(* 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
Basic proof reconstruction from ATP proofs.
*)
signature ATP_PROOF_RECONSTRUCT =
sig
type 'a atp_type = 'a ATP_Problem.atp_type
type ('a, 'b) atp_term = ('a, 'b) ATP_Problem.atp_term
type ('a, 'b, 'c, 'd) atp_formula = ('a, 'b, 'c, 'd) ATP_Problem.atp_formula
type stature = ATP_Problem_Generate.stature
type atp_step_name = ATP_Proof.atp_step_name
type ('a, 'b) atp_step = ('a, 'b) ATP_Proof.atp_step
type 'a atp_proof = 'a ATP_Proof.atp_proof
val metisN : 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 default_metis_lam_trans : string
val leo2_extcnf_equal_neg_rule : string
val satallax_tab_rule_prefix : string
val forall_of : term -> term -> term
val exists_of : term -> term -> term
val simplify_bool : term -> term
val var_name_of_typ : typ -> string
val rename_bound_vars : term -> term
val type_enc_aliases : (string * string list) list
val unalias_type_enc : string -> string list
val term_of_atp : Proof.context -> ATP_Problem.atp_format -> ATP_Problem_Generate.type_enc ->
bool -> int Symtab.table -> typ option -> (string, string atp_type) atp_term -> term
val prop_of_atp : Proof.context -> ATP_Problem.atp_format -> ATP_Problem_Generate.type_enc ->
bool -> int Symtab.table ->
(string, string, (string, string atp_type) atp_term, string) atp_formula -> term
val used_facts_in_atp_proof : Proof.context -> (string * stature) list -> string atp_proof ->
(string * stature) list
val used_facts_in_unsound_atp_proof : Proof.context -> (string * stature) list -> 'a atp_proof ->
string list option
val atp_proof_prefers_lifting : string atp_proof -> bool
val is_typed_helper_used_in_atp_proof : string atp_proof -> bool
val replace_dependencies_in_line : atp_step_name * atp_step_name list -> ('a, 'b) atp_step ->
('a, 'b) atp_step
val termify_atp_proof : Proof.context -> string -> ATP_Problem.atp_format ->
ATP_Problem_Generate.type_enc -> string Symtab.table -> (string * term) list ->
int Symtab.table -> string atp_proof -> (term, string) atp_step list
val repair_waldmeister_endgame : (term, 'a) atp_step list -> (term, 'a) atp_step list
val infer_formulas_types : Proof.context -> term list -> term list
val introduce_spassy_skolems : (term, string) atp_step list -> (term, string) atp_step list
val factify_atp_proof : (string * 'a) list -> term list -> term -> (term, string) atp_step list ->
(term, string) atp_step list
end;
structure ATP_Proof_Reconstruct : ATP_PROOF_RECONSTRUCT =
struct
open ATP_Util
open ATP_Problem
open ATP_Proof
open ATP_Problem_Generate
val metisN = "metis"
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 default_metis_lam_trans = combsN
val leo2_extcnf_equal_neg_rule = "extcnf_equal_neg"
val satallax_tab_rule_prefix = "tab_"
fun term_name' (Var ((s, _), _)) = perhaps (try Name.dest_skolem) s
| term_name' _ = ""
fun lambda' v = Term.lambda_name (term_name' v, v)
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_tfree ctxt w =
let val ww = "'" ^ w in
TFree (ww, the_default \<^sort>\<open>type\<close> (Variable.def_sort ctxt (ww, ~1)))
end
fun simplify_bool ((all as Const (\<^const_name>\<open>All\<close>, _)) $ Abs (s, T, t)) =
let val t' = simplify_bool t in
if loose_bvar1 (t', 0) then all $ Abs (s, T, t') else t'
end
| simplify_bool (Const (\<^const_name>\<open>Not\<close>, _) $ t) = s_not (simplify_bool t)
| simplify_bool (Const (\<^const_name>\<open>conj\<close>, _) $ t $ u) =
s_conj (simplify_bool t, simplify_bool u)
