diff --git a/src/HOL/Tools/Meson/meson.ML b/src/HOL/Tools/Meson/meson.ML --- a/src/HOL/Tools/Meson/meson.ML +++ b/src/HOL/Tools/Meson/meson.ML @@ -1,793 +1,793 @@ (* 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 type simp_options = {if_simps : bool, let_simps : bool} val simp_options_all_true : simp_options val trace : bool Config.T val max_clauses : int Config.T val first_order_resolve : Proof.context -> thm -> thm -> thm val size_of_subgoals: thm -> int val has_too_many_clauses: Proof.context -> term -> bool val make_cnf: thm list -> thm -> Proof.context -> thm list * Proof.context val finish_cnf: thm list -> thm list val presimplified_consts : string list val presimplify: simp_options -> Proof.context -> thm -> thm val make_nnf: simp_options -> Proof.context -> thm -> thm val choice_theorems : theory -> thm list val skolemize_with_choice_theorems : simp_options -> Proof.context -> thm list -> thm -> thm val skolemize : simp_options -> Proof.context -> thm -> thm val cong_extensionalize_thm : Proof.context -> thm -> thm val abs_extensionalize_conv : Proof.context -> conv val abs_extensionalize_thm : Proof.context -> thm -> thm 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: Proof.context -> (thm -> int) -> thm list -> tactic val depth_prolog_tac: Proof.context -> thm list -> tactic val gocls: thm list -> thm list val skolemize_prems_tac : simp_options -> 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': Proof.context -> thm list -> int -> tactic val iter_deepen_prolog_tac: Proof.context -> thm list -> tactic val iter_deepen_meson_tac: Proof.context -> thm list -> int -> tactic val make_meta_clause: Proof.context -> thm -> thm val make_meta_clauses: Proof.context -> thm list -> thm list val meson_tac: Proof.context -> thm list -> int -> tactic end structure Meson : MESON = struct type simp_options = {if_simps : bool, let_simps : bool} val simp_options_all_true = {if_simps = true, let_simps = true} 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 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 **) (*FIXME: currently does not "rename variables apart"*) fun first_order_resolve ctxt thA thB = (case \<^try>\ let val thy = Proof_Context.theory_of ctxt val tmA = Thm.concl_of thA val \<^Const_>\Pure.imp for tmB _\ = Thm.prop_of thB val tenv = Pattern.first_order_match thy (tmB, tmA) (Vartab.empty, Vartab.empty) |> snd val insts = Vartab.fold (fn (xi, (_, t)) => cons (xi, Thm.cterm_of ctxt t)) tenv []; in thA RS (infer_instantiate ctxt insts 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 for t\ = add_names (flip_quant quant) t | add_names quant \<^Const_>\implies for 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 = Thm.concl_of th val r = Thm.concl_of rl val th' = th RS Thm.rename_boundvars r (protect_bound_var_names r) rl val t' = Thm.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; (* Special version of "resolve_tac" that works around an explosion in the unifier. If the goal has the form "?P c", the danger is that resolving it against a property of the form "... c ... c ... c ..." will lead to a huge unification problem, due to the (spurious) choices between projection and imitation. The workaround is to instantiate "?P := (%c. ... c ... c ... c ...)" manually. *) fun quant_resolve_tac ctxt th i st = case (Thm.concl_of st, Thm.prop_of th) of (\<^Const_>\Trueprop for \Var _ $ (c as Free _)\\, \<^Const_>\Trueprop for _\) => let val cc = Thm.cterm_of ctxt c val ct = Thm.dest_arg (Thm.cprop_of th) in resolve_tac ctxt [th] i (Thm.instantiate' [] [SOME (Thm.lambda cc ct)] st) end | _ => resolve_tac ctxt [th] i st (*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" + "\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 tacf [prem] = quant_resolve_tac ctxt (nf prem) 1 | tacf prems = error (cat_lines ("Bad proof state in forward_res, please inform lcp@cl.cam.ac.uk:" :: Thm.string_of_thm ctxt st :: "Premises:" :: map (Thm.string_of_thm ctxt) prems)) in case Seq.pull (ALLGOALS (Misc_Legacy.METAHYPS ctxt 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_>\Trueprop for p\ = has p | has \<^Const_>\Not for p\ = has p | has \<^Const_>\disj for p q\ = member (op =) bs \<^const_name>\disj\ orelse has p orelse has q | has \<^Const_>\conj for p q\ = member (op =) bs \<^const_name>\conj\ orelse has p orelse has q | has \<^Const_>\All _ for \Abs(_,_,p)\\ = member (op =) bs \<^const_name>\All\ orelse has p | has \<^Const_>\Ex _ for \Abs(_,_,p)\\ = member (op =) bs \<^const_name>\Ex\ orelse has p | has _ = false in has end; (**** Clause handling ****) fun literals \<^Const_>\Trueprop for P\ = literals P | literals \<^Const_>\disj for P Q\ = literals P @ literals Q | literals \<^Const_>\Not for 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_>\disj for P Q\ (poslits, neglits) = signed_lits_aux Q (signed_lits_aux P (poslits, neglits)) | signed_lits_aux \<^Const_>\Not for 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 (Thm.concl_of th)) ([],[]); (*Literals like X=X are tautologous*) fun taut_poslit \<^Const_>\HOL.eq _ for t u\ = t aconv u | taut_poslit \<^Const_>\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) (\<^term>\False\ :: 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 (Thm.concl_of th) of \<^Const_>\disj for \<^Const_>\disj for _ _\ _\ => refl_clause_aux n (th RS disj_assoc) (*isolate an atom as first disjunct*) | \<^Const_>\disj for \<^Const_>\Not for \<^Const_>\HOL.eq _ for 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>\disj for _ _\ => refl_clause_aux n (th RS disj_comm) | _ => (*not a disjunction*) th; fun notequal_lits_count \<^Const_>\disj for P Q\ = notequal_lits_count P + notequal_lits_count Q | notequal_lits_count \<^Const_>\Not for \<^Const_>\HOL.eq _ for _ _\\ = 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 (Thm.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 ctxt nf hyps st = case Seq.pull (REPEAT (Misc_Legacy.METAHYPS ctxt (fn major::minors => resolve_tac ctxt [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 +(*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 ctxt (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 (Thm.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_>\Trueprop for t\ = signed_nclauses b t | signed_nclauses b \<^Const_>\Not for t\ = signed_nclauses (not b) t | signed_nclauses b \<^Const_>\conj for 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_>\disj for 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_>\implies for 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_>\HOL.eq \T\ for 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_>\Ex _ for \Abs (_,_,t)\\ = signed_nclauses b t | signed_nclauses b \<^Const_>\All _ for \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)))) |> Thm.term_of |> dest_Var; fun name_of \<^Const_>\All _ for \Abs(x, _, _)\\ = x | name_of _ = Name.uu; in fun freeze_spec th ctxt = let val ([x], ctxt') = Variable.variant_fixes [name_of (HOLogic.dest_Trueprop (Thm.concl_of th))] ctxt; val spec' = spec |> Thm.instantiate (TVars.empty, Vars.make [(spec_var, Thm.cterm_of ctxt' (Free (x, snd spec_var)))]); in (th RS spec', ctxt') end end; fun apply_skolem_theorem ctxt (th, rls) = let fun tryall [] = raise THM ("apply_skolem_theorem", 0, th::rls) | tryall (rl :: rls) = first_order_resolve ctxt 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 (th, ths) = let val ctxt_ref = Unsynchronized.ref ctxt (* FIXME ??? *) fun cnf_aux (th,ths) = if not (can HOLogic.dest_Trueprop (Thm.prop_of th)) then ths (*meta-level: ignore*) else if not (has_conns [\<^const_name>\All\, \<^const_name>\Ex\, \<^const_name>\HOL.conj\] (Thm.prop_of