : PROOFTERM =
   128 struct
   129 
   130 open Envir;
   131 
   132 
   133 (***** datatype proof *****)
   134 
   135 datatype proof =
   136    MinProof
   137  | PBound of int
   138  | Abst of string * typ option * proof
   139  | AbsP of string * term option * proof
   140  | op % of proof * term option
   141  | op %% of proof * proof
   142  | Hyp of term
   143  | PAxm of string * term * typ list option
   144  | Oracle of string * term * typ list option
   145  | Promise of serial * term * typ list
   146  | PThm of serial * ((string * term * typ list option) * proof_body lazy)
   147 and proof_body = PBody of
   148   {oracles: (string * term) OrdList.T,
   149    thms: (serial * (string * term * proof_body lazy)) OrdList.T,
   150    proof: proof};
   151 
   152 type oracle = string * term;
   153 type pthm = serial * (string * term * proof_body lazy);
   154 
   155 val force_body = Lazy.force #> (fn PBody args => args);
   156 val force_proof = #proof o force_body;
   157 
   158 fun proof_of (PBody {proof, ...}) = proof;
   159 
   160 
   161 (***** proof atoms *****)
   162 
   163 fun fold_body_thms f =
   164   let
   165     fun app (PBody {thms, ...}) = thms |> fold (fn (i, (name, prop, body)) => fn (x, seen) =>
   166       if Inttab.defined seen i then (x, seen)
   167       else
   168         let
   169           val body' = Lazy.force body;
   170           val (x', seen') = app body' (x, Inttab.update (i, ()) seen);
   171         in (f (name, prop, body') x', seen') end);
   172   in fn bodies => fn x => #1 (fold app bodies (x, Inttab.empty)) end;
   173 
   174 fun fold_proof_atoms all f =
   175   let
   176     fun app (Abst (_, _, prf)) = app prf
   177       | app (AbsP (_, _, prf)) = app prf
   178       | app (prf % _) = app prf
   179       | app (prf1 %% prf2) = app prf1 #> app prf2
   180       | app (prf as PThm (i, (_, body))) = (fn (x, seen) =>
   181           if Inttab.defined seen i then (x, seen)
   182           else
   183             let val (x', seen') =
   184               (if all then app (force_proof body) else I) (x, Inttab.update (i, ()) seen)
   185             in (f prf x', seen') end)
   186       | app prf = (fn (x, seen) => (f prf x, seen));
   187   in fn prfs => fn x => #1 (fold app prfs (x, Inttab.empty)) end;
   188 
   189 
   190 (* proof body *)
   191 
   192 val oracle_ord = prod_ord fast_string_ord Term.fast_term_ord;
   193 fun thm_ord ((i, _): pthm, (j, _)) = int_ord (j, i);
   194 
   195 fun make_body prf =
   196   let
   197     val (oracles, thms) = fold_proof_atoms false
   198       (fn Oracle (s, prop, _) => apfst (cons (s, prop))
   199         | PThm (i, ((name, prop, _), body)) => apsnd (cons (i, (name, prop, body)))
   200         | _ => I) [prf] ([], []);
   201   in (OrdList.make oracle_ord oracles, OrdList.make thm_ord thms) end;
   202 
   203 fun make_proof_body prf =
   204   let val (oracles, thms) = make_body prf
   205   in PBody {oracles = oracles, thms = thms, proof = prf} end;
   206 
   207 val make_oracles = #1 o make_body;
   208 val make_thms = #2 o make_body;
   209 
   210 val merge_oracles = OrdList.union oracle_ord;
   211 val merge_thms = OrdList.union thm_ord;
   212 
   213 fun merge_body (oracles1, thms1) (oracles2, thms2) =
   214   (merge_oracles oracles1 oracles2, merge_thms thms1 thms2);
   215 
   216 
   217 (***** proof objects with different levels of detail *****)
   218 
   219 fun (prf %> t) = prf % SOME t;
   220 
   221 val proof_combt = Library.foldl (op %>);
   222 val proof_combt' = Library.foldl (op %);
   223 val proof_combP = Library.foldl (op %%);
   224 
   225 fun strip_combt prf =
   226     let fun stripc (prf % t, ts) = stripc (prf, t::ts)
   227           | stripc  x =  x
   228     in  stripc (prf, [])  end;
   229 
   230 fun strip_combP prf =
   231     let fun stripc (prf %% prf', prfs) = stripc (prf, prf'::prfs)
   232           | stripc  x =  x
   233     in  stripc (prf, [])  end;
   234 
   235 fun strip_thm (body as PBody {proof, ...}) =
   236   (case strip_combt (fst (strip_combP proof)) of
   237     (PThm (_, (_, body')), _) => Lazy.force body'
   238   | _ => body);
   239 
   240 val mk_Abst = fold_rev (fn (s, T:typ) => fn prf => Abst (s, NONE, prf));
   241 fun mk_AbsP (i, prf) = funpow i (fn prf => AbsP ("H", NONE, prf)) prf;
   242 
   243 fun apsome f NONE = raise SAME
   244   | apsome f (SOME x) = (case f x of NONE => raise SAME | some => some);
   245 
   246 fun apsome' f NONE = raise SAME
   247   | apsome' f (SOME x) = SOME (f x);
   248 
   249 fun map_proof_terms_option f g =
   250   let
   251     fun map_typs (T :: Ts) =
   252           (case g T of
   253             NONE => T :: map_typs Ts
   254           | SOME T' => T' :: (map_typs Ts handle SAME => Ts))
   255       | map_typs [] = raise SAME;
   256 
   257     fun mapp (Abst (s, T, prf)) = (Abst (s, apsome g T, mapph prf)
   258           handle SAME => Abst (s, T, mapp prf))
   259       | mapp (AbsP (s, t, prf)) = (AbsP (s, apsome f t, mapph prf)
   260           handle SAME => AbsP (s, t, mapp prf))
   261       | mapp (prf % t) = (mapp prf % (apsome f t handle SAME => t)
   262           handle SAME => prf % apsome f t)
   263       | mapp (prf1 %% prf2) = (mapp prf1 %% mapph prf2
   264           handle SAME => prf1 %% mapp prf2)
   265       | mapp (PAxm (a, prop, SOME Ts)) = PAxm (a, prop, SOME (map_typs Ts))
   266       | mapp (Oracle (a, prop, SOME Ts)) = Oracle (a, prop, SOME (map_typs Ts))
   267       | mapp (Promise (i, prop, Ts)) = Promise (i, prop, map_typs Ts)
   268       | mapp (PThm (i, ((a, prop, SOME Ts), body))) =
   269           PThm (i, ((a, prop, SOME (map_typs Ts)), body))
   270       | mapp _ = raise SAME
   271     and mapph prf = (mapp prf handle SAME => prf)
   272 
   273   in mapph end;
   274 
   275 fun same eq f x =
   276   let val x' = f x
   277   in if eq (x, x') then raise SAME else x' end;
   278 
   279 fun map_proof_terms f g =
   280   map_proof_terms_option
   281    (fn t => SOME (same (op =) f t) handle SAME => NONE)
   282    (fn T => SOME (same (op =) g T) handle SAME => NONE);
   283 
   284 fun fold_proof_terms f g (Abst (_, SOME T, prf)) = g T #> fold_proof_terms f g prf
   285   | fold_proof_terms f g (Abst (_, NONE, prf)) = fold_proof_terms f g prf
   286   | fold_proof_terms f g (AbsP (_, SOME t, prf)) = f t #> fold_proof_terms f g prf
   287   | fold_proof_terms f g (AbsP (_, NONE, prf)) = fold_proof_terms f g prf
   288   | fold_proof_terms f g (prf % SOME t) = fold_proof_terms f g prf #> f t
   289   | fold_proof_terms f g (prf % NONE) = fold_proof_terms f g prf
   290   | fold_proof_terms f g (prf1 %% prf2) =
   291       fold_proof_terms f g prf1 #> fold_proof_terms f g prf2
   292   | fold_proof_terms _ g (PAxm (_, _, SOME Ts)) = fold g Ts
   293   | fold_proof_terms _ g (Oracle (_, _, SOME Ts)) = fold g Ts
   294   | fold_proof_terms _ g (Promise (_, _, Ts)) = fold g Ts
   295   | fold_proof_terms _ g (PThm (_, ((_, _, SOME Ts), _))) = fold g Ts
   296   | fold_proof_terms _ _ _ = I;
   297 
   298 fun maxidx_proof prf = fold_proof_terms Term.maxidx_term Term.maxidx_typ prf;
   299 
   300 fun size_of_proof (Abst (_, _, prf)) = 1 + size_of_proof prf
   301   | size_of_proof (AbsP (_, t, prf)) = 1 + size_of_proof prf
   302   | size_of_proof (prf % _) = 1 + size_of_proof prf
   303   | size_of_proof (prf1 %% prf2) = size_of_proof prf1 + size_of_proof prf2
   304   | size_of_proof _ = 1;
   305 
   306 fun change_type opTs (PAxm (name, prop, _)) = PAxm (name, prop, opTs)
   307   | change_type opTs (Oracle (name, prop, _)) = Oracle (name, prop, opTs)
   308   | change_type opTs (Promise _) = error "change_type: unexpected promise"
   309   | change_type opTs (PThm (i, ((name, prop, _), body))) = PThm (i, ((name, prop, opTs), body))
   310   | change_type _ prf = prf;
   311 
   312 
   313 (***** utilities *****)
   314 
   315 fun strip_abs (_::Ts) (Abs (_, _, t)) = strip_abs Ts t
   316   | strip_abs _ t = t;
   317 
   318 fun mk_abs Ts t = Library.foldl (fn (t', T) => Abs ("", T, t')) (t, Ts);
   319 
   320 
   321 (*Abstraction of a proof term over its occurrences of v,
   322     which must contain no loose bound variables.
