(* Title: HOL/Tools/Metis/metis_tactics.ML
Author: Kong W. Susanto, Cambridge University Computer Laboratory
Author: Lawrence C. Paulson, Cambridge University Computer Laboratory
Author: Jasmin Blanchette, TU Muenchen
Copyright Cambridge University 2007
HOL setup for the Metis prover.
*)
signature METIS_TACTICS =
sig
type type_sys = ATP_Translate.type_sys
val metisN : string
val metisF_N : string
val metisFT_N : string
val metisX_N : string
val trace : bool Config.T
val verbose : bool Config.T
val new_skolemizer : bool Config.T
val metis_tac : Proof.context -> thm list -> int -> tactic
val metisF_tac : Proof.context -> thm list -> int -> tactic
val metisH_tac : Proof.context -> thm list -> int -> tactic
val metisFT_tac : Proof.context -> thm list -> int -> tactic
val metisX_tac : Proof.context -> type_sys option -> thm list -> int -> tactic
val setup : theory -> theory
end
structure Metis_Tactics : METIS_TACTICS =
struct
open ATP_Translate
open Metis_Translate
open Metis_Reconstruct
fun method_binding_for_mode HO = @{binding metis}
| method_binding_for_mode FO = @{binding metisF}
| method_binding_for_mode FT = @{binding metisFT}
| method_binding_for_mode MX = @{binding metisX}
val metisN = Binding.qualified_name_of (method_binding_for_mode HO)
val metisF_N = Binding.qualified_name_of (method_binding_for_mode FO)
val metisFT_N = Binding.qualified_name_of (method_binding_for_mode FT)
val metisX_N = Binding.qualified_name_of (method_binding_for_mode MX)
val new_skolemizer =
Attrib.setup_config_bool @{binding metis_new_skolemizer} (K false)
(* Designed to work also with monomorphic instances of polymorphic theorems. *)
fun have_common_thm ths1 ths2 =
exists (member (untyped_aconv o pairself prop_of) ths1)
(map Meson.make_meta_clause ths2)
(*Determining which axiom clauses are actually used*)
fun used_axioms axioms (th, Metis_Proof.Axiom _) = SOME (lookth axioms th)
| used_axioms _ _ = NONE
(* Lightweight predicate type information comes in two flavors, "t = t'" and
"t => t'", where "t" and "t'" are the same term modulo type tags.
In Isabelle, type tags are stripped away, so we are left with "t = t" or
"t => t". Type tag idempotence is also handled this way. *)
fun reflexive_or_trivial_from_metis ctxt sym_tab old_skolems mth =
let val thy = Proof_Context.theory_of ctxt in
case hol_clause_from_metis ctxt sym_tab old_skolems mth of
Const (@{const_name HOL.eq}, _) $ _ $ t =>
t |> cterm_of thy |> Thm.reflexive RS @{thm meta_eq_to_obj_eq}
| Const (@{const_name disj}, _) $ t1 $ t2 =>
(if can HOLogic.dest_not t1 then t2 else t1)
|> HOLogic.mk_Trueprop |> cterm_of thy |> Thm.trivial
| _ => raise Fail "unexpected tags sym clause"
end
|> Meson.make_meta_clause
val clause_params =
{ordering = Metis_KnuthBendixOrder.default,
orderLiterals = Metis_Clause.UnsignedLiteralOrder,
orderTerms = true}
val active_params =
{clause = clause_params,
prefactor = #prefactor Metis_Active.default,
postfactor = #postfactor Metis_Active.default}
val waiting_params =
{symbolsWeight = 1.0,
variablesWeight = 0.0,
literalsWeight = 0.0,
models = []}
val resolution_params = {active = active_params, waiting = waiting_params}
(* Main function to start Metis proof and reconstruction *)
fun FOL_SOLVE type_sys (mode :: fallback_modes) ctxt cls ths0 =
let val thy = Proof_Context.theory_of ctxt
val new_skolemizer =
Config.get ctxt new_skolemizer orelse null (Meson.choice_theorems thy)
val th_cls_pairs =
map2 (fn j => fn th =>
(Thm.get_name_hint th,
Meson_Clausify.cnf_axiom ctxt new_skolemizer j th))
(0 upto length ths0 - 1) ths0
val ths = maps (snd o snd) th_cls_pairs
val dischargers = map (fst o snd) th_cls_pairs
val _ = trace_msg ctxt (fn () => "FOL_SOLVE: CONJECTURE CLAUSES")
val _ = app (fn th => trace_msg ctxt (fn () => Display.string_of_thm ctxt th)) cls
val _ = trace_msg ctxt (fn () => "THEOREM CLAUSES")
val _ = app (fn th => trace_msg ctxt (fn () => Display.string_of_thm ctxt th)) ths
val (mode, sym_tab, {axioms, old_skolems, ...}) =
prepare_metis_problem ctxt mode type_sys cls ths
fun get_isa_thm mth Isa_Reflexive_or_Trivial =
reflexive_or_trivial_from_metis ctxt sym_tab old_skolems mth
| get_isa_thm _ (Isa_Raw ith) = ith
val axioms = axioms |> map (fn (mth, ith) => (mth, get_isa_thm mth ith))
val _ = trace_msg ctxt (fn () => "CLAUSES GIVEN TO METIS")
val thms = axioms |> map fst
val _ = app (fn th => trace_msg ctxt (fn () => Metis_Thm.toString th)) thms
val _ = trace_msg ctxt (fn () => "mode = " ^ string_of_mode mode)
val _ = trace_msg ctxt (fn () => "START METIS PROVE PROCESS")
in
case filter (fn t => prop_of t aconv @{prop False}) cls of
