(*  Title:      Pure/proofterm.ML

    Author:     Stefan Berghofer, TU Muenchen

LF style proof terms.

*)

infix 8 % %% %>;

signature BASIC_PROOFTERM =

sig

  val proofs: int Unsynchronized.ref

  datatype proof =

     MinProof

   | PBound of int

   | Abst of string * typ option * proof

   | AbsP of string * term option * proof

   | op % of proof * term option

   | op %% of proof * proof

   | Hyp of term

   | PAxm of string * term * typ list option

   | OfClass of typ * class

   | Oracle of string * term * typ list option

   | Promise of serial * term * typ list

   | PThm of serial * ((string * term * typ list option) * proof_body future)

  and proof_body = PBody of

    {oracles: (string * term) Ord_List.T,

     thms: (serial * (string * term * proof_body future)) Ord_List.T,

     proof: proof}

  val %> : proof * term -> proof

end;

signature PROOFTERM =

sig

  include BASIC_PROOFTERM

  type oracle = string * term

  type pthm = serial * (string * term * proof_body future)

  val proof_of: proof_body -> proof

  val join_proof: proof_body future -> proof

  val fold_proof_atoms: bool -> (proof -> 'a -> 'a) -> proof list -> 'a -> 'a

  val fold_body_thms: (string * term * proof_body -> 'a -> 'a) -> proof_body list -> 'a -> 'a

  val join_bodies: proof_body list -> unit

  val status_of: proof_body list -> {failed: bool, oracle: bool, unfinished: bool}

  val oracle_ord: oracle * oracle -> order

  val thm_ord: pthm * pthm -> order

  val merge_oracles: oracle Ord_List.T -> oracle Ord_List.T -> oracle Ord_List.T

  val merge_thms: pthm Ord_List.T -> pthm Ord_List.T -> pthm Ord_List.T

  val all_oracles_of: proof_body -> oracle Ord_List.T

  val approximate_proof_body: proof -> proof_body

  (** primitive operations **)

  val proofs_enabled: unit -> bool

  val proof_combt: proof * term list -> proof

  val proof_combt': proof * term option list -> proof

  val proof_combP: proof * proof list -> proof

  val strip_combt: proof -> proof * term option list

  val strip_combP: proof -> proof * proof list

  val strip_thm: proof_body -> proof_body

  val map_proof_same: term Same.operation -> typ Same.operation

    -> (typ * class -> proof) -> proof Same.operation

  val map_proof_terms_same: term Same.operation -> typ Same.operation -> proof Same.operation

  val map_proof_types_same: typ Same.operation -> proof Same.operation

  val map_proof_terms: (term -> term) -> (typ -> typ) -> proof -> proof

  val map_proof_types: (typ -> typ) -> proof -> proof

  val fold_proof_terms: (term -> 'a -> 'a) -> (typ -> 'a -> 'a) -> proof -> 'a -> 'a

  val maxidx_proof: proof -> int -> int

  val size_of_proof: proof -> int

  val change_type: typ list option -> proof -> proof

  val prf_abstract_over: term -> proof -> proof

  val prf_incr_bv: int -> int -> int -> int -> proof -> proof

  val incr_pboundvars: int -> int -> proof -> proof

  val prf_loose_bvar1: proof -> int -> bool

  val prf_loose_Pbvar1: proof -> int -> bool

  val prf_add_loose_bnos: int -> int -> proof -> int list * int list -> int list * int list

  val norm_proof: Envir.env -> proof -> proof

  val norm_proof': Envir.env -> proof -> proof

  val prf_subst_bounds: term list -> proof -> proof

  val prf_subst_pbounds: proof list -> proof -> proof

  val freeze_thaw_prf: proof -> proof * (proof -> proof)

  (** proof terms for specific inference rules **)

  val implies_intr_proof: term -> proof -> proof

  val implies_intr_proof': term -> proof -> proof

  val forall_intr_proof: term -> string -> proof -> proof

  val forall_intr_proof': term -> proof -> proof

  val varify_proof: term -> (string * sort) list -> proof -> proof

  val legacy_freezeT: term -> proof -> proof

  val rotate_proof: term list -> term -> int -> proof -> proof

  val permute_prems_proof: term list -> int -> int -> proof -> proof

  val generalize: string list * string list -> int -> proof -> proof

  val instantiate: ((indexname * sort) * typ) list * ((indexname * typ) * term) list

    -> proof -> proof

  val lift_proof: term -> int -> term -> proof -> proof

  val incr_indexes: int -> proof -> proof

  val assumption_proof: term list -> term -> int -> proof -> proof

  val bicompose_proof: bool -> term list -> term list -> term list -> term option ->

    int -> int -> proof -> proof -> proof

  val equality_axms: (string * term) list

  val reflexive_axm: proof

  val symmetric_axm: proof

  val transitive_axm: proof

  val equal_intr_axm: proof

  val equal_elim_axm: proof

  val abstract_rule_axm: proof

  val combination_axm: proof

  val reflexive: proof

  val symmetric: proof -> proof

  val transitive: term -> typ -> proof -> proof -> proof

  val abstract_rule: term -> string -> proof -> proof

  val combination: term -> term -> term -> term -> typ -> proof -> proof -> proof

  val equal_intr: term -> term -> proof -> proof -> proof

  val equal_elim: term -> term -> proof -> proof -> proof

  val strip_shyps_proof: Sorts.algebra -> (typ * sort) list -> (typ * sort) list ->

    sort list -> proof -> proof

  val classrel_proof: theory -> class * class -> proof

  val arity_proof: theory -> string * sort list * class -> proof

  val of_sort_proof: theory -> (typ * class -> proof) -> typ * sort -> proof list

  val install_axclass_proofs:

