author | wenzelm |
Fri, 04 Jan 2019 23:22:53 +0100 | |
changeset 69593 | 3dda49e08b9d |
parent 69204 | d5ab1636660b |
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permissions | -rw-r--r-- |
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(* Title: HOL/Tools/SMT/smt_replay_methods.ML |
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Author: Sascha Boehme, TU Muenchen |
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Author: Jasmin Blanchette, TU Muenchen |
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Author: Mathias Fleury, MPII |
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Proof methods for replaying SMT proofs. |
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*) |
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signature SMT_REPLAY_METHODS = |
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sig |
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val pretty_goal: Proof.context -> string -> string -> thm list -> term -> Pretty.T |
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val trace_goal: Proof.context -> string -> thm list -> term -> unit |
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val trace: Proof.context -> (unit -> string) -> unit |
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val replay_error: Proof.context -> string -> string -> thm list -> term -> 'a |
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val replay_rule_error: Proof.context -> string -> thm list -> term -> 'a |
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(*theory lemma methods*) |
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type th_lemma_method = Proof.context -> thm list -> term -> thm |
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val add_th_lemma_method: string * th_lemma_method -> Context.generic -> |
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Context.generic |
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val get_th_lemma_method: Proof.context -> th_lemma_method Symtab.table |
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val discharge: int -> thm list -> thm -> thm |
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val match_instantiate: Proof.context -> term -> thm -> thm |
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val prove: Proof.context -> term -> (Proof.context -> int -> tactic) -> thm |
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(*abstraction*) |
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type abs_context = int * term Termtab.table |
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type 'a abstracter = term -> abs_context -> 'a * abs_context |
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val add_arith_abstracter: (term abstracter -> term option abstracter) -> |
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Context.generic -> Context.generic |
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val abstract_lit: term -> abs_context -> term * abs_context |
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val abstract_conj: term -> abs_context -> term * abs_context |
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val abstract_disj: term -> abs_context -> term * abs_context |
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val abstract_not: (term -> abs_context -> term * abs_context) -> |
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term -> abs_context -> term * abs_context |
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val abstract_unit: term -> abs_context -> term * abs_context |
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val abstract_prop: term -> abs_context -> term * abs_context |
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val abstract_term: term -> abs_context -> term * abs_context |
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val abstract_arith: Proof.context -> term -> abs_context -> term * abs_context |
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val prove_abstract: Proof.context -> thm list -> term -> |
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(Proof.context -> thm list -> int -> tactic) -> |
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(abs_context -> (term list * term) * abs_context) -> thm |
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val prove_abstract': Proof.context -> term -> (Proof.context -> thm list -> int -> tactic) -> |
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(abs_context -> term * abs_context) -> thm |
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val try_provers: Proof.context -> string -> (string * (term -> 'a)) list -> thm list -> term -> |
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'a |
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(*shared tactics*) |
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val cong_basic: Proof.context -> thm list -> term -> thm |
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val cong_full: Proof.context -> thm list -> term -> thm |
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val cong_unfolding_first: Proof.context -> thm list -> term -> thm |
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val certify_prop: Proof.context -> term -> cterm |
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end; |
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structure SMT_Replay_Methods: SMT_REPLAY_METHODS = |
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struct |
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(* utility functions *) |
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fun trace ctxt f = SMT_Config.trace_msg ctxt f () |
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fun pretty_thm ctxt thm = Syntax.pretty_term ctxt (Thm.concl_of thm) |
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fun pretty_goal ctxt msg rule thms t = |
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let |
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val full_msg = msg ^ ": " ^ quote rule |