   323   The resulting proof term is ready to become the body of an Abst.*)
   324 
   325 fun prf_abstract_over v =
   326   let
   327     fun abst' lev u = if v aconv u then Bound lev else
   328       (case u of
   329          Abs (a, T, t) => Abs (a, T, abst' (lev + 1) t)
   330        | f $ t => (abst' lev f $ absth' lev t handle SAME => f $ abst' lev t)
   331        | _ => raise SAME)
   332     and absth' lev t = (abst' lev t handle SAME => t);
   333 
   334     fun abst lev (AbsP (a, t, prf)) =
   335           (AbsP (a, apsome' (abst' lev) t, absth lev prf)
   336            handle SAME => AbsP (a, t, abst lev prf))
   337       | abst lev (Abst (a, T, prf)) = Abst (a, T, abst (lev + 1) prf)
   338       | abst lev (prf1 %% prf2) = (abst lev prf1 %% absth lev prf2
   339           handle SAME => prf1 %% abst lev prf2)
   340       | abst lev (prf % t) = (abst lev prf % Option.map (absth' lev) t
   341           handle SAME => prf % apsome' (abst' lev) t)
   342       | abst _ _ = raise SAME
   343     and absth lev prf = (abst lev prf handle SAME => prf)
   344 
   345   in absth 0 end;
   346 
   347 
   348 (*increments a proof term's non-local bound variables
   349   required when moving a proof term within abstractions
   350      inc is  increment for bound variables
   351      lev is  level at which a bound variable is considered 'loose'*)
   352 
   353 fun incr_bv' inct tlev t = incr_bv (inct, tlev, t);
   354 
   355 fun prf_incr_bv' incP inct Plev tlev (PBound i) =
   356       if i >= Plev then PBound (i+incP) else raise SAME
   357   | prf_incr_bv' incP inct Plev tlev (AbsP (a, t, body)) =
   358       (AbsP (a, apsome' (same (op =) (incr_bv' inct tlev)) t,
   359          prf_incr_bv incP inct (Plev+1) tlev body) handle SAME =>
   360            AbsP (a, t, prf_incr_bv' incP inct (Plev+1) tlev body))
   361   | prf_incr_bv' incP inct Plev tlev (Abst (a, T, body)) =
   362       Abst (a, T, prf_incr_bv' incP inct Plev (tlev+1) body)
   363   | prf_incr_bv' incP inct Plev tlev (prf %% prf') =
   364       (prf_incr_bv' incP inct Plev tlev prf %% prf_incr_bv incP inct Plev tlev prf'
   365        handle SAME => prf %% prf_incr_bv' incP inct Plev tlev prf')
   366   | prf_incr_bv' incP inct Plev tlev (prf % t) =
   367       (prf_incr_bv' incP inct Plev tlev prf % Option.map (incr_bv' inct tlev) t
   368        handle SAME => prf % apsome' (same (op =) (incr_bv' inct tlev)) t)
   369   | prf_incr_bv' _ _ _ _ _ = raise SAME
   370 and prf_incr_bv incP inct Plev tlev prf =
   371       (prf_incr_bv' incP inct Plev tlev prf handle SAME => prf);
   372 
   373 fun incr_pboundvars  0 0 prf = prf
   374   | incr_pboundvars incP inct prf = prf_incr_bv incP inct 0 0 prf;
   375 
   376 
   377 fun prf_loose_bvar1 (prf1 %% prf2) k = prf_loose_bvar1 prf1 k orelse prf_loose_bvar1 prf2 k
   378   | prf_loose_bvar1 (prf % SOME t) k = prf_loose_bvar1 prf k orelse loose_bvar1 (t, k)
   379   | prf_loose_bvar1 (_ % NONE) _ = true
   380   | prf_loose_bvar1 (AbsP (_, SOME t, prf)) k = loose_bvar1 (t, k) orelse prf_loose_bvar1 prf k
   381   | prf_loose_bvar1 (AbsP (_, NONE, _)) k = true
   382   | prf_loose_bvar1 (Abst (_, _, prf)) k = prf_loose_bvar1 prf (k+1)
   383   | prf_loose_bvar1 _ _ = false;
   384 
   385 fun prf_loose_Pbvar1 (PBound i) k = i = k
   386   | prf_loose_Pbvar1 (prf1 %% prf2) k = prf_loose_Pbvar1 prf1 k orelse prf_loose_Pbvar1 prf2 k
   387   | prf_loose_Pbvar1 (prf % _) k = prf_loose_Pbvar1 prf k
   388   | prf_loose_Pbvar1 (AbsP (_, _, prf)) k = prf_loose_Pbvar1 prf (k+1)
   389   | prf_loose_Pbvar1 (Abst (_, _, prf)) k = prf_loose_Pbvar1 prf k
   390   | prf_loose_Pbvar1 _ _ = false;
   391 
   392 fun prf_add_loose_bnos plev tlev (PBound i) (is, js) =
   393       if i < plev then (is, js) else (insert (op =) (i-plev) is, js)
   394   | prf_add_loose_bnos plev tlev (prf1 %% prf2) p =
   395       prf_add_loose_bnos plev tlev prf2
   396         (prf_add_loose_bnos plev tlev prf1 p)
   397   | prf_add_loose_bnos plev tlev (prf % opt) (is, js) =
   398       prf_add_loose_bnos plev tlev prf (case opt of
   399           NONE => (is, insert (op =) ~1 js)
   400         | SOME t => (is, add_loose_bnos (t, tlev, js)))
   401   | prf_add_loose_bnos plev tlev (AbsP (_, opt, prf)) (is, js) =
   402       prf_add_loose_bnos (plev+1) tlev prf (case opt of
   403           NONE => (is, insert (op =) ~1 js)
   404         | SOME t => (is, add_loose_bnos (t, tlev, js)))
   405   | prf_add_loose_bnos plev tlev (Abst (_, _, prf)) p =
   406       prf_add_loose_bnos plev (tlev+1) prf p
   407   | prf_add_loose_bnos _ _ _ _ = ([], []);
   408 
   409 
   410 (**** substitutions ****)
   411 
   412 fun del_conflicting_tvars envT T = TermSubst.instantiateT
   413   (map_filter (fn ixnS as (_, S) =>
   414      (Type.lookup envT ixnS; NONE) handle TYPE _ =>
   415         SOME (ixnS, TFree ("'dummy", S))) (typ_tvars T)) T;
   416 
   417 fun del_conflicting_vars env t = TermSubst.instantiate
   418   (map_filter (fn ixnS as (_, S) =>
   419      (Type.lookup (type_env env) ixnS; NONE) handle TYPE _ =>
   420         SOME (ixnS, TFree ("'dummy", S))) (term_tvars t),
   421    map_filter (fn Var (ixnT as (_, T)) =>
   422      (Envir.lookup (env, ixnT); NONE) handle TYPE _ =>
   423         SOME (ixnT, Free ("dummy", T))) (term_vars t)) t;
   424 
   425 fun norm_proof env =
   426   let
   427     val envT = type_env env;
   428     fun msg s = warning ("type conflict in norm_proof:\n" ^ s);
   429     fun htype f t = f env t handle TYPE (s, _, _) =>
   430       (msg s; f env (del_conflicting_vars env t));
   431     fun htypeT f T = f envT T handle TYPE (s, _, _) =>
   432       (msg s; f envT (del_conflicting_tvars envT T));
   433     fun htypeTs f Ts = f envT Ts handle TYPE (s, _, _) =>
   434       (msg s; f envT (map (del_conflicting_tvars envT) Ts));
   435     fun norm (Abst (s, T, prf)) = (Abst (s, apsome' (htypeT norm_type_same) T, normh prf)
   436           handle SAME => Abst (s, T, norm prf))
   437       | norm (AbsP (s, t, prf)) = (AbsP (s, apsome' (htype norm_term_same) t, normh prf)
   438           handle SAME => AbsP (s, t, norm prf))
   439       | norm (prf % t) = (norm prf % Option.map (htype norm_term) t
   440           handle SAME => prf % apsome' (htype norm_term_same) t)
   441       | norm (prf1 %% prf2) = (norm prf1 %% normh prf2
   442           handle SAME => prf1 %% norm prf2)
   443       | norm (PAxm (s, prop, Ts)) = PAxm (s, prop, apsome' (htypeTs norm_types_same) Ts)
