Quellcode-Bibliothek

^© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei: z3_replay_methods.ML Sprache: SML

Original von: Isabelle^©

(*  Title:      HOL/Tools/SMT/z3_replay_methods.ML
    Author:     Sascha Boehme, TU Muenchen
    Author:     Jasmin Blanchette, TU Muenchen

Proof methods for replaying Z3 proofs.
*)

signature Z3_REPLAY_METHODS =
sig

  (*methods for Z3 proof rules*)
  type z3_method = Proof.context -> thm list -> term -> thm
  val true_axiom: z3_method
  val mp: z3_method
  val refl: z3_method
  val symm: z3_method
  val trans: z3_method
  val cong: z3_method
  val quant_intro: z3_method
  val distrib: z3_method
  val and_elim: z3_method
  val not_or_elim: z3_method
  val rewrite: z3_method
  val rewrite_star: z3_method
  val pull_quant: z3_method
  val push_quant: z3_method
  val elim_unused: z3_method
  val dest_eq_res: z3_method
  val quant_inst: z3_method
  val lemma: z3_method
  val unit_res: z3_method
  val iff_true: z3_method
  val iff_false: z3_method
  val comm: z3_method
  val def_axiom: z3_method
  val apply_def: z3_method
  val iff_oeq: z3_method
  val nnf_pos: z3_method
  val nnf_neg: z3_method
  val mp_oeq: z3_method
  val arith_th_lemma: z3_method
  val th_lemma: string -> z3_method
  val method_for: Z3_Proof.z3_rule -> z3_method
end;

structure Z3_Replay_Methods: Z3_REPLAY_METHODS =
struct

type z3_method = Proof.context -> thm list -> term -> thm

(* utility functions *)

fun replay_error ctxt msg rule thms t =
  SMT_Replay_Methods.replay_error ctxt msg (Z3_Proof.string_of_rule rule) thms t

fun replay_rule_error ctxt rule thms t =
  SMT_Replay_Methods.replay_rule_error "Z3" ctxt (Z3_Proof.string_of_rule rule) thms t

fun trace_goal ctxt rule thms t =
  SMT_Replay_Methods.trace_goal ctxt (Z3_Proof.string_of_rule rule) thms t

fun dest_prop (Const (\<^const_name>\<open>Trueprop\<close>, _) $ t) = t
  | dest_prop t = t

fun dest_thm thm = dest_prop (Thm.concl_of thm)

val try_provers = SMT_Replay_Methods.try_provers "Z3"

fun prop_tac ctxt prems =
  Method.insert_tac ctxt prems
  THEN' SUBGOAL (fn (prop, i) =>
    if Term.size_of_term prop > 100 then SAT.satx_tac ctxt i
    else (Classical.fast_tac ctxt ORELSE' Clasimp.force_tac ctxt) i)

fun quant_tac ctxt = Blast.blast_tac ctxt

fun apply_rule ctxt t =
  (case Z3_Replay_Rules.apply ctxt (SMT_Replay_Methods.certify_prop ctxt t) of
    SOME thm => thm
  | NONE => raise Fail "apply_rule")

(*theory-lemma*)

fun arith_th_lemma_tac ctxt prems =
  Method.insert_tac ctxt prems
  THEN' SELECT_GOAL (Local_Defs.unfold0_tac ctxt @{thms z3div_def z3mod_def})
  THEN' Arith_Data.arith_tac ctxt

fun arith_th_lemma ctxt thms t =
  SMT_Replay_Methods.prove_abstract ctxt thms t arith_th_lemma_tac (
    fold_map (SMT_Replay_Methods.abstract_arith ctxt o dest_thm) thms ##>>
    SMT_Replay_Methods.abstract_arith ctxt (dest_prop t))

fun th_lemma name ctxt thms =
  (case Symtab.lookup (SMT_Replay_Methods.get_th_lemma_method ctxt) name of
    SOME method => method ctxt thms
  | NONE => replay_error ctxt "Bad theory" (Z3_Proof.Th_Lemma name) thms)

