[{"type":"thinking","thinking":"From NatInt.lean's doc: \"formula `∑ i ∈ range k, f i + ∑' i, f (i + k) = ∑' i, f i`, ∈ `sum_add_tsum_nat_add`\" — so the lemma is `∑ i ∈ range k, f i + ∑' i, f (i + k) = ∑' i, f i` (range-first). And the LiDigammaLeft usage: their hsplit is `∑' = ∑ range + ∑' shifted` with `.symm` ✓ — exactly my form. So `(hsumh.sum_add_tsum_nat_add 4).symm` ✓ correct.\n\nNote also their shifted term: `(fun m : ℕ => ...) (n + N)` — they state the shifted tsum with the explicit lambda application. And the important gotcha (banked session 027): \"sum_add_tsum_nat_add's shifted terms are beta-redexes (state goals in redex form)\". So in MY hsplit, the shifted term should be written as `∑' n : ℕ, (fun m : ℕ => ...) (n + 4)` in redex form to match! My current hsplit: `∑' n : ℕ, h (n + 4)` — h is a set-variable (a function variable), so h (n+4) is NOT a redex — it matches (fun m => h m) (n+4)? The lemma's shifted form: ∑' i, f (i + k) — with f := h (the set var): ∑' n, h (n+4) ✓ exactly (f (i+k) with f = h — the application f (i + k) = h (n + 4) ✓ syntactic match!). Since h is an opaque variable (not a lambda), no redex issue ✓.\n\nBut then htail's h1 and the congr steps DO have h as a lambda after rw [hh]... in htail I rw [hh] exposing the lambda — the show-form issue at line 366: my show `(if n + 4 = 0 then 0 else Real.log (↑n + 4) / ↑(n + 4) ^ 2) = Real.log (↑n + 4) / (↑n + 4) ^ 2` vs target `Real.log (↑n + 4) / (↑n + 4) ^ 2 = (fun n => ...) (n + 4)` — the target's LHS/RHS are flipped vs my show AND the target has the redex `(fun n => ...) (n+4)`. The goal in htail's h1 was: after `refine tsum_congr fun n => ?_` — goal: `(fun n => Real.log (↑n+4)/(↑n+4)²) n`? hmm — the error shows target `Real.log (↑n + 4) / (↑n + 4) ^ 2 = (fun n => if ...) (n + 4)` — so the tsum_congr goal was (tail-fn) n = (shifted-h) n — i.e. the tsum_congr's pointwise direction is my-tail-fn = h(n+4) — my `rw [h1]`-style — let me rewrite htail's h1 block:\n\n```lean\n  have htail : ∑' n : ℕ, h (n + 4) ≤ (Real.log 3 + 1) / 3 := by\n    have h1 : (∑' n : ℕ, h (n + 4))\n        = ∑' n : ℕ, Real.log ((n : ℝ) + 4) / ((n : ℝ) + 4) ^ 2 := by\n      refine tsum_congr fun n => ?_\n      rw [hh]\n      show (fun n => if n = 0 then (0 : ℝ) else Real.log (n : ℝ) / (n : ℝ) ^ 2) (n + 4)\n          = Real.log ((n : ℝ) + 4) / ((n : ℝ) + 4) ^ 2\n      show (if (n + 4 : ℕ) = 0 then (0 : ℝ) else Real.log (↑(n + 4)) / (↑(n + 4) : ℝ) ^ 2)\n          = Real.log ((n : ℝ) + 4) / ((n : ℝ) + 4) ^ 2\n      rw [if_neg (by omega : (n + 4 : ℕ) ≠ 0)]\n      push_cast\n      ring_nf\n    rw [h1]\n    exact tsum_log_div_sq_tail_le\n```\nHmm — the double show is ugly. The issue: after rw [hh], h (n+4) becomes (fun n => if n = 0 then ...) (n + 4) — a redex. The first show: `(fun n => ...) (n+4) = tail-value` — is that defeq to the goal `h (n+4) = tail-value`? After rw [hh], the goal IS (fun n => ...) (n+4) = tail — the first show is syntactic-identical ✓ (fine, no-op). Then the second show with the beta-reduced form: defeq ✓ (beta). Then rw [if_neg ...] on the reduced form ✓ then push_cast: ↑(n+4) → ↑n+4 ✓ ring_nf closes.