th)) then nodups ctxt th :: ths (*no work to do, terminate*) else case head_of (HOLogic.dest_Trueprop (Thm.concl_of th)) of \<^Const_>\conj\ => (*conjunction*) cnf_aux (th RS conjunct1, cnf_aux (th RS conjunct2, ths)) | \<^Const_>\All _\ => (*universal quantifier*) let val (th', ctxt') = freeze_spec th (! ctxt_ref) in ctxt_ref := ctxt'; cnf_aux (th', ths) end | \<^Const_>\Ex _\ => (*existential quantifier: Insert Skolem functions*) cnf_aux (apply_skolem_theorem (! ctxt_ref) (th, old_skolem_ths), ths) | \<^Const_>\disj\ => (*Disjunction of P, Q: Create new goal of proving ?P | ?Q and solve it using all combinations of converting P, Q to CNF.*) (*There is one assumption, which gets bound to prem and then normalized via cnf_nil. The normal form is given to resolve_tac, instantiate a Boolean variable created by resolution with disj_forward. Since (cnf_nil prem) returns a LIST of theorems, we can backtrack to get all combinations.*) let val tac = Misc_Legacy.METAHYPS ctxt (fn [prem] => resolve_tac ctxt (cnf_nil prem) 1) 1 in Seq.list_of ((tac THEN tac) (th RS disj_forward)) @ ths end | _ => nodups ctxt th :: ths (*no work to do*) and cnf_nil th = cnf_aux (th, []) val cls = if has_too_many_clauses ctxt (Thm.concl_of th) then (trace_msg ctxt (fn () => "cnf is ignoring: " ^ Thm.string_of_thm ctxt th); ths) else cnf_aux (th, ths) in (cls, !ctxt_ref) end fun make_cnf old_skolem_ths th ctxt = cnf old_skolem_ths ctxt (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_>\Trueprop for \<^Const_>\disj for \<^Const_>\disj for _ _\ _\\ = true | is_left _ = false; (*Associate disjuctions to right -- make leftmost disjunct a LITERAL*) fun assoc_right th = if is_left (Thm.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]); fun rigid t = not (is_Var (head_of t)); fun ok4horn \<^Const_>\Trueprop for \<^Const_>\disj for t _\\ = rigid t | ok4horn \<^Const_>\Trueprop for t\ = rigid t | ok4horn _ = false; (*Create a meta-level Horn clause*) fun make_horn crules th = if ok4horn (Thm.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(Thm.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 (Thm.prop_of th)); val neg_clauses = filter is_negative; (***** MESON PROOF PROCEDURE *****) fun rhyps (\<^Const_>\Pure.imp for \<^Const_>\Trueprop for 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 (Thm.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,[]))) (Thm.prems_of st) then Seq.empty else Seq.single st; (* resolve_from_net_tac actually made it slower... *) fun prolog_step_tac ctxt horns i = (assume_tac ctxt i APPEND resolve_tac ctxt 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 (Thm.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_>\Trueprop for \<^Const_>\Not for t\\ = rigid t | ok4nnf \<^Const_>\Trueprop for t\ = rigid t | ok4nnf _ = false; fun make_nnf1 ctxt th = if ok4nnf (Thm.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 (*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} fun nnf_extra_simps ({if_simps, ...} : simp_options) = (if if_simps then @{thms split_ifs} else []) @ @{thms ex_simps all_simps simp_thms} (* FIXME: "let_simp" is probably redundant now that we also rewrite with "Let_def [abs_def]". *) fun nnf_ss simp_options = simpset_of (put_simpset HOL_basic_ss \<^context> addsimps (nnf_extra_simps simp_options) addsimprocs [\<^simproc>\defined_All\, \<^simproc>\defined_Ex\, \<^simproc>\neq\, \<^simproc>\let_simp\]) val presimplified_consts = [\<^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\] fun presimplify (simp_options as {let_simps, ...} : simp_options) ctxt = rewrite_rule ctxt (map safe_mk_meta_eq nnf_simps) #> simplify (put_simpset (nnf_ss simp_options) ctxt) #> let_simps ? rewrite_rule ctxt @{thms Let_def [abs_def]} fun make_nnf simp_options ctxt th = (case Thm.prems_of th of [] => th |> presimplify simp_options 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 simp_options ctxt choice_ths = let fun aux th = if not (has_conns [\<^const_name>\Ex\] (Thm.