src/Pure/proofterm.ML
author wenzelm
Sun Mar 01 23:36:12 2009 +0100 (2009-03-01)
changeset 30190 479806475f3c
parent 30146 a77fc0209723
child 30711 952fdbee1b48
permissions -rw-r--r--
use long names for old-style fold combinators;
     1 (*  Title:      Pure/proofterm.ML
     2     Author:     Stefan Berghofer, TU Muenchen
     3 
     4 LF style proof terms.
     5 *)
     6 
     7 infix 8 % %% %>;
     8 
     9 signature BASIC_PROOFTERM =
    10 sig
    11   val proofs: int ref
    12 
    13   datatype proof =
    14      MinProof
    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    | PAxm of string * term * typ list option
    22    | Oracle of string * term * typ list option
    23    | Promise of serial * term * typ list
    24    | PThm of serial * ((string * term * typ list option) * proof_body future)
    25   and proof_body = PBody of
    26     {oracles: (string * term) OrdList.T,
    27      thms: (serial * (string * term * proof_body future)) OrdList.T,
    28      proof: proof}
    29 
    30   val %> : proof * term -> proof
    31 end;
    32 
    33 signature PROOFTERM =
    34 sig
    35   include BASIC_PROOFTERM
    36 
    37   type oracle = string * term
    38   type pthm = serial * (string * term * proof_body future)
    39   val join_body: proof_body future ->
    40     {oracles: oracle OrdList.T, thms: pthm OrdList.T, proof: proof}
    41   val join_proof: proof_body future -> proof
    42   val proof_of: proof_body -> proof
    43   val fold_body_thms: (string * term * proof_body -> 'a -> 'a) -> proof_body list -> 'a -> 'a
    44   val fold_proof_atoms: bool -> (proof -> 'a -> 'a) -> proof list -> 'a -> 'a
    45 
    46   val oracle_ord: oracle * oracle -> order
    47   val thm_ord: pthm * pthm -> order
    48   val make_proof_body: proof -> proof_body
    49   val merge_oracles: oracle OrdList.T -> oracle OrdList.T -> oracle OrdList.T
    50   val make_oracles: proof -> oracle OrdList.T
    51   val merge_thms: pthm OrdList.T -> pthm OrdList.T -> pthm OrdList.T
    52   val make_thms: proof -> pthm OrdList.T
    53 
    54   (** primitive operations **)
    55   val proof_combt: proof * term list -> proof
    56   val proof_combt': proof * term option list -> proof
    57   val proof_combP: proof * proof list -> proof
    58   val strip_combt: proof -> proof * term option list
    59   val strip_combP: proof -> proof * proof list
    60   val strip_thm: proof_body -> proof_body
    61   val map_proof_terms_option: (term -> term option) -> (typ -> typ option) -> proof -> proof
    62   val map_proof_terms: (term -> term) -> (typ -> typ) -> proof -> proof
    63   val fold_proof_terms: (term -> 'a -> 'a) -> (typ -> 'a -> 'a) -> proof -> 'a -> 'a
    64   val maxidx_proof: proof -> int -> int
    65   val size_of_proof: proof -> int
    66   val change_type: typ list option -> proof -> proof
    67   val prf_abstract_over: term -> proof -> proof
    68   val prf_incr_bv: int -> int -> int -> int -> proof -> proof
    69   val incr_pboundvars: int -> int -> proof -> proof
    70   val prf_loose_bvar1: proof -> int -> bool
    71   val prf_loose_Pbvar1: proof -> int -> bool
    72   val prf_add_loose_bnos: int -> int -> proof -> int list * int list -> int list * int list
    73   val norm_proof: Envir.env -> proof -> proof
    74   val norm_proof': Envir.env -> proof -> proof
    75   val prf_subst_bounds: term list -> proof -> proof
    76   val prf_subst_pbounds: proof list -> proof -> proof
    77   val freeze_thaw_prf: proof -> proof * (proof -> proof)
    78 
    79   (** proof terms for specific inference rules **)
    80   val implies_intr_proof: term -> proof -> proof
    81   val forall_intr_proof: term -> string -> proof -> proof
    82   val varify_proof: term -> (string * sort) list -> proof -> proof
    83   val freezeT: term -> proof -> proof
    84   val rotate_proof: term list -> term -> int -> proof -> proof
    85   val permute_prems_prf: term list -> int -> int -> proof -> proof
    86   val generalize: string list * string list -> int -> proof -> proof
    87   val instantiate: ((indexname * sort) * typ) list * ((indexname * typ) * term) list
    88     -> proof -> proof
    89   val lift_proof: term -> int -> term -> proof -> proof
    90   val assumption_proof: term list -> term -> int -> proof -> proof
    91   val bicompose_proof: bool -> term list -> term list -> term list -> term option ->
    92     int -> int -> proof -> proof -> proof
    93   val equality_axms: (string * term) list
    94   val reflexive_axm: proof
    95   val symmetric_axm: proof
    96   val transitive_axm: proof
    97   val equal_intr_axm: proof
    98   val equal_elim_axm: proof
    99   val abstract_rule_axm: proof
   100   val combination_axm: proof
   101   val reflexive: proof
   102   val symmetric: proof -> proof
   103   val transitive: term -> typ -> proof -> proof -> proof
   104   val abstract_rule: term -> string -> proof -> proof
   105   val combination: term -> term -> term -> term -> typ -> proof -> proof -> proof
   106   val equal_intr: term -> term -> proof -> proof -> proof
   107   val equal_elim: term -> term -> proof -> proof -> proof
   108   val axm_proof: string -> term -> proof
   109   val oracle_proof: string -> term -> proof
   110   val promise_proof: theory -> serial -> term -> proof
   111   val fulfill_proof: theory -> (serial * proof) list -> proof_body -> proof_body
   112   val thm_proof: theory -> string -> term list -> term ->
   113     (serial * proof future) list -> proof_body -> pthm * proof
   114   val get_name: term list -> term -> proof -> string
   115 
   116   (** rewriting on proof terms **)
   117   val add_prf_rrule: proof * proof -> theory -> theory
   118   val add_prf_rproc: (typ list -> proof -> proof option) -> theory -> theory
   119   val rewrite_proof: theory -> (proof * proof) list *
   120     (typ list -> proof -> proof option) list -> proof -> proof
   121   val rewrite_proof_notypes: (proof * proof) list *
   122     (typ list -> proof -> proof option) list -> proof -> proof
   123   val rew_proof: theory -> proof -> proof
   124 end
   125 
   126 structure Proofterm : PROOFTERM =
   127 struct
   128 
   129 open Envir;
   130 
   131 
   132 (***** datatype proof *****)
   133 
   134 datatype proof =
   135    MinProof
   136  | PBound of int
   137  | Abst of string * typ option * proof
   138  | AbsP of string * term option * proof
   139  | op % of proof * term option
   140  | op %% of proof * proof
   141  | Hyp of term
   142  | PAxm of string * term * typ list option
   143  | Oracle of string * term * typ list option
   144  | Promise of serial * term * typ list
   145  | PThm of serial * ((string * term * typ list option) * proof_body future)
   146 and proof_body = PBody of
   147   {oracles: (string * term) OrdList.T,
   148    thms: (serial * (string * term * proof_body future)) OrdList.T,
   149    proof: proof};
   150 
   151 type oracle = string * term;
   152 type pthm = serial * (string * term * proof_body future);
   153 
   154 val join_body = Future.join #> (fn PBody args => args);
   155 val join_proof = #proof o join_body;
   156 
   157 fun proof_of (PBody {proof, ...}) = proof;
