(* 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_PROOF_RECONSTRUCT =
sig
type 'a proof = 'a ATP_Proof.proof
type stature = ATP_Problem_Generate.stature
datatype reconstructor =
Metis of string * string |
SMT
datatype play =
Played of reconstructor * Time.time |
Trust_Playable of reconstructor * Time.time option |
Failed_to_Play of reconstructor
type minimize_command = string list -> string
type one_line_params =
play * string * (string * stature) list * minimize_command * int * int
type isar_params =
bool * bool * Time.time * real * string Symtab.table
* (string * stature) list vector * int Symtab.table * string proof * thm
val smtN : string
val string_for_reconstructor : reconstructor -> string
val thms_of_name : Proof.context -> string -> thm list
val lam_trans_from_atp_proof : string proof -> string -> string
val is_typed_helper_used_in_atp_proof : string proof -> bool
val used_facts_in_atp_proof :
Proof.context -> (string * stature) list vector -> string proof ->
(string * stature) list
val used_facts_in_unsound_atp_proof :
Proof.context -> (string * stature) list vector -> 'a proof ->
string list option
val one_line_proof_text : int -> one_line_params -> string
val isar_proof_text :
Proof.context -> bool -> isar_params -> one_line_params -> string
val proof_text :
Proof.context -> bool -> isar_params -> int -> one_line_params -> string
end;
structure Sledgehammer_Reconstruct : SLEDGEHAMMER_PROOF_RECONSTRUCT =
struct
open ATP_Util
open ATP_Problem
open ATP_Proof
open ATP_Problem_Generate
open ATP_Proof_Reconstruct
open Sledgehammer_Util
structure String_Redirect = ATP_Proof_Redirect(
type key = step_name
val ord = fn ((s, _ : string list), (s', _)) => fast_string_ord (s, s')
val string_of = fst)
open String_Redirect
(** reconstructors **)
datatype reconstructor =
Metis of string * string |
SMT
datatype play =
Played of reconstructor * Time.time |
Trust_Playable of reconstructor * Time.time option |
Failed_to_Play of reconstructor
val smtN = "smt"
fun string_for_reconstructor (Metis (type_enc, lam_trans)) =
metis_call type_enc lam_trans
| string_for_reconstructor SMT = smtN
fun thms_of_name ctxt name =
let
val lex = Keyword.get_lexicons
val get = maps (Proof_Context.get_fact ctxt o fst)
in
Source.of_string name
|> Symbol.source
|> Token.source {do_recover = SOME false} lex Position.start
|> Token.source_proper
|> Source.source Token.stopper (Parse_Spec.xthms1 >> get) NONE
|> Source.exhaust
end
(** 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 Inference_Step ((_, ss), _, _, _, []) => exists pred ss
| _ => false)
fun lam_trans_from_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_ext = "extcnf_equal_neg"
val leo2_unfold_def = "unfold_def"
fun add_fact ctxt fact_names (Inference_Step ((_, ss), _, _, rule, deps)) =
(if rule = leo2_ext then
insert (op =) (ext_name ctxt, (Global, General))
else if rule = leo2_unfold_def 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 = satallax_coreN then
(fn [] =>
(* Satallax doesn't include definitions in its 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)
| add_fact _ _ _ = 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
(** one-liner reconstructor proofs **)
fun string_for_label (s, num) = s ^ string_of_int num
fun show_time NONE = ""
| show_time (SOME ext_time) = " (" ^ string_from_ext_time ext_time ^ ")"
(* FIXME: Various bugs, esp. with "unfolding"
fun unusing_chained_facts _ 0 = ""
| unusing_chained_facts used_chaineds num_chained =
if length used_chaineds = num_chained then ""
else if null used_chaineds then "(* using no facts *) "
else "(* using only " ^ space_implode " " used_chaineds ^ " *) "
*)
fun apply_on_subgoal _ 1 = "by "
| apply_on_subgoal 1 _ = "apply "
| apply_on_subgoal i n =
"prefer " ^ string_of_int i ^ " " ^ apply_on_subgoal 1 n
fun using_labels [] = ""
| using_labels ls =
"using " ^ space_implode " " (map string_for_label ls) ^ " "
fun command_call name [] =
name |> not (Lexicon.is_identifier name) ? enclose "(" ")"
| command_call name args = "(" ^ name ^ " " ^ space_implode " " args ^ ")"
fun reconstructor_command reconstr i n used_chaineds num_chained (ls, ss) =
(* unusing_chained_facts used_chaineds num_chained ^ *)
using_labels ls ^ apply_on_subgoal i n ^
command_call (string_for_reconstructor reconstr) ss
fun try_command_line banner time command =
banner ^ ": " ^ Markup.markup Isabelle_Markup.sendback command ^
show_time time ^ "."
