[{"type":"thinking","thinking":"`Nat.cast_analyticOrderNatAt (hf : analyticOrderAt f z₀ ≠ ⊤) : ↑(analyticOrderNatAt f z₀) = analyticOrderAt f z₀` (with the cast implicit in the display — it's `((analyticOrderNatAt f z₀ : ℕ) : ℕ∞) = analyticOrderAt f z₀`).\n\nSo the final chain for the order invariance:\n```lean\n  have hF_top : analyticOrderAt (fun w => riemannZeta (discCenter t + (5 / 2 : ℂ) * w)) w₀ ≠ ⊤ := by\n    rw [horder]\n    exact ENat.natCast_ne_top _  -- hmm — with horder : analyticOrderAt F w₀ = ↑(m): rw makes goal ↑(m) ≠ ⊤: ENat.natCast_ne_top : (↑n : ℕ∞) ≠ ⊤? or (n : ℕ∞) ≠ ⊤... `ENat.natCast_ne_top (n : ℕ) : (↑n : ℕ∞) ≠ ⊤`? hmm — the form: (↑(m) : ℕ∞) ≠ ⊤: ENat.natCast_ne_top _ ✓ or `by simp`.\n  have h1 := Nat.cast_analyticOrderNatAt hF_top  -- ↑(analyticOrderNatAt F w₀) = analyticOrderAt F w₀\n  have h2 := Nat.cast_analyticOrderNatAt (zeta_order_ne_top_of_ne hz₀ne)  -- ↑(mζ) = analyticOrderAt ζ z₀\n  rw [← Nat.cast_inj (R := ℕ∞)]  -- goal ↑(mF) = ↑(mζ)\n  rw [h1, h2, horder]  -- goal: ↑(mζ) = analyticOrderAt ζ z₀?? — no: after h1: analyticOrderAt F w₀ = ...; h2: RHS ↑(mζ) → analyticOrderAt ζ z₀: goal analyticOrderAt F w₀ = analyticOrderAt ζ z₀: horder : analyticOrderAt F w₀ = ↑(mζ): rw [horder]: goal ↑(mζ) = analyticOrderAt ζ z₀: close via h2.symm: exact h2.symm\n```\nSo: rw [h1, h2, horder]; exact h2.symm — hmm after rw [h1, h2, horder]: the goal becomes ↑(mζ) = analyticOrderAt ζ z₀?? wait — h2 : ↑(mζ) = analyticOrderAt ζ z₀ — rw [h2] rewrites ↑(mζ) → analyticOrderAt ζ z₀ in the goal — the goal at that point: analyticOrderAt F w₀ = ↑(mζ) — rw [h2] targets ↑(mζ)?? — the ↑(mζ) appears on the RHS ✓ becomes analyticOrderAt ζ z₀: goal analyticOrderAt F w₀ = analyticOrderAt ζ z₀: then rw [horder]: LHS → ↑(mζ): goal ↑(mζ) = analyticOrderAt ζ z₀ — hmm — then `exact h2.symm`? h2.symm : analyticOrderAt ζ z₀ = ↑(mζ) — goal is ↑(mζ) = analyticOrderAt ζ z₀: that's h2 ✓ exact h2. Wait I flipped: goal after rw [horder]: ↑(mζ) = analyticOrderAt ζ z₀ and h2 : ↑(mζ) = analyticOrderAt ζ z₀ — EXACT MATCH: exact h2. Hmm but then the earlier rw [h2] already consumed it... — after `rw [h1, h2, horder]`: rw [h1]: ↑(mF) → analyticOrderAt F w₀; rw [h2]: ↑(mζ) → analyticOrderAt ζ z₀: goal becomes analyticOrderAt F w₀ = analyticOrderAt ζ z₀; rw [horder]: analyticOrderAt F w₀ → ↑(mζ): goal ↑(mζ) = analyticOrderAt ζ z₀ — which is h2 — but h2 was already used in the rw chain... it's still available as a hypothesis ✓ `exact h2`. Hmm wait — does rw [h2] at that point work? After rw [h1]: goal: analyticOrderAt F w₀ = ↑(mζ): rw [h2] rewrites ↑(mζ) → analyticOrderAt ζ z₀ ✓: goal analyticOrderAt F w₀ = analyticOrderAt ζ z₀: then horder : analyticOrderAt F w₀ = ↑(mζ) — rw [horder] rewrites the analyticOrderAt F w₀ → ↑(mζ): goal ↑(mζ) = analyticOrderAt ζ z₀ = h2 ✓ exact h2. OK.