1 (* Title: HOL/Tools/ATP/atp_proof_reconstruct.ML
2 Author: Lawrence C. Paulson, Cambridge University Computer Laboratory
3 Author: Claire Quigley, Cambridge University Computer Laboratory
4 Author: Jasmin Blanchette, TU Muenchen
6 Proof reconstruction from ATP proofs.
9 signature ATP_PROOF_RECONSTRUCT =
11 type ('a, 'b) ho_term = ('a, 'b) ATP_Problem.ho_term
12 type ('a, 'b, 'c) formula = ('a, 'b, 'c) ATP_Problem.formula
13 type 'a proof = 'a ATP_Proof.proof
14 type stature = ATP_Problem_Generate.stature
16 datatype reconstructor =
17 Metis of string * string |
21 Played of reconstructor * Time.time |
22 Trust_Playable of reconstructor * Time.time option |
23 Failed_to_Play of reconstructor
25 type minimize_command = string list -> string
26 type one_line_params =
27 play * string * (string * stature) list * minimize_command * int * int
29 bool * int * string Symtab.table * (string * stature) list vector
30 * int Symtab.table * string proof * thm
34 val full_typesN : string
35 val partial_typesN : string
36 val no_typesN : string
37 val really_full_type_enc : string
38 val full_type_enc : string
39 val partial_type_enc : string
40 val no_type_enc : string
41 val full_type_encs : string list
42 val partial_type_encs : string list
43 val metis_default_lam_trans : string
44 val metis_call : string -> string -> string
45 val string_for_reconstructor : reconstructor -> string
46 val used_facts_in_atp_proof :
47 Proof.context -> (string * stature) list vector -> string proof
48 -> (string * stature) list
49 val lam_trans_from_atp_proof : string proof -> string -> string
50 val is_typed_helper_used_in_atp_proof : string proof -> bool
51 val used_facts_in_unsound_atp_proof :
52 Proof.context -> (string * stature) list vector -> 'a proof
54 val unalias_type_enc : string -> string list
55 val one_line_proof_text : one_line_params -> string
56 val make_tvar : string -> typ
57 val make_tfree : Proof.context -> string -> typ
59 Proof.context -> bool -> int Symtab.table -> typ option
60 -> (string, string) ho_term -> term
62 Proof.context -> bool -> int Symtab.table
63 -> (string, string, (string, string) ho_term) formula -> term
65 Proof.context -> bool -> isar_params -> one_line_params -> string
67 Proof.context -> bool -> isar_params -> one_line_params -> string
70 structure ATP_Proof_Reconstruct : ATP_PROOF_RECONSTRUCT =
76 open ATP_Problem_Generate
78 structure String_Redirect = ATP_Proof_Redirect(
80 val ord = fn ((s, _ : string list), (s', _)) => fast_string_ord (s, s')
85 datatype reconstructor =
86 Metis of string * string |
90 Played of reconstructor * Time.time |
91 Trust_Playable of reconstructor * Time.time option |
92 Failed_to_Play of reconstructor
94 type minimize_command = string list -> string
95 type one_line_params =
96 play * string * (string * stature) list * minimize_command * int * int
98 bool * int * string Symtab.table * (string * stature) list vector
99 * int Symtab.table * string proof * thm
104 val full_typesN = "full_types"
105 val partial_typesN = "partial_types"
106 val no_typesN = "no_types"
108 val really_full_type_enc = "mono_tags"
109 val full_type_enc = "poly_guards_query"
110 val partial_type_enc = "poly_args"
111 val no_type_enc = "erased"
113 val full_type_encs = [full_type_enc, really_full_type_enc]
114 val partial_type_encs = partial_type_enc :: full_type_encs
116 val type_enc_aliases =
117 [(full_typesN, full_type_encs),
118 (partial_typesN, partial_type_encs),
119 (no_typesN, [no_type_enc])]
121 fun unalias_type_enc s =
122 AList.lookup (op =) type_enc_aliases s |> the_default [s]
124 val metis_default_lam_trans = combsN
126 fun metis_call type_enc lam_trans =
129 case AList.find (fn (enc, encs) => enc = hd encs) type_enc_aliases
133 val opts = [] |> type_enc <> partial_typesN ? cons type_enc
134 |> lam_trans <> metis_default_lam_trans ? cons lam_trans
135 in metisN ^ (if null opts then "" else " (" ^ commas opts ^ ")") end
137 fun string_for_reconstructor (Metis (type_enc, lam_trans)) =
138 metis_call type_enc lam_trans
