src/Pure/proofterm.ML

tuned signature;

(*  Title:      Pure/proofterm.ML
    Author:     Stefan Berghofer, TU Muenchen

LF style proof terms.
*)

infix 8 % %% %>;

signature PROOFTERM =
sig
  type thm_header =
    {serial: serial, pos: Position.T list, theory_name: string, name: string,
      prop: term, types: typ list option}
  type thm_body
  type thm_node
  datatype proof =
     MinProof
   | PBound of int
   | Abst of string * typ option * proof
   | AbsP of string * term option * proof
   | % of proof * term option
   | %% of proof * proof
   | Hyp of term
   | PAxm of string * term * typ list option
   | OfClass of typ * class
   | Oracle of string * term * typ list option
   | PThm of thm_header * thm_body
  and proof_body = PBody of
    {oracles: ((string * Position.T) * term option) Ord_List.T,
     thms: (serial * thm_node) Ord_List.T,
     proof: proof}
  type oracle = (string * Position.T) * term option
  type thm = serial * thm_node
  exception MIN_PROOF of unit
  val proof_of: proof_body -> proof
  val join_proof: proof_body future -> proof
  val map_proof_of: (proof -> proof) -> proof_body -> proof_body
  val thm_header: serial -> Position.T list -> string -> string -> term -> typ list option ->
    thm_header
  val thm_body: proof_body -> thm_body
  val thm_body_proof_raw: thm_body -> proof
  val thm_body_proof_open: thm_body -> proof
  val thm_node_theory_name: thm_node -> string
  val thm_node_name: thm_node -> string
  val thm_node_prop: thm_node -> term
  val thm_node_body: thm_node -> proof_body future
  val thm_node_thms: thm_node -> thm list
  val join_thms: thm list -> proof_body list
  val make_thm: thm_header -> thm_body -> thm
  val fold_proof_atoms: bool -> (proof -> 'a -> 'a) -> proof list -> 'a -> 'a
  val fold_body_thms:
    ({serial: serial, name: string, prop: term, body: proof_body} -> 'a -> 'a) ->
    proof_body list -> 'a -> 'a
  val oracle_ord: oracle ord
  val thm_ord: thm ord
  val unions_oracles: oracle Ord_List.T list -> oracle Ord_List.T
  val unions_thms: thm Ord_List.T list -> thm Ord_List.T
  val no_proof_body: proof -> proof_body
  val no_thm_names: proof -> proof
  val no_thm_proofs: proof -> proof
  val no_body_proofs: proof -> proof

  val encode: Consts.T -> proof XML.Encode.T
  val encode_body: Consts.T -> proof_body XML.Encode.T
  val encode_standard_term: Consts.T -> term XML.Encode.T
  val encode_standard_proof: Consts.T -> proof XML.Encode.T
  val decode: Consts.T -> proof XML.Decode.T
  val decode_body: Consts.T -> proof_body XML.Decode.T

  val %> : proof * term -> proof

  (*primitive operations*)
  val proofs: int Unsynchronized.ref
  val proofs_enabled: unit -> bool
  val atomic_proof: proof -> bool
  val compact_proof: proof -> 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_body: 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) -> proof -> 'a -> 'a
  val fold_proof_terms_types: (term -> 'a -> 'a) -> (typ -> 'a -> 'a) -> proof -> 'a -> 'a
  val maxidx_proof: proof -> int -> int
  val size_of_proof: proof -> int
  val change_types: 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 trivial_proof: proof
  val implies_intr_proof: term -> proof -> proof
  val implies_intr_proof': term -> proof -> proof
  val forall_intr_proof: string * term -> typ option -> 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 -> (string * typ) list -> term list -> int -> proof -> proof
  val permute_prems_proof: term list -> int -> int -> proof -> proof
  val generalize_proof: string list * string list -> int -> term -> 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 option -> term list ->
    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: proof
  val symmetric_proof: proof -> proof
  val transitive_proof: typ -> term -> proof -> proof -> proof
  val equal_intr_proof: term -> term -> proof -> proof -> proof
  val equal_elim_proof: term -> term -> proof -> proof -> proof
  val abstract_rule_proof: string * term -> proof -> proof
  val combination_proof: term -> term -> term -> term -> proof -> proof -> proof
  val strip_shyps_proof: Sorts.algebra -> (typ * sort) list -> (typ * sort) list ->
    sort list -> proof -> proof
  val of_sort_proof: Sorts.algebra ->
    (class * class -> proof) ->
    (string * class list list * class -> proof) ->
    (typ * class -> proof) -> typ * sort -> proof list
  val axm_proof: string -> term -> proof
  val oracle_proof: string -> term -> proof
  val shrink_proof: proof -> 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 set_preproc: (theory -> proof -> proof) -> theory -> theory
  val apply_preproc: theory -> proof -> proof
  val forall_intr_variables_term: term -> term
  val forall_intr_variables: term -> proof -> proof
  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 reconstruct_proof: theory -> term -> proof -> proof
  val prop_of': term list -> proof -> term
  val prop_of: proof -> term
  val expand_name_empty: thm_header -> string option
  val expand_proof: theory -> (thm_header -> string option) -> proof -> proof