   158 
   159 
   160 (***** proof atoms *****)
   161 
   162 fun fold_body_thms f =
   163   let
   164     fun app (PBody {thms, ...}) = thms |> fold (fn (i, (name, prop, body)) => fn (x, seen) =>
   165       if Inttab.defined seen i then (x, seen)
   166       else
   167         let
   168           val body' = Future.join body;
   169           val (x', seen') = app body' (x, Inttab.update (i, ()) seen);
   170         in (f (name, prop, body') x', seen') end);
   171   in fn bodies => fn x => #1 (fold app bodies (x, Inttab.empty)) end;
   172 
   173 fun fold_proof_atoms all f =
   174   let
   175     fun app (Abst (_, _, prf)) = app prf
   176       | app (AbsP (_, _, prf)) = app prf
   177       | app (prf % _) = app prf
   178       | app (prf1 %% prf2) = app prf1 #> app prf2
   179       | app (prf as PThm (i, (_, body))) = (fn (x, seen) =>
   180           if Inttab.defined seen i then (x, seen)
   181           else
   182             let val (x', seen') =
   183               (if all then app (join_proof body) else I) (x, Inttab.update (i, ()) seen)
   184             in (f prf x', seen') end)
   185       | app prf = (fn (x, seen) => (f prf x, seen));
   186   in fn prfs => fn x => #1 (fold app prfs (x, Inttab.empty)) end;
   187 
   188 
   189 (* proof body *)
   190 
   191 val oracle_ord = prod_ord fast_string_ord TermOrd.fast_term_ord;
   192 fun thm_ord ((i, _): pthm, (j, _)) = int_ord (j, i);
   193 
   194 fun make_body prf =
   195   let
   196     val (oracles, thms) = fold_proof_atoms false
   197       (fn Oracle (s, prop, _) => apfst (cons (s, prop))
   198         | PThm (i, ((name, prop, _), body)) => apsnd (cons (i, (name, prop, body)))
   199         | _ => I) [prf] ([], []);
   200   in (OrdList.make oracle_ord oracles, OrdList.make thm_ord thms) end;
   201 
   202 fun make_proof_body prf =
   203   let val (oracles, thms) = make_body prf
   204   in PBody {oracles = oracles, thms = thms, proof = prf} end;
   205 
   206 val make_oracles = #1 o make_body;
   207 val make_thms = #2 o make_body;
   208 
   209 val merge_oracles = OrdList.union oracle_ord;
   210 val merge_thms = OrdList.union thm_ord;
   211 
   212 fun merge_body (oracles1, thms1) (oracles2, thms2) =
   213   (merge_oracles oracles1 oracles2, merge_thms thms1 thms2);
   214 
   215 
   216 (***** proof objects with different levels of detail *****)
   217 
   218 fun (prf %> t) = prf % SOME t;
   219 
   220 val proof_combt = Library.foldl (op %>);
   221 val proof_combt' = Library.foldl (op %);
   222 val proof_combP = Library.foldl (op %%);
   223 
   224 fun strip_combt prf =
   225     let fun stripc (prf % t, ts) = stripc (prf, t::ts)
   226           | stripc  x =  x
   227     in  stripc (prf, [])  end;
   228 
   229 fun strip_combP prf =
   230     let fun stripc (prf %% prf', prfs) = stripc (prf, prf'::prfs)
   231           | stripc  x =  x
   232     in  stripc (prf, [])  end;
   233 
   234 fun strip_thm (body as PBody {proof, ...}) =
   235   (case strip_combt (fst (strip_combP proof)) of
   236     (PThm (_, (_, body')), _) => Future.join body'
   237   | _ => body);
   238 
   239 val mk_Abst = fold_rev (fn (s, T:typ) => fn prf => Abst (s, NONE, prf));
   240 fun mk_AbsP (i, prf) = funpow i (fn prf => AbsP ("H", NONE, prf)) prf;
   241 
   242 fun apsome f NONE = raise SAME
   243   | apsome f (SOME x) = (case f x of NONE => raise SAME | some => some);
   244 
   245 fun apsome' f NONE = raise SAME
   246   | apsome' f (SOME x) = SOME (f x);
   247 
   248 fun map_proof_terms_option f g =
   249   let
   250     fun map_typs (T :: Ts) =
   251           (case g T of
   252             NONE => T :: map_typs Ts
   253           | SOME T' => T' :: (map_typs Ts handle SAME => Ts))
   254       | map_typs [] = raise SAME;
   255 
   256     fun mapp (Abst (s, T, prf)) = (Abst (s, apsome g T, mapph prf)
   257           handle SAME => Abst (s, T, mapp prf))
   258       | mapp (AbsP (s, t, prf)) = (AbsP (s, apsome f t, mapph prf)
   259           handle SAME => AbsP (s, t, mapp prf))
   260       | mapp (prf % t) = (mapp prf % (apsome f t handle SAME => t)
   261           handle SAME => prf % apsome f t)
   262       | mapp (prf1 %% prf2) = (mapp prf1 %% mapph prf2
   263           handle SAME => prf1 %% mapp prf2)
   264       | mapp (PAxm (a, prop, SOME Ts)) = PAxm (a, prop, SOME (map_typs Ts))
   265       | mapp (Oracle (a, prop, SOME Ts)) = Oracle (a, prop, SOME (map_typs Ts))
   266       | mapp (Promise (i, prop, Ts)) = Promise (i, prop, map_typs Ts)
   267       | mapp (PThm (i, ((a, prop, SOME Ts), body))) =
   268           PThm (i, ((a, prop, SOME (map_typs Ts)), body))
   269       | mapp _ = raise SAME
   270     and mapph prf = (mapp prf handle SAME => prf)
   271 
   272   in mapph end;
   273 
   274 fun same eq f x =
   275   let val x' = f x
   276   in if eq (x, x') then raise SAME else x' end;
   277 
   278 fun map_proof_terms f g =
   279   map_proof_terms_option
   280    (fn t => SOME (same (op =) f t) handle SAME => NONE)
   281    (fn T => SOME (same (op =) g T) handle SAME => NONE);
   282 
   283 fun fold_proof_terms f g (Abst (_, SOME T, prf)) = g T #> fold_proof_terms f g prf
   284   | fold_proof_terms f g (Abst (_, NONE, prf)) = fold_proof_terms f g prf
   285   | fold_proof_terms f g (AbsP (_, SOME t, prf)) = f t #> fold_proof_terms f g prf
   286   | fold_proof_terms f g (AbsP (_, NONE, prf)) = fold_proof_terms f g prf
   287   | fold_proof_terms f g (prf % SOME t) = fold_proof_terms f g prf #> f t
   288   | fold_proof_terms f g (prf % NONE) = fold_proof_terms f g prf
   289   | fold_proof_terms f g (prf1 %% prf2) =
   290       fold_proof_terms f g prf1 #> fold_proof_terms f g prf2
   291   | fold_proof_terms _ g (PAxm (_, _, SOME Ts)) = fold g Ts
   292   | fold_proof_terms _ g (Oracle (_, _, SOME Ts)) = fold g Ts
   293   | fold_proof_terms _ g (Promise (_, _, Ts)) = fold g Ts
   294   | fold_proof_terms _ g (PThm (_, ((_, _, SOME Ts), _))) = fold g Ts
   295   | fold_proof_terms _ _ _ = I;
   296 
   297 fun maxidx_proof prf = fold_proof_terms Term.maxidx_term Term.maxidx_typ prf;
   298 
   299 fun size_of_proof (Abst (_, _, prf)) = 1 + size_of_proof prf
   300   | size_of_proof (AbsP (_, t, prf)) = 1 + size_of_proof prf
   301   | size_of_proof (prf % _) = 1 + size_of_proof prf
   302   | size_of_proof (prf1 %% prf2) = size_of_proof prf1 + size_of_proof prf2
   303   | size_of_proof _ = 1;
   304 
   305 fun change_type opTs (PAxm (name, prop, _)) = PAxm (name, prop, opTs)
   306   | change_type opTs (Oracle (name, prop, _)) = Oracle (name, prop, opTs)
   307   | change_type opTs (Promise _) = error "change_type: unexpected promise"
   308   | change_type opTs (PThm (i, ((name, prop, _), body))) = PThm (i, ((name, prop, opTs), body))
   309   | change_type _ prf = prf;
   310 
   311 
   312 (***** utilities *****)
   313 
   314 fun strip_abs (_::Ts) (Abs (_, _, t)) = strip_abs Ts t
   315   | strip_abs _ t = t;
   316 
   317 fun mk_abs Ts t = Library.foldl (fn (t', T) => Abs ("", T, t')) (t, Ts);
   318 
   319 
   320 (*Abstraction of a proof term over its occurrences of v,
   321     which must contain no loose bound variables.
