src/Pure/Proof/proof_syntax.ML

eliminated polymorphic equality;

(*  Title:      Pure/Proof/proof_syntax.ML
    ID:         $Id$
    Author:     Stefan Berghofer, TU Muenchen

Function for parsing and printing proof terms.
*)

signature PROOF_SYNTAX =
sig
  val proofT: typ
  val add_proof_syntax: theory -> theory
  val disambiguate_names: theory -> Proofterm.proof ->
    Proofterm.proof * Proofterm.proof Symtab.table
  val proof_of_term: theory -> Proofterm.proof Symtab.table ->
    bool -> term -> Proofterm.proof
  val term_of_proof: Proofterm.proof -> term
  val cterm_of_proof: theory -> Proofterm.proof -> cterm * (cterm -> Proofterm.proof)
  val read_term: theory -> typ -> string -> term
  val read_proof: theory -> bool -> string -> Proofterm.proof
  val proof_syntax: Proofterm.proof -> theory -> theory
  val proof_of: bool -> thm -> Proofterm.proof
  val pretty_proof: Proof.context -> Proofterm.proof -> Pretty.T
  val pretty_proof_of: Proof.context -> bool -> thm -> Pretty.T
end;

structure ProofSyntax : PROOF_SYNTAX =
struct

open Proofterm;

(**** add special syntax for embedding proof terms ****)

val proofT = Type ("proof", []);
val paramT = Type ("param", []);
val paramsT = Type ("params", []);
val idtT = Type ("idt", []);
val aT = TFree (Name.aT, []);

(** constants for theorems and axioms **)

fun add_proof_atom_consts names thy =
  thy
  |> Sign.absolute_path
  |> Sign.add_consts_i (map (fn name => (name, proofT, NoSyn)) names);

(** constants for application and abstraction **)

fun add_proof_syntax thy =
  thy
  |> Theory.copy
  |> Sign.root_path
  |> Sign.add_defsort_i []
  |> Sign.add_types [("proof", 0, NoSyn)]
  |> Sign.add_consts_i
      [("Appt", [proofT, aT] ---> proofT, Mixfix ("(1_ %/ _)", [4, 5], 4)),
       ("AppP", [proofT, proofT] ---> proofT, Mixfix ("(1_ %%/ _)", [4, 5], 4)),
       ("Abst", (aT --> proofT) --> proofT, NoSyn),
       ("AbsP", [propT, proofT --> proofT] ---> proofT, NoSyn),
       ("Hyp", propT --> proofT, NoSyn),
       ("Oracle", propT --> proofT, NoSyn),
       ("MinProof", proofT, Delimfix "?")]
  |> Sign.add_nonterminals ["param", "params"]
  |> Sign.add_syntax_i
      [("_Lam", [paramsT, proofT] ---> proofT, Mixfix ("(1Lam _./ _)", [0, 3], 3)),
       ("_Lam0", [paramT, paramsT] ---> paramsT, Mixfix ("_/ _", [1, 0], 0)),
       ("_Lam0", [idtT, paramsT] ---> paramsT, Mixfix ("_/ _", [1, 0], 0)),
       ("_Lam1", [idtT, propT] ---> paramT, Mixfix ("_: _", [0, 0], 0)),
       ("", paramT --> paramT, Delimfix "'(_')"),
       ("", idtT --> paramsT, Delimfix "_"),
       ("", paramT --> paramsT, Delimfix "_")]
  |> Sign.add_modesyntax_i ("xsymbols", true)
      [("_Lam", [paramsT, proofT] ---> proofT, Mixfix ("(1\\<Lambda>_./ _)", [0, 3], 3)),
       ("Appt", [proofT, aT] ---> proofT, Mixfix ("(1_ \\<cdot>/ _)", [4, 5], 4)),
       ("AppP", [proofT, proofT] ---> proofT, Mixfix ("(1_ \\<bullet>/ _)", [4, 5], 4))]
  |> Sign.add_modesyntax_i ("latex", false)
      [("_Lam", [paramsT, proofT] ---> proofT, Mixfix ("(1\\<^bold>\\<lambda>_./ _)", [0, 3], 3))]
  |> Sign.add_trrules_i (map Syntax.ParsePrintRule
      [(Syntax.mk_appl (Constant "_Lam")
          [Syntax.mk_appl (Constant "_Lam0") [Variable "l", Variable "m"], Variable "A"],
        Syntax.mk_appl (Constant "_Lam")
          [Variable "l", Syntax.mk_appl (Constant "_Lam") [Variable "m", Variable "A"]]),
       (Syntax.mk_appl (Constant "_Lam")
          [Syntax.mk_appl (Constant "_Lam1") [Variable "x", Variable "A"], Variable "B"],
        Syntax.mk_appl (Constant "AbsP") [Variable "A",
          (Syntax.mk_appl (Constant "_abs") [Variable "x", Variable "B"])]),
       (Syntax.mk_appl (Constant "_Lam") [Variable "x", Variable "A"],
        Syntax.mk_appl (Constant "Abst")
          [(Syntax.mk_appl (Constant "_abs") [Variable "x", Variable "A"])])]);