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val assms = |
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if null thms then [] |
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else [Pretty.big_list "assumptions:" (map (pretty_thm ctxt) thms)] |
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val concl = Pretty.big_list "proposition:" [Syntax.pretty_term ctxt t] |
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in Pretty.big_list full_msg (assms @ [concl]) end |
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fun replay_error ctxt msg rule thms t = error (Pretty.string_of (pretty_goal ctxt msg rule thms t)) |
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fun replay_rule_error ctxt = replay_error ctxt "Failed to replay Z3 proof step" |
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fun trace_goal ctxt rule thms t = |
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trace ctxt (fn () => Pretty.string_of (pretty_goal ctxt "Goal" rule thms t)) |
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fun as_prop (t as Const (\<^const_name>\<open>Trueprop\<close>, _) $ _) = t |
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| as_prop t = HOLogic.mk_Trueprop t |
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fun dest_prop (Const (\<^const_name>\<open>Trueprop\<close>, _) $ t) = t |
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| dest_prop t = t |
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fun dest_thm thm = dest_prop (Thm.concl_of thm) |
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(* plug-ins *) |
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type abs_context = int * term Termtab.table |
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type 'a abstracter = term -> abs_context -> 'a * abs_context |
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type th_lemma_method = Proof.context -> thm list -> term -> thm |
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fun id_ord ((id1, _), (id2, _)) = int_ord (id1, id2) |
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structure Plugins = Generic_Data |
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( |
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type T = |
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(int * (term abstracter -> term option abstracter)) list * |
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th_lemma_method Symtab.table |
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val empty = ([], Symtab.empty) |
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val extend = I |
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fun merge ((abss1, ths1), (abss2, ths2)) = ( |
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Ord_List.merge id_ord (abss1, abss2), |
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Symtab.merge (K true) (ths1, ths2)) |
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) |
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fun add_arith_abstracter abs = Plugins.map (apfst (Ord_List.insert id_ord (serial (), abs))) |
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fun get_arith_abstracters ctxt = map snd (fst (Plugins.get (Context.Proof ctxt))) |
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|
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fun add_th_lemma_method method = Plugins.map (apsnd (Symtab.update_new method)) |
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fun get_th_lemma_method ctxt = snd (Plugins.get (Context.Proof ctxt)) |
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|
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fun match ctxt pat t = |
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(Vartab.empty, Vartab.empty) |
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|> Pattern.first_order_match (Proof_Context.theory_of ctxt) (pat, t) |
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|
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fun gen_certify_inst sel cert ctxt thm t = |
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let |
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val inst = match ctxt (dest_thm thm) (dest_prop t) |
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fun cert_inst (ix, (a, b)) = ((ix, a), cert b) |
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in Vartab.fold (cons o cert_inst) (sel inst) [] end |
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|
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fun match_instantiateT ctxt t thm = |
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if Term.exists_type (Term.exists_subtype Term.is_TVar) (dest_thm thm) then |
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Thm.instantiate (gen_certify_inst fst (Thm.ctyp_of ctxt) ctxt thm t, []) thm |
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else thm |
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|
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fun match_instantiate ctxt t thm = |
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let val thm' = match_instantiateT ctxt t thm in |
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Thm.instantiate ([], gen_certify_inst snd (Thm.cterm_of ctxt) ctxt thm' t) thm' |
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end |
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|
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fun discharge _ [] thm = thm |
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| discharge i (rule :: rules) thm = discharge (i + Thm.nprems_of rule) rules (rule RSN (i, thm)) |
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|
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fun by_tac ctxt thms ns ts t tac = |
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Goal.prove ctxt [] (map as_prop ts) (as_prop t) |
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(fn {context, prems} => HEADGOAL (tac context prems)) |