   322   The resulting proof term is ready to become the body of an Abst.*)
   323 
   324 fun prf_abstract_over v =
   325   let
   326     fun abst' lev u = if v aconv u then Bound lev else
   327       (case u of
   328          Abs (a, T, t) => Abs (a, T, abst' (lev + 1) t)
   329        | f $ t => (abst' lev f $ absth' lev t handle SAME => f $ abst' lev t)
   330        | _ => raise SAME)
   331     and absth' lev t = (abst' lev t handle SAME => t);
   332 
   333     fun abst lev (AbsP (a, t, prf)) =
   334           (AbsP (a, apsome' (abst' lev) t, absth lev prf)
   335            handle SAME => AbsP (a, t, abst lev prf))
   336       | abst lev (Abst (a, T, prf)) = Abst (a, T, abst (lev + 1) prf)
   337       | abst lev (prf1 %% prf2) = (abst lev prf1 %% absth lev prf2
   338           handle SAME => prf1 %% abst lev prf2)
   339       | abst lev (prf % t) = (abst lev prf % Option.map (absth' lev) t
   340           handle SAME => prf % apsome' (abst' lev) t)
   341       | abst _ _ = raise SAME
   342     and absth lev prf = (abst lev prf handle SAME => prf)
   343 
   344   in absth 0 end;
   345 
   346 
   347 (*increments a proof term's non-local bound variables
   348   required when moving a proof term within abstractions
   349      inc is  increment for bound variables
   350      lev is  level at which a bound variable is considered 'loose'*)
   351 
   352 fun incr_bv' inct tlev t = incr_bv (inct, tlev, t);
   353 
   354 fun prf_incr_bv' incP inct Plev tlev (PBound i) =
   355       if i >= Plev then PBound (i+incP) else raise SAME
   356   | prf_incr_bv' incP inct Plev tlev (AbsP (a, t, body)) =
   357       (AbsP (a, apsome' (same (op =) (incr_bv' inct tlev)) t,
   358          prf_incr_bv incP inct (Plev+1) tlev body) handle SAME =>
   359            AbsP (a, t, prf_incr_bv' incP inct (Plev+1) tlev body))
   360   | prf_incr_bv' incP inct Plev tlev (Abst (a, T, body)) =
   361       Abst (a, T, prf_incr_bv' incP inct Plev (tlev+1) body)
   362   | prf_incr_bv' incP inct Plev tlev (prf %% prf') =
   363       (prf_incr_bv' incP inct Plev tlev prf %% prf_incr_bv incP inct Plev tlev prf'
   364        handle SAME => prf %% prf_incr_bv' incP inct Plev tlev prf')
   365   | prf_incr_bv' incP inct Plev tlev (prf % t) =
   366       (prf_incr_bv' incP inct Plev tlev prf % Option.map (incr_bv' inct tlev) t
   367        handle SAME => prf % apsome' (same (op =) (incr_bv' inct tlev)) t)
   368   | prf_incr_bv' _ _ _ _ _ = raise SAME
   369 and prf_incr_bv incP inct Plev tlev prf =
   370       (prf_incr_bv' incP inct Plev tlev prf handle SAME => prf);
   371 
   372 fun incr_pboundvars  0 0 prf = prf
   373   | incr_pboundvars incP inct prf = prf_incr_bv incP inct 0 0 prf;
   374 
   375 
   376 fun prf_loose_bvar1 (prf1 %% prf2) k = prf_loose_bvar1 prf1 k orelse prf_loose_bvar1 prf2 k
   377   | prf_loose_bvar1 (prf % SOME t) k = prf_loose_bvar1 prf k orelse loose_bvar1 (t, k)
   378   | prf_loose_bvar1 (_ % NONE) _ = true
   379   | prf_loose_bvar1 (AbsP (_, SOME t, prf)) k = loose_bvar1 (t, k) orelse prf_loose_bvar1 prf k
   380   | prf_loose_bvar1 (AbsP (_, NONE, _)) k = true
   381   | prf_loose_bvar1 (Abst (_, _, prf)) k = prf_loose_bvar1 prf (k+1)
   382   | prf_loose_bvar1 _ _ = false;
   383 
   384 fun prf_loose_Pbvar1 (PBound i) k = i = k
   385   | prf_loose_Pbvar1 (prf1 %% prf2) k = prf_loose_Pbvar1 prf1 k orelse prf_loose_Pbvar1 prf2 k
   386   | prf_loose_Pbvar1 (prf % _) k = prf_loose_Pbvar1 prf k
   387   | prf_loose_Pbvar1 (AbsP (_, _, prf)) k = prf_loose_Pbvar1 prf (k+1)
   388   | prf_loose_Pbvar1 (Abst (_, _, prf)) k = prf_loose_Pbvar1 prf k
   389   | prf_loose_Pbvar1 _ _ = false;
   390 
   391 fun prf_add_loose_bnos plev tlev (PBound i) (is, js) =
   392       if i < plev then (is, js) else (insert (op =) (i-plev) is, js)
   393   | prf_add_loose_bnos plev tlev (prf1 %% prf2) p =
   394       prf_add_loose_bnos plev tlev prf2
   395         (prf_add_loose_bnos plev tlev prf1 p)
   396   | prf_add_loose_bnos plev tlev (prf % opt) (is, js) =
   397       prf_add_loose_bnos plev tlev prf (case opt of
   398           NONE => (is, insert (op =) ~1 js)
   399         | SOME t => (is, add_loose_bnos (t, tlev, js)))
   400   | prf_add_loose_bnos plev tlev (AbsP (_, opt, prf)) (is, js) =
   401       prf_add_loose_bnos (plev+1) tlev prf (case opt of
   402           NONE => (is, insert (op =) ~1 js)
   403         | SOME t => (is, add_loose_bnos (t, tlev, js)))
   404   | prf_add_loose_bnos plev tlev (Abst (_, _, prf)) p =
   405       prf_add_loose_bnos plev (tlev+1) prf p
   406   | prf_add_loose_bnos _ _ _ _ = ([], []);
   407 
   408 
   409 (**** substitutions ****)
   410 
   411 fun del_conflicting_tvars envT T = TermSubst.instantiateT
   412   (map_filter (fn ixnS as (_, S) =>
   413      (Type.lookup envT ixnS; NONE) handle TYPE _ =>
   414         SOME (ixnS, TFree ("'dummy", S))) (OldTerm.typ_tvars T)) T;
   415 
   416 fun del_conflicting_vars env t = TermSubst.instantiate
   417   (map_filter (fn ixnS as (_, S) =>
   418      (Type.lookup (type_env env) ixnS; NONE) handle TYPE _ =>
   419         SOME (ixnS, TFree ("'dummy", S))) (OldTerm.term_tvars t),
   420    map_filter (fn Var (ixnT as (_, T)) =>
   421      (Envir.lookup (env, ixnT); NONE) handle TYPE _ =>
   422         SOME (ixnT, Free ("dummy", T))) (OldTerm.term_vars t)) t;
   423 
   424 fun norm_proof env =
   425   let
   426     val envT = type_env env;
   427     fun msg s = warning ("type conflict in norm_proof:\n" ^ s);
   428     fun htype f t = f env t handle TYPE (s, _, _) =>
   429       (msg s; f env (del_conflicting_vars env t));
   430     fun htypeT f T = f envT T handle TYPE (s, _, _) =>
   431       (msg s; f envT (del_conflicting_tvars envT T));
   432     fun htypeTs f Ts = f envT Ts handle TYPE (s, _, _) =>
   433       (msg s; f envT (map (del_conflicting_tvars envT) Ts));
   434     fun norm (Abst (s, T, prf)) = (Abst (s, apsome' (htypeT norm_type_same) T, normh prf)
   435           handle SAME => Abst (s, T, norm prf))
   436       | norm (AbsP (s, t, prf)) = (AbsP (s, apsome' (htype norm_term_same) t, normh prf)
   437           handle SAME => AbsP (s, t, norm prf))
   438       | norm (prf % t) = (norm prf % Option.map (htype norm_term) t
   439           handle SAME => prf % apsome' (htype norm_term_same) t)
   440       | norm (prf1 %% prf2) = (norm prf1 %% normh prf2
   441           handle SAME => prf1 %% norm prf2)
   442       | norm (PAxm (s, prop, Ts)) = PAxm (s, prop, apsome' (htypeTs norm_types_same) Ts)
   443       | norm (Oracle (s, prop, Ts)) = Oracle (s, prop, apsome' (htypeTs norm_types_same) Ts)
   444       | norm (Promise (i, prop, Ts)) = Promise (i, prop, htypeTs norm_types_same Ts)
   445       | norm (PThm (i, ((s, t, Ts), body))) =
   446           PThm (i, ((s, t, apsome' (htypeTs norm_types_same) Ts), body))
   447       | norm _ = raise SAME
   448     and normh prf = (norm prf handle SAME => prf);
   449   in normh end;
   450 
   451 
   452 (***** Remove some types in proof term (to save space) *****)
   453 
   454 fun remove_types (Abs (s, _, t)) = Abs (s, dummyT, remove_types t)