(**** create unambiguous theorem names ****)

fun disambiguate_names thy prf =
  let
    val thms = thms_of_proof prf Symtab.empty;
    val thms' = map (apsnd Thm.full_prop_of) (PureThy.all_thms_of thy);

    val tab = Symtab.fold (fn (key, ps) => fn tab =>
      let val prop = the_default (Bound 0) (AList.lookup (op =) thms' key)
      in fst (fold_rev (fn (prop', prf) => fn x as (tab, i) =>
        if prop <> prop' then
          (Symtab.update (key ^ "_" ^ string_of_int i, prf) tab, i+1)
        else x) ps (tab, 1))
      end) thms Symtab.empty;

    fun rename (Abst (s, T, prf)) = Abst (s, T, rename prf)
      | rename (AbsP (s, t, prf)) = AbsP (s, t, rename prf)
      | rename (prf1 %% prf2) = rename prf1 %% rename prf2
      | rename (prf % t) = rename prf % t
      | rename (prf' as PThm (s, prf, prop, Ts)) =
          let
            val prop' = the_default (Bound 0) (AList.lookup (op =) thms' s);
            val ps = remove (op =) prop' (map fst (the (Symtab.lookup thms s)))
          in if prop = prop' then prf' else
            PThm (s ^ "_" ^ string_of_int (length ps - find_index (fn p => p = prop) ps),
              prf, prop, Ts)
          end
      | rename prf = prf

  in (rename prf, tab) end;


(**** translation between proof terms and pure terms ****)

fun proof_of_term thy tab ty =
  let
    val thms = PureThy.all_thms_of thy;
    val axms = Theory.all_axioms_of thy;

    fun mk_term t = (if ty then I else map_types (K dummyT))
      (Term.no_dummy_patterns t);

    fun prf_of [] (Bound i) = PBound i
      | prf_of Ts (Const (s, Type ("proof", _))) =
          change_type (if ty then SOME Ts else NONE)
            (case NameSpace.explode s of
               "axm" :: xs =>
                 let
                   val name = NameSpace.implode xs;
                   val prop = (case AList.lookup (op =) axms name of
                       SOME prop => prop
                     | NONE => error ("Unknown axiom " ^ quote name))
                 in PAxm (name, prop, NONE) end
             | "thm" :: xs =>
                 let val name = NameSpace.implode xs;
                 in (case AList.lookup (op =) thms name of
                     SOME thm => fst (strip_combt (Thm.proof_of thm))
                   | NONE => (case Symtab.lookup tab name of
                         SOME prf => prf
                       | NONE => error ("Unknown theorem " ^ quote name)))
                 end
             | _ => error ("Illegal proof constant name: " ^ quote s))
      | prf_of Ts (Const ("Hyp", _) $ prop) = Hyp prop
      | prf_of Ts (v as Var ((_, Type ("proof", _)))) = Hyp v
      | prf_of [] (Const ("Abst", _) $ Abs (s, T, prf)) =
          if T = proofT then
            error ("Term variable abstraction may not bind proof variable " ^ quote s)
          else Abst (s, if ty then SOME T else NONE,
            incr_pboundvars (~1) 0 (prf_of [] prf))
      | prf_of [] (Const ("AbsP", _) $ t $ Abs (s, _, prf)) =
          AbsP (s, case t of
                Const ("dummy_pattern", _) => NONE
              | _ $ Const ("dummy_pattern", _) => NONE
              | _ => SOME (mk_term t),
            incr_pboundvars 0 (~1) (prf_of [] prf))
      | prf_of [] (Const ("AppP", _) $ prf1 $ prf2) =
          prf_of [] prf1 %% prf_of [] prf2
      | prf_of Ts (Const ("Appt", _) $ prf $ Const ("TYPE", Type (_, [T]))) =
          prf_of (T::Ts) prf
      | prf_of [] (Const ("Appt", _) $ prf $ t) = prf_of [] prf %
          (case t of Const ("dummy_pattern", _) => NONE | _ => SOME (mk_term t))
      | prf_of _ t = error ("Not a proof term:\n" ^
          Syntax.string_of_term_global thy t)

  in prf_of [] end;


val AbsPt = Const ("AbsP", [propT, proofT --> proofT] ---> proofT);
val AppPt = Const ("AppP", [proofT, proofT] ---> proofT);
val Hypt = Const ("Hyp", propT --> proofT);
val Oraclet = Const ("Oracle", propT --> proofT);
val MinProoft = Const ("MinProof", proofT);

