[{"type":"thinking","thinking":"I left sorries again (hFsum and the final). Let me complete the proof properly. After hFle:\n```lean\n  have hGsum : Summable (fun j => (4 * A0count / X) * G j) := (dyadicLog_summable X).mul_left _\n  have hFsum : Summable F := by\n    apply Summable.of_norm_bounded hGsum\n    intro j\n    rw [Real.norm_eq_abs, abs_of_nonneg (hFnn j), Real.norm_eq_abs, abs_of_nonneg (hGnn j)]\n    exact hFle j\n  -- the real total bound\n  have htotal : ∑' j : ℕ, F j ≤ 16 * A0count * Real.log X / X := by\n    calc ∑' j : ℕ, F j ≤ ∑' j : ℕ, (4 * A0count / X) * G j :=\n          Summable.tsum_le_tsum hFle hFsum hGsum\n      _ = (4 * A0count / X) * ∑' j : ℕ, G j := tsum_mul_left\n      _ = (4 * A0count / X) * (10 * Real.log 2 + 2 * Real.log X) := by\n          rw [hGdef]  -- the G tsum = tsum_dyadic_log's LHS ✓ defeq\n          rw [tsum_dyadic_log]\n      _ ≤ 16 * A0count * Real.log X / X := dyadic_tail_constant hX\n  -- the fiberwise comparison\n  calc ∑' j : ℕ, ∑' b : (dyadClass X) ⁻¹' {j}, ENNReal.ofReal (tailTerm X b.1)\n      ≤ ∑' j : ℕ, ENNReal.ofReal (F j) := by\n        apply ENNReal.tsum_le_tsum\n        intro j\n        exact fiber_tail_le hX j\n    _ = ENNReal.ofReal (∑' j : ℕ, F j) := (ENNReal.ofReal_tsum_of_nonneg hFnn hFsum).symm\n    _ ≤ ENNReal.ofReal (16 * A0count * Real.log X / X) := ENNReal.ofReal_le_ofReal htotal\n```\nWait — the tsum_dyadic_log's LHS form: ∑' j, ((j+4)·log2 + logX)/2^j — my G := fun j => ((j:ℝ)+4)·log2 + logX)/2^j — hmm the form: tsum_dyadic_log's summand: `((j + 4) * Real.log 2 + Real.log X) / (2 : ℝ) ^ j` where (j+4) has j : ℕ cast — my G's `((j : ℝ) + 4)` — same ✓. So ∑' G = 10log2+2logX via tsum_dyadic_log — after rw [hGdef] the goal's tsum over the G-LAMBDA: ∑' j, G j vs ∑' j, (the lambda body): the rw [hGdef] turns G into the lambda — then the goal's ∑' j, (fun j => ...) j — a beta-redex — tsum_dyadic_log's LHS is ∑' j, (body) — hmm the redex: (fun j => body) j — tsum_dyadic_log : ∑' j, body = ...: the rw [tsum_dyadic_log] would need the pattern ∑' j, body — but the goal has ∑' j, (fun j => body) j — redex mismatch (journal-027 gotcha!). Fix: `show ∑' j : ℕ, (((j : ℝ) + 4) * Real.log 2 + Real.log X) / (2:ℝ)^j = _` then rw [tsum_dyadic_log] — the show's LHS is the non-reduced body form ✓ — wait the goal after `rw [hGdef]`... let me restructure: the calc step `_ = (4A₀/X)·(10log2+2logX)`: proof: `by rw [tsum_mul_left]; rw [show ∑' j : ℕ, G j = 10*log2 + 2*logX from ...]