  val standard_vars: Name.context -> term * proof option -> term * proof option
  val standard_vars_term: Name.context -> term -> term
  val add_standard_vars: proof -> (string * typ) list -> (string * typ) list
  val add_standard_vars_term: term -> (string * typ) list -> (string * typ) list

  val export_enabled: unit -> bool
  val export_standard_enabled: unit -> bool
  val export_proof_boxes_required: theory -> bool
  val export_proof_boxes: proof_body list -> unit
  val fulfill_norm_proof: theory -> (serial * proof_body) list -> proof_body -> proof_body
  val thm_proof: theory -> (class * class -> proof) ->
    (string * class list list * class -> proof) -> string * Position.T -> sort list ->
    term list -> term -> (serial * proof_body future) list -> proof_body -> thm * proof
  val unconstrain_thm_proof: theory -> (class * class -> proof) ->
    (string * class list list * class -> proof) -> sort list -> term ->
    (serial * proof_body future) list -> proof_body -> thm * proof
  val get_identity: sort list -> term list -> term -> proof ->
    {serial: serial, theory_name: string, name: string} option
  val get_approximative_name: sort list -> term list -> term -> proof -> string
  type thm_id = {serial: serial, theory_name: string}
  val make_thm_id: serial * string -> thm_id
  val thm_header_id: thm_header -> thm_id
  val thm_id: thm -> thm_id
  val get_id: sort list -> term list -> term -> proof -> thm_id option
  val this_id: thm_id option -> thm_id -> bool
  val proof_boxes: {excluded: thm_id -> bool, included: thm_id -> bool} ->
    proof list -> (thm_header * proof) list  (*exception MIN_PROOF*)
end

structure Proofterm : PROOFTERM =
struct

(** datatype proof **)

type thm_header =
  {serial: serial, pos: Position.T list, theory_name: string, name: string,
    prop: term, types: typ list option};

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
 | PThm of thm_header * thm_body
and proof_body = PBody of
  {oracles: ((string * Position.T) * term option) Ord_List.T,
   thms: (serial * thm_node) Ord_List.T,
   proof: proof}
and thm_body =
  Thm_Body of {open_proof: proof -> proof, body: proof_body future}
and thm_node =
  Thm_Node of {theory_name: string, name: string, prop: term,
    body: proof_body future, export: unit lazy, consolidate: unit lazy};

type oracle = (string * Position.T) * term option;
val oracle_ord: oracle ord =
  prod_ord (prod_ord fast_string_ord Position.ord) (option_ord Term_Ord.fast_term_ord);

type thm = serial * thm_node;
val thm_ord: thm ord = fn ((i, _), (j, _)) => int_ord (j, i);


exception MIN_PROOF of unit;

fun proof_of (PBody {proof, ...}) = proof;
val join_proof = Future.join #> proof_of;

fun map_proof_of f (PBody {oracles, thms, proof}) =
  PBody {oracles = oracles, thms = thms, proof = f proof};

fun thm_header serial pos theory_name name prop types : thm_header =
  {serial = serial, pos = pos, theory_name = theory_name, name = name, prop = prop, types = types};

fun thm_body body = Thm_Body {open_proof = I, body = Future.value body};
fun thm_body_proof_raw (Thm_Body {body, ...}) = join_proof body;
fun thm_body_proof_open (Thm_Body {open_proof, body, ...}) = open_proof (join_proof body);

