(* Title: HOL/Tools/ATP/atp_waldmeister.ML
Author: Albert Steckermeier, TU Muenchen
Author: Jasmin Blanchette, TU Muenchen
General-purpose functions used by the Sledgehammer modules.
*)
exception FailureMessage of string
signature ATP_WALDMEISTER_SKOLEMIZER =
sig
val skolemize : bool -> Proof.context -> term -> (Proof.context * (term list * term))
end;
signature ATP_WALDMEISTER_TYPE_ENCODER =
sig
val encode_type : typ -> string
val decode_type_string : string -> typ
val encode_types : typ list -> string
val decode_types : string -> typ list
val encode_const : string * typ list -> string
val decode_const : string -> string * typ list
end;
signature ATP_WALDMEISTER =
sig
type 'a atp_problem = 'a ATP_Problem.atp_problem
type ('a, 'b) atp_step = ('a, 'b) ATP_Proof.atp_step
type 'a atp_proof = 'a ATP_Proof.atp_proof
type stature = ATP_Problem_Generate.stature
type waldmeister_info = (string * (term list * (term option * term))) list
val waldmeister_skolemize_rule : string
val generate_waldmeister_problem : Proof.context -> term list -> term ->
((string * stature) * term) list ->
string atp_problem * string Symtab.table * (string * term) list * int Symtab.table *
waldmeister_info
val termify_waldmeister_proof : Proof.context -> string Symtab.table -> string atp_proof ->
(term, string) atp_step list
val introduce_waldmeister_skolems : waldmeister_info -> (term, string) atp_step list ->
(term, string) atp_step list
end;
structure ATP_Waldmeister_Skolemizer : ATP_WALDMEISTER_SKOLEMIZER =
struct
open HOLogic
fun contains_quantor (Const (@{const_name Ex}, _) $ _) = true
| contains_quantor (Const (@{const_name All}, _) $ _) = true
| contains_quantor (t1 $ t2) = contains_quantor t1 orelse contains_quantor t2
| contains_quantor _ = false
fun mk_fun_for_bvar ctxt1 ctxt2 arg_trms (bound_name, ty) =
let
val fun_type = (map type_of arg_trms) ---> ty
val (fun_name, _) = singleton (Variable.variant_frees ctxt2 []) ("sko_" ^ bound_name,fun_type)
val (_, ctxt1_new) = Variable.add_fixes [fun_name] ctxt1
val (_, ctxt2_new) = Variable.add_fixes [fun_name] ctxt2
in
(Term.list_comb (Free (fun_name,fun_type), arg_trms), ctxt1_new, ctxt2_new)
end
fun skolem_free ctxt1 ctxt2 vars (bound_name, ty, trm) =
let
val (fun_trm, ctxt1_new, ctxt2_new) =
mk_fun_for_bvar ctxt1 ctxt2 (List.rev vars) (bound_name,ty)
in
(Term.subst_bounds ([fun_trm], trm), ctxt1_new, ctxt2_new)
end
fun skolem_var ctxt (bound_name, ty, trm) =
let
val (var_name, _) = singleton (Variable.variant_frees ctxt []) (bound_name, ty)
val (_, ctxt') = Variable.add_fixes [var_name] ctxt
val var = Var ((var_name, 0), ty)
in
(Term.subst_bounds ([var], trm), ctxt', var)
end
fun skolem_bound is_free ctxt1 ctxt2 spets vars x =
if is_free then
let
val (trm', ctxt1', ctxt2') = skolem_free ctxt1 ctxt2 vars x
in
(ctxt1', ctxt2',spets, trm', vars)
end
else
let
val (trm', ctxt2', var) = skolem_var ctxt2 x
in
(ctxt1, ctxt2', spets, trm', var :: vars)
end
fun skolemize' pos ctxt1 ctxt2 spets vars (Const (@{const_name Not}, _) $ trm') =
let
val (ctxt1', ctxt2', spets', trm'') = skolemize' (not pos) ctxt1 ctxt2 spets vars trm'
in
(ctxt1', ctxt2', map mk_not spets', mk_not trm'')
end
| skolemize' pos ctxt1 ctxt2 spets vars (trm as (Const (@{const_name HOL.eq}, t) $ a $ b)) =