fun minimize_line _ [] = ""
| minimize_line minimize_command ss =
case minimize_command ss of
"" => ""
| command =>
"\nTo minimize: " ^ Markup.markup Isabelle_Markup.sendback command ^ "."
fun split_used_facts facts =
facts |> List.partition (fn (_, (sc, _)) => sc = Chained)
|> pairself (sort_distinct (string_ord o pairself fst))
type minimize_command = string list -> string
type one_line_params =
play * string * (string * stature) list * minimize_command * int * int
fun one_line_proof_text num_chained
(preplay, banner, used_facts, minimize_command, subgoal,
subgoal_count) =
let
val (chained, extra) = split_used_facts used_facts
val (failed, reconstr, ext_time) =
case preplay of
Played (reconstr, time) => (false, reconstr, (SOME (false, time)))
| Trust_Playable (reconstr, time) =>
(false, reconstr,
case time of
NONE => NONE
| SOME time =>
if time = Time.zeroTime then NONE else SOME (true, time))
| Failed_to_Play reconstr => (true, reconstr, NONE)
val try_line =
([], map fst extra)
|> reconstructor_command reconstr subgoal subgoal_count (map fst chained)
num_chained
|> (if failed then
enclose "One-line proof reconstruction failed: "
".\n(Invoking \"sledgehammer\" with \"[strict]\" might \
\solve this.)"
else
try_command_line banner ext_time)
in try_line ^ minimize_line minimize_command (map fst (extra @ chained)) end
(** Isar proof construction and manipulation **)
type label = string * int
type facts = label list * string list
datatype isar_qualifier = Show | Then | Moreover | Ultimately
datatype isar_step =
Fix of (string * typ) list |
Let of term * term |
Assume of label * term |
Prove of isar_qualifier list * label * term * byline
and byline =
By_Metis of facts |
Case_Split of isar_step list list * facts
val assume_prefix = "a"
val have_prefix = "f"
val raw_prefix = "x"
fun raw_label_for_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_for_name name
| label_of_clause c = (space_implode "___" (map fst c), 0)
fun add_fact_from_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_from_dependencies fact_names names =
apfst (insert (op =) (label_of_clause names))
fun repair_name "$true" = "c_True"
| repair_name "$false" = "c_False"
| repair_name "$$e" = tptp_equal (* seen in Vampire proofs *)
| repair_name s =
if is_tptp_equal s orelse
(* seen in Vampire proofs *)
(String.isPrefix "sQ" s andalso String.isSuffix "_eqProxy" s) then
tptp_equal
else
s
fun unvarify_term (Var ((s, 0), T)) = Free (s, T)
| unvarify_term t = raise TERM ("unvarify_term: non-Var", [t])
fun infer_formula_types ctxt =
Type.constraint HOLogic.boolT
#> Syntax.check_term
(Proof_Context.set_mode Proof_Context.mode_schematic ctxt)
val combinator_table =
[(@{const_name Meson.COMBI}, @{thm Meson.COMBI_def [abs_def]}),
(@{const_name Meson.COMBK}, @{thm Meson.COMBK_def [abs_def]}),
(@{const_name Meson.COMBB}, @{thm Meson.COMBB_def [abs_def]}),
(@{const_name Meson.COMBC}, @{thm Meson.COMBC_def [abs_def]}),
(@{const_name Meson.COMBS}, @{thm Meson.COMBS_def [abs_def]})]
fun uncombine_term thy =
let
fun aux (t1 $ t2) = betapply (pairself aux (t1, t2))
| aux (Abs (s, T, t')) = Abs (s, T, aux t')
| aux (t as Const (x as (s, _))) =
(case AList.lookup (op =) combinator_table s of
SOME thm => thm |> prop_of |> specialize_type thy x
|> Logic.dest_equals |> snd
| NONE => t)
| aux t = t
in aux end
fun decode_line sym_tab (Definition_Step (name, phi1, phi2)) ctxt =
let
val thy = Proof_Context.theory_of ctxt
val t1 = prop_from_atp ctxt true sym_tab phi1
val vars = snd (strip_comb t1)
val frees = map unvarify_term vars
val unvarify_args = subst_atomic (vars ~~ frees)
val t2 = prop_from_atp ctxt true sym_tab phi2
val (t1, t2) =
HOLogic.eq_const HOLogic.typeT $ t1 $ t2
|> unvarify_args |> uncombine_term thy |> infer_formula_types ctxt
|> HOLogic.dest_eq
in
(Definition_Step (name, t1, t2),
fold Variable.declare_term (maps Misc_Legacy.term_frees [t1, t2]) ctxt)
