# Equality using Coercion considered harmful?

Currently, Sage specifies that, upon comparing with `a==b` two objects a and b that do not have the same parent, a coercion is attempted to put them in the same parent. To prepare a discussion on whether that specification is viable in the long run, we are gathering here (borderline?) use cases where this behaviour is harmful. (Note that there is also the related problem of broken notions of equality for elements with the same parent which is discussed at EqualityUnhashable.)

sage: bool(pi == 3.14159265358979323) True sage: hash(pi) 2943215493 sage: hash(3.14159265358979323) 1826823505

However Python 2.7 documentation specifies:

object.__hash__(self) Called by built-in function hash() and for operations on members of hashed collections including set, frozenset, and dict. __hash__() should return an integer. The only required property is that objects which compare equal have the same hash value;

As a first consequence of the above behavior:

sage: {3.1415926535897932: 'approx', pi: 'exact'} {3.1415926535897932: 'approx', pi: 'exact'} sage: {0:"exact", 0.0000000000000000000:"approx"} {0: 'approx'}

And also:

sage: pii = 3.14159265358979323 sage: bool(pii == pi) True sage: dd = {pi: "exact"} sage: pi in dd True sage: pii in dd False sage: pii in dd.keys() True

### pi is an integer (and transitivity of equality)

sage: R1 = RealField(2) sage: R2 = RealField(56) sage: pi1 = R1(pi); pi1 3.0 sage: pi2 = R2(pi); pi2 3.141592653589793 sage: pi1 == pi2 True sage: bool(pi == pi2) True sage: pi1 == 3 True

And we get a nice proof that pi = 3.

### More strange consequences when using UniqueRepresentation

sage: F1 = FiniteEnumeratedSet([0.000000]) sage: F2 = FiniteEnumeratedSet([0]) sage: F1 is F2 True sage: F2.list() [0.000000000000000] sage:

### An example showing that Sage's specifications clashes with Python's specifications

sage: S = SymmetricFunctions(QQ) sage: x = S.s()[5] sage: y = S.p()(x) sage: x == y True sage: hash(x), hash(y) (-1840429907820881728, 5178019317311573726)

It's surely syntactically nice to have x == y evaluate True after a coercion. However enforcing that the two hash functions be the same would be simply impossible: this would force to systematically coerce any symmetric function to some fixed base for computing the hash function, and we just can't afford that.

Yet another example (#22594)

sage: bring=ZZ sage: for a in cartesian_product([[bring.one()]]): ....: print a[0].parent() ....: Integer Ring sage: bring=QQ sage: for a in cartesian_product([[bring.one()]]): ....: print a[0].parent() ....: Integer Ring