(* Title: HOL/Tools/SMT/z3_proof_reconstruction.ML
Author: Sascha Boehme, TU Muenchen
Proof reconstruction for proofs found by Z3.
*)
signature Z3_PROOF_RECONSTRUCTION =
sig
val add_z3_rule: thm -> Context.generic -> Context.generic
val reconstruct: Proof.context -> SMT_Translate.recon -> string list ->
(int list * thm) * Proof.context
val setup: theory -> theory
end
structure Z3_Proof_Reconstruction: Z3_PROOF_RECONSTRUCTION =
struct
structure P = Z3_Proof_Parser
structure T = Z3_Proof_Tools
structure L = Z3_Proof_Literals
fun z3_exn msg = raise SMT_Failure.SMT (SMT_Failure.Other_Failure
("Z3 proof reconstruction: " ^ msg))
(** net of schematic rules **)
val z3_ruleN = "z3_rule"
local
val description = "declaration of Z3 proof rules"
val eq = Thm.eq_thm
structure Z3_Rules = Generic_Data
(
type T = thm Net.net
val empty = Net.empty
val extend = I
val merge = Net.merge eq
)
val prep = `Thm.prop_of o Simplifier.rewrite_rule [L.rewrite_true]
fun ins thm net = Net.insert_term eq (prep thm) net handle Net.INSERT => net
fun del thm net = Net.delete_term eq (prep thm) net handle Net.DELETE => net
val add = Thm.declaration_attribute (Z3_Rules.map o ins)
val del = Thm.declaration_attribute (Z3_Rules.map o del)
in
val add_z3_rule = Z3_Rules.map o ins
fun by_schematic_rule ctxt ct =
the (T.net_instance (Z3_Rules.get (Context.Proof ctxt)) ct)
val z3_rules_setup =
Attrib.setup (Binding.name z3_ruleN) (Attrib.add_del add del) description #>
Global_Theory.add_thms_dynamic (Binding.name z3_ruleN, Net.content o Z3_Rules.get)
end
(** proof tools **)
fun named ctxt name prover ct =
let val _ = SMT_Config.trace_msg ctxt I ("Z3: trying " ^ name ^ " ...")
in prover ct end
fun NAMED ctxt name tac i st =
let val _ = SMT_Config.trace_msg ctxt I ("Z3: trying " ^ name ^ " ...")
in tac i st end
fun pretty_goal ctxt thms t =
[Pretty.block [Pretty.str "proposition: ", Syntax.pretty_term ctxt t]]
|> not (null thms) ? cons (Pretty.big_list "assumptions:"
(map (Display.pretty_thm ctxt) thms))
fun try_apply ctxt thms =
let
fun try_apply_err ct = Pretty.string_of (Pretty.chunks [
Pretty.big_list ("Z3 found a proof," ^
" but proof reconstruction failed at the following subgoal:")
(pretty_goal ctxt thms (Thm.term_of ct)),
Pretty.str ("Adding a rule to the lemma group " ^ quote z3_ruleN ^
" might solve this problem.")])
fun apply [] ct = error (try_apply_err ct)
| apply (prover :: provers) ct =
(case try prover ct of
SOME thm => (SMT_Config.trace_msg ctxt I "Z3: succeeded"; thm)
| NONE => apply provers ct)
in apply o cons (named ctxt "schematic rules" (by_schematic_rule ctxt)) end
local
val rewr_if =
@{lemma "(if P then Q1 else Q2) = ((P --> Q1) & (~P --> Q2))" by simp}
in
val simp_fast_tac =
Simplifier.simp_tac (HOL_ss addsimps [rewr_if])
THEN_ALL_NEW Classical.fast_tac HOL_cs
end
(** theorems and proofs **)
(* theorem incarnations *)
datatype theorem =
Thm of thm | (* theorem without special features *)
MetaEq of thm | (* meta equality "t == s" *)
Literals of thm * L.littab
(* "P1 & ... & Pn" and table of all literals P1, ..., Pn *)
fun thm_of (Thm thm) = thm
| thm_of (MetaEq thm) = thm COMP @{thm meta_eq_to_obj_eq}
| thm_of (Literals (thm, _)) = thm
fun meta_eq_of (MetaEq thm) = thm
| meta_eq_of p = mk_meta_eq (thm_of p)
