src/HOL/Tools/ATP/atp_satallax.ML

tuned;

(*  Title:      HOL/Tools/ATP/atp_satallax.ML
    Author:     Mathias Fleury, ENS Rennes
    Author:     Jasmin Blanchette, TU Muenchen

Satallax proof parser and transformation for Sledgehammer.
*)

signature ATP_SATALLAX =
sig
  val atp_proof_of_tstplike_proof : string -> string ATP_Proof.atp_problem -> string ->
    string ATP_Proof.atp_proof
end;

structure ATP_Satallax : ATP_SATALLAX =
struct

open ATP_Proof
open ATP_Util
open ATP_Problem

(*Undocumented format:
thf (number, plain | Axiom | ..., inference(rule, [status(thm), assumptions ([hypotheses list]),
detailed_rule(discharge,used_hypothese_list) *], used_hypotheses_list, premises))
(seen by tab_mat)

Also seen -- but we can ignore it:
"tab_inh (a) __11." meaning that the type a is inhabited usueful of variable eigen__11,
*)
fun parse_satallax_tstp_information x =
  ((Symbol.scan_ascii_id || ($$ "$" |-- Symbol.scan_ascii_id))
  -- Scan.option ( $$ "("
    |-- (Scan.option (Symbol.scan_ascii_id --| $$ ",")
       -- ((Scan.option (($$ "[" |-- (Scan.option ((scan_general_id
         -- Scan.repeat ($$ "," |-- scan_general_id)) >> op ::) --| $$ "]"))
         || (scan_general_id) >> (fn x => SOME [x]))
       >> (fn (SOME x) => x | NONE => NONE)) --| $$ ")"))
  || (skip_term >> K (NONE, NONE)))) x

fun parse_prem x =
  ((Symbol.scan_ascii_id || scan_general_id) --| Scan.option ($$ ":" -- skip_term)) x

fun parse_prems x =
  (($$ "[" |-- parse_prem -- Scan.repeat ($$ "," |-- parse_prem) --| $$ "]")
     >> op ::) x

fun parse_tstp_satallax_extra_arguments x =
  ($$ "," |-- scan_general_id -- (($$ "(" |-- Symbol.scan_ascii_id --| $$ "," )
  -- ($$ "[" |-- Scan.option ((parse_satallax_tstp_information
  -- Scan.repeat ($$ "," |-- parse_satallax_tstp_information)) >> op ::)
  --| $$ "]") --
  (Scan.optional ($$ "," |-- parse_prems) [] -- Scan.optional ($$ "," |-- parse_prems) []
    >> (fn (x, []) => x | (_, x) => x))
   --| $$ ")")) x

val dummy_satallax_step = ((("~1", "tab_inh"), AAtom (ATerm(("False", []), []))), NONE)
fun parse_tstp_thf0_satallax_line x =
  (((Scan.this_string tptp_thf
  -- $$ "(") |-- scan_general_id --| $$ "," -- Symbol.scan_ascii_id --| $$ ","
  -- parse_thf0_formula -- Scan.option parse_tstp_satallax_extra_arguments --| $$ ")" --| $$ ".")
  || (Scan.this_string "tab_inh" |-- skip_term --| $$ ".")
     >> K dummy_satallax_step) x

datatype satallax_step = Satallax_Step of {
  id: string,
  role: string,
  theorem: (string, string, (string, string ATP_Problem.atp_type) ATP_Problem.atp_term, string)
    ATP_Problem.atp_formula,
  assumptions: string list,
  rule: string,
  used_assumptions: string list,
  detailed_eigen: string,
  generated_goal_assumptions: (string * string list) list}

fun mk_satallax_step id role theorem assumptions rule used_assumptions
    generated_goal_assumptions detailed_eigen =
  Satallax_Step {id = id, role = role, theorem = theorem, assumptions = assumptions, rule = rule,
    used_assumptions = used_assumptions, generated_goal_assumptions = generated_goal_assumptions,
    detailed_eigen = detailed_eigen}

fun get_assumptions (("assumptions", SOME (_ , assumptions)) :: _) =
    the_default [] assumptions
  | get_assumptions (_ :: l) = get_assumptions l
  | get_assumptions [] = []

