(*  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;