(* Title: HOL/Tools/ATP/atp_proof_reconstruct.ML
Author: Lawrence C. Paulson, Cambridge University Computer Laboratory
Author: Claire Quigley, Cambridge University Computer Laboratory
Author: Jasmin Blanchette, TU Muenchen
Proof reconstruction from ATP proofs.
*)
signature ATP_PROOF_RECONSTRUCT =
sig
type ('a, 'b) ho_term = ('a, 'b) ATP_Problem.ho_term
type ('a, 'b, 'c) formula = ('a, 'b, 'c) ATP_Problem.formula
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 * int * string Symtab.table * (string * stature) list vector
* int Symtab.table * string proof * thm
val metisN : string
val smtN : string
val full_typesN : string
val partial_typesN : string
val no_typesN : string
val really_full_type_enc : string
val full_type_enc : string
val partial_type_enc : string
val no_type_enc : string
val full_type_encs : string list
val partial_type_encs : string list
val metis_default_lam_trans : string
val metis_call : string -> string -> string
val string_for_reconstructor : reconstructor -> string
val used_facts_in_atp_proof :
Proof.context -> (string * stature) list vector -> string proof
-> (string * stature) 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_unsound_atp_proof :
Proof.context -> (string * stature) list vector -> 'a proof
-> string list option
val unalias_type_enc : string -> string list
val one_line_proof_text : one_line_params -> string
val make_tvar : string -> typ
val make_tfree : Proof.context -> string -> typ
val term_from_atp :
Proof.context -> bool -> int Symtab.table -> typ option
-> (string, string) ho_term -> term
val prop_from_atp :
Proof.context -> bool -> int Symtab.table
-> (string, string, (string, string) ho_term) formula -> term
val isar_proof_text :
Proof.context -> bool -> isar_params -> one_line_params -> string
val proof_text :
Proof.context -> bool -> isar_params -> one_line_params -> string
end;
structure ATP_Proof_Reconstruct : ATP_PROOF_RECONSTRUCT =
struct
open ATP_Util
open ATP_Problem
open ATP_Proof
open ATP_Problem_Generate
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
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 * int * string Symtab.table * (string * stature) list vector
* int Symtab.table * string proof * thm
val metisN = "metis"
val smtN = "smt"
val full_typesN = "full_types"
val partial_typesN = "partial_types"
val no_typesN = "no_types"
val really_full_type_enc = "mono_tags"
val full_type_enc = "poly_guards_query"
val partial_type_enc = "poly_args"
val no_type_enc = "erased"
val full_type_encs = [full_type_enc, really_full_type_enc]
val partial_type_encs = partial_type_enc :: full_type_encs
val type_enc_aliases =
[(full_typesN, full_type_encs),
(partial_typesN, partial_type_encs),
(no_typesN, [no_type_enc])]
fun unalias_type_enc s =
AList.lookup (op =) type_enc_aliases s |> the_default [s]
val metis_default_lam_trans = combsN
fun metis_call type_enc lam_trans =
let
val type_enc =
case AList.find (fn (enc, encs) => enc = hd encs) type_enc_aliases
type_enc of
[alias] => alias
| _ => type_enc
val opts = [] |> type_enc <> partial_typesN ? cons type_enc
|> lam_trans <> metis_default_lam_trans ? cons lam_trans
in metisN ^ (if null opts then "" else " (" ^ commas opts ^ ")") end
fun string_for_reconstructor (Metis (type_enc, lam_trans)) =
metis_call type_enc lam_trans
| string_for_reconstructor SMT = smtN
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
fun is_axiom_used_in_proof pred =
exists (fn Inference_Step ((_, ss), _, _, []) => exists pred ss | _ => false)
val is_combinator_def = String.isPrefix (helper_prefix ^ combinator_prefix)
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
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
val leo2_ext = "extcnf_equal_neg"
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
fun add_fact ctxt fact_names (Inference_Step ((_, ss), _, rule, deps)) =
(if rule = leo2_ext orelse rule = satallax_unsat_coreN then
insert (op =) (ext_name ctxt, (Global, General))
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
(** Soft-core proof reconstruction: one-liners **)
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 ^ ")"
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 command_call name [] =
name |> not (Lexicon.is_identifier name) ? enclose "(" ")"
| command_call name args = "(" ^ name ^ " " ^ space_implode " " args ^ ")"
fun try_command_line banner time command =
banner ^ ": " ^ Markup.markup Isabelle_Markup.sendback command ^ show_time time ^ "."
