Integrating a choice-free Finset.sum_le_sum into mathlib

Determine a method for integrating into the Lean mathlib library a choice-free version of Finset.sum_le_sum (the standard inequality comparing sums over finite sets) that assumes decidable equality on the index type, given that the current mathlib proof uses the axiom of choice to obtain DecidableEq ι and the available choice-free alternative is technically weaker due to this additional assumption.

Background

A core goal of the paper is to carry out synthetic differential geometry in a constructive setting, avoiding uses of the axiom of choice within Lean and mathlib. This requires replacing or reproving certain standard results whose existing proofs rely on choice. One such result is Finset.sum_le_sum, where the mathlib proof uses choice to obtain DecidableEq for the index type.

The authors note that in their applications the index types already have decidable equality, allowing them to supply a choice-free alternative to Finset.sum_le_sum tailored to their setting. However, integrating this weaker, assumption-dependent version directly into mathlib is nontrivial, because doing so would ripple through many dependent theorems and definitions and potentially require adding decidable equality assumptions widely. The authors therefore flag the integration question as unresolved and, for their project, provide a local replacement rather than modifying mathlib.