fun rep_thm_node (Thm_Node args) = args;
val thm_node_theory_name = #theory_name o rep_thm_node;
val thm_node_name = #name o rep_thm_node;
val thm_node_prop = #prop o rep_thm_node;
val thm_node_body = #body o rep_thm_node;
val thm_node_thms = thm_node_body #> Future.join #> (fn PBody {thms, ...} => thms);
val thm_node_export = #export o rep_thm_node;
val thm_node_consolidate = #consolidate o rep_thm_node;

fun join_thms (thms: thm list) =
  Future.joins (map (thm_node_body o #2) thms);

val consolidate_bodies =
  maps (fn PBody {thms, ...} => map (thm_node_consolidate o #2) thms)
  #> Lazy.consolidate #> map Lazy.force #> ignore;

fun make_thm_node theory_name name prop body export =
  let
    val consolidate =
      Lazy.lazy_name "Proofterm.make_thm_node" (fn () =>
        let val PBody {thms, ...} = Future.join body
        in consolidate_bodies (join_thms thms) end);
  in
    Thm_Node {theory_name = theory_name, name = name, prop = prop, body = body,
      export = export, consolidate = consolidate}
  end;

val no_export = Lazy.value ();

fun make_thm ({serial, theory_name, name, prop, ...}: thm_header) (Thm_Body {body, ...}) =
  (serial, make_thm_node theory_name name prop body no_export);


(* 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 ({serial = i, ...}, Thm_Body {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, ...}) =
      tap join_thms thms |> fold (fn (i, thm_node) => fn (x, seen) =>
        if Inttab.defined seen i then (x, seen)
        else
          let
            val name = thm_node_name thm_node;
            val prop = thm_node_prop thm_node;
            val body = Future.join (thm_node_body thm_node);
            val (x', seen') = app body (x, Inttab.update (i, ()) seen);
          in (f {serial = i, name = name, prop = prop, body = body} x', seen') end);
  in fn bodies => fn x => #1 (fold app bodies (x, Inttab.empty)) end;


(* proof body *)

val unions_oracles = Ord_List.unions oracle_ord;
val unions_thms = Ord_List.unions thm_ord;

fun no_proof_body proof = PBody {oracles = [], thms = [], proof = proof};
val no_thm_body = thm_body (no_proof_body MinProof);

fun no_thm_names (Abst (x, T, prf)) = Abst (x, T, no_thm_names prf)
  | no_thm_names (AbsP (x, t, prf)) = AbsP (x, t, no_thm_names prf)
  | no_thm_names (prf % t) = no_thm_names prf % t
  | no_thm_names (prf1 %% prf2) = no_thm_names prf1 %% no_thm_names prf2
  | no_thm_names (PThm ({serial, pos, theory_name, name = _, prop, types}, thm_body)) =
      PThm (thm_header serial pos theory_name "" prop types, thm_body)
  | no_thm_names a = a;

fun no_thm_proofs (Abst (x, T, prf)) = Abst (x, T, no_thm_proofs prf)
  | no_thm_proofs (AbsP (x, t, prf)) = AbsP (x, t, no_thm_proofs prf)
  | no_thm_proofs (prf % t) = no_thm_proofs prf % t
  | no_thm_proofs (prf1 %% prf2) = no_thm_proofs prf1 %% no_thm_proofs prf2
  | no_thm_proofs (PThm (header, _)) = PThm (header, no_thm_body)
  | no_thm_proofs a = a;

fun no_body_proofs (Abst (x, T, prf)) = Abst (x, T, no_body_proofs prf)
  | no_body_proofs (AbsP (x, t, prf)) = AbsP (x, t, no_body_proofs prf)
  | no_body_proofs (prf % t) = no_body_proofs prf % t
  | no_body_proofs (prf1 %% prf2) = no_body_proofs prf1 %% no_body_proofs prf2
  | no_body_proofs (PThm (header, Thm_Body {open_proof, body})) =
      let
        val body' = Future.value (no_proof_body (join_proof body));
        val thm_body' = Thm_Body {open_proof = open_proof, body = body'};
      in PThm (header, thm_body') end
  | no_body_proofs a = a;



(** XML data representation **)

