Functional induction does not preserve match cases syntactically

Prerequisites

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Description

If I define a function on natural numbers using the equation compiler and have a case for n + 1, then the corresponding case in the functional induction principle is spelled n.succ.

Context

In #3629 we changed cases and induction such that they generate goals with n + 1 instead of n.succ. This is one case where functional induction is less nice to use than induction or the equation compiler.

Steps to Reproduce

def f : Nat → Nat
  | 0 => 0
  | n + 1 => f n

#check f.induct

Expected behavior: f.induct (motive : Nat → Prop) (case1 : motive 0) (case2 : ∀ (n : Nat), motive n → motive (n + 1)) : ∀ (a : Nat), motive a

Actual behavior: f.induct (motive : Nat → Prop) (case1 : motive 0) (case2 : ∀ (n : Nat), motive n → motive n.succ) : ∀ (a : Nat), motive a

Versions

4.12.0-nightly-2024-09-07 on live.lean-lang.org

Impact

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Reproduce the minimal example with the equation compiler and #check f.induct, then trace how the functional induction principle is generated. The fix is done when the induction case for n + 1 is presented syntactically as motive (n + 1) rather than motive n.succ, with the example still type-checking.