[{"type":"thinking","thinking":"The signature is: `Finset.sum_image [DecidableEq ι] {s : Finset κ} {g : κ → ι} (h : Set.InjOn g s) : ∑ x ∈ s.image g, f x = ∑ x ∈ s, f (g x)` — the VALUE function is `f` (implicit, from the outer section variables), the MAP is `g`, and the hypothesis is `Set.InjOn g s` (not the ∀∈∀∈ form — though Set.InjOn g s unfolds to exactly that). So my named args were wrong: the map is `g`, the summand function is `f`, hypothesis is `Set.InjOn oddEmb (Icc 1 K)`.\n\nAlso note: `Set.InjOn g s : ∀ a ∈ s, ∀ b ∈ s, g a = g b → a = b` — definitionally that. My hinj has the right shape ✓.\n\nFix:\n```lean\n        have hsi := Finset.sum_image (s := Finset.Icc 1 K) (g := oddEmb)\n          (f := fun m : {m : ℕ // Odd m ∧ 3 ≤ m} =>\n            (1 - 1 / ((m : ℕ) : ℝ)) ^ n - 1 + (n : ℝ) / ((m : ℕ) : ℝ)) hinj\n```\nhinj : Set.InjOn oddEmb ↑(Icc 1 K)? hinj's type: ∀ x ∈ Finset.Icc 1 K, ∀ y ∈ ..., oddEmb x = oddEmb y → x = y — Set.InjOn is defined on a SET with ∈ membership: `Set.InjOn f s := ∀ ⦃a⦄, a ∈ s → ∀ ⦃b⦄, b ∈ s → f a = f b → a = b`. My hinj quantifies over Finset membership — defeq? Finset.mem_coe bridges. The earlier elaboration accepted hinj at argument position when the lemma's hypothesis was the ∀∈ form... now the expected type is Set.InjOn — the elaborator will unfold Set.InjOn and match ∀ x, x ∈ ↑s → ... — my hinj uses Finset membership `x ∈ Finset.Icc 1 K` which is `Finset.Mem` instance vs Set.Mem for the coerced set — defeq but maybe not syntactic. Provide explicitly: `(show Set.InjOn oddEmb ↑(Finset.Icc 1 K) from hinj)`? Membership x ∈ (↑(Finset.Icc 1 K) : Set ℕ) is defeq to x ∈ Finset.Icc 1 K (via Finset.mem_coe — an iff, membership instances: Set.Mem vs Finset.Mem — coerced set membership is DEFINED as finset membership ✓ defeq, `from hinj` should typecheck).","signature":""},{"type":"text","text":"`Finset.sum_image`'s hypothesis is `Set.InjOn` with the map named `g` and summand `f`. Fixing the call."},{"type":"tool_use","id":"Edit_69","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":"        rw [hconv]\n        have hsi := Finset.sum_image (s := Finset.Icc 1 K) (f := oddEmb)\n          (g := fun m : {m : ℕ // Odd m ∧ 3 ≤ m} =>\n            (1 - 1 / ((m : ℕ) : ℝ)) ^ n - 1 + (n : ℝ) / ((m : ℕ) : ℝ)) hinj\n        rw [← hsi]","new_string":"        rw [hconv]\n        have hsi := Finset.sum_image (s := Finset.Icc 1 K) (g := oddEmb)\n          (f := fun m : {m : ℕ // Odd m ∧ 3 ≤ m} =>\n            (1 - 1 / ((m : ℕ) : ℝ)) ^ n - 1 + (n : ℝ) / ((m : ℕ) : ℝ))\n          (show Set.InjOn oddEmb ↑(Finset.Icc 1 K) from hinj)\n        rw [← hsi]"},"extra_content":null}]