author | blanchet |
Wed, 29 Sep 2010 23:06:02 +0200 | |
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parent 39886 | 8a9f0c97d550 |
child 39890 | a1695e2169d0 |
permissions | -rw-r--r-- |
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(* Title: HOL/Tools/Sledgehammer/metis_tactics.ML |
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Author: Kong W. Susanto, Cambridge University Computer Laboratory |
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Author: Lawrence C. Paulson, Cambridge University Computer Laboratory |
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Author: Jasmin Blanchette, TU Muenchen |
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The Metis prover (slightly modified version from Larry);
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Copyright Cambridge University 2007 |
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HOL setup for the Metis prover. |
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*) |
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signature METIS_TACTICS = |
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sig |
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val trace : bool Unsynchronized.ref |
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val type_lits : bool Config.T |
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val metis_tac : Proof.context -> thm list -> int -> tactic |
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val metisF_tac : Proof.context -> thm list -> int -> tactic |
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val metisFT_tac : Proof.context -> thm list -> int -> tactic |
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val setup : theory -> theory |
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end |
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structure Metis_Tactics : METIS_TACTICS = |
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struct |
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open Metis_Translate |
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open Metis_Reconstruct |
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fun trace_msg msg = if !trace then tracing (msg ()) else () |
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val (type_lits, type_lits_setup) = Attrib.config_bool "metis_type_lits" (K true); |
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fun is_false t = t aconv (HOLogic.mk_Trueprop HOLogic.false_const); |
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fun have_common_thm ths1 ths2 = |
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exists (member Thm.eq_thm ths1) (map Meson.make_meta_clause ths2) |
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(*Determining which axiom clauses are actually used*) |
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fun used_axioms axioms (th, Metis_Proof.Axiom _) = SOME (lookth axioms th) |
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| used_axioms _ _ = NONE; |
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val clause_params = |
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{ordering = Metis_KnuthBendixOrder.default, |
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orderLiterals = Metis_Clause.UnsignedLiteralOrder, |
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orderTerms = true} |
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val active_params = |
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{clause = clause_params, |
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prefactor = #prefactor Metis_Active.default, |
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postfactor = #postfactor Metis_Active.default} |
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val waiting_params = |
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{symbolsWeight = 1.0, |
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variablesWeight = 0.0, |
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literalsWeight = 0.0, |
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models = []} |
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val resolution_params = {active = active_params, waiting = waiting_params} |
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(* In principle, it should be sufficient to apply "assume_tac" to unify the |
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conclusion with one of the premises. However, in practice, this fails |
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horribly because of the mildly higher-order nature of the unification |
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problems. Typical constraints are of the form "?x a b =?= b", where "a" and |
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"b" are goal parameters. *) |
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fun unify_one_prem_with_concl thy i th = |
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let |
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val goal = Logic.get_goal (prop_of th) i |> Envir.beta_eta_contract |
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val prems = Logic.strip_assums_hyp goal |
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val concl = Logic.strip_assums_concl goal |
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fun add_types Tp instT = |
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if exists (curry (op =) Tp) instT then instT |
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else Tp :: map (apsnd (typ_subst_atomic [Tp])) instT |
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fun unify_types (T, U) = |
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if T = U then |
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I |
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else case (T, U) of |
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(TVar _, _) => add_types (T, U) |
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| (_, TVar _) => add_types (U, T) |
