Source file intItv.ml
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(**
IntItv - Intervals for arbitrary precision integers.
We rely on Zarith for arithmetic operations, and IntBounds to
represent unbounded intervals.
*)
open Bot
module B = IntBound
(** {2 Types} *)
type t = B.t (** lower bound *) * B.t (** upper bound *)
(**
The type of non-empty intervals: a lower bound and an upper bound.
The lower bound can be MINF, but not PINF; the upper bound can be PINF, but not MINF.
Moreover, lower bound ≤ upper bound.
*)
type t_with_bot = t with_bot
(** The type of possibly empty intervals. *)
let is_valid ((a,b):t) : bool =
B.leq a b && a <> B.PINF && b <> B.MINF
(** {2 Constructors} *)
let of_bound (a:B.t) (b:B.t) : t =
if a = B.PINF || b = B.MINF || B.gt a b then
invalid_arg (Printf.sprintf "IntItv.of_bound [%s,%s]" (B.to_string a) (B.to_string b));
a, b
let of_z (a:Z.t) (b:Z.t) : t =
if Z.gt a b then
invalid_arg (Printf.sprintf "IntItv.of_z [%s,%s]" (Z.to_string a) (Z.to_string b));
B.Finite a, B.Finite b
let of_int (a:int) (b:int) : t =
if a > b then
invalid_arg (Printf.sprintf "IntItv.of_int [%i,%i]" a b);
B.of_int a, B.of_int b
let of_int64 (a:int64) (b:int64) : t =
if a > b then
invalid_arg (Printf.sprintf "IntItv.of_int64 [%Li,%Li]" a b);
B.of_int64 a, B.of_int64 b
(** Constructs a non-empty interval. *)
let of_float (a:float) (b:float) : t =
if a > b || a = infinity || b = neg_infinity then
invalid_arg (Printf.sprintf "IntItv.of_float [%f,%f]" a b);
B.of_float a, B.of_float b
(** Constructs a non-empty interval. *)
let of_range = of_z
let of_bound_bot (a:B.t) (b:B.t) : t_with_bot =
if B.gt a b || a = B.PINF || b = B.MINF then BOT
else Nb (a, b)
let of_range_bot (a:Z.t) (b:Z.t) : t_with_bot =
if Z.gt a b then BOT
else Nb (B.Finite a, B.Finite b)
let of_int_bot (a:int) (b:int) : t_with_bot =
if a > b then BOT
else Nb (B.of_int a, B.of_int b)
let of_int64_bot (a:int64) (b:int64) : t_with_bot =
if a > b then BOT
else Nb (B.of_int64 a, B.of_int64 b)
(** Constructs a possibly empty interval. *)
let of_float_bot (a:float) (b:float) : t_with_bot =
if a > b || a = infinity || b = neg_infinity then BOT
else Nb (B.of_float a, B.of_float b)
(** Constructs a possibly empty interval. *)
let hull (a:B.t) (b:B.t) : t =
B.min a b, B.max a b
(** Constructs the smallest interval containing a and b. *)
let cst (c:Z.t) : t =
B.Finite c, B.Finite c
(** Singleton interval. *)
let cst_int (c:int) : t = cst (Z.of_int c)
let cst_int64 (c:int64) : t = cst (Z.of_int64 c)
let zero : t = cst Z.zero
(** [0,0] *)
let one : t = cst Z.one
(** [1,1] *)
let mone : t = cst Z.minus_one
(** [-1,-1] *)
let zero_one : t = B.Finite Z.zero, B.Finite Z.one
(** [0,1] *)
let mone_zero : t = B.Finite Z.minus_one, B.Finite Z.zero
(** [-1,0] *)
let mone_one : t = B.Finite Z.minus_one, B.Finite Z.one
(** [-1,1] *)
let zero_inf : t = B.Finite Z.zero, B.PINF
(** [0,+∞] *)
let minf_zero : t = B.MINF, B.Finite Z.zero
(** [-∞,0] *)
let minf_inf : t = B.MINF, B.PINF
(** [-∞,+∞] *)
let unsigned (bits:int) : t = B.zero, B.pred (B.pow2 bits)
let unsigned8 : t = unsigned 8
let unsigned16 : t = unsigned 16
let unsigned32 : t = unsigned 32
let unsigned64 : t = unsigned 64