fun using_labels [] = ""
| using_labels ls =
"using " ^ space_implode " " (map string_for_label ls) ^ " "
fun reconstructor_command reconstr i n (ls, ss) =
using_labels ls ^ apply_on_subgoal i n ^
command_call (string_for_reconstructor reconstr) ss
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))
fun one_line_proof_text (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
|> (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
(** Hard-core proof reconstruction: structured Isar proofs **)
fun forall_of v t = HOLogic.all_const (fastype_of v) $ lambda v t
fun exists_of v t = HOLogic.exists_const (fastype_of v) $ lambda v t
fun make_tvar s = TVar (("'" ^ s, 0), HOLogic.typeS)
fun make_tfree ctxt w =
let val ww = "'" ^ w in
TFree (ww, the_default HOLogic.typeS (Variable.def_sort ctxt (ww, ~1)))
end
val indent_size = 2
val no_label = ("", ~1)
val raw_prefix = "x"
val assum_prefix = "a"
val have_prefix = "f"
fun raw_label_for_name (num, ss) =
case resolve_conjecture ss of
[j] => (conjecture_prefix, j)
| _ => (raw_prefix ^ ascii_of num, 0)
(**** INTERPRETATION OF TSTP SYNTAX TREES ****)
exception HO_TERM of (string, string) ho_term list
exception FORMULA of (string, string, (string, string) ho_term) formula list
exception SAME of unit
(* Type variables are given the basic sort "HOL.type". Some will later be
constrained by information from type literals, or by type inference. *)
fun typ_from_atp ctxt (u as ATerm (a, us)) =
let val Ts = map (typ_from_atp ctxt) us in
case unprefix_and_unascii type_const_prefix a of
SOME b => Type (invert_const b, Ts)
| NONE =>
if not (null us) then
raise HO_TERM [u] (* only "tconst"s have type arguments *)
else case unprefix_and_unascii tfree_prefix a of
SOME b => make_tfree ctxt b
| NONE =>
(* Could be an Isabelle variable or a variable from the ATP, say "X1"
or "_5018". Sometimes variables from the ATP are indistinguishable
from Isabelle variables, which forces us to use a type parameter in
all cases. *)
(a |> perhaps (unprefix_and_unascii tvar_prefix), HOLogic.typeS)
|> Type_Infer.param 0
end
(* Type class literal applied to a type. Returns triple of polarity, class,
type. *)
fun type_constraint_from_term ctxt (u as ATerm (a, us)) =
case (unprefix_and_unascii class_prefix a, map (typ_from_atp ctxt) us) of
(SOME b, [T]) => (b, T)
| _ => raise HO_TERM [u]
(* Accumulate type constraints in a formula: negative type literals. *)
fun add_var (key, z) = Vartab.map_default (key, []) (cons z)
fun add_type_constraint false (cl, TFree (a ,_)) = add_var ((a, ~1), cl)
| add_type_constraint false (cl, TVar (ix, _)) = add_var (ix, cl)
| add_type_constraint _ _ = I
fun repair_variable_name f s =
let
fun subscript_name s n = s ^ nat_subscript n
val s = String.map f s
in
case space_explode "_" s of
[_] => (case take_suffix Char.isDigit (String.explode s) of
(cs1 as _ :: _, cs2 as _ :: _) =>
subscript_name (String.implode cs1)
(the (Int.fromString (String.implode cs2)))
| (_, _) => s)
| [s1, s2] => (case Int.fromString s2 of
SOME n => subscript_name s1 n
| NONE => s)
| _ => s
end
(* The number of type arguments of a constant, zero if it's monomorphic. For
(instances of) Skolem pseudoconstants, this information is encoded in the
constant name. *)
fun num_type_args thy s =
if String.isPrefix skolem_const_prefix s then
s |> Long_Name.explode |> List.last |> Int.fromString |> the
else if String.isPrefix lam_lifted_prefix s then
if String.isPrefix lam_lifted_poly_prefix s then 2 else 0
else
(s, Sign.the_const_type thy s) |> Sign.const_typargs thy |> length
fun slack_fastype_of t = fastype_of t handle TERM _ => HOLogic.typeT
(* First-order translation. No types are known for variables. "HOLogic.typeT"
should allow them to be inferred. *)
fun term_from_atp ctxt textual sym_tab =
let
val thy = Proof_Context.theory_of ctxt
(* For Metis, we use 1 rather than 0 because variable references in clauses
