Mathematica not simplifing when using FullSimplify

Question

I am trying to simplify the following $(Z_i - Z_{i+1})^4$ using

FullSimplify[(Subscript[Z, i] - Subscript[Z, 1 + i])^4 , 
Assumptions -> {\[CapitalDelta]z > 0, 
Subscript[Z, i] \[Element] Reals, 
Subscript[Z, i - 1] < Subscript[Z, i] < Subscript[Z, i + 1], 
Subscript[Z, i + 1] - Subscript[Z, i ] == \[CapitalDelta]z, 
Subscript[Z, i] - Subscript[Z, i - 1 ] == \[CapitalDelta]z}]

The answer should be $\Delta z^4$ but it just returns the same thing that I put in. Is there something that I am missing?

Answer 1

You can use a custom transformation function:

t[e_] := Collect[e /. Subscript[z, i_] :> Subscript[z, 0] + i dz, dz]
FullSimplify[
 (Subscript[z, i] - Subscript[z, i + 1] + Subscript[z, i + 10] -Subscript[z, i + 2])^4, 
 TransformationFunctions -> {t, Automatic}]

Or define z to have that property:

Clear@z
z[i_] := z[0] + i dz /; i =!= 0
Simplify@(z[i] - z[i + 1] + z[i + 10] - z[i + 2])^4

A bit cleaner:

Clear@z
z[i_] := z0 + i dz