(* Title: HOL/Tools/SMT/z3_replay.ML
Author: Sascha Boehme, TU Muenchen
Author: Jasmin Blanchette, TU Muenchen
Z3 proof parsing and replay.
*)
signature Z3_REPLAY =
sig
val parse_proof: Proof.context -> SMT_Translate.replay_data ->
((string * ATP_Problem_Generate.stature) * thm) list -> term list -> term -> string list ->
SMT_Solver.parsed_proof
val replay: Proof.context -> SMT_Translate.replay_data -> string list -> thm
end;
structure Z3_Replay: Z3_REPLAY =
struct
fun params_of t = Term.strip_qnt_vars @{const_name Pure.all} t
fun varify ctxt thm =
let
val maxidx = Thm.maxidx_of thm + 1
val vs = params_of (Thm.prop_of thm)
val vars = map_index (fn (i, (n, T)) => Var ((n, i + maxidx), T)) vs
in Drule.forall_elim_list (map (Proof_Context.cterm_of ctxt) vars) thm end
fun add_paramTs names t =
fold2 (fn n => fn (_, T) => AList.update (op =) (n, T)) names (params_of t)
fun new_fixes ctxt nTs =
let
val (ns, ctxt') = Variable.variant_fixes (replicate (length nTs) "") ctxt
fun mk (n, T) n' = (n, Proof_Context.cterm_of ctxt' (Free (n', T)))
in (ctxt', Symtab.make (map2 mk nTs ns)) end
fun forall_elim_term ct (Const (@{const_name Pure.all}, _) $ (a as Abs _)) =
Term.betapply (a, Thm.term_of ct)
| forall_elim_term _ qt = raise TERM ("forall_elim'", [qt])
fun apply_fixes elim env = fold (elim o the o Symtab.lookup env)
val apply_fixes_prem = uncurry o apply_fixes Thm.forall_elim
val apply_fixes_concl = apply_fixes forall_elim_term
fun export_fixes env names = Drule.forall_intr_list (map (the o Symtab.lookup env) names)
fun under_fixes f ctxt (prems, nthms) names concl =
let
val thms1 = map (varify ctxt) prems
val (ctxt', env) =
add_paramTs names concl []
|> fold (uncurry add_paramTs o apsnd Thm.prop_of) nthms
|> new_fixes ctxt
val thms2 = map (apply_fixes_prem env) nthms
val t = apply_fixes_concl env names concl
in export_fixes env names (f ctxt' (thms1 @ thms2) t) end
fun replay_thm ctxt assumed nthms (Z3_Proof.Z3_Step {id, rule, concl, fixes, is_fix_step, ...}) =
if Z3_Proof.is_assumption rule then
(case Inttab.lookup assumed id of
SOME (_, thm) => thm
| NONE => Thm.assume (Proof_Context.cterm_of ctxt concl))
else
under_fixes (Z3_Replay_Methods.method_for rule) ctxt
(if is_fix_step then (map snd nthms, []) else ([], nthms)) fixes concl
fun replay_step ctxt assumed (step as Z3_Proof.Z3_Step {id, rule, prems, fixes, ...}) state =
let
val (proofs, stats) = state
val nthms = map (the o Inttab.lookup proofs) prems
val replay = Timing.timing (replay_thm ctxt assumed nthms)
val ({elapsed, ...}, thm) =
SMT_Config.with_time_limit ctxt SMT_Config.reconstruction_step_timeout replay step
handle TimeLimit.TimeOut => raise SMT_Failure.SMT SMT_Failure.Time_Out
val stats' = Symtab.cons_list (Z3_Proof.string_of_rule rule, Time.toMilliseconds elapsed) stats
in (Inttab.update (id, (fixes, thm)) proofs, stats') end
local
val remove_trigger = mk_meta_eq @{thm trigger_def}
val remove_fun_app = mk_meta_eq @{thm fun_app_def}
fun rewrite_conv _ [] = Conv.all_conv
| rewrite_conv ctxt eqs = Simplifier.full_rewrite (empty_simpset ctxt addsimps eqs)
val rewrite_true_rule = @{lemma "True == ~ False" by simp}