if t = @{typ "bool \<Rightarrow> bool \<Rightarrow> bool"} andalso contains_quantor trm then
skolemize' pos ctxt1 ctxt2 (trm :: spets) vars (mk_conj (mk_imp (a, b), mk_imp (b, a)))
else
(ctxt1, ctxt2, spets, trm)
| skolemize' pos ctxt1 ctxt2 spets vars (trm as (Const (name, _) $ Abs x)) =
if name = @{const_name Ex} orelse name = @{const_name All} then
let
val is_free = (name = @{const_name Ex} andalso pos)
orelse (name = @{const_name All} andalso not pos)
val (ctxt1', ctxt2', spets', trm', vars') =
skolem_bound is_free ctxt1 ctxt2 (if is_free then trm :: spets else spets) vars x
in
skolemize' pos ctxt1' ctxt2' spets' vars' trm'
end
else
(ctxt1, ctxt2, spets, trm)
| skolemize' pos ctxt1 ctxt2 spets vars ((c as Const (name, _)) $ a $ b) =
if name = @{const_name conj} orelse name = @{const_name disj} orelse
name = @{const_name implies} then
let
val pos_a = if name = @{const_name implies} then not pos else pos
val (ctxt1', ctxt2', spets', a') = skolemize' pos_a ctxt1 ctxt2 [] vars a
val (ctxt1'', ctxt2'', spets'', b') = skolemize' pos ctxt1' ctxt2' [] vars b
in
(ctxt1'', ctxt2'',
map (fn trm => c $ a' $ trm) spets'' @ map (fn trm => c $ trm $ b) spets' @ spets,
c $ a' $ b')
end
else
(ctxt1,ctxt2,spets,c $ a $ b)
| skolemize' _ ctxt1 ctxt2 spets _ trm = (ctxt1, ctxt2, spets, trm)
fun skolemize positve ctxt trm =
let
val (ctxt1, _, spets, skolemized_trm) = skolemize' positve ctxt ctxt [] [] trm
in
(ctxt1, (trm :: List.rev spets, skolemized_trm))
end
end;
structure ATP_Waldmeister_Type_Encoder : ATP_WALDMEISTER_TYPE_ENCODER =
struct
val delimiter = ";"
val open_paranthesis = "["
val close_parathesis = "]"
val type_prefix = "Type"
val tfree_prefix = "TFree"
val tvar_prefix = "TVar"
val identifier_character = not o member (op =) [delimiter, open_paranthesis, close_parathesis]
fun encode_type (Type (name, types)) =
type_prefix ^ open_paranthesis ^ name ^ delimiter ^
(map encode_type types |> String.concatWith delimiter) ^ close_parathesis
| encode_type (TFree (name, sorts)) =
tfree_prefix ^ open_paranthesis ^ name ^ delimiter ^ (String.concatWith delimiter sorts) ^
close_parathesis
| encode_type (TVar ((name, i), sorts)) =
tvar_prefix ^ open_paranthesis ^ open_paranthesis ^ name ^ delimiter ^ Int.toString i ^
close_parathesis ^ delimiter ^ (String.concatWith delimiter sorts) ^ close_parathesis
fun encode_types types = (String.concatWith delimiter (map encode_type types))
fun parse_identifier x =
(Scan.many identifier_character >> implode) x
fun parse_star delim scanner x =
(Scan.optional (scanner ::: Scan.repeat ($$ delim |-- scanner)) []) x
fun parse_type x = (Scan.this_string type_prefix |-- $$ open_paranthesis |-- parse_identifier --|
$$ delimiter -- parse_star delimiter parse_any_type --| $$ close_parathesis >> Type) x
and parse_tfree x = (Scan.this_string tfree_prefix |-- $$ open_paranthesis |-- parse_identifier --|
$$ delimiter -- parse_star delimiter parse_identifier --| $$ close_parathesis >> TFree) x
and parse_tvar x = (Scan.this_string tvar_prefix |-- $$ open_paranthesis |-- $$ open_paranthesis
|-- parse_identifier --| $$ delimiter -- (parse_identifier >> (Int.fromString #> the)) --| $$
close_parathesis --| $$ delimiter -- parse_star delimiter parse_identifier --|
$$ close_parathesis >> TVar) x
