[{"type":"thinking","thinking":"`pow_le_pow_left₀ (ha : 0 ≤ a) (hab : a ≤ b) (n) : a ^ n ≤ b ^ n` — the current name in this Mathlib (v4.33). ✓\n\nAll names verified:\n- `pow_le_pow_left₀` (not pow_le_pow_left)\n- `Real.log_le_sub_one_of_pos` ✓\n- `Complex.norm_natCast` (used in LiCircleBound:3410) ✓\n- `Real.log_le_rpow_div` ✓\n- `Real.pow_rpow_inv_natCast (hx : 0 ≤ x) (hn : n ≠ 0) : (x ^ n) ^ (n⁻¹ : ℝ) = x` ✓\n- `Real.rpow_inv_natCast_pow (hx : 0 ≤ x) (hn : n ≠ 0) : (x ^ (n⁻¹ : ℝ)) ^ n = x` ✓\n- `Complex.dist_le_div_mul_dist_of_mapsTo_ball` ✓\n- `AnalyticOn.norm_le_of_norm_le_on_sphere` (Zeta23 StrongPNTPrefix) ✓ — LiDisc imports StrongPNTPrefix ✓ transitively available.\n- `AnalyticAt.differentiableAt` ✓\n\nMark todo 2 done, todo 3 in progress. Now write LiSqueeze.lean. Plan the structure:\n\n```lean\n/-\nNguyenRH1/LiSqueeze.lean — tier ii′ sub-stage ii′-5e-v-d (D-068 §5):\nthe Schwarz squeeze and the partial-fraction capstone.\n\n...\n-/\nimport Mathlib\nimport NguyenRH1.LiCircleBound\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n```\n\nWait — LiCircleBound's open line: `open Zeta23 Complex Filter Topology`. Is there also a `noncomputable section` + namespace? Check LiCircleBound line 31: `noncomputable section` and presumably `namespace NguyenRH1`. Let me check the exact header to mirror.\n\nAlso the end: `end NguyenRH1` then `end` (the noncomputable section end). Mirror that.\n\nNow write the whole file. Let me be careful about a few more details:\n\n1. In §3 `gentire_circle_le`, Fexpr_circle_le's signature: `{k : ℕ} (hk : 4000 ≤ k) {R : ℝ} (hRlo : ...) (_hRhi : ...) (hgap : ...) (hgapInt : ...) {s : ℂ} (hs : ‖s‖ = R)`. ✓.\n\n2. `not_isNontrivialZero_on_gapCircle` exact signature — grep it to be sure about argument order (hk hRlo hgap hs).\n\n3. `nine_le_log_of_16002_le` ✓ used in LiCircleBound:3826.\n\n4. `two_le_C0hat {t : ℝ} (ht : 1 ≤ t) : 2 ≤ C0hat t` ✓.\n\n5. `A0count_pos` — used at line 3828: `A0count_pos.le` ✓ exists.\n\n6. `lowZeroConst_nonneg`, `dirichletC2_nonneg` — used at 3922 ✓.\n\n7. `gentire_eq_Fexpr` — signature `(hygood : ¬ badPoint y) : Gentire y = Fexpr y`? From line 1109: `rw [gentire_eq_Fexpr hygood]` where the goal mentions Gentire y... rw with the equation Gentire y = Fexpr y rewrites Gentire y → Fexpr y ✓. And in my step: I have goal ‖Gentire z‖ ≤ RB; rw [gentire_eq_Fexpr hgood] turns goal into ‖Fexpr z‖ ≤ RB ✓.\n\n8. The capstone: `badPoint` disjunct 4 is `∃ n : ℕ, 1 ≤ n ∧ s = -2 * (n : ℂ)`. In LiFEntire line 75: `∃ n : ℕ, 1 ≤ n ∧ s = -2 * (n : ℂ)` ✓.\n\n9. In the capstone statement, D-068 §5 writes `digamma (s/2)/2`. Fexpr def line 70: `Complex.digamma (s / 2) / 2` ✓.\n\n10. For `exists_gapCircle_radius hZC (k := k)`: signature `(hZC : ZeroCountProp) {k : ℕ} (hk : 1 ≤ k)`. k implicit → use (k := k). \n\n11. `Real.rpow_le_rpow`: signature `(hx : 0 ≤ x) (hxy : x ≤ y) (hz : 0 ≤ z) : x ^ z ≤ y ^ z` ✓ standard.\n\n12. `Real.rpow_pos_of_pos (hx : 0 < x) (r : ℝ) : 0 < x ^ r` ✓. `Real.rpow_nonneg (hx : 0 ≤ x) (r : ℝ) : 0 ≤ x ^ r` ✓.