val prep_rules = [@{thm Let_def}, remove_trigger, remove_fun_app, rewrite_true_rule]
fun rewrite _ [] = I
| rewrite ctxt eqs = Conv.fconv_rule (rewrite_conv ctxt eqs)
fun lookup_assm assms_net ct =
Z3_Replay_Util.net_instances assms_net ct
|> map (fn ithm as (_, thm) => (ithm, Thm.cprop_of thm aconvc ct))
in
fun add_asserted outer_ctxt rewrite_rules assms steps ctxt =
let
val eqs = map (rewrite ctxt [rewrite_true_rule]) rewrite_rules
val eqs' = union Thm.eq_thm eqs prep_rules
val assms_net =
assms
|> map (apsnd (rewrite ctxt eqs'))
|> map (apsnd (Conv.fconv_rule Thm.eta_conversion))
|> Z3_Replay_Util.thm_net_of snd
fun revert_conv ctxt = rewrite_conv ctxt eqs' then_conv Thm.eta_conversion
fun assume thm ctxt =
let
val ct = Thm.cprem_of thm 1
val (thm', ctxt') = yield_singleton Assumption.add_assumes ct ctxt
in (thm' RS thm, ctxt') end
fun add1 id fixes thm1 ((i, th), exact) ((iidths, thms), (ctxt, ptab)) =
let
val (thm, ctxt') = if exact then (Thm.implies_elim thm1 th, ctxt) else assume thm1 ctxt
val thms' = if exact then thms else th :: thms
in (((i, (id, th)) :: iidths, thms'), (ctxt', Inttab.update (id, (fixes, thm)) ptab)) end
fun add (Z3_Proof.Z3_Step {id, rule, concl, fixes, ...})
(cx as ((iidths, thms), (ctxt, ptab))) =
if Z3_Proof.is_assumption rule andalso rule <> Z3_Proof.Hypothesis then
let
val ct = Proof_Context.cterm_of ctxt concl
val thm1 = Thm.trivial ct |> Conv.fconv_rule (Conv.arg1_conv (revert_conv outer_ctxt))
val thm2 = singleton (Variable.export ctxt outer_ctxt) thm1
in
(case lookup_assm assms_net (Thm.cprem_of thm2 1) of
[] =>
let val (thm, ctxt') = assume thm1 ctxt
in ((iidths, thms), (ctxt', Inttab.update (id, (fixes, thm)) ptab)) end
| ithms => fold (add1 id fixes thm1) ithms cx)
end
else
cx
in fold add steps (([], []), (ctxt, Inttab.empty)) end
end
(* |- (EX x. P x) = P c |- ~ (ALL x. P x) = ~ P c *)
local
val sk_rules = @{lemma
"c = (SOME x. P x) ==> (EX x. P x) = P c"
"c = (SOME x. ~ P x) ==> (~ (ALL x. P x)) = (~ P c)"
by (metis someI_ex)+}
in
fun discharge_sk_tac ctxt i st =
(rtac @{thm trans} i
THEN resolve_tac ctxt sk_rules i
THEN (rtac @{thm refl} ORELSE' discharge_sk_tac ctxt) (i+1)
THEN rtac @{thm refl} i) st
end
val true_thm = @{lemma "~False" by simp}
fun make_discharge_rules rules = rules @ [@{thm allI}, @{thm refl}, @{thm reflexive}, true_thm]
val intro_def_rules = @{lemma
"(~ P | P) & (P | ~ P)"
"(P | ~ P) & (~ P | P)"
by fast+}
fun discharge_assms_tac ctxt rules =
REPEAT
(HEADGOAL (resolve_tac ctxt (intro_def_rules @ rules) ORELSE'
SOLVED' (discharge_sk_tac ctxt)))
fun discharge_assms ctxt rules thm =
(if Thm.nprems_of thm = 0 then
thm
else
(case Seq.pull (discharge_assms_tac ctxt rules thm) of
SOME (thm', _) => thm'
| NONE => raise THM ("failed to discharge premise", 1, [thm])))
|> Goal.norm_result ctxt
fun discharge rules outer_ctxt inner_ctxt =
singleton (Proof_Context.export inner_ctxt outer_ctxt)
#> discharge_assms outer_ctxt (make_discharge_rules rules)
fun parse_proof outer_ctxt
({context = ctxt, typs, terms, ll_defs, rewrite_rules, assms} : SMT_Translate.replay_data)
xfacts prems concl output =
let
val (steps, ctxt2) = Z3_Proof.parse typs terms output ctxt