fun get_detailled_eigen ((_, SOME (SOME "eigenvar" , var)) :: _) =
    hd (the_default [""] var)
  | get_detailled_eigen (_ :: l) = get_detailled_eigen l
  | get_detailled_eigen [] = ""

fun seperate_dependices dependencies =
  let
    fun sep_dep [] used_assumptions new_goals generated_assumptions _ =
        (used_assumptions, new_goals, generated_assumptions)
      | sep_dep (x :: l) used_assumptions new_goals generated_assumptions state =
        if hd (raw_explode x) = "h" orelse Int.fromString x = NONE then
          if state = 0 then
            sep_dep l (x :: used_assumptions) new_goals generated_assumptions state
          else
            sep_dep l used_assumptions new_goals (x :: generated_assumptions) 3
        else
          if state = 1 orelse state = 0 then
            sep_dep l used_assumptions (x :: new_goals) generated_assumptions 1
          else
            raise Fail ("incorrect Satallax proof: " ^ @{make_string} l)
  in
    sep_dep dependencies [] [] [] 0
  end

fun create_grouped_goal_assumption rule new_goals generated_assumptions =
  let
    val number_of_new_goals = length new_goals
    val number_of_new_assms = length generated_assumptions
  in
    if number_of_new_goals = number_of_new_assms then
      new_goals ~~ (map single generated_assumptions)
    else if 2 * number_of_new_goals = number_of_new_assms then
      let
        fun group_by_pair (new_goal :: new_goals) (assumpt1 :: assumpt2 :: generated_assumptions) =
            (new_goal, [assumpt1, assumpt2]) :: group_by_pair new_goals generated_assumptions
          | group_by_pair [] [] = []
      in
        group_by_pair new_goals generated_assumptions
      end
    else
      raise Fail ("the rule " ^ rule ^" is not supported in the reconstruction")
  end
fun to_satallax_step (((id, role), formula), (SOME (_,((rule, l), used_rules)))) =
    let
      val (used_assumptions, new_goals, generated_assumptions) = seperate_dependices used_rules
    in
      mk_satallax_step id role formula (get_assumptions (the_default [] l)) rule used_assumptions
        (create_grouped_goal_assumption rule new_goals generated_assumptions)
        (get_detailled_eigen (the_default [] l))
    end
  | to_satallax_step (((id, role), formula), NONE) =
      mk_satallax_step id role formula [] "assumption" [] [] ""

fun is_assumption (Satallax_Step {role, ...}) = role = "assumption" orelse role = "axiom" orelse
  role = "negated_conjecture" orelse role = "conjecture"

fun seperate_assumptions_and_steps l =
  let
    fun seperate_assumption [] l l' = (l, l')
      | seperate_assumption (step :: steps) l l' =
        if is_assumption step then
          seperate_assumption steps (step :: l) l'
        else
          seperate_assumption steps l (step :: l')
  in
    seperate_assumption l [] []
  end

datatype satallax_proof_graph =
  Linear_Part of {node: satallax_step, succs: satallax_proof_graph list} |
  Tree_Part of {node: satallax_step, deps: satallax_proof_graph list}

fun find_proof_step ((x as Satallax_Step {id, ...}) :: l) h =
    if h = id then x else find_proof_step l h
  | find_proof_step [] h = raise Fail ("not_found: " ^ @{make_string} h ^ " (probably a parsing \
    \error)")

fun remove_not_not (x as ATerm ((op1, _), [ATerm ((op2, _), [th])])) =
    if op1 = op2 andalso op1 = tptp_not then th else x
  | remove_not_not th = th

fun tptp_term_equal (ATerm((op1, _), l1), ATerm((op2, _), l2)) = op1 = op2 andalso
    fold2 (fn t1 => fn t2 => fn c => c andalso t1 = t2) l1 l2 true
  | tptp_term_equal (_, _) = false

val dummy_true_aterm = ATerm (("$true", [dummy_atype]), [])

fun find_assumptions_to_inline ths (assm :: assms) to_inline no_inline =
    (case List.find (curry (op =) assm o fst) no_inline of
      SOME _ => find_assumptions_to_inline ths assms to_inline no_inline
    | NONE =>
      (case List.find (curry (op =) assm o fst) to_inline of
        NONE => find_assumptions_to_inline ths assms to_inline no_inline
      | SOME (_, th) =>
        let
          val simplified_ths_with_inlined_asms =
            (case List.partition (curry tptp_term_equal th o remove_not_not) ths of
              ([], ths) => ATerm ((tptp_not, [dummy_atype]), [th]) :: ths
            | (_, _) => [])
        in
          find_assumptions_to_inline simplified_ths_with_inlined_asms assms to_inline no_inline
        end))
  | find_assumptions_to_inline ths [] _ _ = ths