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 simp_options ctxt end fun skolemize simp_options ctxt = let val thy = Proof_Context.theory_of ctxt in skolemize_with_choice_theorems simp_options ctxt (choice_theorems thy) end exception NO_F_PATTERN of unit fun get_F_pattern T t u = let fun pat t u = let val ((head1, args1), (head2, args2)) = (t, u) |> apply2 strip_comb in if head1 = head2 then let val pats = map2 pat args1 args2 in case filter (is_some o fst) pats of [(SOME T, _)] => (SOME T, list_comb (head1, map snd pats)) | [] => (NONE, t) | _ => raise NO_F_PATTERN () end else let val T = fastype_of t in if can dest_funT T then (SOME T, Bound 0) else raise NO_F_PATTERN () end end in if T = \<^Type>\bool\ then NONE else case pat t u of (SOME T, p as _ $ _) => SOME (Abs (Name.uu, T, p)) | _ => NONE end handle NO_F_PATTERN () => NONE val ext_cong_neq = @{thm ext_cong_neq} -(* Strengthens "f g ~= f h" to "f g ~= f h & (EX x. g x ~= h x)". *) +(* Strengthens "f g \ f h" to "f g \ f h \ (\x. g x \ h x)". *) fun cong_extensionalize_thm ctxt th = (case Thm.concl_of th of \<^Const_>\Trueprop for \<^Const_>\Not for \<^Const_>\HOL.eq T for \t as _ $ _\ \u as _ $ _\\\\ => (case get_F_pattern T t u of SOME p => th RS infer_instantiate ctxt [(("F", 0), Thm.cterm_of ctxt p)] ext_cong_neq | NONE => th) | _ => th) (* 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 abs_extensionalize_conv ctxt ct = (case Thm.term_of ct of \<^Const_>\HOL.eq _ for _ \Abs _\\ => ct |> (Conv.rewr_conv @{thm fun_eq_iff [THEN eq_reflection]} then_conv abs_extensionalize_conv ctxt) | _ $ _ => Conv.comb_conv (abs_extensionalize_conv ctxt) ct | Abs _ => Conv.abs_conv (abs_extensionalize_conv o snd) ctxt ct | _ => Conv.all_conv ct) val abs_extensionalize_thm = Conv.fconv_rule o abs_extensionalize_conv fun try_skolemize_etc simp_options ctxt th = let val th = th |> cong_extensionalize_thm ctxt in [th] (* Extensionalize lambdas in "th", because that makes sense and that's what Sledgehammer does, but also keep an unextensionalized version of "th" for backward compatibility. *) |> insert Thm.eq_thm_prop (abs_extensionalize_thm ctxt th) |> map_filter (fn th => th |> try (skolemize simp_options ctxt) |> tap (fn NONE => trace_msg ctxt (fn () => "Failed to skolemize " ^ Thm.string_of_thm ctxt th) | _ => ())) end fun add_clauses ctxt th cls = let val (cnfs, ctxt') = ctxt |> Variable.declare_thm th |> make_cnf [] th; in Variable.export ctxt' ctxt cnfs @ cls end; (*Sort clauses by number of literals*) fun fewerlits (th1, th2) = nliterals (Thm.prop_of th1) < nliterals (Thm.prop_of th2) (*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 (make_ord fewerlits) 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 ctxt sizef horns = BEST_FIRST (has_fewer_prems 1, sizef) (prolog_step_tac ctxt horns 1); fun depth_prolog_tac ctxt horns = DEPTH_FIRST (has_fewer_prems 1) (prolog_step_tac ctxt 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 simp_options ctxt prems = cut_facts_tac (maps (try_skolemize_etc simp_options ctxt) prems) THEN' REPEAT o eresolve_tac ctxt [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 ctxt 1, resolve_tac ctxt @{thms ccontr} 1, preskolem_tac, Subgoal.FOCUS (fn {context = ctxt', prems = negs, ...