(*  Title:      HOL/Tools/Sledgehammer/sledgehammer_reconstruct.ML

    Author:     Jasmin Blanchette, TU Muenchen

    Author:     Steffen Juilf Smolka, TU Muenchen

Isar proof reconstruction from ATP proofs.

*)

signature SLEDGEHAMMER_PROOF_RECONSTRUCT =

sig

  type 'a proof = 'a ATP_Proof.proof

  type stature = ATP_Problem_Generate.stature

  datatype reconstructor =

    Metis of string * string |

    SMT

  datatype play =

    Played of reconstructor * Time.time |

    Trust_Playable of reconstructor * Time.time option |

    Failed_to_Play of reconstructor

  type minimize_command = string list -> string

  type one_line_params =

    play * string * (string * stature) list * minimize_command * int * int

  type isar_params =

    bool * bool * Time.time * real * string Symtab.table

    * (string * stature) list vector * int Symtab.table * string proof * thm

  val smtN : string

  val string_for_reconstructor : reconstructor -> string

  val thms_of_name : Proof.context -> string -> thm list

  val lam_trans_from_atp_proof : string proof -> string -> string

  val is_typed_helper_used_in_atp_proof : string proof -> bool

  val used_facts_in_atp_proof :

    Proof.context -> (string * stature) list vector -> string proof ->

    (string * stature) list

  val used_facts_in_unsound_atp_proof :

    Proof.context -> (string * stature) list vector -> 'a proof ->

    string list option

  val one_line_proof_text : int -> one_line_params -> string

  val isar_proof_text :

    Proof.context -> bool -> isar_params -> one_line_params -> string

  val proof_text :

    Proof.context -> bool -> isar_params -> int -> one_line_params -> string

end;

structure Sledgehammer_Reconstruct : SLEDGEHAMMER_PROOF_RECONSTRUCT =

struct

open ATP_Util

open ATP_Problem

open ATP_Proof

open ATP_Problem_Generate

open ATP_Proof_Reconstruct

open Sledgehammer_Util

structure String_Redirect = ATP_Proof_Redirect(

  type key = step_name

  val ord = fn ((s, _ : string list), (s', _)) => fast_string_ord (s, s')

  val string_of = fst)

open String_Redirect

(** reconstructors **)

datatype reconstructor =

  Metis of string * string |

  SMT

datatype play =

  Played of reconstructor * Time.time |

  Trust_Playable of reconstructor * Time.time option |

  Failed_to_Play of reconstructor

val smtN = "smt"

fun string_for_reconstructor (Metis (type_enc, lam_trans)) =

    metis_call type_enc lam_trans

  | string_for_reconstructor SMT = smtN

fun thms_of_name ctxt name =

  let

    val lex = Keyword.get_lexicons

    val get = maps (Proof_Context.get_fact ctxt o fst)

  in

    Source.of_string name

    |> Symbol.source

    |> Token.source {do_recover = SOME false} lex Position.start

    |> Token.source_proper

    |> Source.source Token.stopper (Parse_Spec.xthms1 >> get) NONE

    |> Source.exhaust

  end

(** fact extraction from ATP proofs **)

fun find_first_in_list_vector vec key =

  Vector.foldl (fn (ps, NONE) => AList.lookup (op =) ps key

                 | (_, value) => value) NONE vec

val unprefix_fact_number = space_implode "_" o tl o space_explode "_"

fun resolve_one_named_fact fact_names s =

  case try (unprefix fact_prefix) s of

    SOME s' =>

    let val s' = s' |> unprefix_fact_number |> unascii_of in

      s' |> find_first_in_list_vector fact_names |> Option.map (pair s')

    end

  | NONE => NONE

fun resolve_fact fact_names = map_filter (resolve_one_named_fact fact_names)

fun is_fact fact_names = not o null o resolve_fact fact_names

fun resolve_one_named_conjecture s =

  case try (unprefix conjecture_prefix) s of

    SOME s' => Int.fromString s'

  | NONE => NONE

val resolve_conjecture = map_filter resolve_one_named_conjecture

val is_conjecture = not o null o resolve_conjecture

val ascii_of_lam_fact_prefix = ascii_of lam_fact_prefix

(* overapproximation (good enough) *)

fun is_lam_lifted s =

  String.isPrefix fact_prefix s andalso

  String.isSubstring ascii_of_lam_fact_prefix s

val is_combinator_def = String.isPrefix (helper_prefix ^ combinator_prefix)

fun is_axiom_used_in_proof pred =

  exists (fn Inference_Step ((_, ss), _, _, _, []) => exists pred ss

           | _ => false)

fun lam_trans_from_atp_proof atp_proof default =

  case (is_axiom_used_in_proof is_combinator_def atp_proof,

        is_axiom_used_in_proof is_lam_lifted atp_proof) of

    (false, false) => default

  | (false, true) => liftingN

(*  | (true, true) => combs_and_liftingN -- not supported by "metis" *)

  | (true, _) => combsN

val is_typed_helper_name =

  String.isPrefix helper_prefix andf String.isSuffix typed_helper_suffix

fun is_typed_helper_used_in_atp_proof atp_proof =

  is_axiom_used_in_proof is_typed_helper_name atp_proof

fun add_non_rec_defs fact_names accum =

  Vector.foldl (fn (facts, facts') =>

      union (op =) (filter (fn (_, (_, status)) => status = Non_Rec_Def) facts)