fun literals_of (Literals (_, lits)) = lits
| literals_of p = L.make_littab [thm_of p]
(* proof representation *)
datatype proof = Unproved of P.proof_step | Proved of theorem
(** core proof rules **)
(* assumption *)
local
val remove_trigger = @{lemma "trigger t p == p"
by (rule eq_reflection, rule trigger_def)}
val prep_rules = [@{thm Let_def}, remove_trigger, L.rewrite_true]
fun rewrite_conv ctxt eqs = Simplifier.full_rewrite
(Simplifier.context ctxt Simplifier.empty_ss addsimps eqs)
fun rewrites f ctxt eqs = map (f (Conv.fconv_rule (rewrite_conv ctxt eqs)))
fun burrow_snd_option f (i, thm) = Option.map (pair i) (f thm)
fun lookup_assm ctxt assms ct =
(case T.net_instance' burrow_snd_option assms ct of
SOME ithm => ithm
| _ => z3_exn ("not asserted: " ^
quote (Syntax.string_of_term ctxt (Thm.term_of ct))))
in
fun prepare_assms ctxt unfolds assms =
let
val unfolds' = rewrites I ctxt [L.rewrite_true] unfolds
val assms' =
rewrites apsnd ctxt (union Thm.eq_thm unfolds' prep_rules) assms
in (unfolds', T.thm_net_of snd assms') end
fun asserted ctxt (unfolds, assms) ct =
let val revert_conv = rewrite_conv ctxt unfolds
in Thm (T.with_conv revert_conv (snd o lookup_assm ctxt assms) ct) end
fun find_assm ctxt (unfolds, assms) ct =
fst (lookup_assm ctxt assms (Thm.rhs_of (rewrite_conv ctxt unfolds ct)))
end
(* P = Q ==> P ==> Q or P --> Q ==> P ==> Q *)
local
val meta_iffD1 = @{lemma "P == Q ==> P ==> (Q::bool)" by simp}
val meta_iffD1_c = T.precompose2 Thm.dest_binop meta_iffD1
val iffD1_c = T.precompose2 (Thm.dest_binop o Thm.dest_arg) @{thm iffD1}
val mp_c = T.precompose2 (Thm.dest_binop o Thm.dest_arg) @{thm mp}
in
fun mp (MetaEq thm) p = Thm (Thm.implies_elim (T.compose meta_iffD1_c thm) p)
| mp p_q p =
let
val pq = thm_of p_q
val thm = T.compose iffD1_c pq handle THM _ => T.compose mp_c pq
in Thm (Thm.implies_elim thm p) end
end
(* and_elim: P1 & ... & Pn ==> Pi *)
(* not_or_elim: ~(P1 | ... | Pn) ==> ~Pi *)
local
fun is_sublit conj t = L.exists_lit conj (fn u => u aconv t)
fun derive conj t lits idx ptab =
let
val lit = the (L.get_first_lit (is_sublit conj t) lits)
val ls = L.explode conj false false [t] lit
val lits' = fold L.insert_lit ls (L.delete_lit lit lits)
fun upd (Proved thm) = Proved (Literals (thm_of thm, lits'))
| upd p = p
in (the (L.lookup_lit lits' t), Inttab.map_entry idx upd ptab) end
fun lit_elim conj (p, idx) ct ptab =
let val lits = literals_of p
in
(case L.lookup_lit lits (T.term_of ct) of
SOME lit => (Thm lit, ptab)
| NONE => apfst Thm (derive conj (T.term_of ct) lits idx ptab))
end
in
val and_elim = lit_elim true
val not_or_elim = lit_elim false
end
(* P1, ..., Pn |- False ==> |- ~P1 | ... | ~Pn *)
local
fun step lit thm =
Thm.implies_elim (Thm.implies_intr (Thm.cprop_of lit) thm) lit
val explode_disj = L.explode false false false
fun intro hyps thm th = fold step (explode_disj hyps th) thm
fun dest_ccontr ct = [Thm.dest_arg (Thm.dest_arg (Thm.dest_arg1 ct))]
val ccontr = T.precompose dest_ccontr @{thm ccontr}
in
fun lemma thm ct =
let
val cu = Thm.capply @{cterm Not} (Thm.dest_arg ct)
val hyps = map_filter (try HOLogic.dest_Trueprop) (#hyps (Thm.rep_thm thm))
in Thm (T.compose ccontr (T.under_assumption (intro hyps thm) cu)) end