\n\nHmm — but does the `show` with `Real.log (↑(n + 4))`... the lambda body's `Real.log n` with n : ℕ coerced: Real.log ↑n — at n+4: Real.log ↑(n+4) ✓.\n\nActually — simpler: skip the double show; use `beta_reduce` or `simp only []`: after rw [hh]: `show` directly the reduced form:\n```lean\n      rw [hh]\n      show (if (n + 4 : ℕ) = 0 then (0 : ℝ) else Real.log (↑(n + 4)) / (↑(n + 4) : ℝ) ^ 2)\n          = Real.log ((n : ℝ) + 4) / ((n : ℝ) + 4) ^ 2\n      rw [if_neg (by omega : (n + 4 : ℕ) ≠ 0)]\n      push_cast\n      ring_nf\n```\nThe show is defeq to the redex goal ✓ (show accepts defeq). The previous failure was because my show had the sides flipped AND mismatched cast forms. This version: LHS reduced-if form with ↑(n+4) casts, RHS the (↑n+4) form. After if_neg + push_cast (turns ↑(n+4) into ↑n+4): both sides log (↑n+4)/(↑n+4)² — ring_nf closes.\n\nHmm wait — will push_cast normalize `↑(n + 4)` to `↑n + 4`? Nat.cast_add: ↑(n+4) = ↑n + ↑4; Nat.cast_ofNat: ↑4 = 4 ✓ push_cast does these ✓.\n\nAlso the same fix needed in hsumh's h1 block (line ~366). Let me look at my current text there and fix both, plus hle_pointwise, hderiv, the tail proof. Let me make the edits.","signature":""},{"type":"text","text":"Applying the round-4 fixes (deriv bridge via Pi.div form, the cast handling in the tail, hle_pointwise show, htail congr, hhead simp, sum_add_tsum_nat_add method form)."},{"type":"tool_use","id":"Edit_194","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean","old_string":"    have hderiv : deriv (fun y => Real.log y / y ^ 2) x\n        = (1 - 2 * Real.log x) / x ^ 3 := by\n      have h := (Real.hasDerivAt_log hx0).div (hasDerivAt_pow 2 x) (pow_ne_zero 2 hx0)\n      have h' : HasDerivAt (fun y => Real.log y / y ^ 2)\n          ((x⁻¹ * x ^ 2 - Real.log x * (↑2 * x ^ (2 - 1))) / (x ^ 2) ^ 2) x := by\n        convert h using 1\n        ext y\n        rw [Pi.div_apply]\n      rw [h'.deriv]\n      field_simp\n      ring\n    rw [hderiv]\n    exact div_nonpos_of_nonpos_of_nonneg (by linarith [hlogx]) (by positivity)","new_string":"    have hderiv : deriv (fun y => Real.log y / y ^ 2) x\n        = (1 - 2 * Real.log x) / x ^ 3 := by\n      have h := (Real.hasDerivAt_log hx0).div (hasDerivAt_pow 2 x) (pow_ne_zero 2 hx0)\n      have h'' : (fun y => Real.log y / y ^ 2) = Real.log / (fun y => y ^ 2) := by\n        ext y\n        rw [Pi.div_apply]\n      rw [h'', h.deriv]\n      have hsimp : (↑2 : ℝ) * x ^ (2 - 1) = 2 * x := by\n        rw [show (2 : ℕ) - 1 = 1 from rfl, pow_one]\n        push_cast\n        ring\n      rw [hsimp]\n      field_simp\n      ring\n    rw [hderiv]\n    exact div_nonpos_of_nonpos_of_nonneg (by linarith [hlogx]) (by positivity)"},"extra_content":null}]