end
| decode_line sym_tab (Inference_Step (name, role, u, rule, deps)) ctxt =
let
val thy = Proof_Context.theory_of ctxt
val t = u |> prop_from_atp ctxt true sym_tab
|> uncombine_term thy |> infer_formula_types ctxt
in
(Inference_Step (name, role, t, rule, deps),
fold Variable.declare_term (Misc_Legacy.term_frees t) ctxt)
end
fun decode_lines ctxt sym_tab lines =
fst (fold_map (decode_line sym_tab) lines ctxt)
fun replace_one_dependency (old, new) dep =
if is_same_atp_step dep old then new else [dep]
fun replace_dependencies_in_line _ (line as Definition_Step _) = line
| replace_dependencies_in_line p
(Inference_Step (name, role, t, rule, deps)) =
Inference_Step (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 is_same_inference _ (Definition_Step _) = false
| is_same_inference t (Inference_Step (_, _, t', _, _)) = t aconv t'
(* Discard facts; consolidate adjacent lines that prove the same formula, since
they differ only in type information.*)
fun add_line _ (line as Definition_Step _) lines = line :: lines
| add_line fact_names (Inference_Step (name as (_, ss), role, t, rule, []))
lines =
(* No dependencies: fact, conjecture, or (for Vampire) internal facts or
definitions. *)
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
(* Is there a repetition? If so, replace later line by earlier one. *)
else case take_prefix (not o is_same_inference t) lines of
(_, []) => lines (* no repetition of proof line *)
| (pre, Inference_Step (name', _, _, _, _) :: post) =>
pre @ map (replace_dependencies_in_line (name', [name])) post
| _ => raise Fail "unexpected inference"
else if is_conjecture ss then
Inference_Step (name, role, t, rule, []) :: lines
else
map (replace_dependencies_in_line (name, [])) lines
| add_line _ (Inference_Step (name, role, t, rule, deps)) lines =
(* Type information will be deleted later; skip repetition test. *)
if is_only_type_information t then
Inference_Step (name, role, t, rule, deps) :: lines
(* Is there a repetition? If so, replace later line by earlier one. *)
else case take_prefix (not o is_same_inference t) lines of
(* FIXME: Doesn't this code risk conflating proofs involving different
types? *)
(_, []) => Inference_Step (name, role, t, rule, deps) :: lines
| (pre, Inference_Step (name', role, t', rule, _) :: post) =>
Inference_Step (name, role, t', rule, deps) ::
pre @ map (replace_dependencies_in_line (name', [name])) post
| _ => raise Fail "unexpected inference"
val waldmeister_conjecture_num = "1.0.0.0"
val repair_waldmeister_endgame =
let
fun do_tail (Inference_Step (name, role, t, rule, deps)) =
Inference_Step (name, role, s_not t, rule, deps)
| do_tail line = line
fun do_body [] = []
| do_body ((line as Inference_Step ((num, _), _, _, _, _)) :: lines) =
if num = waldmeister_conjecture_num then map do_tail (line :: lines)
else line :: do_body lines
| do_body (line :: lines) = line :: do_body lines
in do_body end
(* Recursively delete empty lines (type information) from the proof. *)
fun add_nontrivial_line (line as Inference_Step (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) []
(* ATPs sometimes reuse free variable names in the strangest ways. Removing
offending lines often does the trick. *)
fun is_bad_free frees (Free x) = not (member (op =) frees x)
| is_bad_free _ _ = false
fun add_desired_line _ _ (line as Definition_Step (name, _, _)) (j, lines) =
(j, line :: map (replace_dependencies_in_line (name, [])) lines)
| add_desired_line fact_names frees
(Inference_Step (name as (_, ss), role, t, rule, deps)) (j, lines) =
(j + 1,
if is_fact fact_names ss orelse
is_conjecture ss orelse
(* the last line must be kept *)
j = 0 orelse
(not (is_only_type_information t) andalso
null (Term.add_tvars t []) andalso
not (exists_subterm (is_bad_free frees) t) andalso
length deps >= 2 andalso
(* kill next to last line, which usually results in a trivial step *)