   {classrel_proof: theory -> class * class -> proof,

    arity_proof: theory -> string * sort list * class -> proof} -> unit

  val axm_proof: string -> term -> proof

  val oracle_proof: string -> term -> oracle * proof

  (** rewriting on proof terms **)

  val add_prf_rrule: proof * proof -> theory -> theory

  val add_prf_rproc: (typ list -> term option list -> proof -> (proof * proof) option) -> theory -> theory

  val no_skel: proof

  val normal_skel: proof

  val rewrite_proof: theory -> (proof * proof) list *

    (typ list -> term option list -> proof -> (proof * proof) option) list -> proof -> proof

  val rewrite_proof_notypes: (proof * proof) list *

    (typ list -> term option list -> proof -> (proof * proof) option) list -> proof -> proof

  val rew_proof: theory -> proof -> proof

  val promise_proof: theory -> serial -> term -> proof

  val fulfill_norm_proof: theory -> (serial * proof_body) list -> proof_body -> proof_body

  val thm_proof: theory -> string -> sort list -> term list -> term ->

    (serial * proof_body future) list -> proof_body -> pthm * proof

  val unconstrain_thm_proof: theory -> sort list -> term ->

    (serial * proof_body future) list -> proof_body -> pthm * proof

  val get_name: sort list -> term list -> term -> proof -> string

  val guess_name: proof -> string

end

structure Proofterm : PROOFTERM =

struct

(***** datatype proof *****)

datatype proof =

   MinProof

 | PBound of int

 | Abst of string * typ option * proof

 | AbsP of string * term option * proof

 | op % of proof * term option

 | op %% of proof * proof

 | Hyp of term

 | PAxm of string * term * typ list option

 | OfClass of typ * class

 | Oracle of string * term * typ list option

 | Promise of serial * term * typ list

 | PThm of serial * ((string * term * typ list option) * proof_body future)

and proof_body = PBody of

  {oracles: (string * term) Ord_List.T,

   thms: (serial * (string * term * proof_body future)) Ord_List.T,

   proof: proof};

type oracle = string * term;

type pthm = serial * (string * term * proof_body future);

fun proof_of (PBody {proof, ...}) = proof;

val join_proof = Future.join #> proof_of;

(***** proof atoms *****)

fun fold_proof_atoms all f =

  let

    fun app (Abst (_, _, prf)) = app prf

      | app (AbsP (_, _, prf)) = app prf

      | app (prf % _) = app prf

      | app (prf1 %% prf2) = app prf1 #> app prf2

      | app (prf as PThm (i, (_, body))) = (fn (x, seen) =>

          if Inttab.defined seen i then (x, seen)

          else

            let val (x', seen') =

              (if all then app (join_proof body) else I) (x, Inttab.update (i, ()) seen)

            in (f prf x', seen') end)

      | app prf = (fn (x, seen) => (f prf x, seen));

  in fn prfs => fn x => #1 (fold app prfs (x, Inttab.empty)) end;

fun fold_body_thms f =

  let

    fun app (PBody {thms, ...}) =

     (Future.join_results (map (#3 o #2) thms);

      thms |> fold (fn (i, (name, prop, body)) => fn (x, seen) =>

        if Inttab.defined seen i then (x, seen)

        else

          let

            val body' = Future.join body;

            val (x', seen') = app body' (x, Inttab.update (i, ()) seen);

          in (f (name, prop, body') x', seen') end));

  in fn bodies => fn x => #1 (fold app bodies (x, Inttab.empty)) end;

fun join_bodies bodies = fold_body_thms (fn _ => fn () => ()) bodies ();

fun status_of bodies =

  let

    fun status (PBody {oracles, thms, ...}) x =

      let

        val ((oracle, unfinished, failed), seen) =

          (thms, x) |-> fold (fn (i, (_, _, body)) => fn (st, seen) =>

            if Inttab.defined seen i then (st, seen)

            else

              let val seen' = Inttab.update (i, ()) seen in

                (case Future.peek body of

                  SOME (Exn.Result body') => status body' (st, seen')

                | SOME (Exn.Exn _) =>

                    let val (oracle, unfinished, _) = st

                    in ((oracle, unfinished, true), seen') end

                | NONE =>

                    let val (oracle, _, failed) = st

                    in ((oracle, true, failed), seen') end)

              end);

      in ((oracle orelse not (null oracles), unfinished, failed), seen) end;

    val (oracle, unfinished, failed) =

      #1 (fold status bodies ((false, false, false), Inttab.empty));

  in {oracle = oracle, unfinished = unfinished, failed = failed} end;

(* proof body *)

val oracle_ord = prod_ord fast_string_ord Term_Ord.fast_term_ord;

fun thm_ord ((i, _): pthm, (j, _)) = int_ord (j, i);

val merge_oracles = Ord_List.union oracle_ord;

val merge_thms = Ord_List.union thm_ord;

val all_oracles_of =

  let

    fun collect (PBody {oracles, thms, ...}) =

     (Future.join_results (map (#3 o #2) thms);

      thms |> fold (fn (i, (_, _, body)) => fn (x, seen) =>

        if Inttab.defined seen i then (x, seen)

        else

          let

            val body' = Future.join body;

            val (x', seen') = collect body' (x, Inttab.update (i, ()) seen);

          in (merge_oracles oracles x', seen') end));

  in fn body => #1 (collect body ([], Inttab.empty)) end;

fun approximate_proof_body prf =

  let

    val (oracles, thms) = fold_proof_atoms false

      (fn Oracle (s, prop, _) => apfst (cons (s, prop))

        | PThm (i, ((name, prop, _), body)) => apsnd (cons (i, (name, prop, body)))

        | _ => I) [prf] ([], []);

  in

    PBody

     {oracles = Ord_List.make oracle_ord oracles,

      thms = Ord_List.make thm_ord thms,

      proof = prf}

  end;