and parse_any_type x = (parse_type || parse_tfree || parse_tvar) x
fun parse_types x = parse_star delimiter parse_any_type x
fun decode_type_string s = Scan.finite Symbol.stopper
(Scan.error (!! (fn _ => raise FailureMessage ("unrecognized type encoding" ^
quote s)) parse_type)) (Symbol.explode s) |> fst
fun decode_types s = Scan.finite Symbol.stopper
(Scan.error (!! (fn _ => raise FailureMessage ("unrecognized type encoding" ^
quote s))) parse_types) (Symbol.explode s) |> fst
fun encode_const (name,tys) = name ^ delimiter ^ encode_types tys
fun parse_const s = (parse_identifier --| $$ delimiter -- parse_types) s
fun decode_const s = Scan.finite Symbol.stopper
(Scan.error (!! (fn _ => raise FailureMessage ("unrecognized const encoding" ^
quote s))) parse_const) (Symbol.explode s) |> fst
end;
structure ATP_Waldmeister (*** : ATP_WALDMEISTER *) =
struct
open ATP_Util
open ATP_Problem
open ATP_Problem_Generate
open ATP_Proof
open ATP_Proof_Reconstruct
open ATP_Waldmeister_Skolemizer
open ATP_Waldmeister_Type_Encoder
open HOLogic
type ('a, 'b) atp_term = ('a, 'b) ATP_Problem.atp_term
type atp_connective = ATP_Problem.atp_connective
type ('a, 'b, 'c, 'd) atp_formula = ('a, 'b, 'c, 'd) ATP_Problem.atp_formula
type atp_format = ATP_Problem.atp_format
type atp_formula_role = ATP_Problem.atp_formula_role
type 'a atp_problem = 'a ATP_Problem.atp_problem
type waldmeister_info = (string * (term list * (term option * term))) list
val const_prefix = #"c"
val var_prefix = #"V"
val free_prefix = #"v"
val conjecture_condition_name = "condition"
val waldmeister_equals = "eq"
val waldmeister_true = "true"
val waldmeister_false = "false"
val waldmeister_skolemize_rule = "waldmeister_skolemize"
val lam_lift_waldmeister_prefix = "lambda_wm"
val waldmeister_apply = "wm_apply"
val factsN = "Relevant facts"
val helpersN = "Helper facts"
val conjN = "Conjecture"
val conj_identifier = conjecture_prefix ^ "0"
val WM_ERROR_MSG = "Waldmeister problem generator failed: "
(*
Some utilitary functions for translation.
*)
fun gen_ascii_tuple str = (str, ascii_of str)
fun mk_eq_true (trm as (Const (@{const_name HOL.eq}, _) $ _ $ _)) = (NONE,trm)
| mk_eq_true trm = (SOME trm,HOLogic.mk_eq (trm, @{term True}))
val is_lambda_name = String.isPrefix lam_lifted_poly_prefix
fun lookup table k =
List.find (fn (key, _) => key = k) table
fun dest_list' (f $ t) =
let
val (function, trms) = dest_list' f
in
(function, t :: trms)
end
| dest_list' t = (t,[]);
fun dest_list trm = dest_list' trm ||> List.rev
fun list_update x [] = [x]
| list_update (a,b) ((c,d) :: xs) =
if a = c andalso b < d then
(a,b) :: xs
else
(c,d) :: list_update (a,b) xs
(*
Hiding partial applications in terms
*)
fun map_minimal_app' info (trm :: trms) =
map_minimal_app' (minimal_app' info trm) trms
| map_minimal_app' info _ = info
and minimal_app' info (trm as _ $ _) =
let
val (function, trms) = dest_list trm
val info' = map_minimal_app' info trms
in
case function of
(Const _) => list_update (function, length trms) info' |
(Free _) => list_update (function, length trms) info' |
_ => info
end
| minimal_app' info (trm as Const _) =
list_update (trm, 0) info
| minimal_app' info (trm as Free _) =
list_update (trm, 0) info
| minimal_app' info _ = info;
fun map_minimal_app trms = map_minimal_app' [] trms
