(*  Title:      HOL/Tools/Meson/meson.ML

    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

    Author:     Jasmin Blanchette, TU Muenchen

The MESON resolution proof procedure for HOL.

When making clauses, avoids using the rewriter -- instead uses RS recursively.

*)

signature MESON =

sig

  val trace : bool Config.T

  val unfold_set_consts : bool Config.T

  val max_clauses : int Config.T

  val term_pair_of: indexname * (typ * 'a) -> term * 'a

  val size_of_subgoals: thm -> int

  val has_too_many_clauses: Proof.context -> term -> bool

  val make_cnf:

    thm list -> thm -> Proof.context

    -> Proof.context -> thm list * Proof.context

  val finish_cnf: thm list -> thm list

  val unfold_set_const_simps : Proof.context -> thm list

  val presimplified_consts : Proof.context -> string list

  val presimplify: Proof.context -> thm -> thm

  val make_nnf: Proof.context -> thm -> thm

  val choice_theorems : theory -> thm list

  val skolemize_with_choice_theorems : Proof.context -> thm list -> thm -> thm

  val skolemize : Proof.context -> thm -> thm

  val extensionalize_conv : Proof.context -> conv

  val extensionalize_theorem : Proof.context -> thm -> thm

  val is_fol_term: theory -> term -> bool

  val make_clauses_unsorted: Proof.context -> thm list -> thm list

  val make_clauses: Proof.context -> thm list -> thm list

  val make_horns: thm list -> thm list

  val best_prolog_tac: (thm -> int) -> thm list -> tactic

  val depth_prolog_tac: thm list -> tactic

  val gocls: thm list -> thm list

  val skolemize_prems_tac : Proof.context -> thm list -> int -> tactic

  val MESON:

    tactic -> (thm list -> thm list) -> (thm list -> tactic) -> Proof.context

    -> int -> tactic

  val best_meson_tac: (thm -> int) -> Proof.context -> int -> tactic

  val safe_best_meson_tac: Proof.context -> int -> tactic

  val depth_meson_tac: Proof.context -> int -> tactic

  val prolog_step_tac': thm list -> int -> tactic

  val iter_deepen_prolog_tac: thm list -> tactic

  val iter_deepen_meson_tac: Proof.context -> thm list -> int -> tactic

  val make_meta_clause: thm -> thm

  val make_meta_clauses: thm list -> thm list

  val meson_tac: Proof.context -> thm list -> int -> tactic

end

structure Meson : MESON =

struct

val trace = Attrib.setup_config_bool @{binding meson_trace} (K false)

fun trace_msg ctxt msg = if Config.get ctxt trace then tracing (msg ()) else ()

val unfold_set_consts =

  Attrib.setup_config_bool @{binding meson_unfold_set_consts} (K false)

val max_clauses = Attrib.setup_config_int @{binding meson_max_clauses} (K 60)

(*No known example (on 1-5-2007) needs even thirty*)

val iter_deepen_limit = 50;

val disj_forward = @{thm disj_forward};

val disj_forward2 = @{thm disj_forward2};

val make_pos_rule = @{thm make_pos_rule};

val make_pos_rule' = @{thm make_pos_rule'};

val make_pos_goal = @{thm make_pos_goal};

val make_neg_rule = @{thm make_neg_rule};

val make_neg_rule' = @{thm make_neg_rule'};

val make_neg_goal = @{thm make_neg_goal};

val conj_forward = @{thm conj_forward};

val all_forward = @{thm all_forward};

val ex_forward = @{thm ex_forward};

val not_conjD = @{thm not_conjD};

val not_disjD = @{thm not_disjD};

val not_notD = @{thm not_notD};

val not_allD = @{thm not_allD};

val not_exD = @{thm not_exD};

val imp_to_disjD = @{thm imp_to_disjD};

val not_impD = @{thm not_impD};

val iff_to_disjD = @{thm iff_to_disjD};

val not_iffD = @{thm not_iffD};

val conj_exD1 = @{thm conj_exD1};

val conj_exD2 = @{thm conj_exD2};

val disj_exD = @{thm disj_exD};

val disj_exD1 = @{thm disj_exD1};

val disj_exD2 = @{thm disj_exD2};

val disj_assoc = @{thm disj_assoc};

val disj_comm = @{thm disj_comm};

val disj_FalseD1 = @{thm disj_FalseD1};

val disj_FalseD2 = @{thm disj_FalseD2};

(**** Operators for forward proof ****)

(** First-order Resolution **)

fun term_pair_of (ix, (ty,t)) = (Var (ix,ty), t);

(*FIXME: currently does not "rename variables apart"*)

fun first_order_resolve thA thB =

  (case

    try (fn () =>

      let val thy = theory_of_thm thA

          val tmA = concl_of thA

          val Const("==>",_) $ tmB $ _ = prop_of thB

          val tenv =

            Pattern.first_order_match thy (tmB, tmA)