   455   | remove_types (t $ u) = remove_types t $ remove_types u
   456   | remove_types (Const (s, _)) = Const (s, dummyT)
   457   | remove_types t = t;
   458 
   459 fun remove_types_env (Envir.Envir {iTs, asol, maxidx}) =
   460   Envir.Envir {iTs = iTs, asol = Vartab.map (apsnd remove_types) asol,
   461     maxidx = maxidx};
   462 
   463 fun norm_proof' env prf = norm_proof (remove_types_env env) prf;
   464 
   465 
   466 (**** substitution of bound variables ****)
   467 
   468 fun prf_subst_bounds args prf =
   469   let
   470     val n = length args;
   471     fun subst' lev (Bound i) =
   472          (if i<lev then raise SAME    (*var is locally bound*)
   473           else  incr_boundvars lev (nth args (i-lev))
   474                   handle Subscript => Bound (i-n))  (*loose: change it*)
   475       | subst' lev (Abs (a, T, body)) = Abs (a, T,  subst' (lev+1) body)
   476       | subst' lev (f $ t) = (subst' lev f $ substh' lev t
   477           handle SAME => f $ subst' lev t)
   478       | subst' _ _ = raise SAME
   479     and substh' lev t = (subst' lev t handle SAME => t);
   480 
   481     fun subst lev (AbsP (a, t, body)) = (AbsP (a, apsome' (subst' lev) t, substh lev body)
   482           handle SAME => AbsP (a, t, subst lev body))
   483       | subst lev (Abst (a, T, body)) = Abst (a, T, subst (lev+1) body)
   484       | subst lev (prf %% prf') = (subst lev prf %% substh lev prf'
   485           handle SAME => prf %% subst lev prf')
   486       | subst lev (prf % t) = (subst lev prf % Option.map (substh' lev) t
   487           handle SAME => prf % apsome' (subst' lev) t)
   488       | subst _ _ = raise SAME
   489     and substh lev prf = (subst lev prf handle SAME => prf)
   490   in case args of [] => prf | _ => substh 0 prf end;
   491 
   492 fun prf_subst_pbounds args prf =
   493   let
   494     val n = length args;
   495     fun subst (PBound i) Plev tlev =
   496          (if i < Plev then raise SAME    (*var is locally bound*)
   497           else incr_pboundvars Plev tlev (nth args (i-Plev))
   498                  handle Subscript => PBound (i-n)  (*loose: change it*))
   499       | subst (AbsP (a, t, body)) Plev tlev = AbsP (a, t, subst body (Plev+1) tlev)
   500       | subst (Abst (a, T, body)) Plev tlev = Abst (a, T, subst body Plev (tlev+1))
   501       | subst (prf %% prf') Plev tlev = (subst prf Plev tlev %% substh prf' Plev tlev
   502           handle SAME => prf %% subst prf' Plev tlev)
   503       | subst (prf % t) Plev tlev = subst prf Plev tlev % t
   504       | subst  prf _ _ = raise SAME
   505     and substh prf Plev tlev = (subst prf Plev tlev handle SAME => prf)
   506   in case args of [] => prf | _ => substh prf 0 0 end;
   507 
   508 
   509 (**** Freezing and thawing of variables in proof terms ****)
   510 
   511 fun frzT names =
   512   map_type_tvar (fn (ixn, xs) => TFree ((the o AList.lookup (op =) names) ixn, xs));
   513 
   514 fun thawT names =
   515   map_type_tfree (fn (s, xs) => case AList.lookup (op =) names s of
   516       NONE => TFree (s, xs)
   517     | SOME ixn => TVar (ixn, xs));
   518 
   519 fun freeze names names' (t $ u) =
   520       freeze names names' t $ freeze names names' u
   521   | freeze names names' (Abs (s, T, t)) =
   522       Abs (s, frzT names' T, freeze names names' t)
   523   | freeze names names' (Const (s, T)) = Const (s, frzT names' T)
   524   | freeze names names' (Free (s, T)) = Free (s, frzT names' T)
   525   | freeze names names' (Var (ixn, T)) =
   526       Free ((the o AList.lookup (op =) names) ixn, frzT names' T)
   527   | freeze names names' t = t;
   528 
   529 fun thaw names names' (t $ u) =
   530       thaw names names' t $ thaw names names' u
   531   | thaw names names' (Abs (s, T, t)) =
   532       Abs (s, thawT names' T, thaw names names' t)
   533   | thaw names names' (Const (s, T)) = Const (s, thawT names' T)
   534   | thaw names names' (Free (s, T)) =
   535       let val T' = thawT names' T
   536       in case AList.lookup (op =) names s of
   537           NONE => Free (s, T')
   538         | SOME ixn => Var (ixn, T')
   539       end
   540   | thaw names names' (Var (ixn, T)) = Var (ixn, thawT names' T)
   541   | thaw names names' t = t;
   542 
   543 fun freeze_thaw_prf prf =
   544   let
   545     val (fs, Tfs, vs, Tvs) = fold_proof_terms
   546       (fn t => fn (fs, Tfs, vs, Tvs) =>
   547          (Term.add_free_names t fs, Term.add_tfree_names t Tfs,
   548           Term.add_var_names t vs, Term.add_tvar_names t Tvs))
   549       (fn T => fn (fs, Tfs, vs, Tvs) =>
   550          (fs, Term.add_tfree_namesT T Tfs,
   551           vs, Term.add_tvar_namesT T Tvs))
   552       prf ([], [], [], []);
   553     val names = vs ~~ Name.variant_list fs (map fst vs);
   554     val names' = Tvs ~~ Name.variant_list Tfs (map fst Tvs);
   555     val rnames = map swap names;
   556     val rnames' = map swap names';
   557   in
   558     (map_proof_terms (freeze names names') (frzT names') prf,
   559      map_proof_terms (thaw rnames rnames') (thawT rnames'))
   560   end;
   561 
   562 
   563 (***** implication introduction *****)
   564 
   565 fun implies_intr_proof h prf =
   566   let
   567     fun abshyp i (Hyp t) = if h aconv t then PBound i else raise SAME
   568       | abshyp i (Abst (s, T, prf)) = Abst (s, T, abshyp i prf)
   569       | abshyp i (AbsP (s, t, prf)) = AbsP (s, t, abshyp (i+1) prf)
   570       | abshyp i (prf % t) = abshyp i prf % t
   571       | abshyp i (prf1 %% prf2) = (abshyp i prf1 %% abshyph i prf2
   572           handle SAME => prf1 %% abshyp i prf2)
   573       | abshyp _ _ = raise SAME
   574     and abshyph i prf = (abshyp i prf handle SAME => prf)
   575   in
   576     AbsP ("H", NONE (*h*), abshyph 0 prf)
   577   end;
   578 
   579 
   580 (***** forall introduction *****)
   581 
   582 fun forall_intr_proof x a prf = Abst (a, NONE, prf_abstract_over x prf);
   583 
   584 
   585 (***** varify *****)
   586 
   587 fun varify_proof t fixed prf =
   588   let
   589     val fs = Term.fold_types (Term.fold_atyps
   590       (fn TFree v => if member (op =) fixed v then I else insert (op =) v | _ => I)) t [];
   591     val used = Name.context
   592       |> fold_types (fold_atyps (fn TVar ((a, _), _) => Name.declare a | _ => I)) t;
   593     val fmap = fs ~~ (#1 (Name.variants (map fst fs) used));
   594     fun thaw (f as (a, S)) =
   595       (case AList.lookup (op =) fmap f of
   596         NONE => TFree f
   597       | SOME b => TVar ((b, 0), S));
   598   in map_proof_terms (map_types (map_type_tfree thaw)) (map_type_tfree thaw) prf end;
   599 
   600 
   601 local
   602 
   603 fun new_name (ix, (pairs,used)) =
   604   let val v = Name.variant used (string_of_indexname ix)
   605   in  ((ix, v) :: pairs, v :: used)  end;
   606 
   607 fun freeze_one alist (ix, sort) = (case AList.lookup (op =) alist ix of
   608     NONE => TVar (ix, sort)
   609   | SOME name => TFree (name, sort));
   610 
   611 in
   612 
   613 fun freezeT t prf =
   614   let
   615     val used = OldTerm.it_term_types OldTerm.add_typ_tfree_names (t, [])
