(* 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 * string * string * Time.time option * real * bool * (term, string) atp_step list *
thm
val isar_proof_text :
Proof.context -> bool option -> isar_params -> one_line_params -> string
val proof_text :
Proof.context -> bool option -> (unit -> 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
fun maybe_mk_Trueprop t = t |> fastype_of t = HOLogic.boolT ? HOLogic.mk_Trueprop
val maybe_dest_Trueprop = perhaps (try HOLogic.dest_Trueprop)
val e_skolemize_rule = "skolemize"
val vampire_skolemisation_rule = "skolemisation"
(* TODO: Use "Z3_Proof.string_of_rule" once it is moved to Isabelle *)
val z3_apply_def_rule = "apply-def"
val z3_hypothesis_rule = "hypothesis"
val z3_intro_def_rule = "intro-def"
val z3_lemma_rule = "lemma"
val z3_skolemize_rule = "sk"
val z3_th_lemma_rule = "th-lemma"
val is_skolemize_rule =
member (op =) [e_skolemize_rule, vampire_skolemisation_rule, z3_skolemize_rule]
fun raw_label_of_num num = (num, 0)
fun label_of_clause [(num, _)] = raw_label_of_num num
| label_of_clause c = (space_implode "___" (map (fst o raw_label_of_num o fst) c), 0)
fun add_fact_of_dependencies [(_, ss as _ :: _)] = apsnd (union (op =) ss)
| 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 [])
fun inline_z3_defs _ [] = []
| inline_z3_defs defs ((line as (name, role, t, rule, deps)) :: lines) =
if rule = z3_intro_def_rule then
let val def = t |> maybe_dest_Trueprop |> HOLogic.dest_eq |> swap in
inline_z3_defs (insert (op =) def defs)
(map (replace_dependencies_in_line (name, [])) lines)
end
else if rule = z3_apply_def_rule then
inline_z3_defs defs (map (replace_dependencies_in_line (name, [])) lines)
else
(name, role, Term.subst_atomic defs t, rule, deps) :: inline_z3_defs defs lines
fun alist_cons_list eq (k, v) = AList.map_default eq (k, []) (cons v)
(* FIXME: use "prop_of_clause" defined below *)
fun add_z3_hypotheses [] = I
| add_z3_hypotheses hyps =
HOLogic.dest_Trueprop
#> curry HOLogic.mk_imp (Library.foldr1 HOLogic.mk_conj (map HOLogic.dest_Trueprop hyps))
#> HOLogic.mk_Trueprop
fun inline_z3_hypotheses _ _ [] = []
| inline_z3_hypotheses hyp_names hyps ((name, role, t, rule, deps) :: lines) =
if rule = z3_hypothesis_rule then
inline_z3_hypotheses (name :: hyp_names) (alist_cons_list (op =) (t, name) hyps) lines
else
let val deps' = subtract (op =) hyp_names deps in
if rule = z3_lemma_rule then
(name, role, t, rule, deps') :: inline_z3_hypotheses hyp_names hyps lines
else
let
val add_hyps = filter_out (null o inter (op =) deps o snd) hyps
val t' = add_z3_hypotheses (map fst add_hyps) t
val deps' = subtract (op =) hyp_names deps
val hyps' = fold (AList.update (op =) o apsnd (insert (op =) name)) add_hyps hyps
in
(name, role, t', rule, deps') :: inline_z3_hypotheses hyp_names hyps' lines
end
end
(* No "real" literals means only type information (tfree_tcs, clsrel, or
clsarity). *)
fun is_only_type_information t = t aconv @{term True}
(* Discard facts; consolidate adjacent lines that prove the same formula, since
they differ only in type information.*)
fun add_line (line as (name as (_, ss), role, t, rule, [])) lines =
(* No dependencies: lemma (for Z3), fact, conjecture, or (for Vampire)
internal facts or definitions. *)
if role = Lemma orelse role = Conjecture orelse role = Negated_Conjecture orelse
role = Hypothesis then
line :: lines
else if role = Axiom then
(* Facts are not proof lines. *)
lines |> is_only_type_information t ? map (replace_dependencies_in_line (name, []))
else
map (replace_dependencies_in_line (name, [])) lines
| add_line line lines = line :: lines
(* 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) []
fun add_desired_lines res [] = rev res
| add_desired_lines res ((name as (_, ss), role, t, rule, deps) :: lines) =
if role <> Plain orelse is_skolemize_rule rule orelse
(* the last line must be kept *)
null lines orelse
(not (is_only_type_information t) andalso null (Term.add_tvars t [])
andalso length deps >= 2 andalso
(* don't keep next to last line, which usually results in a trivial step *)
not (can the_single lines)) then
add_desired_lines ((name, role, t, rule, deps) :: res) lines
else
add_desired_lines res (map (replace_dependencies_in_line (name, deps)) lines)
val add_labels_of_proof =
steps_of_proof
#> fold_isar_steps (byline_of_step #> (fn SOME ((ls, _), _) => union (op =) ls | _ => I))
fun kill_useless_labels_in_proof proof =
let
val used_ls = add_labels_of_proof proof []
fun kill_label l = if member (op =) used_ls l then l else no_label
fun kill_assms assms = map (apfst kill_label) assms
fun kill_step (Prove (qs, xs, l, t, subproofs, by)) =
Prove (qs, xs, kill_label l, t, map kill_proof subproofs, by)
| kill_step step = step
and kill_proof (Proof (fix, assms, steps)) =
Proof (fix, kill_assms assms, map kill_step steps)
in
kill_proof proof
end
fun prefix_of_depth n = replicate_string (n + 1)
val assume_prefix = "a"
val have_prefix = "f"
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 relabel_facts subst = apfst (maps (the_list o AList.lookup (op =) subst))