j <> 1) then
Inference_Step (name, role, t, rule, deps) :: lines (* keep line *)
else
map (replace_dependencies_in_line (name, deps)) lines) (* drop line *)
(** Type annotations **)
fun post_traverse_term_type' f _ (t as Const (_, T)) s = f t T s
| post_traverse_term_type' f _ (t as Free (_, T)) s = f t T s
| post_traverse_term_type' f _ (t as Var (_, T)) s = f t T s
| post_traverse_term_type' f env (t as Bound i) s = f t (nth env i) s
| post_traverse_term_type' f env (Abs (x, T1, b)) s =
let
val ((b', s'), T2) = post_traverse_term_type' f (T1 :: env) b s
in f (Abs (x, T1, b')) (T1 --> T2) s' end
| post_traverse_term_type' f env (u $ v) s =
let
val ((u', s'), Type (_, [_, T])) = post_traverse_term_type' f env u s
val ((v', s''), _) = post_traverse_term_type' f env v s'
in f (u' $ v') T s'' end
fun post_traverse_term_type f s t =
post_traverse_term_type' (fn t => fn T => fn s => (f t T s, T)) [] t s |> fst
fun post_fold_term_type f s t =
post_traverse_term_type (fn t => fn T => fn s => (t, f t T s)) s t |> snd
(* Data structures, orders *)
val cost_ord = prod_ord int_ord (prod_ord int_ord int_ord)
structure Var_Set_Tab = Table(
type key = indexname list
val ord = list_ord Term_Ord.fast_indexname_ord)
(* (1) Generalize Types *)
fun generalize_types ctxt t =
t |> map_types (fn _ => dummyT)
|> Syntax.check_term
(Proof_Context.set_mode Proof_Context.mode_pattern ctxt)
(* (2) Typing-spot Table *)
local
fun key_of_atype (TVar (z, _)) =
Ord_List.insert Term_Ord.fast_indexname_ord z
| key_of_atype _ = I
fun key_of_type T = fold_atyps key_of_atype T []
fun update_tab t T (tab, pos) =
(case key_of_type T of
[] => tab
| key =>
let val cost = (size_of_typ T, (size_of_term t, pos)) in
case Var_Set_Tab.lookup tab key of
NONE => Var_Set_Tab.update_new (key, cost) tab
| SOME old_cost =>
(case cost_ord (cost, old_cost) of
LESS => Var_Set_Tab.update (key, cost) tab
| _ => tab)
end,
pos + 1)
in
val typing_spot_table =
post_fold_term_type update_tab (Var_Set_Tab.empty, 0) #> fst
end
(* (3) Reverse-Greedy *)
fun reverse_greedy typing_spot_tab =
let
fun update_count z =
fold (fn tvar => fn tab =>
let val c = Vartab.lookup tab tvar |> the_default 0 in
Vartab.update (tvar, c + z) tab
end)
fun superfluous tcount =
forall (fn tvar => the (Vartab.lookup tcount tvar) > 1)
fun drop_superfluous (tvars, (_, (_, spot))) (spots, tcount) =
if superfluous tcount tvars then (spots, update_count ~1 tvars tcount)
else (spot :: spots, tcount)
val (typing_spots, tvar_count_tab) =
Var_Set_Tab.fold
(fn kv as (k, _) => apfst (cons kv) #> apsnd (update_count 1 k))
typing_spot_tab ([], Vartab.empty)
|>> sort_distinct (rev_order o cost_ord o pairself snd)
in fold drop_superfluous typing_spots ([], tvar_count_tab) |> fst end
(* (4) Introduce Annotations *)
fun introduce_annotations thy spots t t' =
let
val get_types = post_fold_term_type (K cons) []
fun match_types tp =
fold (Sign.typ_match thy) (op ~~ (pairself get_types tp)) Vartab.empty
fun unica' b x [] = if b then [x] else []
| unica' b x (y :: ys) =
if x = y then unica' false x ys
else unica' true y ys |> b ? cons x
fun unica ord xs =
case sort ord xs of x :: ys => unica' true x ys | [] => []
val add_all_tfree_namesT = fold_atyps (fn TFree (x, _) => cons x | _ => I)
fun erase_unica_tfrees env =
let
val unica =
Vartab.fold (add_all_tfree_namesT o snd o snd) env []
|> unica fast_string_ord
val erase_unica = map_atyps
(fn T as TFree (s, _) =>
if Ord_List.member fast_string_ord unica s then dummyT else T
| T => T)
in Vartab.map (K (apsnd erase_unica)) env end
val env = match_types (t', t) |> erase_unica_tfrees
fun get_annot env (TFree _) = (false, (env, dummyT))
| get_annot env (T as TVar (v, S)) =
let val T' = Envir.subst_type env T in
if T' = dummyT then (false, (env, dummyT))
else (true, (Vartab.update (v, (S, dummyT)) env, T'))
end
| get_annot env (Type (S, Ts)) =
