(* Title: HOL/Tools/Nitpick/nitpick_preproc.ML
Author: Jasmin Blanchette, TU Muenchen
Copyright 2008, 2009, 2010
Nitpick's HOL preprocessor.
*)
signature NITPICK_PREPROC =
sig
type hol_context = Nitpick_HOL.hol_context
val preprocess_term :
hol_context -> term
-> ((term list * term list) * (bool * bool)) * term * bool
end
structure Nitpick_Preproc : NITPICK_PREPROC =
struct
open Nitpick_Util
open Nitpick_HOL
(* polarity -> string -> bool *)
fun is_positive_existential polar quant_s =
(polar = Pos andalso quant_s = @{const_name Ex}) orelse
(polar = Neg andalso quant_s <> @{const_name Ex})
(** Binary coding of integers **)
(* If a formula contains a numeral whose absolute value is more than this
threshold, the unary coding is likely not to work well and we prefer the
binary coding. *)
val binary_int_threshold = 3
(* term -> bool *)
fun may_use_binary_ints (t1 $ t2) =
may_use_binary_ints t1 andalso may_use_binary_ints t2
| may_use_binary_ints (t as Const (s, _)) =
t <> @{const Suc} andalso
not (member (op =) [@{const_name Abs_Frac}, @{const_name Rep_Frac},
@{const_name nat_gcd}, @{const_name nat_lcm},
@{const_name Frac}, @{const_name norm_frac}] s)
| may_use_binary_ints (Abs (_, _, t')) = may_use_binary_ints t'
| may_use_binary_ints _ = true
fun should_use_binary_ints (t1 $ t2) =
should_use_binary_ints t1 orelse should_use_binary_ints t2
| should_use_binary_ints (Const (s, T)) =
((s = @{const_name times} orelse s = @{const_name div}) andalso
is_integer_type (body_type T)) orelse
(String.isPrefix numeral_prefix s andalso
let val n = the (Int.fromString (unprefix numeral_prefix s)) in
n < ~ binary_int_threshold orelse n > binary_int_threshold
end)
| should_use_binary_ints (Abs (_, _, t')) = should_use_binary_ints t'
| should_use_binary_ints _ = false
(** Uncurrying **)
(* theory -> term -> int Termtab.tab -> int Termtab.tab *)
fun add_to_uncurry_table thy t =
let
(* term -> term list -> int Termtab.tab -> int Termtab.tab *)
fun aux (t1 $ t2) args table =
let val table = aux t2 [] table in aux t1 (t2 :: args) table end
| aux (Abs (_, _, t')) _ table = aux t' [] table
| aux (t as Const (x as (s, _))) args table =
if is_built_in_const thy [(NONE, true)] true x orelse
is_constr_like thy x orelse
is_sel s orelse s = @{const_name Sigma} then
table
else
Termtab.map_default (t, 65536) (curry Int.min (length args)) table
| aux _ _ table = table
in aux t [] end
(* int -> int -> string *)
fun uncurry_prefix_for k j =
uncurry_prefix ^ string_of_int k ^ "@" ^ string_of_int j ^ name_sep
(* int Termtab.tab term -> term *)
fun uncurry_term table t =
let
(* term -> term list -> term *)
fun aux (t1 $ t2) args = aux t1 (aux t2 [] :: args)
| aux (Abs (s, T, t')) args = betapplys (Abs (s, T, aux t' []), args)
| aux (t as Const (s, T)) args =
(case Termtab.lookup table t of
SOME n =>
if n >= 2 then
let
val arg_Ts = strip_n_binders n T |> fst
val j =
if is_iterator_type (hd arg_Ts) then
1
else case find_index (not_equal bool_T) arg_Ts of
~1 => n
| j => j
val ((before_args, tuple_args), after_args) =
args |> chop n |>> chop j
val ((before_arg_Ts, tuple_arg_Ts), rest_T) =
T |> strip_n_binders n |>> chop j
val tuple_T = HOLogic.mk_tupleT tuple_arg_Ts
in
if n - j < 2 then
betapplys (t, args)
else
betapplys (Const (uncurry_prefix_for (n - j) j ^ s,
before_arg_Ts ---> tuple_T --> rest_T),
before_args @ [mk_flat_tuple tuple_T tuple_args] @
after_args)
end
else
betapplys (t, args)
| NONE => betapplys (t, args))
| aux t args = betapplys (t, args)
in aux t [] end
(** Boxing **)
(* hol_context -> typ -> term -> term *)
fun constr_expand (hol_ctxt as {thy, stds, ...}) T t =
(case head_of t of
Const x => if is_constr_like thy x then t else raise SAME ()
| _ => raise SAME ())
handle SAME () =>
let
val x' as (_, T') =
if is_pair_type T then
let val (T1, T2) = HOLogic.dest_prodT T in
(@{const_name Pair}, T1 --> T2 --> T)
end
else
datatype_constrs hol_ctxt T |> hd
val arg_Ts = binder_types T'
in
list_comb (Const x', map2 (select_nth_constr_arg thy stds x' t)
(index_seq 0 (length arg_Ts)) arg_Ts)
end
(* hol_context -> bool -> term -> term *)
fun box_fun_and_pair_in_term (hol_ctxt as {thy, stds, fast_descrs, ...}) def
orig_t =
let
(* typ -> typ *)
fun box_relational_operator_type (Type ("fun", Ts)) =
Type ("fun", map box_relational_operator_type Ts)
| box_relational_operator_type (Type ("*", Ts)) =
Type ("*", map (box_type hol_ctxt InPair) Ts)
| box_relational_operator_type T = T
(* (term -> term) -> int -> term -> term *)
fun coerce_bound_no f j t =
case t of
t1 $ t2 => coerce_bound_no f j t1 $ coerce_bound_no f j t2
| Abs (s, T, t') => Abs (s, T, coerce_bound_no f (j + 1) t')
| Bound j' => if j' = j then f t else t
| _ => t
(* typ -> typ -> term -> term *)
fun coerce_bound_0_in_term new_T old_T =
old_T <> new_T ? coerce_bound_no (coerce_term [new_T] old_T new_T) 0
(* typ list -> typ -> term -> term *)
and coerce_term Ts new_T old_T t =
if old_T = new_T then
t
else
case (new_T, old_T) of
(Type (new_s, new_Ts as [new_T1, new_T2]),
Type ("fun", [old_T1, old_T2])) =>
(case eta_expand Ts t 1 of
Abs (s, _, t') =>
Abs (s, new_T1,
t' |> coerce_bound_0_in_term new_T1 old_T1
|> coerce_term (new_T1 :: Ts) new_T2 old_T2)
|> Envir.eta_contract
|> new_s <> "fun"
? construct_value thy stds
(@{const_name FunBox}, Type ("fun", new_Ts) --> new_T)
o single
| t' => raise TERM ("Nitpick_Preproc.box_fun_and_pair_in_term.\
\coerce_term", [t']))
| (Type (new_s, new_Ts as [new_T1, new_T2]),
Type (old_s, old_Ts as [old_T1, old_T2])) =>
if old_s = @{type_name fun_box} orelse
old_s = @{type_name pair_box} orelse old_s = "*" then
case constr_expand hol_ctxt old_T t of
Const (@{const_name FunBox}, _) $ t1 =>
if new_s = "fun" then
coerce_term Ts new_T (Type ("fun", old_Ts)) t1
else
construct_value thy stds
(@{const_name FunBox}, Type ("fun", new_Ts) --> new_T)
[coerce_term Ts (Type ("fun", new_Ts))
(Type ("fun", old_Ts)) t1]
| Const _ $ t1 $ t2 =>
construct_value thy stds
(if new_s = "*" then @{const_name Pair}