   444       | norm (Oracle (s, prop, Ts)) = Oracle (s, prop, apsome' (htypeTs norm_types_same) Ts)
   445       | norm (Promise (i, prop, Ts)) = Promise (i, prop, htypeTs norm_types_same Ts)
   446       | norm (PThm (i, ((s, t, Ts), body))) =
   447           PThm (i, ((s, t, apsome' (htypeTs norm_types_same) Ts), body))
   448       | norm _ = raise SAME
   449     and normh prf = (norm prf handle SAME => prf);
   450   in normh end;
   451 
   452 
   453 (***** Remove some types in proof term (to save space) *****)
   454 
   455 fun remove_types (Abs (s, _, t)) = Abs (s, dummyT, remove_types t)
   456   | remove_types (t $ u) = remove_types t $ remove_types u
   457   | remove_types (Const (s, _)) = Const (s, dummyT)
   458   | remove_types t = t;
   459 
   460 fun remove_types_env (Envir.Envir {iTs, asol, maxidx}) =
   461   Envir.Envir {iTs = iTs, asol = Vartab.map (apsnd remove_types) asol,
   462     maxidx = maxidx};
   463 
   464 fun norm_proof' env prf = norm_proof (remove_types_env env) prf;
   465 
   466 
   467 (**** substitution of bound variables ****)
   468 
   469 fun prf_subst_bounds args prf =
   470   let
   471     val n = length args;
   472     fun subst' lev (Bound i) =
   473          (if i<lev then raise SAME    (*var is locally bound*)
   474           else  incr_boundvars lev (List.nth (args, i-lev))
   475                   handle Subscript => Bound (i-n)  (*loose: change it*))
   476       | subst' lev (Abs (a, T, body)) = Abs (a, T,  subst' (lev+1) body)
   477       | subst' lev (f $ t) = (subst' lev f $ substh' lev t
   478           handle SAME => f $ subst' lev t)
   479       | subst' _ _ = raise SAME
   480     and substh' lev t = (subst' lev t handle SAME => t);
   481 
   482     fun subst lev (AbsP (a, t, body)) = (AbsP (a, apsome' (subst' lev) t, substh lev body)
   483           handle SAME => AbsP (a, t, subst lev body))
   484       | subst lev (Abst (a, T, body)) = Abst (a, T, subst (lev+1) body)
   485       | subst lev (prf %% prf') = (subst lev prf %% substh lev prf'
   486           handle SAME => prf %% subst lev prf')
   487       | subst lev (prf % t) = (subst lev prf % Option.map (substh' lev) t
   488           handle SAME => prf % apsome' (subst' lev) t)
   489       | subst _ _ = raise SAME
   490     and substh lev prf = (subst lev prf handle SAME => prf)
   491   in case args of [] => prf | _ => substh 0 prf end;
   492 
   493 fun prf_subst_pbounds args prf =
   494   let
   495     val n = length args;
   496     fun subst (PBound i) Plev tlev =
   497          (if i < Plev then raise SAME    (*var is locally bound*)
   498           else incr_pboundvars Plev tlev (List.nth (args, i-Plev))
   499                  handle Subscript => PBound (i-n)  (*loose: change it*))
   500       | subst (AbsP (a, t, body)) Plev tlev = AbsP (a, t, subst body (Plev+1) tlev)
   501       | subst (Abst (a, T, body)) Plev tlev = Abst (a, T, subst body Plev (tlev+1))
   502       | subst (prf %% prf') Plev tlev = (subst prf Plev tlev %% substh prf' Plev tlev
   503           handle SAME => prf %% subst prf' Plev tlev)
   504       | subst (prf % t) Plev tlev = subst prf Plev tlev % t
   505       | subst  prf _ _ = raise SAME
   506     and substh prf Plev tlev = (subst prf Plev tlev handle SAME => prf)
   507   in case args of [] => prf | _ => substh prf 0 0 end;
   508 
   509 
   510 (**** Freezing and thawing of variables in proof terms ****)
   511 
   512 fun frzT names =
   513   map_type_tvar (fn (ixn, xs) => TFree ((the o AList.lookup (op =) names) ixn, xs));
   514 
   515 fun thawT names =
   516   map_type_tfree (fn (s, xs) => case AList.lookup (op =) names s of
   517       NONE => TFree (s, xs)
   518     | SOME ixn => TVar (ixn, xs));
   519 
   520 fun freeze names names' (t $ u) =
   521       freeze names names' t $ freeze names names' u
   522   | freeze names names' (Abs (s, T, t)) =
   523       Abs (s, frzT names' T, freeze names names' t)
   524   | freeze names names' (Const (s, T)) = Const (s, frzT names' T)
   525   | freeze names names' (Free (s, T)) = Free (s, frzT names' T)
   526   | freeze names names' (Var (ixn, T)) =
   527       Free ((the o AList.lookup (op =) names) ixn, frzT names' T)
   528   | freeze names names' t = t;
   529 
   530 fun thaw names names' (t $ u) =
   531       thaw names names' t $ thaw names names' u
   532   | thaw names names' (Abs (s, T, t)) =
   533       Abs (s, thawT names' T, thaw names names' t)
   534   | thaw names names' (Const (s, T)) = Const (s, thawT names' T)
   535   | thaw names names' (Free (s, T)) =
   536       let val T' = thawT names' T
   537       in case AList.lookup (op =) names s of
   538           NONE => Free (s, T')
   539         | SOME ixn => Var (ixn, T')
   540       end
   541   | thaw names names' (Var (ixn, T)) = Var (ixn, thawT names' T)
   542   | thaw names names' t = t;
   543 
   544 fun freeze_thaw_prf prf =
   545   let
   546     val (fs, Tfs, vs, Tvs) = fold_proof_terms
   547       (fn t => fn (fs, Tfs, vs, Tvs) =>
   548          (add_term_frees (t, fs), add_term_tfree_names (t, Tfs),
   549           add_term_vars (t, vs), add_term_tvar_ixns (t, Tvs)))
   550       (fn T => fn (fs, Tfs, vs, Tvs) =>
   551          (fs, add_typ_tfree_names (T, Tfs),
   552           vs, add_typ_ixns (Tvs, T)))
   553       prf ([], [], [], []);
   554     val fs' = map (fst o dest_Free) fs;
   555     val vs' = map (fst o dest_Var) vs;
   556     val names = vs' ~~ Name.variant_list fs' (map fst vs');
   557     val names' = Tvs ~~ Name.variant_list Tfs (map fst Tvs);
   558     val rnames = map swap names;
   559     val rnames' = map swap names';
   560   in
   561     (map_proof_terms (freeze names names') (frzT names') prf,
   562      map_proof_terms (thaw rnames rnames') (thawT rnames'))
   563   end;
   564 
   565 
   566 (***** implication introduction *****)
   567 
   568 fun implies_intr_proof h prf =
   569   let
   570     fun abshyp i (Hyp t) = if h aconv t then PBound i else raise SAME
   571       | abshyp i (Abst (s, T, prf)) = Abst (s, T, abshyp i prf)
   572       | abshyp i (AbsP (s, t, prf)) = AbsP (s, t, abshyp (i+1) prf)
   573       | abshyp i (prf % t) = abshyp i prf % t
   574       | abshyp i (prf1 %% prf2) = (abshyp i prf1 %% abshyph i prf2
   575           handle SAME => prf1 %% abshyp i prf2)
   576       | abshyp _ _ = raise SAME
   577     and abshyph i prf = (abshyp i prf handle SAME => prf)
