Module CongUtils.IntCongSource

IntCong - Integer congruences.

We rely on Zarith for arithmetic operations.

Types

Sourcetype t = Z.t * Z.t

offset

The type of possibly empty congruence sets.

Sourceval is_valid : t -> bool
module I = ItvUtils.IntItv

Arithmetic utilities

Sourceval gcd : Z.t -> Z.t -> Z.t

Greatest common divisor of |a| and |b|. 0 is neutral.

Sourceval gcd3 : Z.t -> Z.t -> Z.t -> Z.t
Sourceval gcd_ext : Z.t -> Z.t -> Z.t * Z.t * Z.t * Z.t

Returns the gcd, lcm and cofactors u, v such that ua+vb=gcd. Undefined if a or b is 0.

Sourceval divides : Z.t -> Z.t -> bool

Wheter b is a multiple of a. Always true if b = 0.

Sourceval rem_zero : Z.t -> Z.t -> Z.t

As Z.erem, but rem_zero a 0 = a.

Constructors

Sourceval of_z : Z.t -> Z.t -> t

Returns aℤ + b.

Sourceval of_int : int -> int -> t
Sourceval of_int64 : int64 -> int64 -> t
Sourceval cst : Z.t -> t

Returns 0ℤ + b

Sourceval cst_int : int -> t
Sourceval cst_int64 : int64 -> t
Sourceval zero : t

0ℤ+0

Sourceval one : t

0ℤ+1

Sourceval mone : t

0ℤ-1

Sourceval minf_inf : t

1ℤ+0

Sourceval of_range : Z.t -> Z.t -> t
Sourceval of_range_bot : Z.t -> Z.t -> t Utils_core.Bot.with_bot
Sourceval of_bound : B.t -> B.t -> t
Sourceval of_bound_bot : B.t -> B.t -> t Utils_core.Bot.with_bot

Congruence overapproximating an interval.

Predicates

Sourceval equal : t -> t -> bool

Equality. = also works.

Sourceval equal_bot : t_with_bot -> t_with_bot -> bool
Sourceval included : t -> t -> bool

Set ordering.

Sourceval included_bot : t_with_bot -> t_with_bot -> bool
Sourceval intersect : t -> t -> bool

Whether the intersection is non-empty.

Sourceval intersect_bot : t_with_bot -> t_with_bot -> bool
Sourceval contains : Z.t -> t -> bool

Whether the set contains some value x.

Sourceval compare : t -> t -> int

A total ordering (lexical ordering) returning -1, 0, or 1. Can be used as compare for sets, maps, etc.

Total ordering on possibly empty congruences.

Sourceval contains_zero : t -> bool

Whether the congruence contains zero.

Sourceval contains_one : t -> bool

Whether the congruence contains one.

Sourceval contains_nonzero : t -> bool

Whether the congruence contains a non-zero value.

Sourceval is_zero : t -> bool
Sourceval is_positive : t -> bool
Sourceval is_negative : t -> bool
Sourceval is_positive_strict : t -> bool
Sourceval is_negative_strict : t -> bool
Sourceval is_nonzero : t -> bool

Sign.

Sourceval is_minf_inf : t -> bool

The congruence represents -∞,+∞.

Sourceval is_singleton : t -> bool

Whether the congruence contains a single element.

Sourceval is_bounded : t -> bool

Whether the congruence contains a finite number of elements.

Sourceval is_in_range : t -> Z.t -> Z.t -> bool

Whether the congruence is included in the range lo,up.

Printing

Sourceval to_string : t -> string
Sourceval print : Stdlib.out_channel -> t -> unit
Sourceval fprint : Stdlib.Format.formatter -> t -> unit
Sourceval bprint : Stdlib.Buffer.t -> t -> unit
Sourceval to_string_bot : t Utils_core.Bot.with_bot -> string
Sourceval print_bot : Stdlib.out_channel -> t Utils_core.Bot.with_bot -> unit
Sourceval fprint_bot : Stdlib.Format.formatter -> t Utils_core.Bot.with_bot -> unit
Sourceval bprint_bot : Stdlib.Buffer.t -> t Utils_core.Bot.with_bot -> unit

Set operations

Sourceval join : t -> t -> t

Abstract union.

Sourceval join_list : t list -> t_with_bot
Sourceval meet : t -> t -> t_with_bot

Abstract intersection.

Sourceval meet_list : t list -> t_with_bot
Sourceval meet_range : t -> Z.t -> Z.t -> t_with_bot

Abstract intersection with lo,up.

Sourceval positive : t -> t_with_bot
Sourceval negative : t -> t_with_bot

Positive and negative part.

Sourceval meet_zero : t -> t_with_bot

Intersects with

.

Sourceval meet_nonzero : t -> t_with_bot

Keeps only non-zero elements.