else @{const_name PairBox}, new_Ts ---> new_T)
[coerce_term Ts new_T1 old_T1 t1,
coerce_term Ts new_T2 old_T2 t2]
| t' => raise TERM ("Nitpick_Preproc.box_fun_and_pair_in_term.\
\coerce_term", [t'])
else
raise TYPE ("coerce_term", [new_T, old_T], [t])
| _ => raise TYPE ("coerce_term", [new_T, old_T], [t])
(* indexname * typ -> typ * term -> typ option list -> typ option list *)
fun add_boxed_types_for_var (z as (_, T)) (T', t') =
case t' of
Var z' => z' = z ? insert (op =) T'
| Const (@{const_name Pair}, _) $ t1 $ t2 =>
(case T' of
Type (_, [T1, T2]) =>
fold (add_boxed_types_for_var z) [(T1, t1), (T2, t2)]
| _ => raise TYPE ("Nitpick_Preproc.box_fun_and_pair_in_term.\
\add_boxed_types_for_var", [T'], []))
| _ => exists_subterm (curry (op =) (Var z)) t' ? insert (op =) T
(* typ list -> typ list -> term -> indexname * typ -> typ *)
fun box_var_in_def new_Ts old_Ts t (z as (_, T)) =
case t of
@{const Trueprop} $ t1 => box_var_in_def new_Ts old_Ts t1 z
| Const (s0, _) $ t1 $ _ =>
if s0 = @{const_name "=="} orelse s0 = @{const_name "op ="} then
let
val (t', args) = strip_comb t1
val T' = fastype_of1 (new_Ts, do_term new_Ts old_Ts Neut t')
in
case fold (add_boxed_types_for_var z)
(fst (strip_n_binders (length args) T') ~~ args) [] of
[T''] => T''
| _ => T
end
else
T
| _ => T
(* typ list -> typ list -> polarity -> string -> typ -> string -> typ
-> term -> term *)
and do_quantifier new_Ts old_Ts polar quant_s quant_T abs_s abs_T t =
let
val abs_T' =
if polar = Neut orelse is_positive_existential polar quant_s then
box_type hol_ctxt InFunLHS abs_T
else
abs_T
val body_T = body_type quant_T
in
Const (quant_s, (abs_T' --> body_T) --> body_T)
$ Abs (abs_s, abs_T',
t |> do_term (abs_T' :: new_Ts) (abs_T :: old_Ts) polar)
end
(* typ list -> typ list -> string -> typ -> term -> term -> term *)
and do_equals new_Ts old_Ts s0 T0 t1 t2 =
let
val (t1, t2) = pairself (do_term new_Ts old_Ts Neut) (t1, t2)
val (T1, T2) = pairself (curry fastype_of1 new_Ts) (t1, t2)
val T = [T1, T2] |> sort TermOrd.typ_ord |> List.last
in
list_comb (Const (s0, T --> T --> body_type T0),
map2 (coerce_term new_Ts T) [T1, T2] [t1, t2])
end
(* string -> typ -> term *)
and do_description_operator s T =
let val T1 = box_type hol_ctxt InFunLHS (range_type T) in
Const (s, (T1 --> bool_T) --> T1)
end
(* typ list -> typ list -> polarity -> term -> term *)
and do_term new_Ts old_Ts polar t =
case t of
Const (s0 as @{const_name all}, T0) $ Abs (s1, T1, t1) =>
do_quantifier new_Ts old_Ts polar s0 T0 s1 T1 t1
| Const (s0 as @{const_name "=="}, T0) $ t1 $ t2 =>
do_equals new_Ts old_Ts s0 T0 t1 t2
| @{const "==>"} $ t1 $ t2 =>
@{const "==>"} $ do_term new_Ts old_Ts (flip_polarity polar) t1
$ do_term new_Ts old_Ts polar t2
| @{const Pure.conjunction} $ t1 $ t2 =>
@{const Pure.conjunction} $ do_term new_Ts old_Ts polar t1
$ do_term new_Ts old_Ts polar t2
| @{const Trueprop} $ t1 =>
@{const Trueprop} $ do_term new_Ts old_Ts polar t1
| @{const Not} $ t1 =>
@{const Not} $ do_term new_Ts old_Ts (flip_polarity polar) t1
| Const (s0 as @{const_name All}, T0) $ Abs (s1, T1, t1) =>
do_quantifier new_Ts old_Ts polar s0 T0 s1 T1 t1
| Const (s0 as @{const_name Ex}, T0) $ Abs (s1, T1, t1) =>
do_quantifier new_Ts old_Ts polar s0 T0 s1 T1 t1
| Const (s0 as @{const_name "op ="}, T0) $ t1 $ t2 =>
do_equals new_Ts old_Ts s0 T0 t1 t2
| @{const "op &"} $ t1 $ t2 =>
@{const "op &"} $ do_term new_Ts old_Ts polar t1
$ do_term new_Ts old_Ts polar t2
| @{const "op |"} $ t1 $ t2 =>
@{const "op |"} $ do_term new_Ts old_Ts polar t1
$ do_term new_Ts old_Ts polar t2
| @{const "op -->"} $ t1 $ t2 =>
@{const "op -->"} $ do_term new_Ts old_Ts (flip_polarity polar) t1
$ do_term new_Ts old_Ts polar t2
| Const (s as @{const_name The}, T) => do_description_operator s T
| Const (s as @{const_name Eps}, T) => do_description_operator s T
| Const (@{const_name quot_normal}, Type ("fun", [_, T2])) =>
let val T' = box_type hol_ctxt InSel T2 in
Const (@{const_name quot_normal}, T' --> T')
end
| Const (s as @{const_name Tha}, T) => do_description_operator s T
| Const (x as (s, T)) =>
Const (s, if s = @{const_name converse} orelse
s = @{const_name trancl} then
box_relational_operator_type T
else if is_built_in_const thy stds fast_descrs x orelse
s = @{const_name Sigma} then
T
else if is_constr_like thy x then
box_type hol_ctxt InConstr T
else if is_sel s
orelse is_rep_fun thy x then
box_type hol_ctxt InSel T
else
box_type hol_ctxt InExpr T)
| t1 $ Abs (s, T, t2') =>
let
val t1 = do_term new_Ts old_Ts Neut t1
val T1 = fastype_of1 (new_Ts, t1)
val (s1, Ts1) = dest_Type T1
val T' = hd (snd (dest_Type (hd Ts1)))
val t2 = Abs (s, T', do_term (T' :: new_Ts) (T :: old_Ts) Neut t2')
val T2 = fastype_of1 (new_Ts, t2)
val t2 = coerce_term new_Ts (hd Ts1) T2 t2
in
betapply (if s1 = "fun" then
t1
else
select_nth_constr_arg thy stds
(@{const_name FunBox}, Type ("fun", Ts1) --> T1) t1 0
(Type ("fun", Ts1)), t2)
end
| t1 $ t2 =>
let
val t1 = do_term new_Ts old_Ts Neut t1
val T1 = fastype_of1 (new_Ts, t1)
val (s1, Ts1) = dest_Type T1
val t2 = do_term new_Ts old_Ts Neut t2
val T2 = fastype_of1 (new_Ts, t2)
val t2 = coerce_term new_Ts (hd Ts1) T2 t2
in
betapply (if s1 = "fun" then
t1
else
select_nth_constr_arg thy stds
(@{const_name FunBox}, Type ("fun", Ts1) --> T1) t1 0
(Type ("fun", Ts1)), t2)
end
| Free (s, T) => Free (s, box_type hol_ctxt InExpr T)
| Var (z as (x, T)) =>
Var (x, if def then box_var_in_def new_Ts old_Ts orig_t z
else box_type hol_ctxt InExpr T)
| Bound _ => t
| Abs (s, T, t') =>
Abs (s, T, do_term (T :: new_Ts) (T :: old_Ts) Neut t')
in do_term [] [] Pos orig_t end
(** Destruction of constructors **)
val val_var_prefix = nitpick_prefix ^ "v"
(* typ list -> int -> int -> int -> term -> term *)
fun fresh_value_var Ts k n j t =
Var ((val_var_prefix ^ nat_subscript (n - j), k), fastype_of1 (Ts, t))
(* typ list -> term -> bool *)
fun has_heavy_bounds_or_vars Ts t =
let
(* typ list -> bool *)
fun aux [] = false