(* truth axiom *)

fun true_axiom _ _ _ = @{thm TrueI}

(* modus ponens *)

fun mp _ [p, p_eq_q] _ = SMT_Replay_Methods.discharge 1 [p_eq_q, p] iffD1
  | mp ctxt thms t = replay_rule_error ctxt Z3_Proof.Modus_Ponens thms t

val mp_oeq = mp

(* reflexivity *)

fun refl ctxt _ t = SMT_Replay_Methods.match_instantiate ctxt t @{thm refl}

(* symmetry *)

fun symm _ [thm] _ = thm RS @{thm sym}
  | symm ctxt thms t = replay_rule_error ctxt Z3_Proof.Reflexivity thms t

(* transitivity *)

fun trans _ [thm1, thm2] _ = thm1 RSN (1, thm2 RSN (2, @{thm trans}))
  | trans ctxt thms t = replay_rule_error ctxt Z3_Proof.Transitivity thms t

(* congruence *)

fun cong ctxt thms = try_provers ctxt
    (Z3_Proof.string_of_rule Z3_Proof.Monotonicity) [
  ("basic", SMT_Replay_Methods.cong_basic ctxt thms),
  ("full", SMT_Replay_Methods.cong_full ctxt thms)] thms

(* quantifier introduction *)

val quant_intro_rules = @{lemma
  "(\x. P x = Q x) ==> (\x. P x) = (\x. Q x)"
  "(\x. P x = Q x) ==> (\x. P x) = (\x. Q x)"
  "(!!x. (\ P x) = Q x) \ (\(\x. P x)) = (\x. Q x)"
  "(\x. (\ P x) = Q x) ==> (\(\x. P x)) = (\x. Q x)"
  by fast+}

fun quant_intro ctxt [thm] t =
    SMT_Replay_Methods.prove ctxt t (K (REPEAT_ALL_NEW (resolve_tac ctxt (thm :: quant_intro_rules))))
  | quant_intro ctxt thms t = replay_rule_error ctxt Z3_Proof.Quant_Intro thms t

(* distributivity of conjunctions and disjunctions *)

(* TODO: there are no tests with this proof rule *)
fun distrib ctxt _ t =
  SMT_Replay_Methods.prove_abstract' ctxt t prop_tac
    (SMT_Replay_Methods.abstract_prop (dest_prop t))

(* elimination of conjunctions *)

fun and_elim ctxt [thm] t =
      SMT_Replay_Methods.prove_abstract ctxt [thm] t prop_tac (
        SMT_Replay_Methods.abstract_lit (dest_prop t) ##>>
        SMT_Replay_Methods.abstract_conj (dest_thm thm) #>>
        apfst single o swap)
  | and_elim ctxt thms t = replay_rule_error ctxt Z3_Proof.And_Elim thms t

(* elimination of negated disjunctions *)

fun not_or_elim ctxt [thm] t =
      SMT_Replay_Methods.prove_abstract ctxt [thm] t prop_tac (
        SMT_Replay_Methods.abstract_lit (dest_prop t) ##>>
        SMT_Replay_Methods.abstract_not SMT_Replay_Methods.abstract_disj (dest_thm thm) #>>
        apfst single o swap)
  | not_or_elim ctxt thms t =
      replay_rule_error ctxt Z3_Proof.Not_Or_Elim thms t

(* rewriting *)

local

fun dest_all (Const (\<^const_name>\<open>HOL.All\<close>, _) $ Abs (_, T, t)) nctxt =
      let
        val (n, nctxt') = Name.variant "" nctxt
        val f = Free (n, T)
        val t' = Term.subst_bound (f, t)
      in dest_all t' nctxt' |>> cons f end
  | dest_all t _ = ([], t)

fun dest_alls t =
  let
    val nctxt = Name.make_context (Term.add_free_names t [])
    val (lhs, rhs) = HOLogic.dest_eq (dest_prop t)
    val (ls, lhs') = dest_all lhs nctxt
    val (rs, rhs') = dest_all rhs nctxt
  in
    if eq_list (op aconv) (ls, rs) then SOME (ls, (HOLogic.mk_eq (lhs', rhs')))
    else NONE
  end

fun forall_intr ctxt t thm =
  let val ct = Thm.cterm_of ctxt t
  in Thm.forall_intr ct thm COMP_INCR @{thm iff_allI} end