\n\n13. `div_mul_cancel₀ : a / b * b = a` needs b ≠ 0: `div_mul_cancel₀ (a : α) (h : b ≠ 0) : a / b * b = a`? Signature check: in Mathlib `div_mul_cancel₀ (a : G₀) (h : b ≠ 0) : a / b * b = a`. Hmm it might be `div_mul_cancel₀ (h : b ≠ 0) : a / b * b = a` with a explicit... Let me just use `by field_simp` or the calc with explicit forms. Actually for `(128 * C₁) * ‖z₀‖ / ε * ε = (128*C₁)*‖z₀‖`: `div_mul_cancel₀` — current form: `theorem div_mul_cancel₀ (a : G₀) {b : G₀} (h : b ≠ 0) : a / b * b = a`. I'll write `(div_mul_cancel₀ _ hε.ne').symm` hmm wait direction: I want (128*C₁*‖z₀‖) = (128*C₁*‖z₀‖/ε)*ε as the first calc step \"X = X/ε*ε\": `(div_mul_cancel₀ _ hε.ne').symm : a = a / ε * ε` ✓.\n\n14. `div_le_div_of_nonneg_right` — session-032 gotcha says it EXISTS ✓: `div_le_div_of_nonneg_right (hab : a ≤ b) (hc : 0 ≤ c) : a / c ≤ b / c`? Hmm — actually wait, that gotcha said \"div_le_div_of_nonneg_right exists / div_le_div_right does NOT\". But what are its args? Might be (h : a ≤ b) (hc : 0 ≤ c). I'll trust and adjust if the build complains.\n\nHmm wait, actually in current Mathlib, div_le_div_of_nonneg_right... Let me double check it's for a/c ≤ b/c given a ≤ b, c ≥ 0 — \"right\" refers to c on the right ✓.\n\n15. In hc1's first have: `(128 * C₁ + 2 * C₀') * ‖z₀‖ / ε ≤ c` where c = 2 + (128*C₁+2*C₀')*‖z₀‖/ε. With `set c := ... with hc`, rw [hc] changes goal c → the expr? NO — hc : c = expr; rw [hc] rewrites occurrences of c with expr: goal becomes expr ≤ expr-ish: (128*C₁+2*C₀')*‖z₀‖/ε ≤ 2 + (128*C₁+2*C₀')*‖z₀‖/ε — linarith closes (needs no hypotheses; the atom (128*C₁+2*C₀')*‖z₀‖/ε appears both sides) ✓.\n\nHmm wait — after `set c := 2 + X with hc`, the goal (if any) has c substituted. In `have h1 : X ≤ c := by rw [hc]; linarith` — the have statement is written by me with c; rw [hc] rewrites c → 2 + X: goal X ≤ 2 + X; linarith — X is the atom `(128 * C₁ + 2 * C₀') * ‖z₀‖ / ε`; linarith treats nonlinear/div as atom ✓ closes: X ≤ 2 + X ⟸ 0 ≤ 2 ✓.\n\n16. hcz : 2 ≤ c (I need c ≥ 2 for hQ3): rw [hc]; goal 2 ≤ 2 + X with X ≥ 0: have hX : 0 ≤ (128*C₁ + 2*C₀')*‖z₀‖/ε := div_nonneg (mul_nonneg (add_nonneg (mul_nonneg (by norm_num) hC₁nn) (mul_nonneg (by norm_num) hC₀'nn)) (norm_nonneg _)) hε.le; linarith [hX].\n\n17. For `hcnn : 0 ≤ c` (needed for pow_rpow_inv_natCast): linarith [hcz].\n\n18. hQc proof:\n```\n    have hQc : c ≤ Q := by\n      have hc4 : c ^ 4 ≤ R := by\n        have h : c ^ 4 ≤ M := le_max_left _ _\n        linarith [hMR]\n      have h1 : c = (c ^ 4) ^ ((4:ℝ)⁻¹) := (Real.pow_rpow_inv_natCast hcnn (by norm_num : (4:ℕ) ≠ 0)).symm\n```\nCareful: pow_rpow_inv_natCast (hx : 0 ≤ x) (hn : n ≠ 0) : (x ^ n) ^ (↑n)⁻¹ = x — n implicit {n : ℕ}? The statement: `theorem pow_rpow_inv_natCast (hx : 0 ≤ x) (hn : n ≠ 0) : (x ^ n) ^ (n⁻¹ : ℝ) = x`. The n⁻¹ : ℝ is (↑n)⁻¹ — with n := 4: (c^4)^((↑(4:ℕ))⁻¹ : ℝ)... the displayed form (n⁻¹ : ℝ) means ((n : ℝ))⁻¹. So h1's RHS is (c^4)^(((4:ℕ):ℝ))⁻¹; my Q is R^((4:ℝ)⁻¹). The cast forms differ syntactically: ((4:ℕ):ℝ) vs (4:ℝ). rpow_le_rpow will produce R^(((4:ℕ):ℝ))⁻¹ if I feed... no wait — rpow_le_rpow's exponent is whatever I write in the goal. Since goal mentions Q := R^((4:ℝ)⁻¹), h2 must match: (c^4)^((4:ℝ)⁻¹) ≤ R^((4:ℝ)⁻¹). And h1 : c = (c^4)^(((4:ℕ):ℝ))⁻¹. Mismatch (4:ℕ):ℝ vs (4:ℝ). push_cast at h1 turns ((4:ℕ):ℝ) into (4:ℝ)? push_cast with Nat.cast_ofNat simp lemma: ((4:ℕ):ℝ) → (4:ℝ) ✓ push_cast handles. So:\n```\n      have h1 : c = (c ^ 4) ^ ((4:ℝ)⁻¹) := by\n        have h := (Real.pow_rpow_inv_natCast (x := c) (n := 4) hcnn (by norm_num)).symm\n        push_cast at h\n        exact h\n```\nHmm, can I pass (n := 4) if n is implicit? `Real.pow_rpow_inv_natCast (hx : 0 ≤ x) (hn : n ≠ 0)` — n appears in hn's type so it's determined by hn : (4:ℕ) ≠ 0 — wait hn : n ≠ 0 with n : ℕ implicit; passing (by norm_num : (4:ℕ) ≠ 0) fixes n := 4 ✓ no need for named (n := 4).