(** Intervals of unsigned integers with the specified bitsize. *)
let signed (bits:int) : t = B.neg (B.pow2 (bits-1)), B.pred (B.pow2 (bits-1))
let signed8 : t = signed 8
let signed16 : t = signed 16
let signed32 : t = signed 32
let signed64 : t = signed 64
(** Intervals of two compement's integers with the specified bitsize. *)
(** {2 Predicates} *)
let equal ((a,b):t) ((a',b'):t) : bool =
B.eq a a' && B.eq b b'
(** Equality. = also works *)
let equal_bot : t_with_bot -> t_with_bot -> bool =
bot_equal equal
let included ((a,b):t) ((a',b'):t) : bool =
B.geq a a' && B.leq b b'
(** Set ordering. *)
let included_bot : t_with_bot -> t_with_bot -> bool =
bot_included included
let intersect ((a,b):t) ((a',b'):t) : bool =
B.leq a b' && B.leq a' b
(** Whether the intervals have an non-empty intersection. *)
let intersect_bot : t_with_bot -> t_with_bot -> bool =
bot_dfl2 false intersect
let contains (x:Z.t) ((a,b):t) : bool =
B.leq a (B.Finite x) && B.leq (B.Finite x) b
(** Whether the interval contains a (finite) value. *)
let compare ((a,b):t) ((a',b'):t) : int =
if B.eq a a' then B.compare b b' else B.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 intervals. *)
let contains_zero ((a,b):t) : bool =
B.sign a <= 0 && B.sign b >= 0
(** [a,b] contains 0. *)
let contains_one ((a,b):t) : bool =
B.leq a B.one && B.geq b B.one
(** [a,b] contains 1. *)
let contains_nonzero ((a,b):t) : bool =
B.neq a B.zero || B.neq b B.zero
(** [a,b] contains a non-zero value. *)
let is_zero (ab:t) : bool = ab = zero
let is_one (ab:t) : bool = ab = one
let is_positive ((a,b):t) : bool = B.is_positive a
let is_negative ((a,b):t) : bool = B.is_negative b
let is_positive_strict ((a,b):t) : bool = B.is_positive_strict a
let is_negative_strict ((a,b):t) : bool = B.is_negative_strict b
let is_nonzero ((a,b):t) : bool = B.gt a B.zero || B.lt b B.zero
(** Interval sign. *)
let is_singleton ((a,b):t) : bool = B.eq a b
(** [a,b] contains a single element. *)
let is_bounded ((a,b):t) : bool = a <> B.MINF && b <> B.PINF
(** [a,b] has finite bounds. *)
let is_minf_inf ((a,b):t) : bool = a = B.MINF && b = B.PINF
(** [a,b] represents [-∞,+∞]. *)
let is_in_range (a:t) (lo:Z.t) (up:Z.t) =
included a (B.Finite lo, B.Finite up)
(** Whether the interval is included in the range [lo,up]. *)
(** {2 Printing} *)
let to_string ((a,b):t) : string = "["^(B.to_string a)^","^(B.to_string b)^"]"
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 Enumeration} *)
let size ((a,b):t) =
match a,b with
| B.Finite x, B.Finite y -> Z.succ (Z.sub y x)
| _ -> invalid_arg (Printf.sprintf "IntItv.size: unbounded interval %s" (to_string (a,b)))
(** Number of elements. Raises an invalid argument if it is unbounded. *)
let to_list ((a,b):t) =
let rec doit l h acc =
if l=h then l :: acc
else doit l (Z.pred h) (h::acc)
in
match a,b with
| B.Finite x, B.Finite y -> doit x y []
| _ -> invalid_arg (Printf.sprintf "IntItv.to_list: unbounded interval %s" (to_string (a,b)))
(** List of elements, in increasing order. Raises an invalid argument if it is unbounded. *)
(** {2 Set operations} *)
let join ((a,b):t) ((a',b'):t) : t =
B.min a a', B.max b b'
(** Join of non-empty intervals. *)