            facts')

    accum fact_names

val isa_ext = Thm.get_name_hint @{thm ext}

val isa_short_ext = Long_Name.base_name isa_ext

fun ext_name ctxt =

  if Thm.eq_thm_prop (@{thm ext},

         singleton (Attrib.eval_thms ctxt) (Facts.named isa_short_ext, [])) then

    isa_short_ext

  else

    isa_ext

val leo2_ext = "extcnf_equal_neg"

val leo2_unfold_def = "unfold_def"

fun add_fact ctxt fact_names (Inference_Step ((_, ss), _, _, rule, deps)) =

    (if rule = leo2_ext then

       insert (op =) (ext_name ctxt, (Global, General))

     else if rule = leo2_unfold_def then

       (* LEO 1.3.3 does not record definitions properly, leading to missing

         dependencies in the TSTP proof. Remove the next line once this is

         fixed. *)

       add_non_rec_defs fact_names

     else if rule = satallax_coreN then

       (fn [] =>

           (* Satallax doesn't include definitions in its unsatisfiable cores,

              so we assume the worst and include them all here. *)

           [(ext_name ctxt, (Global, General))] |> add_non_rec_defs fact_names

         | facts => facts)

     else

       I)

    #> (if null deps then union (op =) (resolve_fact fact_names ss)

        else I)

  | add_fact _ _ _ = I

fun used_facts_in_atp_proof ctxt fact_names atp_proof =

  if null atp_proof then Vector.foldl (uncurry (union (op =))) [] fact_names

  else fold (add_fact ctxt fact_names) atp_proof []

fun used_facts_in_unsound_atp_proof _ _ [] = NONE

  | used_facts_in_unsound_atp_proof ctxt fact_names atp_proof =

    let val used_facts = used_facts_in_atp_proof ctxt fact_names atp_proof in

      if forall (fn (_, (sc, _)) => sc = Global) used_facts andalso

         not (is_axiom_used_in_proof (is_conjecture o single) atp_proof) then

        SOME (map fst used_facts)

      else

        NONE

    end

(** one-liner reconstructor proofs **)

fun string_for_label (s, num) = s ^ string_of_int num

fun show_time NONE = ""

  | show_time (SOME ext_time) = " (" ^ string_from_ext_time ext_time ^ ")"

(* FIXME: Various bugs, esp. with "unfolding"

fun unusing_chained_facts _ 0 = ""

  | unusing_chained_facts used_chaineds num_chained =

    if length used_chaineds = num_chained then ""

    else if null used_chaineds then "(* using no facts *) "

    else "(* using only " ^ space_implode " " used_chaineds ^ " *) "

*)

fun apply_on_subgoal _ 1 = "by "

  | apply_on_subgoal 1 _ = "apply "

  | apply_on_subgoal i n =

    "prefer " ^ string_of_int i ^ " " ^ apply_on_subgoal 1 n

fun using_labels [] = ""

  | using_labels ls =

    "using " ^ space_implode " " (map string_for_label ls) ^ " "

fun command_call name [] =

    name |> not (Lexicon.is_identifier name) ? enclose "(" ")"

  | command_call name args = "(" ^ name ^ " " ^ space_implode " " args ^ ")"

fun reconstructor_command reconstr i n used_chaineds num_chained (ls, ss) =

  (* unusing_chained_facts used_chaineds num_chained ^ *)

  using_labels ls ^ apply_on_subgoal i n ^

  command_call (string_for_reconstructor reconstr) ss

fun try_command_line banner time command =

  banner ^ ": " ^ Markup.markup Isabelle_Markup.sendback command ^

  show_time time ^ "."

fun minimize_line _ [] = ""

  | minimize_line minimize_command ss =

    case minimize_command ss of

      "" => ""

    | command =>

      "\nTo minimize: " ^ Markup.markup Isabelle_Markup.sendback command ^ "."

fun split_used_facts facts =

  facts |> List.partition (fn (_, (sc, _)) => sc = Chained)

        |> pairself (sort_distinct (string_ord o pairself fst))

type minimize_command = string list -> string

type one_line_params =

  play * string * (string * stature) list * minimize_command * int * int

fun one_line_proof_text num_chained

        (preplay, banner, used_facts, minimize_command, subgoal,

         subgoal_count) =

  let

    val (chained, extra) = split_used_facts used_facts

    val (failed, reconstr, ext_time) =

      case preplay of

        Played (reconstr, time) => (false, reconstr, (SOME (false, time)))

      | Trust_Playable (reconstr, time) =>

        (false, reconstr,

         case time of

           NONE => NONE

         | SOME time =>

           if time = Time.zeroTime then NONE else SOME (true, time))

      | Failed_to_Play reconstr => (true, reconstr, NONE)

    val try_line =

      ([], map fst extra)

      |> reconstructor_command reconstr subgoal subgoal_count (map fst chained)

                               num_chained

      |> (if failed then

            enclose "One-line proof reconstruction failed: "

                     ".\n(Invoking \"sledgehammer\" with \"[strict]\" might \

                     \solve this.)"

