[goldenkey]

Forgive me but if we actually have to implement mathematics from pure computational theory, wouldn't type-linear functions be equivalent to actual mathematical linear functions?

[waltpad]

Well, if copies may be borrowed, as it is the case in Rust, I suppose that a type-linear function, as I called it, doesn't have to be computationally linear. The `mult` example we discussed could not be applied twice to the same value, unless it is declared as borrowing its parameters from the caller.

Elaborating on this, with a function signature like the following:

    fn mult(x : Vec<f32>, y : Vec<f32>) -> Vec<f32> {
       // matrix product implementation 
    }

arguments x and y cannot point to the same value, because a value cannot be moved twice. A call like

    mult(a,a)

will generate an error.

Conversely, with the following signature:

    fn mult(x : &Vec<f32>, y : &Vec<f32>) -> Vec<f32> {
       // ...
    }

the call:

    mult(&a,&a)

will typecheck, because values are passed by reference.

Now, nothing prevents one to implement the function square, based on the second version of mult:

    fn square(x : Vec<f32>) -> Vec<f32> {
        return mult(&x,&x)
    }

which is type-linear on its parameter x, but computationally is not linear.

Clearly your original question was spot on.

[goldenkey]

I see what you mean with the copying. I just thought based on OPs comments there would have been a strong correspondence between lambda calculus numbers (Church encoding) and the operations on them, and actual type-linear functions.