   578   in
   579     AbsP ("H", NONE (*h*), abshyph 0 prf)
   580   end;
   581 
   582 
   583 (***** forall introduction *****)
   584 
   585 fun forall_intr_proof x a prf = Abst (a, NONE, prf_abstract_over x prf);
   586 
   587 
   588 (***** varify *****)
   589 
   590 fun varify_proof t fixed prf =
   591   let
   592     val fs = Term.fold_types (Term.fold_atyps
   593       (fn TFree v => if member (op =) fixed v then I else insert (op =) v | _ => I)) t [];
   594     val ixns = add_term_tvar_ixns (t, []);
   595     val fmap = fs ~~ Name.variant_list (map #1 ixns) (map fst fs);
   596     fun thaw (f as (a, S)) =
   597       (case AList.lookup (op =) fmap f of
   598         NONE => TFree f
   599       | SOME b => TVar ((b, 0), S));
   600   in map_proof_terms (map_types (map_type_tfree thaw)) (map_type_tfree thaw) prf end;
   601 
   602 
   603 local
   604 
   605 fun new_name (ix, (pairs,used)) =
   606   let val v = Name.variant used (string_of_indexname ix)
   607   in  ((ix, v) :: pairs, v :: used)  end;
   608 
   609 fun freeze_one alist (ix, sort) = (case AList.lookup (op =) alist ix of
   610     NONE => TVar (ix, sort)
   611   | SOME name => TFree (name, sort));
   612 
   613 in
   614 
   615 fun freezeT t prf =
   616   let
   617     val used = it_term_types add_typ_tfree_names (t, [])
   618     and tvars = map #1 (it_term_types add_typ_tvars (t, []));
   619     val (alist, _) = List.foldr new_name ([], used) tvars;
   620   in
   621     (case alist of
   622       [] => prf (*nothing to do!*)
   623     | _ =>
   624       let val frzT = map_type_tvar (freeze_one alist)
   625       in map_proof_terms (map_types frzT) frzT prf end)
   626   end;
   627 
   628 end;
   629 
   630 
   631 (***** rotate assumptions *****)
   632 
   633 fun rotate_proof Bs Bi m prf =
   634   let
   635     val params = Term.strip_all_vars Bi;
   636     val asms = Logic.strip_imp_prems (Term.strip_all_body Bi);
   637     val i = length asms;
   638     val j = length Bs;
   639   in
   640     mk_AbsP (j+1, proof_combP (prf, map PBound
   641       (j downto 1) @ [mk_Abst params (mk_AbsP (i,
   642         proof_combP (proof_combt (PBound i, map Bound ((length params - 1) downto 0)),
   643           map PBound (((i-m-1) downto 0) @ ((i-1) downto (i-m))))))]))
   644   end;
   645 
   646 
   647 (***** permute premises *****)
   648 
   649 fun permute_prems_prf prems j k prf =
   650   let val n = length prems
   651   in mk_AbsP (n, proof_combP (prf,
   652     map PBound ((n-1 downto n-j) @ (k-1 downto 0) @ (n-j-1 downto k))))
   653   end;
   654 
   655 
   656 (***** generalization *****)
   657 
   658 fun generalize (tfrees, frees) idx =
   659   map_proof_terms_option
   660     (TermSubst.generalize_option (tfrees, frees) idx)
   661     (TermSubst.generalizeT_option tfrees idx);
   662 
   663 
   664 (***** instantiation *****)
   665 
   666 fun instantiate (instT, inst) =
   667   map_proof_terms_option
   668     (TermSubst.instantiate_option (instT, map (apsnd remove_types) inst))
   669     (TermSubst.instantiateT_option instT);
   670 
   671 
   672 (***** lifting *****)
   673 
   674 fun lift_proof Bi inc prop prf =
   675   let
   676     fun lift'' Us Ts t = strip_abs Ts (Logic.incr_indexes (Us, inc) (mk_abs Ts t));
   677 
   678     fun lift' Us Ts (Abst (s, T, prf)) =
   679           (Abst (s, apsome' (same (op =) (Logic.incr_tvar inc)) T, lifth' Us (dummyT::Ts) prf)
   680            handle SAME => Abst (s, T, lift' Us (dummyT::Ts) prf))
   681       | lift' Us Ts (AbsP (s, t, prf)) =
   682           (AbsP (s, apsome' (same (op =) (lift'' Us Ts)) t, lifth' Us Ts prf)
   683            handle SAME => AbsP (s, t, lift' Us Ts prf))
   684       | lift' Us Ts (prf % t) = (lift' Us Ts prf % Option.map (lift'' Us Ts) t
   685           handle SAME => prf % apsome' (same (op =) (lift'' Us Ts)) t)
   686       | lift' Us Ts (prf1 %% prf2) = (lift' Us Ts prf1 %% lifth' Us Ts prf2
   687           handle SAME => prf1 %% lift' Us Ts prf2)
   688       | lift' _ _ (PAxm (s, prop, Ts)) =
   689           PAxm (s, prop, apsome' (same (op =) (map (Logic.incr_tvar inc))) Ts)
   690       | lift' _ _ (Oracle (s, prop, Ts)) =
   691           Oracle (s, prop, apsome' (same (op =) (map (Logic.incr_tvar inc))) Ts)
   692       | lift' _ _ (Promise (i, prop, Ts)) =
   693           Promise (i, prop, same (op =) (map (Logic.incr_tvar inc)) Ts)
   694       | lift' _ _ (PThm (i, ((s, prop, Ts), body))) =
   695           PThm (i, ((s, prop, apsome' (same (op =) (map (Logic.incr_tvar inc))) Ts), body))
   696       | lift' _ _ _ = raise SAME
   697     and lifth' Us Ts prf = (lift' Us Ts prf handle SAME => prf);
   698 
   699     val ps = map (Logic.lift_all inc Bi) (Logic.strip_imp_prems prop);
   700     val k = length ps;
   701 
   702     fun mk_app b (i, j, prf) =
   703           if b then (i-1, j, prf %% PBound i) else (i, j-1, prf %> Bound j);
   704 
   705     fun lift Us bs i j (Const ("==>", _) $ A $ B) =
   706             AbsP ("H", NONE (*A*), lift Us (true::bs) (i+1) j B)
   707       | lift Us bs i j (Const ("all", _) $ Abs (a, T, t)) =
   708             Abst (a, NONE (*T*), lift (T::Us) (false::bs) i (j+1) t)
   709       | lift Us bs i j _ = proof_combP (lifth' (rev Us) [] prf,
   710             map (fn k => (#3 (fold_rev mk_app bs (i-1, j-1, PBound k))))
   711               (i + k - 1 downto i));
   712   in
   713     mk_AbsP (k, lift [] [] 0 0 Bi)
   714   end;
   715 
   716 
   717 (***** proof by assumption *****)
   718 
   719 fun mk_asm_prf t i m =
   720   let
   721     fun imp_prf _ i 0 = PBound i
   722       | imp_prf (Const ("==>", _) $ A $ B) i m = AbsP ("H", NONE (*A*), imp_prf B (i+1) (m-1))
   723       | imp_prf _ i _ = PBound i;
   724     fun all_prf (Const ("all", _) $ Abs (a, T, t)) = Abst (a, NONE (*T*), all_prf t)
   725       | all_prf t = imp_prf t (~i) m
   726   in all_prf t end;
   727 
   728 fun assumption_proof Bs Bi n prf =
   729   mk_AbsP (length Bs, proof_combP (prf,
   730     map PBound (length Bs - 1 downto 0) @ [mk_asm_prf Bi n ~1]));
   731 
   732 
   733 (***** Composition of object rule with proof state *****)
   734 
   735 fun flatten_params_proof i j n (Const ("==>", _) $ A $ B, k) =
   736       AbsP ("H", NONE (*A*), flatten_params_proof (i+1) j n (B, k))
   737   | flatten_params_proof i j n (Const ("all", _) $ Abs (a, T, t), k) =