} => EVERY1 [skolemize_prems_tac simp_options_all_true 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 ctxt (gocls cls) 1) (has_fewer_prems 1, sizef) (prolog_step_tac ctxt (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 ctxt (gocls cls) 1, depth_prolog_tac ctxt (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' ctxt horns = let val horn0s = (*0 subgoals vs 1 or more*) take_prefix Thm.no_prems horns val nrtac = resolve_from_net_tac ctxt (Tactic.build_net horns) in fn i => eq_assume_tac i ORELSE match_tac ctxt horn0s i ORELSE (*no backtracking if unit MATCHES*) ((assume_tac ctxt i APPEND nrtac i) THEN check_tac) end; fun iter_deepen_prolog_tac ctxt horns = ITER_DEEPEN iter_deepen_limit (has_fewer_prems 1) (prolog_step_tac' ctxt 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 (Thm.string_of_thm ctxt) (cls @ ths) @ ["clauses:"] @ map (Thm.string_of_thm ctxt) horns)) in THEN_ITER_DEEPEN iter_deepen_limit (resolve_tac ctxt goes 1) (has_fewer_prems 1) (prolog_step_tac' ctxt 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 = @{lemma "\ P \ P \ False" by (rule notE)}; val notEfalse' = @{lemma "P \ \ P \ False" by (rule notE)}; 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 ctxt th = let val (fth, thaw) = Misc_Legacy.freeze_thaw_robust ctxt 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 ctxt ths = name_thms "MClause#" (distinct Thm.eq_thm_prop (map (make_meta_clause ctxt) ths)); end; diff --git a/src/HOL/Tools/Meson/meson_clausify.ML b/src/HOL/Tools/Meson/meson_clausify.ML --- a/src/HOL/Tools/Meson/meson_clausify.ML +++ b/src/HOL/Tools/Meson/meson_clausify.ML @@ -1,388 +1,388 @@ (* Title: HOL/Tools/Meson/meson_clausify.ML Author: Jia Meng, Cambridge University Computer Laboratory and NICTA Author: Jasmin Blanchette, TU Muenchen Transformation of HOL theorems into CNF forms. *) signature MESON_CLAUSIFY = sig val new_skolem_var_prefix : string val new_nonskolem_var_prefix : string val is_zapped_var_name : string -> bool val is_quasi_lambda_free : term -> bool val introduce_combinators_in_cterm : Proof.context -> cterm -> thm val introduce_combinators_in_theorem : Proof.context -> thm -> thm val cluster_of_zapped_var_name : string -> (int * (int * int)) * bool val ss_only : thm list -> Proof.context -> Proof.context val cnf_axiom : Meson.simp_options -> Proof.context -> bool -> bool -> int -> thm -> (thm * term) option * thm list end; structure Meson_Clausify : MESON_CLAUSIFY = struct (* the extra "Meson" helps prevent clashes (FIXME) *) val new_skolem_var_prefix = "MesonSK" val new_nonskolem_var_prefix = "MesonV" fun is_zapped_var_name s = exists (fn prefix => String.isPrefix prefix s) [new_skolem_var_prefix, new_nonskolem_var_prefix] (**** Transformation of Elimination Rules into First-Order Formulas****) val cfalse = Thm.cterm_of \<^theory_context>\HOL\ \<^term>\False\; val ctp_false = Thm.cterm_of \<^theory_context>\HOL\ (HOLogic.mk_Trueprop \<^term>\False\); (* Converts an elim-rule into an equivalent theorem that does not have the predicate variable. Leaves other theorems unchanged. We simply instantiate the conclusion variable to False. (Cf. "transform_elim_prop" in "Sledgehammer_Util".) *) fun transform_elim_theorem th = (case Thm.concl_of th of (*conclusion variable*) \<^Const_>\Trueprop for \Var (v as (_, \<^Type>\bool\))\\ => Thm.instantiate (TVars.empty, Vars.make [(v, cfalse)]) th | Var (v as (_, \<^Type>\prop\)) => Thm.instantiate (TVars.empty, Vars.make [(v, ctp_false)]) th | _ => th) (**** SKOLEMIZATION BY INFERENCE (lcp) ****) fun mk_old_skolem_term_wrapper t = let val T = fastype_of t in \<^Const>\Meson.skolem T for t\ end fun beta_eta_in_abs_body (Abs (s, T, t')) = Abs (s, T, beta_eta_in_abs_body t') | beta_eta_in_abs_body t = Envir.beta_eta_contract t (*Traverse a theorem, accumulating Skolem function definitions.