fun inline_if_needed_and_simplify th assms to_inline no_inline =
  (case find_assumptions_to_inline [th] assms to_inline no_inline of
    [] => dummy_true_aterm
  | l => foldl1 (fn (a, b) =>
    (case b of
      ATerm (("$false", _), _) => a
    | ATerm (("~", _ ), [ATerm (("$true", _), _)]) => a
    | _ => ATerm ((tptp_or, [dummy_atype]), [a, b]))) l)

fun get_conclusion (Satallax_Step {theorem = AAtom theorem, ...}) = theorem

fun add_assumptions new_used_assumptions (Satallax_Step {id, role, theorem, assumptions,
    rule, generated_goal_assumptions, used_assumptions, detailed_eigen}) =
  mk_satallax_step id role theorem assumptions rule (new_used_assumptions @ used_assumptions)
    generated_goal_assumptions detailed_eigen

fun set_rule new_rule (Satallax_Step {id, role, theorem, assumptions,
    generated_goal_assumptions, used_assumptions, detailed_eigen, ...}) =
  mk_satallax_step id role theorem assumptions new_rule used_assumptions
    generated_goal_assumptions detailed_eigen

fun add_detailled_eigen eigen (Satallax_Step {id, role, theorem, assumptions,
    rule, generated_goal_assumptions, used_assumptions, detailed_eigen}) =
  mk_satallax_step id role theorem assumptions rule used_assumptions
    generated_goal_assumptions (if detailed_eigen <> "" then detailed_eigen else eigen)

fun transform_inline_assumption hypotheses_step proof_sketch =
  let
    fun inline_in_step (Linear_Part {node as Satallax_Step {generated_goal_assumptions,
          used_assumptions, rule, detailed_eigen, ...}, succs}) =
        if generated_goal_assumptions = [] then
          Linear_Part {node = node, succs = []}
        else
          let
            (*one singel goal is created, two hypothesis can be created, for the "and" rule:
              (A /\ B) create two hypotheses A, and B.*)
            fun set_hypotheses_as_goal [hypothesis] succs =
                Linear_Part {node = add_detailled_eigen detailed_eigen
                    (set_rule rule (add_assumptions used_assumptions
                    (find_proof_step hypotheses_step hypothesis))),
                  succs = map inline_in_step succs}
              | set_hypotheses_as_goal (hypothesis :: new_goal_hypotheses) succs =
                Linear_Part {node = set_rule rule (add_assumptions used_assumptions
                    (find_proof_step hypotheses_step hypothesis)),
                  succs = [set_hypotheses_as_goal new_goal_hypotheses succs]}
          in
            set_hypotheses_as_goal (snd (hd generated_goal_assumptions)) succs
          end
      | inline_in_step (Tree_Part {node = node, deps}) =
        Tree_Part {node = node, deps = map inline_in_step deps}

    fun inline_contradictory_assumptions (Linear_Part {node as Satallax_Step{id, theorem, ...},
       succs}) (to_inline, no_inline) =
      let
        val (succs, inliner) = fold_map inline_contradictory_assumptions succs
          (to_inline, (id, theorem) :: no_inline)
      in
        (Linear_Part {node = node, succs = succs}, inliner)
      end
    | inline_contradictory_assumptions (Tree_Part {node = Satallax_Step {id, role,
        theorem = AAtom theorem, assumptions, rule, generated_goal_assumptions,
        used_assumptions, detailed_eigen}, deps}) (to_inline, no_inline) =
      let
        val (dep', (to_inline', no_inline')) = fold_map inline_contradictory_assumptions deps
          (to_inline, no_inline)
        val to_inline'' =
          map (fn s => (s, get_conclusion (find_proof_step hypotheses_step s)))
            (filter (fn s => (nth_string s 0 = "h") andalso List.find (curry (op =) s o fst)
              no_inline' = NONE) (used_assumptions @ (map snd generated_goal_assumptions |> flat)))
          @ to_inline'
        val node' = Satallax_Step {id = id, role = role, theorem =
            AAtom (inline_if_needed_and_simplify theorem assumptions to_inline'' no_inline'),
          assumptions = assumptions, rule = rule,
          generated_goal_assumptions = generated_goal_assumptions, detailed_eigen = detailed_eigen,
          used_assumptions =
            filter (fn s => List.find (curry (op =) s o fst) to_inline'' = NONE)
            used_assumptions}
      in
        (Tree_Part {node = node', deps = dep'}, (to_inline'', no_inline'))
      end
  in
    fst (inline_contradictory_assumptions (inline_in_step proof_sketch) ([], []))
  end