(***** proof objects with different levels of detail *****)

fun (prf %> t) = prf % SOME t;

val proof_combt = Library.foldl (op %>);

val proof_combt' = Library.foldl (op %);

val proof_combP = Library.foldl (op %%);

fun strip_combt prf =

    let fun stripc (prf % t, ts) = stripc (prf, t::ts)

          | stripc  x =  x

    in  stripc (prf, [])  end;

fun strip_combP prf =

    let fun stripc (prf %% prf', prfs) = stripc (prf, prf'::prfs)

          | stripc  x =  x

    in  stripc (prf, [])  end;

fun strip_thm (body as PBody {proof, ...}) =

  (case strip_combt (fst (strip_combP proof)) of

    (PThm (_, (_, body')), _) => Future.join body'

  | _ => body);

val mk_Abst = fold_rev (fn (s, T:typ) => fn prf => Abst (s, NONE, prf));

fun mk_AbsP (i, prf) = funpow i (fn prf => AbsP ("H", NONE, prf)) prf;

fun map_proof_same term typ ofclass =

  let

    val typs = Same.map typ;

    fun proof (Abst (s, T, prf)) =

          (Abst (s, Same.map_option typ T, Same.commit proof prf)

            handle Same.SAME => Abst (s, T, proof prf))

      | proof (AbsP (s, t, prf)) =

          (AbsP (s, Same.map_option term t, Same.commit proof prf)

            handle Same.SAME => AbsP (s, t, proof prf))

      | proof (prf % t) =

          (proof prf % Same.commit (Same.map_option term) t

            handle Same.SAME => prf % Same.map_option term t)

      | proof (prf1 %% prf2) =

          (proof prf1 %% Same.commit proof prf2

            handle Same.SAME => prf1 %% proof prf2)

      | proof (PAxm (a, prop, SOME Ts)) = PAxm (a, prop, SOME (typs Ts))

      | proof (OfClass T_c) = ofclass T_c

      | proof (Oracle (a, prop, SOME Ts)) = Oracle (a, prop, SOME (typs Ts))

      | proof (Promise (i, prop, Ts)) = Promise (i, prop, typs Ts)

      | proof (PThm (i, ((a, prop, SOME Ts), body))) =

          PThm (i, ((a, prop, SOME (typs Ts)), body))

      | proof _ = raise Same.SAME;

  in proof end;

fun map_proof_terms_same term typ = map_proof_same term typ (fn (T, c) => OfClass (typ T, c));

fun map_proof_types_same typ = map_proof_terms_same (Term_Subst.map_types_same typ) typ;

fun same eq f x =

  let val x' = f x

  in if eq (x, x') then raise Same.SAME else x' end;

fun map_proof_terms f g = Same.commit (map_proof_terms_same (same (op =) f) (same (op =) g));

fun map_proof_types f = Same.commit (map_proof_types_same (same (op =) f));

fun fold_proof_terms f g (Abst (_, SOME T, prf)) = g T #> fold_proof_terms f g prf

  | fold_proof_terms f g (Abst (_, NONE, prf)) = fold_proof_terms f g prf

  | fold_proof_terms f g (AbsP (_, SOME t, prf)) = f t #> fold_proof_terms f g prf

  | fold_proof_terms f g (AbsP (_, NONE, prf)) = fold_proof_terms f g prf

  | fold_proof_terms f g (prf % SOME t) = fold_proof_terms f g prf #> f t

  | fold_proof_terms f g (prf % NONE) = fold_proof_terms f g prf

  | fold_proof_terms f g (prf1 %% prf2) =

      fold_proof_terms f g prf1 #> fold_proof_terms f g prf2

  | fold_proof_terms _ g (PAxm (_, _, SOME Ts)) = fold g Ts

  | fold_proof_terms _ g (OfClass (T, _)) = g T

  | fold_proof_terms _ g (Oracle (_, _, SOME Ts)) = fold g Ts

  | fold_proof_terms _ g (Promise (_, _, Ts)) = fold g Ts

  | fold_proof_terms _ g (PThm (_, ((_, _, SOME Ts), _))) = fold g Ts

  | fold_proof_terms _ _ _ = I;

fun maxidx_proof prf = fold_proof_terms Term.maxidx_term Term.maxidx_typ prf;

fun size_of_proof (Abst (_, _, prf)) = 1 + size_of_proof prf

  | size_of_proof (AbsP (_, t, prf)) = 1 + size_of_proof prf

  | size_of_proof (prf % _) = 1 + size_of_proof prf

  | size_of_proof (prf1 %% prf2) = size_of_proof prf1 + size_of_proof prf2

  | size_of_proof _ = 1;

fun change_type opTs (PAxm (name, prop, _)) = PAxm (name, prop, opTs)

  | change_type (SOME [T]) (OfClass (_, c)) = OfClass (T, c)

  | change_type opTs (Oracle (name, prop, _)) = Oracle (name, prop, opTs)

  | change_type opTs (Promise _) = raise Fail "change_type: unexpected promise"

  | change_type opTs (PThm (i, ((name, prop, _), body))) =

      PThm (i, ((name, prop, opTs), body))

  | change_type _ prf = prf;

(***** utilities *****)

fun strip_abs (_::Ts) (Abs (_, _, t)) = strip_abs Ts t

  | strip_abs _ t = t;

fun mk_abs Ts t = Library.foldl (fn (t', T) => Abs ("", T, t')) (t, Ts);

(*Abstraction of a proof term over its occurrences of v,

    which must contain no loose bound variables.