(case fold_rev (fn T => fn (b, (env, Ts)) =>
let
val (b', (env', T)) = get_annot env T
in (b orelse b', (env', T :: Ts)) end)
Ts (false, (env, [])) of
(true, (env', Ts)) => (true, (env', Type (S, Ts)))
| (false, (env', _)) => (false, (env', dummyT)))
fun post1 _ T (env, cp, ps as p :: ps', annots) =
if p <> cp then
(env, cp + 1, ps, annots)
else
let val (_, (env', T')) = get_annot env T in
(env', cp + 1, ps', (p, T') :: annots)
end
| post1 _ _ accum = accum
val (_, _, _, annots) = post_fold_term_type post1 (env, 0, spots, []) t'
fun post2 t _ (cp, annots as (p, T) :: annots') =
if p <> cp then (t, (cp + 1, annots))
else (Type.constraint T t, (cp + 1, annots'))
| post2 t _ x = (t, x)
in post_traverse_term_type post2 (0, rev annots) t |> fst end
(* (5) Annotate *)
fun annotate_types ctxt t =
let
val thy = Proof_Context.theory_of ctxt
val t' = generalize_types ctxt t
val typing_spots =
t' |> typing_spot_table
|> reverse_greedy
|> sort int_ord
in introduce_annotations thy typing_spots t t' end
val indent_size = 2
val no_label = ("", ~1)
fun string_for_proof ctxt type_enc lam_trans i n =
let
fun fix_print_mode f x =
Print_Mode.setmp (filter (curry (op =) Symbol.xsymbolsN)
(print_mode_value ())) f x
fun do_indent ind = replicate_string (ind * indent_size) " "
fun do_free (s, T) =
maybe_quote s ^ " :: " ^
maybe_quote (fix_print_mode (Syntax.string_of_typ ctxt) T)
fun do_label l = if l = no_label then "" else string_for_label l ^ ": "
fun do_have qs =
(if member (op =) qs Moreover then "moreover " else "") ^
(if member (op =) qs Ultimately then "ultimately " else "") ^
(if member (op =) qs Then then
if member (op =) qs Show then "thus" else "hence"
else
if member (op =) qs Show then "show" else "have")
val do_term =
maybe_quote o fix_print_mode (Syntax.string_of_term ctxt)
o annotate_types ctxt
val reconstr = Metis (type_enc, lam_trans)
fun do_facts ind (ls, ss) =
"\n" ^ do_indent (ind + 1) ^
reconstructor_command reconstr 1 1 [] 0
(ls |> sort_distinct (prod_ord string_ord int_ord),
ss |> sort_distinct string_ord)
and do_step ind (Fix xs) =
do_indent ind ^ "fix " ^ space_implode " and " (map do_free xs) ^ "\n"
| do_step ind (Let (t1, t2)) =
do_indent ind ^ "let " ^ do_term t1 ^ " = " ^ do_term t2 ^ "\n"
| do_step ind (Assume (l, t)) =
do_indent ind ^ "assume " ^ do_label l ^ do_term t ^ "\n"
| do_step ind (Prove (qs, l, t, By_Metis facts)) =
do_indent ind ^ do_have qs ^ " " ^
do_label l ^ do_term t ^ do_facts ind facts ^ "\n"
| do_step ind (Prove (qs, l, t, Case_Split (proofs, facts))) =
implode (map (prefix (do_indent ind ^ "moreover\n") o do_block ind)
proofs) ^
do_indent ind ^ do_have qs ^ " " ^ do_label l ^ do_term t ^
do_facts ind facts ^ "\n"
and do_steps prefix suffix ind steps =
let val s = implode (map (do_step ind) steps) in
replicate_string (ind * indent_size - size prefix) " " ^ prefix ^
String.extract (s, ind * indent_size,
SOME (size s - ind * indent_size - 1)) ^
suffix ^ "\n"
end
and do_block ind proof = do_steps "{ " " }" (ind + 1) proof
(* One-step proofs are pointless; better use the Metis one-liner
directly. *)
and do_proof [Prove (_, _, _, By_Metis _)] = ""
| do_proof proof =
(if i <> 1 then "prefer " ^ string_of_int i ^ "\n" else "") ^
do_indent 0 ^ "proof -\n" ^ do_steps "" "" 1 proof ^ do_indent 0 ^
(if n <> 1 then "next" else "qed")
in do_proof end
fun used_labels_of_step (Prove (_, _, _, by)) =
(case by of
By_Metis (ls, _) => ls
| Case_Split (proofs, (ls, _)) =>
fold (union (op =) o used_labels_of) proofs ls)
| used_labels_of_step _ = []
and used_labels_of proof = fold (union (op =) o used_labels_of_step) proof []
fun kill_useless_labels_in_proof proof =
let
val used_ls = used_labels_of proof
fun do_label l = if member (op =) used_ls l then l else no_label
fun do_step (Assume (l, t)) = Assume (do_label l, t)
| do_step (Prove (qs, l, t, by)) =
Prove (qs, do_label l, t,
case by of