(* encode *)

local

open XML.Encode Term_XML.Encode;

fun proof consts prf = prf |> variant
 [fn MinProof => ([], []),
  fn PBound a => ([], int a),
  fn Abst (a, b, c) => ([a], pair (option typ) (proof consts) (b, c)),
  fn AbsP (a, b, c) => ([a], pair (option (term consts)) (proof consts) (b, c)),
  fn a % b => ([], pair (proof consts) (option (term consts)) (a, b)),
  fn a %% b => ([], pair (proof consts) (proof consts) (a, b)),
  fn Hyp a => ([], term consts a),
  fn PAxm (a, b, c) => ([a], pair (term consts) (option (list typ)) (b, c)),
  fn OfClass (a, b) => ([b], typ a),
  fn Oracle (a, b, c) => ([a], pair (term consts) (option (list typ)) (b, c)),
  fn PThm ({serial, pos, theory_name, name, prop, types}, Thm_Body {open_proof, body, ...}) =>
    ([int_atom serial, theory_name, name],
      pair (list properties) (pair (term consts) (pair (option (list typ)) (proof_body consts)))
        (map Position.properties_of pos, (prop, (types, map_proof_of open_proof (Future.join body)))))]
and proof_body consts (PBody {oracles, thms, proof = prf}) =
  triple (list (pair (pair string (properties o Position.properties_of))
      (option (term consts)))) (list (thm consts)) (proof consts) (oracles, thms, prf)
and thm consts (a, thm_node) =
  pair int (pair string (pair string (pair (term consts) (proof_body consts))))
    (a, (thm_node_theory_name thm_node, (thm_node_name thm_node, (thm_node_prop thm_node,
      (Future.join (thm_node_body thm_node))))));

fun standard_term consts t = t |> variant
 [fn Const (a, b) => ([a], list typ (Consts.typargs consts (a, b))),
  fn Free (a, _) => ([a], []),
  fn Var (a, _) => (indexname a, []),
  fn Bound a => ([], int a),
  fn Abs (a, b, c) => ([a], pair typ (standard_term consts) (b, c)),
  fn op $ a => ([], pair (standard_term consts) (standard_term consts) a)];

fun standard_proof consts prf = prf |> variant
 [fn MinProof => ([], []),
  fn PBound a => ([], int a),
  fn Abst (a, SOME b, c) => ([a], pair typ (standard_proof consts) (b, c)),
  fn AbsP (a, SOME b, c) => ([a], pair (standard_term consts) (standard_proof consts) (b, c)),
  fn a % SOME b => ([], pair (standard_proof consts) (standard_term consts) (a, b)),
  fn a %% b => ([], pair (standard_proof consts) (standard_proof consts) (a, b)),
  fn Hyp a => ([], standard_term consts a),
  fn PAxm (name, _, SOME Ts) => ([name], list typ Ts),
  fn OfClass (T, c) => ([c], typ T),
  fn Oracle (name, prop, SOME Ts) => ([name], pair (standard_term consts) (list typ) (prop, Ts)),
  fn PThm ({serial, theory_name, name, types = SOME Ts, ...}, _) =>
    ([int_atom serial, theory_name, name], list typ Ts)];

in

val encode = proof;
val encode_body = proof_body;
val encode_standard_term = standard_term;
val encode_standard_proof = standard_proof;

end;


(* decode *)

local

open XML.Decode Term_XML.Decode;

fun proof consts prf = prf |> variant
 [fn ([], []) => MinProof,
  fn ([], a) => PBound (int a),
  fn ([a], b) => let val (c, d) = pair (option typ) (proof consts) b in Abst (a, c, d) end,
  fn ([a], b) => let val (c, d) = pair (option (term consts)) (proof consts) b in AbsP (a, c, d) end,
  fn ([], a) => op % (pair (proof consts) (option (term consts)) a),
  fn ([], a) => op %% (pair (proof consts) (proof consts) a),
  fn ([], a) => Hyp (term consts a),
  fn ([a], b) => let val (c, d) = pair (term consts) (option (list typ)) b in PAxm (a, c, d) end,
  fn ([b], a) => OfClass (typ a, b),
  fn ([a], b) => let val (c, d) = pair (term consts) (option (list typ)) b in Oracle (a, c, d) end,
  fn ([a, b, c], d) =>
    let
      val ((e, (f, (g, h)))) =
        pair (list properties) (pair (term consts) (pair (option (list typ)) (proof_body consts))) d;
      val header = thm_header (int_atom a) (map Position.of_properties e) b c f g;
    in PThm (header, thm_body h) end]
and proof_body consts x =
  let
    val (a, b, c) =
      triple (list (pair (pair string (Position.of_properties o properties))
        (option (term consts)))) (list (thm consts)) (proof consts) x;
  in PBody {oracles = a, thms = b, proof = c} end
and thm consts x =
  let
    val (a, (b, (c, (d, e)))) =
      pair int (pair string (pair string (pair (term consts) (proof_body consts)))) x
  in (a, make_thm_node b c d (Future.value e) no_export) end;

in

val decode = proof;
val decode_body = proof_body;

end;