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|> Drule.generalize ([], ns) |
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|> discharge 1 thms |
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|
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fun prove ctxt t tac = by_tac ctxt [] [] [] t (K o tac) |
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|
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|
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(* abstraction *) |
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|
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fun prove_abstract ctxt thms t tac f = |
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let |
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val ((prems, concl), (_, ts)) = f (1, Termtab.empty) |
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val ns = Termtab.fold (fn (_, v) => cons (fst (Term.dest_Free v))) ts [] |
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in |
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by_tac ctxt [] ns prems concl tac |
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|> match_instantiate ctxt t |
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|> discharge 1 thms |
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end |
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|
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fun prove_abstract' ctxt t tac f = |
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prove_abstract ctxt [] t tac (f #>> pair []) |
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168 |
|
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fun lookup_term (_, terms) t = Termtab.lookup terms t |
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|
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fun abstract_sub t f cx = |
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(case lookup_term cx t of |
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SOME v => (v, cx) |
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| NONE => f cx) |
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175 |
|
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fun mk_fresh_free t (i, terms) = |
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let val v = Free ("t" ^ string_of_int i, fastype_of t) |
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in (v, (i + 1, Termtab.update (t, v) terms)) end |
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179 |
|
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fun apply_abstracters _ [] _ cx = (NONE, cx) |
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| apply_abstracters abs (abstracter :: abstracters) t cx = |
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(case abstracter abs t cx of |
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(NONE, _) => apply_abstracters abs abstracters t cx |
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| x as (SOME _, _) => x) |
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185 |
|
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fun abstract_term (t as _ $ _) = abstract_sub t (mk_fresh_free t) |
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| abstract_term (t as Abs _) = abstract_sub t (mk_fresh_free t) |
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| abstract_term t = pair t |
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189 |
|
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fun abstract_bin abs f t t1 t2 = abstract_sub t (abs t1 ##>> abs t2 #>> f) |
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191 |
|
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fun abstract_ter abs f t t1 t2 t3 = |
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abstract_sub t (abs t1 ##>> abs t2 ##>> abs t3 #>> (Scan.triple1 #> f)) |
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194 |
|
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fun abstract_lit (@{const HOL.Not} $ t) = abstract_term t #>> HOLogic.mk_not |
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| abstract_lit t = abstract_term t |
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197 |
|
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fun abstract_not abs (t as @{const HOL.Not} $ t1) = |
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abstract_sub t (abs t1 #>> HOLogic.mk_not) |
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| abstract_not _ t = abstract_lit t |
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201 |
|
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fun abstract_conj (t as @{const HOL.conj} $ t1 $ t2) = |
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abstract_bin abstract_conj HOLogic.mk_conj t t1 t2 |
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| abstract_conj t = abstract_lit t |
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|
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fun abstract_disj (t as @{const HOL.disj} $ t1 $ t2) = |
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abstract_bin abstract_disj HOLogic.mk_disj t t1 t2 |
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| abstract_disj t = abstract_lit t |
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209 |
|
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fun abstract_prop (t as (c as @{const If (bool)}) $ t1 $ t2 $ t3) = |
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abstract_ter abstract_prop (fn (t1, t2, t3) => c $ t1 $ t2 $ t3) t t1 t2 t3 |
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| abstract_prop (t as @{const HOL.disj} $ t1 $ t2) = |
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abstract_bin abstract_prop HOLogic.mk_disj t t1 t2 |
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| abstract_prop (t as @{const HOL.conj} $ t1 $ t2) = |
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abstract_bin abstract_prop HOLogic.mk_conj t t1 t2 |