val mk_tyapp = fold (fn T => fn prf => Const ("Appt",
  [proofT, Term.itselfT T] ---> proofT) $ prf $ Logic.mk_type T);

fun term_of _ (PThm (name, _, _, NONE)) =
      Const (NameSpace.append "thm" name, proofT)
  | term_of _ (PThm (name, _, _, SOME Ts)) =
      mk_tyapp Ts (Const (NameSpace.append "thm" name, proofT))
  | term_of _ (PAxm (name, _, NONE)) = Const (NameSpace.append "axm" name, proofT)
  | term_of _ (PAxm (name, _, SOME Ts)) =
      mk_tyapp Ts (Const (NameSpace.append "axm" name, proofT))
  | term_of _ (PBound i) = Bound i
  | term_of Ts (Abst (s, opT, prf)) =
      let val T = the_default dummyT opT
      in Const ("Abst", (T --> proofT) --> proofT) $
        Abs (s, T, term_of (T::Ts) (incr_pboundvars 1 0 prf))
      end
  | term_of Ts (AbsP (s, t, prf)) =
      AbsPt $ the_default (Term.dummy_pattern propT) t $
        Abs (s, proofT, term_of (proofT::Ts) (incr_pboundvars 0 1 prf))
  | term_of Ts (prf1 %% prf2) =
      AppPt $ term_of Ts prf1 $ term_of Ts prf2
  | term_of Ts (prf % opt) =
      let val t = the_default (Term.dummy_pattern dummyT) opt
      in Const ("Appt",
        [proofT, fastype_of1 (Ts, t) handle TERM _ => dummyT] ---> proofT) $
          term_of Ts prf $ t
      end
  | term_of Ts (Hyp t) = Hypt $ t
  | term_of Ts (Oracle (_, t, _)) = Oraclet $ t
  | term_of Ts (MinProof _) = MinProoft;

val term_of_proof = term_of [];

fun cterm_of_proof thy prf =
  let
    val (prf', tab) = disambiguate_names thy prf;
    val thm_names = filter_out (fn s => s = "")
      (map fst (PureThy.all_thms_of thy) @ map fst (Symtab.dest tab));
    val axm_names = map fst (Theory.all_axioms_of thy);
    val thy' = thy
      |> add_proof_syntax
      |> add_proof_atom_consts
        (map (NameSpace.append "axm") axm_names @ map (NameSpace.append "thm") thm_names)
  in
    (cterm_of thy' (term_of_proof prf'),
     proof_of_term thy tab true o Thm.term_of)
  end;

fun read_term thy =
  let
    val thm_names = filter_out (fn s => s = "") (map fst (PureThy.all_thms_of thy));
    val axm_names = map fst (Theory.all_axioms_of thy);
    val ctxt = thy
      |> add_proof_syntax
      |> add_proof_atom_consts
        (map (NameSpace.append "axm") axm_names @ map (NameSpace.append "thm") thm_names)
      |> ProofContext.init
      |> ProofContext.allow_dummies
      |> ProofContext.set_mode ProofContext.mode_schematic;
  in
    fn ty => fn s =>
      (if ty = propT then Syntax.parse_prop else Syntax.parse_term) ctxt s
      |> TypeInfer.constrain ty |> Syntax.check_term ctxt
  end;

fun read_proof thy =
  let val rd = read_term thy proofT
  in fn ty => fn s => proof_of_term thy Symtab.empty ty (Logic.varify (rd s)) end;

fun proof_syntax prf =
  let
    val thm_names = filter_out (fn s => s = "")
      (map fst (Symtab.dest (thms_of_proof prf Symtab.empty)));
    val axm_names = map fst (Symtab.dest (axms_of_proof prf Symtab.empty));
  in
    add_proof_syntax #>
    add_proof_atom_consts
      (map (NameSpace.append "thm") thm_names @ map (NameSpace.append "axm") axm_names)
  end;

fun proof_of full thm =
  let
    val thy = Thm.theory_of_thm thm;
    val prop = Thm.full_prop_of thm;
    val prf = Thm.proof_of thm;
    val prf' = (case strip_combt (fst (strip_combP prf)) of
        (PThm (_, prf', prop', _), _) => if prop = prop' then prf' else prf
      | _ => prf)
  in if full then Reconstruct.reconstruct_proof thy prop prf' else prf' end;

fun pretty_proof ctxt prf =
  ProofContext.pretty_term_abbrev
    (ProofContext.transfer_syntax (proof_syntax prf (ProofContext.theory_of ctxt)) ctxt)
    (term_of_proof prf);

fun pretty_proof_of ctxt full th =
  pretty_proof ctxt (proof_of full th);

end;