let join_bot (a:t_with_bot) (b:t_with_bot) : t_with_bot =
bot_neutral2 join a b
(** Join of possibly empty intervals. *)
let join_list (l:t list) : t_with_bot =
List.fold_left (fun a b -> join_bot a (Nb b)) BOT l
(** Join of a list of (non-empty) intervals. *)
let meet ((a,b):t) ((a',b'):t) : t_with_bot =
of_bound_bot (B.max a a') (B.min b b')
(** Intersection of non-emtpty intervals (possibly empty) *)
let meet_bot (a:t_with_bot) (b:t_with_bot) : t_with_bot =
bot_absorb2 meet a b
(** Intersection of possibly empty intervals. *)
let meet_list (l:t list) : t_with_bot =
List.fold_left (fun a b -> meet_bot a (Nb b)) (Nb minf_inf) l
(** Meet of a list of (non-empty) intervals. *)
let widen ((a,b):t) ((a',b'):t) : t =
(if B.lt a' a then B.MINF else a),
(if B.gt b' b then B.PINF else b)
(** Basic widening: put unstable bounds to infinity. *)
let widen_bot (a:t_with_bot) (b:t_with_bot) : t_with_bot =
bot_neutral2 widen a b
let positive (a:t) : t_with_bot = meet a zero_inf
let negative (a:t) : t_with_bot = meet a minf_zero
(** Positive and negative part. *)
let meet_zero (a:t) : t_with_bot =
meet a zero
(** Intersects with {0}. *)
let meet_nonzero ((a,b):t) : t_with_bot =
match B.is_zero a, B.is_zero b with
| true, true -> BOT
| true, false -> Nb (B.succ a, b)
| false, true -> Nb (a, B.pred b)
| false, false -> Nb (a,b)
(** Keeps only non-zero elements. *)
(** {2 Forward operations} *)
(** Given one or two interval argument(s), return the interval result. *)
let neg ((a,b):t) : t =
B.neg b, B.neg a
(** Negation. *)
let abs ((a,b):t) : t =
if B.sign a <= 0 then
if B.sign b <= 0 then neg (a,b)
else B.zero, B.max (B.neg a) b
else a,b
(** Absolute value. *)
let succ ((a,b):t) : t =
B.succ a, B.succ b
(** Add 1. *)
let pred ((a,b):t) : t =
B.pred a, B.pred b
(** Subtract 1. *)
let add ((a,b):t) ((a',b'):t) : t =
B.add a a', B.add b b'
(** Addition. *)
let sub ((a,b):t) ((a',b'):t) : t =
B.sub a b', B.sub b a'
(** Subtraction. *)
let minmax4 op (a,b) (c,d) =
let x,y,z,t = op a c, op a d, op b c, op b d in
B.min (B.min x y) (B.min z t), B.max (B.max x y) (B.max z t)
let mul (ab:t) (ab':t) : t =
minmax4 B.mul ab ab'
(** Multiplication. *)
let div_unmerged (ab:t) ((a',b'):t) : t list =
let div_pos ab ab' = minmax4 B.div ab ab' in
(if B.is_positive_strict b' then [div_pos ab (B.max a' B.one, b')] else [])@
(if B.is_negative_strict a' then [div_pos ab (a', B.min b' B.minus_one)] else [])
(** Division (with truncation).
Returns a list of 0, 1, or 2 intervals to remain precise.
*)
let ediv_unmerged (ab:t) ((a',b'):t) : t list =
let div_pos ab ab' = minmax4 B.ediv ab ab' in
(if B.is_positive_strict b' then [div_pos ab (B.max a' B.one, b')] else [])@
(if B.is_negative_strict a' then [div_pos ab (a', B.min b' B.minus_one)] else [])
(** Euclidian division (towards -oo).
Returns a list of 0, 1, or 2 intervals to remain precise.
*)
let div (a:t) (b:t) : t_with_bot =
join_list (div_unmerged a b)
(** Division (with truncation).
Returns a single (possibly empty) overapproximating interval.
*)
let ediv (a:t) (b:t) : t_with_bot =
join_list (ediv_unmerged a b)
(** Division (euclidian, towards -oo)
Returns a single (possibly empty) overapproximating interval.
