(* Title: HOL/Tools/Sledgehammer/sledgehammer_isar.ML
Author: Jasmin Blanchette, TU Muenchen
Author: Steffen Juilf Smolka, TU Muenchen
Isar proof reconstruction from ATP proofs.
*)
signature SLEDGEHAMMER_ISAR =
sig
type atp_step_name = ATP_Proof.atp_step_name
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 * string * string * Time.time * real * bool * (term, string) atp_step list * thm
val proof_text : Proof.context -> bool -> bool option -> (unit -> isar_params) -> int ->
one_line_params -> string
end;
structure Sledgehammer_Isar : SLEDGEHAMMER_ISAR =
struct
open ATP_Util
open ATP_Problem
open ATP_Proof
open ATP_Proof_Reconstruct
open Sledgehammer_Util
open Sledgehammer_Reconstructor
open Sledgehammer_Isar_Proof
open Sledgehammer_Isar_Preplay
open Sledgehammer_Isar_Compress
open Sledgehammer_Isar_Try0
open Sledgehammer_Isar_Minimize
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
val e_skolemize_rules = ["skolemize", "shift_quantors"]
val spass_pirate_datatype_rule = "DT"
val vampire_skolemisation_rule = "skolemisation"
(* TODO: Use "Z3_Proof.string_of_rule" once it is moved to Isabelle *)
val z3_skolemize_rule = "sk"
val z3_th_lemma_rule = "th-lemma"
val skolemize_rules =
e_skolemize_rules @ [spass_skolemize_rule, vampire_skolemisation_rule, z3_skolemize_rule]
val is_skolemize_rule = member (op =) skolemize_rules
val is_arith_rule = String.isPrefix z3_th_lemma_rule
val is_datatype_rule = String.isPrefix spass_pirate_datatype_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))
(* No "real" literals means only type information (tfree_tcs, clsrel, or clsarity). *)
fun is_only_type_information t = t aconv @{prop True}
(* Discard facts; consolidate adjacent lines that prove the same formula, since they differ only in
type information. *)
fun add_line_pass1 (line as (name, 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 orelse is_arith_rule rule 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_pass1 line lines = line :: lines
fun add_lines_pass2 res [] = rev res
| add_lines_pass2 res ((name, role, t, rule, deps) :: lines) =
let
val is_last_line = null lines
fun looks_interesting () =
not (is_only_type_information t) andalso null (Term.add_tvars t [])
andalso length deps >= 2 andalso not (can the_single lines)
fun is_skolemizing_line (_, _, _, rule', deps') =
is_skolemize_rule rule' andalso member (op =) deps' name
fun is_before_skolemize_rule () = exists is_skolemizing_line lines
in
if role <> Plain orelse is_skolemize_rule rule orelse is_arith_rule rule orelse
is_datatype_rule rule orelse is_last_line orelse looks_interesting () orelse
is_before_skolemize_rule () then
add_lines_pass2 ((name, role, t, rule, deps) :: res) lines
else
add_lines_pass2 res (map (replace_dependencies_in_line (name, deps)) lines)
end
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
val assume_prefix = "a"
val have_prefix = "f"
val relabel_proof =
let
fun fresh_label depth prefix (accum as (l, subst, next)) =
if l = no_label then
accum
else
let val l' = (replicate_string (depth + 1) 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 = apfst (relabel_facts subst) by
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_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), methss))) =
let val (qs', lfs) = chain_qs_lfs lbl lfs in
Prove (qs' @ qs, xs, l, t, chain_proofs subproofs, ((lfs, gfs), methss))
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 * string * string * Time.time * real * bool * (term, string) atp_step list * thm
val arith_methodss =
[[Arith_Method, Simp_Method, Auto_Method, Fastforce_Method, Blast_Method, Force_Method,
Metis_Method], [Meson_Method]]
val datatype_methodss = [[Simp_Size_Method, Simp_Method]]
val metislike_methodss =
[[Metis_Method, Simp_Method, Auto_Method, Arith_Method, Blast_Method, Fastforce_Method,
Force_Method], [Meson_Method]]
val rewrite_methodss =
[[Auto_Method, Simp_Method, Fastforce_Method, Force_Method, Metis_Method], [Meson_Method]]
val skolem_methodss = [[Metis_Method, Blast_Method], [Meson_Method]]