139 | string_for_reconstructor SMT = smtN
141 fun find_first_in_list_vector vec key =
142 Vector.foldl (fn (ps, NONE) => AList.lookup (op =) ps key
143 | (_, value) => value) NONE vec
145 val unprefix_fact_number = space_implode "_" o tl o space_explode "_"
147 fun resolve_one_named_fact fact_names s =
148 case try (unprefix fact_prefix) s of
150 let val s' = s' |> unprefix_fact_number |> unascii_of in
151 s' |> find_first_in_list_vector fact_names |> Option.map (pair s')
154 fun resolve_fact fact_names = map_filter (resolve_one_named_fact fact_names)
155 fun is_fact fact_names = not o null o resolve_fact fact_names
157 fun resolve_one_named_conjecture s =
158 case try (unprefix conjecture_prefix) s of
159 SOME s' => Int.fromString s'
162 val resolve_conjecture = map_filter resolve_one_named_conjecture
163 val is_conjecture = not o null o resolve_conjecture
165 fun is_axiom_used_in_proof pred =
166 exists (fn Inference_Step ((_, ss), _, _, []) => exists pred ss | _ => false)
168 val is_combinator_def = String.isPrefix (helper_prefix ^ combinator_prefix)
170 val ascii_of_lam_fact_prefix = ascii_of lam_fact_prefix
172 (* overapproximation (good enough) *)
173 fun is_lam_lifted s =
174 String.isPrefix fact_prefix s andalso
175 String.isSubstring ascii_of_lam_fact_prefix s
177 fun lam_trans_from_atp_proof atp_proof default =
178 case (is_axiom_used_in_proof is_combinator_def atp_proof,
179 is_axiom_used_in_proof is_lam_lifted atp_proof) of
180 (false, false) => default
181 | (false, true) => liftingN
182 (* | (true, true) => combs_and_liftingN -- not supported by "metis" *)
183 | (true, _) => combsN
185 val is_typed_helper_name =
186 String.isPrefix helper_prefix andf String.isSuffix typed_helper_suffix
187 fun is_typed_helper_used_in_atp_proof atp_proof =
188 is_axiom_used_in_proof is_typed_helper_name atp_proof
190 val leo2_ext = "extcnf_equal_neg"
191 val isa_ext = Thm.get_name_hint @{thm ext}
192 val isa_short_ext = Long_Name.base_name isa_ext
195 if Thm.eq_thm_prop (@{thm ext},
196 singleton (Attrib.eval_thms ctxt) (Facts.named isa_short_ext, [])) then
201 fun add_fact _ fact_names (Inference_Step ((_, ss), _, _, [])) =
202 union (op =) (resolve_fact fact_names ss)
203 | add_fact ctxt _ (Inference_Step (_, _, rule, _)) =
204 if rule = leo2_ext then insert (op =) (ext_name ctxt, (Global, General))
208 fun used_facts_in_atp_proof ctxt fact_names atp_proof =
209 if null atp_proof then Vector.foldl (uncurry (union (op =))) [] fact_names
210 else fold (add_fact ctxt fact_names) atp_proof []
212 fun used_facts_in_unsound_atp_proof _ _ [] = NONE
213 | used_facts_in_unsound_atp_proof ctxt fact_names atp_proof =
214 let val used_facts = used_facts_in_atp_proof ctxt fact_names atp_proof in
215 if forall (fn (_, (sc, _)) => sc = Global) used_facts andalso
216 not (is_axiom_used_in_proof (is_conjecture o single) atp_proof) then
217 SOME (map fst used_facts)
223 (** Soft-core proof reconstruction: one-liners **)
225 fun string_for_label (s, num) = s ^ string_of_int num
227 fun show_time NONE = ""
228 | show_time (SOME ext_time) = " (" ^ string_from_ext_time ext_time ^ ")"
230 fun apply_on_subgoal _ 1 = "by "
231 | apply_on_subgoal 1 _ = "apply "
232 | apply_on_subgoal i n =
233 "prefer " ^ string_of_int i ^ " " ^ apply_on_subgoal 1 n
234 fun command_call name [] =
235 name |> not (Lexicon.is_identifier name) ? enclose "(" ")"
236 | command_call name args = "(" ^ name ^ " " ^ space_implode " " args ^ ")"
237 fun try_command_line banner time command =
238 banner ^ ": " ^ Markup.markup Isabelle_Markup.sendback command ^ show_time time ^ "."
239 fun using_labels [] = ""
241 "using " ^ space_implode " " (map string_for_label ls) ^ " "
242 fun reconstructor_command reconstr i n (ls, ss) =
243 using_labels ls ^ apply_on_subgoal i n ^
244 command_call (string_for_reconstructor reconstr) ss
245 fun minimize_line _ [] = ""
246 | minimize_line minimize_command ss =
247 case minimize_command ss of
250 "\nTo minimize: " ^ Markup.markup Isabelle_Markup.sendback command ^ "."