   738       Abst (a, NONE (*T*), flatten_params_proof i (j+1) n (t, k))
   739   | flatten_params_proof i j n (_, k) = proof_combP (proof_combt (PBound (k+i),
   740       map Bound (j-1 downto 0)), map PBound (remove (op =) (i-n) (i-1 downto 0)));
   741 
   742 fun bicompose_proof flatten Bs oldAs newAs A n m rprf sprf =
   743   let
   744     val la = length newAs;
   745     val lb = length Bs;
   746   in
   747     mk_AbsP (lb+la, proof_combP (sprf,
   748       map PBound (lb + la - 1 downto la)) %%
   749         proof_combP (rprf, (if n>0 then [mk_asm_prf (the A) n m] else []) @
   750           map (if flatten then flatten_params_proof 0 0 n else PBound o snd)
   751             (oldAs ~~ (la - 1 downto 0))))
   752   end;
   753 
   754 
   755 (***** axioms for equality *****)
   756 
   757 val aT = TFree ("'a", []);
   758 val bT = TFree ("'b", []);
   759 val x = Free ("x", aT);
   760 val y = Free ("y", aT);
   761 val z = Free ("z", aT);
   762 val A = Free ("A", propT);
   763 val B = Free ("B", propT);
   764 val f = Free ("f", aT --> bT);
   765 val g = Free ("g", aT --> bT);
   766 
   767 local open Logic in
   768 
   769 val equality_axms =
   770   [("reflexive", mk_equals (x, x)),
   771    ("symmetric", mk_implies (mk_equals (x, y), mk_equals (y, x))),
   772    ("transitive", list_implies ([mk_equals (x, y), mk_equals (y, z)], mk_equals (x, z))),
   773    ("equal_intr", list_implies ([mk_implies (A, B), mk_implies (B, A)], mk_equals (A, B))),
   774    ("equal_elim", list_implies ([mk_equals (A, B), A], B)),
   775    ("abstract_rule", mk_implies
   776       (all x (mk_equals (f $ x, g $ x)), mk_equals (lambda x (f $ x), lambda x (g $ x)))),
   777    ("combination", list_implies
   778       ([mk_equals (f, g), mk_equals (x, y)], mk_equals (f $ x, g $ y)))];
   779 
   780 val [reflexive_axm, symmetric_axm, transitive_axm, equal_intr_axm,
   781   equal_elim_axm, abstract_rule_axm, combination_axm] =
   782     map (fn (s, t) => PAxm ("Pure." ^ s, varify t, NONE)) equality_axms;
   783 
   784 end;
   785 
   786 val reflexive = reflexive_axm % NONE;
   787 
   788 fun symmetric (prf as PAxm ("Pure.reflexive", _, _) % _) = prf
   789   | symmetric prf = symmetric_axm % NONE % NONE %% prf;
   790 
   791 fun transitive _ _ (PAxm ("Pure.reflexive", _, _) % _) prf2 = prf2
   792   | transitive _ _ prf1 (PAxm ("Pure.reflexive", _, _) % _) = prf1
   793   | transitive u (Type ("prop", [])) prf1 prf2 =
   794       transitive_axm % NONE % SOME (remove_types u) % NONE %% prf1 %% prf2
   795   | transitive u T prf1 prf2 =
   796       transitive_axm % NONE % NONE % NONE %% prf1 %% prf2;
   797 
   798 fun abstract_rule x a prf =
   799   abstract_rule_axm % NONE % NONE %% forall_intr_proof x a prf;
   800 
   801 fun check_comb (PAxm ("Pure.combination", _, _) % f % g % _ % _ %% prf %% _) =
   802       is_some f orelse check_comb prf
   803   | check_comb (PAxm ("Pure.transitive", _, _) % _ % _ % _ %% prf1 %% prf2) =
   804       check_comb prf1 andalso check_comb prf2
   805   | check_comb (PAxm ("Pure.symmetric", _, _) % _ % _ %% prf) = check_comb prf
   806   | check_comb _ = false;
   807 
   808 fun combination f g t u (Type (_, [T, U])) prf1 prf2 =
   809   let
   810     val f = Envir.beta_norm f;
   811     val g = Envir.beta_norm g;
   812     val prf =  if check_comb prf1 then
   813         combination_axm % NONE % NONE
   814       else (case prf1 of
   815           PAxm ("Pure.reflexive", _, _) % _ =>
   816             combination_axm %> remove_types f % NONE
   817         | _ => combination_axm %> remove_types f %> remove_types g)
   818   in
   819     (case T of
   820        Type ("fun", _) => prf %
   821          (case head_of f of
   822             Abs _ => SOME (remove_types t)
   823           | Var _ => SOME (remove_types t)
   824           | _ => NONE) %
   825          (case head_of g of
   826             Abs _ => SOME (remove_types u)
   827           | Var _ => SOME (remove_types u)
   828           | _ => NONE) %% prf1 %% prf2
   829      | _ => prf % NONE % NONE %% prf1 %% prf2)
   830   end;
   831 
   832 fun equal_intr A B prf1 prf2 =
   833   equal_intr_axm %> remove_types A %> remove_types B %% prf1 %% prf2;
   834 
   835 fun equal_elim A B prf1 prf2 =
   836   equal_elim_axm %> remove_types A %> remove_types B %% prf1 %% prf2;
   837 
   838 
   839 (***** axioms and theorems *****)
   840 
   841 val proofs = ref 2;
   842 
   843 fun vars_of t = map Var (rev (Term.add_vars t []));
   844 fun frees_of t = map Free (rev (Term.add_frees t []));
   845 
   846 fun test_args _ [] = true
   847   | test_args is (Bound i :: ts) =
   848       not (member (op =) is i) andalso test_args (i :: is) ts
   849   | test_args _ _ = false;
   850 
   851 fun is_fun (Type ("fun", _)) = true
   852   | is_fun (TVar _) = true
   853   | is_fun _ = false;
   854 
   855 fun add_funvars Ts (vs, t) =
   856   if is_fun (fastype_of1 (Ts, t)) then
   857     vs union map_filter (fn Var (ixn, T) =>
   858       if is_fun T then SOME ixn else NONE | _ => NONE) (vars_of t)
   859   else vs;
   860 
   861 fun add_npvars q p Ts (vs, Const ("==>", _) $ t $ u) =
   862       add_npvars q p Ts (add_npvars q (not p) Ts (vs, t), u)
   863   | add_npvars q p Ts (vs, Const ("all", Type (_, [Type (_, [T, _]), _])) $ t) =
   864       add_npvars q p Ts (vs, if p andalso q then betapply (t, Var (("",0), T)) else t)
   865   | add_npvars q p Ts (vs, Abs (_, T, t)) = add_npvars q p (T::Ts) (vs, t)
   866   | add_npvars _ _ Ts (vs, t) = add_npvars' Ts (vs, t)
   867 and add_npvars' Ts (vs, t) = (case strip_comb t of
   868     (Var (ixn, _), ts) => if test_args [] ts then vs
   869       else Library.foldl (add_npvars' Ts)
   870         (AList.update (op =) (ixn,
   871           Library.foldl (add_funvars Ts) ((these ooo AList.lookup) (op =) vs ixn, ts)) vs, ts)
   872   | (Abs (_, T, u), ts) => Library.foldl (add_npvars' (T::Ts)) (vs, u :: ts)
   873   | (_, ts) => Library.foldl (add_npvars' Ts) (vs, ts));
   874 
   875 fun prop_vars (Const ("==>", _) $ P $ Q) = prop_vars P union prop_vars Q
   876   | prop_vars (Const ("all", _) $ Abs (_, _, t)) = prop_vars t
   877   | prop_vars t = (case strip_comb t of
   878       (Var (ixn, _), _) => [ixn] | _ => []);
   879 
   880 fun is_proj t =
   881   let
   882     fun is_p i t = (case strip_comb t of
   883         (Bound j, []) => false
   884       | (Bound j, ts) => j >= i orelse exists (is_p i) ts