\n\nBut actually — is it (x := c) explicit-named? x implicit {x}? The statement has (hx : 0 ≤ x) explicit so x implicit. Just: `Real.pow_rpow_inv_natCast hcnn (show (4:ℕ) ≠ 0 by norm_num)` : (c ^ 4) ^ (((4:ℕ):ℝ))⁻¹ = c — hmm wait, will Lean elaborate `(n⁻¹ : ℝ)` with n a metavariable until hn given? hn : n ≠ 0 — the `show (4:ℕ) ≠ 0` fixes it ✓. Then .symm : c = (c^4)^(↑4)⁻¹. push_cast at h: normalizes ((4:ℕ):ℝ)⁻¹... push_cast rewrites Nat.cast_ofNat: ↑(4:ℕ) = (4:ℝ)? The simp lemma `Nat.cast_ofNat : ((OfNat.ofNat n : ℕ) : α) = OfNat.ofNat n` ✓ push_cast uses it. So h : c = (c^4)^((4:ℝ)⁻¹) ✓.\n\nBut hold on — is the exponent in pow_rpow_inv_natCast exactly `(↑n)⁻¹`? Source line 506-508: `theorem pow_rpow_inv_natCast (hx : 0 ≤ x) (hn : n ≠ 0) : (x ^ n) ^ (n⁻¹ : ℝ) = x`. In Lean source `(n⁻¹ : ℝ)` means ((n : ℝ))⁻¹ ✓.\n\nThen:\n```\n      have h2 : (c ^ 4) ^ ((4:ℝ)⁻¹) ≤ Q := Real.rpow_le_rpow (pow_nonneg (by linarith [hcz] : (0:ℝ) ≤ c) 4) hc4 (by positivity : (0:ℝ) ≤ (4:ℝ)⁻¹)\n```\nwait — Q is set to R^((4:ℝ)⁻¹); h2's statement I write as ≤ Q directly (rpow_le_rpow produces ≤ R^((4:ℝ)⁻¹); with set, the goal/have type I write `... ≤ Q` and the proof produces `... ≤ R^((4:ℝ)⁻¹)` — after `set`, R^((4:ℝ)⁻¹) is defeq-displayed as Q; does `exact`/`show` accept? `set` uses `let`-like local definition? `set x := e` introduces x as a local definition and h : x = e; the tactic state goal is rewritten, and terms with e are NOT auto-folded in my own haves — but defeq holds (x is a local hyp with value? `set` introduces x := e as a let-binding in the context; typechecking h2 : ... ≤ Q against proof of ... ≤ R^(4⁻¹) — is it defeq? set adds `x : α := e` as a let var; Q unfolds to R^(4⁻¹) definitionally ✓ exact works). To be safe I can rw [← hQ] to convert. I'll write h2 with Q in the statement and give proof via rpow_le_rpow ... and if elaboration gripes, add `rw [← hQ]; exact ...`. Then finish: `rw [h1]; exact h2` — goal c ≤ Q; rw [h1] — h1 : c = (c^4)^(4⁻¹); rw rewrites c → RHS in goal: (c^4)^(4⁻¹) ≤ Q ✓ exact h2.\n\nHmm, rw [h1] rewrites ALL c occurrences — the goal c ≤ Q has only one c ✓.\n\n19. hR4 : R = Q^4:\n```\n    have hR4 : R = Q ^ 4 := by\n      have h := Real.rpow_inv_natCast_pow (x := R) ... \n```\nrpow_inv_natCast_pow (hx : 0 ≤ x) (hn : n ≠ 0) : (x ^ (n⁻¹ : ℝ)) ^ n = x: `Real.rpow_inv_natCast_pow hRpos.le (show (4:ℕ) ≠ 0 by norm_num)` : (R ^ ((4:ℕ):ℝ)⁻¹)^4 = R. push_cast at h; then h : (R^((4:ℝ)⁻¹))^4 = R; rw [← hQ] at h: h : Q^4 = R; exact h.symm.\nCareful: rw [← hQ] at h rewrites R^((4:ℝ)⁻¹) → Q in h ✓ (hQ : Q = R^(4⁻¹); ← direction rewrites RHS→LHS: R^(4⁻¹) → Q ✓). But it would also rewrite R^(4⁻¹) inside... h : (R^((4:ℝ)⁻¹))^4 = R — the outer R in \"= R\" is not R^(4⁻¹) ✓ fine.\n\n20. div_mul_eq_mul_div : a / b * c = a * c / b ✓ standard name.