(* Title: HOL/Tools/Sledgehammer/sledgehammer_reconstruct.ML
Author: Jasmin Blanchette, TU Muenchen
Author: Steffen Juilf Smolka, TU Muenchen
Isar proof reconstruction from ATP proofs.
*)
signature SLEDGEHAMMER_RECONSTRUCT =
sig
type ('a, 'b) atp_step = ('a, 'b) ATP_Proof.atp_step
type 'a atp_proof = 'a ATP_Proof.atp_proof
type stature = ATP_Problem_Generate.stature
type one_line_params = Sledgehammer_Reconstructor.one_line_params
type isar_params =
bool * bool * Time.time option * real * bool * (string * stature) list vector
* (unit -> (term, string) atp_step list) * thm
val lam_trans_of_atp_proof : string atp_proof -> string -> string
val is_typed_helper_used_in_atp_proof : string atp_proof -> bool
val used_facts_in_atp_proof :
Proof.context -> (string * stature) list vector -> string atp_proof ->
(string * stature) list
val used_facts_in_unsound_atp_proof :
Proof.context -> (string * stature) list vector -> 'a atp_proof ->
string list option
val isar_proof_text :
Proof.context -> bool option -> isar_params -> one_line_params -> string
val proof_text :
Proof.context -> bool option -> isar_params -> int -> one_line_params
-> string
end;
structure Sledgehammer_Reconstruct : SLEDGEHAMMER_RECONSTRUCT =
struct
open ATP_Util
open ATP_Problem
open ATP_Proof
open ATP_Problem_Generate
open ATP_Proof_Reconstruct
open Sledgehammer_Util
open Sledgehammer_Reconstructor
open Sledgehammer_Proof
open Sledgehammer_Annotate
open Sledgehammer_Print
open Sledgehammer_Preplay
open Sledgehammer_Compress
open Sledgehammer_Try0
open Sledgehammer_Minimize_Isar
structure String_Redirect = ATP_Proof_Redirect(
type key = atp_step_name
val ord = fn ((s, _ : string list), (s', _)) => fast_string_ord (s, s')
val string_of = fst)
open String_Redirect
(** fact extraction from ATP proofs **)
fun find_first_in_list_vector vec key =
Vector.foldl (fn (ps, NONE) => AList.lookup (op =) ps key
| (_, value) => value) NONE vec
val unprefix_fact_number = space_implode "_" o tl o space_explode "_"
fun resolve_one_named_fact fact_names s =
case try (unprefix fact_prefix) s of
SOME s' =>
let val s' = s' |> unprefix_fact_number |> unascii_of in
s' |> find_first_in_list_vector fact_names |> Option.map (pair s')
end
| NONE => NONE
fun resolve_fact fact_names = map_filter (resolve_one_named_fact fact_names)
fun is_fact fact_names = not o null o resolve_fact fact_names
fun resolve_one_named_conjecture s =
case try (unprefix conjecture_prefix) s of
SOME s' => Int.fromString s'
| NONE => NONE
val resolve_conjecture = map_filter resolve_one_named_conjecture
val is_conjecture = not o null o resolve_conjecture
val ascii_of_lam_fact_prefix = ascii_of lam_fact_prefix
(* overapproximation (good enough) *)
fun is_lam_lifted s =
String.isPrefix fact_prefix s andalso
String.isSubstring ascii_of_lam_fact_prefix s
val is_combinator_def = String.isPrefix (helper_prefix ^ combinator_prefix)
fun is_axiom_used_in_proof pred =
exists (fn ((_, ss), _, _, _, []) => exists pred ss | _ => false)
fun lam_trans_of_atp_proof atp_proof default =
case (is_axiom_used_in_proof is_combinator_def atp_proof,
is_axiom_used_in_proof is_lam_lifted atp_proof) of
(false, false) => default
| (false, true) => liftingN
(* | (true, true) => combs_and_liftingN -- not supported by "metis" *)
| (true, _) => combsN
val is_typed_helper_name =
String.isPrefix helper_prefix andf String.isSuffix typed_helper_suffix
fun is_typed_helper_used_in_atp_proof atp_proof =
is_axiom_used_in_proof is_typed_helper_name atp_proof
fun add_non_rec_defs fact_names accum =
Vector.foldl (fn (facts, facts') =>
union (op =) (filter (fn (_, (_, status)) => status = Non_Rec_Def) facts)