                                          (Vartab.empty, Vartab.empty) |> snd

          val ct_pairs = map (pairself (cterm_of thy) o term_pair_of) (Vartab.dest tenv)

      in  thA RS (cterm_instantiate ct_pairs thB)  end) () of

    SOME th => th

  | NONE => raise THM ("first_order_resolve", 0, [thA, thB]))

(* Hack to make it less likely that we lose our precious bound variable names in

   "rename_bound_vars_RS" below, because of a clash. *)

val protect_prefix = "Meson_xyzzy"

fun protect_bound_var_names (t $ u) =

    protect_bound_var_names t $ protect_bound_var_names u

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

    Abs (protect_prefix ^ s, T, protect_bound_var_names t')

  | protect_bound_var_names t = t

fun fix_bound_var_names old_t new_t =

  let

    fun quant_of @{const_name All} = SOME true

      | quant_of @{const_name Ball} = SOME true

      | quant_of @{const_name Ex} = SOME false

      | quant_of @{const_name Bex} = SOME false

      | quant_of _ = NONE

    val flip_quant = Option.map not

    fun some_eq (SOME x) (SOME y) = x = y

      | some_eq _ _ = false

    fun add_names quant (Const (quant_s, _) $ Abs (s, _, t')) =

        add_names quant t' #> some_eq quant (quant_of quant_s) ? cons s

      | add_names quant (@{const Not} $ t) = add_names (flip_quant quant) t

      | add_names quant (@{const implies} $ t1 $ t2) =

        add_names (flip_quant quant) t1 #> add_names quant t2

      | add_names quant (t1 $ t2) = fold (add_names quant) [t1, t2]

      | add_names _ _ = I

    fun lost_names quant =

      subtract (op =) (add_names quant new_t []) (add_names quant old_t [])

    fun aux ((t1 as Const (quant_s, _)) $ (Abs (s, T, t'))) =

      t1 $ Abs (s |> String.isPrefix protect_prefix s

                   ? perhaps (try (fn _ => hd (lost_names (quant_of quant_s)))),

                T, aux t')

      | aux (t1 $ t2) = aux t1 $ aux t2

      | aux t = t

  in aux new_t end

(* Forward proof while preserving bound variables names *)

fun rename_bound_vars_RS th rl =

  let

    val t = concl_of th

    val r = concl_of rl

    val th' = th RS Thm.rename_boundvars r (protect_bound_var_names r) rl

    val t' = concl_of th'

  in Thm.rename_boundvars t' (fix_bound_var_names t t') th' end

(*raises exception if no rules apply*)

fun tryres (th, rls) =

  let fun tryall [] = raise THM("tryres", 0, th::rls)

        | tryall (rl::rls) =

          (rename_bound_vars_RS th rl handle THM _ => tryall rls)

  in  tryall rls  end;

(*Permits forward proof from rules that discharge assumptions. The supplied proof state st,

  e.g. from conj_forward, should have the form

    "[| P' ==> ?P; Q' ==> ?Q |] ==> ?P & ?Q"

  and the effect should be to instantiate ?P and ?Q with normalized versions of P' and Q'.*)

fun forward_res ctxt nf st =

  let fun forward_tacf [prem] = rtac (nf prem) 1

        | forward_tacf prems =

            error (cat_lines

              ("Bad proof state in forward_res, please inform lcp@cl.cam.ac.uk:" ::

                Display.string_of_thm ctxt st ::

                "Premises:" :: map (Display.string_of_thm ctxt) prems))

  in

    case Seq.pull (ALLGOALS (Misc_Legacy.METAHYPS forward_tacf) st)

    of SOME(th,_) => th

     | NONE => raise THM("forward_res", 0, [st])

  end;

(*Are any of the logical connectives in "bs" present in the term?*)

fun has_conns bs =

  let fun has (Const _) = false

        | has (Const(@{const_name Trueprop},_) $ p) = has p

        | has (Const(@{const_name Not},_) $ p) = has p

        | has (Const(@{const_name HOL.disj},_) $ p $ q) = member (op =) bs @{const_name HOL.disj} orelse has p orelse has q

        | has (Const(@{const_name HOL.conj},_) $ p $ q) = member (op =) bs @{const_name HOL.conj} orelse has p orelse has q

        | has (Const(@{const_name All},_) $ Abs(_,_,p)) = member (op =) bs @{const_name All} orelse has p

        | has (Const(@{const_name Ex},_) $ Abs(_,_,p)) = member (op =) bs @{const_name Ex} orelse has p

        | has _ = false

  in  has  end;

