(* 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 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 [])
(* 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 (name as (_, ss), role, t, rule, []) lines =
(* No dependencies: fact, conjecture, or (for Vampire) internal facts or
definitions. *)
if role = Conjecture orelse role = Negated_Conjecture orelse role = Hypothesis then
(name, role, t, rule, []) :: 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 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 (name as (_, ss), role, t, rule, deps) (j, lines) =
(j + 1,
if role <> Plain 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 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 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) |>> Assume ||> 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
fun maybe_mk_Trueprop t = t |> fastype_of t = HOLogic.boolT ? HOLogic.mk_Trueprop
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
fun isar_proof_of () =
let
val atp_proof =
atp_proof
|> rpair [] |-> fold_rev add_line
|> rpair [] |-> fold_rev add_nontrivial_line
|> rpair (0, [])
|-> fold_rev add_desired_line
|> snd
val conjs =
atp_proof |> map_filter (fn (name, role, _, _, _) =>
if role = Conjecture orelse role = Negated_Conjecture then SOME name else NONE)
val assms =
atp_proof
|> map_filter (try (fn ((num, _), Hypothesis, t, _, _) => (raw_label_of_num num, t)))
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 [(num, _)] =
Symtab.lookup steps num |> the |> snd |> maybe_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 (([the predecessor], []), MetisM)))
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 (fold add_fact_of_dependencies gamma no_facts, MetisM)
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 (no_facts, MetisM))] []
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 ((the_list predecessor, []), MetisM))
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
(*
|> tap (tracing o prefix "REFUTE: " o string_of_refute_graph)
*)
|> redirect_graph axioms tainted bot
(*
|> tap (tracing o prefix "DIRECT: " o string_of_direct_proof)
*)
|> isar_proof_of_direct_proof
|> 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
|> 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 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;