(* 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
(*abstraction*)
type abs_context = int * term Termtab.table
type 'a abstracter = term -> abs_context -> 'a * abs_context
val add_arith_abstracter: (term abstracter -> term option abstracter) ->
Context.generic -> Context.generic
(*theory lemma methods*)
type th_lemma_method = Proof.context -> thm list -> term -> thm
val add_th_lemma_method: string * th_lemma_method -> Context.generic ->
Context.generic
(*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 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 trace ctxt f = SMT_Config.trace_msg ctxt f ()
fun pretty_thm ctxt thm = Syntax.pretty_term ctxt (Thm.concl_of thm)
fun pretty_goal ctxt msg rule thms t =
let
val full_msg = msg ^ ": " ^ quote (Z3_Proof.string_of_rule rule)
val assms =
if null thms then []
else [Pretty.big_list "assumptions:" (map (pretty_thm ctxt) thms)]
val concl = Pretty.big_list "proposition:" [Syntax.pretty_term ctxt t]
in Pretty.big_list full_msg (assms @ [concl]) end
fun replay_error ctxt msg rule thms t = error (Pretty.string_of (pretty_goal ctxt msg rule thms t))
fun replay_rule_error ctxt = replay_error ctxt "Failed to replay Z3 proof step"
fun trace_goal ctxt rule thms t =
trace ctxt (fn () => Pretty.string_of (pretty_goal ctxt "Goal" rule thms t))
fun as_prop (t as Const (@{const_name Trueprop}, _) $ _) = t
| as_prop t = HOLogic.mk_Trueprop t
fun dest_prop (Const (@{const_name Trueprop}, _) $ t) = t
| dest_prop t = t
fun dest_thm thm = dest_prop (Thm.concl_of thm)
fun certify_prop ctxt t = SMT_Util.certify ctxt (as_prop t)
fun try_provers ctxt rule [] thms t = replay_rule_error ctxt rule thms t
| try_provers ctxt rule ((name, prover) :: named_provers) thms t =
(case (trace ctxt (K ("Trying prover " ^ quote name)); try prover t) of
SOME thm => thm
| NONE => try_provers ctxt rule named_provers thms t)
fun match ctxt pat t =
(Vartab.empty, Vartab.empty)
|> Pattern.first_order_match (Proof_Context.theory_of ctxt) (pat, t)
fun gen_certify_inst sel mk cert ctxt thm t =
let
val inst = match ctxt (dest_thm thm) (dest_prop t)
fun cert_inst (ix, (a, b)) = (cert (mk (ix, a)), cert b)
in Vartab.fold (cons o cert_inst) (sel inst) [] end
fun match_instantiateT ctxt t thm =
if Term.exists_type (Term.exists_subtype Term.is_TVar) (dest_thm thm) then
let val certT = Thm.ctyp_of (Proof_Context.theory_of ctxt)
in Thm.instantiate (gen_certify_inst fst TVar certT ctxt thm t, []) thm end
else thm
fun match_instantiate ctxt t thm =
let
val cert = SMT_Util.certify ctxt
val thm' = match_instantiateT ctxt t thm
in Thm.instantiate ([], gen_certify_inst snd Var cert ctxt thm' t) thm' end
fun apply_rule ctxt t =
(case Z3_Replay_Rules.apply ctxt (certify_prop ctxt t) of
SOME thm => thm
| NONE => raise Fail "apply_rule")
fun discharge _ [] thm = thm
| discharge i (rule :: rules) thm = discharge (i + Thm.nprems_of rule) rules (rule RSN (i, thm))
fun by_tac ctxt thms ns ts t tac =
Goal.prove ctxt [] (map as_prop ts) (as_prop t)
(fn {context, prems} => HEADGOAL (tac context prems))
|> Drule.generalize ([], ns)
|> discharge 1 thms
fun prove ctxt t tac = by_tac ctxt [] [] [] t (K o tac)
fun prop_tac ctxt prems =
Method.insert_tac 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
(* plug-ins *)
type abs_context = int * term Termtab.table
type 'a abstracter = term -> abs_context -> 'a * abs_context
type th_lemma_method = Proof.context -> thm list -> term -> thm
fun id_ord ((id1, _), (id2, _)) = int_ord (id1, id2)