(**** Clause handling ****)

fun literals (Const(@{const_name Trueprop},_) $ P) = literals P

  | literals (Const(@{const_name HOL.disj},_) $ P $ Q) = literals P @ literals Q

  | literals (Const(@{const_name Not},_) $ P) = [(false,P)]

  | literals P = [(true,P)];

(*number of literals in a term*)

val nliterals = length o literals;

(*** Tautology Checking ***)

fun signed_lits_aux (Const (@{const_name HOL.disj}, _) $ P $ Q) (poslits, neglits) =

      signed_lits_aux Q (signed_lits_aux P (poslits, neglits))

  | signed_lits_aux (Const(@{const_name Not},_) $ P) (poslits, neglits) = (poslits, P::neglits)

  | signed_lits_aux P (poslits, neglits) = (P::poslits, neglits);

fun signed_lits th = signed_lits_aux (HOLogic.dest_Trueprop (concl_of th)) ([],[]);

(*Literals like X=X are tautologous*)

fun taut_poslit (Const(@{const_name HOL.eq},_) $ t $ u) = t aconv u

  | taut_poslit (Const(@{const_name True},_)) = true

  | taut_poslit _ = false;

fun is_taut th =

  let val (poslits,neglits) = signed_lits th

  in  exists taut_poslit poslits

      orelse

      exists (member (op aconv) neglits) (HOLogic.false_const :: poslits)

  end

  handle TERM _ => false;       (*probably dest_Trueprop on a weird theorem*)

(*** To remove trivial negated equality literals from clauses ***)

(*They are typically functional reflexivity axioms and are the converses of

  injectivity equivalences*)

val not_refl_disj_D = @{thm not_refl_disj_D};

(*Is either term a Var that does not properly occur in the other term?*)

fun eliminable (t as Var _, u) = t aconv u orelse not (Logic.occs(t,u))

  | eliminable (u, t as Var _) = t aconv u orelse not (Logic.occs(t,u))

  | eliminable _ = false;

fun refl_clause_aux 0 th = th

  | refl_clause_aux n th =

       case HOLogic.dest_Trueprop (concl_of th) of

          (Const (@{const_name HOL.disj}, _) $ (Const (@{const_name HOL.disj}, _) $ _ $ _) $ _) =>

            refl_clause_aux n (th RS disj_assoc)    (*isolate an atom as first disjunct*)

        | (Const (@{const_name HOL.disj}, _) $ (Const(@{const_name Not},_) $ (Const(@{const_name HOL.eq},_) $ t $ u)) $ _) =>

            if eliminable(t,u)

            then refl_clause_aux (n-1) (th RS not_refl_disj_D)  (*Var inequation: delete*)

            else refl_clause_aux (n-1) (th RS disj_comm)  (*not between Vars: ignore*)

        | (Const (@{const_name HOL.disj}, _) $ _ $ _) => refl_clause_aux n (th RS disj_comm)

        | _ => (*not a disjunction*) th;

fun notequal_lits_count (Const (@{const_name HOL.disj}, _) $ P $ Q) =

      notequal_lits_count P + notequal_lits_count Q

  | notequal_lits_count (Const(@{const_name Not},_) $ (Const(@{const_name HOL.eq},_) $ _ $ _)) = 1

  | notequal_lits_count _ = 0;

(*Simplify a clause by applying reflexivity to its negated equality literals*)

fun refl_clause th =

  let val neqs = notequal_lits_count (HOLogic.dest_Trueprop (concl_of th))

  in  zero_var_indexes (refl_clause_aux neqs th)  end

  handle TERM _ => th;  (*probably dest_Trueprop on a weird theorem*)

(*** Removal of duplicate literals ***)

(*Forward proof, passing extra assumptions as theorems to the tactic*)

fun forward_res2 nf hyps st =

  case Seq.pull

        (REPEAT

         (Misc_Legacy.METAHYPS (fn major::minors => rtac (nf (minors@hyps) major) 1) 1)

         st)

  of SOME(th,_) => th

   | NONE => raise THM("forward_res2", 0, [st]);

(*Remove duplicates in P|Q by assuming ~P in Q

  rls (initially []) accumulates assumptions of the form P==>False*)

fun nodups_aux ctxt rls th = nodups_aux ctxt rls (th RS disj_assoc)

    handle THM _ => tryres(th,rls)

    handle THM _ => tryres(forward_res2 (nodups_aux ctxt) rls (th RS disj_forward2),

                           [disj_FalseD1, disj_FalseD2, asm_rl])

    handle THM _ => th;

(*Remove duplicate literals, if there are any*)

fun nodups ctxt th =

  if has_duplicates (op =) (literals (prop_of th))

    then nodups_aux ctxt [] th

    else th;