\n\n21. The `key` have: after intro ε hε, the final calc:\n```\n    have hdz0 : dist z₀ (0:ℂ) = ‖z₀‖ := by rw [dist_eq_norm, sub_zero]\n    calc dist (Gentire z₀) (Gentire 0)\n        ≤ (RB + ‖Gentire 0‖) / R * dist z₀ 0 := hschwarz\n      _ = (RB + ‖Gentire 0‖) / R * ‖z₀‖ := by rw [hdz0]\n      _ ≤ ε := hmain\n```\nhschwarz's type: dist (Gentire z₀) (Gentire 0) ≤ (RB + ‖Gentire 0‖)/R * dist z₀ (0) — wait the Schwarz statement: dist (f z) (f c) ≤ R₂ / R₁ * dist z c with c := (0:ℂ) ✓.\n\nHmm — one subtlety: `Metric.ball` vs `ball` in Schwarz.lean — the file has `open Metric` presumably; the lemma's ball is Metric.ball ✓. And in Mathlib Schwarz.lean the quantifiers: {f : E → F} {c z : E} {R₁ R₂ : ℝ}. E := ℂ ✓ (DifferentiableOn ℂ f requires E a normed space over ℂ ✓).\n\n22. hschwarz call: `Complex.dist_le_div_mul_dist_of_mapsTo_ball hdiff hmaps hz0mem`. With `open Complex` in scope (LiCircleBound opens Complex), might clash-free either way; write fully qualified ✓.\n\n23. Also — IMPORTANT: does the file's `open Complex` shadow `dist_le_div_mul_dist_of_mapsTo_ball`? No, fine.\n\n24. In hmain I need hQ4pos and the conversion h1. But hR4 : R = Q^4: rw [hR4] inside h1's proof: goal (RB + ‖Gentire 0‖)/R*‖z₀‖ = (RB+‖Gentire 0‖)*‖z₀‖/Q^4; rw [hR4] changes R → Q^4 ✓ then ring: (X)/(Q^4)*Y = X*Y/(Q^4) — ring handles division? ring works in fields ✓ (Q^4 a nonzero atom? ring treats division as mul by inv, field identities hold formally — ring can prove a/b*c = a*c/b in a field without nonzero side conditions? `ring` works in commutative (semi)rings; with division it needs `field_simp` semantics — actually ring DOES handle field expressions with / as inv-mul, and a/b*c = a*c/b is a valid commutative-group identity (b⁻¹ commutes) — yes ring proves it without nonzero hyps since inv is total (0⁻¹ = 0) and the identity a*b⁻¹*c = a*c*b⁻¹ holds in any commutative group-with-zero ✓).\n\n25. hsum assembly detail: (RB + ‖Gentire 0‖)*‖z₀‖ ≤ ε*Q^4.\nh1 : (RB + ‖G0‖)*‖z₀‖ ≤ (C₀' + 64*C₁*Q^3 + ‖G0‖)*‖z₀‖ via mul_le_mul_of_nonneg_right (add_le_add_right hRBle _)... wait: RB ≤ C₀' + 64*C₁*Q^3 (hRBle) ⟹ RB + ‖G0‖ ≤ C₀' + 64*C₁*Q^3 + ‖G0‖ (add_le_add_right hRBle _) ✓ then mul right.\nh2 : (C₀' + 64*C₁*Q^3 + ‖G0‖)*‖z₀‖ = C₀'*‖z₀‖ + 64*C₁*Q^3*‖z₀‖ + ‖G0‖*‖z₀‖ by ring ✓.\nh3 : ‖G0‖*‖z₀‖ ≤ C₀'*‖z₀‖: mul_le_mul_of_nonneg_right (h : ‖G0‖ ≤ C₀') (norm_nonneg _): ‖G0‖ ≤ C₀' = 2*lowZeroConst + dirichletC2 + ‖G0‖: linarith [lowZeroConst_nonneg, dirichletC2_nonneg] after rw [hC0'] — goal ‖Gentire 0‖ ≤ C₀'; rw [hC0'] : ‖G0‖ ≤ 2*lowZeroConst + dirichletC2 + ‖G0‖; linarith [h1nn, h2nn] ✓.\nFinal: linarith [h1, h2, h3, hA, hB] — goal (RB+‖G0‖)*‖z₀‖ ≤ ε*Q⁴. Atoms: (RB+‖G0‖)*‖z₀‖, the products... linarith sees h1 : A ≤ B, h2 : B = C (equation), h3 : D ≤ E, hA : F ≤ (ε/2)*Q⁴, hB : G ≤ (ε/2)*Q⁴ where C = E + F + D... linarith: A ≤ B = C = C₀'‖z₀‖ + 64C₁Q³‖z₀‖ + ‖G0‖‖z₀‖ ≤ C₀'‖z₀‖ + (ε/2)Q⁴ + C₀'‖z₀‖ — hmm need 2·C₀'‖z₀‖ bound but hB : C₀'·‖z₀‖ ≤ (ε/2)·Q⁴ and h3: ‖G0‖‖z₀‖ ≤ C₀'‖z₀‖ ⟹ sum: C₀'‖z₀‖ + 64C₁Q³‖z₀‖ + ‖G0‖‖z₀‖ ≤ (ε/2)Q⁴ + (ε/2)Q⁴ + (ε/2)Q⁴ = 1.5·εQ⁴ — TOO BIG! \n\nRedo: the decomposition should be (C₀' + 64C₁Q³)*‖z₀‖ — WITHOUT the extra ‖G0‖. Fold ‖Gentire 0‖ into C₀' — wait C₀' := 2*lowZeroConst + dirichletC2 + ‖Gentire 0‖ ALREADY includes