          else

            try_command_line banner ext_time)

  in try_line ^ minimize_line minimize_command (map fst (extra @ chained)) end

(** Isar proof construction and manipulation **)

type label = string * int

type facts = label list * string list

datatype isar_qualifier = Show | Then | Moreover | Ultimately

datatype isar_step =

  Fix of (string * typ) list |

  Let of term * term |

  Assume of label * term |

  Prove of isar_qualifier list * label * term * byline

and byline =

  By_Metis of facts |

  Case_Split of isar_step list list * facts

val assume_prefix = "a"

val have_prefix = "f"

val raw_prefix = "x"

fun raw_label_for_name (num, ss) =

  case resolve_conjecture ss of

    [j] => (conjecture_prefix, j)

  | _ => (raw_prefix ^ ascii_of num, 0)

fun label_of_clause [name] = raw_label_for_name name

  | label_of_clause c = (space_implode "___" (map fst c), 0)

fun add_fact_from_dependencies fact_names (names as [(_, ss)]) =

    if is_fact fact_names ss then

      apsnd (union (op =) (map fst (resolve_fact fact_names ss)))

    else

      apfst (insert (op =) (label_of_clause names))

  | add_fact_from_dependencies fact_names names =

    apfst (insert (op =) (label_of_clause names))

fun repair_name "$true" = "c_True"

  | repair_name "$false" = "c_False"

  | repair_name "$$e" = tptp_equal (* seen in Vampire proofs *)

  | repair_name s =

    if is_tptp_equal s orelse

       (* seen in Vampire proofs *)

       (String.isPrefix "sQ" s andalso String.isSuffix "_eqProxy" s) then

      tptp_equal

    else

      s

fun unvarify_term (Var ((s, 0), T)) = Free (s, T)

  | unvarify_term t = raise TERM ("unvarify_term: non-Var", [t])

fun infer_formula_types ctxt =

  Type.constraint HOLogic.boolT

  #> Syntax.check_term

         (Proof_Context.set_mode Proof_Context.mode_schematic ctxt)

val combinator_table =

  [(@{const_name Meson.COMBI}, @{thm Meson.COMBI_def [abs_def]}),

   (@{const_name Meson.COMBK}, @{thm Meson.COMBK_def [abs_def]}),

   (@{const_name Meson.COMBB}, @{thm Meson.COMBB_def [abs_def]}),

   (@{const_name Meson.COMBC}, @{thm Meson.COMBC_def [abs_def]}),

   (@{const_name Meson.COMBS}, @{thm Meson.COMBS_def [abs_def]})]

fun uncombine_term thy =

  let

    fun aux (t1 $ t2) = betapply (pairself aux (t1, t2))

      | aux (Abs (s, T, t')) = Abs (s, T, aux t')

      | aux (t as Const (x as (s, _))) =

        (case AList.lookup (op =) combinator_table s of

           SOME thm => thm |> prop_of |> specialize_type thy x

                           |> Logic.dest_equals |> snd

         | NONE => t)

      | aux t = t

  in aux end

fun decode_line sym_tab (Definition_Step (name, phi1, phi2)) ctxt =

    let

      val thy = Proof_Context.theory_of ctxt

      val t1 = prop_from_atp ctxt true sym_tab phi1

      val vars = snd (strip_comb t1)

      val frees = map unvarify_term vars

      val unvarify_args = subst_atomic (vars ~~ frees)

      val t2 = prop_from_atp ctxt true sym_tab phi2

      val (t1, t2) =

        HOLogic.eq_const HOLogic.typeT $ t1 $ t2

        |> unvarify_args |> uncombine_term thy |> infer_formula_types ctxt

        |> HOLogic.dest_eq

    in

      (Definition_Step (name, t1, t2),

       fold Variable.declare_term (maps Misc_Legacy.term_frees [t1, t2]) ctxt)

    end

  | decode_line sym_tab (Inference_Step (name, role, u, rule, deps)) ctxt =

    let

      val thy = Proof_Context.theory_of ctxt

      val t = u |> prop_from_atp ctxt true sym_tab

                |> uncombine_term thy |> infer_formula_types ctxt

    in

      (Inference_Step (name, role, t, rule, deps),

       fold Variable.declare_term (Misc_Legacy.term_frees t) ctxt)

    end

fun decode_lines ctxt sym_tab lines =

  fst (fold_map (decode_line sym_tab) lines ctxt)

fun replace_one_dependency (old, new) dep =

  if is_same_atp_step dep old then new else [dep]

fun replace_dependencies_in_line _ (line as Definition_Step _) = line

  | replace_dependencies_in_line p

        (Inference_Step (name, role, t, rule, deps)) =

    Inference_Step (name, role, t, rule,

                    fold (union (op =) o replace_one_dependency p) deps [])

(* No "real" literals means only type information (tfree_tcs, clsrel, or

   clsarity). *)

fun is_only_type_information t = t aconv @{term True}

fun is_same_inference _ (Definition_Step _) = false

  | is_same_inference t (Inference_Step (_, _, t', _, _)) = t aconv t'