fun mk_waldmeister_app function [] = function
| mk_waldmeister_app function (a :: args) =
let
val funT = type_of function
val argT = type_of a
val resT = dest_funT funT |> snd
val newT = funT --> argT --> resT
in
mk_waldmeister_app (Const (waldmeister_apply ^ "," ^
encode_types [resT, argT], newT) $ function $ a) args
end
fun hide_partial_applications info (trm as (_ $ _)) =
let
val (function, trms) = dest_list trm
val trms' = map (hide_partial_applications info) trms
in
case function of
Var _ => mk_waldmeister_app function trms' |
_ =>
let
val min_args = lookup info function |> the |> snd
val args0 = List.take (trms',min_args)
val args1 = List.drop (trms',min_args)
val function' = list_comb (function,args0)
in
mk_waldmeister_app function' args1
end
end
| hide_partial_applications _ t = t;
fun remove_waldmeister_app ((c as Const (name, _)) $ x $ y) =
if String.isPrefix waldmeister_apply name then
remove_waldmeister_app x $ remove_waldmeister_app y
else
c $ remove_waldmeister_app x $ remove_waldmeister_app y
| remove_waldmeister_app (x $ y) = remove_waldmeister_app x $ remove_waldmeister_app y
| remove_waldmeister_app x = x
(*
Translation from Isabelle terms to ATP terms.
*)
fun trm_to_atp'' thy (Const (x, ty)) args =
let
val ty_args = if is_lambda_name x orelse String.isPrefix waldmeister_apply x then
[] else Sign.const_typargs thy (x, ty)
in
[ATerm ((gen_ascii_tuple (String.str const_prefix ^ encode_const (x, ty_args)), []), args)]
end
| trm_to_atp'' _ (Free (x, _)) args =
[ATerm ((gen_ascii_tuple (String.str free_prefix ^ x), []), args)]
| trm_to_atp'' _ (Var ((x, _), _)) args =
[ATerm ((gen_ascii_tuple (String.str var_prefix ^ x), []), args)]
| trm_to_atp'' thy (trm1 $ trm2) args = trm_to_atp'' thy trm1 (trm_to_atp'' thy trm2 [] @ args)
| trm_to_atp'' _ _ _ = raise FailureMessage (WM_ERROR_MSG ^ "Unexpected term")
fun trm_to_atp' thy trm = trm_to_atp'' thy trm [] |> hd
fun eq_trm_to_atp thy (Const (@{const_name HOL.eq}, _) $ lhs $ rhs) =
ATerm ((("equal", "equal"), []), [trm_to_atp' thy lhs, trm_to_atp' thy rhs])
| eq_trm_to_atp _ _ = raise FailureMessage (WM_ERROR_MSG ^ "Non-eq term")
(* Translation from ATP terms to Isabelle terms. *)
fun construct_term thy (name, args) =
let
val prefix = String.sub (name, 0)
val encoded_name = String.extract(name, 1, NONE)
fun dummy_fun_type () = replicate (length args) dummyT ---> dummyT
in
if prefix = const_prefix then
let
val (const_name, ty_args) = if String.isPrefix waldmeister_apply encoded_name then
(waldmeister_apply, []) else decode_const encoded_name
val const_trans_name =
if is_lambda_name const_name then
lam_lift_waldmeister_prefix ^ (* ?? *)
String.extract(const_name, size lam_lifted_poly_prefix, NONE)
else
const_name
in
Const (const_trans_name,
if is_lambda_name const_name orelse String.isPrefix waldmeister_apply const_name then
dummyT
else
Sign.const_instance thy (const_name, ty_args))
end
else if prefix = free_prefix then
Free (encoded_name, dummy_fun_type ())
else if Char.isUpper prefix then
Var ((name, 0), dummy_fun_type ())
(* Use name instead of encoded_name because Waldmeister renames free variables. *)
else if name = waldmeister_equals then
(case args of
[_, _] => eq_const dummyT
| _ => raise FailureMessage