252 fun split_used_facts facts =
253 facts |> List.partition (fn (_, (sc, _)) => sc = Chained)
254 |> pairself (sort_distinct (string_ord o pairself fst))
256 fun one_line_proof_text (preplay, banner, used_facts, minimize_command,
257 subgoal, subgoal_count) =
259 val (chained, extra) = split_used_facts used_facts
260 val (failed, reconstr, ext_time) =
262 Played (reconstr, time) => (false, reconstr, (SOME (false, time)))
263 | Trust_Playable (reconstr, time) =>
268 if time = Time.zeroTime then NONE else SOME (true, time))
269 | Failed_to_Play reconstr => (true, reconstr, NONE)
272 |> reconstructor_command reconstr subgoal subgoal_count
274 enclose "One-line proof reconstruction failed: "
275 ".\n(Invoking \"sledgehammer\" with \"[strict]\" might \
278 try_command_line banner ext_time)
279 in try_line ^ minimize_line minimize_command (map fst (extra @ chained)) end
281 (** Hard-core proof reconstruction: structured Isar proofs **)
283 fun forall_of v t = HOLogic.all_const (fastype_of v) $ lambda v t
284 fun exists_of v t = HOLogic.exists_const (fastype_of v) $ lambda v t
286 fun make_tvar s = TVar (("'" ^ s, 0), HOLogic.typeS)
287 fun make_tfree ctxt w =
288 let val ww = "'" ^ w in
289 TFree (ww, the_default HOLogic.typeS (Variable.def_sort ctxt (ww, ~1)))
293 val no_label = ("", ~1)
296 val assum_prefix = "a"
297 val have_prefix = "f"
299 fun raw_label_for_name (num, ss) =
300 case resolve_conjecture ss of
301 [j] => (conjecture_prefix, j)
302 | _ => (raw_prefix ^ ascii_of num, 0)
304 (**** INTERPRETATION OF TSTP SYNTAX TREES ****)
306 exception HO_TERM of (string, string) ho_term list
307 exception FORMULA of (string, string, (string, string) ho_term) formula list
308 exception SAME of unit
310 (* Type variables are given the basic sort "HOL.type". Some will later be
311 constrained by information from type literals, or by type inference. *)
312 fun typ_from_atp ctxt (u as ATerm (a, us)) =
313 let val Ts = map (typ_from_atp ctxt) us in
314 case unprefix_and_unascii type_const_prefix a of
315 SOME b => Type (invert_const b, Ts)
317 if not (null us) then
318 raise HO_TERM [u] (* only "tconst"s have type arguments *)
319 else case unprefix_and_unascii tfree_prefix a of
320 SOME b => make_tfree ctxt b
322 (* Could be an Isabelle variable or a variable from the ATP, say "X1"
323 or "_5018". Sometimes variables from the ATP are indistinguishable
324 from Isabelle variables, which forces us to use a type parameter in
326 (a |> perhaps (unprefix_and_unascii tvar_prefix), HOLogic.typeS)
327 |> Type_Infer.param 0
330 (* Type class literal applied to a type. Returns triple of polarity, class,
332 fun type_constraint_from_term ctxt (u as ATerm (a, us)) =
333 case (unprefix_and_unascii class_prefix a, map (typ_from_atp ctxt) us) of
334 (SOME b, [T]) => (b, T)
335 | _ => raise HO_TERM [u]
337 (* Accumulate type constraints in a formula: negative type literals. *)
338 fun add_var (key, z) = Vartab.map_default (key, []) (cons z)
339 fun add_type_constraint false (cl, TFree (a ,_)) = add_var ((a, ~1), cl)
340 | add_type_constraint false (cl, TVar (ix, _)) = add_var (ix, cl)
341 | add_type_constraint _ _ = I
343 fun repair_variable_name f s =
345 fun subscript_name s n = s ^ nat_subscript n
346 val s = String.map f s
348 case space_explode "_" s of
349 [_] => (case take_suffix Char.isDigit (String.explode s) of
350 (cs1 as _ :: _, cs2 as _ :: _) =>
351 subscript_name (String.implode cs1)
352 (the (Int.fromString (String.implode cs2)))
354 | [s1, s2] => (case Int.fromString s2 of
355 SOME n => subscript_name s1 n
360 (* The number of type arguments of a constant, zero if it's monomorphic. For
361 (instances of) Skolem pseudoconstants, this information is encoded in the
363 fun num_type_args thy s =
364 if String.isPrefix skolem_const_prefix s then
365 s |> Long_Name.explode |> List.last |> Int.fromString |> the
366 else if String.isPrefix lam_lifted_prefix s then
367 if String.isPrefix lam_lifted_poly_prefix s then 2 else 0
369 (s, Sign.the_const_type thy s) |> Sign.const_typargs thy |> length