fun correct_dependencies (Linear_Part {node, succs}) =
    Linear_Part {node = node, succs = map correct_dependencies succs}
  | correct_dependencies (Tree_Part {node, deps}) =
    let
      val new_used_assumptions =
        map (fn Linear_Part {node = (Satallax_Step{id, ...}), ...} => id
              | Tree_Part {node = (Satallax_Step{id, ...}), ...} => id) deps
    in
      Tree_Part {node = add_assumptions new_used_assumptions node,
        deps = map correct_dependencies deps}
    end

fun create_satallax_proof_graph (hypotheses_step, proof_sketch) =
  let
    fun create_step (step as Satallax_Step {generated_goal_assumptions, ...}) =
      Linear_Part {node = step,
        succs = map (create_step o (find_proof_step (hypotheses_step @ proof_sketch)))
          (map fst generated_goal_assumptions)}
    fun reverted_discharged_steps is_branched (Linear_Part {node as
          Satallax_Step {generated_goal_assumptions, ...}, succs}) =
        if is_branched orelse length generated_goal_assumptions > 1 then
          Tree_Part {node = node, deps = map (reverted_discharged_steps true) succs}
        else
          Linear_Part {node = node, succs = map (reverted_discharged_steps is_branched) succs}
    val proof_with_correct_sense =
        correct_dependencies (reverted_discharged_steps false
          (create_step (find_proof_step proof_sketch "0" )))
  in
    transform_inline_assumption hypotheses_step proof_with_correct_sense
  end

val satallax_known_rules = ["eq_ind", "eq_trans2", "eq_trans3", "eq_neg_neg_id", "eq_true",
  "eq_and_nor", "eq_or_nand", "eq_or_imp", "eq_and_imp", "eq_imp_or", "eq_iff", "eq_sym_eq",
  "eq_forall_nexists", "eq_exists_nforall", "eq_leib1", "eq_leib2", "eq_leib3", "eq_leib4",
  "eq_eta", "SinhE", "eq_forall_Seps", "eq_SPi_Seps", "eq_exists_Seps"]
val is_special_satallax_rule = member (op =) satallax_known_rules

fun terms_to_upper_case var (AAbs (((var', ty), b), ts)) =
    let
      val bdy = terms_to_upper_case var b
      val ts' = map (terms_to_upper_case var) ts
    in
      AAbs (((((var = var' ? String.implode o (map Char.toUpper) o String.explode) var'), ty),
        bdy), ts')
    end
  | terms_to_upper_case var (ATerm ((var', ty), ts)) =
    ATerm ((((var = var' ? String.implode o (map Char.toUpper) o String.explode) var'),
      ty), map (terms_to_upper_case var) ts)

fun add_quantifier_in_tree_part var_rule_to_add assumption_to_add
      (Linear_Part {node as Satallax_Step {detailed_eigen, rule, ...} , succs}) =
    Linear_Part {node = node, succs = map (add_quantifier_in_tree_part
      ((detailed_eigen <> "" ? cons (detailed_eigen, rule)) var_rule_to_add) assumption_to_add)
      succs}
  | add_quantifier_in_tree_part var_rule_to_add assumption_to_add
      (Tree_Part {node = Satallax_Step {rule, detailed_eigen, id, role, theorem = AAtom th,
      assumptions, used_assumptions, generated_goal_assumptions}, deps}) =
    let
      val theorem' = fold (fn v => fn body => terms_to_upper_case (fst v) body) var_rule_to_add th
      fun add_quantified_var (var, rule) = fn body =>
        let
          val quant = if rule = "tab_ex" then tptp_ho_exists else tptp_ho_forall
          val upperVar = (String.implode o (map Char.toUpper) o String.explode) var
          val quant_bdy = if rule = "tab_ex"
            then ATerm ((quant, []), [AAbs (((upperVar, dummy_atype), body), []) ]) else body
        in
          quant_bdy
        end
      val theorem'' = fold add_quantified_var var_rule_to_add theorem'
      val node' = mk_satallax_step id role (AAtom theorem'') assumptions rule
        (used_assumptions @ (filter (curry (op <=) (the (Int.fromString id)) o size)
        assumption_to_add)) generated_goal_assumptions detailed_eigen
    in
      if detailed_eigen <> "" then
        Tree_Part {node = node',
          deps = map (add_quantifier_in_tree_part ((detailed_eigen, rule) :: var_rule_to_add)
          (used_assumptions @ assumption_to_add)) deps}
      else
        Tree_Part {node = node',
          deps = map (add_quantifier_in_tree_part var_rule_to_add assumption_to_add) deps}
    end