| simplify_bool (Const (\<^const_name>\<open>disj\<close>, _) $ t $ u) =
s_disj (simplify_bool t, simplify_bool u)
| simplify_bool (Const (\<^const_name>\<open>implies\<close>, _) $ t $ u) =
s_imp (simplify_bool t, simplify_bool u)
| simplify_bool ((t as Const (\<^const_name>\<open>HOL.eq\<close>, _)) $ u $ v) =
(case (u, v) of
(Const (\<^const_name>\<open>True\<close>, _), _) => v
| (u, Const (\<^const_name>\<open>True\<close>, _)) => u
| (Const (\<^const_name>\<open>False\<close>, _), v) => s_not v
| (u, Const (\<^const_name>\<open>False\<close>, _)) => s_not u
| _ => if u aconv v then \<^const>\<open>True\<close> else t $ simplify_bool u $ simplify_bool v)
| simplify_bool (t $ u) = simplify_bool t $ simplify_bool u
| simplify_bool (Abs (s, T, t)) = Abs (s, T, simplify_bool t)
| simplify_bool t = t
fun single_letter upper s =
let val s' = if String.isPrefix "o" s orelse String.isPrefix "O" s then "z" else s in
String.extract (Name.desymbolize (SOME upper) (Long_Name.base_name s'), 0, SOME 1)
end
fun var_name_of_typ (Type (\<^type_name>\<open>fun\<close>, [_, T])) =
if body_type T = HOLogic.boolT then "p" else "f"
| var_name_of_typ (Type (\<^type_name>\<open>set\<close>, [T])) =
let fun default () = single_letter true (var_name_of_typ T) in
(case T of
Type (s, [T1, T2]) => if String.isSuffix "prod" s andalso T1 = T2 then "r" else default ()
| _ => default ())
end
| var_name_of_typ (Type (s, Ts)) =
if String.isSuffix "list" s then var_name_of_typ (the_single Ts) ^ "s"
else single_letter false (Long_Name.base_name s)
| var_name_of_typ (TFree (s, _)) = single_letter false (perhaps (try (unprefix "'")) s)
| var_name_of_typ (TVar ((s, _), T)) = var_name_of_typ (TFree (s, T))
fun rename_bound_vars (t $ u) = rename_bound_vars t $ rename_bound_vars u
| rename_bound_vars (Abs (_, T, t)) = Abs (var_name_of_typ T, T, rename_bound_vars t)
| rename_bound_vars t = t
exception ATP_CLASS of string list
exception ATP_TYPE of string atp_type list
exception ATP_TERM of (string, string atp_type) atp_term list
exception ATP_FORMULA of
(string, string, (string, string atp_type) atp_term, string) atp_formula list
exception SAME of unit
fun class_of_atp_class cls =
(case unprefix_and_unascii class_prefix cls of
SOME s => s
| NONE => raise ATP_CLASS [cls])
fun atp_type_of_atp_term (ATerm ((s, _), us)) =
let val tys = map atp_type_of_atp_term us in
if s = tptp_fun_type then
(case tys of
[ty1, ty2] => AFun (ty1, ty2)
| _ => raise ATP_TYPE tys)
else
AType ((s, []), tys)
end
(* 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_of_atp_type ctxt (ty as AType ((a, clss), tys)) =
let val Ts = map (typ_of_atp_type ctxt) tys in
(case unprefix_and_unascii native_type_prefix a of
SOME b => typ_of_atp_type ctxt (atp_type_of_atp_term (unmangled_type b))
| NONE =>
(case unprefix_and_unascii type_const_prefix a of
SOME b => Type (invert_const b, Ts)
| NONE =>
if not (null tys) then
raise ATP_TYPE [ty] (* only "tconst"s have type arguments *)
else
(case unprefix_and_unascii tfree_prefix a of
SOME b => make_tfree ctxt b
| NONE =>
(* The term 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. *)
Type_Infer.param 0 ("'" ^ perhaps (unprefix_and_unascii tvar_prefix) a,
(if null clss then \<^sort>\<open>type\<close> else map class_of_atp_class clss)))))
end
| typ_of_atp_type ctxt (AFun (ty1, ty2)) = typ_of_atp_type ctxt ty1 --> typ_of_atp_type ctxt ty2
fun typ_of_atp_term ctxt = typ_of_atp_type ctxt o atp_type_of_atp_term
(* Type class literal applied to a type. Returns triple of polarity, class, type. *)
