Source file intCong.ml
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(**
IntCong - Integer congruences.
We rely on Zarith for arithmetic operations.
*)
open Bot
(** {2 Types} *)
type t = Z.t (** stride *) * Z.t (** offset *)
(**
The type of non-empty congruence sets (a,b) represents aℤ + b.
a must be positive, and 0 ≤ b < a when a > 0.
0ℤ+b represents the singleton {b}.
1ℤ+0 represents the whole set of integers.
aℤ+b represents the set { ak + b | k ∊ ℕ }.
*)
type t_with_bot = t with_bot
(** The type of possibly empty congruence sets. *)
let is_valid ((a,b):t) : bool =
a = Z.zero || (a > Z.zero && Z.zero <= b && b < a)
module I = ItvUtils.IntItv
module B = ItvUtils.IntBound
(** {2 Arithmetic utilities} *)
let gcd (a:Z.t) (b:Z.t) : Z.t =
if a = Z.zero then b
else if b = Z.zero then a
else Z.gcd a b
(** Greatest common divisor of |a| and |b|. 0 is neutral. *)
let gcd3 (a:Z.t) (b:Z.t) (c:Z.t) : Z.t =
gcd (gcd a b) c
let gcd_ext (a:Z.t) (b:Z.t) : Z.t * Z.t * Z.t * Z.t =
let gcd, u, v = Z.gcdext a b in
gcd, Z.div (Z.mul a b) gcd, u, v
(** Returns the gcd, lcm and cofactors u, v such that ua+vb=gcd.
Undefined if a or b is 0.
*)
let divides (a:Z.t) (b:Z.t) : bool =
if a = Z.zero then b = Z.zero
else Z.rem b a = Z.zero
(** Wheter b is a multiple of a. Always true if b = 0. *)
let rem_zero (a:Z.t) (b:Z.t) : Z.t =
if b = Z.zero then a else Z.erem a b
(** As Z.erem, but rem_zero a 0 = a. *)
(** {2 Constructors} *)
let of_z (a:Z.t) (b:Z.t) : t =
if a = Z.zero then a, b
else
let a = Z.abs a in
a, Z.erem b a
(** Returns aℤ + b. *)
let of_int (a:int) (b:int) : t =
of_z (Z.of_int a) (Z.of_int b)
let of_int64 (a:int64) (b:int64) : t =
of_z (Z.of_int64 a) (Z.of_int64 b)
let cst (b:Z.t) : t = Z.zero, b
(** Returns 0ℤ + b *)
let cst_int (b:int) : t = cst (Z.of_int b)
let cst_int64 (b:int64) : t = cst (Z.of_int64 b)
let zero : t = cst_int 0
(** 0ℤ+0 *)
let one : t = cst_int 1
(** 0ℤ+1 *)
let mone : t = cst_int (-1)
(** 0ℤ-1 *)
let minf_inf : t = of_int 1 0
(** 1ℤ+0 *)
let of_range (lo:Z.t) (hi:Z.t) : t =
if lo = hi then cst lo else minf_inf
let of_range_bot (lo:Z.t) (hi:Z.t) : t with_bot =
if lo = hi then Nb (cst lo)
else if lo > hi then BOT
else Nb minf_inf
let of_bound (lo:B.t) (hi:B.t) : t =
match lo,hi with
| B.Finite l, B.Finite h -> of_range l h
| _ -> minf_inf
let of_bound_bot (lo:B.t) (hi:B.t) : t with_bot =
match lo,hi with
| B.Finite l, B.Finite h -> of_range_bot l h
| _ -> Nb minf_inf
(** Congruence overapproximating an interval. *)
(** {2 Predicates} *)
let equal (a:t) (b:t) : bool =
a = b
(** Equality. = also works. *)
let equal_bot : t_with_bot -> t_with_bot -> bool =
bot_equal equal
let included ((a,b):t) ((c,d):t) : bool =
divides c a && rem_zero b c = d
(** Set ordering. *)
let included_bot : t_with_bot -> t_with_bot -> bool =
bot_included included
let intersect ((a,b):t) ((a',b'):t) : bool =
rem_zero (Z.sub b b') (gcd a a') = Z.zero
(** Whether the intersection is non-empty. *)
let intersect_bot : t_with_bot -> t_with_bot -> bool =
bot_dfl2 false intersect
let contains (x:Z.t) ((a,b):t) : bool =
rem_zero x a = b
(** Whether the set contains some value x. *)
let compare ((a,b):t) ((a',b'):t) : int =
if a = a' then Z.compare b b' else Z.compare a a'
(**
A total ordering (lexical ordering) returning -1, 0, or 1.
Can be used as compare for sets, maps, etc.