facts')
accum fact_names
val isa_ext = Thm.get_name_hint @{thm ext}
val isa_short_ext = Long_Name.base_name isa_ext
fun ext_name ctxt =
if Thm.eq_thm_prop (@{thm ext},
singleton (Attrib.eval_thms ctxt) (Facts.named isa_short_ext, [])) then
isa_short_ext
else
isa_ext
val leo2_extcnf_equal_neg_rule = "extcnf_equal_neg"
val leo2_unfold_def_rule = "unfold_def"
fun add_fact ctxt fact_names ((_, ss), _, _, rule, deps) =
(if rule = leo2_extcnf_equal_neg_rule then
insert (op =) (ext_name ctxt, (Global, General))
else if rule = leo2_unfold_def_rule then
(* LEO 1.3.3 does not record definitions properly, leading to missing
dependencies in the TSTP proof. Remove the next line once this is
fixed. *)
add_non_rec_defs fact_names
else if rule = agsyhol_coreN orelse rule = satallax_coreN then
(fn [] =>
(* agsyHOL and Satallax don't include definitions in their
unsatisfiable cores, so we assume the worst and include them all
here. *)
[(ext_name ctxt, (Global, General))] |> add_non_rec_defs fact_names
| facts => facts)
else
I)
#> (if null deps then union (op =) (resolve_fact fact_names ss) else I)
fun used_facts_in_atp_proof ctxt fact_names atp_proof =
if null atp_proof then Vector.foldl (uncurry (union (op =))) [] fact_names
else fold (add_fact ctxt fact_names) atp_proof []
fun used_facts_in_unsound_atp_proof _ _ [] = NONE
| used_facts_in_unsound_atp_proof ctxt fact_names atp_proof =
let val used_facts = used_facts_in_atp_proof ctxt fact_names atp_proof in
if forall (fn (_, (sc, _)) => sc = Global) used_facts andalso
not (is_axiom_used_in_proof (is_conjecture o single) atp_proof) then
SOME (map fst used_facts)
else
NONE
end
(** Isar proof construction and manipulation **)
val assume_prefix = "a"
val have_prefix = "f"
val raw_prefix = "x"
fun raw_label_of_name (num, ss) =
case resolve_conjecture ss of
[j] => (conjecture_prefix, j)
| _ => (raw_prefix ^ ascii_of num, 0)
fun label_of_clause [name] = raw_label_of_name name
| label_of_clause c =
(space_implode "___" (map (fst o raw_label_of_name) c), 0)
fun add_fact_of_dependencies fact_names (names as [(_, ss)]) =
if is_fact fact_names ss then
apsnd (union (op =) (map fst (resolve_fact fact_names ss)))
else
apfst (insert (op =) (label_of_clause names))
| add_fact_of_dependencies _ names =
apfst (insert (op =) (label_of_clause names))
fun replace_one_dependency (old, new) dep =
if is_same_atp_step dep old then new else [dep]
fun replace_dependencies_in_line p (name, role, t, rule, deps) =
(name, role, t, rule, fold (union (op =) o replace_one_dependency p) deps [])
(* No "real" literals means only type information (tfree_tcs, clsrel, or
clsarity). *)
fun is_only_type_information t = t aconv @{term True}
fun s_maybe_not role = role <> Conjecture ? s_not
fun is_same_inference (role, t) (_, role', t', _, _) =
s_maybe_not role t aconv s_maybe_not role' t'
(* Discard facts; consolidate adjacent lines that prove the same formula, since
they differ only in type information.*)
fun add_line fact_names (name as (_, ss), role, t, rule, []) lines =
(* No dependencies: fact, conjecture, or (for Vampire) internal facts or
definitions. *)
if is_conjecture ss then
(name, role, t, rule, []) :: lines
else if is_fact fact_names ss then
(* Facts are not proof lines. *)
if is_only_type_information t then
map (replace_dependencies_in_line (name, [])) lines
else
lines
else
map (replace_dependencies_in_line (name, [])) lines
| add_line _ (line as (name, role, t, _, _)) lines =
(* Type information will be deleted later; skip repetition test. *)
if is_only_type_information t then
line :: lines
(* Is there a repetition? If so, replace later line by earlier one. *)