371 fun slack_fastype_of t = fastype_of t handle TERM _ => HOLogic.typeT
373 (* First-order translation. No types are known for variables. "HOLogic.typeT"
374 should allow them to be inferred. *)
375 fun term_from_atp ctxt textual sym_tab =
377 val thy = Proof_Context.theory_of ctxt
378 (* For Metis, we use 1 rather than 0 because variable references in clauses
379 may otherwise conflict with variable constraints in the goal. At least,
380 type inference often fails otherwise. See also "axiom_inference" in
381 "Metis_Reconstruct". *)
382 val var_index = if textual then 0 else 1
383 fun do_term extra_ts opt_T u =
386 if String.isPrefix native_type_prefix s then
387 @{const True} (* ignore TPTP type information *)
388 else if s = tptp_equal then
389 let val ts = map (do_term [] NONE) us in
390 if textual andalso length ts = 2 andalso
391 hd ts aconv List.last ts then
392 (* Vampire is keen on producing these. *)
395 list_comb (Const (@{const_name HOL.eq}, HOLogic.typeT), ts)
397 else case unprefix_and_unascii const_prefix s of
400 val ((s', s''), mangled_us) =
401 s' |> unmangled_const |>> `invert_const
403 if s' = type_tag_name then
404 case mangled_us @ us of
406 do_term extra_ts (SOME (typ_from_atp ctxt typ_u)) term_u
407 | _ => raise HO_TERM us
408 else if s' = predicator_name then
409 do_term [] (SOME @{typ bool}) (hd us)
410 else if s' = app_op_name then
411 let val extra_t = do_term [] NONE (List.last us) in
412 do_term (extra_t :: extra_ts)
414 SOME T => SOME (slack_fastype_of extra_t --> T)
416 (nth us (length us - 2))
418 else if s' = type_guard_name then
419 @{const True} (* ignore type predicates *)
422 val new_skolem = String.isPrefix new_skolem_const_prefix s''
424 length us - the_default 0 (Symtab.lookup sym_tab s)
425 val (type_us, term_us) =
426 chop num_ty_args us |>> append mangled_us
427 val term_ts = map (do_term [] NONE) term_us
429 (if not (null type_us) andalso
430 num_type_args thy s' = length type_us then
431 let val Ts = type_us |> map (typ_from_atp ctxt) in
433 SOME (Type_Infer.paramify_vars (tl Ts ---> hd Ts))
435 try (Sign.const_instance thy) (s', Ts)
442 | NONE => map slack_fastype_of term_ts --->
445 | NONE => HOLogic.typeT))
448 Var ((new_skolem_var_name_from_const s'', var_index), T)
450 Const (unproxify_const s', T)
451 in list_comb (t, term_ts @ extra_ts) end
453 | NONE => (* a free or schematic variable *)
455 val term_ts = map (do_term [] NONE) us
456 val ts = term_ts @ extra_ts
459 SOME T => map slack_fastype_of term_ts ---> T
460 | NONE => map slack_fastype_of ts ---> HOLogic.typeT
462 case unprefix_and_unascii fixed_var_prefix s of
463 SOME s => Free (s, T)
465 case unprefix_and_unascii schematic_var_prefix s of
466 SOME s => Var ((s, var_index), T)
468 Var ((s |> textual ? repair_variable_name Char.toLower,
470 in list_comb (t, ts) end
473 fun term_from_atom ctxt textual sym_tab pos (u as ATerm (s, _)) =
474 if String.isPrefix class_prefix s then
475 add_type_constraint pos (type_constraint_from_term ctxt u)
476 #> pair @{const True}
478 pair (term_from_atp ctxt textual sym_tab (SOME @{typ bool}) u)
480 val combinator_table =
481 [(@{const_name Meson.COMBI}, @{thm Meson.COMBI_def [abs_def]}),
482 (@{const_name Meson.COMBK}, @{thm Meson.COMBK_def [abs_def]}),
483 (@{const_name Meson.COMBB}, @{thm Meson.COMBB_def [abs_def]}),
484 (@{const_name Meson.COMBC}, @{thm Meson.COMBC_def [abs_def]}),
485 (@{const_name Meson.COMBS}, @{thm Meson.COMBS_def [abs_def]})]
487 fun uncombine_term thy =
489 fun aux (t1 $ t2) = betapply (pairself aux (t1, t2))
490 | aux (Abs (s, T, t')) = Abs (s, T, aux t')
491 | aux (t as Const (x as (s, _))) =
492 (case AList.lookup (op =) combinator_table s of
493 SOME thm => thm |> prop_of |> specialize_type thy x
494 |> Logic.dest_equals |> snd
499 (* Update schematic type variables with detected sort constraints. It's not
500 totally clear whether this code is necessary. *)
501 fun repair_tvar_sorts (t, tvar_tab) =
503 fun do_type (Type (a, Ts)) = Type (a, map do_type Ts)
504 | do_type (TVar (xi, s)) =
505 TVar (xi, the_default s (Vartab.lookup tvar_tab xi))
506 | do_type (TFree z) = TFree z