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

val proofs = Unsynchronized.ref 2;
fun proofs_enabled () = ! proofs >= 2;

fun atomic_proof prf =
  (case prf of
    Abst _ => false
  | AbsP _ => false
  | op % _ => false
  | op %% _ => false
  | MinProof => false
  | _ => true);

fun compact_proof (prf % _) = compact_proof prf
  | compact_proof (prf1 %% prf2) = atomic_proof prf2 andalso compact_proof prf1
  | compact_proof prf = atomic_proof prf;

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 (body as PBody {proof, ...}) =
  (case fst (strip_combt (fst (strip_combP proof))) of
    PThm (_, Thm_Body {body = body', ...}) => Future.join body'
  | _ => body);

val mk_Abst = fold_rev (fn (x, _: typ) => fn prf => Abst (x, NONE, prf));
val mk_AbsP = fold_rev (fn _: term => fn prf => AbsP ("H", NONE, 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 (PThm ({serial, pos, theory_name, name, prop, types = SOME Ts}, thm_body)) =
          PThm (thm_header serial pos theory_name name prop (SOME (typs Ts)), thm_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 (Abst (_, _, prf)) = fold_proof_terms f prf
  | fold_proof_terms f (AbsP (_, SOME t, prf)) = f t #> fold_proof_terms f prf
  | fold_proof_terms f (AbsP (_, NONE, prf)) = fold_proof_terms f prf
  | fold_proof_terms f (prf % SOME t) = fold_proof_terms f prf #> f t
  | fold_proof_terms f (prf % NONE) = fold_proof_terms f prf
  | fold_proof_terms f (prf1 %% prf2) = fold_proof_terms f prf1 #> fold_proof_terms f prf2
  | fold_proof_terms _ _ = I;

fun fold_proof_terms_types f g (Abst (_, SOME T, prf)) = g T #> fold_proof_terms_types f g prf
  | fold_proof_terms_types f g (Abst (_, NONE, prf)) = fold_proof_terms_types f g prf
  | fold_proof_terms_types f g (AbsP (_, SOME t, prf)) = f t #> fold_proof_terms_types f g prf
  | fold_proof_terms_types f g (AbsP (_, NONE, prf)) = fold_proof_terms_types f g prf
  | fold_proof_terms_types f g (prf % SOME t) = fold_proof_terms_types f g prf #> f t
  | fold_proof_terms_types f g (prf % NONE) = fold_proof_terms_types f g prf
  | fold_proof_terms_types f g (prf1 %% prf2) =
      fold_proof_terms_types f g prf1 #> fold_proof_terms_types f g prf2
  | fold_proof_terms_types _ g (PAxm (_, _, SOME Ts)) = fold g Ts
  | fold_proof_terms_types _ g (OfClass (T, _)) = g T
  | fold_proof_terms_types _ g (Oracle (_, _, SOME Ts)) = fold g Ts
  | fold_proof_terms_types _ g (PThm ({types = SOME Ts, ...}, _)) = fold g Ts
  | fold_proof_terms_types _ _ _ = I;

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

fun size_of_proof (Abst (_, _, prf)) = 1 + size_of_proof prf
  | size_of_proof (AbsP (_, _, 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_types types (PAxm (name, prop, _)) = PAxm (name, prop, types)
  | change_types (SOME [T]) (OfClass (_, c)) = OfClass (T, c)
  | change_types types (Oracle (name, prop, _)) = Oracle (name, prop, types)
  | change_types types (PThm ({serial, pos, theory_name, name, prop, types = _}, thm_body)) =
      PThm (thm_header serial pos theory_name name prop types, thm_body)
  | change_types _ 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 _ Plev _ (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, _)) _ = 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 _ (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, Logic.dummy_tfree S)) (Term.add_tvarsT 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, Logic.dummy_tfree S)) (Term.add_tvars t []),
   map_filter (fn (ixnT as (_, T)) =>
     (Envir.lookup env ixnT; NONE) handle TYPE _ =>
        SOME (ixnT, Free ("dummy", T))) (Term.add_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 (PThm ({serial = i, pos = p, theory_name, name = a, prop = t, types = Ts}, thm_body)) =
          PThm (thm_header i p theory_name a t
            (Same.map_option (htypeTs Envir.norm_types_same) Ts), thm_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  _ _ _ = 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 *)