end
(* \/{P1, ..., Pn, Q1, ..., Qn}, ~P1, ..., ~Pn ==> \/{Q1, ..., Qn} *)
local
val explode_disj = L.explode false true false
val join_disj = L.join false
fun unit thm thms th =
let val t = @{term Not} $ T.prop_of thm and ts = map T.prop_of thms
in join_disj (L.make_littab (thms @ explode_disj ts th)) t end
fun dest_arg2 ct = Thm.dest_arg (Thm.dest_arg ct)
fun dest ct = pairself dest_arg2 (Thm.dest_binop ct)
val contrapos = T.precompose2 dest @{lemma "(~P ==> ~Q) ==> Q ==> P" by fast}
in
fun unit_resolution thm thms ct =
Thm.capply @{cterm Not} (Thm.dest_arg ct)
|> T.under_assumption (unit thm thms)
|> Thm o T.discharge thm o T.compose contrapos
end
(* P ==> P == True or P ==> P == False *)
local
val iff1 = @{lemma "P ==> P == (~ False)" by simp}
val iff2 = @{lemma "~P ==> P == False" by simp}
in
fun iff_true thm = MetaEq (thm COMP iff1)
fun iff_false thm = MetaEq (thm COMP iff2)
end
(* distributivity of | over & *)
fun distributivity ctxt = Thm o try_apply ctxt [] [
named ctxt "fast" (T.by_tac (Classical.fast_tac HOL_cs))]
(* FIXME: not very well tested *)
(* Tseitin-like axioms *)
local
val disjI1 = @{lemma "(P ==> Q) ==> ~P | Q" by fast}
val disjI2 = @{lemma "(~P ==> Q) ==> P | Q" by fast}
val disjI3 = @{lemma "(~Q ==> P) ==> P | Q" by fast}
val disjI4 = @{lemma "(Q ==> P) ==> P | ~Q" by fast}
fun prove' conj1 conj2 ct2 thm =
let val lits = L.true_thm :: L.explode conj1 true (conj1 <> conj2) [] thm
in L.join conj2 (L.make_littab lits) (Thm.term_of ct2) end
fun prove rule (ct1, conj1) (ct2, conj2) =
T.under_assumption (prove' conj1 conj2 ct2) ct1 COMP rule
fun prove_def_axiom ct =
let val (ct1, ct2) = Thm.dest_binop (Thm.dest_arg ct)
in
(case Thm.term_of ct1 of
@{term Not} $ (@{term HOL.conj} $ _ $ _) =>
prove disjI1 (Thm.dest_arg ct1, true) (ct2, true)
| @{term HOL.conj} $ _ $ _ =>
prove disjI3 (Thm.capply @{cterm Not} ct2, false) (ct1, true)
| @{term Not} $ (@{term HOL.disj} $ _ $ _) =>
prove disjI3 (Thm.capply @{cterm Not} ct2, false) (ct1, false)
| @{term HOL.disj} $ _ $ _ =>
prove disjI2 (Thm.capply @{cterm Not} ct1, false) (ct2, true)
| Const (@{const_name SMT.distinct}, _) $ _ =>
let
fun dis_conv cv = Conv.arg_conv (Conv.arg1_conv cv)
fun prv cu =
let val (cu1, cu2) = Thm.dest_binop (Thm.dest_arg cu)
in prove disjI4 (Thm.dest_arg cu2, true) (cu1, true) end
in T.with_conv (dis_conv T.unfold_distinct_conv) prv ct end
| @{term Not} $ (Const (@{const_name SMT.distinct}, _) $ _) =>
let
fun dis_conv cv = Conv.arg_conv (Conv.arg1_conv (Conv.arg_conv cv))
fun prv cu =
let val (cu1, cu2) = Thm.dest_binop (Thm.dest_arg cu)
in prove disjI1 (Thm.dest_arg cu1, true) (cu2, true) end
in T.with_conv (dis_conv T.unfold_distinct_conv) prv ct end
| _ => raise CTERM ("prove_def_axiom", [ct]))
end
in
fun def_axiom ctxt = Thm o try_apply ctxt [] [
named ctxt "conj/disj/distinct" prove_def_axiom,
T.by_abstraction (true, false) ctxt [] (fn ctxt' =>
named ctxt' "simp+fast" (T.by_tac simp_fast_tac))]
end
(* local definitions *)
local
val intro_rules = [
@{lemma "n == P ==> (~n | P) & (n | ~P)" by simp},
@{lemma "n == (if P then s else t) ==> (~P | n = s) & (P | n = t)"
by simp},
@{lemma "n == P ==> n = P" by (rule meta_eq_to_obj_eq)} ]
val apply_rules = [