| aux [T] = is_fun_type T orelse is_pair_type T
| aux _ = true
in aux (map snd (Term.add_vars t []) @ map (nth Ts) (loose_bnos t)) end
(* hol_context -> typ list -> bool -> int -> int -> term -> term list
-> term list -> term * term list *)
fun pull_out_constr_comb ({thy, stds, ...} : hol_context) Ts relax k level t
args seen =
let val t_comb = list_comb (t, args) in
case t of
Const x =>
if not relax andalso is_constr thy stds x andalso
not (is_fun_type (fastype_of1 (Ts, t_comb))) andalso
has_heavy_bounds_or_vars Ts t_comb andalso
not (loose_bvar (t_comb, level)) then
let
val (j, seen) = case find_index (curry (op =) t_comb) seen of
~1 => (0, t_comb :: seen)
| j => (j, seen)
in (fresh_value_var Ts k (length seen) j t_comb, seen) end
else
(t_comb, seen)
| _ => (t_comb, seen)
end
(* (term -> term) -> typ list -> int -> term list -> term list *)
fun equations_for_pulled_out_constrs mk_eq Ts k seen =
let val n = length seen in
map2 (fn j => fn t => mk_eq (fresh_value_var Ts k n j t, t))
(index_seq 0 n) seen
end
(* hol_context -> bool -> term -> term *)
fun pull_out_universal_constrs hol_ctxt def t =
let
val k = maxidx_of_term t + 1
(* typ list -> bool -> term -> term list -> term list -> term * term list *)
fun do_term Ts def t args seen =
case t of
(t0 as Const (@{const_name "=="}, _)) $ t1 $ t2 =>
do_eq_or_imp Ts true def t0 t1 t2 seen
| (t0 as @{const "==>"}) $ t1 $ t2 =>
if def then (t, []) else do_eq_or_imp Ts false def t0 t1 t2 seen
| (t0 as Const (@{const_name "op ="}, _)) $ t1 $ t2 =>
do_eq_or_imp Ts true def t0 t1 t2 seen
| (t0 as @{const "op -->"}) $ t1 $ t2 =>
do_eq_or_imp Ts false def t0 t1 t2 seen
| Abs (s, T, t') =>
let val (t', seen) = do_term (T :: Ts) def t' [] seen in
(list_comb (Abs (s, T, t'), args), seen)
end
| t1 $ t2 =>
let val (t2, seen) = do_term Ts def t2 [] seen in
do_term Ts def t1 (t2 :: args) seen
end
| _ => pull_out_constr_comb hol_ctxt Ts def k 0 t args seen
(* typ list -> bool -> bool -> term -> term -> term -> term list
-> term * term list *)
and do_eq_or_imp Ts eq def t0 t1 t2 seen =
let
val (t2, seen) = if eq andalso def then (t2, seen)
else do_term Ts false t2 [] seen
val (t1, seen) = do_term Ts false t1 [] seen
in (t0 $ t1 $ t2, seen) end
val (concl, seen) = do_term [] def t [] []
in
Logic.list_implies (equations_for_pulled_out_constrs Logic.mk_equals [] k
seen, concl)
end
(* term -> term -> term *)
fun mk_exists v t =
HOLogic.exists_const (fastype_of v) $ lambda v (incr_boundvars 1 t)
(* hol_context -> term -> term *)
fun pull_out_existential_constrs hol_ctxt t =
let
val k = maxidx_of_term t + 1
(* typ list -> int -> term -> term list -> term list -> term * term list *)
fun aux Ts num_exists t args seen =
case t of
(t0 as Const (@{const_name Ex}, _)) $ Abs (s1, T1, t1) =>
let
val (t1, seen') = aux (T1 :: Ts) (num_exists + 1) t1 [] []
val n = length seen'
(* unit -> term list *)
fun vars () = map2 (fresh_value_var Ts k n) (index_seq 0 n) seen'
in
(equations_for_pulled_out_constrs HOLogic.mk_eq Ts k seen'
|> List.foldl s_conj t1 |> fold mk_exists (vars ())
|> curry3 Abs s1 T1 |> curry (op $) t0, seen)
end
| t1 $ t2 =>
let val (t2, seen) = aux Ts num_exists t2 [] seen in
aux Ts num_exists t1 (t2 :: args) seen
end
| Abs (s, T, t') =>
let
val (t', seen) = aux (T :: Ts) 0 t' [] (map (incr_boundvars 1) seen)
in (list_comb (Abs (s, T, t'), args), map (incr_boundvars ~1) seen) end
| _ =>
if num_exists > 0 then
pull_out_constr_comb hol_ctxt Ts false k num_exists t args seen
else
(list_comb (t, args), seen)
in aux [] 0 t [] [] |> fst end
(* hol_context -> bool -> term -> term *)
fun destroy_pulled_out_constrs (hol_ctxt as {thy, stds, ...}) axiom t =
let
(* styp -> int *)
val num_occs_of_var =
fold_aterms (fn Var z => (fn f => fn z' => f z' |> z = z' ? Integer.add 1)
| _ => I) t (K 0)
(* bool -> term -> term *)
fun aux careful ((t0 as Const (@{const_name "=="}, _)) $ t1 $ t2) =
aux_eq careful true t0 t1 t2
| aux careful ((t0 as @{const "==>"}) $ t1 $ t2) =
t0 $ aux false t1 $ aux careful t2
| aux careful ((t0 as Const (@{const_name "op ="}, _)) $ t1 $ t2) =
aux_eq careful true t0 t1 t2
| aux careful ((t0 as @{const "op -->"}) $ t1 $ t2) =
t0 $ aux false t1 $ aux careful t2
| aux careful (Abs (s, T, t')) = Abs (s, T, aux careful t')
| aux careful (t1 $ t2) = aux careful t1 $ aux careful t2
| aux _ t = t
(* bool -> bool -> term -> term -> term -> term *)
and aux_eq careful pass1 t0 t1 t2 =
((if careful then
raise SAME ()
else if axiom andalso is_Var t2 andalso
num_occs_of_var (dest_Var t2) = 1 then
@{const True}
else case strip_comb t2 of
(* The first case is not as general as it could be. *)
(Const (@{const_name PairBox}, _),
[Const (@{const_name fst}, _) $ Var z1,
Const (@{const_name snd}, _) $ Var z2]) =>
if z1 = z2 andalso num_occs_of_var z1 = 2 then @{const True}
else raise SAME ()
| (Const (x as (s, T)), args) =>
let val arg_Ts = binder_types T in
if length arg_Ts = length args andalso
(is_constr thy stds x orelse s = @{const_name Pair}) andalso
(not careful orelse not (is_Var t1) orelse
String.isPrefix val_var_prefix (fst (fst (dest_Var t1)))) then
discriminate_value hol_ctxt x t1 ::
map3 (sel_eq x t1) (index_seq 0 (length args)) arg_Ts args
|> foldr1 s_conj
else
raise SAME ()
end
| _ => raise SAME ())
|> body_type (type_of t0) = prop_T ? HOLogic.mk_Trueprop)
handle SAME () => if pass1 then aux_eq careful false t0 t2 t1
else t0 $ aux false t2 $ aux false t1
(* styp -> term -> int -> typ -> term -> term *)
and sel_eq x t n nth_T nth_t =
HOLogic.eq_const nth_T $ nth_t
$ select_nth_constr_arg thy stds x t n nth_T
|> aux false
in aux axiom t end
(** Destruction of universal and existential equalities **)
(* term -> term *)
fun curry_assms (@{const "==>"} $ (@{const Trueprop}
$ (@{const "op &"} $ t1 $ t2)) $ t3) =
curry_assms (Logic.list_implies ([t1, t2] |> map HOLogic.mk_Trueprop, t3))
| curry_assms (@{const "==>"} $ t1 $ t2) =
@{const "==>"} $ curry_assms t1 $ curry_assms t2
| curry_assms t = t
(* term -> term *)