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| abstract_prop (t as @{const HOL.implies} $ t1 $ t2) = |
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217 |
abstract_bin abstract_prop HOLogic.mk_imp t t1 t2 |
69593 | 218 |
| abstract_prop (t as \<^term>\<open>HOL.eq :: bool => _\<close> $ t1 $ t2) = |
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abstract_bin abstract_prop HOLogic.mk_eq t t1 t2 |
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| abstract_prop t = abstract_not abstract_prop t |
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221 |
|
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222 |
fun abstract_arith ctxt u = |
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223 |
let |
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fun abs (t as (c as Const (\<^const_name>\<open>Hilbert_Choice.Eps\<close>, _) $ Abs (s, T, t'))) = |
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225 |
abstract_sub t (abstract_term t) |
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226 |
| abs (t as (c as Const _) $ Abs (s, T, t')) = |
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abstract_sub t (abs t' #>> (fn u' => c $ Abs (s, T, u'))) |
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| abs (t as (c as Const (\<^const_name>\<open>If\<close>, _)) $ t1 $ t2 $ t3) = |
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229 |
abstract_ter abs (fn (t1, t2, t3) => c $ t1 $ t2 $ t3) t t1 t2 t3 |
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| abs (t as @{const HOL.Not} $ t1) = abstract_sub t (abs t1 #>> HOLogic.mk_not) |
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| abs (t as @{const HOL.disj} $ t1 $ t2) = |
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abstract_sub t (abs t1 ##>> abs t2 #>> HOLogic.mk_disj) |
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| abs (t as (c as Const (\<^const_name>\<open>uminus_class.uminus\<close>, _)) $ t1) = |
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abstract_sub t (abs t1 #>> (fn u => c $ u)) |
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| abs (t as (c as Const (\<^const_name>\<open>plus_class.plus\<close>, _)) $ t1 $ t2) = |
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abstract_sub t (abs t1 ##>> abs t2 #>> (fn (u1, u2) => c $ u1 $ u2)) |
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| abs (t as (c as Const (\<^const_name>\<open>minus_class.minus\<close>, _)) $ t1 $ t2) = |
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abstract_sub t (abs t1 ##>> abs t2 #>> (fn (u1, u2) => c $ u1 $ u2)) |
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| abs (t as (c as Const (\<^const_name>\<open>times_class.times\<close>, _)) $ t1 $ t2) = |
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abstract_sub t (abs t1 ##>> abs t2 #>> (fn (u1, u2) => c $ u1 $ u2)) |
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| abs (t as (c as Const (\<^const_name>\<open>z3div\<close>, _)) $ t1 $ t2) = |
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abstract_sub t (abs t1 ##>> abs t2 #>> (fn (u1, u2) => c $ u1 $ u2)) |
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| abs (t as (c as Const (\<^const_name>\<open>z3mod\<close>, _)) $ t1 $ t2) = |
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abstract_sub t (abs t1 ##>> abs t2 #>> (fn (u1, u2) => c $ u1 $ u2)) |
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| abs (t as (c as Const (\<^const_name>\<open>HOL.eq\<close>, _)) $ t1 $ t2) = |
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abstract_sub t (abs t1 ##>> abs t2 #>> (fn (u1, u2) => c $ u1 $ u2)) |
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| abs (t as (c as Const (\<^const_name>\<open>ord_class.less\<close>, _)) $ t1 $ t2) = |
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abstract_sub t (abs t1 ##>> abs t2 #>> (fn (u1, u2) => c $ u1 $ u2)) |
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| abs (t as (c as Const (\<^const_name>\<open>ord_class.less_eq\<close>, _)) $ t1 $ t2) = |
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abstract_sub t (abs t1 ##>> abs t2 #>> (fn (u1, u2) => c $ u1 $ u2)) |
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| abs t = abstract_sub t (fn cx => |
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if can HOLogic.dest_number t then (t, cx) |
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else |
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(case apply_abstracters abs (get_arith_abstracters ctxt) t cx of |
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(SOME u, cx') => (u, cx') |
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| (NONE, _) => abstract_term t cx)) |
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in abs u end |
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|
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fun abstract_unit (t as (@{const HOL.Not} $ (@{const HOL.disj} $ t1 $ t2))) = |
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abstract_sub t (abstract_unit t1 ##>> abstract_unit t2 #>> |
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HOLogic.mk_not o HOLogic.mk_disj) |
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| abstract_unit (t as (@{const HOL.disj} $ t1 $ t2)) = |
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abstract_sub t (abstract_unit t1 ##>> abstract_unit t2 #>> |
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HOLogic.mk_disj) |
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| abstract_unit (t as (Const(\<^const_name>\<open>HOL.eq\<close>, _) $ t1 $ t2)) = |
266 |
if fastype_of t1 = \<^typ>\<open>bool\<close> then |
|
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abstract_sub t (abstract_unit t1 ##>> abstract_unit t2 #>> |
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HOLogic.mk_eq) |
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else abstract_lit t |