local

fun frzT names =
  map_type_tvar (fn (ixn, S) => TFree (the (AList.lookup (op =) names ixn), S));

fun thawT names =
  map_type_tfree (fn (a, S) =>
    (case AList.lookup (op =) names a of
      NONE => TFree (a, S)
    | SOME ixn => TVar (ixn, S)));

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' (Const (s, T)) = Const (s, frzT names' T)
  | freeze _ names' (Free (s, T)) = Free (s, frzT names' T)
  | freeze names names' (Var (ixn, T)) =
      Free (the (AList.lookup (op =) names ixn), frzT names' T)
  | freeze _ _ 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' (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' (Var (ixn, T)) = Var (ixn, thawT names' T)
  | thaw _ _ t = t;

in

fun freeze_thaw_prf prf =
  let
    val (fs, Tfs, vs, Tvs) = fold_proof_terms_types
      (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;

end;



(** inference rules **)

(* trivial implication *)

val trivial_proof = AbsP ("H", NONE, PBound 0);


(* 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 (a, v) opt_T prf = Abst (a, opt_T, prf_abstract_over v prf);

fun forall_intr_proof' v prf =
  let val (a, T) = (case v of Var ((a, _), T) => (a, T) | Free (a, T) => (a, T))
  in forall_intr_proof (a, v) (SOME 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 (a, S) =
      (case AList.lookup (op =) fmap (a, S) of
        NONE => TFree (a, S)
      | 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 = Term.add_tfree_names t [];
    val (alist, _) = fold_rev new_name (map #1 (Term.add_tvars t [])) ([], used);
  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' params asms m prf =
  let
    val i = length asms;
    val j = length Bs;
  in
    mk_AbsP (Bs @ [Bi']) (proof_combP (prf, map PBound
      (j downto 1) @ [mk_Abst params (mk_AbsP asms
        (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 prems'
      (proof_combP (prf, map PBound ((n-1 downto n-j) @ (k-1 downto 0) @ (n-j-1 downto k))))
  end;


(* generalization *)

fun generalize_proof (tfrees, frees) idx prop prf =
  let
    val gen =
      if null frees then []
      else
        fold_aterms (fn Free (x, T) => member (op =) frees x ? insert (op =) (x, T) | _ => I)
          (Term_Subst.generalize (tfrees, []) idx prop) [];
  in
    prf
    |> Same.commit (map_proof_terms_same
        (Term_Subst.generalize_same (tfrees, []) idx)
        (Term_Subst.generalizeT_same tfrees idx))
    |> fold (fn (x, T) => forall_intr_proof (x, Free (x, T)) NONE) gen
    |> fold_rev (fn (x, T) => fn prf' => prf' %> Var (Name.clean_index (x, idx), T)) gen
  end;


(* 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' _ _ (PThm ({serial = i, pos = p, theory_name, name = s, prop, types = Ts}, thm_body)) =
          PThm (thm_header i p theory_name s prop
            ((Same.map_option o Same.map) (Logic.incr_tvar inc) Ts), thm_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 ("Pure.imp", _) $ A $ B) =
            AbsP ("H", NONE (*A*), lift Us (true::bs) (i+1) j B)
      | lift Us bs i j (Const ("Pure.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 ps (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 ("Pure.imp", _) $ A $ B) i m = AbsP ("H", NONE (*A*), imp_prf B (i+1) (m-1))
      | imp_prf _ i _ = PBound i;
    fun all_prf (Const ("Pure.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 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 ("Pure.imp", _) $ A $ B, k) =
      AbsP ("H", NONE (*A*), flatten_params_proof (i+1) j n (B, k))
  | flatten_params_proof i j n (Const ("Pure.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 As A oldAs n m rprf sprf =
  let
    val lb = length Bs;
    val la = length As;
  in
    mk_AbsP (Bs @ As) (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;