*) fun old_skolem_defs th = let fun dec_sko \<^Const_>\Ex _ for \body as Abs (_, T, p)\\ rhss = (*Existential: declare a Skolem function, then insert into body and continue*) let val args = Misc_Legacy.term_frees body (* Forms a lambda-abstraction over the formal parameters *) val rhs = fold_rev (absfree o dest_Free) args (HOLogic.choice_const T $ beta_eta_in_abs_body body) |> mk_old_skolem_term_wrapper val comb = list_comb (rhs, args) in dec_sko (subst_bound (comb, p)) (rhs :: rhss) end | dec_sko \<^Const_>\All _ for \Abs (a, T, p)\\ rhss = (*Universal quant: insert a free variable into body and continue*) let val fname = singleton (Name.variant_list (Misc_Legacy.add_term_names (p, []))) a in dec_sko (subst_bound (Free(fname,T), p)) rhss end | dec_sko \<^Const_>\conj for p q\ rhss = rhss |> dec_sko p |> dec_sko q | dec_sko \<^Const_>\disj for p q\ rhss = rhss |> dec_sko p |> dec_sko q | dec_sko \<^Const_>\Trueprop for p\ rhss = dec_sko p rhss | dec_sko _ rhss = rhss in dec_sko (Thm.prop_of th) [] end; (**** REPLACING ABSTRACTIONS BY COMBINATORS ****) fun is_quasi_lambda_free \<^Const_>\Meson.skolem _ for _\ = true | is_quasi_lambda_free (t1 $ t2) = is_quasi_lambda_free t1 andalso is_quasi_lambda_free t2 | is_quasi_lambda_free (Abs _) = false | is_quasi_lambda_free _ = true fun abstract ctxt ct = let val Abs (_, _, body) = Thm.term_of ct val (x, cbody) = Thm.dest_abs NONE ct val (A, cbodyT) = Thm.dest_funT (Thm.ctyp_of_cterm ct) fun makeK () = Thm.instantiate' [SOME A, SOME cbodyT] [SOME cbody] @{thm abs_K} in case body of Const _ => makeK() | Free _ => makeK() | Var _ => makeK() (*though Var isn't expected*) | Bound 0 => Thm.instantiate' [SOME A] [] @{thm abs_I} (*identity: I*) | rator$rand => if Term.is_dependent rator then (*C or S*) if Term.is_dependent rand then (*S*) let val crator = Thm.lambda x (Thm.dest_fun cbody) val crand = Thm.lambda x (Thm.dest_arg cbody) val (C, B) = Thm.dest_funT (Thm.dest_ctyp1 (Thm.ctyp_of_cterm crator)) val abs_S' = @{thm abs_S} |> instantiate'_normalize [SOME A, SOME B, SOME C] [SOME crator, SOME crand] val (_,rhs) = Thm.dest_equals (Thm.cprop_of abs_S') in Thm.transitive abs_S' (Conv.binop_conv (abstract ctxt) rhs) end else (*C*) let val crator = Thm.lambda x (Thm.dest_fun cbody) val crand = Thm.dest_arg cbody val (C, B) = Thm.dest_funT (Thm.dest_ctyp1 (Thm.ctyp_of_cterm crator)) val abs_C' = @{thm abs_C} |> instantiate'_normalize [SOME A, SOME B, SOME C] [SOME crator, SOME crand] val (_,rhs) = Thm.dest_equals (Thm.cprop_of abs_C') in Thm.transitive abs_C' (Conv.fun_conv (Conv.arg_conv (abstract ctxt)) rhs) end else if Term.is_dependent rand then (*B or eta*) if rand = Bound 0 then Thm.eta_conversion ct else (*B*) let val crator = Thm.dest_fun cbody val crand = Thm.lambda x (Thm.dest_arg cbody) val (C, B) = Thm.dest_funT (Thm.ctyp_of_cterm crator) val abs_B' = @{thm abs_B} |> instantiate'_normalize [SOME A, SOME B, SOME C] [SOME crator, SOME crand] val (_,rhs) = Thm.dest_equals (Thm.cprop_of abs_B') in Thm.transitive abs_B' (Conv.arg_conv (abstract ctxt) rhs) end else makeK () | _ => raise Fail "abstract: Bad term" end; (* Traverse a theorem, remplacing lambda-abstractions with combinators. *) fun introduce_combinators_in_cterm ctxt ct = if is_quasi_lambda_free (Thm.term_of ct) then Thm.reflexive ct else case Thm.term_of ct of Abs _ => let val (cv, cta) = Thm.dest_abs NONE ct val (v, _) = dest_Free (Thm.term_of cv) val u_th = introduce_combinators_in_cterm ctxt cta val cu = Thm.rhs_of u_th val comb_eq = abstract ctxt (Thm.lambda cv cu) in Thm.transitive (Thm.abstract_rule v cv u_th) comb_eq end | _ $ _ => let val (ct1, ct2) = Thm.dest_comb ct in Thm.combination (introduce_combinators_in_cterm ctxt ct1) (introduce_combinators_in_cterm ctxt ct2) end fun introduce_combinators_in_theorem ctxt th = if is_quasi_lambda_free (Thm.prop_of th) then th else let val th = Drule.eta_contraction_rule th val eqth = introduce_combinators_in_cterm ctxt (Thm.cprop_of th) in Thm.equal_elim eqth th end handle THM (msg, _, _) => (warning ("Error in the combinator translation of " ^ Thm.string_of_thm ctxt th ^ "\nException message: " ^ msg); (* A type variable of sort "{}" will make "abstraction" fail. *) TrueI) (*cterms are used throughout for efficiency*) val cTrueprop = Thm.cterm_of \<^theory_context>\HOL\ HOLogic.Trueprop; (*Given an abstraction over n variables, replace the bound variables by free ones. Return the body, along with the list of free variables.