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| abstract_unit (t as (@{const HOL.Not} $ Const(\<^const_name>\<open>HOL.eq\<close>, _) $ t1 $ t2)) = |
271 |
if fastype_of t1 = \<^typ>\<open>bool\<close> then |
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abstract_sub t (abstract_unit t1 ##>> abstract_unit t2 #>> |
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HOLogic.mk_eq #>> HOLogic.mk_not) |
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else abstract_lit t |
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| abstract_unit (t as (@{const HOL.Not} $ t1)) = |
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abstract_sub t (abstract_unit t1 #>> HOLogic.mk_not) |
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| abstract_unit t = abstract_lit t |
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|
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|
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(* theory lemmas *) |
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|
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fun try_provers ctxt rule [] thms t = replay_rule_error ctxt rule thms t |
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| try_provers ctxt rule ((name, prover) :: named_provers) thms t = |
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(case (trace ctxt (K ("Trying prover " ^ quote name)); try prover t) of |
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SOME thm => thm |
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| NONE => try_provers ctxt rule named_provers thms t) |
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287 |
|
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|
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(* congruence *) |
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|
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fun certify_prop ctxt t = Thm.cterm_of ctxt (as_prop t) |
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|
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fun ctac ctxt prems i st = st |> ( |
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resolve_tac ctxt (@{thm refl} :: prems) i |
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ORELSE (cong_tac ctxt i THEN ctac ctxt prems (i + 1) THEN ctac ctxt prems i)) |
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|
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fun cong_basic ctxt thms t = |
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let val st = Thm.trivial (certify_prop ctxt t) |
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in |
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(case Seq.pull (ctac ctxt thms 1 st) of |
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SOME (thm, _) => thm |
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| NONE => raise THM ("cong", 0, thms @ [st])) |
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end |
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304 |
|
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val cong_dest_rules = @{lemma |
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"(\<not> P \<or> Q) \<and> (P \<or> \<not> Q) \<Longrightarrow> P = Q" |
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"(P \<or> \<not> Q) \<and> (\<not> P \<or> Q) \<Longrightarrow> P = Q" |
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308 |
by fast+} |
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309 |
|
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fun cong_full_core_tac ctxt = |
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311 |
eresolve_tac ctxt @{thms subst} |
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THEN' resolve_tac ctxt @{thms refl} |
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ORELSE' Classical.fast_tac ctxt |
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314 |
|
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fun cong_full ctxt thms t = prove ctxt t (fn ctxt' => |
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Method.insert_tac ctxt thms |
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THEN' (cong_full_core_tac ctxt' |
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ORELSE' dresolve_tac ctxt cong_dest_rules |
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THEN' cong_full_core_tac ctxt')) |
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320 |
|
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fun cong_unfolding_first ctxt thms t = |
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let val reorder_for_simp = try (fn thm => |
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323 |
let val t = Thm.prop_of ( @{thm eq_reflection} OF [thm]) |
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val thm = (case Logic.dest_equals t of |
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(t1, t2) => if Term.size_of_term t1 > Term.size_of_term t2 then @{thm eq_reflection} OF [thm] |
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else @{thm eq_reflection} OF [thm OF @{thms sym}]) |
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handle TERM("dest_equals", _) => @{thm eq_reflection} OF [thm] |
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in thm end) |
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329 |
in |
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330 |
prove ctxt t (fn ctxt => |
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331 |
Raw_Simplifier.rewrite_goal_tac ctxt |
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(map_filter reorder_for_simp thms) |
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THEN' Method.insert_tac ctxt thms |
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THEN' K (Clasimp.auto_tac ctxt)) |
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335 |
end |
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|
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end; |