@{lemma "(~n | P) & (n | ~P) ==> P == n" by (atomize(full)) fast},
@{lemma "(~P | n = s) & (P | n = t) ==> (if P then s else t) == n"
by (atomize(full)) fastsimp} ]
val inst_rule = T.match_instantiate Thm.dest_arg
fun apply_rule ct =
(case get_first (try (inst_rule ct)) intro_rules of
SOME thm => thm
| NONE => raise CTERM ("intro_def", [ct]))
in
fun intro_def ct = T.make_hyp_def (apply_rule ct) #>> Thm
fun apply_def thm =
get_first (try (fn rule => MetaEq (thm COMP rule))) apply_rules
|> the_default (Thm thm)
end
(* negation normal form *)
local
val quant_rules1 = ([
@{lemma "(!!x. P x == Q) ==> ALL x. P x == Q" by simp},
@{lemma "(!!x. P x == Q) ==> EX x. P x == Q" by simp}], [
@{lemma "(!!x. P x == Q x) ==> ALL x. P x == ALL x. Q x" by simp},
@{lemma "(!!x. P x == Q x) ==> EX x. P x == EX x. Q x" by simp}])
val quant_rules2 = ([
@{lemma "(!!x. ~P x == Q) ==> ~(ALL x. P x) == Q" by simp},
@{lemma "(!!x. ~P x == Q) ==> ~(EX x. P x) == Q" by simp}], [
@{lemma "(!!x. ~P x == Q x) ==> ~(ALL x. P x) == EX x. Q x" by simp},
@{lemma "(!!x. ~P x == Q x) ==> ~(EX x. P x) == ALL x. Q x" by simp}])
fun nnf_quant_tac thm (qs as (qs1, qs2)) i st = (
Tactic.rtac thm ORELSE'
(Tactic.match_tac qs1 THEN' nnf_quant_tac thm qs) ORELSE'
(Tactic.match_tac qs2 THEN' nnf_quant_tac thm qs)) i st
fun nnf_quant vars qs p ct =
T.as_meta_eq ct
|> T.by_tac (nnf_quant_tac (T.varify vars (meta_eq_of p)) qs)
fun prove_nnf ctxt = try_apply ctxt [] [
named ctxt "conj/disj" L.prove_conj_disj_eq,
T.by_abstraction (true, false) ctxt [] (fn ctxt' =>
named ctxt' "simp+fast" (T.by_tac simp_fast_tac))]
in
fun nnf ctxt vars ps ct =
(case T.term_of ct of
_ $ (l as Const _ $ Abs _) $ (r as Const _ $ Abs _) =>
if l aconv r
then MetaEq (Thm.reflexive (Thm.dest_arg (Thm.dest_arg ct)))
else MetaEq (nnf_quant vars quant_rules1 (hd ps) ct)
| _ $ (@{term Not} $ (Const _ $ Abs _)) $ (Const _ $ Abs _) =>
MetaEq (nnf_quant vars quant_rules2 (hd ps) ct)
| _ =>
let
val nnf_rewr_conv = Conv.arg_conv (Conv.arg_conv
(T.unfold_eqs ctxt (map (Thm.symmetric o meta_eq_of) ps)))
in Thm (T.with_conv nnf_rewr_conv (prove_nnf ctxt) ct) end)
end
(** equality proof rules **)
(* |- t = t *)
fun refl ct = MetaEq (Thm.reflexive (Thm.dest_arg (Thm.dest_arg ct)))
(* s = t ==> t = s *)
local
val symm_rule = @{lemma "s = t ==> t == s" by simp}
in
fun symm (MetaEq thm) = MetaEq (Thm.symmetric thm)
| symm p = MetaEq (thm_of p COMP symm_rule)
end
(* s = t ==> t = u ==> s = u *)
local
val trans1 = @{lemma "s == t ==> t = u ==> s == u" by simp}
val trans2 = @{lemma "s = t ==> t == u ==> s == u" by simp}
val trans3 = @{lemma "s = t ==> t = u ==> s == u" by simp}
in
fun trans (MetaEq thm1) (MetaEq thm2) = MetaEq (Thm.transitive thm1 thm2)
| trans (MetaEq thm) q = MetaEq (thm_of q COMP (thm COMP trans1))
| trans p (MetaEq thm) = MetaEq (thm COMP (thm_of p COMP trans2))
| trans p q = MetaEq (thm_of q COMP (thm_of p COMP trans3))
end
(* t1 = s1 ==> ... ==> tn = sn ==> f t1 ... tn = f s1 .. sn
(reflexive antecendents are droppped) *)
local
exception MONO
fun prove_refl (ct, _) = Thm.reflexive ct
fun prove_comb f g cp =
let val ((ct1, ct2), (cu1, cu2)) = pairself Thm.dest_comb cp
in Thm.combination (f (ct1, cu1)) (g (ct2, cu2)) end
fun prove_arg f = prove_comb prove_refl f