fun transform_to_standard_atp_step already_transformed proof =
  let
    fun create_fact_step id =
      ((id, [id]), Axiom, AAtom (ATerm((id, []), [])), "", [])
    fun transform_one_step already_transformed (Satallax_Step {id, theorem, used_assumptions, role,
        rule, ...}) =
      let
        val role' = role_of_tptp_string role
        val new_transformed = filter
          (fn s => size s >=4 andalso not (is_special_satallax_rule s) andalso not
          (member (op =) already_transformed s)) used_assumptions
      in
        (map create_fact_step new_transformed
        @ [((id, []), if role' = Unknown then Plain else role', theorem, rule,
           map (fn s => (s, [])) (filter_out is_special_satallax_rule used_assumptions))],
        new_transformed @ already_transformed)
      end
    fun transform_steps (Linear_Part {node, succs}) already_transformed =
        transform_one_step already_transformed node
        ||> (fold_map transform_steps succs)
        ||> apfst flat
        |> (fn (a, (b, transformed)) => (a @ b, transformed))
      | transform_steps (Tree_Part {node, deps}) already_transformed =
        fold_map transform_steps deps already_transformed
        |>> flat
        ||> (fn transformed => transform_one_step transformed node)
        |> (fn (a, (b, transformed)) => (a @ b, transformed))
  in
    fst (transform_steps proof already_transformed)
  end

fun remove_unused_dependency steps =
  let
    fun find_all_ids [] = []
      | find_all_ids (((id, _), _, _, _, _) :: steps) = id :: find_all_ids steps
    fun keep_only_used used_ids steps =
      let
        fun remove_unused (((id, ids), role, theorem, rule, deps) :: steps) =
            (((id, ids), role, theorem, rule, filter (member (op =) used_ids o fst) deps) :: steps)
          | remove_unused [] = []
      in
        remove_unused steps
      end
  in
    keep_only_used (find_all_ids steps) steps
  end

fun parse_proof local_name problem =
  strip_spaces_except_between_idents
  #> raw_explode
  #>
    (if local_name <> satallaxN then
      (Scan.error (!! (fn _ => raise UNRECOGNIZED_ATP_PROOF ())
        (Scan.finite Symbol.stopper (Scan.repeats1 (parse_line local_name problem))))
         #> fst)
    else
      (Scan.error (!! (fn _ => raise UNRECOGNIZED_ATP_PROOF ())
        (Scan.finite Symbol.stopper (Scan.repeat1 (parse_tstp_thf0_satallax_line))))
        #> fst
        #> rev
        #> map to_satallax_step
        #> seperate_assumptions_and_steps
        #> create_satallax_proof_graph
        #> add_quantifier_in_tree_part [] []
        #> transform_to_standard_atp_step []
        #> remove_unused_dependency))

fun atp_proof_of_tstplike_proof _ _ "" = []
  | atp_proof_of_tstplike_proof local_prover problem tstp =
    (case core_of_agsyhol_proof tstp of
      SOME facts => facts |> map (core_inference agsyhol_core_rule)
    | NONE =>
      tstp ^ "$" (* the $ sign acts as a sentinel (FIXME: needed?) *)
      |> parse_proof local_prover problem
      |> local_prover = vampireN ? perhaps (try (sort (vampire_step_name_ord o apply2 #1)))
      |> local_prover = zipperpositionN ? rev)

end;