*)
let rem ((a,b):t) (ab':t) : t_with_bot =
let a',b' = abs ab' in
if B.is_zero b' then BOT else
let m = B.pred b' in
if B.gt a (B.neg a') && B.lt b a' then
Nb (a,b)
else if B.equal a' b' && B.equal (B.div a a') (B.div b a') then
Nb (B.rem a a', B.rem b a')
else if B.is_positive a then
Nb (B.zero, m)
else if B.is_negative b then
Nb (B.neg m, B.zero)
else
Nb (B.neg m, m)
(** Remainder. Uses the C semantics for remainder (%). *)
let erem ((a,b):t) (ab':t) : t_with_bot =
let a',b' = abs ab' in
if B.is_zero b' then BOT else
let m = B.pred b' in
if B.equal a' b' && B.equal (B.ediv a a') (B.ediv b a') then
Nb (B.erem a a', B.erem b a')
else
Nb (B.zero, m)
(** Euclidian remainder. rounding towards -oo *)
let pow (ab:t) (ab':t) : t =
minmax4 B.pow ab ab'
(** Power. *)
let wrap ((a,b):t) (lo:Z.t) (up:Z.t) : t =
match a,b with
| B.MINF,_ | _,B.PINF -> B.Finite lo, B.Finite up
| B.Finite aa, B.Finite bb ->
if aa >= lo && bb <= up then a,b
else
let w = Z.succ (Z.sub up lo) in
let (aq,ar), (bq,br) = Z.ediv_rem (Z.sub aa lo) w, Z.ediv_rem (Z.sub bb lo) w in
if aq = bq then
B.Finite (Z.add ar lo), B.Finite (Z.add br lo)
else
B.Finite lo, B.Finite up
| _ -> invalid_arg (Printf.sprintf "IntItv.wrap %s in [%s,%s]" (to_string (a,b)) (Z.to_string lo) (Z.to_string up))
(** Put back the interval inside [lo,up] by modular arithmetics.
Useful to model the effect of arithmetic or conversion overflow. *)
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 -> zero_one
| _ -> failwith "unreachable case encountered in IntItv.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). *)
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.
*)
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_leq ((a,b):t) ((a',b'):t) : t = to_bool (B.gt b a') (B.leq a b')
let log_geq ((a,b):t) ((a',b'):t) : t = to_bool (B.lt a b') (B.geq b a')
let log_lt ((a,b):t) ((a',b'):t) : t = to_bool (B.geq b a') (B.lt a b')
let log_gt ((a,b):t) ((a',b'):t) : t = to_bool (B.leq a b') (B.gt b a')
let log_neq (ab:t) (ab':t) : t = to_bool (intersect ab ab') (not (equal ab ab' && is_singleton ab))
(** 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_leq ((a,b):t) ((a',b'):t) : bool = B.leq a b'
let is_log_geq ((a,b):t) ((a',b'):t) : bool = B.geq b a'
let is_log_lt ((a,b):t) ((a',b'):t) : bool = B.lt a b'
let is_log_gt ((a,b):t) ((a',b'):t) : bool = B.gt b a'
let is_log_neq (ab:t) (ab':t) : bool = not (equal ab ab' && is_singleton ab)
(** C comparison tests. Returns a boolean if the test may succeed *)
(** {2 Bit operations} *)
let shift_left (ab:t) (ab':t) : t_with_bot =
match positive ab' with
| BOT -> BOT
| Nb ab'' -> Nb (minmax4 B.shift_left ab ab'')
(** Bitshift left: multiplication by a power of 2. *)
let shift_right (ab:t) (ab':t) : t_with_bot =
match positive ab' with
| BOT -> BOT
| Nb ab'' -> Nb (minmax4 B.shift_right ab ab'')
(** Bitshift right: division by a power of 2 rounding towards -∞. *)
let shift_right_trunc (ab:t) (ab':t) : t_with_bot =
match positive ab' with
| BOT -> BOT
| Nb ab'' -> Nb (minmax4 B.shift_right_trunc ab ab'')
(** 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 *)
(** Internal functions *)
let min_or (a:Z.t) (b:Z.t) (c:Z.t) (d:Z.t) : Z.t =
let rec doit i =
if i < 0 then Z.logor a c else
let ai, ci = Z.shift_right a i, Z.shift_right c i in