val destroy_universal_equalities =
let
(* term list -> (indexname * typ) list -> term -> term *)
fun aux prems zs t =
case t of
@{const "==>"} $ t1 $ t2 => aux_implies prems zs t1 t2
| _ => Logic.list_implies (rev prems, t)
(* term list -> (indexname * typ) list -> term -> term -> term *)
and aux_implies prems zs t1 t2 =
case t1 of
Const (@{const_name "=="}, _) $ Var z $ t' => aux_eq prems zs z t' t1 t2
| @{const Trueprop} $ (Const (@{const_name "op ="}, _) $ Var z $ t') =>
aux_eq prems zs z t' t1 t2
| @{const Trueprop} $ (Const (@{const_name "op ="}, _) $ t' $ Var z) =>
aux_eq prems zs z t' t1 t2
| _ => aux (t1 :: prems) (Term.add_vars t1 zs) t2
(* term list -> (indexname * typ) list -> indexname * typ -> term -> term
-> term -> term *)
and aux_eq prems zs z t' t1 t2 =
if not (member (op =) zs z) andalso
not (exists_subterm (curry (op =) (Var z)) t') then
aux prems zs (subst_free [(Var z, t')] t2)
else
aux (t1 :: prems) (Term.add_vars t1 zs) t2
in aux [] [] end
(* theory -> (typ option * bool) list -> int -> term list -> term list
-> (term * term list) option *)
fun find_bound_assign thy stds j =
let
(* term list -> term list -> (term * term list) option *)
fun do_term _ [] = NONE
| do_term seen (t :: ts) =
let
(* bool -> term -> term -> (term * term list) option *)
fun do_eq pass1 t1 t2 =
(if loose_bvar1 (t2, j) then
if pass1 then do_eq false t2 t1 else raise SAME ()
else case t1 of
Bound j' => if j' = j then SOME (t2, ts @ seen) else raise SAME ()
| Const (s, Type ("fun", [T1, T2])) $ Bound j' =>
if j' = j andalso
s = nth_sel_name_for_constr_name @{const_name FunBox} 0 then
SOME (construct_value thy stds (@{const_name FunBox}, T2 --> T1)
[t2], ts @ seen)
else
raise SAME ()
| _ => raise SAME ())
handle SAME () => do_term (t :: seen) ts
in
case t of
Const (@{const_name "op ="}, _) $ t1 $ t2 => do_eq true t1 t2
| _ => do_term (t :: seen) ts
end
in do_term end
(* int -> term -> term -> term *)
fun subst_one_bound j arg t =
let
(* term * int -> term *)
fun aux (Bound i, lev) =
if i < lev then raise SAME ()
else if i = lev then incr_boundvars (lev - j) arg
else Bound (i - 1)
| aux (Abs (a, T, body), lev) = Abs (a, T, aux (body, lev + 1))
| aux (f $ t, lev) =
(aux (f, lev) $ (aux (t, lev) handle SAME () => t)
handle SAME () => f $ aux (t, lev))
| aux _ = raise SAME ()
in aux (t, j) handle SAME () => t end
(* hol_context -> term -> term *)
fun destroy_existential_equalities ({thy, stds, ...} : hol_context) =
let
(* string list -> typ list -> term list -> term *)
fun kill [] [] ts = foldr1 s_conj ts
| kill (s :: ss) (T :: Ts) ts =
(case find_bound_assign thy stds (length ss) [] ts of
SOME (_, []) => @{const True}
| SOME (arg_t, ts) =>
kill ss Ts (map (subst_one_bound (length ss)
(incr_bv (~1, length ss + 1, arg_t))) ts)
| NONE =>
Const (@{const_name Ex}, (T --> bool_T) --> bool_T)
$ Abs (s, T, kill ss Ts ts))
| kill _ _ _ = raise UnequalLengths
(* string list -> typ list -> term -> term *)
fun gather ss Ts (Const (@{const_name Ex}, _) $ Abs (s1, T1, t1)) =
gather (ss @ [s1]) (Ts @ [T1]) t1
| gather [] [] (Abs (s, T, t1)) = Abs (s, T, gather [] [] t1)
| gather [] [] (t1 $ t2) = gather [] [] t1 $ gather [] [] t2
| gather [] [] t = t
| gather ss Ts t = kill ss Ts (conjuncts_of (gather [] [] t))
in gather [] [] end
(** Skolemization **)
(* int -> int -> string *)
fun skolem_prefix_for k j =
skolem_prefix ^ string_of_int k ^ "@" ^ string_of_int j ^ name_sep
(* hol_context -> int -> term -> term *)
fun skolemize_term_and_more (hol_ctxt as {thy, def_table, skolems, ...})
skolem_depth =
let
(* int list -> int list *)
val incrs = map (Integer.add 1)
(* string list -> typ list -> int list -> int -> polarity -> term -> term *)
fun aux ss Ts js depth polar t =
let
(* string -> typ -> string -> typ -> term -> term *)
fun do_quantifier quant_s quant_T abs_s abs_T t =
if not (loose_bvar1 (t, 0)) then
aux ss Ts js depth polar (incr_boundvars ~1 t)
else if depth <= skolem_depth andalso
is_positive_existential polar quant_s then
let
val j = length (!skolems) + 1
val sko_s = skolem_prefix_for (length js) j ^ abs_s
val _ = Unsynchronized.change skolems (cons (sko_s, ss))
val sko_t = list_comb (Const (sko_s, rev Ts ---> abs_T),
map Bound (rev js))
val abs_t = Abs (abs_s, abs_T, aux ss Ts (incrs js) depth polar t)
in
if null js then betapply (abs_t, sko_t)
else Const (@{const_name Let}, abs_T --> quant_T) $ sko_t $ abs_t
end
else
Const (quant_s, quant_T)
$ Abs (abs_s, abs_T,
if is_higher_order_type abs_T then
t
else
aux (abs_s :: ss) (abs_T :: Ts) (0 :: incrs js)
(depth + 1) polar t)
in
case t of
Const (s0 as @{const_name all}, T0) $ Abs (s1, T1, t1) =>
do_quantifier s0 T0 s1 T1 t1
| @{const "==>"} $ t1 $ t2 =>
@{const "==>"} $ aux ss Ts js depth (flip_polarity polar) t1
$ aux ss Ts js depth polar t2
| @{const Pure.conjunction} $ t1 $ t2 =>
@{const Pure.conjunction} $ aux ss Ts js depth polar t1
$ aux ss Ts js depth polar t2
| @{const Trueprop} $ t1 =>
@{const Trueprop} $ aux ss Ts js depth polar t1
| @{const Not} $ t1 =>
@{const Not} $ aux ss Ts js depth (flip_polarity polar) t1
| Const (s0 as @{const_name All}, T0) $ Abs (s1, T1, t1) =>
do_quantifier s0 T0 s1 T1 t1
| Const (s0 as @{const_name Ex}, T0) $ Abs (s1, T1, t1) =>
do_quantifier s0 T0 s1 T1 t1
| @{const "op &"} $ t1 $ t2 =>
@{const "op &"} $ aux ss Ts js depth polar t1
$ aux ss Ts js depth polar t2
| @{const "op |"} $ t1 $ t2 =>
@{const "op |"} $ aux ss Ts js depth polar t1
$ aux ss Ts js depth polar t2
| @{const "op -->"} $ t1 $ t2 =>
@{const "op -->"} $ aux ss Ts js depth (flip_polarity polar) t1
$ aux ss Ts js depth polar t2
| (t0 as Const (@{const_name Let}, _)) $ t1 $ t2 =>
t0 $ t1 $ aux ss Ts js depth polar t2
| Const (x as (s, T)) =>
if is_inductive_pred hol_ctxt x andalso
not (is_well_founded_inductive_pred hol_ctxt x) then
let
val gfp = (fixpoint_kind_of_const thy def_table x = Gfp)
val (pref, connective, set_oper) =
if gfp then
(lbfp_prefix, @{const "op |"},