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| (Type (s, Ts), Type (t, Us)) => |
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if s = t andalso length Ts = length Us then fold unify_types (Ts ~~ Us) |
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else raise TYPE ("unify_types", [T, U], []) |
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| _ => raise TYPE ("unify_types", [T, U], []) |
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fun pair_untyped_aconv (t1, t2) (u1, u2) = |
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untyped_aconv t1 u1 andalso untyped_aconv t2 u2 |
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fun add_terms tp inst = |
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if exists (pair_untyped_aconv tp) inst then inst |
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else tp :: map (apsnd (subst_atomic [tp])) inst |
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fun is_flex t = |
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case strip_comb t of |
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(Var _, args) => forall (is_Bound orf is_Var orf is_Free) args |
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| _ => false |
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fun unify_flex flex rigid = |
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case strip_comb flex of |
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(Var (z as (_, T)), args) => |
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add_terms (Var z, |
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(* FIXME: reindex bound variables *) |
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fold_rev (curry absdummy) (take (length args) (binder_types T)) rigid) |
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| _ => raise TERM ("unify_flex: expected flex", [flex]) |
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fun unify_potential_flex comb atom = |
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if is_flex comb then unify_flex comb atom |
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else if is_Var atom then add_terms (atom, comb) |
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else raise TERM ("unify_terms", [comb, atom]) |
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fun unify_terms (t, u) = |
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case (t, u) of |
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(t1 $ t2, u1 $ u2) => |
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if is_flex t then unify_flex t u |
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else if is_flex u then unify_flex u t |
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else fold unify_terms [(t1, u1), (t2, u2)] |
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| (_ $ _, _) => unify_potential_flex t u |
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| (_, _ $ _) => unify_potential_flex u t |
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| (Var _, _) => add_terms (t, u) |
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| (_, Var _) => add_terms (u, t) |
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| _ => if untyped_aconv t u then I else raise TERM ("unify_terms", [t, u]) |
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fun unify_prem prem = |
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let |
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val inst = [] |> unify_terms (prem, concl) |
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val instT = fold (unify_types o pairself fastype_of) inst [] |
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val inst = inst |> map (pairself (subst_atomic_types instT)) |
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val cinstT = instT |> map (pairself (ctyp_of thy)) |
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val cinst = inst |> map (pairself (cterm_of thy)) |
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in th |> Thm.instantiate (cinstT, []) |> Thm.instantiate ([], cinst) end |
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in |
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case prems of |
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[prem] => unify_prem prem |
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| _ => |
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case fold (fn prem => fn th as SOME _ => th |
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| NONE => try unify_prem prem) prems NONE of |
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122 |
SOME th => th |
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123 |
| NONE => raise Fail "unify_one_prem_with_concl" |
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124 |
end |
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125 |
|
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126 |
(* Attempts to derive the theorem "False" from a theorem of the form |
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127 |
"P1 ==> ... ==> Pn ==> False", where the "Pi"s are to be discharged using the |
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128 |
specified axioms. The axioms have leading "All" and "Ex" quantifiers, which |
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129 |
must be eliminated first. *) |
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130 |
fun discharge_skolem_premises ctxt axioms premises_imp_false = |
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131 |
if prop_of premises_imp_false aconv @{prop False} then |
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132 |
premises_imp_false |
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133 |
else |
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134 |
let val thy = ProofContext.theory_of ctxt in |
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135 |
Goal.prove ctxt [] [] @{prop False} |
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136 |
(K (cut_rules_tac axioms 1 |
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137 |