*) fun c_variant_abs_multi (ct0, vars) = let val (cv,ct) = Thm.dest_abs NONE ct0 in c_variant_abs_multi (ct, cv::vars) end handle CTERM _ => (ct0, rev vars); (* Given the definition of a Skolem function, return a theorem to replace an existential formula by a use of that function. - Example: "EX x. x : A & x ~: B ==> sko A B : A & sko A B ~: B" [.] *) + Example: "\x. x \ A \ x \ B \ sko A B \ A \ sko A B \ B" *) fun old_skolem_theorem_of_def ctxt rhs0 = let val rhs = rhs0 |> Type.legacy_freeze_thaw |> #1 |> Thm.cterm_of ctxt val rhs' = rhs |> Thm.dest_comb |> snd val (ch, frees) = c_variant_abs_multi (rhs', []) val (hilbert, cabs) = ch |> Thm.dest_comb |>> Thm.term_of val T = case hilbert of Const (_, Type (\<^type_name>\fun\, [_, T])) => T | _ => raise TERM ("old_skolem_theorem_of_def: expected \"Eps\"", [hilbert]) val cex = Thm.cterm_of ctxt (HOLogic.exists_const T) val ex_tm = Thm.apply cTrueprop (Thm.apply cex cabs) val conc = Drule.list_comb (rhs, frees) |> Drule.beta_conv cabs |> Thm.apply cTrueprop fun tacf [prem] = rewrite_goals_tac ctxt @{thms skolem_def [abs_def]} THEN resolve_tac ctxt [(prem |> rewrite_rule ctxt @{thms skolem_def [abs_def]}) RS Global_Theory.get_thm (Proof_Context.theory_of ctxt) "Hilbert_Choice.someI_ex"] 1 in Goal.prove_internal ctxt [ex_tm] conc tacf |> forall_intr_list frees |> Thm.forall_elim_vars 0 (*Introduce Vars, but don't discharge defs.*) |> Thm.varifyT_global end fun to_definitional_cnf_with_quantifiers ctxt th = let val eqth = CNF.make_cnfx_thm ctxt (HOLogic.dest_Trueprop (Thm.prop_of th)) val eqth = eqth RS @{thm eq_reflection} val eqth = eqth RS @{thm TruepropI} in Thm.equal_elim eqth th end fun zapped_var_name ((ax_no, cluster_no), skolem) index_no s = (if skolem then new_skolem_var_prefix else new_nonskolem_var_prefix) ^ "_" ^ string_of_int ax_no ^ "_" ^ string_of_int cluster_no ^ "_" ^ string_of_int index_no ^ "_" ^ Name.desymbolize (SOME false) s fun cluster_of_zapped_var_name s = let val get_int = the o Int.fromString o nth (space_explode "_" s) in ((get_int 1, (get_int 2, get_int 3)), String.isPrefix new_skolem_var_prefix s) end fun rename_bound_vars_to_be_zapped ax_no = let fun aux (cluster as (cluster_no, cluster_skolem)) index_no pos t = case t of (t1 as Const (s, _)) $ Abs (s', T, t') => if s = \<^const_name>\Pure.all\ orelse s = \<^const_name>\All\ orelse s = \<^const_name>\Ex\ then let val skolem = (pos = (s = \<^const_name>\Ex\)) val (cluster, index_no) = if skolem = cluster_skolem then (cluster, index_no) else ((cluster_no ||> cluster_skolem ? Integer.add 1, skolem), 0) val s' = zapped_var_name cluster index_no s' in t1 $ Abs (s', T, aux cluster (index_no + 1) pos t') end else t | (t1 as Const (s, _)) $ t2 $ t3 => if s = \<^const_name>\Pure.imp\ orelse s = \<^const_name>\HOL.implies\ then t1 $ aux cluster index_no (not pos) t2 $ aux cluster index_no pos t3 else if s = \<^const_name>\HOL.conj\ orelse s = \<^const_name>\HOL.disj\ then t1 $ aux cluster index_no pos t2 $ aux cluster index_no pos t3 else t | (t1 as Const (s, _)) $ t2 => if s = \<^const_name>\Trueprop\ then t1 $ aux cluster index_no pos t2 else if s = \<^const_name>\Not\ then t1 $ aux cluster index_no (not pos) t2 else t | _ => t in aux ((ax_no, 0), true) 0 true end fun zap pos ct = ct |> (case Thm.term_of ct of Const (s, _) $ Abs (s', _, _) => if s = \<^const_name>\Pure.all\ orelse s = \<^const_name>\All\ orelse s = \<^const_name>\Ex\ then Thm.dest_comb #> snd #> Thm.dest_abs (SOME s') #> snd #> zap pos else Conv.all_conv | Const (s, _) $ _ $ _ => if s = \<^const_name>\Pure.imp\ orelse s = \<^const_name>\implies\ then Conv.combination_conv (Conv.arg_conv (zap (not pos))) (zap pos) else if s = \<^const_name>\conj\ orelse s = \<^const_name>\disj\ then Conv.combination_conv (Conv.arg_conv (zap pos)) (zap pos) else Conv.all_conv | Const (s, _) $ _ => if s = \<^const_name>\Trueprop\ then Conv.arg_conv (zap pos) else if s = \<^const_name>\Not\ then Conv.arg_conv (zap (not pos)) else Conv.all_conv | _ => Conv.all_conv) fun ss_only ths ctxt = clear_simpset (put_simpset HOL_basic_ss ctxt) addsimps ths val cheat_choice = \<^prop>\\x. \y. Q x y \ \f. \x. Q x (f x)\ |> Logic.varify_global |> Skip_Proof.make_thm \<^theory> (* Converts an Isabelle theorem into NNF. *) fun nnf_axiom simp_options choice_ths new_skolem ax_no th ctxt = let val thy = Proof_Context.theory_of ctxt val th = th |> transform_elim_theorem |> zero_var_indexes |> new_skolem ? Thm.forall_intr_vars val (th, ctxt) = Variable.import true [th] ctxt |>> snd |>> the_single val th = th |> Conv.fconv_rule (Object_Logic.atomize ctxt) |> Meson.cong_extensionalize_thm ctxt |> Meson.abs_extensionalize_thm ctxt |> Meson.make_nnf simp_options ctxt in if new_skolem then let fun skolemize choice_ths = Meson.skolemize_with_choice_theorems simp_options ctxt choice_ths #> simplify (ss_only @{thms all_simps[symmetric]} ctxt) val no_choice = null choice_ths val pull_out = if no_choice then simplify (ss_only @{thms all_simps[symmetric] ex_simps[symmetric]} ctxt) else skolemize choice_ths val discharger_th = th |> pull_out val discharger_th = discharger_th |> Meson.has_too_many_clauses ctxt (Thm.concl_of discharger_th) ? (to_definitional_cnf_with_quantifiers ctxt #> pull_out) val zapped_th = discharger_th |> Thm.prop_of |> rename_bound_vars_to_be_zapped ax_no |> (if no_choice then Skip_Proof.make_thm thy #> skolemize [cheat_choice] #> Thm.cprop_of else Thm.cterm_of ctxt) |> zap true val fixes = [] |> Term.add_free_names (Thm.prop_of zapped_th) |> filter is_zapped_var_name val ctxt' = ctxt |> Variable.add_fixes_direct fixes val fully_skolemized_t = zapped_th |> singleton (Variable.export ctxt' ctxt) |> Thm.cprop_of |> Thm.dest_equals |> snd |> Thm.term_of in if exists_subterm (fn Var ((s, _), _) => String.isPrefix new_skolem_var_prefix s | _ => false) fully_skolemized_t then let val (fully_skolemized_ct, ctxt) = yield_singleton (Variable.import_terms true) fully_skolemized_t ctxt |>> Thm.cterm_of ctxt in (SOME (discharger_th, fully_skolemized_ct), (Thm.assume fully_skolemized_ct, ctxt)) end else (NONE, (th, ctxt)) end else (NONE, (th |> Meson.has_too_many_clauses ctxt (Thm.concl_of th) ? to_definitional_cnf_with_quantifiers ctxt, ctxt)) end (* Convert a theorem to CNF, with additional premises due to skolemization. *) fun cnf_axiom simp_options ctxt0 new_skolem combinators ax_no th = let val choice_ths = Meson.choice_theorems (Proof_Context.theory_of ctxt0) val (opt, (nnf_th, ctxt1)) = nnf_axiom simp_options choice_ths new_skolem ax_no th ctxt0 fun clausify th = Meson.make_cnf (if new_skolem orelse null choice_ths then [] else map (old_skolem_theorem_of_def ctxt1) (old_skolem_defs th)) th ctxt1 val (cnf_ths, ctxt2) = clausify nnf_th fun intr_imp ct th = Thm.instantiate (TVars.empty, Vars.make [(\<^var>\?i::nat\, Thm.cterm_of ctxt2 (HOLogic.mk_nat ax_no))]) (zero_var_indexes @{thm skolem_COMBK_D}) RS Thm.implies_intr ct th in (opt |> Option.map (I #>> singleton (Variable.export ctxt2 ctxt0) ##> (Thm.term_of #> HOLogic.dest_Trueprop #> singleton (Variable.export_terms ctxt2 ctxt0))), cnf_ths |> map (combinators ? introduce_combinators_in_theorem ctxt2 #> (case opt of SOME (_, ct) => intr_imp ct | NONE => I)) |> Variable.export ctxt2 ctxt0 |> Meson.finish_cnf |> map (Thm.close_derivation \<^here>)) end handle THM _ => (NONE, []) end;