   616     and tvars = map #1 (OldTerm.it_term_types OldTerm.add_typ_tvars (t, []));
   617     val (alist, _) = List.foldr new_name ([], used) tvars;
   618   in
   619     (case alist of
   620       [] => prf (*nothing to do!*)
   621     | _ =>
   622       let val frzT = map_type_tvar (freeze_one alist)
   623       in map_proof_terms (map_types frzT) frzT prf end)
   624   end;
   625 
   626 end;
   627 
   628 
   629 (***** rotate assumptions *****)
   630 
   631 fun rotate_proof Bs Bi m prf =
   632   let
   633     val params = Term.strip_all_vars Bi;
   634     val asms = Logic.strip_imp_prems (Term.strip_all_body Bi);
   635     val i = length asms;
   636     val j = length Bs;
   637   in
   638     mk_AbsP (j+1, proof_combP (prf, map PBound
   639       (j downto 1) @ [mk_Abst params (mk_AbsP (i,
   640         proof_combP (proof_combt (PBound i, map Bound ((length params - 1) downto 0)),
   641           map PBound (((i-m-1) downto 0) @ ((i-1) downto (i-m))))))]))
   642   end;
   643 
   644 
   645 (***** permute premises *****)
   646 
   647 fun permute_prems_prf prems j k prf =
   648   let val n = length prems
   649   in mk_AbsP (n, proof_combP (prf,
   650     map PBound ((n-1 downto n-j) @ (k-1 downto 0) @ (n-j-1 downto k))))
   651   end;
   652 
   653 
   654 (***** generalization *****)
   655 
   656 fun generalize (tfrees, frees) idx =
   657   map_proof_terms_option
   658     (TermSubst.generalize_option (tfrees, frees) idx)
   659     (TermSubst.generalizeT_option tfrees idx);
   660 
   661 
   662 (***** instantiation *****)
   663 
   664 fun instantiate (instT, inst) =
   665   map_proof_terms_option
   666     (TermSubst.instantiate_option (instT, map (apsnd remove_types) inst))
   667     (TermSubst.instantiateT_option instT);
   668 
   669 
   670 (***** lifting *****)
   671 
   672 fun lift_proof Bi inc prop prf =
   673   let
   674     fun lift'' Us Ts t = strip_abs Ts (Logic.incr_indexes (Us, inc) (mk_abs Ts t));
   675 
   676     fun lift' Us Ts (Abst (s, T, prf)) =
   677           (Abst (s, apsome' (same (op =) (Logic.incr_tvar inc)) T, lifth' Us (dummyT::Ts) prf)
   678            handle SAME => Abst (s, T, lift' Us (dummyT::Ts) prf))
   679       | lift' Us Ts (AbsP (s, t, prf)) =
   680           (AbsP (s, apsome' (same (op =) (lift'' Us Ts)) t, lifth' Us Ts prf)
   681            handle SAME => AbsP (s, t, lift' Us Ts prf))
   682       | lift' Us Ts (prf % t) = (lift' Us Ts prf % Option.map (lift'' Us Ts) t
   683           handle SAME => prf % apsome' (same (op =) (lift'' Us Ts)) t)
   684       | lift' Us Ts (prf1 %% prf2) = (lift' Us Ts prf1 %% lifth' Us Ts prf2
   685           handle SAME => prf1 %% lift' Us Ts prf2)
   686       | lift' _ _ (PAxm (s, prop, Ts)) =
   687           PAxm (s, prop, apsome' (same (op =) (map (Logic.incr_tvar inc))) Ts)
   688       | lift' _ _ (Oracle (s, prop, Ts)) =
   689           Oracle (s, prop, apsome' (same (op =) (map (Logic.incr_tvar inc))) Ts)
   690       | lift' _ _ (Promise (i, prop, Ts)) =
   691           Promise (i, prop, same (op =) (map (Logic.incr_tvar inc)) Ts)
   692       | lift' _ _ (PThm (i, ((s, prop, Ts), body))) =
   693           PThm (i, ((s, prop, apsome' (same (op =) (map (Logic.incr_tvar inc))) Ts), body))
   694       | lift' _ _ _ = raise SAME
   695     and lifth' Us Ts prf = (lift' Us Ts prf handle SAME => prf);
   696 
   697     val ps = map (Logic.lift_all inc Bi) (Logic.strip_imp_prems prop);
   698     val k = length ps;
   699 
   700     fun mk_app b (i, j, prf) =
   701           if b then (i-1, j, prf %% PBound i) else (i, j-1, prf %> Bound j);
   702 
   703     fun lift Us bs i j (Const ("==>", _) $ A $ B) =
   704             AbsP ("H", NONE (*A*), lift Us (true::bs) (i+1) j B)
   705       | lift Us bs i j (Const ("all", _) $ Abs (a, T, t)) =
   706             Abst (a, NONE (*T*), lift (T::Us) (false::bs) i (j+1) t)
   707       | lift Us bs i j _ = proof_combP (lifth' (rev Us) [] prf,
   708             map (fn k => (#3 (fold_rev mk_app bs (i-1, j-1, PBound k))))
   709               (i + k - 1 downto i));
   710   in
   711     mk_AbsP (k, lift [] [] 0 0 Bi)
   712   end;
   713 
   714 
   715 (***** proof by assumption *****)
   716 
   717 fun mk_asm_prf t i m =
   718   let
   719     fun imp_prf _ i 0 = PBound i
   720       | imp_prf (Const ("==>", _) $ A $ B) i m = AbsP ("H", NONE (*A*), imp_prf B (i+1) (m-1))
   721       | imp_prf _ i _ = PBound i;
   722     fun all_prf (Const ("all", _) $ Abs (a, T, t)) = Abst (a, NONE (*T*), all_prf t)
   723       | all_prf t = imp_prf t (~i) m
   724   in all_prf t end;
   725 
   726 fun assumption_proof Bs Bi n prf =
   727   mk_AbsP (length Bs, proof_combP (prf,
   728     map PBound (length Bs - 1 downto 0) @ [mk_asm_prf Bi n ~1]));
   729 
   730 
   731 (***** Composition of object rule with proof state *****)
   732 
   733 fun flatten_params_proof i j n (Const ("==>", _) $ A $ B, k) =
   734       AbsP ("H", NONE (*A*), flatten_params_proof (i+1) j n (B, k))
   735   | flatten_params_proof i j n (Const ("all", _) $ Abs (a, T, t), k) =
   736       Abst (a, NONE (*T*), flatten_params_proof i (j+1) n (t, k))
   737   | flatten_params_proof i j n (_, k) = proof_combP (proof_combt (PBound (k+i),
   738       map Bound (j-1 downto 0)), map PBound (remove (op =) (i-n) (i-1 downto 0)));
   739 
   740 fun bicompose_proof flatten Bs oldAs newAs A n m rprf sprf =
   741   let
   742     val la = length newAs;
   743     val lb = length Bs;
   744   in
   745     mk_AbsP (lb+la, proof_combP (sprf,
   746       map PBound (lb + la - 1 downto la)) %%
   747         proof_combP (rprf, (if n>0 then [mk_asm_prf (the A) n m] else []) @
   748           map (if flatten then flatten_params_proof 0 0 n else PBound o snd)
   749             (oldAs ~~ (la - 1 downto 0))))
   750   end;
   751 
   752 
   753 (***** axioms for equality *****)
   754 
   755 val aT = TFree ("'a", []);
   756 val bT = TFree ("'b", []);
   757 val x = Free ("x", aT);
   758 val y = Free ("y", aT);
   759 val z = Free ("z", aT);
   760 val A = Free ("A", propT);
   761 val B = Free ("B", propT);
   762 val f = Free ("f", aT --> bT);
   763 val g = Free ("g", aT --> bT);
   764 
   765 local open Logic in
   766 
   767 val equality_axms =
   768   [("reflexive", mk_equals (x, x)),
   769    ("symmetric", mk_implies (mk_equals (x, y), mk_equals (y, x))),
   770    ("transitive", list_implies ([mk_equals (x, y), mk_equals (y, z)], mk_equals (x, z))),
   771    ("equal_intr", list_implies ([mk_implies (A, B), mk_implies (B, A)], mk_equals (A, B))),
   772    ("equal_elim", list_implies ([mk_equals (A, B), A], B)),
   773    ("abstract_rule", mk_implies
   774       (all x (mk_equals (f $ x, g $ x)), mk_equals (lambda x (f $ x), lambda x (g $ x)))),
   775    ("combination", list_implies
   776       ([mk_equals (f, g), mk_equals (x, y)], mk_equals (f $ x, g $ y)))];
   777 
   778 val [reflexive_axm, symmetric_axm, transitive_axm, equal_intr_axm,
   779   equal_elim_axm, abstract_rule_axm, combination_axm] =
   780     map (fn (s, t) => PAxm ("Pure." ^ s, varify t, NONE)) equality_axms;
   781 
   782 end;
   783 
   784 val reflexive = reflexive_axm % NONE;
