[{"type":"thinking","thinking":"Multiple errors:\n\n1. `Real.pi_lt_315` doesn't exist in this Mathlib. Check the pi bounds: `Real.pi_gt_314` / `Real.pi_lt_315`? Maybe `Real.pi_lt_315` was renamed. Let me grep for pi bounds.\n\n2. digamma_right_le's hdist linarith fail: ‖w+m‖ ≥ |w.re + m| and I need ≥ 1: |w.re+m| could be... w.re + m ≥ 1.75 (m ≥ 0) ⟹ |w.re+m| = w.re+m ≥ 1.75 ≥ 1 ✓: linarith [hwre, h1, h3] — the atoms: h1 : ‖w+↑m‖ ≥ |w.re + ↑m| (after rw [h2] — WAIT: h1's RHS after rw [h2] at h1: h2 says (w+↑m).re = w.re + ↑m — but h1 was |(w + ↑m).re| — after rw, h1 : ‖w + ↑m‖ ≥ |w.re + ↑m| ✓ (context shows that). Then |w.re + ↑m| = w.re + ↑m (positive: w.re ≥ 1.75, m ≥ 0): linarith needs abs_of_nonneg — NOT automatic: linarith doesn't split |·|. Give: have h4 : |w.re + ↑m| = w.re + ↑m := abs_of_nonneg (by linarith [hwre, h3]); linarith [h1, h4, hwre, h3].\n\n3. `Complex.ofNat_re`/`Complex.ofNat_im` unknown — for (s/2).re: use `Complex.div_ofNat_re`? doesn't exist either maybe. Alternatives: simp [Complex.div_re, Complex.ofNat_re] — ofNat_re unknown. What computes (s/2).re? s/2 with 2 : ℂ OfNat: `Complex.ofNat_re : (OfNat.ofNat n : ℂ).re = n`?? unknown. Try: `Complex.div_re : (z / w).re = (z.re * w.re + z.im * w.im) / (w.re^2 + w.im^2)`?? hmm — for w = 2: messy. There's surely `Complex.ofNat_div_re : (z / ofNat n).re = z.re / n`? Let me grep for div_ofNat_re or similar. Or compute directly: (s/2).re: s/2 = s * (2:ℂ)⁻¹?? Or: have h : (s/2).re = s.re/2 := by rw [Complex.div_re]; simp — try `simp [Complex.div_re]` alone? Let me test variants quickly via a scratch check. Actually — there might be simp lemmas `Complex.div_re_ofNat`... Let me grep for \"ofNat\" in Complex/Basic for re/im of numerals.\n\n4. himsq's nlinarith fail: `s.im ^ 2 < 100` from |s.im| < 10: nlinarith [him10, abs_nonneg s.im] — needs (|s.im|)² < 100 ⟸ |s.im| < 10 and |s.im| ≥ 0: nlinarith should do: (|s.im|)² < 10²·... it multiplies him10 by itself: him10·him10: |s.im|² < 100 ✓ then sq_abs: s.im² = |s.im|² — need sq_abs rewrite! nlinarith doesn't know s.im² = |s.im|². Fix: have h := sq_abs s.im; nlinarith [him10, abs_nonneg s.im, h] — hmm sq_abs : |x|² = x²: gives |s.im|² = s.im²: nlinarith with that ✓ — or: `nlinarith [him10, abs_nonneg s.im, sq_abs s.im]`? Wait — sq_abs s.im : |s.im| ^ 2 = s.im ^ 2: nlinarith multiplies pairs: him10·him10 gives |s.im|² < 100 (needs |s.im| ≥ 0 ✓): then s.im² = |s.im|² < 100 ✓. Let me: have h1 : s.im ^ 2 = |s.im| ^ 2 := (sq_abs s.im).symm; nlinarith [him10, abs_nonneg s.im, h1]. Hmm nlinarith multiplying him10×him10 — him10 : |s.im| < 10: product with itself: |s.im|² < 10·|s.im|... hmm: a < b and a ≥ 