Forward operations

Sourceval neg : t -> t

Negation.

Sourceval abs : t -> t

Absolute value.

Sourceval succ : t -> t

Adding 1.

Sourceval pred : t -> t

Subtracting 1.

Sourceval add : t -> t -> t

Addition.

Sourceval sub : t -> t -> t

Subtraction.

Sourceval mul : t -> t -> t

Multiplication.

Sourceval div : t -> t -> t_with_bot

Division (with truncation).

Sourceval rem : t -> t -> t_with_bot

Remainder. Uses the C semantics for remainder (%).

Sourceval wrap : t -> Z.t -> Z.t -> t

Put back inside lo,up by modular arithmetics.

Sourceval to_bool : bool -> bool -> t
Sourceval log_cast : t -> t

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.

Sourceval log_not : t -> t

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.

Sourceval log_and : t -> t -> t

Logical and.

Sourceval log_or : t -> t -> t

Logical or.

Sourceval log_xor : t -> t -> t

Logical exclusive or.

Sourceval log_eq : t -> t -> t
Sourceval log_neq : t -> t -> t
Sourceval log_sgl : (Z.t -> Z.t -> bool) -> t -> t -> t
Sourceval log_leq : t -> t -> t
Sourceval log_geq : t -> t -> t
Sourceval log_lt : t -> t -> t
Sourceval log_gt : t -> t -> t

C comparison tests. Returns an interval included in 0,1 (a boolean)

Sourceval is_log_eq : t -> t -> bool
Sourceval is_log_neq : t -> t -> bool
Sourceval is_log_sgl : (Z.t -> Z.t -> bool) -> t -> t -> bool
Sourceval is_log_leq : t -> t -> bool
Sourceval is_log_geq : t -> t -> bool
Sourceval is_log_lt : t -> t -> bool
Sourceval is_log_gt : t -> t -> bool

C comparison tests. Returns a boolean if the test may succeed

Sourceval shift_left : t -> t -> t_with_bot

Bitshift left: multiplication by a power of 2.

Sourceval shift_right : t -> t -> t_with_bot

Bitshift right: division by a power of 2 rounding towards -∞.

Sourceval shift_right_trunc : t -> t -> t_with_bot

Unsigned bitshift right: division by a power of 2 with truncation.

Sourceval bit_not : t -> t

Bitwise negation: ~x = -x-1

Filters

Given two interval aruments, return the arguments assuming that the predicate holds.

Sourceval filter_eq : t -> t -> (t * t) Utils_core.Bot.with_bot
Sourceval filter_sgl : (Z.t -> Z.t -> bool) -> t -> t -> (t * t) Utils_core.Bot.with_bot
Sourceval filter_neq : t -> t -> (t * t) Utils_core.Bot.with_bot
Sourceval filter_leq : t -> t -> (t * t) Utils_core.Bot.with_bot
Sourceval filter_geq : t -> t -> (t * t) Utils_core.Bot.with_bot
Sourceval filter_lt : t -> t -> (t * t) Utils_core.Bot.with_bot
Sourceval filter_gt : t -> t -> (t * t) Utils_core.Bot.with_bot

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.

Sourceval bwd_neg : t -> t -> t_with_bot
Sourceval bwd_abs : t -> t -> t_with_bot
Sourceval bwd_succ : t -> t -> t_with_bot
Sourceval bwd_pred : t -> t -> t_with_bot
Sourceval bwd_add : t -> t -> t -> (t * t) Utils_core.Bot.with_bot
Sourceval bwd_sub : t -> t -> t -> (t * t) Utils_core.Bot.with_bot
Sourceval bwd_mul : t -> t -> t -> (t * t) Utils_core.Bot.with_bot
Sourceval bwd_bit_not : t -> t -> t_with_bot
Sourceval bwd_join : t -> t -> t -> (t * t) Utils_core.Bot.with_bot

Backward join: both arguments and intersected with the result.

Sourceval bwd_rem : t -> t -> t -> (t * (Z.t * Z.t)) Utils_core.Bot.with_bot
Sourceval bwd_div : t -> t -> t -> (t * t) Utils_core.Bot.with_bot
Sourceval bwd_wrap : t -> 'a -> t -> t_with_bot
Sourceval bwd_shift_left : t -> t -> t -> (t * t) Utils_core.Bot.with_bot
Sourceval bwd_shift_right : t -> t -> t -> (t * t) Utils_core.Bot.with_bot
Sourceval bwd_shift_right_trunc : t -> t -> t -> (t * t) Utils_core.Bot.with_bot

Reduction

Sourceval meet_inter : t -> I.t -> (t * I.t) Utils_core.Bot.with_bot

Intersects a congruence with an interval, and returns the set represented both as a congruence and as an interval. Useful to implement reductions.