structure Plugins = Generic_Data
(
type T =
(int * (term abstracter -> term option abstracter)) list *
th_lemma_method Symtab.table
val empty = ([], Symtab.empty)
val extend = I
fun merge ((abss1, ths1), (abss2, ths2)) = (
Ord_List.merge id_ord (abss1, abss2),
Symtab.merge (K true) (ths1, ths2))
)
fun add_arith_abstracter abs = Plugins.map (apfst (Ord_List.insert id_ord (serial (), abs)))
fun get_arith_abstracters ctxt = map snd (fst (Plugins.get (Context.Proof ctxt)))
fun add_th_lemma_method method = Plugins.map (apsnd (Symtab.update_new method))
fun get_th_lemma_method ctxt = snd (Plugins.get (Context.Proof ctxt))
(* abstraction *)
fun prove_abstract ctxt thms t tac f =
let
val ((prems, concl), (_, ts)) = f (1, Termtab.empty)
val ns = Termtab.fold (fn (_, v) => cons (fst (Term.dest_Free v))) ts []
in
by_tac ctxt [] ns prems concl tac
|> match_instantiate ctxt t
|> discharge 1 thms
end
fun prove_abstract' ctxt t tac f =
prove_abstract ctxt [] t tac (f #>> pair [])
fun lookup_term (_, terms) t = Termtab.lookup terms t
fun abstract_sub t f cx =
(case lookup_term cx t of
SOME v => (v, cx)
| NONE => f cx)
fun mk_fresh_free t (i, terms) =
let val v = Free ("t" ^ string_of_int i, fastype_of t)
in (v, (i + 1, Termtab.update (t, v) terms)) end
fun apply_abstracters _ [] _ cx = (NONE, cx)
| apply_abstracters abs (abstracter :: abstracters) t cx =
(case abstracter abs t cx of
(NONE, _) => apply_abstracters abs abstracters t cx
| x as (SOME _, _) => x)
fun abstract_term (t as _ $ _) = abstract_sub t (mk_fresh_free t)
| abstract_term (t as Abs _) = abstract_sub t (mk_fresh_free t)
| abstract_term t = pair t
fun abstract_bin abs f t t1 t2 = abstract_sub t (abs t1 ##>> abs t2 #>> f)
fun abstract_ter abs f t t1 t2 t3 =
abstract_sub t (abs t1 ##>> abs t2 ##>> abs t3 #>> (Parse.triple1 #> f))
fun abstract_lit (@{const HOL.Not} $ t) = abstract_term t #>> HOLogic.mk_not
| abstract_lit t = abstract_term t
fun abstract_not abs (t as @{const HOL.Not} $ t1) =
abstract_sub t (abs t1 #>> HOLogic.mk_not)
| abstract_not _ t = abstract_lit t
fun abstract_conj (t as @{const HOL.conj} $ t1 $ t2) =
abstract_bin abstract_conj HOLogic.mk_conj t t1 t2
| abstract_conj t = abstract_lit t
fun abstract_disj (t as @{const HOL.disj} $ t1 $ t2) =
abstract_bin abstract_disj HOLogic.mk_disj t t1 t2
| abstract_disj t = abstract_lit t
fun abstract_prop (t as (c as @{const If (bool)}) $ t1 $ t2 $ t3) =
abstract_ter abstract_prop (fn (t1, t2, t3) => c $ t1 $ t2 $ t3) t t1 t2 t3
| abstract_prop (t as @{const HOL.disj} $ t1 $ t2) =
abstract_bin abstract_prop HOLogic.mk_disj t t1 t2
| abstract_prop (t as @{const HOL.conj} $ t1 $ t2) =
abstract_bin abstract_prop HOLogic.mk_conj t t1 t2
| abstract_prop (t as @{const HOL.implies} $ t1 $ t2) =
abstract_bin abstract_prop HOLogic.mk_imp t t1 t2
| abstract_prop (t as @{term "HOL.eq :: bool => _"} $ t1 $ t2) =
abstract_bin abstract_prop HOLogic.mk_eq t t1 t2
| abstract_prop t = abstract_not abstract_prop t
fun abstract_arith ctxt u =
let
fun abs (t as (c as Const _) $ Abs (s, T, t')) =
abstract_sub t (abs t' #>> (fn u' => c $ Abs (s, T, u')))
| abs (t as (c as Const (@{const_name If}, _)) $ t1 $ t2 $ t3) =
abstract_ter abs (fn (t1, t2, t3) => c $ t1 $ t2 $ t3) t t1 t2 t3
| abs (t as @{const HOL.Not} $ t1) = abstract_sub t (abs t1 #>> HOLogic.mk_not)
| abs (t as @{const HOL.disj} $ t1 $ t2) =
abstract_sub t (abs t1 ##>> abs t2 #>> HOLogic.mk_disj)
| abs (t as (c as Const (@{const_name uminus_class.uminus}, _)) $ t1) =