(* Discard facts; consolidate adjacent lines that prove the same formula, since

   they differ only in type information.*)

fun add_line _ (line as Definition_Step _) lines = line :: lines

  | add_line fact_names (Inference_Step (name as (_, ss), role, t, rule, []))

             lines =

    (* No dependencies: fact, conjecture, or (for Vampire) internal facts or

       definitions. *)

    if is_fact fact_names ss then

      (* Facts are not proof lines. *)

      if is_only_type_information t then

        map (replace_dependencies_in_line (name, [])) lines

      (* Is there a repetition? If so, replace later line by earlier one. *)

      else case take_prefix (not o is_same_inference t) lines of

        (_, []) => lines (* no repetition of proof line *)

      | (pre, Inference_Step (name', _, _, _, _) :: post) =>

        pre @ map (replace_dependencies_in_line (name', [name])) post

      | _ => raise Fail "unexpected inference"

    else if is_conjecture ss then

      Inference_Step (name, role, t, rule, []) :: lines

    else

      map (replace_dependencies_in_line (name, [])) lines

  | add_line _ (Inference_Step (name, role, t, rule, deps)) lines =

    (* Type information will be deleted later; skip repetition test. *)

    if is_only_type_information t then

      Inference_Step (name, role, t, rule, deps) :: lines

    (* Is there a repetition? If so, replace later line by earlier one. *)

    else case take_prefix (not o is_same_inference t) lines of

      (* FIXME: Doesn't this code risk conflating proofs involving different

         types? *)

       (_, []) => Inference_Step (name, role, t, rule, deps) :: lines

     | (pre, Inference_Step (name', role, t', rule, _) :: post) =>

       Inference_Step (name, role, t', rule, deps) ::

       pre @ map (replace_dependencies_in_line (name', [name])) post

     | _ => raise Fail "unexpected inference"

val waldmeister_conjecture_num = "1.0.0.0"

val repair_waldmeister_endgame =

  let

    fun do_tail (Inference_Step (name, role, t, rule, deps)) =

        Inference_Step (name, role, s_not t, rule, deps)

      | do_tail line = line

    fun do_body [] = []

      | do_body ((line as Inference_Step ((num, _), _, _, _, _)) :: lines) =

        if num = waldmeister_conjecture_num then map do_tail (line :: lines)

        else line :: do_body lines

      | do_body (line :: lines) = line :: do_body lines

  in do_body end

(* Recursively delete empty lines (type information) from the proof. *)

fun add_nontrivial_line (line as Inference_Step (name, _, t, _, [])) lines =

    if is_only_type_information t then delete_dependency name lines

    else line :: lines

  | add_nontrivial_line line lines = line :: lines

and delete_dependency name lines =

  fold_rev add_nontrivial_line

           (map (replace_dependencies_in_line (name, [])) lines) []

(* ATPs sometimes reuse free variable names in the strangest ways. Removing

   offending lines often does the trick. *)

fun is_bad_free frees (Free x) = not (member (op =) frees x)

  | is_bad_free _ _ = false

fun add_desired_line _ _ (line as Definition_Step (name, _, _)) (j, lines) =

    (j, line :: map (replace_dependencies_in_line (name, [])) lines)

  | add_desired_line fact_names frees

        (Inference_Step (name as (_, ss), role, t, rule, deps)) (j, lines) =

    (j + 1,

     if is_fact fact_names ss orelse

        is_conjecture ss orelse

        (* the last line must be kept *)

        j = 0 orelse

        (not (is_only_type_information t) andalso

         null (Term.add_tvars t []) andalso

         not (exists_subterm (is_bad_free frees) t) andalso

         length deps >= 2 andalso

         (* kill next to last line, which usually results in a trivial step *)

         j <> 1) then

       Inference_Step (name, role, t, rule, deps) :: lines  (* keep line *)

     else

       map (replace_dependencies_in_line (name, deps)) lines)  (* drop line *)

(** Type annotations **)

fun post_traverse_term_type' f _ (t as Const (_, T)) s = f t T s

  | post_traverse_term_type' f _ (t as Free (_, T)) s = f t T s

  | post_traverse_term_type' f _ (t as Var (_, T)) s = f t T s

  | post_traverse_term_type' f env (t as Bound i) s = f t (nth env i) s

  | post_traverse_term_type' f env (Abs (x, T1, b)) s =

    let

      val ((b', s'), T2) = post_traverse_term_type' f (T1 :: env) b s

    in f (Abs (x, T1, b')) (T1 --> T2) s' end

  | post_traverse_term_type' f env (u $ v) s =

    let

      val ((u', s'), Type (_, [_, T])) = post_traverse_term_type' f env u s

      val ((v', s''), _) = post_traverse_term_type' f env v s'

    in f (u' $ v') T s'' end

fun post_traverse_term_type f s t =

  post_traverse_term_type' (fn t => fn T => fn s => (f t T s, T)) [] t s |> fst

fun post_fold_term_type f s t =

  post_traverse_term_type (fn t => fn T => fn s => (t, f t T s)) s t |> snd

(* Data structures, orders *)

val cost_ord = prod_ord int_ord (prod_ord int_ord int_ord)

structure Var_Set_Tab = Table(

  type key = indexname list

  val ord = list_ord Term_Ord.fast_indexname_ord)

(* (1) Generalize Types *)

fun generalize_types ctxt t =

  t |> map_types (fn _ => dummyT)

    |> Syntax.check_term

         (Proof_Context.set_mode Proof_Context.mode_pattern ctxt)