(*** The basic CNF transformation ***)

fun estimated_num_clauses bound t =

 let

  fun sum x y = if x < bound andalso y < bound then x+y else bound

  fun prod x y = if x < bound andalso y < bound then x*y else bound

  (*Estimate the number of clauses in order to detect infeasible theorems*)

  fun signed_nclauses b (Const(@{const_name Trueprop},_) $ t) = signed_nclauses b t

    | signed_nclauses b (Const(@{const_name Not},_) $ t) = signed_nclauses (not b) t

    | signed_nclauses b (Const(@{const_name HOL.conj},_) $ t $ u) =

        if b then sum (signed_nclauses b t) (signed_nclauses b u)

             else prod (signed_nclauses b t) (signed_nclauses b u)

    | signed_nclauses b (Const(@{const_name HOL.disj},_) $ t $ u) =

        if b then prod (signed_nclauses b t) (signed_nclauses b u)

             else sum (signed_nclauses b t) (signed_nclauses b u)

    | signed_nclauses b (Const(@{const_name HOL.implies},_) $ t $ u) =

        if b then prod (signed_nclauses (not b) t) (signed_nclauses b u)

             else sum (signed_nclauses (not b) t) (signed_nclauses b u)

    | signed_nclauses b (Const(@{const_name HOL.eq}, Type ("fun", [T, _])) $ t $ u) =

        if T = HOLogic.boolT then (*Boolean equality is if-and-only-if*)

            if b then sum (prod (signed_nclauses (not b) t) (signed_nclauses b u))

                          (prod (signed_nclauses (not b) u) (signed_nclauses b t))

                 else sum (prod (signed_nclauses b t) (signed_nclauses b u))

                          (prod (signed_nclauses (not b) t) (signed_nclauses (not b) u))

        else 1

    | signed_nclauses b (Const(@{const_name Ex}, _) $ Abs (_,_,t)) = signed_nclauses b t

    | signed_nclauses b (Const(@{const_name All},_) $ Abs (_,_,t)) = signed_nclauses b t

    | signed_nclauses _ _ = 1; (* literal *)

 in signed_nclauses true t end

fun has_too_many_clauses ctxt t =

  let val max_cl = Config.get ctxt max_clauses in

    estimated_num_clauses (max_cl + 1) t > max_cl

  end

(*Replaces universally quantified variables by FREE variables -- because

  assumptions may not contain scheme variables.  Later, generalize using Variable.export. *)

local  

  val spec_var = Thm.dest_arg (Thm.dest_arg (#2 (Thm.dest_implies (Thm.cprop_of spec))));

  val spec_varT = #T (Thm.rep_cterm spec_var);

  fun name_of (Const (@{const_name All}, _) $ Abs(x,_,_)) = x | name_of _ = Name.uu;

in  

  fun freeze_spec th ctxt =

    let

      val cert = Thm.cterm_of (Proof_Context.theory_of ctxt);

      val ([x], ctxt') = Variable.variant_fixes [name_of (HOLogic.dest_Trueprop (concl_of th))] ctxt;

      val spec' = Thm.instantiate ([], [(spec_var, cert (Free (x, spec_varT)))]) spec;

    in (th RS spec', ctxt') end

end;

(*Used with METAHYPS below. There is one assumption, which gets bound to prem

  and then normalized via function nf. The normal form is given to resolve_tac,

  instantiate a Boolean variable created by resolution with disj_forward. Since

  (nf prem) returns a LIST of theorems, we can backtrack to get all combinations.*)

fun resop nf [prem] = resolve_tac (nf prem) 1;

(* Any need to extend this list with "HOL.type_class", "HOL.eq_class",

   and "Pure.term"? *)

val has_meta_conn = exists_Const (member (op =) ["==", "==>", "=simp=>", "all", "prop"] o #1);

fun apply_skolem_theorem (th, rls) =

  let

    fun tryall [] = raise THM ("apply_skolem_theorem", 0, th::rls)

      | tryall (rl :: rls) =

        first_order_resolve th rl handle THM _ => tryall rls

  in tryall rls end

(* Conjunctive normal form, adding clauses from th in front of ths (for foldr).

   Strips universal quantifiers and breaks up conjunctions.

   Eliminates existential quantifiers using Skolemization theorems. *)

fun cnf old_skolem_ths ctxt ctxt0 (th, ths) =

  let val ctxt0r = Unsynchronized.ref ctxt0   (* FIXME ??? *)

      fun cnf_aux (th,ths) =

        if not (can HOLogic.dest_Trueprop (prop_of th)) then ths (*meta-level: ignore*)

        else if not (has_conns [@{const_name All}, @{const_name Ex}, @{const_name HOL.conj}] (prop_of th))

        then nodups ctxt0 th :: ths (*no work to do, terminate*)

        else case head_of (HOLogic.dest_Trueprop (concl_of th)) of

            Const (@{const_name HOL.conj}, _) => (*conjunction*)

