[{"type":"thinking","thinking":"Error 1070: `rw [hsq] at hmain ⊢` — I wrote `rw [hsq] at hmain ⊢` — TWO targets — hsq : (T − γ)² = |γ−T|² — hmain's statement has `(T−γ)²` inside the integrand ✓ should rewrite there... wait no — the error says it didn't find `(T - (↑ρ).im)^2` in the TARGET (⊢). The target: `∫ σ in -5/2..3, 1/√((σ−β)² + |γ−T|²) ≤ 2·log(16.5/|γ−T|)` — WAIT — the target ALREADY has |γ−T|²! — that's hmain's form after... no — hold on. The hper goal: `∫ σ, 1/√((σ−ρ.1.re)² + (T−ρ.1.im)²) ≤ 2·log(148.5·C0hat)`. And hmain := per_zero_integral_le_two_log ... : `∫ σ, 1/√((σ−β)² + δ²) ≤ 2·log(16.5/δ)` with δ := |γ−T| — hmain's integrand has |γ−T|² (from δ²). \n\nI wrote `rw [hsq] at hmain ⊢` — hsq : (T−γ)² = |γ−T|²:\n- at hmain: hmain has |γ−T|² (not (T−γ)²) — rw [hsq] looks for (T−γ)² — NOT FOUND in hmain either!! — hmm — wait, actually — hmain := per_zero_integral_le_two_log ρ.2.2.1 ρ.2.2.2 hδpos hδ1.le — per_zero's δ is implicit {δ} — determined by hδpos : 0 < |γ−T| — so δ := |γ−T| — hmain : ∫ σ, 1/√((σ−ρ.1.re)² + |γ−T|²) ≤ 2·log(16.5/|γ−T|) — contains |γ−T|² not (T−γ)² — so rw [hsq] at hmain fails (but the error shown is about ⊢). And at ⊢: the goal has (T−γ)² ✓ rw [hsq] at ⊢ should find it... — the error message shows the ⊢ target WITH |γ−T|² already — meaning hsq got rewritten somewhere?? — no — the error says the PATTERN (T−γ)² was not found in the displayed target — which ALREADY shows |γ−T|². \n\nOH — I see — `rw [hsq] at hmain ⊢` — Lean processes the locations left-to-right: `at hmain` FIRST — fails (no (T−γ)² in hmain)... — hmm, but then the error would be about hmain. The displayed target is the GOAL ⊢ — with |γ−T|² — so the rw at ⊢ also ran and... — wait — if `at hmain` fails, the whole rw fails with the error about hmain. The error shows the pattern not found \"in the target expression\" — \"target\" = the GOAL ⊢ — and the displayed goal already has |γ−T|². Hmm — but the goal should have (T−γ)² (from hper's statement)...\n\nWAIT — look at the displayed target: `∫ σ in -5/2..3, 1/√((σ−β)² + |γ−T|²) ≤ 2·log(16.5/|γ−T|)` — the RHS is `2·log(16.5/|γ−T|)` — that's hmain's STATEMENT, not hper's goal (which has `2·log(148.5·C0hat)`)! So this \"target\" IS hmain (the rw failed at hmain first). The error message's \"target expression\" is hmain's type. So: `rw [hsq] at hmain ⊢` — hsq's LHS (T−γ)² not in hmain (hmain has |γ−T|² — hsq's LHS is (T−γ)² — these are DIFFERENT subterms!) — I had the direction BACKWARDS again. \n\nWhat I actually want: hper's goal has (T−γ)²; hmain has δ² = |γ−T|². To connect: rewrite the GOAL's (T−γ)² into |γ−T|² — that's hsq : (T−γ)² = |γ−T|² applied FORWARD at ⊢: `rw [hsq]` — but hsq's own proof needs the direction I just analyzed: `(T−γ)² = |γ−T|²` — prove by: rw [sq_abs] turning |γ−T|² → (γ−T)², then ring (since (T−γ)² = (γ−T)²).\n\nSo:\n```lean\n    have hsq : (T - ρ.1.im) ^ 2 = |ρ.1.im - T| ^ 2 := by\n      rw [sq_abs (ρ.1.im - T)]\n      ring\n    have hmain := per_zero_integral_le_two_log ρ.2.2.1 ρ.2.2.2 hδpos hδ1.le\n    rw [hsq]  -- only at the goal, not at hmain\n```\nThen goal: ∫ 1/√((σ−β)²+|γ−T|²) ≤ 2·log(148.5·C0hat) — matches hmain's LHS ✓ then linarith [hmain, h3] as before ✓.