THEN TRY (REPEAT_ALL_NEW (etac @{thm exE}) 1) |
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138 |
(* FIXME: THEN etac @{lemma "P ==> (P ==> P ==> Q) ==> Q" by fast} 1 *) |
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139 |
THEN TRY (REPEAT_ALL_NEW (etac @{thm allE}) 1) |
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140 |
THEN match_tac [premises_imp_false] 1 |
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141 |
THEN DETERM_UNTIL_SOLVED |
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142 |
(PRIMITIVE (unify_one_prem_with_concl thy 1) |
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143 |
THEN assume_tac 1))) |
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144 |
end |
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145 |
|
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146 |
(* Main function to start Metis proof and reconstruction *) |
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fun FOL_SOLVE mode ctxt cls ths0 = |
148 |
let val thy = ProofContext.theory_of ctxt |
|
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149 |
val type_lits = Config.get ctxt type_lits |
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val th_cls_pairs = |
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map (fn th => (Thm.get_name_hint th, Meson_Clausifier.cnf_axiom thy th)) |
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152 |
ths0 |
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153 |
val thss = map (snd o snd) th_cls_pairs |
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154 |
val dischargers = map_filter (fst o snd) th_cls_pairs |
32956 | 155 |
val _ = trace_msg (fn () => "FOL_SOLVE: CONJECTURE CLAUSES") |
156 |
val _ = app (fn th => trace_msg (fn () => Display.string_of_thm ctxt th)) cls |
|
157 |
val _ = trace_msg (fn () => "THEOREM CLAUSES") |
|
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val _ = app (app (fn th => trace_msg (fn () => Display.string_of_thm ctxt th))) thss |
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159 |
val (mode, {axioms, tfrees, old_skolems}) = |
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160 |
build_logic_map mode ctxt type_lits cls thss |
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val _ = if null tfrees then () |
162 |
else (trace_msg (fn () => "TFREE CLAUSES"); |
|
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163 |
app (fn TyLitFree ((s, _), (s', _)) => |
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164 |
trace_msg (fn () => s ^ "(" ^ s' ^ ")")) tfrees) |
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val _ = trace_msg (fn () => "CLAUSES GIVEN TO METIS") |
166 |
val thms = map #1 axioms |
|
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val _ = app (fn th => trace_msg (fn () => Metis_Thm.toString th)) thms |
32956 | 168 |
val _ = trace_msg (fn () => "mode = " ^ string_of_mode mode) |
169 |
val _ = trace_msg (fn () => "START METIS PROVE PROCESS") |
|
170 |
in |
|
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case filter (is_false o prop_of) cls of |
32956 | 172 |
false_th::_ => [false_th RS @{thm FalseE}] |
173 |
| [] => |
|
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case Metis_Resolution.new resolution_params {axioms = thms, conjecture = []} |
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|> Metis_Resolution.loop of |
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176 |
Metis_Resolution.Contradiction mth => |
32956 | 177 |
let val _ = trace_msg (fn () => "METIS RECONSTRUCTION START: " ^ |
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178 |
Metis_Thm.toString mth) |
32956 | 179 |
val ctxt' = fold Variable.declare_constraints (map prop_of cls) ctxt |
180 |
(*add constraints arising from converting goal to clause form*) |
|
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181 |
val proof = Metis_Proof.proof mth |
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182 |
val result = |
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183 |
fold (replay_one_inference ctxt' mode old_skolems) proof axioms |
32956 | 184 |
and used = map_filter (used_axioms axioms) proof |
185 |
val _ = trace_msg (fn () => "METIS COMPLETED...clauses actually used:") |
|
186 |
val _ = app (fn th => trace_msg (fn () => Display.string_of_thm ctxt th)) used |
|
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187 |
val unused = th_cls_pairs |> map_filter (fn (name, (_, cls)) => |
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if have_common_thm used cls then NONE else SOME name) |
32956 | 189 |
in |
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190 |
if not (null cls) andalso not (have_common_thm used cls) then |
36383 | 191 |
warning "Metis: The assumptions are inconsistent." |
192 |
else |
|
193 |
(); |
|
194 |
if not (null unused) then |
|
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195 |
warning ("Metis: Unused theorems: " ^ commas_quote unused |
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196 |
^ ".") |
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197 |
else |
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198 |
(); |
32956 | 199 |
case result of |
200 |
(_,ith)::_ => |
|
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201 |
(trace_msg (fn () => "Success: " ^ Display.string_of_thm ctxt ith); |
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202 |
[discharge_skolem_premises ctxt dischargers ith]) |
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203 |
| _ => (trace_msg (fn () => "Metis: No result"); []) |
32956 | 204 |
end |
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205 |
| Metis_Resolution.Satisfiable _ => |
32956 | 206 |
(trace_msg (fn () => "Metis: No first-order proof with the lemmas supplied"); |
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207 |
[]) |
32956 | 208 |
end; |
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209 |
|
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210 |
(* Extensionalize "th", because that makes sense and that's what Sledgehammer |
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211 |