fun type_constraint_of_term ctxt (u as ATerm ((a, _), us)) =
(case (unprefix_and_unascii class_prefix a, map (typ_of_atp_term ctxt) us) of
(SOME b, [T]) => (b, T)
| _ => raise ATP_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_var_name s =
(case unprefix_and_unascii schematic_var_prefix s of
SOME s' => s'
| NONE => s)
(* 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 robust_const_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 _ => Type_Infer.anyT \<^sort>\<open>type\<close>
val spass_skolem_prefix = "sk" (* "skc" or "skf" *)
val vampire_skolem_prefix = "sK"
fun var_index_of_textual textual = if textual then 0 else 1
fun quantify_over_var textual quant_of var_s var_T t =
let
val vars =
((var_s, var_index_of_textual textual), var_T) ::
filter (fn ((s, _), _) => s = var_s) (Term.add_vars t [])
val normTs = vars |> AList.group (op =) |> map (apsnd hd)
fun norm_var_types (Var (x, T)) =
Var (x, the_default T (AList.lookup (op =) normTs x))
| norm_var_types t = t
in t |> map_aterms norm_var_types |> fold_rev quant_of (map Var normTs) end
(* This assumes that distinct names are mapped to distinct names by "Variable.variant_frees". This
does not hold in general but should hold for ATP-generated Skolem function names, since these end
with a digit and "variant_frees" appends letters. *)
fun fresh_up ctxt s = fst (hd (Variable.variant_frees ctxt [] [(s, ())]))
(* Higher-order translation. Variables are typed (although we don't use that information). Lambdas
are typed. The code is similar to "term_of_atp_fo". *)
fun term_of_atp_ho ctxt sym_tab =
let
val thy = Proof_Context.theory_of ctxt
val var_index = var_index_of_textual true
fun do_term opt_T u =
(case u of
AAbs (((var, ty), term), []) =>
let
val typ = typ_of_atp_type ctxt ty
val var_name = repair_var_name var
val tm = do_term NONE term
in quantify_over_var true lambda' var_name typ tm end
| ATerm ((s, tys), us) =>
if s = ""
then error "Isar proof reconstruction failed because the ATP proof \
\contains unparsable material"
else if s = tptp_equal then
list_comb (Const (\<^const_name>\<open>HOL.eq\<close>, Type_Infer.anyT \<^sort>\<open>type\<close>),
map (do_term NONE) us)
else if not (null us) then
let
val args = map (do_term NONE) us
val opt_T' = SOME (map slack_fastype_of args ---> the_default dummyT opt_T)
val func = do_term opt_T' (ATerm ((s, tys), []))
in foldl1 (op $) (func :: args) end
else if s = tptp_or then HOLogic.disj
else if s = tptp_and then HOLogic.conj
else if s = tptp_implies then HOLogic.imp
else if s = tptp_iff orelse s = tptp_equal then HOLogic.eq_const dummyT
else if s = tptp_not_iff orelse s = tptp_not_equal then \<^term>\<open>\<lambda>P Q. Q \<noteq> P\<close>
else if s = tptp_if then \<^term>\<open>\<lambda>P Q. Q \<longrightarrow> P\<close>
else if s = tptp_not_and then \<^term>\<open>\<lambda>P Q. \<not> (P \<and> Q)\<close>
else if s = tptp_not_or then \<^term>\<open>\<lambda>P Q. \<not> (P \<or> Q)\<close>
else if s = tptp_not then HOLogic.Not
else if s = tptp_ho_forall then HOLogic.all_const dummyT
else if s = tptp_ho_exists then HOLogic.exists_const dummyT
else if s = tptp_hilbert_choice then HOLogic.choice_const dummyT
else if s = tptp_hilbert_the then \<^term>\<open>The\<close>
else
(case unprefix_and_unascii const_prefix s of
SOME s' =>
let
val ((s', _), mangled_us) = s' |> unmangled_const |>> `invert_const
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 Ts = map (typ_of_atp_type ctxt) tys @ map (typ_of_atp_term ctxt) type_us