@{const_name semilattice_sup_class.sup})
else
(ubfp_prefix, @{const "op &"},
@{const_name semilattice_inf_class.inf})
(* unit -> term *)
fun pos () = unrolled_inductive_pred_const hol_ctxt gfp x
|> aux ss Ts js depth polar
fun neg () = Const (pref ^ s, T)
in
(case polar |> gfp ? flip_polarity of
Pos => pos ()
| Neg => neg ()
| Neut =>
if is_fun_type T then
let
val ((trunk_arg_Ts, rump_arg_T), body_T) =
T |> strip_type |>> split_last
val set_T = rump_arg_T --> body_T
(* (unit -> term) -> term *)
fun app f =
list_comb (f (),
map Bound (length trunk_arg_Ts - 1 downto 0))
in
List.foldr absdummy
(Const (set_oper, set_T --> set_T --> set_T)
$ app pos $ app neg) trunk_arg_Ts
end
else
connective $ pos () $ neg ())
end
else
Const x
| t1 $ t2 =>
betapply (aux ss Ts [] (skolem_depth + 1) polar t1,
aux ss Ts [] depth Neut t2)
| Abs (s, T, t1) => Abs (s, T, aux ss Ts (incrs js) depth polar t1)
| _ => t
end
in aux [] [] [] 0 Pos end
(** Function specialization **)
(* term -> term list *)
fun params_in_equation (@{const "==>"} $ _ $ t2) = params_in_equation t2
| params_in_equation (@{const Trueprop} $ t1) = params_in_equation t1
| params_in_equation (Const (@{const_name "op ="}, _) $ t1 $ _) =
snd (strip_comb t1)
| params_in_equation _ = []
(* styp -> styp -> int list -> term list -> term list -> term -> term *)
fun specialize_fun_axiom x x' fixed_js fixed_args extra_args t =
let
val k = fold Integer.max (map maxidx_of_term (fixed_args @ extra_args)) 0
+ 1
val t = map_aterms (fn Var ((s, i), T) => Var ((s, k + i), T) | t' => t') t
val fixed_params = filter_indices fixed_js (params_in_equation t)
(* term list -> term -> term *)
fun aux args (Abs (s, T, t)) = list_comb (Abs (s, T, aux [] t), args)
| aux args (t1 $ t2) = aux (aux [] t2 :: args) t1
| aux args t =
if t = Const x then
list_comb (Const x', extra_args @ filter_out_indices fixed_js args)
else
let val j = find_index (curry (op =) t) fixed_params in
list_comb (if j >= 0 then nth fixed_args j else t, args)
end
in aux [] t end
(* hol_context -> styp -> (int * term option) list *)
fun static_args_in_term ({ersatz_table, ...} : hol_context) x t =
let
(* term -> term list -> term list -> term list list *)
fun fun_calls (Abs (_, _, t)) _ = fun_calls t []
| fun_calls (t1 $ t2) args = fun_calls t2 [] #> fun_calls t1 (t2 :: args)
| fun_calls t args =
(case t of
Const (x' as (s', T')) =>
x = x' orelse (case AList.lookup (op =) ersatz_table s' of
SOME s'' => x = (s'', T')
| NONE => false)
| _ => false) ? cons args
(* term list list -> term list list -> term list -> term list list *)
fun call_sets [] [] vs = [vs]
| call_sets [] uss vs = vs :: call_sets uss [] []
| call_sets ([] :: _) _ _ = []
| call_sets ((t :: ts) :: tss) uss vs =
OrdList.insert TermOrd.term_ord t vs |> call_sets tss (ts :: uss)
val sets = call_sets (fun_calls t [] []) [] []
val indexed_sets = sets ~~ (index_seq 0 (length sets))
in
fold_rev (fn (set, j) =>
case set of
[Var _] => AList.lookup (op =) indexed_sets set = SOME j
? cons (j, NONE)
| [t as Const _] => cons (j, SOME t)
| [t as Free _] => cons (j, SOME t)
| _ => I) indexed_sets []
end
(* hol_context -> styp -> term list -> (int * term option) list *)
fun static_args_in_terms hol_ctxt x =
map (static_args_in_term hol_ctxt x)
#> fold1 (OrdList.inter (prod_ord int_ord (option_ord TermOrd.term_ord)))
(* (int * term option) list -> (int * term) list -> int list *)
fun overlapping_indices [] _ = []
| overlapping_indices _ [] = []
| overlapping_indices (ps1 as (j1, t1) :: ps1') (ps2 as (j2, t2) :: ps2') =
if j1 < j2 then overlapping_indices ps1' ps2
else if j1 > j2 then overlapping_indices ps1 ps2'
else overlapping_indices ps1' ps2' |> the_default t2 t1 = t2 ? cons j1
(* typ list -> term -> bool *)
fun is_eligible_arg Ts t =
let val bad_Ts = map snd (Term.add_vars t []) @ map (nth Ts) (loose_bnos t) in
null bad_Ts orelse
(is_higher_order_type (fastype_of1 (Ts, t)) andalso
forall (not o is_higher_order_type) bad_Ts)
end
(* int -> string *)
fun special_prefix_for j = special_prefix ^ string_of_int j ^ name_sep
(* If a constant's definition is picked up deeper than this threshold, we
prevent excessive specialization by not specializing it. *)
val special_max_depth = 20
val bound_var_prefix = "b"
(* hol_context -> int -> term -> term *)
fun specialize_consts_in_term (hol_ctxt as {specialize, simp_table,
special_funs, ...}) depth t =
if not specialize orelse depth > special_max_depth then
t
else
let
val blacklist = if depth = 0 then []
else case term_under_def t of Const x => [x] | _ => []
(* term list -> typ list -> term -> term *)
fun aux args Ts (Const (x as (s, T))) =
((if not (member (op =) blacklist x) andalso not (null args) andalso
not (String.isPrefix special_prefix s) andalso
is_equational_fun hol_ctxt x then
let
val eligible_args = filter (is_eligible_arg Ts o snd)
(index_seq 0 (length args) ~~ args)
val _ = not (null eligible_args) orelse raise SAME ()
val old_axs = equational_fun_axioms hol_ctxt x
|> map (destroy_existential_equalities hol_ctxt)
val static_params = static_args_in_terms hol_ctxt x old_axs
val fixed_js = overlapping_indices static_params eligible_args
val _ = not (null fixed_js) orelse raise SAME ()
val fixed_args = filter_indices fixed_js args
val vars = fold Term.add_vars fixed_args []
|> sort (TermOrd.fast_indexname_ord o pairself fst)
val bound_js = fold (fn t => fn js => add_loose_bnos (t, 0, js))
fixed_args []
|> sort int_ord
val live_args = filter_out_indices fixed_js args
val extra_args = map Var vars @ map Bound bound_js @ live_args
val extra_Ts = map snd vars @ filter_indices bound_js Ts
val k = maxidx_of_term t + 1
(* int -> term *)
fun var_for_bound_no j =
Var ((bound_var_prefix ^
nat_subscript (find_index (curry (op =) j) bound_js
+ 1), k),
nth Ts j)
val fixed_args_in_axiom =
map (curry subst_bounds
(map var_for_bound_no (index_seq 0 (length Ts))))
fixed_args
in
case AList.lookup (op =) (!special_funs)
(x, fixed_js, fixed_args_in_axiom) of
SOME x' => list_comb (Const x', extra_args)