match Z.is_even ai, Z.is_even ci with
| true, false ->
let a' = Z.shift_left (Z.logor ai Z.one) i in
if a' <= b then Z.logor a' c
else doit (i-1)
| false, true ->
let c' = Z.shift_left (Z.logor ci Z.one) i in
if c' <= d then Z.logor a c'
else doit (i-1)
| _ ->
doit (i-1)
in
let mag =
if (a >= Z.zero) = (c >= Z.zero) then
Z.numbits (Z.logxor a c) - 1
else
max (max (Z.numbits a) (Z.numbits b)) (max (Z.numbits c) (Z.numbits d))
in
doit mag
let max_or (a:Z.t) (b:Z.t) (c:Z.t) (d:Z.t) : Z.t =
let rec doit i =
if i < 0 then Z.logor b d else
let bi, di = Z.shift_right b i, Z.shift_right d i in
if Z.is_odd bi && Z.is_odd di then
let b' = Z.pred (Z.shift_left bi i) in
if a <= b' then Z.logor b' d else
let d' = Z.pred (Z.shift_left di i) in
if c <= d' then Z.logor b d' else
doit (i-1)
else doit (i-1)
in
let mag =
if (a >= Z.zero) || (c >= Z.zero) then
Z.numbits (Z.logand b d) - 1
else
max (max (Z.numbits a) (Z.numbits b)) (max (Z.numbits c) (Z.numbits d))
in
doit mag
let bounds_or (a:Z.t) (b:Z.t) (c:Z.t) (d:Z.t) : Z.t * Z.t =
match a >= Z.zero || b < Z.zero, c >= Z.zero || d < Z.zero with
| true, true ->
min_or a b c d, max_or a b c d
| false, true ->
if c >= Z.zero then min_or a Z.minus_one c d, max_or Z.zero b c d
else c, Z.minus_one
| true, false ->
if a >= Z.zero then min_or a b c Z.minus_one, max_or a b Z.zero d
else a, Z.minus_one
| false, false ->
Z.min a c, max_or Z.zero b Z.zero d
let bounds_and (a:Z.t) (b:Z.t) (c:Z.t) (d:Z.t) : Z.t * Z.t =
let nh,nl = bounds_or (Z.lognot b) (Z.lognot a) (Z.lognot d) (Z.lognot c) in
Z.lognot nl, Z.lognot nh
let min_xor (a:Z.t) (b:Z.t) (c:Z.t) (d:Z.t) : Z.t =
let rec doit a c i =
if i < 0 then Z.logxor a c else
let ai, ci = Z.shift_right a i, Z.shift_right c i in
match Z.is_even ai, Z.is_even ci with
| true, false ->
let a' = Z.shift_left (Z.logor ai Z.one) i in
doit (if a' <= b then a' else a) c (i-1)
| false, true ->
let c' = Z.shift_left (Z.logor ci Z.one) i in
doit a (if c' <= d then c' else c) (i-1)
| _ ->
doit a c (i-1)
in
let mag =
if (a >= Z.zero) = (c >= Z.zero) then Z.numbits (Z.logxor a c) - 1
else max (max (Z.numbits a) (Z.numbits b)) (max (Z.numbits c) (Z.numbits d))
in
doit a c mag
let max_xor (a:Z.t) (b:Z.t) (c:Z.t) (d:Z.t) : Z.t =
let rec doit b d i =
if i < 0 then Z.logxor b d else
let bi, di = Z.shift_right b i, Z.shift_right d i in
if Z.is_odd bi && Z.is_odd di then
let b' = Z.pred (Z.shift_left bi i) in
if a <= b' then doit b' d (i-1)
else
let d' = Z.pred (Z.shift_left di i) in
if c <= d' then doit b d' (i-1)
else doit b d (i-1)
else
doit b d (i-1)
in
let mag =
if (a >= Z.zero) || (c >= Z.zero) then Z.numbits (Z.logand b d) - 1
else max (max (Z.numbits a) (Z.numbits b)) (max (Z.numbits c) (Z.numbits d))
in
doit b d mag
let bounds_xor (a:Z.t) (b:Z.t) (c:Z.t) (d:Z.t) : Z.t * Z.t =
let combine2 a1 b1 c1 d1 a2 b2 c2 d2 =
let l1,h1 = min_xor a1 b1 c1 d1, max_xor a1 b1 c1 d1
and l2,h2 = min_xor a2 b2 c2 d2, max_xor a2 b2 c2 d2 in
Z.min l1 l2, Z.max h1 h2
in
match a >= Z.zero || b < Z.zero, c >= Z.zero || d < Z.zero with
| true,true -> min_xor a b c d, max_xor a b c d
| false,true ->
combine2 a Z.minus_one c d Z.zero b c d
| true,false ->
combine2 a b c Z.minus_one a b Z.zero d
| false,false ->
let l1,h1 = combine2 a Z.minus_one c Z.minus_one Z.zero b c Z.minus_one