` — with G the def: ∑' j, G j = ∑' j, body (defeq) = 10log2+2logX (tsum_dyadic_log): have hGtotal : ∑' j : ℕ, G j = 10*Real.log 2 + 2*Real.log X := tsum_dyadic_log X — the defeq ascription ✓ (G j ≡ body). So:\n```lean\n      _ = (4 * A0count / X) * (10 * Real.log 2 + 2 * Real.log X) := by\n          rw [tsum_mul_left, show ∑' j : ℕ, G j = _ from tsum_dyadic_log X]  -- hmm the show's _ — circular. Use: rw [tsum_mul_left, hGtotal] with hGtotal : ∑' j, G j = 10log2+2logX := tsum_dyadic_log X (ascription ✓)\n```\nLet me finalize with hGtotal. Also the earlier hpow : (2:ℝ)^(j+4) = 16·2^j: my proof: `rw [show (2:ℝ)^(j+4) = (2:ℝ)^4 * 2^j from by rw [← pow_add]]; norm_num` — the show: 2^(j+4) = 2^4·2^j via ← pow_add: pow_add : a^(m+n) = a^m·a^n: ← : 2^4·2^j = 2^(4+j)... hmm the show's proof `by rw [← pow_add]`: goal 2^(j+4) = 2^4·2^j: ← pow_add on the RHS 2^4·2^j → 2^(4+j): goal 2^(j+4) = 2^(4+j): needs j+4 = 4+j: `rw [add_comm]`?? the rw [← pow_add] leaves 2^(j+4) = 2^(4+j) — not rfl (j+4 vs 4+j) — then? Add `; rw [add_comm]`?? the rw [← pow_add] rewrites 2^4*2^j → 2^(4+j): goal 2^(j+4) = 2^(4+j): congr + add_comm: `by rw [← pow_add]; congr 1; omega`? Hmm — `congr 1` on 2^(j+4) = 2^(4+j) leaves j+4 = 4+j: omega ✓. Or simpler: `have hpow : (2:ℝ)^(j+4) = 16 * 2^j := by rw [pow_add]; norm_num` — pow_add : 2^(j+4) = 2^j·2^4: then norm_num : 2^4 = 16: goal 2^j·2^4 = 16·2^j: ring: `by rw [pow_add]; norm_num; ring`? — after rw [pow_add]: goal 2^j * 2^4 = 16*2^j: norm_num simplifies 2^4 → 16: 2^j*16 = 16*2^j: ring... but norm_num might close it entirely (it evaluates 2^4 but the 2^j is opaque — norm_num leaves 2^j*16 = 16*2^j then... it can close via mul_comm? norm_num doesn't do commutativity). Use `by rw [pow_add]; ring_nf; norm_num`?? Let me just: `by rw [pow_add]; ring_nf` — ring_nf normalizes 2^j·2^4 → 16·2^j ✓ (numeral evaluation inside ring_nf ✓) and 16·2^j ✓ closes. Let me use that.\n\nAlso in hfac1 — the nlinarith after rw [div_le_div_iff₀ ...]: the goal: A₀(2^{j+1}X+3)·(X·2^j) ≤ (4A₀/X)·(2^jX)²·... wait — div_le_div_iff₀ (hb : 0 < b) (hd : 0 < d) : a/b ≤ c/d ↔ a*d ≤ c*b: my a = A₀(2^{j+1}X+3), b = (2^jX)², c = 4A₀/X, d = 2^j: goal: A₀(2^{j+1}X+3)·2^j ≤ (4A₀/X)·(2^jX)²: with (2^jX)² = 4^jX²: RHS = 4A₀·4^jX²/X = 4A₀·4^j·X (the /X): LHS = A₀(2^{j+1}X+3)·2^j = A₀·2^j·(2·2^jX + 3) = 2A₀·4^j·X + 3A₀·2^j: need 2A₀·4^jX + 3A₀·2^j ≤ 4A₀·4^jX ⟺ 3A₀·2^j ≤ 2A₀·4^jX ⟺ 3 ≤ 2·2^jX (÷A₀2^j > 0) ✓ (2^jX ≥ 200). nlinarith [hpow, h3le, h2jX1, ...] — the division-by-X in the RHS makes it messy — nlinarith with the /X: the term (4A₀/X)·(2^jX)²: ring_nf normalizes to 4A₀·(2^jX)²/X — a RATIONAL monomial — nlinarith handles field... nlinarith over ordered fields with division — it multiplies through? Risky. Alternative: avoid the /X form on the RHS: write the target as F_j ≤ (4A₀/X)·G_j with the two-factor split where the factors are (A₀(2^{j+1}X+3)/(2^jX)²) ≤ (4A₀)/(X·2^j) — both as plain fractions — div_le_div_iff₀ gives a·d ≤ c·b: A₀(2^{j+1}X+3)·(X·2^j) ≤ 4A₀·(2^jX)² — wait I should set d := X·2^j (not 2^j): c = 4A₀, d = X·2^j: (4A₀)/(X·2^j) ✓: a·d ≤ c·b: A₀(2^{j+1}X+3)·(X·2^j) ≤ 4A₀·(2^jX)² — pure polynomial ✓ nlinarith [hpow, ...]: LHS = A₀(2·2^jX+3)·X·2^j = 2A₀·4^jX² + 3A₀·2^jX: RHS = 4A₀·4^jX²: ⟺ 3A₀·2^jX ≤ 2A₀·4^jX² ⟺ 3 ≤ 2·2^j·X ✓. nlinarith needs the product decomposition — give it the pieces: have hkey : 3 * A0count * (2^j) * X ≤ 2 * A0count * 4^j * X^2: from A₀ > 0, 2^j ≥ 1, X ≥ 200: 2·4^jX² = 2·(2^jX)² ≥ 2·(2^jX)·200·... Let me give nlinarith the explicit product witness: mul_le_mul (3 ≤ 2·2^jX) — no: the cleanest: \n\n3A₀·2^jX ≤ 2A₀·4^jX² ⟺ A₀·2^jX·(3) ≤ A₀·2^jX·(2·2^jX) ⟸ 3 ≤ 2·2^jX (since A₀·2^jX ≥ 0): mul_le_mul_of_nonneg_left (3 ≤ 2·2^jX) (A₀·2^jX ≥ 0) : A₀·2^jX·3 ≤ A₀·2^jX·(2·2^jX) ✓. Then assemble:\n```lean\n      have hkey : A0count * ((2 : ℝ) ^ j * X) * 3\n          ≤ A0count * ((2 : ℝ) ^ j * X) * (2 * ((2 : ℝ) ^ j * X)) :=\n        mul_le_mul_of_nonneg_left (by linarith [h2jX200]) (by positivity)\n      -- where h2jX200 : 200 ≤ 2^jX\n      nlinarith [hkey, hpow, h2jX, h2j1X, A0count_pos.le, h2j]\n```\nwith the goal A₀(2^{j+1}X+3)·(X·2^j) ≤ 4A₀·(2^jX)² — expand both via ring-normal forms + hpow (2^{j+1}X = 2·2^jX): LHS = (2A₀·2^jX + 3A₀)·X·2^j = 2A₀·4^jX² + 3A₀·2^jX — hmm the forms: (2^jX)² vs 4^j·X² — ring_nf handles ✓; hkey : A₀·2^jX·3 ≤ A₀·2^jX·(2·2^jX) = 2A₀·(2^jX)²: so 3A₀·2^jX ≤ 2A₀(2^jX)² — wait hkey is A₀(2^jX)·3 ≤ A₀(2^jX)(2·2^jX) — that's exactly 3A₀·2^jX ≤ 2A₀·(2^jX)² after ring_nf ✓. So goal ⟸ hkey: nlinarith [hkey, hpow] — the LHS expansion needs hpow to eliminate 2^{j+1}X: nlinarith uses hpow as an equation between atoms — the monomial A₀·2^{j+1}X·X·2^j vs A₀·2^jX·X·2^j — hpow multiplied by A₀·X·2^j — NONLINEAR again!! nlinarith DOES multiply hypotheses (it's nonlinear!) — nlinarith can multiply hpow by the nonneg A₀·X·2^j? nlinarith's strategy multiplies PAIRS of hypotheses/atoms including equalities?? nlinarith multiplies hypotheses with hypotheses (both sides must be nonneg for ≤) — equalities can be multiplied freely. So nlinarith [hpow, hkey, ...] should find: hpow·(A₀·X·2^j as atom) — nlinarith considers products of hypotheses with the goal's atoms... it might find it. Alternatively rewrite FIRST: rw [hpow] at the goal — the goal after div_le_div_iff₀: `A0count * (2^(j+1)*X + 3) * (X * 2^j) ≤ 4 * A0count / X * ((2^j*X)^2) * ...