| NONE =>
let
val extra_args_in_axiom =
map Var vars @ map var_for_bound_no bound_js
val x' as (s', _) =
(special_prefix_for (length (!special_funs) + 1) ^ s,
extra_Ts @ filter_out_indices fixed_js (binder_types T)
---> body_type T)
val new_axs =
map (specialize_fun_axiom x x' fixed_js
fixed_args_in_axiom extra_args_in_axiom) old_axs
val _ =
Unsynchronized.change special_funs
(cons ((x, fixed_js, fixed_args_in_axiom), x'))
val _ = add_simps simp_table s' new_axs
in list_comb (Const x', extra_args) end
end
else
raise SAME ())
handle SAME () => list_comb (Const x, args))
| aux args Ts (Abs (s, T, t)) =
list_comb (Abs (s, T, aux [] (T :: Ts) t), args)
| aux args Ts (t1 $ t2) = aux (aux [] Ts t2 :: args) Ts t1
| aux args _ t = list_comb (t, args)
in aux [] [] t end
type special_triple = int list * term list * styp
val cong_var_prefix = "c"
(* typ -> special_triple -> special_triple -> term *)
fun special_congruence_axiom T (js1, ts1, x1) (js2, ts2, x2) =
let
val (bounds1, bounds2) = pairself (map Var o special_bounds) (ts1, ts2)
val Ts = binder_types T
val max_j = fold (fold Integer.max) [js1, js2] ~1
val (eqs, (args1, args2)) =
fold (fn j => case pairself (fn ps => AList.lookup (op =) ps j)
(js1 ~~ ts1, js2 ~~ ts2) of
(SOME t1, SOME t2) => apfst (cons (t1, t2))
| (SOME t1, NONE) => apsnd (apsnd (cons t1))
| (NONE, SOME t2) => apsnd (apfst (cons t2))
| (NONE, NONE) =>
let val v = Var ((cong_var_prefix ^ nat_subscript j, 0),
nth Ts j) in
apsnd (pairself (cons v))
end) (max_j downto 0) ([], ([], []))
in
Logic.list_implies (eqs |> filter_out (op =) |> distinct (op =)
|> map Logic.mk_equals,
Logic.mk_equals (list_comb (Const x1, bounds1 @ args1),
list_comb (Const x2, bounds2 @ args2)))
|> close_form (* TODO: needed? *)
end
(* hol_context -> styp list -> term list *)
fun special_congruence_axioms (hol_ctxt as {special_funs, ...}) xs =
let
val groups =
!special_funs
|> map (fn ((x, js, ts), x') => (x, (js, ts, x')))
|> AList.group (op =)
|> filter_out (is_equational_fun_surely_complete hol_ctxt o fst)
|> map (fn (x, zs) => (x, zs |> member (op =) xs x ? cons ([], [], x)))
(* special_triple -> int *)
fun generality (js, _, _) = ~(length js)
(* special_triple -> special_triple -> bool *)
fun is_more_specific (j1, t1, x1) (j2, t2, x2) =
x1 <> x2 andalso OrdList.subset (prod_ord int_ord TermOrd.term_ord)
(j2 ~~ t2, j1 ~~ t1)
(* typ -> special_triple list -> special_triple list -> special_triple list
-> term list -> term list *)
fun do_pass_1 _ [] [_] [_] = I
| do_pass_1 T skipped _ [] = do_pass_2 T skipped
| do_pass_1 T skipped all (z :: zs) =
case filter (is_more_specific z) all
|> sort (int_ord o pairself generality) of
[] => do_pass_1 T (z :: skipped) all zs
| (z' :: _) => cons (special_congruence_axiom T z z')
#> do_pass_1 T skipped all zs
(* typ -> special_triple list -> term list -> term list *)
and do_pass_2 _ [] = I
| do_pass_2 T (z :: zs) =
fold (cons o special_congruence_axiom T z) zs #> do_pass_2 T zs
in fold (fn ((_, T), zs) => do_pass_1 T [] zs zs) groups [] end
(** Axiom selection **)
(* Similar to "Refute.specialize_type" but returns all matches rather than only
the first (preorder) match. *)
(* theory -> styp -> term -> term list *)
fun multi_specialize_type thy slack (s, T) t =
let
(* term -> (typ * term) list -> (typ * term) list *)
fun aux (Const (s', T')) ys =
if s = s' then
ys |> (if AList.defined (op =) ys T' then
I
else
cons (T', Refute.monomorphic_term
(Sign.typ_match thy (T', T) Vartab.empty) t)
handle Type.TYPE_MATCH => I
| Refute.REFUTE _ =>
if slack then
I
else
raise NOT_SUPPORTED ("too much polymorphism in \
\axiom involving " ^ quote s))
else
ys
| aux _ ys = ys
in map snd (fold_aterms aux t []) end
(* theory -> bool -> const_table -> styp -> term list *)
fun nondef_props_for_const thy slack table (x as (s, _)) =
these (Symtab.lookup table s) |> maps (multi_specialize_type thy slack x)
(* 'a Symtab.table -> 'a list *)
fun all_table_entries table = Symtab.fold (append o snd) table []
(* const_table -> string -> const_table *)
fun extra_table table s = Symtab.make [(s, all_table_entries table)]
(* int -> term -> term *)
fun eval_axiom_for_term j t =
Logic.mk_equals (Const (eval_prefix ^ string_of_int j, fastype_of t), t)
(* term -> bool *)
val is_trivial_equation = the_default false o try (op aconv o Logic.dest_equals)
(* Prevents divergence in case of cyclic or infinite axiom dependencies. *)
val axioms_max_depth = 255
(* hol_context -> term -> (term list * term list) * (bool * bool) *)
fun axioms_for_term
(hol_ctxt as {thy, max_bisim_depth, stds, user_axioms, fast_descrs,
evals, def_table, nondef_table, user_nondefs, ...}) t =
let
type accumulator = styp list * (term list * term list)
(* (term list * term list -> term list)
-> ((term list -> term list) -> term list * term list
-> term list * term list)
-> int -> term -> accumulator -> accumulator *)
fun add_axiom get app depth t (accum as (xs, axs)) =
let
val t = t |> unfold_defs_in_term hol_ctxt
|> skolemize_term_and_more hol_ctxt ~1
in
if is_trivial_equation t then
accum
else
let val t' = t |> specialize_consts_in_term hol_ctxt depth in
if exists (member (op aconv) (get axs)) [t, t'] then accum
else add_axioms_for_term (depth + 1) t' (xs, app (cons t') axs)
end
end
(* int -> term -> accumulator -> accumulator *)
and add_def_axiom depth = add_axiom fst apfst depth
and add_nondef_axiom depth = add_axiom snd apsnd depth
and add_maybe_def_axiom depth t =
(if head_of t <> @{const "==>"} then add_def_axiom
else add_nondef_axiom) depth t
and add_eq_axiom depth t =
(if is_constr_pattern_formula thy t then add_def_axiom
else add_nondef_axiom) depth t
(* int -> term -> accumulator -> accumulator *)
and add_axioms_for_term depth t (accum as (xs, axs)) =
case t of
t1 $ t2 => accum |> fold (add_axioms_for_term depth) [t1, t2]
| Const (x as (s, T)) =>
(if member (op =) xs x orelse
is_built_in_const thy stds fast_descrs x then
accum
else