(** type classes **)

fun strip_shyps_proof algebra present witnessed extra_sorts prf =
  let
    fun get S (T', S') = if Sorts.sort_le algebra (S', S) then SOME T' else NONE;
    val extra = map (`Logic.dummy_tfree) 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;

fun of_sort_proof algebra classrel_proof arity_proof hyps =
  Sorts.of_sort_derivation algebra
   {class_relation = fn _ => fn _ => fn (prf, c1) => fn c2 =>
      if c1 = c2 then prf else classrel_proof (c1, c2) %% prf,
    type_constructor = fn (a, _) => fn dom => fn c =>
      let val Ss = map (map snd) dom and prfs = maps (map fst) dom
      in proof_combP (arity_proof (a, Ss, c), prfs) end,
    type_variable = fn typ => map (fn c => (hyps (typ, c), c)) (Type.sort_of_atyp typ)};



(** axioms and theorems **)

val add_type_variables = (fold_types o fold_atyps) (insert (op =));
fun type_variables_of t = rev (add_type_variables t []);

val add_variables = fold_aterms (fn a => (is_Var a orelse is_Free a) ? insert (op =) a);
fun variables_of t = rev (add_variables 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 vars_of t = map Var (rev (Term.add_vars t []));

fun add_funvars Ts (vs, t) =
  if is_fun (fastype_of1 (Ts, t)) then
    union (op =) vs (map_filter (fn Var (ixn, T) =>
      if is_fun T then SOME ixn else NONE | _ => NONE) (vars_of t))
  else vs;

fun add_npvars q p Ts (vs, Const ("Pure.imp", _) $ t $ u) =
      add_npvars q p Ts (add_npvars q (not p) Ts (vs, t), u)
  | add_npvars q p Ts (vs, Const ("Pure.all", Type (_, [Type (_, [T, _]), _])) $ t) =
      add_npvars q p Ts (vs, if p andalso q then betapply (t, Var (("",0), T)) else t)
  | add_npvars q p Ts (vs, Abs (_, T, t)) = add_npvars q p (T::Ts) (vs, t)
  | add_npvars _ _ Ts (vs, t) = add_npvars' Ts (vs, t)
and add_npvars' Ts (vs, t) =
  (case strip_comb t of
    (Var (ixn, _), ts) => if test_args [] ts then vs
      else Library.foldl (add_npvars' Ts)
        (AList.update (op =) (ixn,
          Library.foldl (add_funvars Ts) ((these ooo AList.lookup) (op =) vs ixn, ts)) vs, ts)
  | (Abs (_, T, u), ts) => Library.foldl (add_npvars' (T::Ts)) (vs, u :: ts)
  | (_, ts) => Library.foldl (add_npvars' Ts) (vs, ts));

fun prop_vars (Const ("Pure.imp", _) $ P $ Q) = union (op =) (prop_vars P) (prop_vars Q)
  | prop_vars (Const ("Pure.all", _) $ Abs (_, _, t)) = prop_vars t
  | prop_vars t = (case strip_comb t of (Var (ixn, _), _) => [ixn] | _ => []);

fun is_proj t =
  let
    fun is_p i t =
      (case strip_comb t of
        (Bound _, []) => false
      | (Bound j, ts) => j >= i orelse exists (is_p i) ts
      | (Abs (_, _, u), _) => is_p (i+1) u
      | (_, ts) => exists (is_p i) ts)
  in
    (case strip_abs_body t of
      Bound _ => true
    | t' => is_p 0 t')
  end;

fun prop_args prop =
  let
    val needed_vars =
      union (op =) (Library.foldl (uncurry (union (op =)))
        ([], map (uncurry (insert (op =))) (add_npvars true true [] ([], prop))))
      (prop_vars prop);
  in
    variables_of prop |> map
      (fn var as Var (ixn, _) => if member (op =) needed_vars ixn then SOME var else NONE
        | free => SOME free)
  end;

fun const_proof mk name prop =
  let
    val args = prop_args prop;
    val ({outer_constraints, ...}, prop1) = Logic.unconstrainT [] prop;
    val head = mk (name, prop1, NONE);
  in proof_combP (proof_combt' (head, args), map OfClass outer_constraints) end;