fun prove f cp = prove_comb (prove f) f cp handle CTERM _ => prove_refl cp
fun prove_nary is_comb f =
let
fun prove (cp as (ct, _)) = f cp handle MONO =>
if is_comb (Thm.term_of ct)
then prove_comb (prove_arg prove) prove cp
else prove_refl cp
in prove end
fun prove_list f n cp =
if n = 0 then prove_refl cp
else prove_comb (prove_arg f) (prove_list f (n-1)) cp
fun with_length f (cp as (cl, _)) =
f (length (HOLogic.dest_list (Thm.term_of cl))) cp
fun prove_distinct f = prove_arg (with_length (prove_list f))
fun prove_eq exn lookup cp =
(case lookup (Logic.mk_equals (pairself Thm.term_of cp)) of
SOME eq => eq
| NONE => if exn then raise MONO else prove_refl cp)
val prove_eq_exn = prove_eq true
and prove_eq_safe = prove_eq false
fun mono f (cp as (cl, _)) =
(case Term.head_of (Thm.term_of cl) of
@{term HOL.conj} => prove_nary L.is_conj (prove_eq_exn f)
| @{term HOL.disj} => prove_nary L.is_disj (prove_eq_exn f)
| Const (@{const_name SMT.distinct}, _) => prove_distinct (prove_eq_safe f)
| _ => prove (prove_eq_safe f)) cp
in
fun monotonicity eqs ct =
let
val lookup = AList.lookup (op aconv) (map (`Thm.prop_of o meta_eq_of) eqs)
val cp = Thm.dest_binop (Thm.dest_arg ct)
in MetaEq (prove_eq_exn lookup cp handle MONO => mono lookup cp) end
end
(* |- f a b = f b a (where f is equality) *)
local
val rule = @{lemma "a = b == b = a" by (atomize(full)) (rule eq_commute)}
in
fun commutativity ct = MetaEq (T.match_instantiate I (T.as_meta_eq ct) rule)
end
(** quantifier proof rules **)
(* P ?x = Q ?x ==> (ALL x. P x) = (ALL x. Q x)
P ?x = Q ?x ==> (EX x. P x) = (EX x. Q x) *)
local
val rules = [
@{lemma "(!!x. P x == Q x) ==> (ALL x. P x) == (ALL x. Q x)" by simp},
@{lemma "(!!x. P x == Q x) ==> (EX x. P x) == (EX x. Q x)" by simp}]
in
fun quant_intro vars p ct =
let
val thm = meta_eq_of p
val rules' = T.varify vars thm :: rules
val cu = T.as_meta_eq ct
in MetaEq (T.by_tac (REPEAT_ALL_NEW (Tactic.match_tac rules')) cu) end
end
(* |- ((ALL x. P x) | Q) = (ALL x. P x | Q) *)
fun pull_quant ctxt = Thm o try_apply ctxt [] [
named ctxt "fast" (T.by_tac (Classical.fast_tac HOL_cs))]
(* FIXME: not very well tested *)
(* |- (ALL x. P x & Q x) = ((ALL x. P x) & (ALL x. Q x)) *)
fun push_quant ctxt = Thm o try_apply ctxt [] [
named ctxt "fast" (T.by_tac (Classical.fast_tac HOL_cs))]
(* FIXME: not very well tested *)
(* |- (ALL x1 ... xn y1 ... yn. P x1 ... xn) = (ALL x1 ... xn. P x1 ... xn) *)
local
val elim_all = @{lemma "(ALL x. P) == P" by simp}
val elim_ex = @{lemma "(EX x. P) == P" by simp}
fun elim_unused_conv ctxt =
Conv.params_conv ~1 (K (Conv.arg_conv (Conv.arg1_conv
(Conv.rewrs_conv [elim_all, elim_ex])))) ctxt
fun elim_unused_tac ctxt =
REPEAT_ALL_NEW (
Tactic.match_tac [@{thm refl}, @{thm iff_allI}, @{thm iff_exI}]
ORELSE' CONVERSION (elim_unused_conv ctxt))
in
fun elim_unused_vars ctxt = Thm o T.by_tac (elim_unused_tac ctxt)
end
(* |- (ALL x1 ... xn. ~(x1 = t1 & ... xn = tn) | P x1 ... xn) = P t1 ... tn *)
fun dest_eq_res ctxt = Thm o try_apply ctxt [] [
named ctxt "fast" (T.by_tac (Classical.fast_tac HOL_cs))]
(* FIXME: not very well tested *)
(* |- ~(ALL x1...xn. P x1...xn) | P a1...an *)
local
val rule = @{lemma "~ P x | Q ==> ~(ALL x. P x) | Q" by fast}
in
val quant_inst = Thm o T.by_tac (