507 fun do_term (Const (a, T)) = Const (a, do_type T)
508 | do_term (Free (a, T)) = Free (a, do_type T)
509 | do_term (Var (xi, T)) = Var (xi, do_type T)
510 | do_term (t as Bound _) = t
511 | do_term (Abs (a, T, t)) = Abs (a, do_type T, do_term t)
512 | do_term (t1 $ t2) = do_term t1 $ do_term t2
513 in t |> not (Vartab.is_empty tvar_tab) ? do_term end
515 fun quantify_over_var quant_of var_s t =
517 val vars = [] |> Term.add_vars t |> filter (fn ((s, _), _) => s = var_s)
519 in fold_rev quant_of vars t end
521 (* Interpret an ATP formula as a HOL term, extracting sort constraints as they
522 appear in the formula. *)
523 fun prop_from_atp ctxt textual sym_tab phi =
525 fun do_formula pos phi =
527 AQuant (_, [], phi) => do_formula pos phi
528 | AQuant (q, (s, _) :: xs, phi') =>
529 do_formula pos (AQuant (q, xs, phi'))
531 #>> quantify_over_var (case q of
533 | AExists => exists_of)
534 (s |> textual ? repair_variable_name Char.toLower)
535 | AConn (ANot, [phi']) => do_formula (not pos) phi' #>> s_not
536 | AConn (c, [phi1, phi2]) =>
537 do_formula (pos |> c = AImplies ? not) phi1
538 ##>> do_formula pos phi2
544 | ANot => raise Fail "impossible connective")
545 | AAtom tm => term_from_atom ctxt textual sym_tab pos tm
546 | _ => raise FORMULA [phi]
547 in repair_tvar_sorts (do_formula true phi Vartab.empty) end
549 fun infer_formula_types ctxt =
550 Type.constraint HOLogic.boolT
552 (Proof_Context.set_mode Proof_Context.mode_schematic ctxt)
554 fun uncombined_etc_prop_from_atp ctxt textual sym_tab =
555 let val thy = Proof_Context.theory_of ctxt in
556 prop_from_atp ctxt textual sym_tab
557 #> textual ? uncombine_term thy #> infer_formula_types ctxt
560 (**** Translation of TSTP files to Isar proofs ****)
562 fun unvarify_term (Var ((s, 0), T)) = Free (s, T)
563 | unvarify_term t = raise TERM ("unvarify_term: non-Var", [t])
565 fun decode_line sym_tab (Definition_Step (name, phi1, phi2)) ctxt =
567 val thy = Proof_Context.theory_of ctxt
568 val t1 = prop_from_atp ctxt true sym_tab phi1
569 val vars = snd (strip_comb t1)
570 val frees = map unvarify_term vars
571 val unvarify_args = subst_atomic (vars ~~ frees)
572 val t2 = prop_from_atp ctxt true sym_tab phi2
574 HOLogic.eq_const HOLogic.typeT $ t1 $ t2
575 |> unvarify_args |> uncombine_term thy |> infer_formula_types ctxt
578 (Definition_Step (name, t1, t2),
579 fold Variable.declare_term (maps Misc_Legacy.term_frees [t1, t2]) ctxt)
581 | decode_line sym_tab (Inference_Step (name, u, rule, deps)) ctxt =
582 let val t = u |> uncombined_etc_prop_from_atp ctxt true sym_tab in
583 (Inference_Step (name, t, rule, deps),
584 fold Variable.declare_term (Misc_Legacy.term_frees t) ctxt)
586 fun decode_lines ctxt sym_tab lines =
587 fst (fold_map (decode_line sym_tab) lines ctxt)
589 fun is_same_inference _ (Definition_Step _) = false
590 | is_same_inference t (Inference_Step (_, t', _, _)) = t aconv t'
592 (* No "real" literals means only type information (tfree_tcs, clsrel, or
594 fun is_only_type_information t = t aconv @{term True}
596 fun replace_one_dependency (old, new) dep =
597 if is_same_atp_step dep old then new else [dep]
598 fun replace_dependencies_in_line _ (line as Definition_Step _) = line
599 | replace_dependencies_in_line p (Inference_Step (name, t, rule, deps)) =
600 Inference_Step (name, t, rule,
601 fold (union (op =) o replace_one_dependency p) deps [])
603 (* Discard facts; consolidate adjacent lines that prove the same formula, since
604 they differ only in type information.*)
605 fun add_line _ (line as Definition_Step _) lines = line :: lines
606 | add_line fact_names (Inference_Step (name as (_, ss), t, rule, [])) lines =
607 (* No dependencies: fact, conjecture, or (for Vampire) internal facts or
609 if is_fact fact_names ss then
610 (* Facts are not proof lines. *)
611 if is_only_type_information t then
612 map (replace_dependencies_in_line (name, [])) lines
613 (* Is there a repetition? If so, replace later line by earlier one. *)
614 else case take_prefix (not o is_same_inference t) lines of
615 (_, []) => lines (* no repetition of proof line *)
