Assignment incompatibility with structurally equivalent generic mapped types

[BigDataSamuli]

TypeScript Version: nightly (2.2.0-dev.20161211)

Code

type Mapped1<T> = {
  [P in keyof T]: T[P];
}

type Mapped2<T> = {
  [P in keyof T]: T[P];
}

function asMapped2<T>(a: Mapped1<T>): Mapped2<T> {
  return a;
}

Expected behavior:
No error.

Actual behavior:

test.ts(10,10): error TS2322: Type 'Mapped1<T>' is not assignable to type 'Mapped2<T>'.
  Type 'T[P]' is not assignable to type 'T[P]'. Two different types with this name exist, but they are 
unrelated.

[ahejlsberg]

Agreed. I think we need a rule that a mapped type { [P1 in K1] : X1 } is related to a mapped type { [P2 in K2]: X2 } when K1 is related to K2 and X1 is related to an instantiation of X2 that substitutes P1 for P2.

[BigDataSamuli]

FWIW, my original use case was similar to the following:

interface Fooify<T> {
  foo: T;
}

interface Barify<T> {
  bar: T;
}

type Mapped1<T> = {
  [P in keyof T]: Fooify<T[P]>;
}

type Mapped2<T> = {
  [P in keyof T]: Barify<T[P]> | Fooify<T[P]>;
}

function asMapped2<T>(a: Mapped1<T>): Mapped2<T> {
  return a;
}

which errors with

test.ts(18,10): error TS2322: Type 'Mapped1<T>' is not assignable to type 'Mapped2<T>'.
  Type 'Fooify<T[P]>' is not assignable to type 'Barify<T[P]> | Fooify<T[P]>'.
    Type 'Fooify<T[P]>' is not assignable to type 'Fooify<T[P]>'. Two different types with this name exist, but they are unrelated.
      Type 'T[P]' is not assignable to type 'T[P]'. Two different types with this name exist, but they are unrelated.

I presumed the issue in the OP is the underlying cause for this.