(* 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 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: 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 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 : 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
val trace = Attrib.setup_config_bool \<^binding>\<open>meson_trace\<close> (K false)
fun trace_msg ctxt msg = if Config.get ctxt trace then tracing (msg ()) else ()
val max_clauses = Attrib.setup_config_int \<^binding>\<open>meson_max_clauses\<close> (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 (fn () =>
let val thy = Proof_Context.theory_of ctxt
val tmA = Thm.concl_of thA
val Const(\<^const_name>\<open>Pure.imp\<close>,_) $ 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>\<open>All\<close> = SOME true
| quant_of \<^const_name>\<open>Ball\<close> = SOME true
| quant_of \<^const_name>\<open>Ex\<close> = SOME false
| quant_of \<^const_name>\<open>Bex\<close> = 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>\<open>Not\<close> $ t) = add_names (flip_quant quant) t
| add_names quant (\<^const>\<open>implies\<close> $ 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>\<open>Trueprop\<close> $ (Var _ $ (c as Free _)), \<^const>\<open>Trueprop\<close> $ _) =>
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"
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 [email protected]:" ::
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(\<^const_name>\<open>Trueprop\<close>,_) $ p) = has p
| has (Const(\<^const_name>\<open>Not\<close>,_) $ p) = has p
| has (Const(\<^const_name>\<open>HOL.disj\<close>,_) $ p $ q) = member (op =) bs \<^const_name>\<open>HOL.disj\<close> orelse has p orelse has q
| has (Const(\<^const_name>\<open>HOL.conj\<close>,_) $ p $ q) = member (op =) bs \<^const_name>\<open>HOL.conj\<close> orelse has p orelse has q
| has (Const(\<^const_name>\<open>All\<close>,_) $ Abs(_,_,p)) = member (op =) bs \<^const_name>\<open>All\<close> orelse has p
| has (Const(\<^const_name>\<open>Ex\<close>,_) $ Abs(_,_,p)) = member (op =) bs \<^const_name>\<open>Ex\<close> orelse has p
| has _ = false
in has end;
(**** Clause handling ****)
fun literals (Const(\<^const_name>\<open>Trueprop\<close>,_) $ P) = literals P
| literals (Const(\<^const_name>\<open>HOL.disj\<close>,_) $ P $ Q) = literals P @ literals Q
| literals (Const(\<^const_name>\<open>Not\<close>,_) $ 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>\<open>HOL.disj\<close>, _) $ P $ Q) (poslits, neglits) =
signed_lits_aux Q (signed_lits_aux P (poslits, neglits))
| signed_lits_aux (Const(\<^const_name>\<open>Not\<close>,_) $ 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(\<^const_name>\<open>HOL.eq\<close>,_) $ t $ u) = t aconv u
| taut_poslit (Const(\<^const_name>\<open>True\<close>,_)) = 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>\<open>False\<close> :: 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 (\<^const_name>\<open>HOL.disj\<close>, _) $ (Const (\<^const_name>\<open>HOL.disj\<close>, _) $ _ $ _) $ _) =>
refl_clause_aux n (th RS disj_assoc) (*isolate an atom as first disjunct*)
| (Const (\<^const_name>\<open>HOL.disj\<close>, _) $ (Const(\<^const_name>\<open>Not\<close>,_) $ (Const(\<^const_name>\<open>HOL.eq\<close>,_) $ 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>\<open>HOL.disj\<close>, _) $ _ $ _) => refl_clause_aux n (th RS disj_comm)
| _ => (*not a disjunction*) th;
fun notequal_lits_count (Const (\<^const_name>\<open>HOL.disj\<close>, _) $ P $ Q) =
notequal_lits_count P + notequal_lits_count Q
| notequal_lits_count (Const(\<^const_name>\<open>Not\<close>,_) $ (Const(\<^const_name>\<open>HOL.eq\<close>,_) $ _ $ _)) = 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
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(\<^const_name>\<open>Trueprop\<close>,_) $ t) = signed_nclauses b t
| signed_nclauses b (Const(\<^const_name>\<open>Not\<close>,_) $ t) = signed_nclauses (not b) t