(* (2) Typing-spot Table *)

local

fun key_of_atype (TVar (z, _)) =

    Ord_List.insert Term_Ord.fast_indexname_ord z

  | key_of_atype _ = I

fun key_of_type T = fold_atyps key_of_atype T []

fun update_tab t T (tab, pos) =

  (case key_of_type T of

     [] => tab

   | key =>

     let val cost = (size_of_typ T, (size_of_term t, pos)) in

       case Var_Set_Tab.lookup tab key of

         NONE => Var_Set_Tab.update_new (key, cost) tab

       | SOME old_cost =>

         (case cost_ord (cost, old_cost) of

            LESS => Var_Set_Tab.update (key, cost) tab

          | _ => tab)

     end,

   pos + 1)

in

val typing_spot_table =

  post_fold_term_type update_tab (Var_Set_Tab.empty, 0) #> fst

end

(* (3) Reverse-Greedy *)

fun reverse_greedy typing_spot_tab =

  let

    fun update_count z =

      fold (fn tvar => fn tab =>

        let val c = Vartab.lookup tab tvar |> the_default 0 in

          Vartab.update (tvar, c + z) tab

        end)

    fun superfluous tcount =

      forall (fn tvar => the (Vartab.lookup tcount tvar) > 1)

    fun drop_superfluous (tvars, (_, (_, spot))) (spots, tcount) =

      if superfluous tcount tvars then (spots, update_count ~1 tvars tcount)

      else (spot :: spots, tcount)

    val (typing_spots, tvar_count_tab) =

      Var_Set_Tab.fold

        (fn kv as (k, _) => apfst (cons kv) #> apsnd (update_count 1 k))

        typing_spot_tab ([], Vartab.empty)

      |>> sort_distinct (rev_order o cost_ord o pairself snd)

  in fold drop_superfluous typing_spots ([], tvar_count_tab) |> fst end

(* (4) Introduce Annotations *)

fun introduce_annotations thy spots t t' =

  let

    val get_types = post_fold_term_type (K cons) []

    fun match_types tp =

      fold (Sign.typ_match thy) (op ~~ (pairself get_types tp)) Vartab.empty

    fun unica' b x [] = if b then [x] else []

      | unica' b x (y :: ys) =

        if x = y then unica' false x ys

        else unica' true y ys |> b ? cons x

    fun unica ord xs =

      case sort ord xs of x :: ys => unica' true x ys | [] => []

    val add_all_tfree_namesT = fold_atyps (fn TFree (x, _) => cons x | _ => I)

    fun erase_unica_tfrees env =

      let

        val unica =

          Vartab.fold (add_all_tfree_namesT o snd o snd) env []

          |> unica fast_string_ord

        val erase_unica = map_atyps

          (fn T as TFree (s, _) =>

              if Ord_List.member fast_string_ord unica s then dummyT else T

            | T => T)

      in Vartab.map (K (apsnd erase_unica)) env end

    val env = match_types (t', t) |> erase_unica_tfrees

    fun get_annot env (TFree _) = (false, (env, dummyT))

      | get_annot env (T as TVar (v, S)) =

        let val T' = Envir.subst_type env T in

          if T' = dummyT then (false, (env, dummyT))

          else (true, (Vartab.update (v, (S, dummyT)) env, T'))

        end

      | get_annot env (Type (S, Ts)) =

        (case fold_rev (fn T => fn (b, (env, Ts)) =>

                  let

                    val (b', (env', T)) = get_annot env T

                  in (b orelse b', (env', T :: Ts)) end)

                Ts (false, (env, [])) of

           (true, (env', Ts)) => (true, (env', Type (S, Ts)))

         | (false, (env', _)) => (false, (env', dummyT)))

    fun post1 _ T (env, cp, ps as p :: ps', annots) =

        if p <> cp then

          (env, cp + 1, ps, annots)

        else

          let val (_, (env', T')) = get_annot env T in

            (env', cp + 1, ps', (p, T') :: annots)

          end

      | post1 _ _ accum = accum

    val (_, _, _, annots) = post_fold_term_type post1 (env, 0, spots, []) t'

    fun post2 t _ (cp, annots as (p, T) :: annots') =

        if p <> cp then (t, (cp + 1, annots))

        else (Type.constraint T t, (cp + 1, annots'))

      | post2 t _ x = (t, x)

  in post_traverse_term_type post2 (0, rev annots) t |> fst end

(* (5) Annotate *)

fun annotate_types ctxt t =

  let

    val thy = Proof_Context.theory_of ctxt

    val t' = generalize_types ctxt t

    val typing_spots =

      t' |> typing_spot_table

         |> reverse_greedy

         |> sort int_ord

  in introduce_annotations thy typing_spots t t' end

val indent_size = 2

val no_label = ("", ~1)

fun string_for_proof ctxt type_enc lam_trans i n =

  let

    fun fix_print_mode f x =

      Print_Mode.setmp (filter (curry (op =) Symbol.xsymbolsN)