fun isar_proof_text ctxt debug isar_proofs isar_params
(one_line_params as (_, _, _, _, subgoal, subgoal_count)) =
let
fun isar_proof_of () =
let
val SOME (verbose, metis_type_enc, metis_lam_trans, preplay_timeout, compress_isar,
try0_isar, atp_proof, goal) = try isar_params ()
val (params, _, 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 do_preplay = preplay_timeout <> Time.zeroTime
val try0_isar = try0_isar andalso do_preplay
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
val atp_proof =
atp_proof
|> rpair [] |-> fold_rev add_line_pass1
|> add_lines_pass2 []
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, methss) =
if is_skolemize_rule rule then (skolems_of t, skolem_methodss)
else if is_arith_rule rule then ([], arith_methodss)
else ([], rewrite_methodss)
in
Prove ([], skos, l, t, [], (([], []), methss))
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
HOLogic.dest_Trueprop
#> role <> Conjecture ? s_not
#> fold exists_of (map Var (Term.add_vars t []))
#> HOLogic.mk_Trueprop
else
I))))
atp_proof
val rule_of_clause_id = fst o the o Symtab.lookup steps o fst
fun prop_of_clause [(num, _)] = Symtab.lookup steps num |> the |> snd |> close_form
| prop_of_clause names =
let
val lits = map (HOLogic.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 eq_set (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_list predecessor, []), metislike_methodss)))
else
I)
|> rev
| isar_steps outer _ accum (Have (id, (gamma, c)) :: infs) =
let
val l = label_of_clause c
val t = prop_of_clause c
val rule = rule_of_clause_id id
val skolem = is_skolemize_rule rule
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
val deps = fold add_fact_of_dependencies gamma no_facts
val methss =
if is_arith_rule rule then arith_methodss
else if is_datatype_rule rule then datatype_methodss
else metislike_methodss
val by = (deps, methss)
in
if is_clause_tainted c then
(case gamma of
[g] =>
if skolem 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, skolem_methodss)] infs
end
else
do_rest l (prove [] by)
| _ => do_rest l (prove [] by))
else
do_rest l (if skolem then Prove ([], skolems_of t, l, t, [], by) else prove [] by)
end
| isar_steps outer predecessor accum (Cases cases :: infs) =
let
fun isar_case (c, subinfs) =
isar_proof false [] [(label_of_clause c, prop_of_clause c)] [] subinfs
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 (filter_out (null o snd) cases),
((the_list predecessor, []), metislike_methodss))
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 (preplay_data as {overall_preplay_outcome, ...}, 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 true params assms lems
|> postprocess_remove_unreferenced_steps I
|> relabel_proof_canonically
|> `(proof_preplay_data debug ctxt metis_type_enc metis_lam_trans do_preplay
preplay_timeout)
val (play_outcome, isar_proof) =
isar_proof
|> compress_proof (if isar_proofs = SOME true then compress_isar else 1000.0) preplay_data
|> try0_isar ? try0_isar_proof preplay_timeout preplay_data
|> postprocess_remove_unreferenced_steps (try0_isar ? minimal_deps_of_step preplay_data)
|> `overall_preplay_outcome
||> (chain_direct_proof #> kill_useless_labels_in_proof #> relabel_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_isar_steps (steps_of_proof isar_proof) 0 in
[string_of_int num_steps ^ " step" ^ plural_s num_steps]
end
else
[]) @
(if do_preplay then [string_of_play_outcome play_outcome] else [])
in
"\n\nStructured proof" ^ (commas msg |> not (null msg) ? enclose " (" ")") ^ ":\n" ^
Active.sendback_markup [Markup.padding_command] isar_text
end)
end
val one_line_proof = one_line_proof_text 0 one_line_params
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 (reconstr, play) =
(case play of
Played _ => reconstr = SMT
| Play_Timed_Out _ => true
| Play_Failed => true
| Not_Played => false)
fun proof_text ctxt debug 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 debug isar_proofs isar_params
else
one_line_proof_text num_chained) one_line_params
end;