Case_Split (proofs, facts) =>
Case_Split (map (map do_step) proofs, facts)
| _ => by)
| do_step step = step
in map do_step proof end
fun prefix_for_depth n = replicate_string (n + 1)
val relabel_proof =
let
fun aux _ _ _ [] = []
| aux subst depth (next_assum, next_have) (Assume (l, t) :: proof) =
if l = no_label then
Assume (l, t) :: aux subst depth (next_assum, next_have) proof
else
let val l' = (prefix_for_depth depth assume_prefix, next_assum) in
Assume (l', t) ::
aux ((l, l') :: subst) depth (next_assum + 1, next_have) proof
end
| aux subst depth (next_assum, next_have)
(Prove (qs, l, t, by) :: proof) =
let
val (l', subst, next_have) =
if l = no_label then
(l, subst, next_have)
else
let val l' = (prefix_for_depth depth have_prefix, next_have) in
(l', (l, l') :: subst, next_have + 1)
end
val relabel_facts =
apfst (maps (the_list o AList.lookup (op =) subst))
val by =
case by of
By_Metis facts => By_Metis (relabel_facts facts)
| Case_Split (proofs, facts) =>
Case_Split (map (aux subst (depth + 1) (1, 1)) proofs,
relabel_facts facts)
in
Prove (qs, l', t, by) :: aux subst depth (next_assum, next_have) proof
end
| aux subst depth nextp (step :: proof) =
step :: aux subst depth nextp proof
in aux [] 0 (1, 1) end
val merge_timeout_slack = 1.2
val label_ord = prod_ord int_ord fast_string_ord o pairself swap
structure Label_Table = Table(
type key = label
val ord = label_ord)
fun shrink_proof debug ctxt type_enc lam_trans preplay
preplay_timeout isar_shrinkage proof =
let
(* clean vector interface *)
fun get i v = Vector.sub (v, i)
fun replace x i v = Vector.update (v, i, x)
fun update f i v = replace (get i v |> f) i v
fun v_fold_index f v s =
Vector.foldl (fn (x, (i, s)) => (i+1, f (i, x) s)) (0, s) v |> snd
(* Queue interface to table *)
fun pop tab key =
let val v = hd (Inttab.lookup_list tab key) in
(v, Inttab.remove_list (op =) (key, v) tab)
end
fun pop_max tab = pop tab (the (Inttab.max_key tab))
val is_empty = Inttab.is_empty
fun add_list tab xs = fold (Inttab.insert_list (op =)) xs tab
(* proof vector *)
val proof_vect = proof |> map SOME |> Vector.fromList
val n = Vector.length proof_vect
val n_target = Real.fromInt n / isar_shrinkage |> Real.round
(* table for mapping from label to proof position *)
fun update_table (i, Prove (_, label, _, _)) =
Label_Table.update_new (label, i)
| update_table _ = I
val label_index_table = fold_index update_table proof Label_Table.empty
(* proof references *)
fun refs (Prove (_, _, _, By_Metis (refs, _))) =
map (the o Label_Table.lookup label_index_table) refs
| refs _ = []
val refed_by_vect =
Vector.tabulate (n, (fn _ => []))
|> fold_index (fn (i, step) => fold (update (cons i)) (refs step)) proof
|> Vector.map rev (* after rev, indices are sorted in ascending order *)
(* candidates for elimination, use table as priority queue (greedy
algorithm) *)
fun cost (Prove (_, _ , t, _)) = Term.size_of_term t
| cost _ = 0
val cand_ord = rev_order o prod_ord int_ord int_ord
val cand_tab =
v_fold_index
(fn (i, [_]) => cons (get i proof_vect |> the |> cost, i)
| _ => I) refed_by_vect []
|> Inttab.make_list
(* Enrich context with local facts *)
val thy = Proof_Context.theory_of ctxt
fun enrich_ctxt' (Prove (_, label, t, _)) ctxt =
Proof_Context.put_thms false
(string_for_label label, SOME [Skip_Proof.make_thm thy t]) ctxt
| enrich_ctxt' _ ctxt = ctxt
val rich_ctxt = fold enrich_ctxt' proof ctxt
(* Timing *)
fun take_time timeout tac arg =
let val timing = Timing.start () in
(TimeLimit.timeLimit timeout tac arg;
Timing.result timing |> #cpu |> SOME)
handle _ => NONE
end
val sum_up_time =
Vector.foldl
((fn (SOME t, (b, s)) => (b, t + s)
| (NONE, (_, s)) => (true, preplay_timeout + s)) o apfst Lazy.force)
(false, seconds 0.0)
(* Metis Preplaying *)
fun try_metis timeout (Prove (_, _, t, By_Metis fact_names)) =
if not preplay then (fn () => SOME (seconds 0.0)) else
let
val facts =