may otherwise conflict with variable constraints in the goal. At least,
type inference often fails otherwise. See also "axiom_inference" in
"Metis_Reconstruct". *)
val var_index = if textual then 0 else 1
fun do_term extra_ts opt_T u =
case u of
ATerm (s, us) =>
if String.isPrefix native_type_prefix s then
@{const True} (* ignore TPTP type information *)
else if s = tptp_equal then
let val ts = map (do_term [] NONE) us in
if textual andalso length ts = 2 andalso
hd ts aconv List.last ts then
(* Vampire is keen on producing these. *)
@{const True}
else
list_comb (Const (@{const_name HOL.eq}, HOLogic.typeT), ts)
end
else case unprefix_and_unascii const_prefix s of
SOME s' =>
let
val ((s', s''), mangled_us) =
s' |> unmangled_const |>> `invert_const
in
if s' = type_tag_name then
case mangled_us @ us of
[typ_u, term_u] =>
do_term extra_ts (SOME (typ_from_atp ctxt typ_u)) term_u
| _ => raise HO_TERM us
else if s' = predicator_name then
do_term [] (SOME @{typ bool}) (hd us)
else if s' = app_op_name then
let val extra_t = do_term [] NONE (List.last us) in
do_term (extra_t :: extra_ts)
(case opt_T of
SOME T => SOME (slack_fastype_of extra_t --> T)
| NONE => NONE)
(nth us (length us - 2))
end
else if s' = type_guard_name then
@{const True} (* ignore type predicates *)
else
let
val new_skolem = String.isPrefix new_skolem_const_prefix s''
val num_ty_args =
length us - the_default 0 (Symtab.lookup sym_tab s)
val (type_us, term_us) =
chop num_ty_args us |>> append mangled_us
val term_ts = map (do_term [] NONE) term_us
val T =
(if not (null type_us) andalso
num_type_args thy s' = length type_us then
let val Ts = type_us |> map (typ_from_atp ctxt) in
if new_skolem then
SOME (Type_Infer.paramify_vars (tl Ts ---> hd Ts))
else if textual then
try (Sign.const_instance thy) (s', Ts)
else
NONE
end
else
NONE)
|> (fn SOME T => T
| NONE => map slack_fastype_of term_ts --->
(case opt_T of
SOME T => T
| NONE => HOLogic.typeT))
val t =
if new_skolem then
Var ((new_skolem_var_name_from_const s'', var_index), T)
else
Const (unproxify_const s', T)
in list_comb (t, term_ts @ extra_ts) end
end
| NONE => (* a free or schematic variable *)
let
val term_ts = map (do_term [] NONE) us
val ts = term_ts @ extra_ts
val T =
case opt_T of
SOME T => map slack_fastype_of term_ts ---> T
| NONE => map slack_fastype_of ts ---> HOLogic.typeT
val t =
case unprefix_and_unascii fixed_var_prefix s of
SOME s => Free (s, T)
| NONE =>
case unprefix_and_unascii schematic_var_prefix s of
SOME s => Var ((s, var_index), T)
| NONE =>
Var ((s |> textual ? repair_variable_name Char.toLower,
var_index), T)
in list_comb (t, ts) end
in do_term [] end
fun term_from_atom ctxt textual sym_tab pos (u as ATerm (s, _)) =
if String.isPrefix class_prefix s then
add_type_constraint pos (type_constraint_from_term ctxt u)
#> pair @{const True}
else
pair (term_from_atp ctxt textual sym_tab (SOME @{typ bool}) u)
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
(* Update schematic type variables with detected sort constraints. It's not
totally clear whether this code is necessary. *)
fun repair_tvar_sorts (t, tvar_tab) =
let
fun do_type (Type (a, Ts)) = Type (a, map do_type Ts)
| do_type (TVar (xi, s)) =
TVar (xi, the_default s (Vartab.lookup tvar_tab xi))
| do_type (TFree z) = TFree z
fun do_term (Const (a, T)) = Const (a, do_type T)
| do_term (Free (a, T)) = Free (a, do_type T)
| do_term (Var (xi, T)) = Var (xi, do_type T)
| do_term (t as Bound _) = t
| do_term (Abs (a, T, t)) = Abs (a, do_type T, do_term t)
| do_term (t1 $ t2) = do_term t1 $ do_term t2
in t |> not (Vartab.is_empty tvar_tab) ? do_term end
fun quantify_over_var quant_of var_s t =
let
val vars = [] |> Term.add_vars t |> filter (fn ((s, _), _) => s = var_s)
|> map Var
in fold_rev quant_of vars t end
(* Interpret an ATP formula as a HOL term, extracting sort constraints as they