else case take_prefix (not o is_same_inference (role, t)) lines of
(_, []) => line :: lines
| (pre, (name', _, _, _, _) :: post) =>
line :: pre @ map (replace_dependencies_in_line (name', [name])) post
(* Recursively delete empty lines (type information) from the proof. *)
fun add_nontrivial_line (line as (name, _, t, _, [])) lines =
if is_only_type_information t then delete_dependency name lines
else line :: lines
| add_nontrivial_line line lines = line :: lines
and delete_dependency name lines =
fold_rev add_nontrivial_line
(map (replace_dependencies_in_line (name, [])) lines) []
val e_skolemize_rule = "skolemize"
val vampire_skolemisation_rule = "skolemisation"
val is_skolemize_rule =
member (op =) [e_skolemize_rule, vampire_skolemisation_rule]
fun add_desired_line fact_names (name as (_, ss), role, t, rule, deps)
(j, lines) =
(j + 1,
if is_fact fact_names ss orelse
is_conjecture ss orelse
is_skolemize_rule rule orelse
(* the last line must be kept *)
j = 0 orelse
(not (is_only_type_information t) andalso
null (Term.add_tvars t []) andalso
length deps >= 2 andalso
(* kill next to last line, which usually results in a trivial step *)
j <> 1) then
(name, role, t, rule, deps) :: lines (* keep line *)
else
map (replace_dependencies_in_line (name, deps)) lines) (* drop line *)
val add_labels_of_proof =
steps_of_proof #> fold_isar_steps
(byline_of_step #> (fn SOME (By ((ls, _), _)) => union (op =) ls | _ => I))
fun kill_useless_labels_in_proof proof =
let
val used_ls = add_labels_of_proof proof []
fun do_label l = if member (op =) used_ls l then l else no_label
fun do_assms (Assume assms) = Assume (map (apfst do_label) assms)
fun do_step (Prove (qs, xs, l, t, subproofs, by)) =
Prove (qs, xs, do_label l, t, map do_proof subproofs, by)
| do_step step = step
and do_proof (Proof (fix, assms, steps)) =
Proof (fix, do_assms assms, map do_step steps)
in do_proof proof end
fun prefix_of_depth n = replicate_string (n + 1)
val relabel_proof =
let
fun fresh_label depth prefix (old as (l, subst, next)) =
if l = no_label then
old
else
let val l' = (prefix_of_depth depth prefix, next) in
(l', (l, l') :: subst, next + 1)
end
fun do_facts subst =
apfst (maps (the_list o AList.lookup (op =) subst))
fun do_assm depth (l, t) (subst, next) =
let
val (l, subst, next) =
(l, subst, next) |> fresh_label depth assume_prefix
in
((l, t), (subst, next))
end
fun do_assms subst depth (Assume assms) =
fold_map (do_assm depth) assms (subst, 1)
|> apfst Assume
|> apsnd fst
fun do_steps _ _ _ [] = []
| do_steps subst depth next (Prove (qs, xs, l, t, sub, by) :: steps) =
let
val (l, subst, next) =
(l, subst, next) |> fresh_label depth have_prefix
val sub = do_proofs subst depth sub
val by = by |> do_byline subst
in Prove (qs, xs, l, t, sub, by) :: do_steps subst depth next steps end
| do_steps subst depth next (step :: steps) =
step :: do_steps subst depth next steps
and do_proof subst depth (Proof (fix, assms, steps)) =
let val (assms, subst) = do_assms subst depth assms in
Proof (fix, assms, do_steps subst depth 1 steps)
end
and do_byline subst byline =
map_facts_of_byline (do_facts subst) byline
and do_proofs subst depth = map (do_proof subst (depth + 1))
in do_proof [] 0 end
val chain_direct_proof =
let
fun do_qs_lfs NONE lfs = ([], lfs)
| do_qs_lfs (SOME l0) lfs =
if member (op =) lfs l0 then ([Then], lfs |> remove (op =) l0)
else ([], lfs)
fun chain_step lbl
(Prove (qs, xs, l, t, subproofs, By ((lfs, gfs), method))) =
let val (qs', lfs) = do_qs_lfs lbl lfs in
Prove (qs' @ qs, xs, l, t, chain_proofs subproofs,
By ((lfs, gfs), method))
end
| chain_step _ step = step
and chain_steps _ [] = []