in

fun focus_eq f ctxt t =
  (case dest_alls t of
    NONE => f ctxt t
  | SOME (vs, t') => fold (forall_intr ctxt) vs (f ctxt t'))

end

fun abstract_eq f (Const (\<^const_name>\<open>HOL.eq\<close>, _) $ t1 $ t2) =
      f t1 ##>> f t2 #>> HOLogic.mk_eq
  | abstract_eq _ t = SMT_Replay_Methods.abstract_term t

fun prove_prop_rewrite ctxt t =
  SMT_Replay_Methods.prove_abstract' ctxt t prop_tac (
    abstract_eq SMT_Replay_Methods.abstract_prop (dest_prop t))

fun arith_rewrite_tac ctxt _ =
  let val backup_tac = Arith_Data.arith_tac ctxt ORELSE' Clasimp.force_tac ctxt in
    (TRY o Simplifier.simp_tac ctxt) THEN_ALL_NEW backup_tac
    ORELSE' backup_tac
  end

fun prove_arith_rewrite ctxt t =
  SMT_Replay_Methods.prove_abstract' ctxt t arith_rewrite_tac (
    abstract_eq (SMT_Replay_Methods.abstract_arith ctxt) (dest_prop t))

val lift_ite_thm = @{thm HOL.if_distrib} RS @{thm eq_reflection}

fun ternary_conv cv = Conv.combination_conv (Conv.binop_conv cv) cv

fun if_context_conv ctxt ct =
  (case Thm.term_of ct of
    Const (\<^const_name>\<open>HOL.If\<close>, _) $ _ $ _ $ _ =>
      ternary_conv (if_context_conv ctxt)
  | _ $ (Const (\<^const_name>\<open>HOL.If\<close>, _) $ _ $ _ $ _) =>
      Conv.rewr_conv lift_ite_thm then_conv ternary_conv (if_context_conv ctxt)
  | _ => Conv.sub_conv (Conv.top_sweep_conv if_context_conv) ctxt) ct

fun lift_ite_rewrite ctxt t =
  SMT_Replay_Methods.prove ctxt t (fn ctxt =>
    CONVERSION (HOLogic.Trueprop_conv (Conv.binop_conv (if_context_conv ctxt)))
    THEN' resolve_tac ctxt @{thms refl})

fun prove_conj_disj_perm ctxt t = SMT_Replay_Methods.prove ctxt t Conj_Disj_Perm.conj_disj_perm_tac

fun rewrite ctxt _ = try_provers ctxt
    (Z3_Proof.string_of_rule Z3_Proof.Rewrite) [
  ("rules", apply_rule ctxt),
  ("conj_disj_perm", prove_conj_disj_perm ctxt),
  ("prop_rewrite", prove_prop_rewrite ctxt),
  ("arith_rewrite", focus_eq prove_arith_rewrite ctxt),
  ("if_rewrite", lift_ite_rewrite ctxt)] []

fun rewrite_star ctxt = rewrite ctxt

(* pulling quantifiers *)

fun pull_quant ctxt _ t = SMT_Replay_Methods.prove ctxt t quant_tac

(* pushing quantifiers *)

fun push_quant _ _ _ = raise Fail "unsupported" (* FIXME *)

(* elimination of unused bound variables *)

val elim_all = @{lemma "P = Q \ (\x. P) = Q" by fast}
val elim_ex = @{lemma "P = Q \ (\x. P) = Q" by fast}

fun elim_unused_tac ctxt i st = (
  match_tac ctxt [@{thm refl}]
  ORELSE' (match_tac ctxt [elim_all, elim_ex] THEN' elim_unused_tac ctxt)
  ORELSE' (
    match_tac ctxt [@{thm iff_allI}, @{thm iff_exI}]
    THEN' elim_unused_tac ctxt)) i st

fun elim_unused ctxt _ t = SMT_Replay_Methods.prove ctxt t elim_unused_tac

(* destructive equality resolution *)

fun dest_eq_res _ _ _ = raise Fail "dest_eq_res" (* FIXME *)

