Assumptions

From SymPy

<source lang="python"> M ... Mathematica S0 ... SymPy, current approach S1 .... SymPy, approach 1 S2 .... SymPy, approach 2

M: Simplify[1/Sqrt[x] - Sqrt[1/x], x > 0] S1: x = Symbol('x', assumptions=IsPositive)

 simplify(1/sqrt(x) - sqrt(1/x))

S2: simplify(1/sqrt(x) - sqrt(1/x), Assumptions(x>0))

M: FunctionExpand[Log[x y], x > 0 && y > 0] S1: x, y = Symbol('x', assumptions=IsPositive), Symbol('y', assumptions=IsPositive)

log(x*y).expand()

S2: log(x*y).expand(Assumptions([x>0, y>0]))

M: Simplify[Sin[n Pi], n \[Element] Integers] S1: n = Symbol('n', assumptions=IsInteger)

 simplify(sin(n*pi))

S2: simplify(sin(n*pi), Assumptions(Element(n, Integer)))

1. we can talk about the syntax of Element(n, Integer)

M: FunctionExpand[EulerPhi[m n], {m, n} \[Element] Integers && GCD[m, n] == 1] S1: n = Symbol('n', assumptions=IsInteger)

 m = Symbol('m', assumptions=IsInteger)
 n.assumptions.add(Eq(gcd(m, n) - 1))
 euler_phi(m, n)

S2: euler_phi(m, n).expand(Assumptions([Element(n, Integer), Element(m, Integer), Eq(gcd(m, n) - 1)]))

1. again we can talk about the syntax of Element(n, Integer)

M: RealQ[x, x \[Element] Real] S0: x = Symbol('x',real=True, integer=True)

 assert x.is_real == True

S1: S2: assert IsElement(x, Real, assumptions=Element(x, Real))

M: Refine[Abs[x], x>0]

  Refine[Abs[x], x<0]

S0: S1: S2: e = abs(x)

 print e.refine(Assumptions(x>0))
 print e.refine(Assumptions(x<0))

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