val axm_proof = const_proof PAxm;
val oracle_proof = const_proof Oracle;

val shrink_proof =
  let
    fun shrink ls lev (prf as Abst (a, T, body)) =
          let val (b, is, ch, body') = shrink ls (lev+1) body
          in (b, is, ch, if ch then Abst (a, T, body') else prf) end
      | shrink ls lev (prf as AbsP (a, t, body)) =
          let val (b, is, ch, body') = shrink (lev::ls) lev body
          in (b orelse member (op =) is 0, map_filter (fn 0 => NONE | i => SOME (i-1)) is,
            ch, if ch then AbsP (a, t, body') else prf)
          end
      | shrink ls lev prf =
          let val (is, ch, _, prf') = shrink' ls lev [] [] prf
          in (false, is, ch, prf') end
    and shrink' ls lev ts prfs (prf as prf1 %% prf2) =
          let
            val p as (_, is', ch', prf') = shrink ls lev prf2;
            val (is, ch, ts', prf'') = shrink' ls lev ts (p::prfs) prf1
          in (union (op =) is is', ch orelse ch', ts',
              if ch orelse ch' then prf'' %% prf' else prf)
          end
      | shrink' ls lev ts prfs (prf as prf1 % t) =
          let val (is, ch, (ch', t')::ts', prf') = shrink' ls lev (t::ts) prfs prf1
          in (is, ch orelse ch', ts',
              if ch orelse ch' then prf' % t' else prf) end
      | shrink' ls lev ts prfs (prf as PBound i) =
          (if exists (fn SOME (Bound j) => lev-j <= nth ls i | _ => true) ts
             orelse has_duplicates (op =)
               (Library.foldl (fn (js, SOME (Bound j)) => j :: js | (js, _) => js) ([], ts))
             orelse exists #1 prfs then [i] else [], false, map (pair false) ts, prf)
      | shrink' _ _ ts _ (Hyp t) = ([], false, map (pair false) ts, Hyp t)
      | shrink' _ _ ts _ (prf as MinProof) = ([], false, map (pair false) ts, prf)
      | shrink' _ _ ts _ (prf as OfClass _) = ([], false, map (pair false) ts, prf)
      | shrink' _ _ ts prfs prf =
          let
            val prop =
              (case prf of
                PAxm (_, prop, _) => prop
              | Oracle (_, prop, _) => prop
              | PThm ({prop, ...}, _) => prop
              | _ => raise Fail "shrink: proof not in normal form");
            val vs = vars_of prop;
            val (ts', ts'') = chop (length vs) ts;
            val insts = take (length ts') (map (fst o dest_Var) vs) ~~ ts';
            val nvs = Library.foldl (fn (ixns', (ixn, ixns)) =>
              insert (op =) ixn
                (case AList.lookup (op =) insts ixn of
                  SOME (SOME t) => if is_proj t then union (op =) ixns ixns' else ixns'
                | _ => union (op =) ixns ixns'))
                  (needed prop ts'' prfs, add_npvars false true [] ([], prop));
            val insts' = map
              (fn (ixn, x as SOME _) => if member (op =) nvs ixn then (false, x) else (true, NONE)
                | (_, x) => (false, x)) insts
          in ([], false, insts' @ map (pair false) ts'', prf) end
    and needed (Const ("Pure.imp", _) $ t $ u) ts ((b, _, _, _)::prfs) =
          union (op =) (if b then map (fst o dest_Var) (vars_of t) else []) (needed u ts prfs)
      | needed (Var (ixn, _)) (_::_) _ = [ixn]
      | needed _ _ _ = [];
  in fn prf => #4 (shrink [] 0 prf) 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_proof = reflexive_axm % NONE;

val is_reflexive_proof = fn PAxm ("Pure.reflexive", _, _) % _ => true | _ => false;

fun symmetric_proof prf =
  if is_reflexive_proof prf then prf
  else symmetric_axm % NONE % NONE %% prf;

fun transitive_proof U u prf1 prf2 =
  if is_reflexive_proof prf1 then prf2
  else if is_reflexive_proof prf2 then prf1
  else if U = propT then transitive_axm % NONE % SOME (remove_types u) % NONE %% prf1 %% prf2
  else transitive_axm % NONE % NONE % NONE %% prf1 %% prf2;

fun equal_intr_proof A B prf1 prf2 =