  The resulting proof term is ready to become the body of an Abst.*)

fun prf_abstract_over v =

  let

    fun abst' lev u = if v aconv u then Bound lev else

      (case u of

         Abs (a, T, t) => Abs (a, T, abst' (lev + 1) t)

       | f $ t => (abst' lev f $ absth' lev t handle Same.SAME => f $ abst' lev t)

       | _ => raise Same.SAME)

    and absth' lev t = (abst' lev t handle Same.SAME => t);

    fun abst lev (AbsP (a, t, prf)) =

          (AbsP (a, Same.map_option (abst' lev) t, absth lev prf)

           handle Same.SAME => AbsP (a, t, abst lev prf))

      | abst lev (Abst (a, T, prf)) = Abst (a, T, abst (lev + 1) prf)

      | abst lev (prf1 %% prf2) = (abst lev prf1 %% absth lev prf2

          handle Same.SAME => prf1 %% abst lev prf2)

      | abst lev (prf % t) = (abst lev prf % Option.map (absth' lev) t

          handle Same.SAME => prf % Same.map_option (abst' lev) t)

      | abst _ _ = raise Same.SAME

    and absth lev prf = (abst lev prf handle Same.SAME => prf);

  in absth 0 end;

(*increments a proof term's non-local bound variables

  required when moving a proof term within abstractions

     inc is  increment for bound variables

     lev is  level at which a bound variable is considered 'loose'*)

fun incr_bv' inct tlev t = incr_bv (inct, tlev, t);

fun prf_incr_bv' incP inct Plev tlev (PBound i) =

      if i >= Plev then PBound (i+incP) else raise Same.SAME

  | prf_incr_bv' incP inct Plev tlev (AbsP (a, t, body)) =

      (AbsP (a, Same.map_option (same (op =) (incr_bv' inct tlev)) t,

         prf_incr_bv incP inct (Plev+1) tlev body) handle Same.SAME =>

           AbsP (a, t, prf_incr_bv' incP inct (Plev+1) tlev body))

  | prf_incr_bv' incP inct Plev tlev (Abst (a, T, body)) =

      Abst (a, T, prf_incr_bv' incP inct Plev (tlev+1) body)

  | prf_incr_bv' incP inct Plev tlev (prf %% prf') =

      (prf_incr_bv' incP inct Plev tlev prf %% prf_incr_bv incP inct Plev tlev prf'

       handle Same.SAME => prf %% prf_incr_bv' incP inct Plev tlev prf')

  | prf_incr_bv' incP inct Plev tlev (prf % t) =

      (prf_incr_bv' incP inct Plev tlev prf % Option.map (incr_bv' inct tlev) t

       handle Same.SAME => prf % Same.map_option (same (op =) (incr_bv' inct tlev)) t)

  | prf_incr_bv' _ _ _ _ _ = raise Same.SAME

and prf_incr_bv incP inct Plev tlev prf =

      (prf_incr_bv' incP inct Plev tlev prf handle Same.SAME => prf);

fun incr_pboundvars  0 0 prf = prf

  | incr_pboundvars incP inct prf = prf_incr_bv incP inct 0 0 prf;

fun prf_loose_bvar1 (prf1 %% prf2) k = prf_loose_bvar1 prf1 k orelse prf_loose_bvar1 prf2 k

  | prf_loose_bvar1 (prf % SOME t) k = prf_loose_bvar1 prf k orelse loose_bvar1 (t, k)

  | prf_loose_bvar1 (_ % NONE) _ = true

  | prf_loose_bvar1 (AbsP (_, SOME t, prf)) k = loose_bvar1 (t, k) orelse prf_loose_bvar1 prf k

  | prf_loose_bvar1 (AbsP (_, NONE, _)) k = true

  | prf_loose_bvar1 (Abst (_, _, prf)) k = prf_loose_bvar1 prf (k+1)

  | prf_loose_bvar1 _ _ = false;

fun prf_loose_Pbvar1 (PBound i) k = i = k

  | prf_loose_Pbvar1 (prf1 %% prf2) k = prf_loose_Pbvar1 prf1 k orelse prf_loose_Pbvar1 prf2 k

  | prf_loose_Pbvar1 (prf % _) k = prf_loose_Pbvar1 prf k

  | prf_loose_Pbvar1 (AbsP (_, _, prf)) k = prf_loose_Pbvar1 prf (k+1)

  | prf_loose_Pbvar1 (Abst (_, _, prf)) k = prf_loose_Pbvar1 prf k

  | prf_loose_Pbvar1 _ _ = false;

fun prf_add_loose_bnos plev tlev (PBound i) (is, js) =

      if i < plev then (is, js) else (insert (op =) (i-plev) is, js)

  | prf_add_loose_bnos plev tlev (prf1 %% prf2) p =

      prf_add_loose_bnos plev tlev prf2

        (prf_add_loose_bnos plev tlev prf1 p)

  | prf_add_loose_bnos plev tlev (prf % opt) (is, js) =

      prf_add_loose_bnos plev tlev prf (case opt of

          NONE => (is, insert (op =) ~1 js)

        | SOME t => (is, add_loose_bnos (t, tlev, js)))

  | prf_add_loose_bnos plev tlev (AbsP (_, opt, prf)) (is, js) =

      prf_add_loose_bnos (plev+1) tlev prf (case opt of

          NONE => (is, insert (op =) ~1 js)

        | SOME t => (is, add_loose_bnos (t, tlev, js)))

  | prf_add_loose_bnos plev tlev (Abst (_, _, prf)) p =

      prf_add_loose_bnos plev (tlev+1) prf p

  | prf_add_loose_bnos _ _ _ _ = ([], []);

(**** substitutions ****)

fun del_conflicting_tvars envT T = Term_Subst.instantiateT

  (map_filter (fn ixnS as (_, S) =>

     (Type.lookup envT ixnS; NONE) handle TYPE _ =>

        SOME (ixnS, TFree ("'dummy", S))) (OldTerm.typ_tvars T)) T;

fun del_conflicting_vars env t = Term_Subst.instantiate

  (map_filter (fn ixnS as (_, S) =>

     (Type.lookup (Envir.type_env env) ixnS; NONE) handle TYPE _ =>

        SOME (ixnS, TFree ("'dummy", S))) (OldTerm.term_tvars t),

   map_filter (fn Var (ixnT as (_, T)) =>

     (Envir.lookup (env, ixnT); NONE) handle TYPE _ =>

        SOME (ixnT, Free ("dummy", T))) (OldTerm.term_vars t)) t;

fun norm_proof env =

  let

    val envT = Envir.type_env env;

    fun msg s = warning ("type conflict in norm_proof:\n" ^ s);

    fun htype f t = f env t handle TYPE (s, _, _) =>

      (msg s; f env (del_conflicting_vars env t));