                               (print_mode_value ())) f x

    fun do_indent ind = replicate_string (ind * indent_size) " "

    fun do_free (s, T) =

      maybe_quote s ^ " :: " ^

      maybe_quote (fix_print_mode (Syntax.string_of_typ ctxt) T)

    fun do_label l = if l = no_label then "" else string_for_label l ^ ": "

    fun do_have qs =

      (if member (op =) qs Moreover then "moreover " else "") ^

      (if member (op =) qs Ultimately then "ultimately " else "") ^

      (if member (op =) qs Then then

         if member (op =) qs Show then "thus" else "hence"

       else

         if member (op =) qs Show then "show" else "have")

    val do_term =

      maybe_quote o fix_print_mode (Syntax.string_of_term ctxt)

      o annotate_types ctxt

    val reconstr = Metis (type_enc, lam_trans)

    fun do_facts ind (ls, ss) =

      "\n" ^ do_indent (ind + 1) ^

      reconstructor_command reconstr 1 1 [] 0

          (ls |> sort_distinct (prod_ord string_ord int_ord),

           ss |> sort_distinct string_ord)

    and do_step ind (Fix xs) =

        do_indent ind ^ "fix " ^ space_implode " and " (map do_free xs) ^ "\n"

      | do_step ind (Let (t1, t2)) =

        do_indent ind ^ "let " ^ do_term t1 ^ " = " ^ do_term t2 ^ "\n"

      | do_step ind (Assume (l, t)) =

        do_indent ind ^ "assume " ^ do_label l ^ do_term t ^ "\n"

      | do_step ind (Prove (qs, l, t, By_Metis facts)) =

        do_indent ind ^ do_have qs ^ " " ^

        do_label l ^ do_term t ^ do_facts ind facts ^ "\n"

      | do_step ind (Prove (qs, l, t, Case_Split (proofs, facts))) =

        implode (map (prefix (do_indent ind ^ "moreover\n") o do_block ind)

                     proofs) ^

        do_indent ind ^ do_have qs ^ " " ^ do_label l ^ do_term t ^

        do_facts ind facts ^ "\n"

    and do_steps prefix suffix ind steps =

      let val s = implode (map (do_step ind) steps) in

        replicate_string (ind * indent_size - size prefix) " " ^ prefix ^

        String.extract (s, ind * indent_size,

                        SOME (size s - ind * indent_size - 1)) ^

        suffix ^ "\n"

      end

    and do_block ind proof = do_steps "{ " " }" (ind + 1) proof

    (* One-step proofs are pointless; better use the Metis one-liner

       directly. *)

    and do_proof [Prove (_, _, _, By_Metis _)] = ""

      | do_proof proof =

        (if i <> 1 then "prefer " ^ string_of_int i ^ "\n" else "") ^

        do_indent 0 ^ "proof -\n" ^ do_steps "" "" 1 proof ^ do_indent 0 ^

        (if n <> 1 then "next" else "qed")

  in do_proof end

fun used_labels_of_step (Prove (_, _, _, by)) =

    (case by of

       By_Metis (ls, _) => ls

     | Case_Split (proofs, (ls, _)) =>

       fold (union (op =) o used_labels_of) proofs ls)

  | used_labels_of_step _ = []

and used_labels_of proof = fold (union (op =) o used_labels_of_step) proof []

fun kill_useless_labels_in_proof proof =

  let

    val used_ls = used_labels_of proof

    fun do_label l = if member (op =) used_ls l then l else no_label

    fun do_step (Assume (l, t)) = Assume (do_label l, t)

      | do_step (Prove (qs, l, t, by)) =

        Prove (qs, do_label l, t,

               case by of

                 Case_Split (proofs, facts) =>

                 Case_Split (map (map do_step) proofs, facts)

               | _ => by)

      | do_step step = step

  in map do_step proof end

fun prefix_for_depth n = replicate_string (n + 1)

val relabel_proof =

  let

    fun aux _ _ _ [] = []

      | aux subst depth (next_assum, next_have) (Assume (l, t) :: proof) =

        if l = no_label then

          Assume (l, t) :: aux subst depth (next_assum, next_have) proof

        else

          let val l' = (prefix_for_depth depth assume_prefix, next_assum) in

            Assume (l', t) ::

            aux ((l, l') :: subst) depth (next_assum + 1, next_have) proof

          end

      | aux subst depth (next_assum, next_have)

            (Prove (qs, l, t, by) :: proof) =

        let

          val (l', subst, next_have) =

            if l = no_label then

              (l, subst, next_have)

            else

              let val l' = (prefix_for_depth depth have_prefix, next_have) in

                (l', (l, l') :: subst, next_have + 1)

              end

          val relabel_facts =

            apfst (maps (the_list o AList.lookup (op =) subst))

          val by =

            case by of

              By_Metis facts => By_Metis (relabel_facts facts)

            | Case_Split (proofs, facts) =>

              Case_Split (map (aux subst (depth + 1) (1, 1)) proofs,

                          relabel_facts facts)

        in

          Prove (qs, l', t, by) :: aux subst depth (next_assum, next_have) proof

        end

      | aux subst depth nextp (step :: proof) =

        step :: aux subst depth nextp proof

  in aux [] 0 (1, 1) end

val merge_timeout_slack = 1.2

val label_ord = prod_ord int_ord fast_string_ord o pairself swap

structure Label_Table = Table(

  type key = label

  val ord = label_ord)