val T =
(if not (null Ts) andalso robust_const_num_type_args thy s' = length Ts then
try (Sign.const_instance thy) (s', Ts)
else
NONE)
|> (fn SOME T => T
| NONE =>
map slack_fastype_of term_ts --->
the_default (Type_Infer.anyT \<^sort>\<open>type\<close>) opt_T)
val t = Const (unproxify_const s', T)
in list_comb (t, term_ts) end
| NONE => (* a free or schematic variable *)
let
val ts = map (do_term NONE) us
val T =
(case tys of
[ty] => typ_of_atp_type ctxt ty
| _ =>
map slack_fastype_of ts --->
(case opt_T of
SOME T => T
| NONE => Type_Infer.anyT \<^sort>\<open>type\<close>))
val t =
(case unprefix_and_unascii fixed_var_prefix s of
SOME s => Free (s, T)
| NONE =>
if not (is_tptp_variable s) then Free (fresh_up ctxt s, T)
else Var ((repair_var_name s, var_index), T))
in list_comb (t, ts) end))
in do_term end
(* First-order translation. No types are known for variables. "Type_Infer.anyT @{sort type}"
should allow them to be inferred. *)
fun term_of_atp_fo 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 = var_index_of_textual textual
fun do_term extra_ts opt_T u =
(case u of
ATerm ((s, tys), us) =>
if s = "" then
error "Isar proof reconstruction failed because the ATP proof contains unparsable \
\material"
else if String.isPrefix native_type_prefix s then
\<^const>\<open>True\<close> (* ignore TPTP type information (needed?) *)
else if s = tptp_equal then
list_comb (Const (\<^const_name>\<open>HOL.eq\<close>, Type_Infer.anyT \<^sort>\<open>type\<close>),
map (do_term [] NONE) us)
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_of_atp_term ctxt typ_u)) term_u
| _ => raise ATP_TERM us)
else if s' = predicator_name then
do_term [] (SOME \<^typ>\<open>bool\<close>) (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>\<open>True\<close> (* 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 Ts = map (typ_of_atp_type ctxt) tys @ map (typ_of_atp_term ctxt) type_us
val T =
(if not (null Ts) andalso robust_const_num_type_args thy s' = length Ts then
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
else
NONE)
|> (fn SOME T => T
| NONE =>
map slack_fastype_of term_ts --->
the_default (Type_Infer.anyT \<^sort>\<open>type\<close>) opt_T)
val t =
if new_skolem then Var ((new_skolem_var_name_of_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
(* SPASS (3.8ds) and Vampire (2.6) pass arguments to Skolem functions in reverse
order, which is incompatible with "metis"'s new skolemizer. *)
|> exists (fn pre => String.isPrefix pre s)
[spass_skolem_prefix, vampire_skolem_prefix] ? rev
val ts = term_ts @ extra_ts
val T =
(case tys of
[ty] => typ_of_atp_type ctxt ty
| _ =>
(case opt_T of
SOME T => map slack_fastype_of term_ts ---> T
| NONE => map slack_fastype_of ts ---> Type_Infer.anyT \<^sort>\<open>type\<close>))
val t =
(case unprefix_and_unascii fixed_var_prefix s of
SOME s => Free (s, T)
| NONE =>
if textual andalso not (is_tptp_variable s) then
Free (s |> textual ? fresh_up ctxt, T)
else
Var ((repair_var_name s, var_index), T))
in list_comb (t, ts) end))
in do_term [] end
fun term_of_atp ctxt (ATP_Problem.THF _) type_enc =
if ATP_Problem_Generate.is_type_enc_higher_order type_enc then K (term_of_atp_ho ctxt)
else error "Unsupported Isar reconstruction"
| term_of_atp ctxt _ type_enc =
if not (ATP_Problem_Generate.is_type_enc_higher_order type_enc) then term_of_atp_fo ctxt
else error "Unsupported Isar reconstruction"
fun term_of_atom ctxt format type_enc textual sym_tab pos (u as ATerm ((s, _), _)) =
if String.isPrefix class_prefix s then
