(* Title: HOL/Tools/SMT2/z3_new_proof_replay.ML
Author: Sascha Boehme, TU Muenchen
Author: Jasmin Blanchette, TU Muenchen
Z3 proof replay.
*)
signature Z3_NEW_PROOF_REPLAY =
sig
val replay: Proof.context -> SMT2_Translate.replay_data -> string list ->
(int list * Z3_New_Proof.z3_step list) * thm
end
structure Z3_New_Proof_Replay: Z3_NEW_PROOF_REPLAY =
struct
fun params_of t = Term.strip_qnt_vars @{const_name 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 (SMT2_Utils.certify 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, SMT2_Utils.certify ctxt' (Free (n', T)))
in (ctxt', Symtab.make (map2 mk nTs ns)) end
fun forall_elim_term ct (Const (@{const_name 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_New_Proof.Z3_Step {id, rule, concl, fixes, is_fix_step, ...}) =
if Z3_New_Proof_Methods.is_assumption rule then
(case Inttab.lookup assumed id of
SOME (_, thm) => thm
| NONE => Thm.assume (SMT2_Utils.certify ctxt concl))
else
under_fixes (Z3_New_Proof_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_New_Proof.Z3_Step {id, prems, fixes, ...}) proofs =
let val nthms = map (the o Inttab.lookup proofs) prems
in Inttab.update (id, (fixes, replay_thm ctxt assumed nthms step)) proofs end
local
val remove_trigger = mk_meta_eq @{thm SMT2.trigger_def}
val remove_weight = mk_meta_eq @{thm SMT2.weight_def}
val remove_fun_app = mk_meta_eq @{thm SMT2.fun_app_def}
fun rewrite_conv _ [] = Conv.all_conv
| rewrite_conv ctxt eqs = Simplifier.full_rewrite (empty_simpset ctxt addsimps eqs)
val prep_rules = [@{thm Let_def}, remove_trigger, remove_weight,
remove_fun_app, Z3_New_Proof_Literals.rewrite_true]
fun rewrite _ [] = I
| rewrite ctxt eqs = Conv.fconv_rule (rewrite_conv ctxt eqs)
fun lookup_assm assms_net ct =
Z3_New_Proof_Tools.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 [Z3_New_Proof_Literals.rewrite_true]) 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_New_Proof_Tools.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) ((is, 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
((insert (op =) i is, thms'),
(ctxt', Inttab.update (id, (fixes, thm)) ptab))
end
fun add (Z3_New_Proof.Z3_Step {id, rule, concl, fixes, ...})
(cx as ((is, thms), (ctxt, ptab))) =
if Z3_New_Proof_Methods.is_assumption rule andalso rule <> Z3_New_Proof.Hypothesis then
let
val ct = SMT2_Utils.certify 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 ((is, 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 i st =
(rtac @{thm trans} i
THEN resolve_tac sk_rules i
THEN (rtac @{thm refl} ORELSE' discharge_sk_tac) (i+1)
THEN rtac @{thm refl} i) st
end
fun make_discharge_rules rules = rules @ [@{thm allI}, @{thm refl},
@{thm reflexive}, Z3_New_Proof_Literals.true_thm]
val intro_def_rules = @{lemma
"(~ P | P) & (P | ~ P)"
"(P | ~ P) & (~ P | P)"
by fast+}
fun discharge_assms_tac rules =
REPEAT (HEADGOAL (resolve_tac (intro_def_rules @ rules) ORELSE' SOLVED' discharge_sk_tac))
fun discharge_assms ctxt rules thm =
(if Thm.nprems_of thm = 0 then
thm
else
(case Seq.pull (discharge_assms_tac 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 replay outer_ctxt
({context=ctxt, typs, terms, rewrite_rules, assms} : SMT2_Translate.replay_data) output =
let
val (steps, ctxt1) = Z3_New_Proof.parse typs terms output ctxt
val ctxt2 = put_simpset (Z3_New_Proof_Tools.make_simpset ctxt1 []) ctxt1
val ((is, rules), (ctxt3, assumed)) = add_asserted outer_ctxt rewrite_rules assms steps ctxt2
val proofs = fold (replay_step ctxt3 assumed) steps assumed
val (_, Z3_New_Proof.Z3_Step {id, ...}) = split_last steps
in
if Config.get ctxt3 SMT2_Config.filter_only_facts then ((is, steps), TrueI)
else (([], steps), Inttab.lookup proofs id |> the |> snd |> discharge rules outer_ctxt ctxt3)
end
end