appear in the formula. *)
fun prop_from_atp ctxt textual sym_tab phi =
let
fun do_formula pos phi =
case phi of
AQuant (_, [], phi) => do_formula pos phi
| AQuant (q, (s, _) :: xs, phi') =>
do_formula pos (AQuant (q, xs, phi'))
(* FIXME: TFF *)
#>> quantify_over_var (case q of
AForall => forall_of
| AExists => exists_of)
(s |> textual ? repair_variable_name Char.toLower)
| AConn (ANot, [phi']) => do_formula (not pos) phi' #>> s_not
| AConn (c, [phi1, phi2]) =>
do_formula (pos |> c = AImplies ? not) phi1
##>> do_formula pos phi2
#>> (case c of
AAnd => s_conj
| AOr => s_disj
| AImplies => s_imp
| AIff => s_iff
| ANot => raise Fail "impossible connective")
| AAtom tm => term_from_atom ctxt textual sym_tab pos tm
| _ => raise FORMULA [phi]
in repair_tvar_sorts (do_formula true phi Vartab.empty) end
fun infer_formula_types ctxt =
Type.constraint HOLogic.boolT
#> Syntax.check_term
(Proof_Context.set_mode Proof_Context.mode_schematic ctxt)
fun uncombined_etc_prop_from_atp ctxt textual sym_tab =
let val thy = Proof_Context.theory_of ctxt in
prop_from_atp ctxt textual sym_tab
#> textual ? uncombine_term thy #> infer_formula_types ctxt
end
(**** Translation of TSTP files to Isar proofs ****)
fun unvarify_term (Var ((s, 0), T)) = Free (s, T)
| unvarify_term t = raise TERM ("unvarify_term: non-Var", [t])
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, u, rule, deps)) ctxt =
let val t = u |> uncombined_etc_prop_from_atp ctxt true sym_tab in
(Inference_Step (name, 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 is_same_inference _ (Definition_Step _) = false
| is_same_inference t (Inference_Step (_, t', _, _)) = t aconv t'
(* No "real" literals means only type information (tfree_tcs, clsrel, or
clsarity). *)
fun is_only_type_information t = t aconv @{term True}
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, t, rule, deps)) =
Inference_Step (name, t, rule,
fold (union (op =) o replace_one_dependency p) deps [])
(* 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), 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, t, rule, []) :: lines
else
map (replace_dependencies_in_line (name, [])) lines
| add_line _ (Inference_Step (name, t, rule, deps)) lines =
(* Type information will be deleted later; skip repetition test. *)
if is_only_type_information t then
Inference_Step (name, 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, t, rule, deps) :: lines
| (pre, Inference_Step (name', t', rule, _) :: post) =>
Inference_Step (name, 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, t, rule, deps)) =
Inference_Step (name, 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 isar_shrink_factor fact_names frees
(Inference_Step (name as (_, ss), 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 j mod isar_shrink_factor = 0 andalso
(* kill next to last line, which usually results in a trivial step *)
j <> 1) then
Inference_Step (name, t, rule, deps) :: lines (* keep line *)
else
map (replace_dependencies_in_line (name, deps)) lines) (* drop line *)
(** 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
fun add_fact_from_dependency fact_names (name as (_, ss)) =
if is_fact fact_names ss then
apsnd (union (op =) (map fst (resolve_fact fact_names ss)))
else
apfst (insert (op =) (raw_label_for_name name))
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
(* FIXME: Still needed? Try with SPASS proofs perhaps. *)
val kill_duplicate_assumptions_in_proof =
let
fun relabel_facts subst =
apfst (map (fn l => AList.lookup (op =) subst l |> the_default l))
fun do_step (step as Assume (l, t)) (proof, subst, assums) =
(case AList.lookup (op aconv) assums t of
SOME l' => (proof, (l, l') :: subst, assums)
| NONE => (step :: proof, subst, (t, l) :: assums))
| do_step (Prove (qs, l, t, by)) (proof, subst, assums) =
(Prove (qs, l, t,
case by of