616 | (pre, Inference_Step (name', _, _, _) :: post) =>
617 pre @ map (replace_dependencies_in_line (name', [name])) post
618 | _ => raise Fail "unexpected inference"
619 else if is_conjecture ss then
620 Inference_Step (name, t, rule, []) :: lines
622 map (replace_dependencies_in_line (name, [])) lines
623 | add_line _ (Inference_Step (name, t, rule, deps)) lines =
624 (* Type information will be deleted later; skip repetition test. *)
625 if is_only_type_information t then
626 Inference_Step (name, t, rule, deps) :: lines
627 (* Is there a repetition? If so, replace later line by earlier one. *)
628 else case take_prefix (not o is_same_inference t) lines of
629 (* FIXME: Doesn't this code risk conflating proofs involving different
631 (_, []) => Inference_Step (name, t, rule, deps) :: lines
632 | (pre, Inference_Step (name', t', rule, _) :: post) =>
633 Inference_Step (name, t', rule, deps) ::
634 pre @ map (replace_dependencies_in_line (name', [name])) post
635 | _ => raise Fail "unexpected inference"
637 val repair_waldmeister_endgame =
639 fun do_tail (Inference_Step (name, t, rule, deps)) =
640 Inference_Step (name, s_not t, rule, deps)
641 | do_tail line = line
643 | do_body ((line as Inference_Step ((_, ss), _, _, _)) :: lines) =
644 if is_conjecture ss then map do_tail (line :: lines)
645 else line :: do_body lines
646 | do_body (line :: lines) = line :: do_body lines
649 (* Recursively delete empty lines (type information) from the proof. *)
650 fun add_nontrivial_line (line as Inference_Step (name, t, _, [])) lines =
651 if is_only_type_information t then delete_dependency name lines
653 | add_nontrivial_line line lines = line :: lines
654 and delete_dependency name lines =
655 fold_rev add_nontrivial_line
656 (map (replace_dependencies_in_line (name, [])) lines) []
658 (* ATPs sometimes reuse free variable names in the strangest ways. Removing
659 offending lines often does the trick. *)
660 fun is_bad_free frees (Free x) = not (member (op =) frees x)
661 | is_bad_free _ _ = false
663 fun add_desired_line _ _ _ (line as Definition_Step (name, _, _)) (j, lines) =
664 (j, line :: map (replace_dependencies_in_line (name, [])) lines)
665 | add_desired_line isar_shrink_factor fact_names frees
666 (Inference_Step (name as (_, ss), t, rule, deps)) (j, lines) =
668 if is_fact fact_names ss orelse
669 is_conjecture ss orelse
670 (* the last line must be kept *)
672 (not (is_only_type_information t) andalso
673 null (Term.add_tvars t []) andalso
674 not (exists_subterm (is_bad_free frees) t) andalso
675 length deps >= 2 andalso j mod isar_shrink_factor = 0 andalso
676 (* kill next to last line, which usually results in a trivial step *)
678 Inference_Step (name, t, rule, deps) :: lines (* keep line *)
680 map (replace_dependencies_in_line (name, deps)) lines) (* drop line *)
682 (** Isar proof construction and manipulation **)
684 type label = string * int
685 type facts = label list * string list
687 datatype isar_qualifier = Show | Then | Moreover | Ultimately
690 Fix of (string * typ) list |
692 Assume of label * term |
693 Prove of isar_qualifier list * label * term * byline
696 Case_Split of isar_step list list * facts
698 fun add_fact_from_dependency fact_names (name as (_, ss)) =
699 if is_fact fact_names ss then
700 apsnd (union (op =) (map fst (resolve_fact fact_names ss)))
702 apfst (insert (op =) (raw_label_for_name name))
704 fun repair_name "$true" = "c_True"
705 | repair_name "$false" = "c_False"
706 | repair_name "$$e" = tptp_equal (* seen in Vampire proofs *)
708 if is_tptp_equal s orelse
709 (* seen in Vampire proofs *)
710 (String.isPrefix "sQ" s andalso String.isSuffix "_eqProxy" s) then
715 (* FIXME: Still needed? Try with SPASS proofs perhaps. *)
716 val kill_duplicate_assumptions_in_proof =
718 fun relabel_facts subst =
719 apfst (map (fn l => AList.lookup (op =) subst l |> the_default l))
720 fun do_step (step as Assume (l, t)) (proof, subst, assums) =
721 (case AList.lookup (op aconv) assums t of
722 SOME l' => (proof, (l, l') :: subst, assums)