   885       | (Abs (_, _, u), _) => is_p (i+1) u
   886       | (_, ts) => exists (is_p i) ts)
   887   in (case strip_abs_body t of
   888         Bound _ => true
   889       | t' => is_p 0 t')
   890   end;
   891 
   892 fun needed_vars prop =
   893   Library.foldl (op union)
   894     ([], map (uncurry (insert (op =))) (add_npvars true true [] ([], prop))) union
   895   prop_vars prop;
   896 
   897 fun gen_axm_proof c name prop =
   898   let
   899     val nvs = needed_vars prop;
   900     val args = map (fn (v as Var (ixn, _)) =>
   901         if member (op =) nvs ixn then SOME v else NONE) (vars_of prop) @
   902       map SOME (frees_of prop);
   903   in
   904     proof_combt' (c (name, prop, NONE), args)
   905   end;
   906 
   907 val axm_proof = gen_axm_proof PAxm;
   908 
   909 val dummy = Const (Term.dummy_patternN, dummyT);
   910 
   911 fun oracle_proof name prop =
   912   if !proofs = 0 then Oracle (name, dummy, NONE)
   913   else gen_axm_proof Oracle name prop;
   914 
   915 fun shrink_proof thy =
   916   let
   917     fun shrink ls lev (prf as Abst (a, T, body)) =
   918           let val (b, is, ch, body') = shrink ls (lev+1) body
   919           in (b, is, ch, if ch then Abst (a, T, body') else prf) end
   920       | shrink ls lev (prf as AbsP (a, t, body)) =
   921           let val (b, is, ch, body') = shrink (lev::ls) lev body
   922           in (b orelse member (op =) is 0, map_filter (fn 0 => NONE | i => SOME (i-1)) is,
   923             ch, if ch then AbsP (a, t, body') else prf)
   924           end
   925       | shrink ls lev prf =
   926           let val (is, ch, _, prf') = shrink' ls lev [] [] prf
   927           in (false, is, ch, prf') end
   928     and shrink' ls lev ts prfs (prf as prf1 %% prf2) =
   929           let
   930             val p as (_, is', ch', prf') = shrink ls lev prf2;
   931             val (is, ch, ts', prf'') = shrink' ls lev ts (p::prfs) prf1
   932           in (is union is', ch orelse ch', ts',
   933               if ch orelse ch' then prf'' %% prf' else prf)
   934           end
   935       | shrink' ls lev ts prfs (prf as prf1 % t) =
   936           let val (is, ch, (ch', t')::ts', prf') = shrink' ls lev (t::ts) prfs prf1
   937           in (is, ch orelse ch', ts',
   938               if ch orelse ch' then prf' % t' else prf) end
   939       | shrink' ls lev ts prfs (prf as PBound i) =
   940           (if exists (fn SOME (Bound j) => lev-j <= List.nth (ls, i) | _ => true) ts
   941              orelse has_duplicates (op =)
   942                (Library.foldl (fn (js, SOME (Bound j)) => j :: js | (js, _) => js) ([], ts))
   943              orelse exists #1 prfs then [i] else [], false, map (pair false) ts, prf)
   944       | shrink' ls lev ts prfs (Hyp t) = ([], false, map (pair false) ts, Hyp t)
   945       | shrink' ls lev ts prfs MinProof = ([], false, map (pair false) ts, MinProof)
   946       | shrink' ls lev ts prfs prf =
   947           let
   948             val prop =
   949               (case prf of
   950                 PAxm (_, prop, _) => prop
   951               | Oracle (_, prop, _) => prop
   952               | Promise (_, prop, _) => prop
   953               | PThm (_, ((_, prop, _), _)) => prop
   954               | _ => error "shrink: proof not in normal form");
   955             val vs = vars_of prop;
   956             val (ts', ts'') = chop (length vs) ts;
   957             val insts = Library.take (length ts', map (fst o dest_Var) vs) ~~ ts';
   958             val nvs = Library.foldl (fn (ixns', (ixn, ixns)) =>
   959               insert (op =) ixn (case AList.lookup (op =) insts ixn of
   960                   SOME (SOME t) => if is_proj t then ixns union ixns' else ixns'
   961                 | _ => ixns union ixns'))
   962                   (needed prop ts'' prfs, add_npvars false true [] ([], prop));
   963             val insts' = map
   964               (fn (ixn, x as SOME _) => if member (op =) nvs ixn then (false, x) else (true, NONE)
   965                 | (_, x) => (false, x)) insts
   966           in ([], false, insts' @ map (pair false) ts'', prf) end
   967     and needed (Const ("==>", _) $ t $ u) ts ((b, _, _, _)::prfs) =
   968           (if b then map (fst o dest_Var) (vars_of t) else []) union needed u ts prfs
   969       | needed (Var (ixn, _)) (_::_) _ = [ixn]
   970       | needed _ _ _ = [];
   971   in shrink end;
   972 
   973 
   974 (**** Simple first order matching functions for terms and proofs ****)
   975 
   976 exception PMatch;
   977 
   978 (** see pattern.ML **)
   979 
   980 fun flt (i: int) = List.filter (fn n => n < i);
   981 
   982 fun fomatch Ts tymatch j =
   983   let
   984     fun mtch (instsp as (tyinsts, insts)) = fn
   985         (Var (ixn, T), t)  =>
   986           if j>0 andalso not (null (flt j (loose_bnos t)))
   987           then raise PMatch
   988           else (tymatch (tyinsts, fn () => (T, fastype_of1 (Ts, t))),
   989             (ixn, t) :: insts)
   990       | (Free (a, T), Free (b, U)) =>
   991           if a=b then (tymatch (tyinsts, K (T, U)), insts) else raise PMatch
   992       | (Const (a, T), Const (b, U))  =>
   993           if a=b then (tymatch (tyinsts, K (T, U)), insts) else raise PMatch
   994       | (f $ t, g $ u) => mtch (mtch instsp (f, g)) (t, u)
   995       | (Bound i, Bound j) => if i=j then instsp else raise PMatch
   996       | _ => raise PMatch
   997   in mtch end;
   998 
   999 fun match_proof Ts tymatch =
  1000   let
  1001     fun optmatch _ inst (NONE, _) = inst
  1002       | optmatch _ _ (SOME _, NONE) = raise PMatch
  1003       | optmatch mtch inst (SOME x, SOME y) = mtch inst (x, y)
  1004 
  1005     fun matcht Ts j (pinst, tinst) (t, u) =
  1006       (pinst, fomatch Ts tymatch j tinst (t, Envir.beta_norm u));
  1007     fun matchT (pinst, (tyinsts, insts)) p =
  1008       (pinst, (tymatch (tyinsts, K p), insts));
  1009     fun matchTs inst (Ts, Us) = Library.foldl (uncurry matchT) (inst, Ts ~~ Us);
  1010 
  1011     fun mtch Ts i j (pinst, tinst) (Hyp (Var (ixn, _)), prf) =
  1012           if i = 0 andalso j = 0 then ((ixn, prf) :: pinst, tinst)
  1013           else (case apfst (flt i) (apsnd (flt j)
  1014                   (prf_add_loose_bnos 0 0 prf ([], []))) of
  1015               ([], []) => ((ixn, incr_pboundvars (~i) (~j) prf) :: pinst, tinst)
  1016             | ([], _) => if j = 0 then