does, but also keep an unextensionalized version of "th" for backward |
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212 |
compatibility. *) |
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213 |
fun also_extensionalize_theorem th = |
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214 |
let val th' = Meson_Clausifier.extensionalize_theorem th in |
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215 |
if Thm.eq_thm (th, th') then [th] |
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216 |
else th :: Meson.make_clauses_unsorted [th'] |
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217 |
end |
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218 |
|
38028 | 219 |
val neg_clausify = |
220 |
single |
|
221 |
#> Meson.make_clauses_unsorted |
|
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222 |
#> maps also_extensionalize_theorem |
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223 |
#> map Meson_Clausifier.introduce_combinators_in_theorem |
38028 | 224 |
#> Meson.finish_cnf |
225 |
||
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|
226 |
fun preskolem_tac ctxt st0 = |
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227 |
(if exists (Meson.has_too_many_clauses ctxt) |
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228 |
(Logic.prems_of_goal (prop_of st0) 1) then |
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|
229 |
cnf.cnfx_rewrite_tac ctxt 1 |
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|
230 |
else |
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|
231 |
all_tac) st0 |
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|
232 |
|
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233 |
val type_has_top_sort = |
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234 |
exists_subtype (fn TFree (_, []) => true | TVar (_, []) => true | _ => false) |
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235 |
|
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|
236 |
fun generic_metis_tac mode ctxt ths i st0 = |
37926
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|
237 |
let |
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|
238 |
val _ = trace_msg (fn () => |
32956 | 239 |
"Metis called with theorems " ^ cat_lines (map (Display.string_of_thm ctxt) ths)) |
240 |
in |
|
37626
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|
241 |
if exists_type type_has_top_sort (prop_of st0) then |
37516
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|
242 |
(warning ("Metis: Proof state contains the universal sort {}"); Seq.empty) |
35568
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|
243 |
else |
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|
244 |
Meson.MESON (preskolem_tac ctxt) (maps neg_clausify) |
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|
245 |
(fn cls => resolve_tac (FOL_SOLVE mode ctxt cls ths) 1) |
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|
246 |
ctxt i st0 |
32956 | 247 |
end |
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|
248 |
|
37516
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|
249 |
val metis_tac = generic_metis_tac HO |
c81c86bfc18a
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|
250 |
val metisF_tac = generic_metis_tac FO |
c81c86bfc18a
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|
251 |
val metisFT_tac = generic_metis_tac FT |
23442
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The Metis prover (slightly modified version from Larry);
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|
252 |
|
38632
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|
253 |
(* Whenever "X" has schematic type variables, we treat "using X by metis" as |
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|
254 |
"by (metis X)", to prevent "Subgoal.FOCUS" from freezing the type variables. |
9cde57cdd0e3
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|
255 |
We don't do it for nonschematic facts "X" because this breaks a few proofs |
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(in the rare and subtle case where a proof relied on extensionality not being |
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applied) and brings few benefits. *) |
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val has_tvar = |
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exists_type (exists_subtype (fn TVar _ => true | _ => false)) o prop_of |
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fun method name mode = |
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Method.setup name (Attrib.thms >> (fn ths => fn ctxt => |
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METHOD (fn facts => |
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let |
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val (schem_facts, nonschem_facts) = |
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List.partition has_tvar facts |
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in |
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HEADGOAL (Method.insert_tac nonschem_facts THEN' |
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CHANGED_PROP |
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o generic_metis_tac mode ctxt (schem_facts @ ths)) |
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end))) |
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|
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val setup = |
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type_lits_setup |
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#> method @{binding metis} HO "Metis for FOL/HOL problems" |
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#> method @{binding metisF} FO "Metis for FOL problems" |
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#> method @{binding metisFT} FT |
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"Metis for FOL/HOL problems with fully-typed translation" |
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|
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end; |