and l2,h2 = combine2 a Z.minus_one Z.zero d Z.zero b Z.zero d in
Z.min l1 l2, Z.max h1 h2
(** Interval functions, based on the previous ones *)
let bit_or (ab:t) (ab':t) : t =
match ab, ab' with
| (B.Finite al, B.Finite ah), (B.Finite bl, B.Finite bh) ->
if al=ah && bl=bh then
cst (Z.logor al bl)
else
let l,h = bounds_or al ah bl bh in
B.Finite l, B.Finite h
| _ ->
if is_positive ab && is_positive ab' then
zero_inf
else
minf_inf
(** Bitwise or. *)
let bit_and (ab:t) (ab':t) : t =
match ab, ab' with
| (B.Finite al, B.Finite ah), (B.Finite bl, B.Finite bh) ->
if al=ah && bl=bh then
cst (Z.logand al bl)
else
let l,h = bounds_and al ah bl bh in
B.Finite l, B.Finite h
| _ ->
if is_positive ab || is_positive ab' then
zero_inf
else
minf_inf
(** Bitwise and. *)
let bit_xor (ab:t) (ab':t) : t =
match ab, ab' with
| (B.Finite al, B.Finite ah), (B.Finite bl, B.Finite bh) ->
if al=ah && bl=bh then
cst (Z.logxor al bl)
else
let l,h = bounds_xor al ah bl bh in
B.Finite l, B.Finite h
| _ ->
if is_positive ab && is_positive ab' then
zero_inf
else
minf_inf
(** Bitwise exclusive or. *)
(** {2 Filters} *)
(** Given two interval aruments, return the arguments assuming that the predicate holds.
*)
let filter_leq ((a,b):t) ((a',b'):t) : (t*t) with_bot =
bot_merge2 (of_bound_bot a (B.min b b')) (of_bound_bot (B.max a a') b')
let filter_geq ((a,b):t) ((a',b'):t) : (t*t) with_bot =
bot_merge2 (of_bound_bot (B.max a a') b) (of_bound_bot a' (B.min b b'))
let filter_lt ((a,b):t) ((a',b'):t) : (t*t) with_bot =
bot_merge2 (of_bound_bot a (B.min b (B.pred b'))) (of_bound_bot (B.max (B.succ a) a') b')
let filter_gt ((a,b):t) ((a',b'):t) : (t*t) with_bot =
bot_merge2 (of_bound_bot (B.max a (B.succ a')) b) (of_bound_bot a' (B.min (B.pred b) b'))
let filter_eq ((a,b):t) ((a',b'):t) : (t*t) with_bot =
match meet (a,b) (a',b') with BOT -> BOT | Nb x -> Nb (x,x)
let filter_neq ((a,b):t) ((a',b'):t) : (t*t) with_bot =
match B.equal a b, B.equal a' b' with
| true, true when B.equal a a' -> BOT
| true, false when B.equal a a' -> bot_merge2 (Nb (a,b)) (of_bound_bot (B.succ a') b')
| true, false when B.equal b b' -> bot_merge2 (Nb (a,b)) (of_bound_bot a' (B.pred b'))
| false, true when B.equal a a' -> bot_merge2 (of_bound_bot (B.succ a) b) (Nb (a',b'))
| false, true when B.equal b b' -> bot_merge2 (of_bound_bot a (B.pred b)) (Nb (a',b'))
| _ -> Nb ((a,b),(a',b'))
(** {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_default_unary (a:t) (r:t) : t_with_bot =
Nb a
(** Fallback for backward unary operators *)
let bwd_default_binary (a:t) (b:t) (r:t) : (t*t) with_bot =
Nb (a,b)
(** Fallback for backward binary operators *)
let bwd_neg (a:t) (r:t) : t_with_bot =
meet a (neg r)
let bwd_abs (a:t) (r:t) : t_with_bot =
join_bot (meet a r) (meet a (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_div ((a,a'):t) ((b,b'):t) (r:t) : (t*t) with_bot =
let m = B.pred (B.max (B.abs b) (B.abs b')) in
let md =
(if B.is_negative_strict a then B.neg m else B.zero),
(if B.is_positive_strict a' then m else B.zero)
in
let aa = meet (a,a') (add (mul r (b,b')) md) in
let ax = sub (a,a') md in
let bb =
if contains_zero ax && contains_zero r then Nb (b,b')
else meet_bot (Nb (b,b')) (div ax r)
in
bot_merge2 aa bb
let bwd_ediv ((a,a'):t) ((b,b'):t) (r:t) : (t*t) with_bot =