add_type_constraint pos (type_constraint_of_term ctxt u)
#> pair \<^const>\<open>True\<close>
else
pair (term_of_atp ctxt format type_enc textual sym_tab (SOME \<^typ>\<open>bool\<close>) u)
(* 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
(* Interpret an ATP formula as a HOL term, extracting sort constraints as they appear in the
formula. *)
fun prop_of_atp ctxt format type_enc 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 textual (case q of AForall => forall_of | AExists => exists_of)
(repair_var_name s) dummyT
| 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_of_atom ctxt format type_enc textual sym_tab pos tm
| _ => raise ATP_FORMULA [phi])
in
repair_tvar_sorts (do_formula true phi Vartab.empty)
end
val unprefix_fact_number = space_implode "_" o tl o space_explode "_"
fun resolve_fact facts s =
(case try (unprefix fact_prefix) s of
SOME s' =>
let val s' = s' |> unprefix_fact_number |> unascii_of in
AList.lookup (op =) facts s' |> Option.map (pair s')
end
| NONE => NONE)
fun resolve_conjecture s =
(case try (unprefix conjecture_prefix) s of
SOME s' => Int.fromString s'
| NONE => NONE)
fun resolve_facts facts = map_filter (resolve_fact facts)
val resolve_conjectures = map_filter resolve_conjecture
fun is_axiom_used_in_proof pred =
exists (fn ((_, ss), _, _, _, []) => exists pred ss | _ => false)
fun add_fact ctxt facts ((_, ss), _, _, rule, deps) =
(if member (op =) [agsyhol_core_rule, leo2_extcnf_equal_neg_rule] rule orelse
String.isPrefix satallax_tab_rule_prefix rule then
insert (op =) (short_thm_name ctxt ext, (Global, General))
else
I)
#> (if null deps then union (op =) (resolve_facts facts ss) else I)
fun used_facts_in_atp_proof ctxt facts atp_proof =
if null atp_proof then facts else fold (add_fact ctxt facts) atp_proof []
fun used_facts_in_unsound_atp_proof _ _ [] = NONE
| used_facts_in_unsound_atp_proof ctxt facts atp_proof =
let val used_facts = used_facts_in_atp_proof ctxt facts atp_proof in
if forall (fn (_, (sc, _)) => sc = Global) used_facts andalso
not (is_axiom_used_in_proof (is_some o resolve_conjecture) atp_proof) then
SOME (map fst used_facts)
else
NONE
end
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 atp_proof_prefers_lifting atp_proof =
(is_axiom_used_in_proof is_combinator_def atp_proof,
is_axiom_used_in_proof is_lam_lifted atp_proof) = (false, true)
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 replace_one_dependency (old, new) dep = if is_same_atp_step dep old then new else [dep]
fun replace_dependencies_in_line old_new (name, role, t, rule, deps) =
(name, role, t, rule, fold (union (op =) o replace_one_dependency old_new) deps [])
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 set_var_index j = map_aterms (fn Var ((s, 0), T) => Var ((s, j), T) | t => t)
fun infer_formulas_types ctxt =
map_index (uncurry (fn j => set_var_index j #> Type.constraint HOLogic.boolT))
#> Syntax.check_terms (Proof_Context.set_mode Proof_Context.mode_schematic ctxt)
#> map (set_var_index 0)
val combinator_table =
[(\<^const_name>\<open>Meson.COMBI\<close>, @{thm Meson.COMBI_def [abs_def]}),
(\<^const_name>\<open>Meson.COMBK\<close>, @{thm Meson.COMBK_def [abs_def]}),
(\<^const_name>\<open>Meson.COMBB\<close>, @{thm Meson.COMBB_def [abs_def]}),
(\<^const_name>\<open>Meson.COMBC\<close>, @{thm Meson.COMBC_def [abs_def]}),
(\<^const_name>\<open>Meson.COMBS\<close>, @{thm Meson.COMBS_def [abs_def]})]
fun uncombine_term thy =
let
fun uncomb (t1 $ t2) = betapply (uncomb t1, uncomb t2)