                cnf_aux (th RS conjunct1, cnf_aux (th RS conjunct2, ths))

          | Const (@{const_name All}, _) => (*universal quantifier*)

                let val (th',ctxt0') = freeze_spec th (!ctxt0r)

                in  ctxt0r := ctxt0'; cnf_aux (th', ths) end

          | Const (@{const_name Ex}, _) =>

              (*existential quantifier: Insert Skolem functions*)

              cnf_aux (apply_skolem_theorem (th, old_skolem_ths), ths)

          | Const (@{const_name HOL.disj}, _) =>

              (*Disjunction of P, Q: Create new goal of proving ?P | ?Q and solve it using

                all combinations of converting P, Q to CNF.*)

              let val tac =

                  Misc_Legacy.METAHYPS (resop cnf_nil) 1 THEN

                   (fn st' => st' |> Misc_Legacy.METAHYPS (resop cnf_nil) 1)

              in  Seq.list_of (tac (th RS disj_forward)) @ ths  end

          | _ => nodups ctxt0 th :: ths  (*no work to do*)

      and cnf_nil th = cnf_aux (th,[])

      val cls =

        if has_too_many_clauses ctxt (concl_of th) then

          (trace_msg ctxt (fn () =>

               "cnf is ignoring: " ^ Display.string_of_thm ctxt0 th); ths)

        else

          cnf_aux (th, ths)

  in (cls, !ctxt0r) end

fun make_cnf old_skolem_ths th ctxt ctxt0 =

  cnf old_skolem_ths ctxt ctxt0 (th, [])

(*Generalization, removal of redundant equalities, removal of tautologies.*)

fun finish_cnf ths = filter (not o is_taut) (map refl_clause ths);

(**** Generation of contrapositives ****)

fun is_left (Const (@{const_name Trueprop}, _) $

               (Const (@{const_name HOL.disj}, _) $ (Const (@{const_name HOL.disj}, _) $ _ $ _) $ _)) = true

  | is_left _ = false;

(*Associate disjuctions to right -- make leftmost disjunct a LITERAL*)

fun assoc_right th =

  if is_left (prop_of th) then assoc_right (th RS disj_assoc)

  else th;

(*Must check for negative literal first!*)

val clause_rules = [disj_assoc, make_neg_rule, make_pos_rule];

(*For ordinary resolution. *)

val resolution_clause_rules = [disj_assoc, make_neg_rule', make_pos_rule'];

(*Create a goal or support clause, conclusing False*)

fun make_goal th =   (*Must check for negative literal first!*)

    make_goal (tryres(th, clause_rules))

  handle THM _ => tryres(th, [make_neg_goal, make_pos_goal]);

(*Sort clauses by number of literals*)

fun fewerlits(th1,th2) = nliterals(prop_of th1) < nliterals(prop_of th2);

fun sort_clauses ths = sort (make_ord fewerlits) ths;

fun has_bool @{typ bool} = true

  | has_bool (Type (_, Ts)) = exists has_bool Ts

  | has_bool _ = false

fun has_fun (Type (@{type_name fun}, _)) = true

  | has_fun (Type (_, Ts)) = exists has_fun Ts

  | has_fun _ = false

(*Is the string the name of a connective? Really only | and Not can remain,

  since this code expects to be called on a clause form.*)

val is_conn = member (op =)

    [@{const_name Trueprop}, @{const_name HOL.conj}, @{const_name HOL.disj},

     @{const_name HOL.implies}, @{const_name Not},

     @{const_name All}, @{const_name Ex}, @{const_name Ball}, @{const_name Bex}];

(*True if the term contains a function--not a logical connective--where the type

  of any argument contains bool.*)

val has_bool_arg_const =

    exists_Const

      (fn (c,T) => not(is_conn c) andalso exists has_bool (binder_types T));

(*A higher-order instance of a first-order constant? Example is the definition of

  one, 1, at a function type in theory Function_Algebras.*)

fun higher_inst_const thy (c,T) =

  case binder_types T of

      [] => false (*not a function type, OK*)

    | Ts => length (binder_types (Sign.the_const_type thy c)) <> length Ts;

(* Returns false if any Vars in the theorem mention type bool.