` wait I set d := X·2^j: c = 4A₀: the goal after div_le_div_iff₀: A₀(2^{j+1}X+3)·(X·2^j) ≤ (4A₀)·(2^jX)²: rw [hpow]?? the goal's LHS has 2^{j+1}X: rw [hpow] turns it into 2·2^jX ✓ — but does the goal contain the /X? c = 4A₀ (no /X since I fold it into d) — wait my hfac1's RHS is `(4 * A0count / X) / (2 : ℝ) ^ j` — c = 4A₀/X, d = 2^j — the c has /X! div_le_div_iff₀ with c = 4A₀/X — c·b = (4A₀/X)·(2^jX)² — has the /X. To avoid: restate hfac1's RHS as (4A₀)/(X·2^j) (a single fraction): `(4 * A0count) / (X * (2:ℝ)^j)`: then div_le_div_iff₀: A₀(2^{j+1}X+3)·(X·2^j) ≤ 4A₀·(2^jX)² ✓ polynomial ✓. And the final form-match: (4A₀/X)/2^j = 4A₀/(X·2^j) ✓ by div_div (div_div : a/b/c = a/(b·c) ✓). So hfac1's statement: `A0count * ((2:ℝ)^(j+1)*X + 3) / ((2:ℝ)^j * X)^2 ≤ (4 * A0count) / (X * (2:ℝ)^j)`. And in the assembly, the (4A₀/X)·G_j factor-form: (4A₀/X)/2^j = 4A₀/(X·2^j) via div_div ✓.\n\nLet me rewrite hfac1 cleanly:\n```lean\n    have hfac1 : A0count * ((2 : ℝ) ^ (j + 1) * X + 3) / ((2 : ℝ) ^ j * X) ^ 2\n        ≤ (4 * A0count) / (X * (2 : ℝ) ^ j) := by\n      rw [div_le_div_iff₀ (by positivity) (by positivity)]\n      have hpow : (2 : ℝ) ^ (j + 1) * X = 2 * ((2 : ℝ) ^ j * X) := by rw [pow_succ']; ring\n      have h2jX200 : (200 : ℝ) ≤ (2 : ℝ) ^ j * X := by\n        have h1 : (1 : ℝ) ≤ (2 : ℝ) ^ j := by exact_mod_cast Nat.one_le_two_pow\n        nlinarith [h1, hX]\n      have hkey : A0count * ((2 : ℝ) ^ j * X) * 3\n          ≤ A0count * ((2 : ℝ) ^ j * X) * (2 * ((2 : ℝ) ^ j * X)) := by\n        apply mul_le_mul_of_nonneg_left _ (by positivity)\n        linarith [h2jX200]\n      rw [hpow]\n      have h2jX : (0 : ℝ) < (2 : ℝ) ^ j * X := by positivity\n      nlinarith [hkey, h2jX, A0count_pos.le]\n```\nWait — after rw [hpow] the goal: A₀(2·2^jX + 3)·(X·2^j) ≤ 4A₀·(2^jX)² — the LHS expands to 2A₀(2^jX)² + 3A₀·2^jX — hmm wait: (2·2^jX + 3)·(X·2^j) = 2·2^jX·X·2^j + 3·X·2^j = 2·(2^jX)² + 3·2^jX — the (2^jX)² vs X²·4^j — ring_nf ✓. Goal ⟸ 3A₀·2^jX ≤ 2A₀(2^jX)² ✓ hkey ✓ and the rest equal: nlinarith [hkey, h2jX, A0count_pos.le] ✓.\n\nAnd the assembly's factor-form: F_j = (1/(2^jX)²)·(A₀(2^{j+1}X+3)·L_j) and (4A₀/X)·G_j = (4A₀/X)·(((j+4)log2+logX)/2^j): write both as products of the two factor-fractions: F_j = [A₀(2^{j+1}X+3)/(2^jX)²]·L_j (ring ✓) and (4A₀/X)·G_j = [4A₀/(X·2^j)]·[(j+4)log2+logX] (ring with div_div ✓). Then mul_le_mul hfac1 hlogle hLnn (nonneg of the second-factor's target: 0 ≤ (j+4)log2+logX ✓ via log2 ≥ 0, logX ≥ 0, j ≥ 0: positivity... log2's nonneg: Real.log_nonneg (1 ≤ 2) ✓ pass).\n\nOK let me rewrite the whole zero_tail_inv_sq_le properly (replace my sorry-ridden draft).","signature":""},{"type":"text","text":"Let me rewrite the `zero_tail_inv_sq_le` block cleanly (removing the sorried draft)."},{"type":"tool_use","id":"Read_447","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean","offset":736,"limit":12},"extra_content":null}]