it! RB + ‖G0‖ ≤ (C₀' + 64C₁Q³) ⟸ RB ≤ C₀' + 64C₁Q³ − ‖G0‖... no: RB = 2*lowZeroConst + dirichletC2 + big; C₀' = 2*lowZeroConst + dirichletC2 + ‖G0‖; so RB + ‖G0‖ = (2low+dir+big) + ‖G0‖ ≤ (2low+dir+‖G0‖) + big' where big ≤ big'... RB + ‖G0‖ = C₀' + big ≤ C₀' + 64C₁Q³ ✓ — i.e., RB + ‖G0‖ ≤ C₀' + 64·C₁·Q³ directly! Because RB = C₀' − ‖G0‖ + big ≤ C₀' + big. So define hRBle' : RB + ‖Gentire 0‖ ≤ C₀' + 64*C₁*Q^3 directly:\n```\n    have hRBle' : RB + ‖Gentire 0‖ ≤ C₀' + 64 * C₁ * Q^3 := by\n      rw [hRB, hC0']\n      -- (2*low + dir + big) + ‖G0‖ ≤ (2*low + dir + ‖G0‖) + 64*C₁*Q^3\n      linarith [hbig]  -- hbig : big ≤ 64*C₁*Q^3\n```\n✓ linarith with atoms big, 64*C₁*Q^3, and the rest canceling.\n\nThen hsum : (RB + ‖G0‖)*‖z₀‖ ≤ (C₀' + 64C₁Q³)*‖z₀‖ = C₀'‖z₀‖ + 64C₁Q³‖z₀‖ ≤ (ε/2)Q⁴ + (ε/2)Q⁴ = εQ⁴:\n```\n    have hsum : (RB + ‖Gentire 0‖) * ‖z₀‖ ≤ ε * Q^4 := by\n      have h1 : (RB + ‖Gentire 0‖) * ‖z₀‖ ≤ (C₀' + 64 * C₁ * Q^3) * ‖z₀‖ :=\n        mul_le_mul_of_nonneg_right hRBle' (norm_nonneg _)\n      have h2 : (C₀' + 64 * C₁ * Q^3) * ‖z₀‖ = C₀' * ‖z₀‖ + 64 * C₁ * Q^3 * ‖z₀‖ := by ring\n      linarith [h1, h2, hA, hB]\n```\nhA : 64*C₁*Q^3*‖z₀‖ ≤ (ε/2)*Q^4, hB : C₀'*‖z₀‖ ≤ (ε/2)*Q^4 ✓ linarith sums ✓.\n\n26. Also drop hRBle (unused) — just build hbig and hRBle'.\n\n27. hbig's proof (big ≤ 64*C₁*Q^3):\n```\n    have hbig : 400000 * A0count * C0hat (R + 2) * (Real.log R)^2 ≤ 64 * C₁ * Q^3 := by\n      have hlogRnn : (0:ℝ) ≤ Real.log R := by linarith [hlogR]\n      have h1 : 400000 * A0count * C0hat (R + 2) * (Real.log R)^2\n          ≤ 400000 * A0count * (5 * Real.log R) * (Real.log R)^2 := by\n        apply mul_le_mul_of_nonneg_right _ (pow_nonneg hlogRnn 2)\n        exact mul_le_mul_of_nonneg_left hĈ5 (mul_nonneg (by norm_num : (0:ℝ) ≤ 400000) hA0nn)\n      have h2 : 400000 * A0count * (5 * Real.log R) * (Real.log R)^2 = C₁ * (Real.log R)^3 := by\n        rw [hC1]; ring\n      have h3 : C₁ * (Real.log R)^3 ≤ 64 * C₁ * Q^3 := by\n        have h := mul_le_mul_of_nonneg_left hL3 hC₁nn  -- C₁*L^3 ≤ C₁*(64*Q^3)\n        have h' : C₁ * (64 * Q^3) = 64 * C₁ * Q^3 := by ring\n        rwa [h'] at h\n      linarith [h1, h2, h3]\n```\n✓.\n\n28. Also `hĈ5 : C0hat (R + 2) ≤ 5 * Real.log R := C0hat_add_two_le hR16002`.\n\n29. In §1, `C0hat_add_two_le`: unfold C0hat — `rw [C0hat]`? C0hat is `def C0hat (t : ℝ) : ℝ := ...`. rw [C0hat] uses equation lemma C0hat t = ... hmm for non-pattern defs `rw [C0hat]` might fail (\"motive\" issues?). In LiSkeleton they must unfold C0hat somewhere — two_le_C0hat's proof in LiContour. Let me check how two_le_C0hat unfolds it. Actually `unfold C0hat` always works; or `show`. Let me check LiContour's usage.\n\nAlso — in hĈ5 the LHS `C0hat (R + 2)` appears in hRB (RB's definition mentions C0hat (R+2)) — no unfolding needed there since I only use hĈ5 as a bound ✓.