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Fungrim entry: 569278

(ab  and  b=2d)        (u=sgn(a))   where (d,u,v)=xgcd ⁣(a,b)\left(\left|a\right| \ne \left|b\right| \;\mathbin{\operatorname{and}}\; \left|b\right| = \left|2 d\right|\right) \;\implies\; \left(u = \operatorname{sgn}(a)\right)\; \text{ where } \left(d, u, v\right) = \operatorname{xgcd}\!\left(a, b\right)
Assumptions:aZ{0}  and  bZ{0}a \in \mathbb{Z} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{Z} \setminus \left\{0\right\}
\left(\left|a\right| \ne \left|b\right| \;\mathbin{\operatorname{and}}\; \left|b\right| = \left|2 d\right|\right) \;\implies\; \left(u = \operatorname{sgn}(a)\right)\; \text{ where } \left(d, u, v\right) = \operatorname{xgcd}\!\left(a, b\right)

a \in \mathbb{Z} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{Z} \setminus \left\{0\right\}
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
Signsgn(z)\operatorname{sgn}(z) Sign function
XGCDxgcd ⁣(a,b)\operatorname{xgcd}\!\left(a, b\right) Extended greatest common divisor
ZZZ\mathbb{Z} Integers
Source code for this entry:
    Formula(Where(Implies(And(NotEqual(Abs(a), Abs(b)), Equal(Abs(b), Abs(Mul(2, d)))), Equal(u, Sign(a))), Equal(Tuple(d, u, v), XGCD(a, b)))),
    Variables(a, b),
    Assumptions(And(Element(a, SetMinus(ZZ, Set(0))), Element(b, SetMinus(ZZ, Set(0))))))

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2021-03-15 19:12:00.328586 UTC