   Also rejects functions whose arguments are Booleans or other functions. *)

fun is_fol_term thy t =

    Term.is_first_order [@{const_name all}, @{const_name All},

                         @{const_name Ex}] t andalso

    not (exists_subterm (fn Var (_, T) => has_bool T orelse has_fun T

                          | _ => false) t orelse

         has_bool_arg_const t orelse

         exists_Const (higher_inst_const thy) t orelse

         has_meta_conn t);

fun rigid t = not (is_Var (head_of t));

fun ok4horn (Const (@{const_name Trueprop},_) $ (Const (@{const_name HOL.disj}, _) $ t $ _)) = rigid t

  | ok4horn (Const (@{const_name Trueprop},_) $ t) = rigid t

  | ok4horn _ = false;

(*Create a meta-level Horn clause*)

fun make_horn crules th =

  if ok4horn (concl_of th)

  then make_horn crules (tryres(th,crules)) handle THM _ => th

  else th;

(*Generate Horn clauses for all contrapositives of a clause. The input, th,

  is a HOL disjunction.*)

fun add_contras crules th hcs =

  let fun rots (0,_) = hcs

        | rots (k,th) = zero_var_indexes (make_horn crules th) ::

                        rots(k-1, assoc_right (th RS disj_comm))

  in case nliterals(prop_of th) of

        1 => th::hcs

      | n => rots(n, assoc_right th)

  end;

(*Use "theorem naming" to label the clauses*)

fun name_thms label =

    let fun name1 th (k, ths) =

          (k-1, Thm.put_name_hint (label ^ string_of_int k) th :: ths)

    in  fn ths => #2 (fold_rev name1 ths (length ths, []))  end;

(*Is the given disjunction an all-negative support clause?*)

fun is_negative th = forall (not o #1) (literals (prop_of th));

val neg_clauses = filter is_negative;

(***** MESON PROOF PROCEDURE *****)

fun rhyps (Const("==>",_) $ (Const(@{const_name Trueprop},_) $ A) $ phi,

           As) = rhyps(phi, A::As)

  | rhyps (_, As) = As;

(** Detecting repeated assumptions in a subgoal **)

(*The stringtree detects repeated assumptions.*)

fun ins_term t net = Net.insert_term (op aconv) (t, t) net;

(*detects repetitions in a list of terms*)

fun has_reps [] = false

  | has_reps [_] = false

  | has_reps [t,u] = (t aconv u)

  | has_reps ts = (fold ins_term ts Net.empty; false) handle Net.INSERT => true;

(*Like TRYALL eq_assume_tac, but avoids expensive THEN calls*)

fun TRYING_eq_assume_tac 0 st = Seq.single st

  | TRYING_eq_assume_tac i st =

       TRYING_eq_assume_tac (i-1) (Thm.eq_assumption i st)

       handle THM _ => TRYING_eq_assume_tac (i-1) st;

fun TRYALL_eq_assume_tac st = TRYING_eq_assume_tac (nprems_of st) st;

(*Loop checking: FAIL if trying to prove the same thing twice

  -- if *ANY* subgoal has repeated literals*)

fun check_tac st =

  if exists (fn prem => has_reps (rhyps(prem,[]))) (prems_of st)

  then  Seq.empty  else  Seq.single st;

(* net_resolve_tac actually made it slower... *)

fun prolog_step_tac horns i =

    (assume_tac i APPEND resolve_tac horns i) THEN check_tac THEN

    TRYALL_eq_assume_tac;

(*Sums the sizes of the subgoals, ignoring hypotheses (ancestors)*)

fun addconcl prem sz = size_of_term (Logic.strip_assums_concl prem) + sz;

fun size_of_subgoals st = fold_rev addconcl (prems_of st) 0;

(*Negation Normal Form*)

val nnf_rls = [imp_to_disjD, iff_to_disjD, not_conjD, not_disjD,

               not_impD, not_iffD, not_allD, not_exD, not_notD];

fun ok4nnf (Const (@{const_name Trueprop},_) $ (Const (@{const_name Not}, _) $ t)) = rigid t

  | ok4nnf (Const (@{const_name Trueprop},_) $ t) = rigid t

  | ok4nnf _ = false;

fun make_nnf1 ctxt th =

  if ok4nnf (concl_of th)

  then make_nnf1 ctxt (tryres(th, nnf_rls))

    handle THM ("tryres", _, _) =>

        forward_res ctxt (make_nnf1 ctxt)

           (tryres(th, [conj_forward,disj_forward,all_forward,ex_forward]))

    handle THM ("tryres", _, _) => th

  else th

fun unfold_set_const_simps ctxt =

  if Config.get ctxt unfold_set_consts then @{thms Collect_def_raw mem_def_raw}

  else []

(*The simplification removes defined quantifiers and occurrences of True and False.