(WM_ERROR_MSG ^ "waldmeister equals needs 2 arguments but has " ^
Int.toString (length args)))
else if name = waldmeister_true then
@{term True}
else if name = waldmeister_false then
@{term False}
else
raise FailureMessage
(WM_ERROR_MSG ^ "Unknown name prefix when parsing Waldmeister proof: name = " ^ name)
end
and atp_to_trm' thy (ATerm ((name,_), args)) =
(case args of
[] => construct_term thy (name, args)
| _ => Term.list_comb (construct_term thy (name, args), map (atp_to_trm' thy) args))
| atp_to_trm' _ _ = raise FailureMessage (WM_ERROR_MSG ^ "atp_to_trm' expects ATerm")
fun atp_to_trm thy (ATerm (("equal", _), [lhs, rhs])) =
mk_eq (atp_to_trm' thy lhs, atp_to_trm' thy rhs)
| atp_to_trm _ (ATerm (("$true", _), _)) = @{term True}
| atp_to_trm _ _ = raise FailureMessage (WM_ERROR_MSG ^ "atp_to_trm expects ATerm")
fun formula_to_trm thy (AAtom aterm) = aterm |> atp_to_trm thy
| formula_to_trm thy (AConn (ANot, [aterm])) =
mk_not (formula_to_trm thy aterm)
| formula_to_trm _ _ =
raise FailureMessage (WM_ERROR_MSG ^ "formula_to_trm expects AAtom or AConn")
(* Abstract translation *)
fun mk_formula prefix_name name atype aterm =
Formula ((prefix_name ^ ascii_of name, name), atype, AAtom aterm, NONE, [])
fun problem_lines_of_fact thy prefix (s, (_, (_, t))) =
mk_formula (prefix ^ "0_") s Axiom (eq_trm_to_atp thy t)
fun make_nice problem = nice_atp_problem true CNF problem
fun mk_conjecture aterm =
let
val formula = mk_anot (AAtom aterm)
in
Formula ((conj_identifier, ""), Hypothesis, formula, NONE, [])
end
fun generate_waldmeister_problem ctxt hyps_t0 concl_t0 facts0 =
let
val thy = Proof_Context.theory_of ctxt
val preproc = Object_Logic.atomize_term thy
val conditions = map preproc hyps_t0
val consequence = preproc concl_t0
val facts = map (apsnd preproc #> apfst fst) facts0 : (string * term) list
fun map_ctxt' _ ctxt [] ys = (ctxt, ys)
| map_ctxt' f ctxt (x :: xs) ys =
let
val (ctxt', x') = f ctxt x
in
map_ctxt' f ctxt' xs (x' :: ys)
end
fun map_ctxt f ctxt xs = map_ctxt' f ctxt xs []
fun skolemize_fact ctxt (name, trm) =
let
val (ctxt', (steps, trm')) = skolemize true ctxt trm
in
(ctxt', (name, (steps, trm')))
end
fun name_list' _ [] _ = []
| name_list' prefix (x :: xs) i = (prefix ^ Int.toString i, x) :: name_list' prefix xs (i + 1)
fun name_list prefix xs = name_list' prefix xs 0
(* Skolemization, hiding lambdas and translating formulas to equations *)
val (ctxt', sko_facts) = map_ctxt skolemize_fact ctxt facts
val (ctxt'', sko_conditions) = map_ctxt (skolemize true) ctxt' conditions
val post_skolem = do_cheaply_conceal_lambdas []
val sko_eq_facts0 = map (apsnd (apsnd (mk_eq_true #> apsnd post_skolem))) sko_facts
val sko_eq_conditions0 = map (apsnd (mk_eq_true #> apsnd post_skolem)) sko_conditions
|> name_list conjecture_condition_name
val (_, eq_conseq as (_, (non_eq_consequence0, eq_consequence0))) =
skolemize false ctxt'' consequence |> apsnd (apsnd (mk_eq_true #> apsnd post_skolem))
val sko_eq_info =
(((conj_identifier, eq_conseq) :: sko_eq_conditions0)
@ map (apfst (fn name => fact_prefix ^ "0_" ^ name)) sko_eq_facts0)
(* Translation of partial function applications *)
val fun_app_info = map_minimal_app (map (snd o snd o snd) sko_eq_info)
fun hide_partial_apps_in_last (x, (y, (z, term))) =
(x, (y, (z, hide_partial_applications fun_app_info term)))