  1017                    ((ixn, incr_pboundvars (~i) (~j) prf) :: pinst, tinst)
  1018                  else raise PMatch
  1019             | _ => raise PMatch)
  1020       | mtch Ts i j inst (prf1 % opt1, prf2 % opt2) =
  1021           optmatch (matcht Ts j) (mtch Ts i j inst (prf1, prf2)) (opt1, opt2)
  1022       | mtch Ts i j inst (prf1 %% prf2, prf1' %% prf2') =
  1023           mtch Ts i j (mtch Ts i j inst (prf1, prf1')) (prf2, prf2')
  1024       | mtch Ts i j inst (Abst (_, opT, prf1), Abst (_, opU, prf2)) =
  1025           mtch (the_default dummyT opU :: Ts) i (j+1)
  1026             (optmatch matchT inst (opT, opU)) (prf1, prf2)
  1027       | mtch Ts i j inst (prf1, Abst (_, opU, prf2)) =
  1028           mtch (the_default dummyT opU :: Ts) i (j+1) inst
  1029             (incr_pboundvars 0 1 prf1 %> Bound 0, prf2)
  1030       | mtch Ts i j inst (AbsP (_, opt, prf1), AbsP (_, opu, prf2)) =
  1031           mtch Ts (i+1) j (optmatch (matcht Ts j) inst (opt, opu)) (prf1, prf2)
  1032       | mtch Ts i j inst (prf1, AbsP (_, _, prf2)) =
  1033           mtch Ts (i+1) j inst (incr_pboundvars 1 0 prf1 %% PBound 0, prf2)
  1034       | mtch Ts i j inst (PAxm (s1, _, opTs), PAxm (s2, _, opUs)) =
  1035           if s1 = s2 then optmatch matchTs inst (opTs, opUs)
  1036           else raise PMatch
  1037       | mtch Ts i j inst (PThm (_, ((name1, prop1, opTs), _)), PThm (_, ((name2, prop2, opUs), _))) =
  1038           if name1 = name2 andalso prop1 = prop2 then
  1039             optmatch matchTs inst (opTs, opUs)
  1040           else raise PMatch
  1041       | mtch _ _ _ inst (PBound i, PBound j) = if i = j then inst else raise PMatch
  1042       | mtch _ _ _ _ _ = raise PMatch
  1043   in mtch Ts 0 0 end;
  1044 
  1045 fun prf_subst (pinst, (tyinsts, insts)) =
  1046   let
  1047     val substT = Envir.typ_subst_TVars tyinsts;
  1048 
  1049     fun subst' lev (t as Var (ixn, _)) = (case AList.lookup (op =) insts ixn of
  1050           NONE => t
  1051         | SOME u => incr_boundvars lev u)
  1052       | subst' lev (Const (s, T)) = Const (s, substT T)
  1053       | subst' lev (Free (s, T)) = Free (s, substT T)
  1054       | subst' lev (Abs (a, T, body)) = Abs (a, substT T, subst' (lev+1) body)
  1055       | subst' lev (f $ t) = subst' lev f $ subst' lev t
  1056       | subst' _ t = t;
  1057 
  1058     fun subst plev tlev (AbsP (a, t, body)) =
  1059           AbsP (a, Option.map (subst' tlev) t, subst (plev+1) tlev body)
  1060       | subst plev tlev (Abst (a, T, body)) =
  1061           Abst (a, Option.map substT T, subst plev (tlev+1) body)
  1062       | subst plev tlev (prf %% prf') = subst plev tlev prf %% subst plev tlev prf'
  1063       | subst plev tlev (prf % t) = subst plev tlev prf % Option.map (subst' tlev) t
  1064       | subst plev tlev (prf as Hyp (Var (ixn, _))) = (case AList.lookup (op =) pinst ixn of
  1065           NONE => prf
  1066         | SOME prf' => incr_pboundvars plev tlev prf')
  1067       | subst _ _ (PAxm (id, prop, Ts)) = PAxm (id, prop, Option.map (map substT) Ts)
  1068       | subst _ _ (Oracle (id, prop, Ts)) = Oracle (id, prop, Option.map (map substT) Ts)
  1069       | subst _ _ (Promise (i, prop, Ts)) = Promise (i, prop, (map substT) Ts)
  1070       | subst _ _ (PThm (i, ((id, prop, Ts), body))) =
  1071           PThm (i, ((id, prop, Option.map (map substT) Ts), body))
  1072       | subst _ _ t = t;
  1073   in subst 0 0 end;
  1074 
  1075 (*A fast unification filter: true unless the two terms cannot be unified.
  1076   Terms must be NORMAL.  Treats all Vars as distinct. *)
  1077 fun could_unify prf1 prf2 =
  1078   let
  1079     fun matchrands (prf1 %% prf2) (prf1' %% prf2') =
  1080           could_unify prf2 prf2' andalso matchrands prf1 prf1'
  1081       | matchrands (prf % SOME t) (prf' % SOME t') =
  1082           Term.could_unify (t, t') andalso matchrands prf prf'
  1083       | matchrands (prf % _) (prf' % _) = matchrands prf prf'
  1084       | matchrands _ _ = true
  1085 
  1086     fun head_of (prf %% _) = head_of prf
  1087       | head_of (prf % _) = head_of prf
  1088       | head_of prf = prf
  1089 
  1090   in case (head_of prf1, head_of prf2) of
  1091         (_, Hyp (Var _)) => true
  1092       | (Hyp (Var _), _) => true
  1093       | (PAxm (a, _, _), PAxm (b, _, _)) => a = b andalso matchrands prf1 prf2
  1094       | (PThm (_, ((a, propa, _), _)), PThm (_, ((b, propb, _), _))) =>
  1095           a = b andalso propa = propb andalso matchrands prf1 prf2
  1096       | (PBound i, PBound j) => i = j andalso matchrands prf1 prf2
  1097       | (AbsP _, _) =>  true   (*because of possible eta equality*)
  1098       | (Abst _, _) =>  true
  1099       | (_, AbsP _) =>  true
  1100       | (_, Abst _) =>  true
  1101       | _ => false
  1102   end;
  1103 
  1104 
  1105 (**** rewriting on proof terms ****)
  1106 
  1107 val skel0 = PBound 0;
  1108 
  1109 fun rewrite_prf tymatch (rules, procs) prf =
  1110   let
  1111     fun rew _ (Abst (_, _, body) % SOME t) = SOME (prf_subst_bounds [t] body, skel0)
  1112       | rew _ (AbsP (_, _, body) %% prf) = SOME (prf_subst_pbounds [prf] body, skel0)
  1113       | rew Ts prf = (case get_first (fn r => r Ts prf) procs of
  1114           SOME prf' => SOME (prf', skel0)
  1115         | NONE => get_first (fn (prf1, prf2) => SOME (prf_subst
  1116             (match_proof Ts tymatch ([], (Vartab.empty, [])) (prf1, prf)) prf2, prf2)
  1117                handle PMatch => NONE) (filter (could_unify prf o fst) rules));
  1118 
  1119     fun rew0 Ts (prf as AbsP (_, _, prf' %% PBound 0)) =
  1120           if prf_loose_Pbvar1 prf' 0 then rew Ts prf
  1121           else
  1122             let val prf'' = incr_pboundvars (~1) 0 prf'
  1123             in SOME (the_default (prf'', skel0) (rew Ts prf'')) end
  1124       | rew0 Ts (prf as Abst (_, _, prf' % SOME (Bound 0))) =
  1125           if prf_loose_bvar1 prf' 0 then rew Ts prf
  1126           else
  1127             let val prf'' = incr_pboundvars 0 (~1) prf'
  1128             in SOME (the_default (prf'', skel0) (rew Ts prf'')) end
  1129       | rew0 Ts prf = rew Ts prf;
  1130 
  1131     fun rew1 _ (Hyp (Var _)) _ = NONE
  1132       | rew1 Ts skel prf = (case rew2 Ts skel prf of
  1133           SOME prf1 => (case rew0 Ts prf1 of
  1134               SOME (prf2, skel') => SOME (the_default prf2 (rew1 Ts skel' prf2))
  1135             | NONE => SOME prf1)
  1136         | NONE => (case rew0 Ts prf of
