Problem

U must be an array. We know U is a value of P, but there is no guarantee that all values of P are arrays. That's because exec depends not on the concrete ICommandNameArgumentTypeMapping interface defined below, but on some P we don't fully know yet. And because we don't know it yet, we cannot trust it to follow the blueprint of ICommandNameArgumentTypeMapping — after all, it might add some properties of its own which are not arrays.

Solution

The solution is to make sure all values — present and future — will always be arrays.

interface ICommandNameArgumentTypeMapping {
    ['cmd1']: [string];
    ['cmd2']: [boolean, number, string];
    ['cmd3']: [boolean, boolean];
    [index: string]: any[]
}

That extra property is called index signature.

You can, of course, be more precise here and say (string | number | boolean)[] instead of any[].

Bonus points

There are a few more mistakes in your code:

• Computed property name cmd2 is duplicated
• The T type parameter is unused
• The P type parameter is incorrectly used (it is not used to describe parameters nor return types)
• exec promises to return U, but it's returning undefined

The corrected solution:

function exec<P extends ICommandNameArgumentTypeMapping, E extends keyof P, U extends P[E]>(mapping: P, command: E, ...rest: U): U{
    return rest;
}

interface ICommandNameArgumentTypeMapping {
    ['cmd1']: [string];
    ['cmd2']: [boolean, number, string];
    ['cmd3']: [boolean, boolean];
    [index: string]: any[]
}

declare const mapping: ICommandNameArgumentTypeMapping;

exec(mapping, 'cmd2', true, 1, 'hello');