\n\n30. After `set RB := ... with hRB`, note the Fexpr_circle_le RHS IS exactly RB's definition — so hsphere/hball statements with RB... I'll write them with the explicit expression (they're proved before set RB, or after — set replaces in goal only, my haves are written by me; I'll order: sphere/ball bounds first with explicit expressions, then set RB, then use hball z hz which has the explicit form; when I need ‖Gentire z‖ ≤ RB from hball z hz (explicit), I rw [← hRB] at that instance... Messy. Alternative: `set RB := ... with hRB` FIRST (right after obtaining R), then state hsphere/hball with RB, and at the Fexpr_circle_le call site the produced term has the explicit form — exact accepts since RB is let-defeq ✓. I'll do set RB early and rely on defeq. If `exact` fails due to let-transparency issues (set's local defs are transparent for exact? Yes — `set` introduces a local definition that's definitionally equal; exact works up to defeq ✓).\n\nOrder inside key:\n- intro ε hε\n- set C₀', C₁, c, M (with haves hC₀'nn, hC₁nn, hcz, hcnn)\n- obtain k, hk4000, hkM\n- obtain R, hRlo, hRhi, hgapInt, hgap (from exists_gapCircle_radius)\n- hR16002, hRpos, hlogR, hMR\n- set RB with hRB; hsphere; hGan; hball; hmaps; hdiff; hz0mem; hschwarz\n- set Q; hQnn, hQpos, hQc, hL, hL3, hR4, hĈ5, hbig, hRBle'\n- hc1, hA, hB, hsum, hmain\n- final calc\n\n31. hz0mem : z₀ ∈ Metric.ball (0:ℂ) R:\n```\n    have hz0mem : z₀ ∈ Metric.ball (0:ℂ) R := by\n      rw [Metric.mem_ball, dist_eq_norm, sub_zero]\n      have h1 : ‖z₀‖ + 1 ≤ M := le_max_right _ _\n      have h2 : (‖z₀‖ + 1 : ℝ) ≤ M := h1  -- M := max (c^4) (‖z₀‖+1)\n      linarith [hMR, norm_nonneg z₀]\n```\nWait — with `set M := max (c^4) (‖z₀‖+1) with hM`, le_max_right gives ‖z₀‖+1 ≤ max (c^4) (‖z₀‖+1); the set-folded M is defeq ✓ `have h1 : ‖z₀‖ + 1 ≤ M := le_max_right _ _` — typechecking le_max_right _ _ : ‖z₀‖+1 ≤ max (c^4) (‖z₀‖+1) against ‖z₀‖+1 ≤ M — defeq via the let ✓ should work.\ngoal after rw: ‖z₀‖ < R. linarith [hMR, h1] where hMR : M ≤ R, h1 : ‖z₀‖+1 ≤ M ⟹ ‖z₀‖ < M ≤ R ✓ (linarith uses norm_nonneg? ‖z₀‖ < ‖z₀‖+1 is linear in atom ‖z₀‖: ‖z₀‖ + 1 ≤ M ⟹ ‖z₀‖ ≤ M − 1 < M ≤ R ✓ pure linarith, no norm_nonneg needed).\n\n32. hkM : M ≤ 4*(k:ℝ) — from the obtain. Then hMR : M ≤ R: R ≥ 4k + 5/2 ≥ 4k ≥ M: linarith [hkM, hRlo] ✓.\n\n33. The k obtain:\n```\n    obtain ⟨k, hk4000, hkM⟩ : ∃ k : ℕ, 4000 ≤ k ∧ M ≤ 4 * (k:ℝ) := by\n      refine ⟨max 4000 ⌈M / 4⌉₊, le_max_left _ _, ?_⟩\n      have h1 : M / 4 ≤ (⌈M / 4⌉₊ : ℝ) := Nat.le_ceil _\n      have h2 : (⌈M / 4⌉₊ : ℝ) ≤ ((max 4000 ⌈M / 4⌉₊ : ℕ) : ℝ) := by\n        exact_mod_cast le_max_right _ _\n      linarith [h1, h2]\n```\nlinarith: M ≤ 4·⌈M/4⌉₊ ≤ 4·max ✓ (atoms the casts; h1: M/4 ≤ c1; h2: c1 ≤ c2; goal M ≤ 4*c2: from M ≤ 4*c1 ≤ 4*c2 — linarith ✓ multiplications by literal 4 are linear ✓).\n\n34. hk4000 : 4000 ≤ k — for Fexpr_circle_le (hk : 4000 ≤ k) ✓, and 1 ≤ k for exists_gapCircle_radius (by omega : 1 ≤ k needs hk4000 in context — after obtain ✓; call: exists_gapCircle_radius hZC (k := k) (by omega)).\n\nWait — exists_gapCircle_radius's hk : 1 ≤ k. With hk4000 : 4000 ≤ k in context, `by omega` closes 1 ≤ k ✓.\n\n35. hgap order check for Fexpr_circle_le call: Fexpr_circle_le hk4000 hRlo hRhi hgap hgapInt hznorm — signature: (hk) (hRlo) (_hRhi) (hgap) (hgapInt) (hs) ✓.\n\nBut CAREFUL: exists_gapCircle_radius returns the conjunction in order ⟨hRlo, hRhi, hgapInt, hgap⟩ (integers third, zeros fourth) ✓ as I wrote.\n\n36. not_badPoint_on_gapCircle's hgap argument order: my own def — I'll mirror Fexpr_circle_le: (hRlo) (hgap) (hgapInt). In gentire_circle_le... actually fold not_badPoint into hsphere directly? Cleaner as its own lemma (also reusable for the capstone? no, capstone's goodness is different). Keep as separate lemma §2.