    fun htypeT f T = f envT T handle TYPE (s, _, _) =>

      (msg s; f envT (del_conflicting_tvars envT T));

    fun htypeTs f Ts = f envT Ts handle TYPE (s, _, _) =>

      (msg s; f envT (map (del_conflicting_tvars envT) Ts));

    fun norm (Abst (s, T, prf)) =

          (Abst (s, Same.map_option (htypeT Envir.norm_type_same) T, Same.commit norm prf)

            handle Same.SAME => Abst (s, T, norm prf))

      | norm (AbsP (s, t, prf)) =

          (AbsP (s, Same.map_option (htype Envir.norm_term_same) t, Same.commit norm prf)

            handle Same.SAME => AbsP (s, t, norm prf))

      | norm (prf % t) =

          (norm prf % Option.map (htype Envir.norm_term) t

            handle Same.SAME => prf % Same.map_option (htype Envir.norm_term_same) t)

      | norm (prf1 %% prf2) =

          (norm prf1 %% Same.commit norm prf2

            handle Same.SAME => prf1 %% norm prf2)

      | norm (PAxm (s, prop, Ts)) =

          PAxm (s, prop, Same.map_option (htypeTs Envir.norm_types_same) Ts)

      | norm (OfClass (T, c)) =

          OfClass (htypeT Envir.norm_type_same T, c)

      | norm (Oracle (s, prop, Ts)) =

          Oracle (s, prop, Same.map_option (htypeTs Envir.norm_types_same) Ts)

      | norm (Promise (i, prop, Ts)) =

          Promise (i, prop, htypeTs Envir.norm_types_same Ts)

      | norm (PThm (i, ((s, t, Ts), body))) =

          PThm (i, ((s, t, Same.map_option (htypeTs Envir.norm_types_same) Ts), body))

      | norm _ = raise Same.SAME;

  in Same.commit norm end;

(***** Remove some types in proof term (to save space) *****)

fun remove_types (Abs (s, _, t)) = Abs (s, dummyT, remove_types t)

  | remove_types (t $ u) = remove_types t $ remove_types u

  | remove_types (Const (s, _)) = Const (s, dummyT)

  | remove_types t = t;

fun remove_types_env (Envir.Envir {maxidx, tenv, tyenv}) =

  Envir.Envir {maxidx = maxidx, tenv = Vartab.map (K (apsnd remove_types)) tenv, tyenv = tyenv};

fun norm_proof' env prf = norm_proof (remove_types_env env) prf;

(**** substitution of bound variables ****)

fun prf_subst_bounds args prf =

  let

    val n = length args;

    fun subst' lev (Bound i) =

         (if i<lev then raise Same.SAME    (*var is locally bound*)

          else  incr_boundvars lev (nth args (i-lev))

                  handle General.Subscript => Bound (i-n))  (*loose: change it*)

      | subst' lev (Abs (a, T, body)) = Abs (a, T,  subst' (lev+1) body)

      | subst' lev (f $ t) = (subst' lev f $ substh' lev t

          handle Same.SAME => f $ subst' lev t)

      | subst' _ _ = raise Same.SAME

    and substh' lev t = (subst' lev t handle Same.SAME => t);

    fun subst lev (AbsP (a, t, body)) =

        (AbsP (a, Same.map_option (subst' lev) t, substh lev body)

          handle Same.SAME => AbsP (a, t, subst lev body))

      | subst lev (Abst (a, T, body)) = Abst (a, T, subst (lev+1) body)

      | subst lev (prf %% prf') = (subst lev prf %% substh lev prf'

          handle Same.SAME => prf %% subst lev prf')

      | subst lev (prf % t) = (subst lev prf % Option.map (substh' lev) t

          handle Same.SAME => prf % Same.map_option (subst' lev) t)

      | subst _ _ = raise Same.SAME

    and substh lev prf = (subst lev prf handle Same.SAME => prf);

  in case args of [] => prf | _ => substh 0 prf end;

fun prf_subst_pbounds args prf =

  let

    val n = length args;

    fun subst (PBound i) Plev tlev =

         (if i < Plev then raise Same.SAME    (*var is locally bound*)

          else incr_pboundvars Plev tlev (nth args (i-Plev))

                 handle General.Subscript => PBound (i-n)  (*loose: change it*))

      | subst (AbsP (a, t, body)) Plev tlev = AbsP (a, t, subst body (Plev+1) tlev)

      | subst (Abst (a, T, body)) Plev tlev = Abst (a, T, subst body Plev (tlev+1))

      | subst (prf %% prf') Plev tlev = (subst prf Plev tlev %% substh prf' Plev tlev

          handle Same.SAME => prf %% subst prf' Plev tlev)

      | subst (prf % t) Plev tlev = subst prf Plev tlev % t

      | subst  prf _ _ = raise Same.SAME

    and substh prf Plev tlev = (subst prf Plev tlev handle Same.SAME => prf)

  in case args of [] => prf | _ => substh prf 0 0 end;

(**** Freezing and thawing of variables in proof terms ****)

fun frzT names =

  map_type_tvar (fn (ixn, xs) => TFree ((the o AList.lookup (op =) names) ixn, xs));

fun thawT names =

  map_type_tfree (fn (s, xs) => case AList.lookup (op =) names s of

      NONE => TFree (s, xs)

    | SOME ixn => TVar (ixn, xs));

fun freeze names names' (t $ u) =

      freeze names names' t $ freeze names names' u

  | freeze names names' (Abs (s, T, t)) =

      Abs (s, frzT names' T, freeze names names' t)

  | freeze names names' (Const (s, T)) = Const (s, frzT names' T)

  | freeze names names' (Free (s, T)) = Free (s, frzT names' T)

  | freeze names names' (Var (ixn, T)) =

      Free ((the o AList.lookup (op =) names) ixn, frzT names' T)

  | freeze names names' t = t;

fun thaw names names' (t $ u) =

      thaw names names' t $ thaw names names' u

  | thaw names names' (Abs (s, T, t)) =

      Abs (s, thawT names' T, thaw names names' t)

  | thaw names names' (Const (s, T)) = Const (s, thawT names' T)

  | thaw names names' (Free (s, T)) =

      let val T' = thawT names' T

      in case AList.lookup (op =) names s of

          NONE => Free (s, T')

        | SOME ixn => Var (ixn, T')

      end

  | thaw names names' (Var (ixn, T)) = Var (ixn, thawT names' T)

  | thaw names names' t = t;

fun freeze_thaw_prf prf =

  let

    val (fs, Tfs, vs, Tvs) = fold_proof_terms

      (fn t => fn (fs, Tfs, vs, Tvs) =>

         (Term.add_free_names t fs, Term.add_tfree_names t Tfs,

          Term.add_var_names t vs, Term.add_tvar_names t Tvs))