| uncomb (Abs (s, T, t)) = Abs (s, T, uncomb t)
| uncomb (t as Const (x as (s, _))) =
(case AList.lookup (op =) combinator_table s of
SOME thm => thm |> Thm.prop_of |> specialize_type thy x |> Logic.dest_equals |> snd
| NONE => t)
| uncomb t = t
in uncomb end
fun unlift_aterm lifted (t as Const (s, _)) =
if String.isPrefix lam_lifted_prefix s then
(* FIXME: do something about the types *)
(case AList.lookup (op =) lifted s of
SOME t' => unlift_term lifted t'
| NONE => t)
else
t
| unlift_aterm _ t = t
and unlift_term lifted =
map_aterms (unlift_aterm lifted)
fun termify_line _ _ _ _ _ (_, Type_Role, _, _, _) = NONE
| termify_line ctxt format type_enc lifted sym_tab (name, role, u, rule, deps) =
let
val thy = Proof_Context.theory_of ctxt
val t = u
|> prop_of_atp ctxt format type_enc true sym_tab
|> unlift_term lifted
|> uncombine_term thy
|> simplify_bool
in
SOME (name, role, t, rule, deps)
end
val waldmeister_conjecture_num = "1.0.0.0"
fun repair_waldmeister_endgame proof =
let
fun repair_tail (name, _, \<^const>\<open>Trueprop\<close> $ t, rule, deps) =
(name, Negated_Conjecture, \<^const>\<open>Trueprop\<close> $ s_not t, rule, deps)
fun repair_body [] = []
| repair_body ((line as ((num, _), _, _, _, _)) :: lines) =
if num = waldmeister_conjecture_num then map repair_tail (line :: lines)
else line :: repair_body lines
in
repair_body proof
end
fun map_proof_terms f (lines : ('a * 'b * 'c * 'd * 'e) list) =
map2 (fn c => fn (a, b, _, d, e) => (a, b, c, d, e)) (f (map #3 lines)) lines
fun termify_atp_proof ctxt local_prover format type_enc pool lifted sym_tab =
nasty_atp_proof pool
#> map_term_names_in_atp_proof repair_name
#> map_filter (termify_line ctxt format type_enc lifted sym_tab)
#> map_proof_terms (infer_formulas_types ctxt #> map HOLogic.mk_Trueprop)
#> local_prover = waldmeisterN ? repair_waldmeister_endgame
fun unskolemize_term skos =
let
val is_skolem_name = member (op =) skos
fun find_argless_skolem (Free _ $ Var _) = NONE
| find_argless_skolem (Free (x as (s, _))) = if is_skolem_name s then SOME x else NONE
| find_argless_skolem (t $ u) =
(case find_argless_skolem t of NONE => find_argless_skolem u | sk => sk)
| find_argless_skolem (Abs (_, _, t)) = find_argless_skolem t
| find_argless_skolem _ = NONE
fun find_skolem_arg (Free (s, _) $ Var v) = if is_skolem_name s then SOME v else NONE
| find_skolem_arg (t $ u) = (case find_skolem_arg t of NONE => find_skolem_arg u | v => v)
| find_skolem_arg (Abs (_, _, t)) = find_skolem_arg t
| find_skolem_arg _ = NONE
fun kill_skolem_arg (t as Free (s, T) $ Var _) =
if is_skolem_name s then Free (s, range_type T) else t
| kill_skolem_arg (t $ u) = kill_skolem_arg t $ kill_skolem_arg u
| kill_skolem_arg (Abs (s, T, t)) = Abs (s, T, kill_skolem_arg t)
| kill_skolem_arg t = t
fun find_var (Var v) = SOME v
| find_var (t $ u) = (case find_var t of NONE => find_var u | v => v)
| find_var (Abs (_, _, t)) = find_var t
| find_var _ = NONE
val safe_abstract_over = abstract_over o apsnd (incr_boundvars 1)
fun unskolem t =
(case find_argless_skolem t of
SOME (x as (s, T)) =>
HOLogic.exists_const T $ Abs (s, T, unskolem (safe_abstract_over (Free x, t)))
| NONE =>
(case find_skolem_arg t of
SOME (v as ((s, _), T)) =>
let
val (haves, have_nots) =
HOLogic.disjuncts t
|> List.partition (exists_subterm (curry (op =) (Var v)))
|> apply2 (fn lits => fold (curry s_disj) lits \<^term>\<open>False\<close>)
in
s_disj (HOLogic.all_const T
$ Abs (s, T, unskolem (safe_abstract_over (Var v, kill_skolem_arg haves))),