| signed_nclauses b (Const(\<^const_name>\<open>HOL.conj\<close>,_) $ 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>\<open>HOL.disj\<close>,_) $ 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>\<open>HOL.implies\<close>,_) $ 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>\<open>HOL.eq\<close>, 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>\<open>Ex\<close>, _) $ Abs (_,_,t)) = signed_nclauses b t
| signed_nclauses b (Const(\<^const_name>\<open>All\<close>,_) $ 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 (\<^const_name>\<open>All\<close>, _) $ 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 ([], [(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>\<open>All\<close>, \<^const_name>\<open>Ex\<close>, \<^const_name>\<open>HOL.conj\<close>] (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 (\<^const_name>\<open>HOL.conj\<close>, _) => (*conjunction*)
cnf_aux (th RS conjunct1, cnf_aux (th RS conjunct2, ths))
| Const (\<^const_name>\<open>All\<close>, _) => (*universal quantifier*)
let val (th', ctxt') = freeze_spec th (! ctxt_ref)
in ctxt_ref := ctxt'; cnf_aux (th', ths) end
| Const (\<^const_name>\<open>Ex\<close>, _) =>
(*existential quantifier: Insert Skolem functions*)
cnf_aux (apply_skolem_theorem (! ctxt_ref) (th, old_skolem_ths), ths)
| Const (\<^const_name>\<open>HOL.disj\<close>, _) =>
(*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 (\<^const_name>\<open>Trueprop\<close>, _) $
(Const (\<^const_name>\<open>HOL.disj\<close>, _) $ (Const (\<^const_name>\<open>HOL.disj\<close>, _) $ _ $ _) $ _)) = 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 (\<^const_name>\<open>Trueprop\<close>,_) $ (Const (\<^const_name>\<open>HOL.disj\<close>, _) $ t $ _)) = rigid t
| ok4horn (Const (\<^const_name>\<open>Trueprop\<close>,_) $ 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(\<^const_name>\<open>Pure.imp\<close>,_) $ (Const(\<^const_name>\<open>Trueprop\<close>,_) $ 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 (\<^const_name>\<open>Trueprop\<close>,_) $ (Const (\<^const_name>\<open>Not\<close>, _) $ t)) = rigid t
| ok4nnf (Const (\<^const_name>\<open>Trueprop\<close>,_) $ 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}
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 [abs_def]". *)
val nnf_ss =
simpset_of (put_simpset HOL_basic_ss \<^context>
addsimps nnf_extra_simps
addsimprocs [\<^simproc>\<open>defined_All\<close>, \<^simproc>\<open>defined_Ex\<close>, \<^simproc>\<open>neq\<close>, \<^simproc>\<open>let_simp\<close>])
val presimplified_consts =
[\<^const_name>\<open>simp_implies\<close>, \<^const_name>\<open>False\<close>, \<^const_name>\<open>True\<close>,
\<^const_name>\<open>Ex1\<close>, \<^const_name>\<open>Ball\<close>, \<^const_name>\<open>Bex\<close>, \<^const_name>\<open>If\<close>,
\<^const_name>\<open>Let\<close>]
fun presimplify ctxt =
rewrite_rule ctxt (map safe_mk_meta_eq nnf_simps)
#> simplify (put_simpset nnf_ss ctxt)
#> rewrite_rule ctxt @{thms Let_def [abs_def]}
fun make_nnf ctxt th =
(case Thm.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>\<open>Ex\<close>] (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 ctxt end
fun skolemize ctxt =
let val thy = Proof_Context.theory_of ctxt in
skolemize_with_choice_theorems 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 = \<^typ>\<open>bool\<close> 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)". *)
fun cong_extensionalize_thm ctxt th =
(case Thm.concl_of th of
\<^const>\<open>Trueprop\<close> $ (\<^const>\<open>Not\<close>
$ (Const (\<^const_name>\<open>HOL.eq\<close>, Type (_, [T, _]))
$ (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 (\<^const_name>\<open>HOL.eq\<close>, _) $ _ $ 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 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 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 ctxt prems =
cut_facts_tac (maps (try_skolemize_etc 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 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;
¤ Dauer der Verarbeitung: 0.33 Sekunden
(vorverarbeitet)
¤
|
Haftungshinweis
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
|