  1137               SOME (prf1, skel') => SOME (the_default prf1 (rew1 Ts skel' prf1))
  1138             | NONE => NONE))
  1139 
  1140     and rew2 Ts skel (prf % SOME t) = (case prf of
  1141             Abst (_, _, body) =>
  1142               let val prf' = prf_subst_bounds [t] body
  1143               in SOME (the_default prf' (rew2 Ts skel0 prf')) end
  1144           | _ => (case rew1 Ts (case skel of skel' % _ => skel' | _ => skel0) prf of
  1145               SOME prf' => SOME (prf' % SOME t)
  1146             | NONE => NONE))
  1147       | rew2 Ts skel (prf % NONE) = Option.map (fn prf' => prf' % NONE)
  1148           (rew1 Ts (case skel of skel' % _ => skel' | _ => skel0) prf)
  1149       | rew2 Ts skel (prf1 %% prf2) = (case prf1 of
  1150             AbsP (_, _, body) =>
  1151               let val prf' = prf_subst_pbounds [prf2] body
  1152               in SOME (the_default prf' (rew2 Ts skel0 prf')) end
  1153           | _ =>
  1154             let val (skel1, skel2) = (case skel of
  1155                 skel1 %% skel2 => (skel1, skel2)
  1156               | _ => (skel0, skel0))
  1157             in case rew1 Ts skel1 prf1 of
  1158                 SOME prf1' => (case rew1 Ts skel2 prf2 of
  1159                     SOME prf2' => SOME (prf1' %% prf2')
  1160                   | NONE => SOME (prf1' %% prf2))
  1161               | NONE => (case rew1 Ts skel2 prf2 of
  1162                     SOME prf2' => SOME (prf1 %% prf2')
  1163                   | NONE => NONE)
  1164             end)
  1165       | rew2 Ts skel (Abst (s, T, prf)) = (case rew1 (the_default dummyT T :: Ts)
  1166               (case skel of Abst (_, _, skel') => skel' | _ => skel0) prf of
  1167             SOME prf' => SOME (Abst (s, T, prf'))
  1168           | NONE => NONE)
  1169       | rew2 Ts skel (AbsP (s, t, prf)) = (case rew1 Ts
  1170               (case skel of AbsP (_, _, skel') => skel' | _ => skel0) prf of
  1171             SOME prf' => SOME (AbsP (s, t, prf'))
  1172           | NONE => NONE)
  1173       | rew2 _ _ _ = NONE
  1174 
  1175   in the_default prf (rew1 [] skel0 prf) end;
  1176 
  1177 fun rewrite_proof thy = rewrite_prf (fn (tyenv, f) =>
  1178   Sign.typ_match thy (f ()) tyenv handle Type.TYPE_MATCH => raise PMatch);
  1179 
  1180 fun rewrite_proof_notypes rews = rewrite_prf fst rews;
  1181 
  1182 
  1183 (**** theory data ****)
  1184 
  1185 structure ProofData = TheoryDataFun
  1186 (
  1187   type T = (stamp * (proof * proof)) list * (stamp * (typ list -> proof -> proof option)) list;
  1188 
  1189   val empty = ([], []);
  1190   val copy = I;
  1191   val extend = I;
  1192   fun merge _ ((rules1, procs1), (rules2, procs2)) : T =
  1193     (AList.merge (op =) (K true) (rules1, rules2),
  1194       AList.merge (op =) (K true) (procs1, procs2));
  1195 );
  1196 
  1197 fun get_data thy = let val (rules, procs) = ProofData.get thy in (map #2 rules, map #2 procs) end;
  1198 fun rew_proof thy = rewrite_prf fst (get_data thy);
  1199 
  1200 fun add_prf_rrule r = (ProofData.map o apfst) (cons (stamp (), r));
  1201 fun add_prf_rproc p = (ProofData.map o apsnd) (cons (stamp (), p));
  1202 
  1203 
  1204 (***** promises *****)
  1205 
  1206 fun promise_proof thy i prop =
  1207   let
  1208     val _ = prop |> Term.exists_subterm (fn t =>
  1209       (Term.is_Free t orelse Term.is_Var t) andalso
  1210         error ("promise_proof: illegal variable " ^ Syntax.string_of_term_global thy t));
  1211     val _ = prop |> Term.exists_type (Term.exists_subtype
  1212       (fn TFree (a, _) => error ("promise_proof: illegal type variable " ^ quote a)
  1213         | _ => false));
  1214   in Promise (i, prop, map TVar (Term.add_tvars prop [])) end;
  1215 
  1216 fun fulfill_proof _ [] body0 = body0
  1217   | fulfill_proof thy promises body0 =
  1218       let
  1219         val PBody {oracles = oracles0, thms = thms0, proof = proof0} = body0;
  1220         val (oracles, thms) = fold (merge_body o make_body o #2) promises (oracles0, thms0);
  1221 
  1222         val tab = Inttab.make promises;
  1223         fun fill (Promise (i, prop, Ts)) =
  1224             (case Inttab.lookup tab i of
  1225               NONE => NONE
  1226             | SOME p => SOME (instantiate (Term.add_tvars prop [] ~~ Ts, []) p))
  1227           | fill _ = NONE;
  1228         val (rules, procs) = get_data thy;
  1229         val proof = rewrite_prf fst (rules, K fill :: procs) proof0;
  1230       in PBody {oracles = oracles, thms = thms, proof = proof} end;
  1231 
  1232 
  1233 (***** theorems *****)
  1234 
  1235 fun thm_proof thy name hyps prop promises body =
  1236   let
  1237     val PBody {oracles = oracles0, thms = thms0, proof = prf} = body;
  1238     val prop = Logic.list_implies (hyps, prop);
  1239     val nvs = needed_vars prop;
  1240     val args = map (fn (v as Var (ixn, _)) =>
  1241         if member (op =) nvs ixn then SOME v else NONE) (vars_of prop) @
  1242       map SOME (frees_of prop);
  1243 
  1244     val proof0 =
  1245       if ! proofs = 2 then
  1246         #4 (shrink_proof thy [] 0 (rew_proof thy (fold_rev implies_intr_proof hyps prf)))
  1247       else MinProof;
  1248 
  1249     fun new_prf () = (serial (), name, prop,
  1250       Lazy.lazy (fn () => fulfill_proof thy (Lazy.force promises)
  1251         (PBody {oracles = oracles0, thms = thms0, proof = proof0})));
  1252 
  1253     val (i, name, prop, body') =
  1254       (case strip_combt (fst (strip_combP prf)) of
  1255         (PThm (i, ((old_name, prop', NONE), body')), args') =>
  1256           if (old_name = "" orelse old_name = name) andalso prop = prop' andalso args = args'
  1257           then (i, name, prop, body')
  1258           else new_prf ()
  1259       | _ => new_prf ());
  1260     val head = PThm (i, ((name, prop, NONE), body'));
  1261   in
  1262     ((i, (name, prop, body')), proof_combP (proof_combt' (head, args), map Hyp hyps))
  1263   end;
  1264 
  1265 fun get_name hyps prop prf =
  1266   let val prop = Logic.list_implies (hyps, prop) in
  1267     (case strip_combt (fst (strip_combP prf)) of
  1268       (PAxm (name, prop', _), _) => if prop = prop' then name else ""   (* FIXME !? *)
  1269     | (PThm (_, ((name, prop', _), _)), _) => if prop = prop' then name else ""
  1270     | _ => "")
  1271   end;
  1272 
  1273 end;
  1274 
  1275 structure BasicProofterm : BASIC_PROOFTERM = Proofterm;
  1276 open BasicProofterm;