REPEAT_ALL_NEW (Tactic.match_tac [rule])
THEN' Tactic.rtac @{thm excluded_middle})
end
(* c = SOME x. P x |- (EX x. P x) = P c
c = SOME x. ~ P x |- ~(ALL x. P x) = ~ P c *)
local
val elim_ex = @{lemma "EX x. P == P" by simp}
val elim_all = @{lemma "~ (ALL x. P) == ~P" by simp}
val sk_ex = @{lemma "c == SOME x. P x ==> EX x. P x == P c"
by simp (intro eq_reflection some_eq_ex[symmetric])}
val sk_all = @{lemma "c == SOME x. ~ P x ==> ~(ALL x. P x) == ~ P c"
by (simp only: not_all) (intro eq_reflection some_eq_ex[symmetric])}
val sk_ex_rule = ((sk_ex, I), elim_ex)
and sk_all_rule = ((sk_all, Thm.dest_arg), elim_all)
fun dest f sk_rule =
Thm.dest_comb (f (Thm.dest_arg (Thm.dest_arg (Thm.cprop_of sk_rule))))
fun type_of f sk_rule = Thm.ctyp_of_term (snd (dest f sk_rule))
fun pair2 (a, b) (c, d) = [(a, c), (b, d)]
fun inst_sk (sk_rule, f) p c =
Thm.instantiate ([(type_of f sk_rule, Thm.ctyp_of_term c)], []) sk_rule
|> (fn sk' => Thm.instantiate ([], (pair2 (dest f sk') (p, c))) sk')
|> Conv.fconv_rule (Thm.beta_conversion true)
fun kind (Const (@{const_name Ex}, _) $ _) = (sk_ex_rule, I, I)
| kind (@{term Not} $ (Const (@{const_name All}, _) $ _)) =
(sk_all_rule, Thm.dest_arg, Thm.capply @{cterm Not})
| kind t = raise TERM ("skolemize", [t])
fun dest_abs_type (Abs (_, T, _)) = T
| dest_abs_type t = raise TERM ("dest_abs_type", [t])
fun bodies_of thy lhs rhs =
let
val (rule, dest, make) = kind (Thm.term_of lhs)
fun dest_body idx cbs ct =
let
val cb = Thm.dest_arg (dest ct)
val T = dest_abs_type (Thm.term_of cb)
val cv = Thm.cterm_of thy (Var (("x", idx), T))
val cu = make (Drule.beta_conv cb cv)
val cbs' = (cv, cb) :: cbs
in
(snd (Thm.first_order_match (cu, rhs)), rev cbs')
handle Pattern.MATCH => dest_body (idx+1) cbs' cu
end
in (rule, dest_body 1 [] lhs) end
fun transitive f thm = Thm.transitive thm (f (Thm.rhs_of thm))
fun sk_step (rule, elim) (cv, mct, cb) ((is, thm), ctxt) =
(case mct of
SOME ct =>
ctxt
|> T.make_hyp_def (inst_sk rule (Thm.instantiate_cterm ([], is) cb) ct)
|>> pair ((cv, ct) :: is) o Thm.transitive thm
| NONE => ((is, transitive (Conv.rewr_conv elim) thm), ctxt))
in
fun skolemize ct ctxt =
let
val (lhs, rhs) = Thm.dest_binop (Thm.dest_arg ct)
val (rule, (ctab, cbs)) = bodies_of (ProofContext.theory_of ctxt) lhs rhs
fun lookup_var (cv, cb) = (cv, AList.lookup (op aconvc) ctab cv, cb)
in
(([], Thm.reflexive lhs), ctxt)
|> fold (sk_step rule) (map lookup_var cbs)
|>> MetaEq o snd
end
end
(** theory proof rules **)
(* theory lemmas: linear arithmetic, arrays *)
fun th_lemma ctxt simpset thms = Thm o try_apply ctxt thms [
T.by_abstraction (false, true) ctxt thms (fn ctxt' => T.by_tac (
NAMED ctxt' "arith" (Arith_Data.arith_tac ctxt')
ORELSE' NAMED ctxt' "simp+arith" (Simplifier.simp_tac simpset THEN_ALL_NEW
Arith_Data.arith_tac ctxt')))]
(* rewriting: prove equalities:
* ACI of conjunction/disjunction
* contradiction, excluded middle
* logical rewriting rules (for negation, implication, equivalence,
distinct)
* normal forms for polynoms (integer/real arithmetic)
* quantifier elimination over linear arithmetic
* ... ? **)
structure Z3_Simps = Named_Thms
(
val name = "z3_simp"
val description = "simplification rules for Z3 proof reconstruction"
)
local