*)
let compare_bot (x:t with_bot) (y:t with_bot) : int =
Bot.bot_compare compare x y
(** Total ordering on possibly empty congruences. *)
let contains_zero ((a,b):t) : bool =
b = Z.zero
(** Whether the congruence contains zero. *)
let contains_one ((a,b):t) : bool =
a = Z.one || b = Z.one
(** Whether the congruence contains one. *)
let contains_nonzero (ab:t) : bool =
ab <> zero
(** Whether the congruence contains a non-zero value. *)
let is_zero (ab:t) : bool = ab = zero
let is_positive ((a,b):t) : bool = a = Z.zero && b >= Z.zero
let is_negative ((a,b):t) : bool = a = Z.zero && b <= Z.zero
let is_positive_strict ((a,b):t) : bool = a = Z.zero && b > Z.zero
let is_negative_strict ((a,b):t) : bool = a = Z.zero && b < Z.zero
let is_nonzero ((a,b):t) : bool = b <> Z.zero
(** Sign. *)
let is_minf_inf ((a,b):t) : bool =
a = Z.one
(** The congruence represents [-∞,+∞]. *)
let is_singleton ((a,b):t) : bool =
a = Z.zero
(** Whether the congruence contains a single element. *)
let is_bounded (ab:t) : bool =
is_singleton ab
(** Whether the congruence contains a finite number of elements. *)
let is_in_range ((a,b):t) (lo:Z.t) (up:Z.t) =
a = Z.zero && b >= lo && b <= up
(** Whether the congruence is included in the range [lo,up]. *)
(** {2 Printing} *)
let to_string ((a,b):t) : string =
if a = Z.zero then Z.to_string b else
let prefix = (if a = Z.one then "" else Z.to_string a) in
match Z.sign b with
| 1 -> prefix^"ℤ+"^(Z.to_string b)
| -1 -> prefix^"ℤ"^(Z.to_string b)
| _ -> prefix^"ℤ"
let print ch (x:t) = output_string ch (to_string x)
let fprint ch (x:t) = Format.pp_print_string ch (to_string x)
let bprint ch (x:t) = Buffer.add_string ch (to_string x)
let to_string_bot = bot_to_string to_string
let print_bot = bot_print print
let fprint_bot = bot_fprint fprint
let bprint_bot = bot_bprint bprint
(** {2 Set operations} *)
let join ((a,b):t) ((a',b'):t) : t =
of_z (gcd3 a a' (Z.sub b b')) b'
(** Abstract union. *)
let join_bot (a:t_with_bot) (b:t_with_bot) : t_with_bot =
bot_neutral2 join a b
let join_list (l:t list) : t_with_bot =
List.fold_left (fun a b -> join_bot a (Nb b)) BOT l
let meet ((a,b):t) ((a',b'):t) : t_with_bot =
match a = Z.zero, a' = Z.zero with
| true, true -> if b = b' then Nb (a,b) else BOT
| true, false -> if contains b (a',b') then Nb (a,b) else BOT
| false, true -> if contains b' (a,b) then Nb (a',b') else BOT
| false, false ->
let gcd, lcm, u, _ = gcd_ext a a' in
if Z.rem (Z.sub b b') gcd = Z.zero
then Nb (of_z lcm (Z.sub b (Z.div (Z.mul a (Z.mul u (Z.sub b b'))) gcd)))
else BOT
(** Abstract intersection. *)
let meet_bot (a:t_with_bot) (b:t_with_bot) : t_with_bot =
bot_absorb2 meet a b
let meet_list (l:t list) : t_with_bot =
List.fold_left (fun a b -> meet_bot a (Nb b)) (Nb minf_inf) l
let meet_range ((a,b):t) (lo:Z.t) (up:Z.t) : t_with_bot =
if a = Z.zero && b < lo || b > up then BOT
else Nb (a,b)
(** Abstract intersection with [lo,up]. *)
let positive (a:t) : t_with_bot =
if is_negative_strict a then BOT else Nb a
let negative (a:t) : t_with_bot =
if is_positive_strict a then BOT else Nb a
(** Positive and negative part. *)
let meet_zero (a:t) : t_with_bot =
meet a zero
(** Intersects with {0}. *)
let meet_nonzero (a:t) : t_with_bot =
if equal a zero then BOT else Nb a
(** Keeps only non-zero elements. *)
(** {2 Forward operations} *)
let neg ((a,b):t) : t =
of_z a (Z.neg b)
(** Negation. *)
let abs ((a,b):t) : t =
if a = Z.zero then a, Z.abs b
else if b = Z.zero then a, b
else join (a,b) (a,Z.sub a b)
(** Absolute value. *)
let succ ((a,b):t) =
of_z a (Z.succ b)
(** Adding 1. *)
let pred ((a,b):t) =
of_z a (Z.pred b)
(** Subtracting 1. *)
let add ((a,b):t) ((a',b'):t) : t =
of_z (gcd a a') (Z.add b b')
(** Addition. *)
let sub ((a,b):t) ((a',b'):t) : t =
of_z (gcd a a') (Z.sub b b')
(** Subtraction. *)
let mul ((a,b):t) ((a',b'):t) : t =
of_z (gcd3 (Z.mul a a') (Z.mul a b') (Z.mul a' b)) (Z.mul b b')
(** Multiplication. *)
let div ((a,b):t) ((a',b'):t) : t_with_bot =
if a' = Z.zero then
if b' = Z.zero then BOT
else if divides b' a && divides b' b
then Nb (of_z (Z.div a b') (Z.div b b'))
else Nb minf_inf