  nnf_ss also includes the one-point simprocs,

  which are needed to avoid the various one-point theorems from generating junk clauses.*)

val nnf_simps =

  @{thms simp_implies_def Ex1_def Ball_def Bex_def if_True if_False if_cancel

         if_eq_cancel cases_simp}

val nnf_extra_simps = @{thms split_ifs ex_simps all_simps simp_thms}

(* FIXME: "let_simp" is probably redundant now that we also rewrite with

  "Let_def_raw". *)

val nnf_ss =

  HOL_basic_ss addsimps nnf_extra_simps

    addsimprocs [@{simproc defined_All}, @{simproc defined_Ex}, @{simproc neq},

                 @{simproc let_simp}]

fun presimplified_consts ctxt =

  [@{const_name simp_implies}, @{const_name False}, @{const_name True},

   @{const_name Ex1}, @{const_name Ball}, @{const_name Bex}, @{const_name If},

   @{const_name Let}]

  |> Config.get ctxt unfold_set_consts

     ? append ([@{const_name Collect}, @{const_name Set.member}])

fun presimplify ctxt =

  rewrite_rule (map safe_mk_meta_eq nnf_simps)

  #> simplify nnf_ss

  (* TODO: avoid introducing "Set.member" in "Ball_def" "Bex_def" above if and

     when "metis_unfold_set_consts" becomes the only mode of operation. *)

  #> Raw_Simplifier.rewrite_rule

         (@{thm Let_def_raw} :: unfold_set_const_simps ctxt)

fun make_nnf ctxt th = case prems_of th of

    [] => th |> presimplify ctxt |> make_nnf1 ctxt

  | _ => raise THM ("make_nnf: premises in argument", 0, [th]);

fun choice_theorems thy =

  try (Global_Theory.get_thm thy) "Hilbert_Choice.choice" |> the_list

(* Pull existential quantifiers to front. This accomplishes Skolemization for

   clauses that arise from a subgoal. *)

fun skolemize_with_choice_theorems ctxt choice_ths =

  let

    fun aux th =

      if not (has_conns [@{const_name Ex}] (prop_of th)) then

        th

      else

        tryres (th, choice_ths @

                    [conj_exD1, conj_exD2, disj_exD, disj_exD1, disj_exD2])

        |> aux

        handle THM ("tryres", _, _) =>

               tryres (th, [conj_forward, disj_forward, all_forward])

               |> forward_res ctxt aux

               |> aux

               handle THM ("tryres", _, _) =>

                      rename_bound_vars_RS th ex_forward

                      |> forward_res ctxt aux

  in aux o make_nnf ctxt end

fun skolemize ctxt =

  let val thy = Proof_Context.theory_of ctxt in

    skolemize_with_choice_theorems ctxt (choice_theorems thy)

  end

(* Removes the lambdas from an equation of the form "t = (%x1 ... xn. u)". It

   would be desirable to do this symmetrically but there's at least one existing

   proof in "Tarski" that relies on the current behavior. *)

fun extensionalize_conv ctxt ct =

  case term_of ct of

    Const (@{const_name HOL.eq}, _) $ _ $ Abs _ =>

    ct |> (Conv.rewr_conv @{thm fun_eq_iff [THEN eq_reflection]}

           then_conv extensionalize_conv ctxt)

  | _ $ _ => Conv.comb_conv (extensionalize_conv ctxt) ct

  | Abs _ => Conv.abs_conv (extensionalize_conv o snd) ctxt ct

  | _ => Conv.all_conv ct

val extensionalize_theorem = Conv.fconv_rule o extensionalize_conv

(* "RS" can fail if "unify_search_bound" is too small. *)

fun try_skolemize_etc ctxt =

  Raw_Simplifier.rewrite_rule (unfold_set_const_simps ctxt)

  (* Extensionalize "th", because that makes sense and that's what Sledgehammer

     does, but also keep an unextensionalized version of "th" for backward

     compatibility. *)

  #> (fn th => insert Thm.eq_thm_prop (extensionalize_theorem ctxt th) [th])

  #> map_filter (fn th => try (skolemize ctxt) th

                          |> tap (fn NONE =>

                                     trace_msg ctxt (fn () =>

                                         "Failed to skolemize " ^

                                          Display.string_of_thm ctxt th)

                                   | _ => ()))

fun add_clauses ctxt th cls =

  let val ctxt0 = Variable.global_thm_context th

      val (cnfs, ctxt) = make_cnf [] th ctxt ctxt0

  in Variable.export ctxt ctxt0 cnfs @ cls end;

(*Make clauses from a list of theorems, previously Skolemized and put into nnf.

  The resulting clauses are HOL disjunctions.*)

fun make_clauses_unsorted ctxt ths = fold_rev (add_clauses ctxt) ths [];

val make_clauses = sort_clauses oo make_clauses_unsorted;

(*Convert a list of clauses (disjunctions) to Horn clauses (contrapositives)*)

fun make_horns ths =

    name_thms "Horn#"

      (distinct Thm.eq_thm_prop (fold_rev (add_contras clause_rules) ths []));

(*Could simply use nprems_of, which would count remaining subgoals -- no

  discrimination as to their size!  With BEST_FIRST, fails for problem 41.*)

fun best_prolog_tac sizef horns =

    BEST_FIRST (has_fewer_prems 1, sizef) (prolog_step_tac horns 1);

fun depth_prolog_tac horns =

    DEPTH_FIRST (has_fewer_prems 1) (prolog_step_tac horns 1);

(*Return all negative clauses, as possible goal clauses*)

fun gocls cls = name_thms "Goal#" (map make_goal (neg_clauses cls));

fun skolemize_prems_tac ctxt prems =

  cut_facts_tac (maps (try_skolemize_etc ctxt) prems) THEN' REPEAT o etac exE

(*Basis of all meson-tactics.  Supplies cltac with clauses: HOL disjunctions.