let val accum = (x :: xs, axs) in
if depth > axioms_max_depth then
raise TOO_LARGE ("Nitpick_Preproc.axioms_for_term.\
\add_axioms_for_term",
"too many nested axioms (" ^
string_of_int depth ^ ")")
else if Refute.is_const_of_class thy x then
let
val class = Logic.class_of_const s
val of_class = Logic.mk_of_class (TVar (("'a", 0), [class]),
class)
val ax1 = try (Refute.specialize_type thy x) of_class
val ax2 = Option.map (Refute.specialize_type thy x o snd)
(Refute.get_classdef thy class)
in
fold (add_maybe_def_axiom depth) (map_filter I [ax1, ax2])
accum
end
else if is_constr thy stds x then
accum
else if is_equational_fun hol_ctxt x then
fold (add_eq_axiom depth) (equational_fun_axioms hol_ctxt x)
accum
else if is_abs_fun thy x then
if is_quot_type thy (range_type T) then
raise NOT_SUPPORTED "\"Abs_\" function of quotient type"
else
accum |> fold (add_nondef_axiom depth)
(nondef_props_for_const thy false nondef_table x)
|> is_funky_typedef thy (range_type T)
? fold (add_maybe_def_axiom depth)
(nondef_props_for_const thy true
(extra_table def_table s) x)
else if is_rep_fun thy x then
if is_quot_type thy (domain_type T) then
raise NOT_SUPPORTED "\"Rep_\" function of quotient type"
else
accum |> fold (add_nondef_axiom depth)
(nondef_props_for_const thy false nondef_table x)
|> is_funky_typedef thy (range_type T)
? fold (add_maybe_def_axiom depth)
(nondef_props_for_const thy true
(extra_table def_table s) x)
|> add_axioms_for_term depth
(Const (mate_of_rep_fun thy x))
|> fold (add_def_axiom depth)
(inverse_axioms_for_rep_fun thy x)
else
accum |> user_axioms <> SOME false
? fold (add_nondef_axiom depth)
(nondef_props_for_const thy false nondef_table x)
end)
|> add_axioms_for_type depth T
| Free (_, T) => add_axioms_for_type depth T accum
| Var (_, T) => add_axioms_for_type depth T accum
| Bound _ => accum
| Abs (_, T, t) => accum |> add_axioms_for_term depth t
|> add_axioms_for_type depth T
(* int -> typ -> accumulator -> accumulator *)
and add_axioms_for_type depth T =
case T of
Type ("fun", Ts) => fold (add_axioms_for_type depth) Ts
| Type ("*", Ts) => fold (add_axioms_for_type depth) Ts
| @{typ prop} => I
| @{typ bool} => I
| @{typ unit} => I
| TFree (_, S) => add_axioms_for_sort depth T S
| TVar (_, S) => add_axioms_for_sort depth T S
| Type (z as (_, Ts)) =>
fold (add_axioms_for_type depth) Ts
#> (if is_pure_typedef thy T then
fold (add_maybe_def_axiom depth) (optimized_typedef_axioms thy z)
else if is_quot_type thy T then
fold (add_def_axiom depth) (optimized_quot_type_axioms thy stds z)
else if max_bisim_depth >= 0 andalso is_codatatype thy T then
fold (add_maybe_def_axiom depth)
(codatatype_bisim_axioms hol_ctxt T)
else
I)
(* int -> typ -> sort -> accumulator -> accumulator *)
and add_axioms_for_sort depth T S =
let
val supers = Sign.complete_sort thy S
val class_axioms =
maps (fn class => map prop_of (AxClass.get_info thy class |> #axioms
handle ERROR _ => [])) supers
val monomorphic_class_axioms =
map (fn t => case Term.add_tvars t [] of
[] => t
| [(x, S)] =>
Refute.monomorphic_term (Vartab.make [(x, (S, T))]) t
| _ => raise TERM ("Nitpick_Preproc.axioms_for_term.\
\add_axioms_for_sort", [t]))
class_axioms
in fold (add_nondef_axiom depth) monomorphic_class_axioms end
val (mono_user_nondefs, poly_user_nondefs) =
List.partition (null o Term.hidden_polymorphism) user_nondefs
val eval_axioms = map2 eval_axiom_for_term (index_seq 0 (length evals))
evals
val (xs, (defs, nondefs)) =
([], ([], [])) |> add_axioms_for_term 1 t
|> fold_rev (add_def_axiom 1) eval_axioms
|> user_axioms = SOME true
? fold (add_nondef_axiom 1) mono_user_nondefs
val defs = defs @ special_congruence_axioms hol_ctxt xs
in
((defs, nondefs), (user_axioms = SOME true orelse null mono_user_nondefs,
null poly_user_nondefs))
end
(** Simplification of constructor/selector terms **)
(* theory -> term -> term *)
fun simplify_constrs_and_sels thy t =
let
(* term -> int -> term *)
fun is_nth_sel_on t' n (Const (s, _) $ t) =
(t = t' andalso is_sel_like_and_no_discr s andalso
sel_no_from_name s = n)
| is_nth_sel_on _ _ _ = false
(* term -> term list -> term *)
fun do_term (Const (@{const_name Rep_Frac}, _)
$ (Const (@{const_name Abs_Frac}, _) $ t1)) [] = do_term t1 []
| do_term (Const (@{const_name Abs_Frac}, _)
$ (Const (@{const_name Rep_Frac}, _) $ t1)) [] = do_term t1 []
| do_term (t1 $ t2) args = do_term t1 (do_term t2 [] :: args)
| do_term (t as Const (x as (s, T))) (args as _ :: _) =
((if is_constr_like thy x then
if length args = num_binder_types T then
case hd args of
Const (_, T') $ t' =>
if domain_type T' = body_type T andalso
forall (uncurry (is_nth_sel_on t'))
(index_seq 0 (length args) ~~ args) then
t'
else
raise SAME ()
| _ => raise SAME ()
else
raise SAME ()
else if is_sel_like_and_no_discr s then
case strip_comb (hd args) of
(Const (x' as (s', T')), ts') =>
if is_constr_like thy x' andalso
constr_name_for_sel_like s = s' andalso
not (exists is_pair_type (binder_types T')) then
list_comb (nth ts' (sel_no_from_name s), tl args)
else
raise SAME ()
| _ => raise SAME ()
else
raise SAME ())
handle SAME () => betapplys (t, args))
| do_term (Abs (s, T, t')) args =
betapplys (Abs (s, T, do_term t' []), args)
| do_term t args = betapplys (t, args)
in do_term t [] end
(** Quantifier massaging: Distributing quantifiers **)
(* term -> term *)
fun distribute_quantifiers t =
case t of
(t0 as Const (@{const_name All}, T0)) $ Abs (s, T1, t1) =>
(case t1 of
(t10 as @{const "op &"}) $ t11 $ t12 =>
t10 $ distribute_quantifiers (t0 $ Abs (s, T1, t11))
$ distribute_quantifiers (t0 $ Abs (s, T1, t12))
| (t10 as @{const Not}) $ t11 =>
t10 $ distribute_quantifiers (Const (@{const_name Ex}, T0)
$ Abs (s, T1, t11))
| t1 =>
if not (loose_bvar1 (t1, 0)) then
distribute_quantifiers (incr_boundvars ~1 t1)
else
t0 $ Abs (s, T1, distribute_quantifiers t1))
| (t0 as Const (@{const_name Ex}, T0)) $ Abs (s, T1, t1) =>
(case distribute_quantifiers t1 of
(t10 as @{const "op |"}) $ t11 $ t12 =>