fact_names
|>> map string_for_label |> op @
|> map (the_single o thms_of_name rich_ctxt)
val goal =
Goal.prove (Config.put Metis_Tactic.verbose debug ctxt) [] [] t
fun tac {context = ctxt, prems = _} =
Metis_Tactic.metis_tac [type_enc] lam_trans ctxt facts 1
in
take_time timeout (fn () => goal tac)
end
(* Lazy metis time vector, cache *)
val metis_time =
Vector.map (Lazy.lazy o try_metis preplay_timeout o the) proof_vect
(* Merging *)
fun merge (Prove (qs1, label1, _, By_Metis (lfs1, gfs1)))
(Prove (qs2, label2 , t, By_Metis (lfs2, gfs2))) =
let
val qs = inter (op =) qs1 qs2 (* FIXME: Is this correct? *)
|> member (op =) (union (op =) qs1 qs2) Ultimately ? cons Ultimately
|> member (op =) qs2 Show ? cons Show
val ls = remove (op =) label1 lfs2 |> union (op =) lfs1
val ss = union (op =) gfs1 gfs2
in Prove (qs, label2, t, By_Metis (ls, ss)) end
fun try_merge metis_time (s1, i) (s2, j) =
(case get i metis_time |> Lazy.force of
NONE => (NONE, metis_time)
| SOME t1 =>
(case get j metis_time |> Lazy.force of
NONE => (NONE, metis_time)
| SOME t2 =>
let
val s12 = merge s1 s2
val timeout =
t1 + t2 |> Time.toReal |> curry Real.* merge_timeout_slack
|> Time.fromReal
in
case try_metis timeout s12 () of
NONE => (NONE, metis_time)
| some_t12 =>
(SOME s12, metis_time
|> replace (seconds 0.0 |> SOME |> Lazy.value) i
|> replace (Lazy.value some_t12) j)
end))
fun merge_steps metis_time proof_vect refed_by cand_tab n' =
if is_empty cand_tab orelse n' <= n_target orelse n'<3 then
(sum_up_time metis_time,
Vector.foldr
(fn (NONE, proof) => proof | (SOME s, proof) => s :: proof)
[] proof_vect)
else
let
val (i, cand_tab) = pop_max cand_tab
val j = get i refed_by |> the_single
val s1 = get i proof_vect |> the
val s2 = get j proof_vect |> the
in
case try_merge metis_time (s1, i) (s2, j) of
(NONE, metis_time) =>
merge_steps metis_time proof_vect refed_by cand_tab n'
| (s, metis_time) => let
val refs = refs s1
val refed_by = refed_by |> fold
(update (Ord_List.remove int_ord i #> Ord_List.insert int_ord j)) refs
val new_candidates =
fold (fn (i, [_]) => cons (cost (get i proof_vect |> the), i)
| _ => I)
(map (fn i => (i, get i refed_by)) refs) []
val cand_tab = add_list cand_tab new_candidates
val proof_vect = proof_vect |> replace NONE i |> replace s j
in
merge_steps metis_time proof_vect refed_by cand_tab (n' - 1)
end
end
in
merge_steps metis_time proof_vect refed_by_vect cand_tab n
end
val chain_direct_proof =
let
fun succedent_of_step (Prove (_, label, _, _)) = SOME label
| succedent_of_step (Assume (label, _)) = SOME label
| succedent_of_step _ = NONE
fun chain_inf (SOME label0)
(step as Prove (qs, label, t, By_Metis (lfs, gfs))) =
if member (op =) lfs label0 then
Prove (Then :: qs, label, t,
By_Metis (filter_out (curry (op =) label0) lfs, gfs))
else
step
| chain_inf _ (Prove (qs, label, t, Case_Split (proofs, facts))) =
Prove (qs, label, t, Case_Split ((map (chain_proof NONE) proofs), facts))
| chain_inf _ step = step
and chain_proof _ [] = []
| chain_proof (prev as SOME _) (i :: is) =
chain_inf prev i :: chain_proof (succedent_of_step i) is
| chain_proof _ (i :: is) =
i :: chain_proof (succedent_of_step i) is
in chain_proof NONE end
type isar_params =
bool * bool * Time.time * real * string Symtab.table
* (string * stature) list vector * int Symtab.table * string proof * thm
fun isar_proof_text ctxt isar_proofs
(debug, verbose, preplay_timeout, isar_shrinkage,
pool, fact_names, sym_tab, atp_proof, goal)
(one_line_params as (_, _, _, _, subgoal, subgoal_count)) =
let
val (params, hyp_ts, concl_t) = strip_subgoal ctxt goal subgoal
val frees = fold Term.add_frees (concl_t :: hyp_ts) []
val one_line_proof = one_line_proof_text 0 one_line_params
val type_enc =
if is_typed_helper_used_in_atp_proof atp_proof then full_typesN
else partial_typesN
val lam_trans = lam_trans_from_atp_proof atp_proof metis_default_lam_trans