| chain_steps (prev as SOME _) (i :: is) =
chain_step prev i :: chain_steps (label_of_step i) is
| chain_steps _ (i :: is) = i :: chain_steps (label_of_step i) is
and chain_proof (Proof (fix, Assume assms, steps)) =
Proof (fix, Assume assms,
chain_steps (try (List.last #> fst) assms) steps)
and chain_proofs proofs = map (chain_proof) proofs
in chain_proof end
type isar_params =
bool * bool * Time.time option * real * bool * (string * stature) list vector
* (unit -> (term, string) atp_step list) * thm
fun isar_proof_text ctxt isar_proofs
(debug, verbose, preplay_timeout, isar_compress, isar_try0, fact_names, atp_proof, goal)
(one_line_params as (_, _, _, _, subgoal, subgoal_count)) =
let
val (params, hyp_ts, concl_t) = strip_subgoal goal subgoal ctxt
val (_, ctxt) =
params
|> map (fn (s, T) => (Binding.name s, SOME T, NoSyn))
|> (fn fixes =>
ctxt |> Variable.set_body false
|> Proof_Context.add_fixes fixes)
val one_line_proof = one_line_proof_text 0 one_line_params
val atp_proof = atp_proof ()
val type_enc =
if is_typed_helper_used_in_atp_proof atp_proof then full_typesN
else partial_typesN
val lam_trans = lam_trans_of_atp_proof atp_proof metis_default_lam_trans
val do_preplay = preplay_timeout <> SOME Time.zeroTime
fun isar_proof_of () =
let
val atp_proof =
atp_proof
|> rpair [] |-> fold_rev (add_line fact_names)
|> rpair [] |-> fold_rev add_nontrivial_line
|> rpair (0, [])
|-> fold_rev (add_desired_line fact_names)
|> snd
val conj_name = conjecture_prefix ^ string_of_int (length hyp_ts)
val conjs =
atp_proof |> map_filter
(fn (name as (_, ss), _, _, _, []) =>
if member (op =) ss conj_name then SOME name else NONE
| _ => NONE)
val assms =
atp_proof |> map_filter
(fn (name as (_, ss), _, _, _, []) =>
(case resolve_conjecture ss of
[j] =>
if j = length hyp_ts then NONE
else SOME (raw_label_of_name name, nth hyp_ts j)
| _ => NONE)
| _ => NONE)
val bot = atp_proof |> List.last |> #1
val refute_graph =
atp_proof
|> map (fn (name, _, _, _, from) => (from, name))
|> make_refute_graph bot
|> fold (Atom_Graph.default_node o rpair ()) conjs
val axioms = axioms_of_refute_graph refute_graph conjs
val tainted = tainted_atoms_of_refute_graph refute_graph conjs
val is_clause_tainted = exists (member (op =) tainted)
val steps =
Symtab.empty
|> fold (fn (name as (s, _), role, t, rule, _) =>
Symtab.update_new (s, (rule,
t |> (if is_clause_tainted [name] then
s_maybe_not role
#> fold exists_of (map Var (Term.add_vars t []))
else
I))))
atp_proof
fun is_clause_skolemize_rule [(s, _)] =
Option.map (is_skolemize_rule o fst) (Symtab.lookup steps s) =
SOME true
| is_clause_skolemize_rule _ = false
(* The assumptions and conjecture are "prop"s; the other formulas are
"bool"s. *)
fun prop_of_clause [(s, ss)] =
(case resolve_conjecture ss of
[j] => if j = length hyp_ts then concl_t else nth hyp_ts j
| _ => the_default ("", @{term False}) (Symtab.lookup steps s)
|> snd |> HOLogic.mk_Trueprop |> close_form)
| prop_of_clause names =
let
val lits = map snd (map_filter (Symtab.lookup steps o fst) names)
in
case List.partition (can HOLogic.dest_not) lits of
(negs as _ :: _, pos as _ :: _) =>
s_imp (Library.foldr1 s_conj (map HOLogic.dest_not negs),
Library.foldr1 s_disj pos)
| _ => fold (curry s_disj) lits @{term False}
end
|> HOLogic.mk_Trueprop |> close_form
fun isar_proof_of_direct_proof infs =
let
fun maybe_show outer c =
(outer andalso length c = 1 andalso subset (op =) (c, conjs))
? cons Show
val is_fixed = Variable.is_declared ctxt orf can Name.dest_skolem
fun skolems_of t =
Term.add_frees t [] |> filter_out (is_fixed o fst) |> rev
fun do_steps outer predecessor accum [] =
accum
|> (if tainted = [] then