t10 $ distribute_quantifiers (t0 $ Abs (s, T1, t11))
$ distribute_quantifiers (t0 $ Abs (s, T1, t12))
| (t10 as @{const "op -->"}) $ t11 $ t12 =>
t10 $ distribute_quantifiers (Const (@{const_name All}, T0)
$ Abs (s, T1, t11))
$ distribute_quantifiers (t0 $ Abs (s, T1, t12))
| (t10 as @{const Not}) $ t11 =>
t10 $ distribute_quantifiers (Const (@{const_name All}, T0)
$ Abs (s, T1, t11))
| t1 =>
if not (loose_bvar1 (t1, 0)) then
distribute_quantifiers (incr_boundvars ~1 t1)
else
t0 $ Abs (s, T1, distribute_quantifiers t1))
| t1 $ t2 => distribute_quantifiers t1 $ distribute_quantifiers t2
| Abs (s, T, t') => Abs (s, T, distribute_quantifiers t')
| _ => t
(** Quantifier massaging: Pushing quantifiers inward **)
(* int -> int -> (int -> int) -> term -> term *)
fun renumber_bounds j n f t =
case t of
t1 $ t2 => renumber_bounds j n f t1 $ renumber_bounds j n f t2
| Abs (s, T, t') => Abs (s, T, renumber_bounds (j + 1) n f t')
| Bound j' =>
Bound (if j' >= j andalso j' < j + n then f (j' - j) + j else j')
| _ => t
(* Maximum number of quantifiers in a cluster for which the exponential
algorithm is used. Larger clusters use a heuristic inspired by Claessen &
Sörensson's polynomial binary splitting procedure (p. 5 of their MODEL 2003
paper). *)
val quantifier_cluster_threshold = 7
(* term -> term *)
val push_quantifiers_inward =
let
(* string -> string list -> typ list -> term -> term *)
fun aux quant_s ss Ts t =
(case t of
Const (s0, _) $ Abs (s1, T1, t1 as _ $ _) =>
if s0 = quant_s then
aux s0 (s1 :: ss) (T1 :: Ts) t1
else if quant_s = "" andalso
(s0 = @{const_name All} orelse s0 = @{const_name Ex}) then
aux s0 [s1] [T1] t1
else
raise SAME ()
| _ => raise SAME ())
handle SAME () =>
case t of
t1 $ t2 =>
if quant_s = "" then
aux "" [] [] t1 $ aux "" [] [] t2
else
let
val typical_card = 4
(* ('a -> ''b list) -> 'a list -> ''b list *)
fun big_union proj ps =
fold (fold (insert (op =)) o proj) ps []
val (ts, connective) = strip_any_connective t
val T_costs =
map (bounded_card_of_type 65536 typical_card []) Ts
val t_costs = map size_of_term ts
val num_Ts = length Ts
(* int -> int *)
val flip = curry (op -) (num_Ts - 1)
val t_boundss = map (map flip o loose_bnos) ts
(* (int list * int) list -> int list
-> (int list * int) list *)
fun merge costly_boundss [] = costly_boundss
| merge costly_boundss (j :: js) =
let
val (yeas, nays) =
List.partition (fn (bounds, _) =>
member (op =) bounds j)
costly_boundss
val yeas_bounds = big_union fst yeas
val yeas_cost = Integer.sum (map snd yeas)
* nth T_costs j
in merge ((yeas_bounds, yeas_cost) :: nays) js end
(* (int list * int) list -> int list -> int *)
val cost = Integer.sum o map snd oo merge
(* (int list * int) list -> int list -> int list *)
fun heuristically_best_permutation _ [] = []
| heuristically_best_permutation costly_boundss js =
let
val (costly_boundss, (j, js)) =
js |> map (`(merge costly_boundss o single))
|> sort (int_ord
o pairself (Integer.sum o map snd o fst))
|> split_list |>> hd ||> pairf hd tl
in
j :: heuristically_best_permutation costly_boundss js
end
val js =
if length Ts <= quantifier_cluster_threshold then
all_permutations (index_seq 0 num_Ts)
|> map (`(cost (t_boundss ~~ t_costs)))
|> sort (int_ord o pairself fst) |> hd |> snd
else
heuristically_best_permutation (t_boundss ~~ t_costs)
(index_seq 0 num_Ts)
val back_js = map (fn j => find_index (curry (op =) j) js)
(index_seq 0 num_Ts)
val ts = map (renumber_bounds 0 num_Ts (nth back_js o flip))
ts
(* (term * int list) list -> term *)
fun mk_connection [] =
raise ARG ("Nitpick_Preproc.push_quantifiers_inward.aux.\
\mk_connection", "")
| mk_connection ts_cum_bounds =
ts_cum_bounds |> map fst
|> foldr1 (fn (t1, t2) => connective $ t1 $ t2)
(* (term * int list) list -> int list -> term *)
fun build ts_cum_bounds [] = ts_cum_bounds |> mk_connection
| build ts_cum_bounds (j :: js) =
let
val (yeas, nays) =
List.partition (fn (_, bounds) =>
member (op =) bounds j)
ts_cum_bounds
||> map (apfst (incr_boundvars ~1))
in
if null yeas then
build nays js
else
let val T = nth Ts (flip j) in
build ((Const (quant_s, (T --> bool_T) --> bool_T)
$ Abs (nth ss (flip j), T,
mk_connection yeas),
big_union snd yeas) :: nays) js
end
end
in build (ts ~~ t_boundss) js end
| Abs (s, T, t') => Abs (s, T, aux "" [] [] t')
| _ => t
in aux "" [] [] end
(** Preprocessor entry point **)
(* hol_context -> term
-> ((term list * term list) * (bool * bool)) * term * bool *)
fun preprocess_term (hol_ctxt as {thy, stds, binary_ints, destroy_constrs,
boxes, skolemize, uncurry, ...}) t =
let
val skolem_depth = if skolemize then 4 else ~1
val (((def_ts, nondef_ts), (got_all_mono_user_axioms, no_poly_user_axioms)),
core_t) = t |> unfold_defs_in_term hol_ctxt
|> close_form
|> skolemize_term_and_more hol_ctxt skolem_depth
|> specialize_consts_in_term hol_ctxt 0
|> `(axioms_for_term hol_ctxt)
val binarize =
is_standard_datatype thy stds nat_T andalso
case binary_ints of
SOME false => false
| _ => forall may_use_binary_ints (core_t :: def_ts @ nondef_ts) andalso
(binary_ints = SOME true orelse
exists should_use_binary_ints (core_t :: def_ts @ nondef_ts))
val box = exists (not_equal (SOME false) o snd) boxes
val table =
Termtab.empty |> uncurry
? fold (add_to_uncurry_table thy) (core_t :: def_ts @ nondef_ts)
(* bool -> term -> term *)
fun do_rest def =
binarize ? binarize_nat_and_int_in_term
#> uncurry ? uncurry_term table
#> box ? box_fun_and_pair_in_term hol_ctxt def
#> destroy_constrs ? (pull_out_universal_constrs hol_ctxt def
#> pull_out_existential_constrs hol_ctxt
#> destroy_pulled_out_constrs hol_ctxt def)
#> curry_assms
#> destroy_universal_equalities
#> destroy_existential_equalities hol_ctxt
#> simplify_constrs_and_sels thy
#> distribute_quantifiers
#> push_quantifiers_inward
#> close_form
#> Term.map_abs_vars shortest_name
in
(((map (do_rest true) def_ts, map (do_rest false) nondef_ts),
(got_all_mono_user_axioms, no_poly_user_axioms)),
do_rest false core_t, binarize)
end
end;