0 ⟹ a² < b²: nlinarith CAN do: (10−|s.im|)·(10+|s.im|) > 0: it considers products of the hypothesis atoms: him10 × him10: |s.im|·|s.im| < 100·... actually nlinarith's certificate: it multiplies PAIRS: (him10) · (him10): |s.im|·|s.im| < 10·10 = 100 — but that's a < b times a < b — the product rule used by nlinarith: (10 − |s.im|) > 0, (10 + |s.im|) > 0: product > 0: 100 − |s.im|² > 0 ✓ nlinarith does consider such products (it multiplies pairs of hypotheses including differences). With h1 in the mix it should close. Let me use: have h1 : s.im ^ 2 = |s.im| ^ 2 := (sq_abs s.im).symm; rw [h1]; nlinarith [him10, abs_nonneg s.im].\n\n5. Line 3231: the h5/h6/h8 mess in hRmin — I overcomplicated. Let me simplify hRmin's proof: R − |s.re| ≥ 0 from |s.re| ≤ R (abs_re_le_norm + hs). My current code has a garbled h5 with leftover h6/h7/h8 nonsense. Rewrite hRmin cleanly:\n```\nhave hRmin : R - |s.re| < 100 / R := by\n  have h9 : |s.re| ≤ R := by\n    have h10 := abs_re_le_norm s\n    rw [hs] at h10\n    exact h10\n  have h1 : (R - |s.re|) * (R + |s.re|) = s.im ^ 2 := by\n    have h2 : (R - |s.re|) * (R + |s.re|) = R ^ 2 - |s.re| ^ 2 := by ring\n    rw [h2, sq_abs]\n    linarith [hre2sq]\n  have h3 : (R - |s.re|) * (R + |s.re|) < 100 := by linarith [h1, himsq]\n  have h4 : (0 : ℝ) < R + |s.re| := by linarith [hRpos, abs_nonneg s.re]\n  have h5 : (R - |s.re|) * R ≤ (R - |s.re|) * (R + |s.re|) :=\n    mul_le_mul_of_nonneg_left (by linarith [abs_nonneg s.re] : R ≤ R + |s.re|)\n      (by linarith [h9] : (0 : ℝ) ≤ R - |s.re|)\n  have h6 : (R - |s.re|) * R < 100 := by linarith [h3, h5]\n  exact (mul_lt_mul_right hRpos).mp (by linarith [h6] : (R - |s.re|) * R < 100)\n```\nHmm — the last step: (R−|s.re|)·R < 100 ⟹ R−|s.re| < 100/R: (mul_lt_mul_right hRpos).mp : a·R < b·R → a < b: with b := 100/R: 100/R·R = 100 (div_mul_cancel₀ R hRpos.ne'): need the RHS as 100/R·R: `have h7 : (100 : ℝ) / R * R = 100 := div_mul_cancel₀ 100 hRpos.ne'`; exact (mul_lt_mul_right hRpos).mp (by linarith [h6, h7]). Wait — h6 : (R−|s.re|)*R < 100 and I want R−|s.re| < 100/R: mul_lt_mul_right (hRpos : 0 < R) : a * R < b * R ↔ a < b: apply with a := R−|s.re|, b := 100/R: need (R−|s.re|)*R < (100/R)*R: from h6 and h7: (100/R)*R = 100 ✓ linarith.\n\nAlso the goal form of hRmin was `<` (strict) ✓ fine.