fun shrink_proof debug ctxt type_enc lam_trans preplay

                 preplay_timeout isar_shrinkage proof =

  let

    (* clean vector interface *)

    fun get i v = Vector.sub (v, i)

    fun replace x i v = Vector.update (v, i, x)

    fun update f i v = replace (get i v |> f) i v

    fun v_fold_index f v s =

      Vector.foldl (fn (x, (i, s)) => (i+1, f (i, x) s)) (0, s) v |> snd

    (* Queue interface to table *)

    fun pop tab key =

      let val v = hd (Inttab.lookup_list tab key) in

        (v, Inttab.remove_list (op =) (key, v) tab)

      end

    fun pop_max tab = pop tab (the (Inttab.max_key tab))

    val is_empty = Inttab.is_empty

    fun add_list tab xs = fold (Inttab.insert_list (op =)) xs tab

    (* proof vector *)

    val proof_vect = proof |> map SOME |> Vector.fromList

    val n = Vector.length proof_vect

    val n_target = Real.fromInt n / isar_shrinkage |> Real.round

    (* table for mapping from label to proof position *)

    fun update_table (i, Prove (_, label, _, _)) =

        Label_Table.update_new (label, i)

      | update_table _ = I

    val label_index_table = fold_index update_table proof Label_Table.empty

    (* proof references *)

    fun refs (Prove (_, _, _, By_Metis (refs, _))) =

      map (the o Label_Table.lookup label_index_table) refs

      | refs _ = []

    val refed_by_vect =

      Vector.tabulate (n, (fn _ => []))

      |> fold_index (fn (i, step) => fold (update (cons i)) (refs step)) proof

      |> Vector.map rev (* after rev, indices are sorted in ascending order *)

    (* candidates for elimination, use table as priority queue (greedy

       algorithm) *)

    fun cost (Prove (_, _ , t, _)) = Term.size_of_term t

      | cost _ = 0

    val cand_ord =  rev_order o prod_ord int_ord int_ord

    val cand_tab =

      v_fold_index

        (fn (i, [_]) => cons (get i proof_vect |> the |> cost, i)

        | _ => I) refed_by_vect []

      |> Inttab.make_list

    (* Enrich context with local facts *)

    val thy = Proof_Context.theory_of ctxt

    fun enrich_ctxt' (Prove (_, label, t, _)) ctxt =

        Proof_Context.put_thms false

            (string_for_label label, SOME [Skip_Proof.make_thm thy t]) ctxt

      | enrich_ctxt' _ ctxt = ctxt

    val rich_ctxt = fold enrich_ctxt' proof ctxt

    (* Timing *)

    fun take_time timeout tac arg =

      let val timing = Timing.start () in

        (TimeLimit.timeLimit timeout tac arg;

         Timing.result timing |> #cpu |> SOME)

        handle _ => NONE

      end

    val sum_up_time =

      Vector.foldl

        ((fn (SOME t, (b, s)) => (b, t + s)

           | (NONE, (_, s)) => (true, preplay_timeout + s)) o apfst Lazy.force)

        (false, seconds 0.0)

    (* Metis Preplaying *)

    fun try_metis timeout (Prove (_, _, t, By_Metis fact_names)) =

      if not preplay then (fn () => SOME (seconds 0.0)) else

        let

          val facts =

            fact_names

            |>> map string_for_label |> op @

            |> map (the_single o thms_of_name rich_ctxt)

          val goal =

            Goal.prove (Config.put Metis_Tactic.verbose debug ctxt) [] [] t

          fun tac {context = ctxt, prems = _} =

            Metis_Tactic.metis_tac [type_enc] lam_trans ctxt facts 1

        in

          take_time timeout (fn () => goal tac)

        end

    (* Lazy metis time vector, cache *)

    val metis_time =

      Vector.map (Lazy.lazy o try_metis preplay_timeout o the) proof_vect

    (* Merging *)

    fun merge (Prove (qs1, label1, _, By_Metis (lfs1, gfs1)))

              (Prove (qs2, label2 , t, By_Metis (lfs2, gfs2))) =

      let

        val qs = inter (op =) qs1 qs2 (* FIXME: Is this correct? *)

          |> member (op =) (union (op =) qs1 qs2) Ultimately ? cons Ultimately

          |> member (op =) qs2 Show ? cons Show

        val ls = remove (op =) label1 lfs2 |> union (op =) lfs1

        val ss = union (op =) gfs1 gfs2

      in Prove (qs, label2, t, By_Metis (ls, ss)) end

    fun try_merge metis_time (s1, i) (s2, j) =

      (case get i metis_time |> Lazy.force of

        NONE => (NONE, metis_time)

      | SOME t1 =>

        (case get j metis_time |> Lazy.force of

          NONE => (NONE, metis_time)

        | SOME t2 =>

          let

            val s12 = merge s1 s2

            val timeout =

              t1 + t2 |> Time.toReal |> curry Real.* merge_timeout_slack

                      |> Time.fromReal

          in

            case try_metis timeout s12 () of

              NONE => (NONE, metis_time)

            | some_t12 =>

              (SOME s12, metis_time

                         |> replace (seconds 0.0 |> SOME |> Lazy.value) i

                         |> replace (Lazy.value some_t12) j)

          end))

    fun merge_steps metis_time proof_vect refed_by cand_tab n' =

      if is_empty cand_tab orelse n' <= n_target orelse n'<3 then

        (sum_up_time metis_time,

         Vector.foldr

           (fn (NONE, proof) => proof | (SOME s, proof) => s :: proof)