By_Metis facts => By_Metis (relabel_facts subst facts)
| Case_Split (proofs, facts) =>
Case_Split (map do_proof proofs,
relabel_facts subst facts)) ::
proof, subst, assums)
| do_step step (proof, subst, assums) = (step :: proof, subst, assums)
and do_proof proof = fold do_step proof ([], [], []) |> #1 |> rev
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_fact) (Assume (l, t) :: proof) =
if l = no_label then
Assume (l, t) :: aux subst depth (next_assum, next_fact) proof
else
let val l' = (prefix_for_depth depth assum_prefix, next_assum) in
Assume (l', t) ::
aux ((l, l') :: subst) depth (next_assum + 1, next_fact) proof
end
| aux subst depth (next_assum, next_fact)
(Prove (qs, l, t, by) :: proof) =
let
val (l', subst, next_fact) =
if l = no_label then
(l, subst, next_fact)
else
let
val l' = (prefix_for_depth depth have_prefix, next_fact)
in (l', (l, l') :: subst, next_fact + 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_fact) proof
end
| aux subst depth nextp (step :: proof) =
step :: aux subst depth nextp proof
in aux [] 0 (1, 1) end
fun string_for_proof ctxt0 type_enc lam_trans i n =
let
val ctxt = ctxt0
(* FIXME: Implement proper handling of type constraints:
|> Config.put show_free_types false
|> Config.put show_types false
|> Config.put show_sorts false
*)
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)
val reconstr = Metis (type_enc, lam_trans)
fun do_facts (ls, ss) =
reconstructor_command reconstr 1 1
(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 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 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 isar_proof_text ctxt isar_proof_requested
(debug, isar_shrink_factor, pool, fact_names, sym_tab, atp_proof, goal)
(one_line_params as (_, _, _, _, subgoal, subgoal_count)) =
let
val isar_shrink_factor =
(if isar_proof_requested then 1 else 2) * isar_shrink_factor
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 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
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 isar_shrink_factor 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)
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 ((s, _), t, _, _) =>
Symtab.update_new (s,
t |> fold forall_of (map Var (Term.add_vars t []))
|> member (op = o apsnd fst) tainted s ? s_not))
atp_proof
fun prop_of_clause c =
fold (curry s_disj) (map_filter (Symtab.lookup props o fst) c)
@{term False}
fun label_of_clause [name] = raw_label_for_name name
| label_of_clause c = (space_implode "___" (map fst c), 0)
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_dependency fact_names
o the_single) gamma ([], [])))
fun do_inf outer (Have z) = do_have outer [] z
| do_inf outer (Hence z) = do_have outer [Then] 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 isar_proof =
(if null params then [] else [Fix params]) @
(ref_graph
|> redirect_graph axioms tainted
|> chain_direct_proof
|> map (do_inf true)
|> kill_duplicate_assumptions_in_proof
|> kill_useless_labels_in_proof
|> relabel_proof)
|> string_for_proof ctxt type_enc lam_trans subgoal subgoal_count
in
case isar_proof of
"" =>
if isar_proof_requested then
"\nNo structured proof available (proof too short)."
else
""
| _ =>
"\n\n" ^ (if isar_proof_requested then "Structured proof"
else "Perhaps this will work") ^
":\n" ^ Markup.markup Isabelle_Markup.sendback isar_proof
end
val isar_proof =
if debug then
isar_proof_of ()
else case try isar_proof_of () of
SOME s => s
| NONE => if isar_proof_requested then
"\nWarning: The Isar proof construction failed."
else
""
in one_line_proof ^ isar_proof end
fun proof_text ctxt isar_proof isar_params
(one_line_params as (preplay, _, _, _, _, _)) =
(if case preplay of Failed_to_Play _ => true | _ => isar_proof then
isar_proof_text ctxt isar_proof isar_params
else
one_line_proof_text) one_line_params
end;