\n\n37. gentire_circle_le (§3):\n```\ntheorem gentire_circle_le {k : ℕ} (hk : 4000 ≤ k) {R : ℝ}\n    (hRlo : 4 * (k:ℝ) + 5/2 ≤ R) (hRhi : R ≤ 4 * (k:ℝ) + 3)\n    (hgap : ∀ ρ : ℂ, IsNontrivialZero ρ →\n      1 / (16 * C0hat (4 * (k:ℝ) + 9/2)) ≤ |ρ.im - R| ∨ 1 ≤ |ρ.im - R|)\n    (hgapInt : ∀ m : ℤ, 1/4 ≤ |R/2 - (m:ℝ)|)\n    {s : ℂ} (hs : ‖s‖ = R) :\n    ‖Gentire s‖ ≤ 2 * lowZeroConst + dirichletC2 +\n      400000 * A0count * C0hat (R + 2) * (Real.log R)^2 := by\n  have hgood : ¬ badPoint s := not_badPoint_on_gapCircle hk hRlo hgap hgapInt hs\n  rw [gentire_eq_Fexpr hgood]\n  exact Fexpr_circle_le hk hRlo hRhi hgap hgapInt hs\n```\n✓.\n\n38. The hsphere in key uses gentire_circle_le hk4000 hRlo hRhi hgap hgapInt hznorm ✓.\n\nOne more check — `Real.log_le_rpow_div` exact form at Pow/Real.lean:883: `lemma log_le_rpow_div {x ε : ℝ} (hx : 0 ≤ x) (hε : 0 < ε) : log x ≤ x ^ ε / ε` — in Real namespace? The file section... it's in `namespace Real` presumably ✓ Real.log_le_rpow_div. With x := R, ε := (4:ℝ)⁻¹: log R ≤ R^((4:ℝ)⁻¹)/(4:ℝ)⁻¹. Then:\n```\n    have hL : Real.log R ≤ 4 * Q := by\n      have h := Real.log_le_rpow_div hRpos.le (show (0:ℝ) < (4:ℝ)⁻¹ by positivity)\n      -- h : log R ≤ R^((4:ℝ)⁻¹)/(4:ℝ)⁻¹\n      have h2 : R ^ ((4:ℝ)⁻¹) / (4:ℝ)⁻¹ = 4 * Q := by\n        rw [← hQ, div_eq_mul_inv, inv_inv]  -- goal becomes... \n        ring\n      rwa [h2] at h\n```\nh2 proof: goal R^((4:ℝ)⁻¹)/(4:ℝ)⁻¹ = 4*Q; rw [← hQ] rewrites R^(4⁻¹) → Q (both occurrences: LHS numerator and... RHS 4*Q has Q already — rw [← hQ] rewrites R^(4⁻¹) inside Q? NO — Q is a local let-var; rw operates syntactically: occurrences of R^(4⁻¹) in the goal get replaced by Q. RHS's Q is already Q (a local variable), not R^(4⁻¹) ✓ untouched. Goal becomes Q/(4⁻¹) = 4*Q. rw [div_eq_mul_inv]: Q*(4⁻¹)⁻¹ = 4*Q. rw [inv_inv]: (4:ℝ)⁻¹⁻¹ = 4: Q*4 = 4*Q: ring ✓.\n\nHmm wait — div_eq_mul_inv : a / b = a * b⁻¹ ✓. inv_inv : a⁻¹⁻¹ = a ✓.\n\nBut there's a subtlety: rw [← hQ] might also rewrite the R^(4⁻¹) appearing... only one occurrence (numerator) ✓.\n\nAlternatively avoid the rw-Q dance: state h2 as `R ^ ((4:ℝ)⁻¹) / (4:ℝ)⁻¹ = 4 * (R ^ ((4:ℝ)⁻¹))` and prove by rw [div_eq_mul_inv, inv_inv]; ring. Then h : log R ≤ 4*R^(4⁻¹); then hL : log R ≤ 4*Q: rwa [h2] at h — h's RHS 4*(R^(4⁻¹)) vs 4*Q — defeq via set-let ✓ exact works if hL's statement uses Q: `exact h` where h : log R ≤ 4*R^(4⁻¹) and goal log R ≤ 4*Q — defeq ✓. Cleaner:\n```\n    have hL : Real.log R ≤ 4 * Q := by\n      have h := Real.log_le_rpow_div hRpos.le (show (0:ℝ) < (4:ℝ)⁻¹ by positivity)\n      have h2 : R ^ ((4:ℝ)⁻¹) / (4:ℝ)⁻¹ = 4 * (R ^ ((4:ℝ)⁻¹)) := by\n        rw [div_eq_mul_inv, inv_inv]; ring\n      rwa [h2] at h  -- h : log R ≤ 4 * R^(4⁻¹); goal ... ≤ 4 * Q; defeq ✓ rwa tries exact after rw\n```\nrwa [h2] at h: rewrites h then exacts — the exact step checks defeq against goal with Q ✓.\n\n39. hL3:\n```\n    have hL3 : (Real.log R)^3 ≤ 64 * Q^3 := by\n      have h1 : (Real.log R)^3 ≤ (4 * Q)^3 :=\n        pow_le_pow_left₀ (by linarith [hlogR]) hL 3\n      have h2 : (4 * Q)^3 = 64 * Q^3 := by ring\n      rwa [h2] at h1\n```\nhL : log R ≤ 4*Q ✓ matches pow_le_pow_left₀'s hab : a ≤ b with b := 4*Q ✓.