      (fn T => fn (fs, Tfs, vs, Tvs) =>

         (fs, Term.add_tfree_namesT T Tfs,

          vs, Term.add_tvar_namesT T Tvs))

      prf ([], [], [], []);

    val names = vs ~~ Name.variant_list fs (map fst vs);

    val names' = Tvs ~~ Name.variant_list Tfs (map fst Tvs);

    val rnames = map swap names;

    val rnames' = map swap names';

  in

    (map_proof_terms (freeze names names') (frzT names') prf,

     map_proof_terms (thaw rnames rnames') (thawT rnames'))

  end;

(***** implication introduction *****)

fun gen_implies_intr_proof f h prf =

  let

    fun abshyp i (Hyp t) = if h aconv t then PBound i else raise Same.SAME

      | abshyp i (Abst (s, T, prf)) = Abst (s, T, abshyp i prf)

      | abshyp i (AbsP (s, t, prf)) = AbsP (s, t, abshyp (i + 1) prf)

      | abshyp i (prf % t) = abshyp i prf % t

      | abshyp i (prf1 %% prf2) =

          (abshyp i prf1 %% abshyph i prf2

            handle Same.SAME => prf1 %% abshyp i prf2)

      | abshyp _ _ = raise Same.SAME

    and abshyph i prf = (abshyp i prf handle Same.SAME => prf);

  in

    AbsP ("H", f h, abshyph 0 prf)

  end;

val implies_intr_proof = gen_implies_intr_proof (K NONE);

val implies_intr_proof' = gen_implies_intr_proof SOME;

(***** forall introduction *****)

fun forall_intr_proof x a prf = Abst (a, NONE, prf_abstract_over x prf);

fun forall_intr_proof' t prf =

  let val (a, T) = (case t of Var ((a, _), T) => (a, T) | Free p => p)

  in Abst (a, SOME T, prf_abstract_over t prf) end;

(***** varify *****)

fun varify_proof t fixed prf =

  let

    val fs = Term.fold_types (Term.fold_atyps

      (fn TFree v => if member (op =) fixed v then I else insert (op =) v | _ => I)) t [];

    val used = Name.context

      |> fold_types (fold_atyps (fn TVar ((a, _), _) => Name.declare a | _ => I)) t;

    val fmap = fs ~~ #1 (fold_map Name.variant (map fst fs) used);

    fun thaw (f as (a, S)) =

      (case AList.lookup (op =) fmap f of

        NONE => TFree f

      | SOME b => TVar ((b, 0), S));

  in map_proof_terms (map_types (map_type_tfree thaw)) (map_type_tfree thaw) prf end;

local

fun new_name (ix, (pairs,used)) =

  let val v = singleton (Name.variant_list used) (string_of_indexname ix)

  in  ((ix, v) :: pairs, v :: used)  end;

fun freeze_one alist (ix, sort) = (case AList.lookup (op =) alist ix of

    NONE => TVar (ix, sort)

  | SOME name => TFree (name, sort));

in

fun legacy_freezeT t prf =

  let

    val used = OldTerm.it_term_types OldTerm.add_typ_tfree_names (t, [])

    and tvars = map #1 (OldTerm.it_term_types OldTerm.add_typ_tvars (t, []));

    val (alist, _) = List.foldr new_name ([], used) tvars;

  in

    (case alist of

      [] => prf (*nothing to do!*)

    | _ =>

      let val frzT = map_type_tvar (freeze_one alist)

      in map_proof_terms (map_types frzT) frzT prf end)

  end;

end;

(***** rotate assumptions *****)

fun rotate_proof Bs Bi m prf =

  let

    val params = Term.strip_all_vars Bi;

    val asms = Logic.strip_imp_prems (Term.strip_all_body Bi);

    val i = length asms;

    val j = length Bs;

  in

    mk_AbsP (j+1, proof_combP (prf, map PBound

      (j downto 1) @ [mk_Abst params (mk_AbsP (i,

        proof_combP (proof_combt (PBound i, map Bound ((length params - 1) downto 0)),

          map PBound (((i-m-1) downto 0) @ ((i-1) downto (i-m))))))]))

  end;

(***** permute premises *****)

fun permute_prems_proof prems j k prf =

  let val n = length prems

  in mk_AbsP (n, proof_combP (prf,

    map PBound ((n-1 downto n-j) @ (k-1 downto 0) @ (n-j-1 downto k))))

  end;

(***** generalization *****)

fun generalize (tfrees, frees) idx =

  Same.commit (map_proof_terms_same

    (Term_Subst.generalize_same (tfrees, frees) idx)

    (Term_Subst.generalizeT_same tfrees idx));

(***** instantiation *****)

fun instantiate (instT, inst) =

  Same.commit (map_proof_terms_same

    (Term_Subst.instantiate_same (instT, map (apsnd remove_types) inst))

    (Term_Subst.instantiateT_same instT));