val sko_eq_facts = map hide_partial_apps_in_last sko_eq_facts0
val sko_eq_conditions = map hide_partial_apps_in_last sko_eq_conditions0
val eq_consequence = hide_partial_applications fun_app_info eq_consequence0
(* Problem creation *)
val fact_lines = map (problem_lines_of_fact thy fact_prefix) sko_eq_facts
val condition_lines =
map (fn (name, (_, (_, trm))) =>
mk_formula fact_prefix name Hypothesis (eq_trm_to_atp thy trm)) sko_eq_conditions
val axiom_lines = fact_lines @ condition_lines
val conj_line = mk_conjecture (eq_trm_to_atp thy eq_consequence)
val helper_lemmas_needed = exists (snd #> snd #> fst #> is_some) sko_eq_facts
orelse exists (snd #> snd #> fst #> is_some) sko_eq_conditions orelse
is_some non_eq_consequence0
val helper_lines =
if helper_lemmas_needed then
[(helpersN,
@{thms waldmeister_fol}
|> map (fn th => (("", (Global, General)), preproc (prop_of th)))
|> map (fn ((s, _) ,t) => mk_formula helper_prefix s Axiom (eq_trm_to_atp thy t)))]
else
[]
val problem = (factsN, axiom_lines) :: helper_lines @ [(conjN, [conj_line])]
val (nice_problem, pool) = make_nice problem
in
(nice_problem, Option.map snd pool |> the_default Symtab.empty, [], Symtab.empty, sko_eq_info)
end
fun termify_line ctxt (name, role, u, rule, deps) =
let
val thy = Proof_Context.theory_of ctxt
val t = u |> formula_to_trm thy |> remove_waldmeister_app
|> singleton (infer_formulas_types ctxt)
|> HOLogic.mk_Trueprop
in
(name, role, t, rule, deps)
end
fun termify_waldmeister_proof ctxt pool =
nasty_atp_proof pool
#> map (termify_line ctxt)
#> repair_waldmeister_endgame
fun get_skolem_info info names = case map (lookup info) names |> List.find is_some of
SOME x => x |
NONE => NONE
fun fix_name name =
if String.isPrefix fact_prefix name andalso String.isSuffix "_J" name then
String.extract(name, size fact_prefix + 2,NONE) |> unascii_of |>
(fn x => fact_prefix ^ "0_" ^ x)
else
name
fun skolemization_steps info
(proof_step as ((waldmeister_name, isabelle_names), _, trm, rule, _)) =
case get_skolem_info info (map fix_name isabelle_names) of
NONE => [proof_step] |
SOME (_, ([], _)) => [proof_step] |
SOME (_, (step :: steps,_)) =>
let
val raw_trm = dest_Trueprop trm
val is_narrowing = raw_trm = @{term "True = False"} orelse raw_trm = @{term "False = True"}
val is_conjecture = String.isPrefix "1.0.0.0" waldmeister_name andalso not is_narrowing
in
if is_narrowing then
[proof_step]
else
let
fun mk_steps _ [] = []
| mk_steps i (x :: xs) = (((waldmeister_name ^ "_" ^ Int.toString i),[]),
Plain, mk_Trueprop ((is_conjecture ? mk_not) x), waldmeister_skolemize_rule,
[(waldmeister_name ^ "_" ^ Int.toString (i-1),
if i = 1 then isabelle_names else [])])
:: mk_steps (i+1) xs
val first_step = ((waldmeister_name ^ "_0", isabelle_names), Unknown,
mk_Trueprop ((is_conjecture ? mk_not) step), rule, [])
val sub_steps = mk_steps 1 steps
val skolem_steps = first_step :: sub_steps
val num_of_steps = length skolem_steps
in
(skolem_steps @
[((waldmeister_name, []), Unknown, trm, waldmeister_skolemize_rule,
[(waldmeister_name ^ "_" ^ Int.toString (num_of_steps - 1),
if num_of_steps = 1 then isabelle_names else [])])])
end
end
fun introduce_waldmeister_skolems info proof_steps = proof_steps
|> maps (skolemization_steps info)
end;