solve_by_elim failure

Prerequisites

• Check that your issue is not already filed:
  https://github.com/leanprover/lean4/issues
• Reduce the issue to a minimal, self-contained, reproducible test case.
  Avoid dependencies to Mathlib or Batteries.
• Test your test case against the latest nightly release, for example on
  https://live.lean-lang.org/#project=lean-nightly
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Description

The solve_by_elim tactic fails on test. It should work.

def injective (f: α → β): Prop := ∀ x y, f x = f y → x = y

def surjective (f: α → β): Prop := ∀ x, ∃ y, f y = x

def bijective (f: α → β): Prop := injective f ∧ surjective f

def prod (f: α → β) (g: γ → σ): α × γ → β × σ := fun (x, y) => (f x, g y)

theorem bijective_id: bijective (id: α → α) := sorry

theorem bijective_compose {f: α → β} {g: β → γ}
  (h₁: bijective f) (h₂: bijective g): bijective (g ∘ f) := sorry

theorem bijective_product {f: α → β} {g: γ → σ} (h₁: bijective f) (h₂: bijective g): 
  bijective (prod f g) := sorry

theorem test {α β γ δ σ: Type}
  (f₀: (α → β) → γ)
  (f₁: (δ → β) → (α → β) × β)
  (f₂: γ × β → σ)
  (h₀: bijective f₀)
  (h₁: bijective f₁)
  (h₂: bijective f₂): bijective (f₂ ∘ prod f₀ id ∘ f₁) := by
   solve_by_elim [bijective_compose, bijective_product, bijective_id]

Context

Discussed during office hours with @kim-em.

Versions

4.30.0
4.29.1

Impact

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Reproduce the minimal test theorem with Lean 4 versions 4.30.0 and 4.29.1, focusing on the solve_by_elim invocation and the supplied bijectivity lemmas. Trace why solve_by_elim [bijective_compose, bijective_product, bijective_id] does not close the goal, then confirm that the original theorem succeeds without changing its intended statement.