abstract_sub t (abs t1 #>> (fn u => c $ u))
| abs (t as (c as Const (@{const_name plus_class.plus}, _)) $ t1 $ t2) =
abstract_sub t (abs t1 ##>> abs t2 #>> (fn (u1, u2) => c $ u1 $ u2))
| abs (t as (c as Const (@{const_name minus_class.minus}, _)) $ t1 $ t2) =
abstract_sub t (abs t1 ##>> abs t2 #>> (fn (u1, u2) => c $ u1 $ u2))
| abs (t as (c as Const (@{const_name times_class.times}, _)) $ t1 $ t2) =
abstract_sub t (abs t1 ##>> abs t2 #>> (fn (u1, u2) => c $ u1 $ u2))
| abs (t as (c as Const (@{const_name z3div}, _)) $ t1 $ t2) =
abstract_sub t (abs t1 ##>> abs t2 #>> (fn (u1, u2) => c $ u1 $ u2))
| abs (t as (c as Const (@{const_name z3mod}, _)) $ t1 $ t2) =
abstract_sub t (abs t1 ##>> abs t2 #>> (fn (u1, u2) => c $ u1 $ u2))
| abs (t as (c as Const (@{const_name HOL.eq}, _)) $ t1 $ t2) =
abstract_sub t (abs t1 ##>> abs t2 #>> (fn (u1, u2) => c $ u1 $ u2))
| abs (t as (c as Const (@{const_name ord_class.less}, _)) $ t1 $ t2) =
abstract_sub t (abs t1 ##>> abs t2 #>> (fn (u1, u2) => c $ u1 $ u2))
| abs (t as (c as Const (@{const_name ord_class.less_eq}, _)) $ t1 $ t2) =
abstract_sub t (abs t1 ##>> abs t2 #>> (fn (u1, u2) => c $ u1 $ u2))
| abs t = abstract_sub t (fn cx =>
if can HOLogic.dest_number t then (t, cx)
else
(case apply_abstracters abs (get_arith_abstracters ctxt) t cx of
(SOME u, cx') => (u, cx')
| (NONE, _) => abstract_term t cx))
in abs u end
(* truth axiom *)
fun true_axiom _ _ _ = @{thm TrueI}
(* modus ponens *)
fun mp _ [p, p_eq_q] _ = 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 = 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 ctac ctxt prems i st = st |> (
resolve_tac (@{thm refl} :: prems) i
ORELSE (cong_tac ctxt i THEN ctac ctxt prems (i + 1) THEN ctac ctxt prems i))
fun cong_basic ctxt thms t =
let val st = Thm.trivial (certify_prop ctxt t)
in
(case Seq.pull (ctac ctxt thms 1 st) of
SOME (thm, _) => thm
| NONE => raise THM ("cong", 0, thms @ [st]))
end
val cong_dest_rules = @{lemma
"(~ P | Q) & (P | ~ Q) ==> P = Q"
"(P | ~ Q) & (~ P | Q) ==> P = Q"
by fast+}
fun cong_full ctxt thms t = prove ctxt t (fn ctxt' =>
Method.insert_tac thms
THEN' (Classical.fast_tac ctxt'
ORELSE' dresolve_tac cong_dest_rules
THEN' Classical.fast_tac ctxt'))
fun cong ctxt thms = try_provers ctxt Z3_Proof.Monotonicity [
("basic", cong_basic ctxt thms),
("full", cong_full ctxt thms)] thms
(* quantifier introduction *)
val quant_intro_rules = @{lemma
"(!!x. P x = Q x) ==> (ALL x. P x) = (ALL x. Q x)"
"(!!x. P x = Q x) ==> (EX x. P x) = (EX x. Q x)"
"(!!x. (~ P x) = Q x) ==> (~ (EX x. P x)) = (ALL x. Q x)"
"(!!x. (~ P x) = Q x) ==> (~ (ALL x. P x)) = (EX x. Q x)"
by fast+}
fun quant_intro ctxt [thm] t =
prove ctxt t (K (REPEAT_ALL_NEW (resolve_tac (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 =
prove_abstract' ctxt t prop_tac (abstract_prop (dest_prop t))
(* elimination of conjunctions *)
fun and_elim ctxt [thm] t =
prove_abstract ctxt [thm] t prop_tac (
abstract_lit (dest_prop t) ##>>
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 =
prove_abstract ctxt [thm] t prop_tac (
abstract_lit (dest_prop t) ##>>
abstract_not 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 HOL.All}, _) $ 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 (Proof_Context.theory_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 HOL.eq}, _) $ t1 $ t2) =
f t1 ##>> f t2 #>> HOLogic.mk_eq
| abstract_eq _ t = abstract_term t
fun prove_prop_rewrite ctxt t =