(***** lifting *****)

fun lift_proof Bi inc prop prf =

  let

    fun lift'' Us Ts t =

      strip_abs Ts (Logic.incr_indexes (Us, inc) (mk_abs Ts t));

    fun lift' Us Ts (Abst (s, T, prf)) =

          (Abst (s, Same.map_option (Logic.incr_tvar_same inc) T, lifth' Us (dummyT::Ts) prf)

           handle Same.SAME => Abst (s, T, lift' Us (dummyT::Ts) prf))

      | lift' Us Ts (AbsP (s, t, prf)) =

          (AbsP (s, Same.map_option (same (op =) (lift'' Us Ts)) t, lifth' Us Ts prf)

           handle Same.SAME => AbsP (s, t, lift' Us Ts prf))

      | lift' Us Ts (prf % t) = (lift' Us Ts prf % Option.map (lift'' Us Ts) t

          handle Same.SAME => prf % Same.map_option (same (op =) (lift'' Us Ts)) t)

      | lift' Us Ts (prf1 %% prf2) = (lift' Us Ts prf1 %% lifth' Us Ts prf2

          handle Same.SAME => prf1 %% lift' Us Ts prf2)

      | lift' _ _ (PAxm (s, prop, Ts)) =

          PAxm (s, prop, (Same.map_option o Same.map) (Logic.incr_tvar_same inc) Ts)

      | lift' _ _ (OfClass (T, c)) =

          OfClass (Logic.incr_tvar_same inc T, c)

      | lift' _ _ (Oracle (s, prop, Ts)) =

          Oracle (s, prop, (Same.map_option o Same.map) (Logic.incr_tvar_same inc) Ts)

      | lift' _ _ (Promise (i, prop, Ts)) =

          Promise (i, prop, Same.map (Logic.incr_tvar_same inc) Ts)

      | lift' _ _ (PThm (i, ((s, prop, Ts), body))) =

          PThm (i, ((s, prop, (Same.map_option o Same.map) (Logic.incr_tvar inc) Ts), body))

      | lift' _ _ _ = raise Same.SAME

    and lifth' Us Ts prf = (lift' Us Ts prf handle Same.SAME => prf);

    val ps = map (Logic.lift_all inc Bi) (Logic.strip_imp_prems prop);

    val k = length ps;

    fun mk_app b (i, j, prf) =

          if b then (i-1, j, prf %% PBound i) else (i, j-1, prf %> Bound j);

    fun lift Us bs i j (Const ("==>", _) $ A $ B) =

            AbsP ("H", NONE (*A*), lift Us (true::bs) (i+1) j B)

      | lift Us bs i j (Const ("all", _) $ Abs (a, T, t)) =

            Abst (a, NONE (*T*), lift (T::Us) (false::bs) i (j+1) t)

      | lift Us bs i j _ = proof_combP (lifth' (rev Us) [] prf,

            map (fn k => (#3 (fold_rev mk_app bs (i-1, j-1, PBound k))))

              (i + k - 1 downto i));

  in

    mk_AbsP (k, lift [] [] 0 0 Bi)

  end;

fun incr_indexes i =

  Same.commit (map_proof_terms_same

    (Logic.incr_indexes_same ([], i)) (Logic.incr_tvar_same i));

(***** proof by assumption *****)

fun mk_asm_prf t i m =

  let

    fun imp_prf _ i 0 = PBound i

      | imp_prf (Const ("==>", _) $ A $ B) i m = AbsP ("H", NONE (*A*), imp_prf B (i+1) (m-1))

      | imp_prf _ i _ = PBound i;

    fun all_prf (Const ("all", _) $ Abs (a, T, t)) = Abst (a, NONE (*T*), all_prf t)

      | all_prf t = imp_prf t (~i) m

  in all_prf t end;

fun assumption_proof Bs Bi n prf =

  mk_AbsP (length Bs, proof_combP (prf,

    map PBound (length Bs - 1 downto 0) @ [mk_asm_prf Bi n ~1]));

(***** Composition of object rule with proof state *****)

fun flatten_params_proof i j n (Const ("==>", _) $ A $ B, k) =

      AbsP ("H", NONE (*A*), flatten_params_proof (i+1) j n (B, k))

  | flatten_params_proof i j n (Const ("all", _) $ Abs (a, T, t), k) =

      Abst (a, NONE (*T*), flatten_params_proof i (j+1) n (t, k))

  | flatten_params_proof i j n (_, k) = proof_combP (proof_combt (PBound (k+i),

      map Bound (j-1 downto 0)), map PBound (remove (op =) (i-n) (i-1 downto 0)));

fun bicompose_proof flatten Bs oldAs newAs A n m rprf sprf =

  let

    val la = length newAs;

    val lb = length Bs;

  in

    mk_AbsP (lb+la, proof_combP (sprf,

      map PBound (lb + la - 1 downto la)) %%

        proof_combP (rprf, (if n>0 then [mk_asm_prf (the A) n m] else []) @

          map (if flatten then flatten_params_proof 0 0 n else PBound o snd)

            (oldAs ~~ (la - 1 downto 0))))

  end;

(***** axioms for equality *****)

val aT = TFree ("'a", []);

val bT = TFree ("'b", []);

val x = Free ("x", aT);

val y = Free ("y", aT);

val z = Free ("z", aT);

val A = Free ("A", propT);

val B = Free ("B", propT);

val f = Free ("f", aT --> bT);

val g = Free ("g", aT --> bT);

val equality_axms =

 [("reflexive", Logic.mk_equals (x, x)),

  ("symmetric", Logic.mk_implies (Logic.mk_equals (x, y), Logic.mk_equals (y, x))),

  ("transitive",

    Logic.list_implies ([Logic.mk_equals (x, y), Logic.mk_equals (y, z)], Logic.mk_equals (x, z))),

  ("equal_intr",

    Logic.list_implies ([Logic.mk_implies (A, B), Logic.mk_implies (B, A)], Logic.mk_equals (A, B))),

  ("equal_elim", Logic.list_implies ([Logic.mk_equals (A, B), A], B)),

  ("abstract_rule",

    Logic.mk_implies

      (Logic.all x

        (Logic.mk_equals (f $ x, g $ x)), Logic.mk_equals (lambda x (f $ x), lambda x (g $ x)))),