unskolem have_nots)
end
| NONE =>
(case find_var t of
SOME (v as ((s, _), T)) =>
HOLogic.all_const T $ Abs (s, T, unskolem (safe_abstract_over (Var v, t)))
| NONE => t)))
in
HOLogic.mk_Trueprop o unskolem o HOLogic.dest_Trueprop
end
fun rename_skolem_args t =
let
fun add_skolem_args (Abs (_, _, t)) = add_skolem_args t
| add_skolem_args t =
(case strip_comb t of
(Free (s, _), args as _ :: _) =>
if String.isPrefix spass_skolem_prefix s then
insert (op =) (s, take_prefix is_Var args)
else
fold add_skolem_args args
| (u, args as _ :: _) => fold add_skolem_args (u :: args)
| _ => I)
fun subst_of_skolem (sk, args) =
map_index (fn (j, Var (z, T)) => (z, Var ((sk ^ "_" ^ string_of_int j, 0), T))) args
val subst = maps subst_of_skolem (add_skolem_args t [])
in
subst_vars ([], subst) t
end
fun introduce_spassy_skolems proof =
let
fun add_skolem (Free (s, _)) = String.isPrefix spass_skolem_prefix s ? insert (op =) s
| add_skolem _ = I
(* union-find would be faster *)
fun add_cycle [] = I
| add_cycle ss =
fold (fn s => Graph.default_node (s, ())) ss
#> fold Graph.add_edge (ss ~~ tl ss @ [hd ss])
val (input_steps, other_steps) = List.partition (null o #5) proof
(* The names of the formulas are added to the Skolem constants, to ensure that formulas giving
rise to several clauses are skolemized together. *)
val skoXss = map (fn ((_, ss), _, t, _, _) => Term.fold_aterms add_skolem t ss) input_steps
val groups0 = Graph.strong_conn (fold add_cycle skoXss Graph.empty)
val groups = filter (exists (String.isPrefix spass_skolem_prefix)) groups0
val skoXss_input_steps = skoXss ~~ input_steps
fun step_name_of_group skoXs = (implode skoXs, [])
fun in_group group = member (op =) group o hd
fun group_of skoX = find_first (fn group => in_group group skoX) groups
fun new_steps (skoXss_steps : (string list * (term, 'a) atp_step) list) group =
let
val name = step_name_of_group group
val name0 = name |>> prefix "0"
val t =
(case map (snd #> #3) skoXss_steps of
[t] => t
| ts => ts
|> map (HOLogic.dest_Trueprop #> rename_skolem_args)
|> Library.foldr1 s_conj
|> HOLogic.mk_Trueprop)
val skos =
fold (union (op =) o filter (String.isPrefix spass_skolem_prefix) o fst) skoXss_steps []
val deps = map (snd #> #1) skoXss_steps
in
[(name0, Unknown, unskolemize_term skos t, spass_pre_skolemize_rule, deps),
(name, Unknown, t, spass_skolemize_rule, [name0])]
end
val sko_steps =
maps (fn group => new_steps (filter (in_group group o fst) skoXss_input_steps) group) groups
val old_news =
map_filter (fn (skoXs, (name, _, _, _, _)) =>
Option.map (pair name o single o step_name_of_group) (group_of skoXs))
skoXss_input_steps
val repair_deps = fold replace_dependencies_in_line old_news
in
input_steps @ sko_steps @ map repair_deps other_steps
end
fun factify_atp_proof facts hyp_ts concl_t atp_proof =
let
fun factify_step ((num, ss), role, t, rule, deps) =
let
val (ss', role', t') =
(case resolve_conjectures ss of
[j] =>
if j = length hyp_ts then ([], Conjecture, concl_t)
else ([], Hypothesis, close_form (nth hyp_ts j))
| _ =>
(case resolve_facts facts ss of
[] => (ss, if role = Lemma then Lemma else Plain, t)
| facts => (map fst facts, Axiom, t)))
in
((num, ss'), role', t', rule, deps)
end
val atp_proof = map factify_step atp_proof
val names = map #1 atp_proof
fun repair_dep (num, ss) = (num, the_default ss (AList.lookup (op =) names num))
fun repair_deps (name, role, t, rule, deps) = (name, role, t, rule, map repair_dep deps)
in
map repair_deps atp_proof
end
end;