prove_abstract' ctxt t prop_tac (
abstract_eq abstract_prop (dest_prop t))
fun arith_rewrite_tac ctxt _ =
TRY o Simplifier.simp_tac ctxt
THEN_ALL_NEW (Arith_Data.arith_tac ctxt ORELSE' Clasimp.force_tac ctxt)
fun prove_arith_rewrite ctxt t =
prove_abstract' ctxt t arith_rewrite_tac (
abstract_eq (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 HOL.If}, _) $ _ $ _ $ _ =>
ternary_conv (if_context_conv ctxt)
| _ $ (Const (@{const_name HOL.If}, _) $ _ $ _ $ _) =>
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 =
prove ctxt t (fn ctxt =>
CONVERSION (HOLogic.Trueprop_conv (Conv.binop_conv (if_context_conv ctxt)))
THEN' rtac @{thm refl})
fun rewrite ctxt _ = try_provers ctxt Z3_Proof.Rewrite [
("rules", apply_rule 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 = 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 ==> (ALL x. P) = Q" by fast}
val elim_ex = @{lemma "P = Q ==> (EX 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 = 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 ==> ~(ALL x. P x) | Q" by fast}
fun quant_inst ctxt _ t = prove ctxt t (fn _ =>
REPEAT_ALL_NEW (rtac quant_inst_rule)
THEN' rtac @{thm 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 HOL.Not} $ t) = HOLogic.mk_Trueprop t
| negated_prop t = HOLogic.mk_Trueprop (HOLogic.mk_not t)
fun intro_hyps tab (t as @{const HOL.disj} $ 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) (#hyps (Thm.crep_thm thm)))
val (thm', terms) = intro_hyps tab (dest_prop t) (thm, [])
in
prove_abstract ctxt [thm'] t prop_tac (
fold (snd oo abstract_lit) terms #>
abstract_disj (dest_thm thm') #>> single ##>>
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 abstract_unit (t as (@{const HOL.Not} $ (@{const HOL.disj} $ t1 $ t2))) =
abstract_sub t (abstract_unit t1 ##>> abstract_unit t2 #>>
HOLogic.mk_not o HOLogic.mk_disj)
| abstract_unit (t as (@{const HOL.disj} $ t1 $ t2)) =
abstract_sub t (abstract_unit t1 ##>> abstract_unit t2 #>>
HOLogic.mk_disj)
| abstract_unit t = abstract_lit t
fun unit_res ctxt thms t =
prove_abstract ctxt thms t prop_tac (
fold_map (abstract_unit o dest_thm) thms ##>>
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 = match_instantiate ctxt t @{thm eq_commute}
(* definitional axioms *)
fun def_axiom_disj ctxt t =
(case dest_prop t of
@{const HOL.disj} $ u1 $ u2 =>
prove_abstract' ctxt t prop_tac (
abstract_prop u2 ##>> abstract_prop u1 #>> HOLogic.mk_disj o swap)
| u => prove_abstract' ctxt t prop_tac (abstract_prop u))
fun def_axiom ctxt _ = try_provers ctxt 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 =
prove_abstract ctxt thms t prop_tac (
fold_map (abstract_prop o dest_thm) thms ##>>
abstract_prop (dest_prop t))
fun nnf ctxt rule thms = try_provers ctxt 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
(* theory lemmas *)
fun arith_th_lemma_tac ctxt prems =
Method.insert_tac prems
THEN' SELECT_GOAL (Local_Defs.unfold_tac ctxt @{thms z3div_def z3mod_def})
THEN' Arith_Data.arith_tac ctxt
fun arith_th_lemma ctxt thms t =
prove_abstract ctxt thms t arith_th_lemma_tac (
fold_map (abstract_arith ctxt o dest_thm) thms ##>>
abstract_arith ctxt (dest_prop t))
val _ = Theory.setup (Context.theory_map (add_th_lemma_method ("arith", arith_th_lemma)))
fun th_lemma name ctxt thms =
(case Symtab.lookup (get_th_lemma_method ctxt) name of
SOME method => method ctxt thms
| NONE => replay_error ctxt "Bad theory" (Z3_Proof.Th_Lemma name) thms)
(* 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;