\n\nError 1101: hsum's Finset.mul_sum step:\n```lean\n      _ = (2 * Real.log (...)) * ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℝ) := by\n          rw [← Finset.mul_sum]\n          apply Finset.sum_congr rfl\n          intro ρ _\n          ring\n```\nThe error: `rw [← Finset.mul_sum]` — pattern `∑ i ∈ ?s, ?a * ?f i` not found in target `∑ ρ, ↑(zeroMult ↑ρ) * (2*log(...)) = (2*log(...)) * ∑ ρ, ↑(zeroMult ↑ρ)`. \n\nHmm — Finset.mul_sum (s) (f) (a) : a * ∑ i ∈ s, f i = ∑ i ∈ s, a * f i — ← direction: pattern `∑ i ∈ ?s, ?a * ?f i` — my LHS terms: `↑(zeroMult ρ) * (2·log(...))` — the constant factor is on the RIGHT (m * B0), not LEFT (a * f i)! The pattern needs `?a * ?f i` — mine is `?f i * ?a` — no match (rw is syntactic).\n\nFix: first rewrite each term's product to constant-left, or use Finset.sum_mul (the constant-RIGHT version): `Finset.sum_mul (s) (f) (a) : (∑ i ∈ s, f i) * a = ∑ i ∈ s, f i * a` — ← Finset.sum_mul : pattern `∑ i ∈ s, ?f i * ?a` ✓ matches my LHS → `(∑ ρ, ↑(zeroMult ρ)) * (2·log)` — then the goal's RHS is `(2·log) * ∑` — mul_comm. So:\n\n```lean\n      _ = (2 * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2))) *\n            ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℝ) := by\n          rw [← Finset.sum_mul, mul_comm]\n```\n\nrw [← Finset.sum_mul]: `∑ ρ, ↑(zeroMult ρ) * B0` → `(∑ ρ, ↑(zeroMult ρ)) * B0` — goal becomes `(∑ m) * B0 = B0 * (∑ m)` — rw [mul_comm] — rewrites the FIRST product found — `(∑ m) * B0` → `B0 * (∑ m)` — goal `B0 * (∑ m) = B0 * (∑ m)` ✓ rfl-closed ✓.\n\nHmm — but mul_comm might rewrite the WRONG product — the goal after ← Finset.sum_mul: `(∑ ρ ∈ nearZerosSub T, ↑(zeroMult ↑ρ)) * (2·log(...)) = (2·log(...)) * ∑ ρ, ↑(zeroMult ↑ρ)` — products: the outer two — mul_comm rewrites the first found (LHS outer) ✓ then rfl ✓. Safe.\n\nLet me check Finset.sum_mul exists: in Mathlib/Algebra/BigOperators/Ring/Finset.lean next to mul_sum ✓ (`lemma sum_mul (s : Finset ι) (f : ι → R) (a : R) : (∑ i ∈ s, f i) * a = ∑ i ∈ s, f i * a` — inside namespace Finset ✓).\n\nApply all three fixes.","signature":""},{"type":"text","text":"Three direction fixes: `sq_abs` forward (not reverse), drop the `at hmain ⊢` double-target, and `Finset.sum_mul` (constant-right) instead of `Finset.mul_sum`."},{"type":"tool_use","id":"Edit_133","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean","old_string":"    have hsq : (T - ρ.1.im) ^ 2 = |ρ.1.im - T| ^ 2 := by\n      rw [← sq_abs (ρ.1.im - T)]\n      ring\n    have hmain := per_zero_integral_le_two_log ρ.2.2.1 ρ.2.2.2 hδpos hδ1.le\n    rw [hsq] at hmain ⊢","new_string":"    have hsq : (T - ρ.1.im) ^ 2 = |ρ.1.im - T| ^ 2 := by\n      rw [sq_abs (ρ.1.im - T)]\n      ring\n    have hmain := per_zero_integral_le_two_log ρ.2.2.1 ρ.2.2.2 hδpos hδ1.le\n    rw [hsq]"},"extra_content":null},{"type":"tool_use","id":"Edit_134","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean","old_string":"      _ = (2 * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2))) *\n            ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℝ) := by\n          rw [← Finset.mul_sum]\n          apply Finset.sum_congr rfl\n          intro ρ _\n          ring","new_string":"      _ = (2 * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2))) *\n            ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℝ) := by\n          rw [← Finset.sum_mul, mul_comm]"},"extra_content":null}]