false_th :: _ => [false_th RS @{thm FalseE}]
| [] =>
case Metis_Resolution.new resolution_params {axioms = thms, conjecture = []}
|> Metis_Resolution.loop of
Metis_Resolution.Contradiction mth =>
let val _ = trace_msg ctxt (fn () => "METIS RECONSTRUCTION START: " ^
Metis_Thm.toString mth)
val ctxt' = fold Variable.declare_constraints (map prop_of cls) ctxt
(*add constraints arising from converting goal to clause form*)
val proof = Metis_Proof.proof mth
val result =
fold (replay_one_inference ctxt' mode old_skolems sym_tab)
proof axioms
val used = map_filter (used_axioms axioms) proof
val _ = trace_msg ctxt (fn () => "METIS COMPLETED...clauses actually used:")
val _ = app (fn th => trace_msg ctxt (fn () => Display.string_of_thm ctxt th)) used
val names = th_cls_pairs |> map fst
val used_names =
th_cls_pairs
|> map_filter (fn (name, (_, cls)) =>
if have_common_thm used cls then SOME name
else NONE)
val unused_names = names |> subtract (op =) used_names
in
if not (null cls) andalso not (have_common_thm used cls) then
verbose_warning ctxt "The assumptions are inconsistent"
else
();
if not (null unused_names) then
"Unused theorems: " ^ commas_quote unused_names
|> verbose_warning ctxt
else
();
case result of
(_,ith)::_ =>
(trace_msg ctxt (fn () => "Success: " ^ Display.string_of_thm ctxt ith);
[discharge_skolem_premises ctxt dischargers ith])
| _ => (trace_msg ctxt (fn () => "Metis: No result"); [])
end
| Metis_Resolution.Satisfiable _ =>
(trace_msg ctxt (fn () => "Metis: No first-order proof with the lemmas supplied");
if null fallback_modes then
()
else
raise METIS ("FOL_SOLVE",
"No first-order proof with the lemmas supplied");
[])
end
handle METIS (loc, msg) =>
case fallback_modes of
[] => error ("Failed to replay Metis proof in Isabelle." ^
(if Config.get ctxt verbose then "\n" ^ loc ^ ": " ^ msg
else ""))
| mode :: _ =>
(verbose_warning ctxt
("Falling back on " ^
quote (Binding.qualified_name_of
(method_binding_for_mode mode)) ^ "...");
FOL_SOLVE type_sys fallback_modes ctxt cls ths0)
val neg_clausify =
single
#> Meson.make_clauses_unsorted
#> map Meson_Clausify.introduce_combinators_in_theorem
#> Meson.finish_cnf
fun preskolem_tac ctxt st0 =
(if exists (Meson.has_too_many_clauses ctxt)
(Logic.prems_of_goal (prop_of st0) 1) then
Simplifier.full_simp_tac (Meson_Clausify.ss_only @{thms not_all not_ex}) 1
THEN cnf.cnfx_rewrite_tac ctxt 1
else
all_tac) st0
val type_has_top_sort =
exists_subtype (fn TFree (_, []) => true | TVar (_, []) => true | _ => false)
fun generic_metis_tac modes type_sys ctxt ths i st0 =
let
val _ = trace_msg ctxt (fn () =>
"Metis called with theorems\n" ^
cat_lines (map (Display.string_of_thm ctxt) ths))
fun tac clause = resolve_tac (FOL_SOLVE type_sys modes ctxt clause ths) 1
in
if exists_type type_has_top_sort (prop_of st0) then
(verbose_warning ctxt "Proof state contains the universal sort {}";
Seq.empty)
else
Meson.MESON (preskolem_tac ctxt) (maps neg_clausify) tac ctxt i st0
end
val metis_modes = [HO, FT]
val metisF_modes = [FO, FT]
val metisH_modes = [HO]
val metisFT_modes = [FT]
val metisX_modes = [MX]
val metis_tac = generic_metis_tac metis_modes NONE
val metisF_tac = generic_metis_tac metisF_modes NONE
val metisH_tac = generic_metis_tac metisH_modes NONE
val metisFT_tac = generic_metis_tac metisFT_modes NONE
fun metisX_tac ctxt type_sys = generic_metis_tac metisX_modes type_sys ctxt
(* Whenever "X" has schematic type variables, we treat "using X by metis" as
"by (metis X)" to prevent "Subgoal.FOCUS" from freezing the type variables.
We don't do it for nonschematic facts "X" because this breaks a few proofs
(in the rare and subtle case where a proof relied on extensionality not being
applied) and brings few benefits. *)
val has_tvar =
exists_type (exists_subtype (fn TVar _ => true | _ => false)) o prop_of
fun method modes (type_sys, ths) ctxt facts =
let
val (schem_facts, nonschem_facts) = List.partition has_tvar facts
val type_sys = type_sys |> Option.map type_sys_from_string
in
HEADGOAL (Method.insert_tac nonschem_facts THEN'
CHANGED_PROP
o generic_metis_tac modes type_sys ctxt (schem_facts @ ths))
end
fun setup_method (modes as mode :: _) =
Method.setup (method_binding_for_mode mode)
((if mode = MX then
Scan.lift (Scan.option (Args.parens Parse.short_ident))
else
Scan.succeed NONE)
-- Attrib.thms >> (METHOD oo method modes))
val setup =
[(metis_modes, "Metis for FOL and HOL problems"),
(metisF_modes, "Metis for FOL problems"),
(metisFT_modes, "Metis for FOL/HOL problems with fully-typed translation"),
(metisX_modes, "Metis for FOL and HOL problems (experimental)")]
|> fold (uncurry setup_method)
end;