fun spec_meta_eq_of thm =
(case try (fn th => th RS @{thm spec}) thm of
SOME thm' => spec_meta_eq_of thm'
| NONE => mk_meta_eq thm)
fun prep (Thm thm) = spec_meta_eq_of thm
| prep (MetaEq thm) = thm
| prep (Literals (thm, _)) = spec_meta_eq_of thm
fun unfold_conv ctxt ths =
Conv.arg_conv (Conv.binop_conv (T.unfold_eqs ctxt (map prep ths)))
fun with_conv _ [] prv = prv
| with_conv ctxt ths prv = T.with_conv (unfold_conv ctxt ths) prv
val unfold_conv =
Conv.arg_conv (Conv.binop_conv (Conv.try_conv T.unfold_distinct_conv))
val prove_conj_disj_eq = T.with_conv unfold_conv L.prove_conj_disj_eq
in
fun rewrite ctxt simpset ths = Thm o with_conv ctxt ths (try_apply ctxt [] [
named ctxt "conj/disj/distinct" prove_conj_disj_eq,
T.by_abstraction (true, false) ctxt [] (fn ctxt' => T.by_tac (
NAMED ctxt' "simp (logic)" (Simplifier.simp_tac simpset)
THEN_ALL_NEW NAMED ctxt' "fast (logic)" (Classical.fast_tac HOL_cs))),
T.by_abstraction (false, true) ctxt [] (fn ctxt' => T.by_tac (
NAMED ctxt' "simp (theory)" (Simplifier.simp_tac simpset)
THEN_ALL_NEW (
NAMED ctxt' "fast (theory)" (Classical.fast_tac HOL_cs)
ORELSE' NAMED ctxt' "arith (theory)" (Arith_Data.arith_tac ctxt')))),
T.by_abstraction (true, true) ctxt [] (fn ctxt' => T.by_tac (
NAMED ctxt' "simp (full)" (Simplifier.simp_tac simpset)
THEN_ALL_NEW (
NAMED ctxt' "fast (full)" (Classical.fast_tac HOL_cs)
ORELSE' NAMED ctxt' "arith (full)" (Arith_Data.arith_tac ctxt'))))])
end
(** proof reconstruction **)
(* tracing and checking *)
local
fun count_rules ptab =
let
fun count (_, Unproved _) (solved, total) = (solved, total + 1)
| count (_, Proved _) (solved, total) = (solved + 1, total + 1)
in Inttab.fold count ptab (0, 0) end
fun header idx r (solved, total) =
"Z3: #" ^ string_of_int idx ^ ": " ^ P.string_of_rule r ^ " (goal " ^
string_of_int (solved + 1) ^ " of " ^ string_of_int total ^ ")"
fun check ctxt idx r ps ct p =
let val thm = thm_of p |> tap (Thm.join_proofs o single)
in
if (Thm.cprop_of thm) aconvc ct then ()
else z3_exn (Pretty.string_of (Pretty.big_list ("proof step failed: " ^
quote (P.string_of_rule r) ^ " (#" ^ string_of_int idx ^ ")")
(pretty_goal ctxt (map (thm_of o fst) ps) (Thm.prop_of thm) @
[Pretty.block [Pretty.str "expected: ",
Syntax.pretty_term ctxt (Thm.term_of ct)]])))
end
in
fun trace_rule idx prove r ps ct (cxp as (ctxt, ptab)) =
let
val _ = SMT_Config.trace_msg ctxt (header idx r o count_rules) ptab
val result as (p, (ctxt', _)) = prove r ps ct cxp
val _ = if not (Config.get ctxt' SMT_Config.trace) then ()
else check ctxt' idx r ps ct p
in result end
end
(* overall reconstruction procedure *)
local
fun not_supported r = raise Fail ("Z3: proof rule not implemented: " ^
quote (P.string_of_rule r))
fun step assms simpset vars r ps ct (cxp as (cx, ptab)) =
(case (r, ps) of
(* core rules *)
(P.TrueAxiom, _) => (Thm L.true_thm, cxp)
| (P.Asserted, _) => (asserted cx assms ct, cxp)
| (P.Goal, _) => (asserted cx assms ct, cxp)
| (P.ModusPonens, [(p, _), (q, _)]) => (mp q (thm_of p), cxp)
| (P.ModusPonensOeq, [(p, _), (q, _)]) => (mp q (thm_of p), cxp)
| (P.AndElim, [(p, i)]) => and_elim (p, i) ct ptab ||> pair cx
| (P.NotOrElim, [(p, i)]) => not_or_elim (p, i) ct ptab ||> pair cx
| (P.Hypothesis, _) => (Thm (Thm.assume ct), cxp)