           [] proof_vect)

      else

        let

          val (i, cand_tab) = pop_max cand_tab

          val j = get i refed_by |> the_single

          val s1 = get i proof_vect |> the

          val s2 = get j proof_vect |> the

        in

          case try_merge metis_time (s1, i) (s2, j) of

            (NONE, metis_time) =>

            merge_steps metis_time proof_vect refed_by cand_tab n'

          | (s, metis_time) => let

            val refs = refs s1

            val refed_by = refed_by |> fold

              (update (Ord_List.remove int_ord i #> Ord_List.insert int_ord j)) refs

            val new_candidates =

              fold (fn (i, [_]) => cons (cost (get i proof_vect |> the), i)

                     | _ => I)

                (map (fn i => (i, get i refed_by)) refs) []

            val cand_tab = add_list cand_tab new_candidates

            val proof_vect = proof_vect |> replace NONE i |> replace s j

          in

            merge_steps metis_time proof_vect refed_by cand_tab (n' - 1)

          end

        end

  in

    merge_steps metis_time proof_vect refed_by_vect cand_tab n

  end

val chain_direct_proof =

  let

    fun succedent_of_step (Prove (_, label, _, _)) = SOME label

      | succedent_of_step (Assume (label, _)) = SOME label

      | succedent_of_step _ = NONE

    fun chain_inf (SOME label0)

                  (step as Prove (qs, label, t, By_Metis (lfs, gfs))) =

        if member (op =) lfs label0 then

          Prove (Then :: qs, label, t,

                 By_Metis (filter_out (curry (op =) label0) lfs, gfs))

        else

          step

      | chain_inf _ (Prove (qs, label, t, Case_Split (proofs, facts))) =

        Prove (qs, label, t, Case_Split ((map (chain_proof NONE) proofs), facts))

      | chain_inf _ step = step

    and chain_proof _ [] = []

      | chain_proof (prev as SOME _) (i :: is) =

        chain_inf prev i :: chain_proof (succedent_of_step i) is

      | chain_proof _ (i :: is) =

        i :: chain_proof (succedent_of_step i) is

  in chain_proof NONE end

type isar_params =

  bool * bool * Time.time * real * string Symtab.table

  * (string * stature) list vector * int Symtab.table * string proof * thm

fun isar_proof_text ctxt isar_proofs

    (debug, verbose, preplay_timeout, isar_shrinkage,

     pool, fact_names, sym_tab, atp_proof, goal)

    (one_line_params as (_, _, _, _, subgoal, subgoal_count)) =

  let

    val (params, hyp_ts, concl_t) = strip_subgoal ctxt goal subgoal

    val frees = fold Term.add_frees (concl_t :: hyp_ts) []

    val one_line_proof = one_line_proof_text 0 one_line_params

    val type_enc =

      if is_typed_helper_used_in_atp_proof atp_proof then full_typesN

      else partial_typesN

    val lam_trans = lam_trans_from_atp_proof atp_proof metis_default_lam_trans

    val preplay = preplay_timeout <> seconds 0.0

    fun isar_proof_of () =

      let

        val atp_proof =

          atp_proof

          |> clean_up_atp_proof_dependencies

          |> nasty_atp_proof pool

          |> map_term_names_in_atp_proof repair_name

          |> decode_lines ctxt sym_tab

          |> rpair [] |-> fold_rev (add_line fact_names)

          |> repair_waldmeister_endgame

          |> rpair [] |-> fold_rev add_nontrivial_line

          |> rpair (0, [])

          |-> fold_rev (add_desired_line fact_names frees)

          |> snd

        val conj_name = conjecture_prefix ^ string_of_int (length hyp_ts)

        val conjs =

          atp_proof |> map_filter

            (fn Inference_Step (name as (_, ss), _, _, _, []) =>

                if member (op =) ss conj_name then SOME name else NONE

              | _ => NONE)

        val assms =

          atp_proof |> map_filter

            (fn Inference_Step (name as (_, ss), _, _, _, []) =>

                (case resolve_conjecture ss of

                   [j] =>

                   if j = length hyp_ts then NONE

                   else SOME (Assume (raw_label_for_name name, nth hyp_ts j))

                 | _ => NONE)

              | _ => NONE)

        fun dep_of_step (Definition_Step _) = NONE

          | dep_of_step (Inference_Step (name, _, _, _, from)) =

            SOME (from, name)

        val ref_graph = atp_proof |> map_filter dep_of_step |> make_ref_graph

        val axioms = axioms_of_ref_graph ref_graph conjs

        val tainted = tainted_atoms_of_ref_graph ref_graph conjs

        val props =

          Symtab.empty

          |> fold (fn Definition_Step _ => I (* FIXME *)

                    | Inference_Step (name as (s, ss), role, t, _, _) =>

                      Symtab.update_new (s,

                        if member (op = o apsnd fst) tainted s then

                          t |> role <> Conjecture ? s_not

                            |> fold exists_of (map Var (Term.add_vars t []))

                        else

                          t))

                  atp_proof