val ((iidths, _), _) = add_asserted outer_ctxt rewrite_rules assms steps ctxt2
fun id_of_index i = the_default ~1 (Option.map fst (AList.lookup (op =) iidths i))
val conjecture_i = 0
val prems_i = 1
val facts_i = prems_i + length prems
val conjecture_id = id_of_index conjecture_i
val prem_ids = map id_of_index (prems_i upto facts_i - 1)
val fact_ids' =
map_filter (fn (i, (id, _)) => try (apsnd (nth xfacts)) (id, i - facts_i)) iidths
val helper_ids' = map_filter (try (fn (~1, idth) => idth)) iidths
val fact_helper_ts =
map (fn (_, th) => (ATP_Util.short_thm_name ctxt th, Thm.prop_of th)) helper_ids' @
map (fn (_, ((s, _), th)) => (s, Thm.prop_of th)) fact_ids'
val fact_helper_ids' =
map (apsnd (ATP_Util.short_thm_name ctxt)) helper_ids' @ map (apsnd (fst o fst)) fact_ids'
in
{outcome = NONE, fact_ids = fact_ids',
atp_proof = fn () => Z3_Isar.atp_proof_of_z3_proof ctxt ll_defs rewrite_rules prems concl
fact_helper_ts prem_ids conjecture_id fact_helper_ids' steps}
end
fun intermediate_statistics ctxt start total =
SMT_Config.statistics_msg ctxt (fn current =>
"Reconstructed " ^ string_of_int current ^ " of " ^ string_of_int total ^ " steps in " ^
string_of_int (Time.toMilliseconds (#elapsed (Timing.result start))) ^ " ms")
fun pretty_statistics total stats =
let
fun mean_of is =
let
val len = length is
val mid = len div 2
in if len mod 2 = 0 then (nth is (mid - 1) + nth is mid) div 2 else nth is mid end
fun pretty_item name p = Pretty.item (Pretty.separate ":" [Pretty.str name, p])
fun pretty (name, milliseconds) = pretty_item name (Pretty.block (Pretty.separate "," [
Pretty.str (string_of_int (length milliseconds) ^ " occurrences") ,
Pretty.str (string_of_int (mean_of milliseconds) ^ " ms mean time"),
Pretty.str (string_of_int (fold Integer.max milliseconds 0) ^ " ms maximum time"),
Pretty.str (string_of_int (fold Integer.add milliseconds 0) ^ " ms total time")]))
in
Pretty.big_list "Z3 proof reconstruction statistics:" (
pretty_item "total time" (Pretty.str (string_of_int total ^ " ms")) ::
map pretty (Symtab.dest stats))
end
fun replay outer_ctxt
({context = ctxt, typs, terms, rewrite_rules, assms, ...} : SMT_Translate.replay_data) output =
let
val (steps, ctxt2) = Z3_Proof.parse typs terms output ctxt
val ((_, rules), (ctxt3, assumed)) = add_asserted outer_ctxt rewrite_rules assms steps ctxt2
val ctxt4 =
ctxt3
|> put_simpset (Z3_Replay_Util.make_simpset ctxt3 [])
|> Config.put SAT.solver (Config.get ctxt3 SMT_Config.sat_solver)
val len = length steps
val start = Timing.start ()
val print_runtime_statistics = intermediate_statistics ctxt4 start len
fun blockwise f (i, x) y =
(if i > 0 andalso i mod 100 = 0 then print_runtime_statistics i else (); f x y)
val (proofs, stats) =
fold_index (blockwise (replay_step ctxt4 assumed)) steps (assumed, Symtab.empty)
val _ = print_runtime_statistics len
val total = Time.toMilliseconds (#elapsed (Timing.result start))
val (_, Z3_Proof.Z3_Step {id, ...}) = split_last steps
val _ = SMT_Config.statistics_msg ctxt4 (Pretty.string_of o pretty_statistics total) stats
in
Inttab.lookup proofs id |> the |> snd |> discharge rules outer_ctxt ctxt4
end
end;