else Nb minf_inf
(** Division (with truncation). *)
let rem ((a,b):t) ((a',b'):t) : t_with_bot =
if a' = Z.zero then
if b' = Z.zero then BOT
else if a = Z.zero then Nb (cst (Z.rem b b'))
else Nb (of_z (gcd a b') (Z.rem b b'))
else Nb (of_z (gcd3 a a' b') b)
(** Remainder. Uses the C semantics for remainder (%). *)
let wrap ((a,b):t) (lo:Z.t) (up:Z.t) : t =
let w = Z.succ (Z.sub up lo) in
if a = Z.zero then of_z Z.zero (Z.add lo (Z.erem (Z.sub b lo) w))
else of_z (gcd a w) (Z.add lo (Z.erem (Z.sub b lo) w))
(** Put back inside [lo,up] by modular arithmetics. *)
let to_bool (can_be_zero:bool) (can_be_one:bool) : t =
match can_be_zero, can_be_one with
| true, false -> zero
| false, true -> one
| true, true -> minf_inf
| _ -> failwith "unreachable case encountered in IntCong.to_bool"
let log_cast (ab:t) : t =
to_bool (contains_zero ab) (contains_nonzero ab)
(** Conversion from integer to boolean in [0,1]: maps 0 to 0 (false) and non-zero to 1 (true).
[0;1] is over-approximated as ℤ.
*)
let log_not (ab:t) : t =
to_bool (contains_nonzero ab) (contains_zero ab)
(** Logical negation.
Logical operation use the C semantics: they accept 0 and non-0 respectively as false and true, but they always return 0 and 1 respectively for false and true.
[0;1] is over-approximated as ℤ.
*)
let log_and (ab:t) (ab':t) : t =
to_bool (contains_zero ab || contains_zero ab') (contains_nonzero ab && contains_nonzero ab')
(** Logical and. *)
let log_or (ab:t) (ab':t) : t =
to_bool (contains_zero ab && contains_zero ab') (contains_nonzero ab || contains_nonzero ab')
(** Logical or. *)
let log_xor (ab:t) (ab':t) : t =
let f,f' = contains_zero ab, contains_zero ab'
and t,t' = contains_nonzero ab, contains_nonzero ab' in
to_bool ((f && f') || (t && t')) ((f && t') || (t && f'))
(** Logical exclusive or. *)
let log_eq (ab:t) (ab':t) : t =
to_bool (not (equal ab ab' && is_singleton ab)) (intersect ab ab')
let log_neq (ab:t) (ab':t) : t =
to_bool (intersect ab ab') (not (equal ab ab' && is_singleton ab))
let log_sgl op ((a,b):t) ((a',b'):t) : t =
if a <> Z.zero || a' <> Z.zero then minf_inf
else if op b b' then one else zero
let log_leq = log_sgl (<=)
let log_geq = log_sgl (>=)
let log_lt = log_sgl (<)
let log_gt = log_sgl (>)
(** C comparison tests. Returns an interval included in [0,1] (a boolean) *)
let is_log_eq (ab:t) (ab':t) : bool =
intersect ab ab'
let is_log_neq (ab:t) (ab':t) : bool =
not (equal ab ab' && is_singleton ab)
let is_log_sgl op ((a,b):t) ((a',b'):t) : bool =
a <> Z.zero || a' <> Z.zero || op b b'
let is_log_leq = is_log_sgl (<=)
let is_log_geq = is_log_sgl (>=)
let is_log_lt = is_log_sgl (<)
let is_log_gt = is_log_sgl (>)
(** C comparison tests. Returns a boolean if the test may succeed *)
let shift_left ((a,b):t) ((a',b'):t) : t_with_bot =
try
if a' = Z.zero then
if b' < Z.zero then BOT
else
let bb' = Z.to_int b' in
Nb (of_z (Z.shift_left a bb') (Z.shift_left b bb'))
else Nb minf_inf
with Z.Overflow -> Nb minf_inf
(** Bitshift left: multiplication by a power of 2. *)
let shift_right ((a,b):t) ((a',b'):t) : t_with_bot =
try
if a' = Z.zero then
if b' < Z.zero then BOT
else
let bb' = Z.to_int b' in
let z = Z.shift_left Z.one bb' in
if a = Z.zero then Nb (cst (Z.shift_right b bb'))
else if divides z a && divides z b then
Nb (of_z (Z.shift_right a bb') (Z.shift_right b bb'))
else Nb minf_inf
else Nb minf_inf
with Z.Overflow -> Nb minf_inf
(** Bitshift right: division by a power of 2 rounding towards -∞. *)
let shift_right_trunc ((a,b):t) ((a',b'):t) : t_with_bot =
try
if a' = Z.zero then
if b' < Z.zero then BOT
else
let bb' = Z.to_int b' in
let z = Z.shift_left Z.one bb' in
if a = Z.zero then Nb (cst (Z.shift_right_trunc b bb'))
else if divides z a && divides z b then
Nb (of_z (Z.shift_right a bb') (Z.shift_right b bb'))
else Nb minf_inf
else Nb minf_inf
with Z.Overflow -> Nb minf_inf
(** Unsigned bitshift right: division by a power of 2 with truncation. *)
let bit_not (ab:t) : t =
pred (neg ab)
(** Bitwise negation: ~x = -x-1 *)
(** {2 Filters} *)
(** Given two interval aruments, return the arguments assuming that the predicate holds.