val preplay = preplay_timeout <> seconds 0.0
fun isar_proof_of () =
let
val atp_proof =
atp_proof
|> clean_up_atp_proof_dependencies
|> nasty_atp_proof pool
|> map_term_names_in_atp_proof repair_name
|> decode_lines ctxt sym_tab
|> rpair [] |-> fold_rev (add_line fact_names)
|> repair_waldmeister_endgame
|> rpair [] |-> fold_rev add_nontrivial_line
|> rpair (0, [])
|-> fold_rev (add_desired_line fact_names frees)
|> snd
val conj_name = conjecture_prefix ^ string_of_int (length hyp_ts)
val conjs =
atp_proof |> map_filter
(fn Inference_Step (name as (_, ss), _, _, _, []) =>
if member (op =) ss conj_name then SOME name else NONE
| _ => NONE)
val assms =
atp_proof |> map_filter
(fn Inference_Step (name as (_, ss), _, _, _, []) =>
(case resolve_conjecture ss of
[j] =>
if j = length hyp_ts then NONE
else SOME (Assume (raw_label_for_name name, nth hyp_ts j))
| _ => NONE)
| _ => NONE)
fun dep_of_step (Definition_Step _) = NONE
| dep_of_step (Inference_Step (name, _, _, _, from)) =
SOME (from, name)
val ref_graph = atp_proof |> map_filter dep_of_step |> make_ref_graph
val axioms = axioms_of_ref_graph ref_graph conjs
val tainted = tainted_atoms_of_ref_graph ref_graph conjs
val props =
Symtab.empty
|> fold (fn Definition_Step _ => I (* FIXME *)
| Inference_Step (name as (s, ss), role, t, _, _) =>
Symtab.update_new (s,
if member (op = o apsnd fst) tainted s then
t |> role <> Conjecture ? s_not
|> fold exists_of (map Var (Term.add_vars t []))
else
t))
atp_proof
(* The assumptions and conjecture are props; the rest are bools. *)
fun prop_of_clause [name as (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 props s)
|> HOLogic.mk_Trueprop |> close_form)
| prop_of_clause names =
let val lits = map_filter (Symtab.lookup props o fst) names in
case List.partition (can HOLogic.dest_not) lits of
(negs as _ :: _, pos as _ :: _) =>
HOLogic.mk_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 do_have outer qs (gamma, c) =
Prove (maybe_show outer c qs, label_of_clause c, prop_of_clause c,
By_Metis (fold (add_fact_from_dependencies fact_names) gamma
([], [])))
fun do_inf outer (Have z) = do_have outer [] z
| do_inf outer (Cases cases) =
let val c = succedent_of_cases cases in
Prove (maybe_show outer c [Ultimately], label_of_clause c,
prop_of_clause c,
Case_Split (map (do_case false) cases, ([], [])))
end
and do_case outer (c, infs) =
Assume (label_of_clause c, prop_of_clause c) ::
map (do_inf outer) infs
val (ext_time, isar_proof) =
ref_graph
|> redirect_graph axioms tainted
|> map (do_inf true)
|> append assms
|> shrink_proof debug ctxt type_enc lam_trans preplay
preplay_timeout
(if isar_proofs then isar_shrinkage else 1000.0)
(* ||> reorder_proof_to_minimize_jumps (* ? *) *)
||> chain_direct_proof
||> kill_useless_labels_in_proof
||> relabel_proof
||> not (null params) ? cons (Fix params)
val num_steps = length isar_proof
val isar_text =
string_for_proof ctxt type_enc lam_trans subgoal subgoal_count
isar_proof
in
case isar_text of
"" =>
if isar_proofs then
"\nNo structured proof available (proof too short)."
else
""
| _ =>
"\n\nStructured proof" ^
(if verbose then
" (" ^ string_of_int num_steps ^ " step" ^ plural_s num_steps ^
(if preplay then ", " ^ string_from_ext_time ext_time
else "") ^ ")"
else if preplay then
" (" ^ string_from_ext_time ext_time ^ ")"
else
"") ^ ":\n" ^ Markup.markup Isabelle_Markup.sendback isar_text
end
val isar_proof =
if debug then
isar_proof_of ()
else case try isar_proof_of () of
SOME s => s
| NONE => if isar_proofs then
"\nWarning: The Isar proof construction failed."
else
""
in one_line_proof ^ isar_proof end
fun proof_text ctxt isar_proofs isar_params num_chained
(one_line_params as (preplay, _, _, _, _, _)) =
(if case preplay of Failed_to_Play _ => true | _ => isar_proofs then
isar_proof_text ctxt isar_proofs isar_params
else
one_line_proof_text num_chained) one_line_params
end;