let m = B.pred (B.max (B.abs b) (B.abs b')) in
let md = B.zero, m in
let aa = meet (a,a') (add (mul r (b,b')) md) in
let ax = sub (a,a') md in
let bb =
if contains_zero ax && contains_zero r then Nb (b,b')
else meet_bot (Nb (b,b')) (ediv ax r)
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 are intersected with the result. *)
let bwd_shift_left (a:t) (b:t) (r:t) : (t*t) with_bot =
match shift_right r b with
| Nb aa -> bot_merge2 (meet a aa) (Nb b)
| BOT -> Nb (a,b)
let bwd_shift_right (a:t) (b:t) (r:t) : (t*t) with_bot =
match shift_left r b with
| Nb (l, h) ->
let aa = l, B.add h (B.pred (B.shift_left B.one (snd b))) in
bot_merge2 (meet a aa) (Nb b)
| BOT -> Nb (a,b)
let bwd_shift_right_trunc (a:t) (b:t) (r:t) : (t*t) with_bot =
match shift_left r b with
| Nb (l, h) ->
let m = B.pred (B.shift_left B.one (snd b)) in
let l = if B.is_negative_strict (fst a) then B.sub l m else l
and h = if B.is_positive_strict (snd a) then B.add h m else h in
bot_merge2 (meet a (l,h)) (Nb b)
| BOT -> Nb (a,b)
let bwd_bit_or (a:t) (b:t) (r:t) : (t*t) with_bot =
let aa = meet a (bit_and r a)
and bb = meet b (bit_and r b) in
bot_merge2 aa bb
let bwd_bit_and (a:t) (b:t) (r:t) : (t*t) with_bot =
let aa = meet a (bit_not( bit_and (bit_not a) (bit_not r)))
and bb = meet b (bit_not (bit_and (bit_not b) (bit_not r))) in
bot_merge2 aa bb
let bwd_bit_xor (a:t) (b:t) (r:t) : (t*t) with_bot =
bot_merge2 (meet a (bit_xor b r)) (meet b (bit_xor a r))
let bwd_convex_join (a:t) (b:t) (r:t) : (t*t) with_bot =
bot_merge2 (meet a r) (meet b r)
let bwd_log_gen if_one if_zero (a:t) (b:t) (r:t) : (t*t) with_bot =
match contains_zero r, contains_one r with
| true, true -> Nb (a,b)
| true, false -> if_zero a b
| false, true -> if_one a b
| false, false -> BOT
let bwd_log_eq = bwd_log_gen filter_eq filter_neq
let bwd_log_neq = bwd_log_gen filter_neq filter_eq
let bwd_log_lt = bwd_log_gen filter_lt filter_geq
let bwd_log_gt = bwd_log_gen filter_gt filter_leq
let bwd_log_leq = bwd_log_gen filter_leq filter_gt
let bwd_log_geq = bwd_log_gen filter_geq filter_lt
let bwd_wrap (a:t) range (r:t) : t_with_bot =
if included a (of_z (fst range) (snd range))
then meet a r
else
match meet r (of_z (fst range) (snd range)) with
| BOT -> BOT
| Nb (Finite lr, Finite ur) ->
let range_size = Z.sub (snd range) (fst range) |> Z.succ in
let compute_bound div a (r1, r2) =
match a with
| B.Finite a ->
let k = div Z.(a - r2) range_size in
B.Finite Z.(r1 + k * range_size)
| MINF | PINF -> a
in
let lower_bound = compute_bound Z.cdiv (fst a) (lr, ur) in
let upper_bound = compute_bound Z.fdiv (snd a) (ur, lr) in
meet_bot (Nb a) (of_bound_bot lower_bound upper_bound)
| Nb _ -> assert false
let pos_mod a b =
let r = IntBound.rem a b in
if IntBound.geq r IntBound.zero then r else IntBound.add b r
let bwd_rem (a:t) (b:t) (r: t) : (t * t) with_bot =
if is_singleton r && is_singleton b && not (contains_zero b) && is_bounded a then
let al, ah = a in
let rs, _ = r in
let bs, _ = b in
let al = IntBound.add al (pos_mod (IntBound.sub rs al) bs) in
let ah = IntBound.sub ah (pos_mod (IntBound.sub ah rs) bs) in
if ah < al then BOT else Nb ((al, ah), b)
else
bwd_default_binary a b r
let bwd_erem : t -> t -> t -> (t*t) with_bot = bwd_default_binary
let bwd_pow = bwd_default_binary