(* quantifier instantiation *)

val quant_inst_rule = @{lemma "\P x \ Q ==> \(\x. P x) \ Q" by fast}

fun quant_inst ctxt _ t = SMT_Replay_Methods.prove ctxt t (fn _ =>
  REPEAT_ALL_NEW (resolve_tac ctxt [quant_inst_rule])
  THEN' resolve_tac ctxt @{thms excluded_middle})

(* propositional lemma *)

exception LEMMA of unit

val intro_hyp_rule1 = @{lemma "(\P \ Q) \ P \ Q" by fast}
val intro_hyp_rule2 = @{lemma "(P \ Q) \ \P \ Q" by fast}

fun norm_lemma thm =
  (thm COMP_INCR intro_hyp_rule1)
  handle THM _ => thm COMP_INCR intro_hyp_rule2

fun negated_prop (\<^const>\<open>HOL.Not\<close> $ t) = HOLogic.mk_Trueprop t
  | negated_prop t = HOLogic.mk_Trueprop (HOLogic.mk_not t)

fun intro_hyps tab (t as \<^const>\<open>HOL.disj\<close> $ t1 $ t2) cx =
      lookup_intro_hyps tab t (fold (intro_hyps tab) [t1, t2]) cx
  | intro_hyps tab t cx =
      lookup_intro_hyps tab t (fn _ => raise LEMMA ()) cx

and lookup_intro_hyps tab t f (cx as (thm, terms)) =
  (case Termtab.lookup tab (negated_prop t) of
    NONE => f cx
  | SOME hyp => (norm_lemma (Thm.implies_intr hyp thm), t :: terms))

fun lemma ctxt (thms as [thm]) t =
    (let
       val tab = Termtab.make (map (`Thm.term_of) (Thm.chyps_of thm))
       val (thm', terms) = intro_hyps tab (dest_prop t) (thm, [])
     in
       SMT_Replay_Methods.prove_abstract ctxt [thm'] t prop_tac (
         fold (snd oo SMT_Replay_Methods.abstract_lit) terms #>
         SMT_Replay_Methods.abstract_disj (dest_thm thm') #>> single ##>>
         SMT_Replay_Methods.abstract_disj (dest_prop t))
     end
     handle LEMMA () => replay_error ctxt "Bad proof state" Z3_Proof.Lemma thms t)
  | lemma ctxt thms t = replay_rule_error ctxt Z3_Proof.Lemma thms t

(* unit resolution *)

fun unit_res ctxt thms t =
  SMT_Replay_Methods.prove_abstract ctxt thms t prop_tac (
    fold_map (SMT_Replay_Methods.abstract_unit o dest_thm) thms ##>>
    SMT_Replay_Methods.abstract_unit (dest_prop t) #>>
    (fn (prems, concl) => (prems, concl)))

(* iff-true *)

val iff_true_rule = @{lemma "P ==> P = True" by fast}

fun iff_true _ [thm] _ = thm RS iff_true_rule
  | iff_true ctxt thms t = replay_rule_error ctxt Z3_Proof.Iff_True thms t

(* iff-false *)

val iff_false_rule = @{lemma "\P \ P = False" by fast}

fun iff_false _ [thm] _ = thm RS iff_false_rule
  | iff_false ctxt thms t = replay_rule_error ctxt Z3_Proof.Iff_False thms t

(* commutativity *)

fun comm ctxt _ t = SMT_Replay_Methods.match_instantiate ctxt t @{thm eq_commute}

(* definitional axioms *)

fun def_axiom_disj ctxt t =
  (case dest_prop t of
    \<^const>\<open>HOL.disj\<close> $ u1 $ u2 =>
      SMT_Replay_Methods.prove_abstract' ctxt t prop_tac (
         SMT_Replay_Methods.abstract_prop u2 ##>>  SMT_Replay_Methods.abstract_prop u1 #>>
         HOLogic.mk_disj o swap)
  | u => SMT_Replay_Methods.prove_abstract' ctxt t prop_tac (SMT_Replay_Methods.abstract_prop u))

fun def_axiom ctxt _ = try_provers ctxt
    (Z3_Proof.string_of_rule Z3_Proof.Def_Axiom) [
  ("rules", apply_rule ctxt),
  ("disj", def_axiom_disj ctxt)] []

