[{"type":"thinking","thinking":"Wait — in summable_mult_vTerm I wrote a strange calc step: `div_le_div_of_nonneg_right hnum (mul_pos hρ0 h1ρ0) |>.trans_eq (one_div _) |>.trans_eq' rfl` — that's garbled (trans_eq' probably doesn't exist, and div_le_div_of_nonneg_right may not be the right lemma). Let me fix that calc step: goal there: mult·(‖num‖/(‖ρ‖‖1−ρ‖)) ≤ mult·(1/(‖ρ‖‖1−ρ‖)): \n```lean\n        apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)\n        exact div_le_div_of_le_left?? \n```\nFor ‖num‖/D ≤ 1/D with D > 0 and ‖num‖ ≤ 1: div_le_div_of_nonneg_right? The right lemma: `div_le_div_right (hD : 0 < D) : a / D ≤ b / D ↔ a ≤ b`? or `div_le_div_of_le_left`? Use: div_le_div_of_nonneg_right?? hmm — simplest: rw [div_le_iff₀ (mul_pos hρ0 h1ρ0)]: goal ‖num‖ ≤ 1·D?? then: ‖num‖ ≤ 1 ≤ D?? NO — D = ‖ρ‖‖1−ρ‖ could be small. WRONG DIRECTION? ‖num‖/D ≤ 1/D ⟺ ‖num‖ ≤ 1 (D > 0) ✓ right direction: `div_le_div_right (hD : 0 < D) : a / D ≤ b / D ↔ a ≤ b` — session-032 banked: \"div_le_div_of_nonneg_right exists / div_le_div_right does NOT\"! So: div_le_div_of_nonneg_right — signature? From the banked note it EXISTS: `div_le_div_of_nonneg_right : a ≤ b → 0 ≤ c → a / c ≤ b / c`? presumably. So:\n```lean\n        apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)\n        exact div_le_div_of_nonneg_right hnum (mul_pos hρ0 h1ρ0).le\n```\nHmm — signature guess: (h : a ≤ b) (hc : 0 ≤ c) : a / c ≤ b / c. If arg order differs, fix at build. Let me simplify that whole step:\n```lean\n  calc (zeroMult ρ : ℝ) * (‖(1 : ℂ) - (ρ : ℂ) - (starRingEnd ℂ) (ρ : ℂ)‖ /\n        (‖(ρ : ℂ)‖ * ‖(1 : ℂ) - (ρ : ℂ)‖))\n      ≤ (zeroMult ρ : ℝ) * (1 / (‖(ρ : ℂ)‖ * ‖(1 : ℂ) - (ρ : ℂ)‖)) :=\n        mul_le_mul_of_nonneg_left (div_le_div_of_nonneg_right hnum (mul_pos hρ0 h1ρ0).le)\n          (Nat.cast_nonneg _)\n    _ = ...\n```\nFix it via edit. Also check the `hid` field_simp;ring — gotcha risk (field_simp might close → ring errors \"no goals\"). I'll see the build errors.\n\nAlso in `two_mul_sigmaP_one`'s h3: `rw [← tsum_sub summable_mult_fst summable_mult_snd]` — tsum_sub signature: tsum_sub (hf : Summable f) (hg : Summable g) : ∑' (f − g) = ∑' f − ∑' g ✓. ← direction: ∑' f − ∑' g → ∑' (f − g) — pattern in goal: (tsum fstFam) − (tsum sndFam): ← tsum_sub rewrites to ∑' ρ, (fstFam ρ − sndFam ρ) ✓ then tsum_congr per-term: mult·(1/ρ + 1/conjρ) − mult·(1/ρ + 1/(1−ρ)) = mult·(1/conjρ − 1/(1−ρ)): ring ✓.\n\nFix the vTerm calc step.","signature":""},{"type":"text","text":"Fixing the garbled calc step in `summable_mult_vTerm` before building."},{"type":"tool_use","id":"Edit_90","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiMoments.lean","old_string":"  calc (zeroMult ρ : ℝ) * (‖(1 : ℂ) - (ρ : ℂ) - (starRingEnd ℂ) (ρ : ℂ)‖ /\n        (‖(ρ : ℂ)‖ * ‖(1 : ℂ) - (ρ : ℂ)‖))\n      ≤ (zeroMult ρ : ℝ) * (1 / (‖(ρ : ℂ)‖ * ‖(1 : ℂ) - (ρ : ℂ)‖)) := by\n        apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)\n        exact div_le_div_of_nonneg_right hnum\n          (mul_pos hρ0 h1ρ0) |>.trans_eq (one_div _) |>.trans_eq' rfl\n    _ = (zeroMult ρ : ℝ) / (‖(ρ : ℂ)‖ * ‖(1 : ℂ) - (ρ : ℂ)‖) := by ring\n    _ ≤ (1 / 2) * ((zeroMult ρ : ℝ) / ‖(ρ : ℂ)‖ ^ 2 +\n          (zeroMult ρ : ℝ) / ‖(1 : ℂ) - (ρ : ℂ)‖ ^ 2) := mult_div_norm_mul_norm_le ρ","new_string":"  calc (zeroMult ρ : ℝ) * (‖(1 : ℂ) - (ρ : ℂ) - (starRingEnd ℂ) (ρ : ℂ)‖ /\n        (‖(ρ : ℂ)‖ * ‖(1 : ℂ) - (ρ : ℂ)‖))\n      ≤ (zeroMult ρ : ℝ) * (1 / (‖(ρ : ℂ)‖ * ‖(1 : ℂ) - (ρ : ℂ)‖)) :=\n        mul_le_mul_of_nonneg_left (div_le_div_of_nonneg_right hnum (mul_pos hρ0 h1ρ0).le)\n          (Nat.cast_nonneg _)\n    _ = (zeroMult ρ : ℝ) / (‖(ρ : ℂ)‖ * ‖(1 : ℂ) - (ρ : ℂ)‖) := by ring\n    _ ≤ (1 / 2) * ((zeroMult ρ : ℝ) / ‖(ρ : ℂ)‖ ^ 2 +\n          (zeroMult ρ : ℝ) / ‖(1 : ℂ) - (ρ : ℂ)‖ ^ 2) := mult_div_norm_mul_norm_le ρ"},"extra_content":null}]