\n\nHmm actually — simpler: skip the natCast_inj dance: the goal analyticOrderNatAt F w₀ = analyticOrderNatAt ζ z₀ (an ℕ equality): \n```lean\n  have : (analyticOrderNatAt (fun w => riemannZeta (discCenter t + (5/2:ℂ)*w)) w₀ : ℕ∞) = analyticOrderNatAt riemannZeta z₀ := by\n    rw [Nat.cast_analyticOrderNatAt hF_top, Nat.cast_analyticOrderNatAt (zeta_order_ne_top_of_ne hz₀ne), horder]\n  exact ENat.natCast_inj.mp this  -- or Nat.cast_inj.mp\n```\n`Nat.cast_inj (R := ℕ∞)`? — (a : ℕ) = (b : ℕ) from (↑a : ℕ∞) = ↑b: `Nat.cast_inj.mp`? WithTop.coe_inj? ENat natCast inj: `ENat.natCast_inj`? or `Nat.cast_inj (R := ℕ∞)`: (↑a : ℕ∞) = ↑b ↔ a = b ✓ `Nat.cast_inj`. So exact Nat.cast_inj.mp this. OK.\n\nLet me apply the three fixes (h3'.trans chain, the final order chain, and div_eq_zero's resolve_right + mem form).","signature":""},{"type":"text","text":"Applying the three fixes."},{"type":"tool_use","id":"Edit_366","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDisc.lean","old_string":"  have h3' := h3.comp_tendsto haffT\n  -- h3' : (fun z => (z − z₀)^m • h' z) ∘ aff =ᶠ ζ ∘ aff; massage the form\n  have hF : (fun w => riemannZeta (discCenter t + (5 / 2 : ℂ) * w)) =ᶠ[𝓝 w₀]\n      fun w => (w - w₀) ^ analyticOrderNatAt riemannZeta z₀ •\n        (((5 / 2 : ℂ) ^ analyticOrderNatAt riemannZeta z₀) *\n          h' (discCenter t + (5 / 2 : ℂ) * w)) := by\n    apply h3'.congr'\n    filter_upwards with w\n    simp only [Function.comp_apply]\n    rw [show discCenter t + (5 / 2 : ℂ) * w - z₀ = (5 / 2 : ℂ) * (w - w₀) by rw [hz₀]; ring]\n    rw [mul_pow, smul_eq_mul, smul_eq_mul]\n    ring\n  have hcof : AnalyticAt ℂ (fun w => ((5 / 2 : ℂ) ^ analyticOrderNatAt riemannZeta z₀) *\n      h' (discCenter t + (5 / 2 : ℂ) * w)) w₀ :=\n    (analyticAt_const.mul (h1.comp_of_eq haff rfl))  -- hmm h1 is AnalyticAt h' z₀\n  have hcof_ne : ((5 / 2 : ℂ) ^ analyticOrderNatAt riemannZeta z₀) * h' z₀ ≠ 0 :=\n    mul_ne_zero (pow_ne_zero _ (by norm_num)) h2\n  have hF_an : AnalyticAt ℂ (fun w => riemannZeta (discCenter t + (5 / 2 : ℂ) * w)) w₀ :=\n    analyticAt_riemannZeta_of_ne_one hz₀ne |>.comp_of_eq haff rfl\n  have horder : analyticOrderAt (fun w => riemannZeta (discCenter t + (5 / 2 : ℂ) * w)) w₀\n      = (analyticOrderNatAt riemannZeta z₀ : ℕ∞) :=\n    (hF_an.analyticOrderAt_eq_natCast).mpr ⟨_, hcof, hcof_ne, hF⟩\n  rw [← Nat.cast_inj (R := ℕ∞), Nat.cast_analyticOrderNatAt\n    (zeta_order_ne_top_of_ne hz₀ne), horder]","new_string":"  have h3' := h3.comp_tendsto haffT\n  -- h3' : ζ ∘ aff =ᶠ series ∘ aff; massage the form\n  have hpt : ∀ w : ℂ,\n      ((fun z => (z - z₀) ^ analyticOrderNatAt riemannZeta z₀ • h' z) ∘\n        fun w => discCenter t + (5 / 2 : ℂ) * w) w\n        = (w - w₀) ^ analyticOrderNatAt riemannZeta z₀ •\n          (((5 / 2 : ℂ) ^ analyticOrderNatAt riemannZeta z₀) *\n            h' (discCenter t + (5 / 2 : ℂ) * w)) := by\n    intro w\n    simp only [Function.comp_apply]\n    rw [show discCenter t + (5 / 2 : ℂ) * w - z₀ = (5 / 2 : ℂ) * (w - w₀) by rw [hz₀]; ring]\n    rw [mul_pow, smul_eq_mul, smul_eq_mul]\n    ring\n  have hF : (fun w => riemannZeta (discCenter t + (5 / 2 : ℂ) * w)) =ᶠ[𝓝 w₀]\n      fun w => (w - w₀) ^ analyticOrderNatAt riemannZeta z₀ •\n        (((5 / 2 : ℂ) ^ analyticOrderNatAt riemannZeta z₀) *\n          h' (discCenter t + (5 / 2 : ℂ) * w)) :=\n    h3'.trans (Filter.Eventually.of_forall hpt)\n  have hcof : AnalyticAt ℂ (fun w => ((5 / 2 : ℂ) ^ analyticOrderNatAt riemannZeta z₀) *\n      h' (discCenter t + (5 / 2 : ℂ) * w)) w₀ :=\n    analyticAt_const.mul (h1.comp_of_eq haff rfl)\n  have hcof_ne : ((5 / 2 : ℂ) ^ analyticOrderNatAt riemannZeta z₀) * h' z₀ ≠ 0 :=\n    mul_ne_zero (pow_ne_zero _ (by norm_num)) h2\n  have hF_an : AnalyticAt ℂ (fun w => riemannZeta (discCenter t + (5 / 2 : ℂ) * w)) w₀ :=\n    analyticAt_riemannZeta_of_ne_one hz₀ne |>.comp_of_eq haff rfl\n  have horder : analyticOrderAt (fun w => riemannZeta (discCenter t + (5 / 2 : ℂ) * w)) w₀\n      = (analyticOrderNatAt riemannZeta z₀ : ℕ∞) :=\n    (hF_an.analyticOrderAt_eq_natCast).mpr ⟨_, hcof, hcof_ne, hF⟩\n  have hF_top : analyticOrderAt (fun w => riemannZeta (discCenter t + (5 / 2 : ℂ) * w)) w₀\n      ≠ ⊤ := by\n    rw [horder]\n    exact ENat.natCast_ne_top _\n  have h1c := Nat.cast_analyticOrderNatAt hF_top\n  have h2c := Nat.cast_analyticOrderNatAt (zeta_order_ne_top_of_ne hz₀ne)\n  have hcast : ((analyticOrderNatAt (fun w => riemannZeta (discCenter t + (5 / 2 : ℂ) * w)) w₀\n      : ℕ) : ℕ∞) = ((analyticOrderNatAt riemannZeta z₀ : ℕ) : ℕ∞) := by\n    rw [h1c, h2c, horder]\n  exact Nat.cast_inj.mp hcast"},"extra_content":null}]