*)
let filter_eq (ab:t) (ab':t) : (t*t) with_bot =
match meet ab ab' with BOT -> BOT | Nb x -> Nb (x,x)
let filter_sgl op ((a,b) as ab:t) ((a',b') as ab':t) : (t*t) with_bot =
if a = Z.zero && a' = Z.zero && not (op b b') then BOT else Nb (ab,ab')
let filter_neq = filter_sgl (<>)
let filter_leq = filter_sgl (<=)
let filter_geq = filter_sgl (>=)
let filter_lt = filter_sgl (<)
let filter_gt = filter_sgl (>)
(** {2 Backward operations} *)
(** Given one or two interval argument(s) and a result interval, return the
argument(s) assuming the result in the operation is in the given result.
*)
let bwd_neg (a:t) (r:t) : t_with_bot =
meet a (neg r)
let bwd_abs (a:t) (r:t) : t_with_bot =
meet a (join r (neg r))
let bwd_succ (a:t) (r:t) : t_with_bot =
meet a (pred r)
let bwd_pred (a:t) (r:t) : t_with_bot =
meet a (succ r)
let bwd_add (a:t) (b:t) (r:t) : (t*t) with_bot =
bot_merge2 (meet a (sub r b)) (meet b (sub r a))
let bwd_sub (a:t) (b:t) (r:t) : (t*t) with_bot =
bot_merge2 (meet a (add b r)) (meet b (sub a r))
let bwd_mul (a:t) (b:t) (r:t) : (t*t) with_bot =
let aa = if contains_zero b && contains_zero r then Nb a else div r b
and bb = if contains_zero a && contains_zero r then Nb b else div r a in
bot_merge2 aa bb
let bwd_bit_not (a:t) (r:t) : t_with_bot =
meet a (bit_not r)
let bwd_join (a:t) (b:t) (r:t) : (t*t) with_bot =
bot_merge2 (meet a r) (meet b r)
(** Backward join: both arguments and intersected with the result. *)
let bwd_rem (a:t) ((b1,b2) as b:t) (r:t) =
bot_merge2 (meet a (add (of_z (gcd b1 b2) Z.zero) r)) (Nb b)
let bwd_div (a:t) (b:t) (r:t) = Nb (a,b)
let bwd_wrap (a:t) range (r:t) : t_with_bot = Nb a
let bwd_shift_left (a:t) (b:t) (r:t) = Nb (a,b)
let bwd_shift_right (a:t) (b:t) (r:t) = Nb (a,b)
let bwd_shift_right_trunc (a:t) (b:t) (r:t) = Nb (a,b)
(** {2 Reduction} *)
let meet_inter ((a,b):t) ((l,h):I.t) : (t * I.t) with_bot =
let l' = match l with
| B.Finite f -> B.Finite (Z.add f (rem_zero (Z.sub b f) a))
| _ -> l
and h' = match h with
| B.Finite f -> B.Finite (Z.sub f (rem_zero (Z.sub f b) a))
| _ -> h
in
match l',h' with
| B.Finite ll, B.Finite hh ->
if ll > hh then BOT
else if ll = hh then Nb (cst ll, (l',h'))
else Nb ((a,b),(l',h'))
| _ -> Nb ((a,b),(l',h'))
(** Intersects a congruence with an interval, and returns the set represented
both as a congruence and as an interval.
Useful to implement reductions.
*)