[{"type":"thinking","thinking":"One error left: `h2 : ((Rfun n) - fun s => A 1 / (s - 1)) = Rfun n` — after rw [hA1]: goal ((Rfun n) − fun s => 0/(s−1)) = Rfun n — simp made progress: (fun s => 0) = 0 remains — the zero function vs lambda — need Pi.zero_apply or funext: `simp` reduced to (fun s => 0) = 0 — add Pi.zero_apply? Actually `simp` should know (fun s => 0) = 0 via `Pi.zero_apply`... hmm the remaining goal (fun s => 0) = 0 — simp lemma `Pi.zero_apply` : (0 : Π i, α i) x = 0 — direction wrong (it rewrites 0 x → 0, not fun → 0). There's `funext; simp` or `rfl`?? (fun s => (0:ℂ)) = 0 — is 0 : ℂ→ℂ defined as fun s => 0? Pi.instZero: 0 = fun _ => 0 ✓ defeq — `rfl` should work! The `simp` left it unsolved — add rfl after, or use funext s; simp. Fix: \n\n```lean\n        have h2 : ((Rfun n) - fun s => A 1 / (s - 1)) = Rfun n := by\n          rw [hA1]\n          simp only [zero_div]\n          -- (Rfun n) − (fun s => 0) = Rfun n\n          rw [show (fun s : ℂ => (0 : ℂ)) = 0 from rfl]\n          exact sub_zero _\n```\nOr simpler: funext s; simp [hA1]. Let me do:\n```lean\n        have h2 : ((Rfun n) - fun s => A 1 / (s - 1)) = Rfun n := by\n          funext s\n          simp [hA1]\n```\nfunext s: goal Rfun n s − (fun s => A 1/(s−1)) s = Rfun n s — beta + hA1: A 1 = 0 — simp [hA1]: (fun..) s → A 1/(s−1) → 0/(s−1) → 0 — then Rfun n s − 0 = Rfun n s — sub_zero ✓ simp closes ✓.","signature":""},{"type":"text","text":"One error left — the function-level `sub_zero` needs pointwise. Fix."},{"type":"tool_use","id":"Edit_48","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSides.lean","old_string":"        have h2 : ((Rfun n) - fun s => A 1 / (s - 1)) = Rfun n := by\n          rw [hA1]; simp","new_string":"        have h2 : ((Rfun n) - fun s => A 1 / (s - 1)) = Rfun n := by\n          funext s\n          simp [hA1]"},"extra_content":null}]