   785 
   786 fun symmetric (prf as PAxm ("Pure.reflexive", _, _) % _) = prf
   787   | symmetric prf = symmetric_axm % NONE % NONE %% prf;
   788 
   789 fun transitive _ _ (PAxm ("Pure.reflexive", _, _) % _) prf2 = prf2
   790   | transitive _ _ prf1 (PAxm ("Pure.reflexive", _, _) % _) = prf1
   791   | transitive u (Type ("prop", [])) prf1 prf2 =
   792       transitive_axm % NONE % SOME (remove_types u) % NONE %% prf1 %% prf2
   793   | transitive u T prf1 prf2 =
   794       transitive_axm % NONE % NONE % NONE %% prf1 %% prf2;
   795 
   796 fun abstract_rule x a prf =
   797   abstract_rule_axm % NONE % NONE %% forall_intr_proof x a prf;
   798 
   799 fun check_comb (PAxm ("Pure.combination", _, _) % f % g % _ % _ %% prf %% _) =
   800       is_some f orelse check_comb prf
   801   | check_comb (PAxm ("Pure.transitive", _, _) % _ % _ % _ %% prf1 %% prf2) =
   802       check_comb prf1 andalso check_comb prf2
   803   | check_comb (PAxm ("Pure.symmetric", _, _) % _ % _ %% prf) = check_comb prf
   804   | check_comb _ = false;
   805 
   806 fun combination f g t u (Type (_, [T, U])) prf1 prf2 =
   807   let
   808     val f = Envir.beta_norm f;
   809     val g = Envir.beta_norm g;
   810     val prf =  if check_comb prf1 then
   811         combination_axm % NONE % NONE
   812       else (case prf1 of
   813           PAxm ("Pure.reflexive", _, _) % _ =>
   814             combination_axm %> remove_types f % NONE
   815         | _ => combination_axm %> remove_types f %> remove_types g)
   816   in
   817     (case T of
   818        Type ("fun", _) => prf %
   819          (case head_of f of
   820             Abs _ => SOME (remove_types t)
   821           | Var _ => SOME (remove_types t)
   822           | _ => NONE) %
   823          (case head_of g of
   824             Abs _ => SOME (remove_types u)
   825           | Var _ => SOME (remove_types u)
   826           | _ => NONE) %% prf1 %% prf2
   827      | _ => prf % NONE % NONE %% prf1 %% prf2)
   828   end;
   829 
   830 fun equal_intr A B prf1 prf2 =
   831   equal_intr_axm %> remove_types A %> remove_types B %% prf1 %% prf2;
   832 
   833 fun equal_elim A B prf1 prf2 =
   834   equal_elim_axm %> remove_types A %> remove_types B %% prf1 %% prf2;
   835 
   836 
   837 (***** axioms and theorems *****)
   838 
   839 val proofs = ref 2;
   840 
   841 fun vars_of t = map Var (rev (Term.add_vars t []));
   842 fun frees_of t = map Free (rev (Term.add_frees t []));
   843 
   844 fun test_args _ [] = true
   845   | test_args is (Bound i :: ts) =
   846       not (member (op =) is i) andalso test_args (i :: is) ts
   847   | test_args _ _ = false;
   848 
   849 fun is_fun (Type ("fun", _)) = true
   850   | is_fun (TVar _) = true
   851   | is_fun _ = false;
   852 
   853 fun add_funvars Ts (vs, t) =
   854   if is_fun (fastype_of1 (Ts, t)) then
   855     vs union map_filter (fn Var (ixn, T) =>
   856       if is_fun T then SOME ixn else NONE | _ => NONE) (vars_of t)
   857   else vs;
   858 
   859 fun add_npvars q p Ts (vs, Const ("==>", _) $ t $ u) =
   860       add_npvars q p Ts (add_npvars q (not p) Ts (vs, t), u)
   861   | add_npvars q p Ts (vs, Const ("all", Type (_, [Type (_, [T, _]), _])) $ t) =
   862       add_npvars q p Ts (vs, if p andalso q then betapply (t, Var (("",0), T)) else t)
   863   | add_npvars q p Ts (vs, Abs (_, T, t)) = add_npvars q p (T::Ts) (vs, t)
   864   | add_npvars _ _ Ts (vs, t) = add_npvars' Ts (vs, t)
   865 and add_npvars' Ts (vs, t) = (case strip_comb t of
   866     (Var (ixn, _), ts) => if test_args [] ts then vs
   867       else Library.foldl (add_npvars' Ts)
   868         (AList.update (op =) (ixn,
   869           Library.foldl (add_funvars Ts) ((these ooo AList.lookup) (op =) vs ixn, ts)) vs, ts)
   870   | (Abs (_, T, u), ts) => Library.foldl (add_npvars' (T::Ts)) (vs, u :: ts)
   871   | (_, ts) => Library.foldl (add_npvars' Ts) (vs, ts));
   872 
   873 fun prop_vars (Const ("==>", _) $ P $ Q) = prop_vars P union prop_vars Q
   874   | prop_vars (Const ("all", _) $ Abs (_, _, t)) = prop_vars t
   875   | prop_vars t = (case strip_comb t of
   876       (Var (ixn, _), _) => [ixn] | _ => []);
   877 
   878 fun is_proj t =
   879   let
   880     fun is_p i t = (case strip_comb t of
   881         (Bound j, []) => false
   882       | (Bound j, ts) => j >= i orelse exists (is_p i) ts
   883       | (Abs (_, _, u), _) => is_p (i+1) u
   884       | (_, ts) => exists (is_p i) ts)
   885   in (case strip_abs_body t of
   886         Bound _ => true
   887       | t' => is_p 0 t')
   888   end;
   889 
   890 fun needed_vars prop =
   891   Library.foldl (op union)
   892     ([], map (uncurry (insert (op =))) (add_npvars true true [] ([], prop))) union
   893   prop_vars prop;
   894 
   895 fun gen_axm_proof c name prop =
   896   let
   897     val nvs = needed_vars prop;
   898     val args = map (fn (v as Var (ixn, _)) =>
   899         if member (op =) nvs ixn then SOME v else NONE) (vars_of prop) @
   900       map SOME (frees_of prop);
   901   in
   902     proof_combt' (c (name, prop, NONE), args)
   903   end;
   904 
   905 val axm_proof = gen_axm_proof PAxm;
   906 
   907 val dummy = Const (Term.dummy_patternN, dummyT);
   908 
   909 fun oracle_proof name prop =
   910   if !proofs = 0 then Oracle (name, dummy, NONE)
   911   else gen_axm_proof Oracle name prop;
   912 
   913 fun shrink_proof thy =
   914   let
   915     fun shrink ls lev (prf as Abst (a, T, body)) =
   916           let val (b, is, ch, body') = shrink ls (lev+1) body
   917           in (b, is, ch, if ch then Abst (a, T, body') else prf) end
   918       | shrink ls lev (prf as AbsP (a, t, body)) =
   919           let val (b, is, ch, body') = shrink (lev::ls) lev body
   920           in (b orelse member (op =) is 0, map_filter (fn 0 => NONE | i => SOME (i-1)) is,
   921             ch, if ch then AbsP (a, t, body') else prf)
   922           end
   923       | shrink ls lev prf =
   924           let val (is, ch, _, prf') = shrink' ls lev [] [] prf
   925           in (false, is, ch, prf') end
   926     and shrink' ls lev ts prfs (prf as prf1 %% prf2) =
   927           let
   928             val p as (_, is', ch', prf') = shrink ls lev prf2;
   929             val (is, ch, ts', prf'') = shrink' ls lev ts (p::prfs) prf1
   930           in (is union is', ch orelse ch', ts',
   931               if ch orelse ch' then prf'' %% prf' else prf)
   932           end
   933       | shrink' ls lev ts prfs (prf as prf1 % t) =
   934           let val (is, ch, (ch', t')::ts', prf') = shrink' ls lev (t::ts) prfs prf1
   935           in (is, ch orelse ch', ts',
   936               if ch orelse ch' then prf' % t' else prf) end
   937       | shrink' ls lev ts prfs (prf as PBound i) =
   938           (if exists (fn SOME (Bound j) => lev-j <= nth ls i | _ => true) ts
   939              orelse has_duplicates (op =)
   940                (Library.foldl (fn (js, SOME (Bound j)) => j :: js | (js, _) => js) ([], ts))
   941              orelse exists #1 prfs then [i] else [], false, map (pair false) ts, prf)
   942       | shrink' ls lev ts prfs (Hyp t) = ([], false, map (pair false) ts, Hyp t)