(* application of definitions *)

fun apply_def _ [thm] _ = thm (* TODO: cover also the missing cases *)
  | apply_def ctxt thms t = replay_rule_error ctxt Z3_Proof.Apply_Def thms t

(* iff-oeq *)

fun iff_oeq _ _ _ = raise Fail "iff_oeq" (* FIXME *)

(* negation normal form *)

fun nnf_prop ctxt thms t =
  SMT_Replay_Methods.prove_abstract ctxt thms t prop_tac (
    fold_map (SMT_Replay_Methods.abstract_prop o dest_thm) thms ##>>
    SMT_Replay_Methods.abstract_prop (dest_prop t))

fun nnf ctxt rule thms = try_provers ctxt (Z3_Proof.string_of_rule rule) [
  ("prop", nnf_prop ctxt thms),
  ("quant", quant_intro ctxt [hd thms])] thms

fun nnf_pos ctxt = nnf ctxt Z3_Proof.Nnf_Pos
fun nnf_neg ctxt = nnf ctxt Z3_Proof.Nnf_Neg

(* mapping of rules to methods *)

fun unsupported rule ctxt = replay_error ctxt "Unsupported" rule
fun assumed rule ctxt = replay_error ctxt "Assumed" rule

fun choose Z3_Proof.True_Axiom = true_axiom
  | choose (r as Z3_Proof.Asserted) = assumed r
  | choose (r as Z3_Proof.Goal) = assumed r
  | choose Z3_Proof.Modus_Ponens = mp
  | choose Z3_Proof.Reflexivity = refl
  | choose Z3_Proof.Symmetry = symm
  | choose Z3_Proof.Transitivity = trans
  | choose (r as Z3_Proof.Transitivity_Star) = unsupported r
  | choose Z3_Proof.Monotonicity = cong
  | choose Z3_Proof.Quant_Intro = quant_intro
  | choose Z3_Proof.Distributivity = distrib
  | choose Z3_Proof.And_Elim = and_elim
  | choose Z3_Proof.Not_Or_Elim = not_or_elim
  | choose Z3_Proof.Rewrite = rewrite
  | choose Z3_Proof.Rewrite_Star = rewrite_star
  | choose Z3_Proof.Pull_Quant = pull_quant
  | choose (r as Z3_Proof.Pull_Quant_Star) = unsupported r
  | choose Z3_Proof.Push_Quant = push_quant
  | choose Z3_Proof.Elim_Unused_Vars = elim_unused
  | choose Z3_Proof.Dest_Eq_Res = dest_eq_res
  | choose Z3_Proof.Quant_Inst = quant_inst
  | choose (r as Z3_Proof.Hypothesis) = assumed r
  | choose Z3_Proof.Lemma = lemma
  | choose Z3_Proof.Unit_Resolution = unit_res
  | choose Z3_Proof.Iff_True = iff_true
  | choose Z3_Proof.Iff_False = iff_false
  | choose Z3_Proof.Commutativity = comm
  | choose Z3_Proof.Def_Axiom = def_axiom
  | choose (r as Z3_Proof.Intro_Def) = assumed r
  | choose Z3_Proof.Apply_Def = apply_def
  | choose Z3_Proof.Iff_Oeq = iff_oeq
  | choose Z3_Proof.Nnf_Pos = nnf_pos
  | choose Z3_Proof.Nnf_Neg = nnf_neg
  | choose (r as Z3_Proof.Nnf_Star) = unsupported r
  | choose (r as Z3_Proof.Cnf_Star) = unsupported r
  | choose (r as Z3_Proof.Skolemize) = assumed r
  | choose Z3_Proof.Modus_Ponens_Oeq = mp_oeq
  | choose (Z3_Proof.Th_Lemma name) = th_lemma name

fun with_tracing rule method ctxt thms t =
  let val _ = trace_goal ctxt rule thms t
  in method ctxt thms t end

fun method_for rule = with_tracing rule (choose rule)

end;

¤ Dauer der Verarbeitung: 0.5 Sekunden (vorverarbeitet) ¤

Download des Quellennavigators
Download des sprechenden Kalenders
in der Quellcodebibliothek suchen

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.