  ("combination", Logic.list_implies

    ([Logic.mk_equals (f, g), Logic.mk_equals (x, y)], Logic.mk_equals (f $ x, g $ y)))];

val [reflexive_axm, symmetric_axm, transitive_axm, equal_intr_axm,

  equal_elim_axm, abstract_rule_axm, combination_axm] =

    map (fn (s, t) => PAxm ("Pure." ^ s, Logic.varify_global t, NONE)) equality_axms;

val reflexive = reflexive_axm % NONE;

fun symmetric (prf as PAxm ("Pure.reflexive", _, _) % _) = prf

  | symmetric prf = symmetric_axm % NONE % NONE %% prf;

fun transitive _ _ (PAxm ("Pure.reflexive", _, _) % _) prf2 = prf2

  | transitive _ _ prf1 (PAxm ("Pure.reflexive", _, _) % _) = prf1

  | transitive u (Type ("prop", [])) prf1 prf2 =

      transitive_axm % NONE % SOME (remove_types u) % NONE %% prf1 %% prf2

  | transitive u T prf1 prf2 =

      transitive_axm % NONE % NONE % NONE %% prf1 %% prf2;

fun abstract_rule x a prf =

  abstract_rule_axm % NONE % NONE %% forall_intr_proof x a prf;

fun check_comb (PAxm ("Pure.combination", _, _) % f % g % _ % _ %% prf %% _) =

      is_some f orelse check_comb prf

  | check_comb (PAxm ("Pure.transitive", _, _) % _ % _ % _ %% prf1 %% prf2) =

      check_comb prf1 andalso check_comb prf2

  | check_comb (PAxm ("Pure.symmetric", _, _) % _ % _ %% prf) = check_comb prf

  | check_comb _ = false;

fun combination f g t u (Type (_, [T, U])) prf1 prf2 =

  let

    val f = Envir.beta_norm f;

    val g = Envir.beta_norm g;

    val prf =  if check_comb prf1 then

        combination_axm % NONE % NONE

      else (case prf1 of

          PAxm ("Pure.reflexive", _, _) % _ =>

            combination_axm %> remove_types f % NONE

        | _ => combination_axm %> remove_types f %> remove_types g)

  in

    (case T of

       Type ("fun", _) => prf %

         (case head_of f of

            Abs _ => SOME (remove_types t)

          | Var _ => SOME (remove_types t)

          | _ => NONE) %

         (case head_of g of

            Abs _ => SOME (remove_types u)

          | Var _ => SOME (remove_types u)

          | _ => NONE) %% prf1 %% prf2

     | _ => prf % NONE % NONE %% prf1 %% prf2)

  end;

fun equal_intr A B prf1 prf2 =

  equal_intr_axm %> remove_types A %> remove_types B %% prf1 %% prf2;

fun equal_elim A B prf1 prf2 =

  equal_elim_axm %> remove_types A %> remove_types B %% prf1 %% prf2;

(**** type classes ****)

fun strip_shyps_proof algebra present witnessed extra_sorts prf =

  let

    fun get S2 (T, S1) = if Sorts.sort_le algebra (S1, S2) then SOME T else NONE;

    val extra = map (fn S => (TFree ("'dummy", S), S)) extra_sorts;

    val replacements = present @ extra @ witnessed;

    fun replace T =

      if exists (fn (T', _) => T' = T) present then raise Same.SAME

      else

        (case get_first (get (Type.sort_of_atyp T)) replacements of

          SOME T' => T'

        | NONE => raise Fail "strip_shyps_proof: bad type variable in proof term");

  in Same.commit (map_proof_types_same (Term_Subst.map_atypsT_same replace)) prf end;

local

type axclass_proofs =

 {classrel_proof: theory -> class * class -> proof,

  arity_proof: theory -> string * sort list * class -> proof};

val axclass_proofs: axclass_proofs Single_Assignment.var =

  Single_Assignment.var "Proofterm.axclass_proofs";

fun axclass_proof which thy x =

  (case Single_Assignment.peek axclass_proofs of

    NONE => raise Fail "Axclass proof operations not installed"

  | SOME prfs => which prfs thy x);

in

val classrel_proof = axclass_proof #classrel_proof;

val arity_proof = axclass_proof #arity_proof;

fun install_axclass_proofs prfs = Single_Assignment.assign axclass_proofs prfs;

end;

local

fun canonical_instance typs =

  let

    val names = Name.invents Name.context Name.aT (length typs);

    val instT = map2 (fn a => fn T => (((a, 0), []), Type.strip_sorts T)) names typs;

  in instantiate (instT, []) end;

in

fun of_sort_proof thy hyps =

  Sorts.of_sort_derivation (Sign.classes_of thy)

   {class_relation = fn typ => fn (prf, c1) => fn c2 =>

      if c1 = c2 then prf

      else canonical_instance [typ] (classrel_proof thy (c1, c2)) %% prf,

    type_constructor = fn (a, typs) => fn dom => fn c =>

      let val Ss = map (map snd) dom and prfs = maps (map fst) dom

      in proof_combP (canonical_instance typs (arity_proof thy (a, Ss, c)), prfs) end,

    type_variable = fn typ => map (fn c => (hyps (typ, c), c)) (Type.sort_of_atyp typ)};

end;

(***** axioms and theorems *****)

val proofs = Unsynchronized.ref 2;

fun proofs_enabled () = ! proofs >= 2;

fun vars_of t = map Var (rev (Term.add_vars t []));

fun frees_of t = map Free (rev (Term.add_frees t []));

fun test_args _ [] = true

  | test_args is (Bound i :: ts) =

      not (member (op =) is i) andalso test_args (i :: is) ts

  | test_args _ _ = false;

fun is_fun (Type ("fun", _)) = true

  | is_fun (TVar _) = true

  | is_fun _ = false;

fun add_funvars Ts (vs, t) =

  if is_fun (fastype_of1 (Ts, t)) then

    union (op =) vs (map_filter (fn Var (ixn, T) =>