   943       | shrink' ls lev ts prfs MinProof = ([], false, map (pair false) ts, MinProof)
   944       | shrink' ls lev ts prfs prf =
   945           let
   946             val prop =
   947               (case prf of
   948                 PAxm (_, prop, _) => prop
   949               | Oracle (_, prop, _) => prop
   950               | Promise (_, prop, _) => prop
   951               | PThm (_, ((_, prop, _), _)) => prop
   952               | _ => error "shrink: proof not in normal form");
   953             val vs = vars_of prop;
   954             val (ts', ts'') = chop (length vs) ts;
   955             val insts = Library.take (length ts', map (fst o dest_Var) vs) ~~ ts';
   956             val nvs = Library.foldl (fn (ixns', (ixn, ixns)) =>
   957               insert (op =) ixn (case AList.lookup (op =) insts ixn of
   958                   SOME (SOME t) => if is_proj t then ixns union ixns' else ixns'
   959                 | _ => ixns union ixns'))
   960                   (needed prop ts'' prfs, add_npvars false true [] ([], prop));
   961             val insts' = map
   962               (fn (ixn, x as SOME _) => if member (op =) nvs ixn then (false, x) else (true, NONE)
   963                 | (_, x) => (false, x)) insts
   964           in ([], false, insts' @ map (pair false) ts'', prf) end
   965     and needed (Const ("==>", _) $ t $ u) ts ((b, _, _, _)::prfs) =
   966           (if b then map (fst o dest_Var) (vars_of t) else []) union needed u ts prfs
   967       | needed (Var (ixn, _)) (_::_) _ = [ixn]
   968       | needed _ _ _ = [];
   969   in shrink end;
   970 
   971 
   972 (**** Simple first order matching functions for terms and proofs ****)
   973 
   974 exception PMatch;
   975 
   976 (** see pattern.ML **)
   977 
   978 fun flt (i: int) = List.filter (fn n => n < i);
   979 
   980 fun fomatch Ts tymatch j =
   981   let
   982     fun mtch (instsp as (tyinsts, insts)) = fn
   983         (Var (ixn, T), t)  =>
   984           if j>0 andalso not (null (flt j (loose_bnos t)))
   985           then raise PMatch
   986           else (tymatch (tyinsts, fn () => (T, fastype_of1 (Ts, t))),
   987             (ixn, t) :: insts)
   988       | (Free (a, T), Free (b, U)) =>
   989           if a=b then (tymatch (tyinsts, K (T, U)), insts) else raise PMatch
   990       | (Const (a, T), Const (b, U))  =>
   991           if a=b then (tymatch (tyinsts, K (T, U)), insts) else raise PMatch
   992       | (f $ t, g $ u) => mtch (mtch instsp (f, g)) (t, u)
   993       | (Bound i, Bound j) => if i=j then instsp else raise PMatch
   994       | _ => raise PMatch
   995   in mtch end;
   996 
   997 fun match_proof Ts tymatch =
   998   let
   999     fun optmatch _ inst (NONE, _) = inst
  1000       | optmatch _ _ (SOME _, NONE) = raise PMatch
  1001       | optmatch mtch inst (SOME x, SOME y) = mtch inst (x, y)
  1002 
  1003     fun matcht Ts j (pinst, tinst) (t, u) =
  1004       (pinst, fomatch Ts tymatch j tinst (t, Envir.beta_norm u));
  1005     fun matchT (pinst, (tyinsts, insts)) p =
  1006       (pinst, (tymatch (tyinsts, K p), insts));
  1007     fun matchTs inst (Ts, Us) = Library.foldl (uncurry matchT) (inst, Ts ~~ Us);
  1008 
  1009     fun mtch Ts i j (pinst, tinst) (Hyp (Var (ixn, _)), prf) =
  1010           if i = 0 andalso j = 0 then ((ixn, prf) :: pinst, tinst)
  1011           else (case apfst (flt i) (apsnd (flt j)
  1012                   (prf_add_loose_bnos 0 0 prf ([], []))) of
  1013               ([], []) => ((ixn, incr_pboundvars (~i) (~j) prf) :: pinst, tinst)
  1014             | ([], _) => if j = 0 then
  1015                    ((ixn, incr_pboundvars (~i) (~j) prf) :: pinst, tinst)
  1016                  else raise PMatch
  1017             | _ => raise PMatch)
  1018       | mtch Ts i j inst (prf1 % opt1, prf2 % opt2) =
  1019           optmatch (matcht Ts j) (mtch Ts i j inst (prf1, prf2)) (opt1, opt2)
  1020       | mtch Ts i j inst (prf1 %% prf2, prf1' %% prf2') =
  1021           mtch Ts i j (mtch Ts i j inst (prf1, prf1')) (prf2, prf2')
  1022       | mtch Ts i j inst (Abst (_, opT, prf1), Abst (_, opU, prf2)) =
  1023           mtch (the_default dummyT opU :: Ts) i (j+1)
  1024             (optmatch matchT inst (opT, opU)) (prf1, prf2)
  1025       | mtch Ts i j inst (prf1, Abst (_, opU, prf2)) =
  1026           mtch (the_default dummyT opU :: Ts) i (j+1) inst
  1027             (incr_pboundvars 0 1 prf1 %> Bound 0, prf2)
  1028       | mtch Ts i j inst (AbsP (_, opt, prf1), AbsP (_, opu, prf2)) =
  1029           mtch Ts (i+1) j (optmatch (matcht Ts j) inst (opt, opu)) (prf1, prf2)
  1030       | mtch Ts i j inst (prf1, AbsP (_, _, prf2)) =
  1031           mtch Ts (i+1) j inst (incr_pboundvars 1 0 prf1 %% PBound 0, prf2)
  1032       | mtch Ts i j inst (PAxm (s1, _, opTs), PAxm (s2, _, opUs)) =
  1033           if s1 = s2 then optmatch matchTs inst (opTs, opUs)
  1034           else raise PMatch
  1035       | mtch Ts i j inst (PThm (_, ((name1, prop1, opTs), _)), PThm (_, ((name2, prop2, opUs), _))) =
  1036           if name1 = name2 andalso prop1 = prop2 then
  1037             optmatch matchTs inst (opTs, opUs)
  1038           else raise PMatch
  1039       | mtch _ _ _ inst (PBound i, PBound j) = if i = j then inst else raise PMatch
  1040       | mtch _ _ _ _ _ = raise PMatch
  1041   in mtch Ts 0 0 end;
  1042 
  1043 fun prf_subst (pinst, (tyinsts, insts)) =
  1044   let
  1045     val substT = Envir.typ_subst_TVars tyinsts;
  1046 
  1047     fun subst' lev (t as Var (ixn, _)) = (case AList.lookup (op =) insts ixn of
  1048           NONE => t
  1049         | SOME u => incr_boundvars lev u)
  1050       | subst' lev (Const (s, T)) = Const (s, substT T)
  1051       | subst' lev (Free (s, T)) = Free (s, substT T)
  1052       | subst' lev (Abs (a, T, body)) = Abs (a, substT T, subst' (lev+1) body)
  1053       | subst' lev (f $ t) = subst' lev f $ subst' lev t
  1054       | subst' _ t = t;
  1055 
  1056     fun subst plev tlev (AbsP (a, t, body)) =
  1057           AbsP (a, Option.map (subst' tlev) t, subst (plev+1) tlev body)
  1058       | subst plev tlev (Abst (a, T, body)) =
  1059           Abst (a, Option.map substT T, subst plev (tlev+1) body)
  1060       | subst plev tlev (prf %% prf') = subst plev tlev prf %% subst plev tlev prf'
  1061       | subst plev tlev (prf % t) = subst plev tlev prf % Option.map (subst' tlev) t
  1062       | subst plev tlev (prf as Hyp (Var (ixn, _))) = (case AList.lookup (op =) pinst ixn of
  1063           NONE => prf
  1064         | SOME prf' => incr_pboundvars plev tlev prf')
  1065       | subst _ _ (PAxm (id, prop, Ts)) = PAxm (id, prop, Option.map (map substT) Ts)
  1066       | subst _ _ (Oracle (id, prop, Ts)) = Oracle (id, prop, Option.map (map substT) Ts)
  1067       | subst _ _ (Promise (i, prop, Ts)) = Promise (i, prop, (map substT) Ts)
  1068       | subst _ _ (PThm (i, ((id, prop, Ts), body))) =
  1069           PThm (i, ((id, prop, Option.map (map substT) Ts), body))
  1070       | subst _ _ t = t;
  1071   in subst 0 0 end;
  1072 
  1073 (*A fast unification filter: true unless the two terms cannot be unified.