fun relabel_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 relabel_assms subst depth assms = fold_map (relabel_assm depth) assms (subst, 1) ||> fst
fun relabel_steps _ _ _ [] = []
| relabel_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 = relabel_proofs subst depth sub
val by = by |> relabel_byline subst
in
Prove (qs, xs, l, t, sub, by) :: relabel_steps subst depth next steps
end
| relabel_steps subst depth next (step :: steps) =
step :: relabel_steps subst depth next steps
and relabel_proof subst depth (Proof (fix, assms, steps)) =
let val (assms, subst) = relabel_assms subst depth assms in
Proof (fix, assms, relabel_steps subst depth 1 steps)
end
and relabel_byline subst byline = apfst (relabel_facts subst) byline
and relabel_proofs subst depth = map (relabel_proof subst (depth + 1))
in
relabel_proof [] 0
end
val chain_direct_proof =
let
fun chain_qs_lfs NONE lfs = ([], lfs)
| chain_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, ((lfs, gfs), method))) =
let val (qs', lfs) = chain_qs_lfs lbl lfs in
Prove (qs' @ qs, xs, l, t, chain_proofs subproofs, ((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, assms, steps)) =
Proof (fix, 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 * string * string * Time.time option * real * bool * (term, string) atp_step list *
thm
fun isar_proof_text ctxt isar_proofs
(debug, verbose, metis_type_enc, metis_lam_trans, preplay_timeout, isar_compress,
isar_try0, 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 do_preplay = preplay_timeout <> SOME Time.zeroTime
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 get_role keep_role ((num, _), role, t, rule, _) =
if keep_role role then SOME ((raw_label_of_num num, t), rule) else NONE
fun isar_proof_of () =
let
val atp_proof =
atp_proof
|> inline_z3_defs []
|> inline_z3_hypotheses [] []
|> rpair [] |-> fold_rev add_line
|> rpair [] |-> fold_rev add_nontrivial_line
|> add_desired_lines []
val conjs =
map_filter (fn (name, role, _, _, _) =>
if member (op =) [Conjecture, Negated_Conjecture] role then SOME name else NONE)
atp_proof
val assms = map_filter (Option.map fst o get_role (curry (op =) Hypothesis)) atp_proof
val lems =
map_filter (get_role (curry (op =) Lemma)) atp_proof
|> map (fn ((l, t), rule) =>
let
val (skos, meth) =
if is_skolemize_rule rule then (skolems_of t, MetisM)
else if rule = z3_th_lemma_rule then ([], ArithM)
else ([], SimpM)
in
Prove ([], skos, l, maybe_mk_Trueprop t, [], (([], []), meth))
end)
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
role <> Conjecture ? (maybe_dest_Trueprop #> s_not)
#> 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 (for ATPs) or
"prop"s (for Z3). TODO: Always use "prop". *)
fun prop_of_clause [(num, _)] =
Symtab.lookup steps num |> the |> snd |> maybe_mk_Trueprop |> close_form
| prop_of_clause names =
let
val lits =
map (maybe_dest_Trueprop o 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 maybe_show outer c =
(outer andalso length c = 1 andalso subset (op =) (c, conjs)) ? cons Show
fun isar_steps outer predecessor accum [] =
accum
|> (if tainted = [] then
cons (Prove (if outer then [Show] else [], [], no_label, concl_t, [],
(([the predecessor], []), MetisM)))
else
I)
|> rev
| isar_steps outer _ accum (Have (gamma, c) :: infs) =
let
val l = label_of_clause c
val t = prop_of_clause c
val by = (fold add_fact_of_dependencies gamma no_facts, MetisM)
fun prove sub by = Prove (maybe_show outer c [], [], l, t, sub, by)
fun do_rest l step = isar_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 (skolems_of (prop_of_clause g), [], rev accum) in
isar_steps outer (SOME l) [prove [subproof] (no_facts, MetisM)] []
end
else
do_rest l (prove [] by)
| _ => do_rest l (prove [] by))
else
(if is_clause_skolemize_rule c then Prove ([], skolems_of t, l, t, [], by)
else prove [] by)
|> do_rest l
end
| isar_steps outer predecessor accum (Cases cases :: infs) =
let
fun isar_case (c, infs) =
isar_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 [], [], l, t, map isar_case cases,
((the_list predecessor, []), MetisM))
in
isar_steps outer (SOME l) (step :: accum) infs
end
and isar_proof outer fix assms lems infs =
Proof (fix, assms, lems @ isar_steps outer NONE [] infs)
val isar_proof_of_direct_proof = isar_proof true params assms lems
(* 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
(*
|> tap (tracing o prefix "Refute graph: " o string_of_refute_graph)
*)
|> redirect_graph axioms tainted bot
(*
|> tap (tracing o prefix "Direct proof: " o string_of_direct_proof)
*)
|> isar_proof_of_direct_proof
|> postprocess_remove_unreferenced_steps I
|> relabel_proof_canonically
|> `(proof_preplay_interface debug ctxt metis_type_enc metis_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
|> postprocess_remove_unreferenced_steps (isar_try0 ? min_deps_of_step preplay_interface)
|> `overall_preplay_stats
||> clean_up_labels_in_proof
val isar_text =
string_of_proof ctxt metis_type_enc metis_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;