  Function mkcl converts theorems to clauses.*)

fun MESON preskolem_tac mkcl cltac ctxt i st =

  SELECT_GOAL

    (EVERY [Object_Logic.atomize_prems_tac 1,

            rtac ccontr 1,

            preskolem_tac,

            Subgoal.FOCUS (fn {context = ctxt', prems = negs, ...} =>

                      EVERY1 [skolemize_prems_tac ctxt negs,

                              Subgoal.FOCUS (cltac o mkcl o #prems) ctxt']) ctxt 1]) i st

  handle THM _ => no_tac st;    (*probably from make_meta_clause, not first-order*)

(** Best-first search versions **)

(*ths is a list of additional clauses (HOL disjunctions) to use.*)

fun best_meson_tac sizef ctxt =

  MESON all_tac (make_clauses ctxt)

    (fn cls =>

         THEN_BEST_FIRST (resolve_tac (gocls cls) 1)

                         (has_fewer_prems 1, sizef)

                         (prolog_step_tac (make_horns cls) 1))

    ctxt

(*First, breaks the goal into independent units*)

fun safe_best_meson_tac ctxt =

  SELECT_GOAL (TRY (safe_tac ctxt) THEN TRYALL (best_meson_tac size_of_subgoals ctxt));

(** Depth-first search version **)

fun depth_meson_tac ctxt =

  MESON all_tac (make_clauses ctxt)

    (fn cls => EVERY [resolve_tac (gocls cls) 1, depth_prolog_tac (make_horns cls)])

    ctxt

(** Iterative deepening version **)

(*This version does only one inference per call;

  having only one eq_assume_tac speeds it up!*)

fun prolog_step_tac' horns =

    let val (horn0s, _) = (*0 subgoals vs 1 or more*)

            take_prefix Thm.no_prems horns

        val nrtac = net_resolve_tac horns

    in  fn i => eq_assume_tac i ORELSE

                match_tac horn0s i ORELSE  (*no backtracking if unit MATCHES*)

                ((assume_tac i APPEND nrtac i) THEN check_tac)

    end;

fun iter_deepen_prolog_tac horns =

    ITER_DEEPEN iter_deepen_limit (has_fewer_prems 1) (prolog_step_tac' horns);

fun iter_deepen_meson_tac ctxt ths = ctxt |> MESON all_tac (make_clauses ctxt)

  (fn cls =>

    (case (gocls (cls @ ths)) of

      [] => no_tac  (*no goal clauses*)

    | goes =>

        let

          val horns = make_horns (cls @ ths)

          val _ = trace_msg ctxt (fn () =>

            cat_lines ("meson method called:" ::

              map (Display.string_of_thm ctxt) (cls @ ths) @

              ["clauses:"] @ map (Display.string_of_thm ctxt) horns))

        in

          THEN_ITER_DEEPEN iter_deepen_limit

            (resolve_tac goes 1) (has_fewer_prems 1) (prolog_step_tac' horns)

        end));

fun meson_tac ctxt ths =

  SELECT_GOAL (TRY (safe_tac ctxt) THEN TRYALL (iter_deepen_meson_tac ctxt ths));

(**** Code to support ordinary resolution, rather than Model Elimination ****)

(*Convert a list of clauses (disjunctions) to meta-level clauses (==>),

  with no contrapositives, for ordinary resolution.*)

(*Rules to convert the head literal into a negated assumption. If the head

  literal is already negated, then using notEfalse instead of notEfalse'

  prevents a double negation.*)

val notEfalse = read_instantiate @{context} [(("R", 0), "False")] notE;

val notEfalse' = rotate_prems 1 notEfalse;

fun negated_asm_of_head th =

    th RS notEfalse handle THM _ => th RS notEfalse';

(*Converting one theorem from a disjunction to a meta-level clause*)

fun make_meta_clause th =

  let val (fth,thaw) = Drule.legacy_freeze_thaw_robust th

  in  

      (zero_var_indexes o Thm.varifyT_global o thaw 0 o 

       negated_asm_of_head o make_horn resolution_clause_rules) fth

  end;

fun make_meta_clauses ths =

    name_thms "MClause#"

      (distinct Thm.eq_thm_prop (map make_meta_clause ths));

end;