| (P.Lemma, [(p, _)]) => (lemma (thm_of p) ct, cxp)
| (P.UnitResolution, (p, _) :: ps) =>
(unit_resolution (thm_of p) (map (thm_of o fst) ps) ct, cxp)
| (P.IffTrue, [(p, _)]) => (iff_true (thm_of p), cxp)
| (P.IffFalse, [(p, _)]) => (iff_false (thm_of p), cxp)
| (P.Distributivity, _) => (distributivity cx ct, cxp)
| (P.DefAxiom, _) => (def_axiom cx ct, cxp)
| (P.IntroDef, _) => intro_def ct cx ||> rpair ptab
| (P.ApplyDef, [(p, _)]) => (apply_def (thm_of p), cxp)
| (P.IffOeq, [(p, _)]) => (p, cxp)
| (P.NnfPos, _) => (nnf cx vars (map fst ps) ct, cxp)
| (P.NnfNeg, _) => (nnf cx vars (map fst ps) ct, cxp)
(* equality rules *)
| (P.Reflexivity, _) => (refl ct, cxp)
| (P.Symmetry, [(p, _)]) => (symm p, cxp)
| (P.Transitivity, [(p, _), (q, _)]) => (trans p q, cxp)
| (P.Monotonicity, _) => (monotonicity (map fst ps) ct, cxp)
| (P.Commutativity, _) => (commutativity ct, cxp)
(* quantifier rules *)
| (P.QuantIntro, [(p, _)]) => (quant_intro vars p ct, cxp)
| (P.PullQuant, _) => (pull_quant cx ct, cxp)
| (P.PushQuant, _) => (push_quant cx ct, cxp)
| (P.ElimUnusedVars, _) => (elim_unused_vars cx ct, cxp)
| (P.DestEqRes, _) => (dest_eq_res cx ct, cxp)
| (P.QuantInst, _) => (quant_inst ct, cxp)
| (P.Skolemize, _) => skolemize ct cx ||> rpair ptab
(* theory rules *)
| (P.ThLemma _, _) => (* FIXME: use arguments *)
(th_lemma cx simpset (map (thm_of o fst) ps) ct, cxp)
| (P.Rewrite, _) => (rewrite cx simpset [] ct, cxp)
| (P.RewriteStar, ps) =>
(rewrite cx simpset (map fst ps) ct, cxp)
| (P.NnfStar, _) => not_supported r
| (P.CnfStar, _) => not_supported r
| (P.TransitivityStar, _) => not_supported r
| (P.PullQuantStar, _) => not_supported r
| _ => raise Fail ("Z3: proof rule " ^ quote (P.string_of_rule r) ^
" has an unexpected number of arguments."))
fun prove ctxt assms vars =
let
val simpset = T.make_simpset ctxt (Z3_Simps.get ctxt)
fun conclude idx rule prop (ps, cxp) =
trace_rule idx (step assms simpset vars) rule ps prop cxp
|-> (fn p => apsnd (Inttab.update (idx, Proved p)) #> pair p)
fun lookup idx (cxp as (_, ptab)) =
(case Inttab.lookup ptab idx of
SOME (Unproved (P.Proof_Step {rule, prems, prop})) =>
fold_map lookup prems cxp
|>> map2 rpair prems
|> conclude idx rule prop
| SOME (Proved p) => (p, cxp)
| NONE => z3_exn ("unknown proof id: " ^ quote (string_of_int idx)))
fun result (p, (cx, _)) = (thm_of p, cx)
in
(fn idx => result o lookup idx o pair ctxt o Inttab.map (K Unproved))
end
fun filter_assms ctxt assms ptab =
let
fun step r ct =
(case r of
P.Asserted => insert (op =) (find_assm ctxt assms ct)
| P.Goal => insert (op =) (find_assm ctxt assms ct)
| _ => I)
fun lookup idx =
(case Inttab.lookup ptab idx of
SOME (P.Proof_Step {rule, prems, prop}) =>
fold lookup prems #> step rule prop
| NONE => z3_exn ("unknown proof id: " ^ quote (string_of_int idx)))
in lookup end
in
fun reconstruct ctxt {typs, terms, unfolds, assms} output =
let
val (idx, (ptab, vars, cx)) = P.parse ctxt typs terms output
val assms' = prepare_assms cx unfolds assms
in
if Config.get cx SMT_Config.filter_only_facts
then ((filter_assms cx assms' ptab idx [], @{thm TrueI}), cx)
else apfst (pair []) (prove cx assms' vars idx ptab)
end
end
val setup = z3_rules_setup #> Z3_Simps.setup
end