  1074   Terms must be NORMAL.  Treats all Vars as distinct. *)
  1075 fun could_unify prf1 prf2 =
  1076   let
  1077     fun matchrands (prf1 %% prf2) (prf1' %% prf2') =
  1078           could_unify prf2 prf2' andalso matchrands prf1 prf1'
  1079       | matchrands (prf % SOME t) (prf' % SOME t') =
  1080           Term.could_unify (t, t') andalso matchrands prf prf'
  1081       | matchrands (prf % _) (prf' % _) = matchrands prf prf'
  1082       | matchrands _ _ = true
  1083 
  1084     fun head_of (prf %% _) = head_of prf
  1085       | head_of (prf % _) = head_of prf
  1086       | head_of prf = prf
  1087 
  1088   in case (head_of prf1, head_of prf2) of
  1089         (_, Hyp (Var _)) => true
  1090       | (Hyp (Var _), _) => true
  1091       | (PAxm (a, _, _), PAxm (b, _, _)) => a = b andalso matchrands prf1 prf2
  1092       | (PThm (_, ((a, propa, _), _)), PThm (_, ((b, propb, _), _))) =>
  1093           a = b andalso propa = propb andalso matchrands prf1 prf2
  1094       | (PBound i, PBound j) => i = j andalso matchrands prf1 prf2
  1095       | (AbsP _, _) =>  true   (*because of possible eta equality*)
  1096       | (Abst _, _) =>  true
  1097       | (_, AbsP _) =>  true
  1098       | (_, Abst _) =>  true
  1099       | _ => false
  1100   end;
  1101 
  1102 
  1103 (**** rewriting on proof terms ****)
  1104 
  1105 val skel0 = PBound 0;
  1106 
  1107 fun rewrite_prf tymatch (rules, procs) prf =
  1108   let
  1109     fun rew _ (Abst (_, _, body) % SOME t) = SOME (prf_subst_bounds [t] body, skel0)
  1110       | rew _ (AbsP (_, _, body) %% prf) = SOME (prf_subst_pbounds [prf] body, skel0)
  1111       | rew Ts prf = (case get_first (fn r => r Ts prf) procs of
  1112           SOME prf' => SOME (prf', skel0)
  1113         | NONE => get_first (fn (prf1, prf2) => SOME (prf_subst
  1114             (match_proof Ts tymatch ([], (Vartab.empty, [])) (prf1, prf)) prf2, prf2)
  1115                handle PMatch => NONE) (filter (could_unify prf o fst) rules));
  1116 
  1117     fun rew0 Ts (prf as AbsP (_, _, prf' %% PBound 0)) =
  1118           if prf_loose_Pbvar1 prf' 0 then rew Ts prf
  1119           else
  1120             let val prf'' = incr_pboundvars (~1) 0 prf'
  1121             in SOME (the_default (prf'', skel0) (rew Ts prf'')) end
  1122       | rew0 Ts (prf as Abst (_, _, prf' % SOME (Bound 0))) =
  1123           if prf_loose_bvar1 prf' 0 then rew Ts prf
  1124           else
  1125             let val prf'' = incr_pboundvars 0 (~1) prf'
  1126             in SOME (the_default (prf'', skel0) (rew Ts prf'')) end
  1127       | rew0 Ts prf = rew Ts prf;
  1128 
  1129     fun rew1 _ (Hyp (Var _)) _ = NONE
  1130       | rew1 Ts skel prf = (case rew2 Ts skel prf of
  1131           SOME prf1 => (case rew0 Ts prf1 of
  1132               SOME (prf2, skel') => SOME (the_default prf2 (rew1 Ts skel' prf2))
  1133             | NONE => SOME prf1)
  1134         | NONE => (case rew0 Ts prf of
  1135               SOME (prf1, skel') => SOME (the_default prf1 (rew1 Ts skel' prf1))
  1136             | NONE => NONE))
  1137 
  1138     and rew2 Ts skel (prf % SOME t) = (case prf of
  1139             Abst (_, _, body) =>
  1140               let val prf' = prf_subst_bounds [t] body
  1141               in SOME (the_default prf' (rew2 Ts skel0 prf')) end
  1142           | _ => (case rew1 Ts (case skel of skel' % _ => skel' | _ => skel0) prf of
  1143               SOME prf' => SOME (prf' % SOME t)
  1144             | NONE => NONE))
  1145       | rew2 Ts skel (prf % NONE) = Option.map (fn prf' => prf' % NONE)
  1146           (rew1 Ts (case skel of skel' % _ => skel' | _ => skel0) prf)
  1147       | rew2 Ts skel (prf1 %% prf2) = (case prf1 of
  1148             AbsP (_, _, body) =>
  1149               let val prf' = prf_subst_pbounds [prf2] body
  1150               in SOME (the_default prf' (rew2 Ts skel0 prf')) end
  1151           | _ =>
  1152             let val (skel1, skel2) = (case skel of
  1153                 skel1 %% skel2 => (skel1, skel2)
  1154               | _ => (skel0, skel0))
  1155             in case rew1 Ts skel1 prf1 of
  1156                 SOME prf1' => (case rew1 Ts skel2 prf2 of
  1157                     SOME prf2' => SOME (prf1' %% prf2')
  1158                   | NONE => SOME (prf1' %% prf2))
  1159               | NONE => (case rew1 Ts skel2 prf2 of
  1160                     SOME prf2' => SOME (prf1 %% prf2')
  1161                   | NONE => NONE)
  1162             end)
  1163       | rew2 Ts skel (Abst (s, T, prf)) = (case rew1 (the_default dummyT T :: Ts)
  1164               (case skel of Abst (_, _, skel') => skel' | _ => skel0) prf of
  1165             SOME prf' => SOME (Abst (s, T, prf'))
  1166           | NONE => NONE)
  1167       | rew2 Ts skel (AbsP (s, t, prf)) = (case rew1 Ts
  1168               (case skel of AbsP (_, _, skel') => skel' | _ => skel0) prf of
  1169             SOME prf' => SOME (AbsP (s, t, prf'))
  1170           | NONE => NONE)
  1171       | rew2 _ _ _ = NONE
  1172 
  1173   in the_default prf (rew1 [] skel0 prf) end;
  1174 
  1175 fun rewrite_proof thy = rewrite_prf (fn (tyenv, f) =>
  1176   Sign.typ_match thy (f ()) tyenv handle Type.TYPE_MATCH => raise PMatch);
  1177 
  1178 fun rewrite_proof_notypes rews = rewrite_prf fst rews;
  1179 
  1180 
  1181 (**** theory data ****)
  1182 
  1183 structure ProofData = TheoryDataFun
  1184 (
  1185   type T = (stamp * (proof * proof)) list * (stamp * (typ list -> proof -> proof option)) list;
  1186 
  1187   val empty = ([], []);
  1188   val copy = I;
  1189   val extend = I;
  1190   fun merge _ ((rules1, procs1), (rules2, procs2)) : T =
  1191     (AList.merge (op =) (K true) (rules1, rules2),
  1192       AList.merge (op =) (K true) (procs1, procs2));
  1193 );
  1194 
  1195 fun get_data thy = let val (rules, procs) = ProofData.get thy in (map #2 rules, map #2 procs) end;
  1196 fun rew_proof thy = rewrite_prf fst (get_data thy);
  1197 
  1198 fun add_prf_rrule r = (ProofData.map o apfst) (cons (stamp (), r));
  1199 fun add_prf_rproc p = (ProofData.map o apsnd) (cons (stamp (), p));
  1200 
  1201 
  1202 (***** promises *****)
  1203 
  1204 fun promise_proof thy i prop =
  1205   let
  1206     val _ = prop |> Term.exists_subterm (fn t =>
  1207       (Term.is_Free t orelse Term.is_Var t) andalso
  1208         error ("promise_proof: illegal variable " ^ Syntax.string_of_term_global thy t));
  1209     val _ = prop |> Term.exists_type (Term.exists_subtype
  1210       (fn TFree (a, _) => error ("promise_proof: illegal type variable " ^ quote a)
  1211         | _ => false));
  1212   in Promise (i, prop, map TVar (Term.add_tvars prop [])) end;
  1213 
  1214 fun fulfill_proof _ [] body0 = body0
  1215   | fulfill_proof thy promises body0 =
  1216       let
  1217         val PBody {oracles = oracles0, thms = thms0, proof = proof0} = body0;
  1218         val (oracles, thms) = fold (merge_body o make_body o #2) promises (oracles0, thms0);
  1219 
  1220         val tab = Inttab.make promises;
  1221         fun fill (Promise (i, prop, Ts)) =
  1222             (case Inttab.lookup tab i of
  1223               NONE => NONE
  1224             | SOME p => SOME (instantiate (Term.add_tvars prop [] ~~ Ts, []) p))
  1225           | fill _ = NONE;
  1226         val (rules, procs) = get_data thy;
  1227         val proof = rewrite_prf fst (rules, K fill :: procs) proof0;
  1228       in PBody {oracles = oracles, thms = thms, proof = proof} end;
  1229 
  1230 fun fulfill_proof_future _ [] body = Future.value body
  1231   | fulfill_proof_future thy promises body =
  1232       Future.fork_deps (map snd promises) (fn () =>
  1233         fulfill_proof thy (map (apsnd Future.join) promises) body);
  1234 
  1235 
  1236 (***** theorems *****)
  1237 
  1238 fun thm_proof thy name hyps prop promises body =
  1239   let
  1240     val PBody {oracles = oracles0, thms = thms0, proof = prf} = body;
  1241     val prop = Logic.list_implies (hyps, prop);
  1242     val nvs = needed_vars prop;
  1243     val args = map (fn (v as Var (ixn, _)) =>
  1244         if member (op =) nvs ixn then SOME v else NONE) (vars_of prop) @
  1245       map SOME (frees_of prop);
  1246 
  1247     val proof0 =
  1248       if ! proofs = 2 then
  1249         #4 (shrink_proof thy [] 0 (rew_proof thy (fold_rev implies_intr_proof hyps prf)))
  1250       else MinProof;
  1251     val body0 = PBody {oracles = oracles0, thms = thms0, proof = proof0};
  1252 
  1253     fun new_prf () = (serial (), name, prop, fulfill_proof_future thy promises body0);
  1254     val (i, name, prop, body') =
  1255       (case strip_combt (fst (strip_combP prf)) of
  1256         (PThm (i, ((old_name, prop', NONE), body')), args') =>
  1257           if (old_name = "" orelse old_name = name) andalso prop = prop' andalso args = args'
  1258           then (i, name, prop, body')
  1259           else new_prf ()
  1260       | _ => new_prf ());
  1261     val head = PThm (i, ((name, prop, NONE), body'));
  1262   in
  1263     ((i, (name, prop, body')), proof_combP (proof_combt' (head, args), map Hyp hyps))
  1264   end;
  1265 
  1266 fun get_name hyps prop prf =
  1267   let val prop = Logic.list_implies (hyps, prop) in
  1268     (case strip_combt (fst (strip_combP prf)) of
  1269       (PAxm (name, prop', _), _) => if prop = prop' then name else ""   (* FIXME !? *)
  1270     | (PThm (_, ((name, prop', _), _)), _) => if prop = prop' then name else ""
  1271     | _ => "")
  1272   end;
  1273 
  1274 end;
  1275 
  1276 structure BasicProofterm : BASIC_PROOFTERM = Proofterm;
  1277 open BasicProofterm;