\n\n40. pow_pos (h : 0 < a) (n : ℕ)? `pow_pos (ha : 0 < a) (n : ℕ) : 0 < a ^ n` ✓.\n\n41. div_le_iff₀ : `div_le_iff₀ (hc : 0 < c) : a / c ≤ b ↔ a ≤ b * c` ✓ standard.\n\n42. Also confirm `div_le_div_of_nonneg_right`: In current Mathlib — `div_le_div_of_nonneg_right (hab : a ≤ b) (hc : 0 ≤ c) : a / c ≤ b / c`. Hmm, actually I have a doubt: it might be in terms of `a / c ≤ b / c ↔`... The session-032 gotcha explicitly banked \"div_le_div_of_nonneg_right exists\" from THIS codebase ✓ trust it.\n\n43. In hc1: `(128 * C₁) * ‖z₀‖ / ε ≤ c` via div_le_div_of_nonneg_right h2 hε.le then .trans h1 where h1 : (128*C₁+2*C₀')*‖z₀‖/ε ≤ c ✓.\n\n44. `dist_eq_zero : dist x y = 0 ↔ x = y` ✓.\n\n45. `Metric.mem_sphere.mp hz : dist z 0 = R`; then `rw [dist_eq_norm, sub_zero] at h`— wait h : dist z (0:ℂ) = R; dist_eq_norm : dist z 0 = ‖z − 0‖; rw at h: h : ‖z − 0‖ = R; sub_zero: h : ‖z‖ = R ✓.\n\nAlso `open Metric`? LiCircleBound opens `Zeta23 Complex Filter Topology` — NOT Metric. So write Metric.ball/Metric.sphere/Metric.mem_ball etc. fully qualified ✓ (as I have been). Note Fexpr_circle_le etc use ‖·‖ directly ✓.\n\n46. `MapsTo` — Function.MapsTo; with no open Function... MapsTo is in root namespace? `Set.MapsTo`! In recent Mathlib, MapsTo is `Set.MapsTo`. Schwarz.lean has `open Set` presumably, its signature `MapsTo f (ball c R₁) (closedBall (f c) R₂)` is Set.MapsTo. So I write `Set.MapsTo Gentire (Metric.ball 0 R) (Metric.closedBall (Gentire 0) (RB + ‖Gentire 0‖))`. intro z hz works for Set.MapsTo (it's ∀ x ∈ s, f x ∈ t) ✓ — Set.MapsTo f s t := ∀ ⦃x⦄, x ∈ s → f x ∈ t; `intro z hz` ✓ works since it's a Pi type.\n\nCheck: does LiCircleBound use Set.MapsTo or MapsTo? Doesn't matter — use Set.MapsTo with the full form; Schwarz's arg expects Set.MapsTo ✓ defeq regardless of how I write `MapsTo` (if MapsTo resolves to Set.MapsTo via open Set — I don't open Set, so write Set.MapsTo explicitly).\n\n47. `mem_closedBall` target: Set.MapsTo ... (Metric.closedBall (Gentire 0) R₂): membership z' ∈ Metric.closedBall c R₂ ↔ dist z' c ≤ R₂ (Metric.mem_closedBall) ✓.\n\n48. norm_sub_le : ‖a − b‖ ≤ ‖a‖ + ‖b‖ ✓.\n\nNow, the hsphere bound needs hznorm : ‖z‖ = R from hz : z ∈ Metric.sphere (0:ℂ) R. ✓ as written.\n\nOne more consideration — `gentire_eq_Fexpr`: let me verify its exact signature/name. From line 1109: `rw [gentire_eq_Fexpr hygood]` where hygood : ¬ badPoint y, goal context — at that point they're proving something about Gentire y... it's `gentire_eq_Fexpr (h : ¬ badPoint s) : Gentire s = Fexpr s` ✓ (line 1116: `gentire_eq_Fexpr hs` used to prove the second component ∀ s, ¬badPoint s → Gentire s = Fexpr s — wait line 1115-1116: `(∀ s, AnalyticAt ...) ∧ ∀ s, ¬ badPoint s → Gentire s = Fexpr s := ⟨fun s => ?_, fun s hs => gentire_eq_Fexpr hs⟩` ✓ confirmed).\n\nFile placement decision: new file LiSqueeze.lean. Also root-import it in NguyenRH1.lean. Let me check the root file's import list.","signature":""},{"type":"text","text":"`pow_le_pow_left₀` confirmed. Checking the root file's import list, then writing LiSqueeze.lean."},{"type":"tool_use","id":"Read_29","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1.lean"},"extra_content":null}]