cons (Prove (if outer then [Show] else [],
Fix [], no_label, concl_t, [],
By_Metis ([the predecessor], [])))
else
I)
|> rev
| do_steps outer _ accum (Have (gamma, c) :: infs) =
let
val l = label_of_clause c
val t = prop_of_clause c
val by =
By_Metis
(fold (add_fact_of_dependencies fact_names) gamma no_facts)
fun prove sub by =
Prove (maybe_show outer c [], Fix [], l, t, sub, by)
fun do_rest l step =
do_steps outer (SOME l) (step :: accum) infs
in
if is_clause_tainted c then
case gamma of
[g] =>
if is_clause_skolemize_rule g andalso
is_clause_tainted g then
let
val subproof =
Proof (Fix (skolems_of (prop_of_clause g)),
Assume [], rev accum)
in
do_steps outer (SOME l)
[prove [subproof] (By_Metis no_facts)] []
end
else
do_rest l (prove [] by)
| _ => do_rest l (prove [] by)
else
if is_clause_skolemize_rule c then
do_rest l (Prove ([], Fix (skolems_of t), l, t, [], by))
else
do_rest l (prove [] by)
end
| do_steps outer predecessor accum (Cases cases :: infs) =
let
fun do_case (c, infs) =
do_proof false [] [(label_of_clause c, prop_of_clause c)]
infs
val c = succedent_of_cases cases
val l = label_of_clause c
val t = prop_of_clause c
val step =
Prove (maybe_show outer c [], Fix [], l, t,
map do_case cases, By_Metis (the_list predecessor, []))
in
do_steps outer (SOME l) (step :: accum) infs
end
and do_proof outer fix assms infs =
Proof (Fix fix, Assume assms, do_steps outer NONE [] infs)
in
do_proof true params assms infs
end
(* 60 seconds seems like a good interpreation of "no timeout" *)
val preplay_timeout = preplay_timeout |> the_default (seconds 60.0)
val clean_up_labels_in_proof =
chain_direct_proof
#> kill_useless_labels_in_proof
#> relabel_proof
val (preplay_interface as {overall_preplay_stats, ...}, isar_proof) =
refute_graph
|> redirect_graph axioms tainted bot
|> isar_proof_of_direct_proof
|> relabel_proof_canonically
|> `(proof_preplay_interface debug ctxt type_enc lam_trans do_preplay
preplay_timeout)
val ((preplay_time, preplay_fail), isar_proof) =
isar_proof
|> compress_proof
(if isar_proofs = SOME true then isar_compress else 1000.0)
preplay_interface
|> isar_try0 ? try0 preplay_timeout preplay_interface
|> minimize_dependencies_and_remove_unrefed_steps isar_try0
preplay_interface
|> `overall_preplay_stats
||> clean_up_labels_in_proof
val isar_text = string_of_proof ctxt type_enc lam_trans subgoal
subgoal_count isar_proof
in
case isar_text of
"" =>
if isar_proofs = SOME true then
"\nNo structured proof available (proof too simple)."
else
""
| _ =>
let
val msg =
(if verbose then
let
val num_steps = add_proof_steps (steps_of_proof isar_proof) 0
in [string_of_int num_steps ^ " step" ^ plural_s num_steps] end
else
[]) @
(if do_preplay then
[(if preplay_fail then "may fail, " else "") ^
string_of_preplay_time preplay_time]
else
[])
in
"\n\nStructured proof"
^ (commas msg |> not (null msg) ? enclose " (" ")")
^ ":\n" ^
Active.sendback_markup [Markup.padding_command] isar_text
end
end
val isar_proof =
if debug then
isar_proof_of ()
else case try isar_proof_of () of
SOME s => s
| NONE => if isar_proofs = SOME true then
"\nWarning: The Isar proof construction failed."
else
""
in one_line_proof ^ isar_proof end
fun isar_proof_would_be_a_good_idea preplay =
case preplay of
Played (reconstr, _) => reconstr = SMT
| Trust_Playable _ => false
| Failed_to_Play _ => true
fun proof_text ctxt isar_proofs isar_params num_chained
(one_line_params as (preplay, _, _, _, _, _)) =
(if isar_proofs = SOME true orelse
(isar_proofs = NONE andalso isar_proof_would_be_a_good_idea preplay) then
isar_proof_text ctxt isar_proofs isar_params
else
one_line_proof_text num_chained) one_line_params
end;