723 | NONE => (step :: proof, subst, (t, l) :: assums))
724 | do_step (Prove (qs, l, t, by)) (proof, subst, assums) =
727 By_Metis facts => By_Metis (relabel_facts subst facts)
728 | Case_Split (proofs, facts) =>
729 Case_Split (map do_proof proofs,
730 relabel_facts subst facts)) ::
731 proof, subst, assums)
732 | do_step step (proof, subst, assums) = (step :: proof, subst, assums)
733 and do_proof proof = fold do_step proof ([], [], []) |> #1 |> rev
736 fun used_labels_of_step (Prove (_, _, _, by)) =
738 By_Metis (ls, _) => ls
739 | Case_Split (proofs, (ls, _)) =>
740 fold (union (op =) o used_labels_of) proofs ls)
741 | used_labels_of_step _ = []
742 and used_labels_of proof = fold (union (op =) o used_labels_of_step) proof []
744 fun kill_useless_labels_in_proof proof =
746 val used_ls = used_labels_of proof
747 fun do_label l = if member (op =) used_ls l then l else no_label
748 fun do_step (Assume (l, t)) = Assume (do_label l, t)
749 | do_step (Prove (qs, l, t, by)) =
750 Prove (qs, do_label l, t,
752 Case_Split (proofs, facts) =>
753 Case_Split (map (map do_step) proofs, facts)
755 | do_step step = step
756 in map do_step proof end
758 fun prefix_for_depth n = replicate_string (n + 1)
762 fun aux _ _ _ [] = []
763 | aux subst depth (next_assum, next_fact) (Assume (l, t) :: proof) =
765 Assume (l, t) :: aux subst depth (next_assum, next_fact) proof
767 let val l' = (prefix_for_depth depth assum_prefix, next_assum) in
769 aux ((l, l') :: subst) depth (next_assum + 1, next_fact) proof
771 | aux subst depth (next_assum, next_fact)
772 (Prove (qs, l, t, by) :: proof) =
774 val (l', subst, next_fact) =
776 (l, subst, next_fact)
779 val l' = (prefix_for_depth depth have_prefix, next_fact)
780 in (l', (l, l') :: subst, next_fact + 1) end
782 apfst (maps (the_list o AList.lookup (op =) subst))
785 By_Metis facts => By_Metis (relabel_facts facts)
786 | Case_Split (proofs, facts) =>
787 Case_Split (map (aux subst (depth + 1) (1, 1)) proofs,
790 Prove (qs, l', t, by) :: aux subst depth (next_assum, next_fact) proof
792 | aux subst depth nextp (step :: proof) =
793 step :: aux subst depth nextp proof
794 in aux [] 0 (1, 1) end
796 fun string_for_proof ctxt0 type_enc lam_trans i n =
799 (* FIXME: Implement proper handling of type constraints:
800 |> Config.put show_free_types false
801 |> Config.put show_types false
802 |> Config.put show_sorts false
804 fun fix_print_mode f x =
805 Print_Mode.setmp (filter (curry (op =) Symbol.xsymbolsN)
806 (print_mode_value ())) f x
807 fun do_indent ind = replicate_string (ind * indent_size) " "
809 maybe_quote s ^ " :: " ^
810 maybe_quote (fix_print_mode (Syntax.string_of_typ ctxt) T)
811 fun do_label l = if l = no_label then "" else string_for_label l ^ ": "
813 (if member (op =) qs Moreover then "moreover " else "") ^
814 (if member (op =) qs Ultimately then "ultimately " else "") ^
815 (if member (op =) qs Then then
816 if member (op =) qs Show then "thus" else "hence"
818 if member (op =) qs Show then "show" else "have")
819 val do_term = maybe_quote o fix_print_mode (Syntax.string_of_term ctxt)
820 val reconstr = Metis (type_enc, lam_trans)
821 fun do_facts (ls, ss) =
822 reconstructor_command reconstr 1 1
823 (ls |> sort_distinct (prod_ord string_ord int_ord),
824 ss |> sort_distinct string_ord)
825 and do_step ind (Fix xs) =
826 do_indent ind ^ "fix " ^ space_implode " and " (map do_free xs) ^ "\n"
827 | do_step ind (Let (t1, t2)) =
828 do_indent ind ^ "let " ^ do_term t1 ^ " = " ^ do_term t2 ^ "\n"
829 | do_step ind (Assume (l, t)) =
830 do_indent ind ^ "assume " ^ do_label l ^ do_term t ^ "\n"
831 | do_step ind (Prove (qs, l, t, By_Metis facts)) =
832 do_indent ind ^ do_have qs ^ " " ^
833 do_label l ^ do_term t ^ " " ^ do_facts facts ^ "\n"
834 | do_step ind (Prove (qs, l, t, Case_Split (proofs, facts))) =
835 implode (map (prefix (do_indent ind ^ "moreover\n") o do_block ind)
837 do_indent ind ^ do_have qs ^ " " ^ do_label l ^ do_term t ^ " " ^
838 do_facts facts ^ "\n"
839 and do_steps prefix suffix ind steps =
840 let val s = implode (map (do_step ind) steps) in
