## Ints

```(theory Ints

:smt-lib-version 2.6
:smt-lib-release "2017-11-24"
:written-by "Cesare Tinelli"
:date "2010-04-17"
:last-updated "2015-04-25"
:update-history
"Note: history only accounts for content changes, not release changes.
2015-04-25 Updated to Version 2.5.
"

:sorts ((Int 0))

:funs ((NUMERAL Int)
(- Int Int)                 ; negation
(- Int Int Int :left-assoc) ; subtraction
(+ Int Int Int :left-assoc)
(* Int Int Int :left-assoc)
(div Int Int Int :left-assoc)
(mod Int Int Int)
(abs Int Int)
(<= Int Int Bool :chainable)
(<  Int Int Bool :chainable)
(>= Int Int Bool :chainable)
(>  Int Int Bool :chainable)
)

:funs_description
"All ranked function symbols of the form
((_ divisible n) Int Bool)
where n is a positive numeral.
"

:values
"The set of values for the sort Int consists of
- all numerals,
- all terms of the form (- n) where n is a numeral other than 0.
"

:definition
"For every expanded signature, the instance of Ints with that
signature is the theory consisting of all Sigma-models that interpret:

- the sort Int as the set of all integer numbers,

- each numeral as the corresponding natural number,

- (_ divisible n) as the function mapping to true all and only
the integers that are divisible by n,

- abs as the absolute value function,

- div and mod according to Boute's Euclidean definition , that is,
so as to satify the formula

(for all ((m Int) (n Int))
(=> (distinct n 0)
(let ((q (div m n)) (r (mod m n)))
(and (= m (+ (* n q) r))
(<= 0 r (- (abs n) 1))))))

- the other function symbols of Ints as expected.

References:
 Boute, Raymond T. (April 1992).
The Euclidean definition of the functions div and mod.
ACM Transactions on Programming Languages and Systems (TOPLAS)
ACM Press. 14 (2): 127 - 144. doi:10.1145/128861.128862.
"

:notes
"Regardless of sign of m,
when n is positive, (div m n) is the floor of the rational number m/n;
when n is negative, (div m n) is the ceiling of m/n.

This contrasts with alternative but less robust definitions of div and mod
where (div m n) is
- always the integer part of m/n (rounding towards 0), or
- always the floor of x/y (rounding towards -infinity).
"

:notes
"See note in the Reals theory declaration about terms of the form (/ t 0).
The same observation applies here to terms of the form (div t 0) and
(mod t 0).
"
)
```
(raw file)