841 replicate_string (ind * indent_size - size prefix) " " ^ prefix ^
842 String.extract (s, ind * indent_size,
843 SOME (size s - ind * indent_size - 1)) ^
846 and do_block ind proof = do_steps "{ " " }" (ind + 1) proof
847 (* One-step proofs are pointless; better use the Metis one-liner
849 and do_proof [Prove (_, _, _, By_Metis _)] = ""
851 (if i <> 1 then "prefer " ^ string_of_int i ^ "\n" else "") ^
852 do_indent 0 ^ "proof -\n" ^ do_steps "" "" 1 proof ^ do_indent 0 ^
853 (if n <> 1 then "next" else "qed")
856 fun isar_proof_text ctxt isar_proof_requested
857 (debug, isar_shrink_factor, pool, fact_names, sym_tab, atp_proof, goal)
858 (one_line_params as (_, _, _, _, subgoal, subgoal_count)) =
860 val isar_shrink_factor =
861 (if isar_proof_requested then 1 else 2) * isar_shrink_factor
862 val (params, hyp_ts, concl_t) = strip_subgoal ctxt goal subgoal
863 val frees = fold Term.add_frees (concl_t :: hyp_ts) []
864 val one_line_proof = one_line_proof_text one_line_params
866 if is_typed_helper_used_in_atp_proof atp_proof then full_typesN
868 val lam_trans = lam_trans_from_atp_proof atp_proof metis_default_lam_trans
870 fun isar_proof_of () =
874 |> clean_up_atp_proof_dependencies
875 |> nasty_atp_proof pool
876 |> map_term_names_in_atp_proof repair_name
877 |> decode_lines ctxt sym_tab
878 |> rpair [] |-> fold_rev (add_line fact_names)
879 |> repair_waldmeister_endgame
880 |> rpair [] |-> fold_rev add_nontrivial_line
882 |-> fold_rev (add_desired_line isar_shrink_factor fact_names frees)
884 val conj_name = conjecture_prefix ^ string_of_int (length hyp_ts)
887 |> map_filter (fn Inference_Step (name as (_, ss), _, _, []) =>
888 if member (op =) ss conj_name then SOME name else NONE
890 fun dep_of_step (Definition_Step _) = NONE
891 | dep_of_step (Inference_Step (name, _, _, from)) = SOME (from, name)
892 val ref_graph = atp_proof |> map_filter dep_of_step |> make_ref_graph
893 val axioms = axioms_of_ref_graph ref_graph conjs
894 val tainted = tainted_atoms_of_ref_graph ref_graph conjs
897 |> fold (fn Definition_Step _ => I (* FIXME *)
898 | Inference_Step ((s, _), t, _, _) =>
899 Symtab.update_new (s,
900 t |> fold forall_of (map Var (Term.add_vars t []))
901 |> member (op = o apsnd fst) tainted s ? s_not))
903 fun prop_of_clause c =
904 fold (curry s_disj) (map_filter (Symtab.lookup props o fst) c)
906 fun label_of_clause [name] = raw_label_for_name name
907 | label_of_clause c = (space_implode "___" (map fst c), 0)
908 fun maybe_show outer c =
909 (outer andalso length c = 1 andalso subset (op =) (c, conjs))
911 fun do_have outer qs (gamma, c) =
912 Prove (maybe_show outer c qs, label_of_clause c, prop_of_clause c,
913 By_Metis (fold (add_fact_from_dependency fact_names
914 o the_single) gamma ([], [])))
915 fun do_inf outer (Have z) = do_have outer [] z
916 | do_inf outer (Hence z) = do_have outer [Then] z
917 | do_inf outer (Cases cases) =
918 let val c = succedent_of_cases cases in
919 Prove (maybe_show outer c [Ultimately], label_of_clause c,
921 Case_Split (map (do_case false) cases, ([], [])))
923 and do_case outer (c, infs) =
924 Assume (label_of_clause c, prop_of_clause c) ::
925 map (do_inf outer) infs
927 (if null params then [] else [Fix params]) @
929 |> redirect_graph axioms tainted
930 |> chain_direct_proof
932 |> kill_duplicate_assumptions_in_proof
933 |> kill_useless_labels_in_proof
935 |> string_for_proof ctxt type_enc lam_trans subgoal subgoal_count
939 if isar_proof_requested then
940 "\nNo structured proof available (proof too short)."
944 "\n\n" ^ (if isar_proof_requested then "Structured proof"
945 else "Perhaps this will work") ^
946 ":\n" ^ Markup.markup Isabelle_Markup.sendback isar_proof
951 else case try isar_proof_of () of
953 | NONE => if isar_proof_requested then
954 "\nWarning: The Isar proof construction failed."
957 in one_line_proof ^ isar_proof end
959 fun proof_text ctxt isar_proof isar_params
960 (one_line_params as (preplay, _, _, _, _, _)) =
961 (if case preplay of Failed_to_Play _ => true | _ => isar_proof then
962 isar_proof_text ctxt isar_proof isar_params
964 one_line_proof_text) one_line_params