\n\n6. hre_neg's proof is also garbled (h10/hsign mess). Rewrite cleanly:\n```\nhave hre_neg : (s / 2).re < 0 := by\n  have h9 : |s.re| ≤ R := by\n    have h10 := abs_re_le_norm s\n    rw [hs] at h10\n    exact h10\n  have h14 : (100 : ℝ) / R < 1 := by\n    rw [div_lt_one hRpos]\n    linarith [hR]\n  have h15 : s.re < 0 := by\n    by_contra hcon\n    push Not at hcon\n    have h16 : s.re / 2 ≥ 0 := by linarith [hcon]\n    -- if re ≥ 0 then re/2 < 1.75 and re > R−100/R (from hRmin + sign) — wait no: hRmin bounds R−|re|: with re ≥ 0: |re| = re: R−re < 100/R: re > R−100/R > R−1 > 1.75·2: re/2 > 1.75 — contradiction with hre2\n    rw [hre] at hre2\n    have h17 : |s.re| = s.re := abs_of_nonneg hcon\n    have h18 : R - s.re < 100 / R := by rw [← h17]; exact hRmin\n    have h19 : (1.75 : ℝ) * 2 < s.re := by linarith [h18, hR, h14]\n    linarith [h19, hre2, h16]\n  rw [hre]\n  linarith [h15]\n```\nHmm wait — the branch analysis: hre2 : (s/2).re < 1.75. If s.re ≥ 0: |s.re| = s.re: hRmin: R−s.re < 100/R ⟹ s.re > R−100/R ≥ R−1 ≥ 16001: s.re/2 > 8000 ≥ 1.75 — contradicts hre2 (rw [hre] at hre2: s.re/2 < 1.75). So s.re < 0 ✓ then (s/2).re = s.re/2 < 0 ✓.\n\nHmm — but actually the branch: hre2 says re(s/2) < 1.75 — with s.re ≥ 0 ⟹ re(s/2) = s.re/2 ≥ 0: could be anywhere in [0, R/2]. The contradiction route works ✓.\n\n7. hlt (the s.re/2 + R/2 < 50/R) is also garbled. Rewrite:\n```\nhave hlt : s.re / 2 + R / 2 < 50 / R := by\n  -- |s.re| = −s.re (s.re < 0 from hre_neg's internals) ⟹ R−(−s.re) = R+s.re < 100/R\n  have h15 : s.re < 0 := by\n    by_contra hcon\n    push Not at hcon\n    have h17 : |s.re| = s.re := abs_of_nonneg hcon\n    have h18 : R - s.re < 100 / R := by rw [← h17]; exact hRmin\n    rw [hre] at hre2\n    have h14 : (100 : ℝ) / R < 1 := by\n      rw [div_lt_one hRpos]\n      linarith [hR]\n    linarith [h18, hR, h14, hre2]\n  have h19 : R + s.re < 100 / R := by\n    have h20 : |s.re| = -s.re := abs_of_neg h15\n    have h21 : R - (-s.re) < 100 / R := by rw [← h20]; exact hRmin\n    linarith [h21]\n  linarith [h19]\n```\nWait — hlt : s.re/2 + R/2 < 50/R: s.re/2 + R/2 = (s.re + R)/2 < (100/R)/2 = 50/R ⟸ R+s.re < 100/R ✓ h19. And hge : 0 ≤ s.re/2+R/2 ⟸ s.re ≥ −R (|s.re| ≤ R ⟹ −R ≤ s.re) ✓.\n\nHmm — but hre_neg's proof ALSO derives s.re < 0 — and hlt needs s.re < 0 too. Since hre_neg comes BEFORE hlt in my layout (hre_neg at line ~3230, hdist later), I can reuse: hre_neg : (s/2).re < 0 ⟹ s.re < 0 (rw [hre] at hre_neg: s.re/2 < 0 ⟹ s.re < 0 ✓). So in hdist's hlt: have h15 : s.re < 0 := by have := hre_neg; rw [hre] at this; linarith. Clean.\n\nBut wait — hre_neg is proved BEFORE hdist ✓ and uses hRmin + hre2 ✓. Then in hdist: hge uses |s.re| ≤ R ✓; hlt uses hre_neg → s.re < 0 + hRmin ✓.\n\nLet me also fix the h4 in hdist (the |x+m| ≥ |m−R/2|−(x+R/2)) — the h6/h7/h8 mess: simplify: \n```\nhave h4 : |(m : ℝ) - R / 2| - (s.re / 2 + R / 2) ≤ |s.re / 2 + (m : ℝ)| := by\n  have h5 : (m : ℝ) - R / 2 = (s.re / 2 + (m : ℝ)) - (s.re / 2 + R / 2) := by ring\n  rw [h5]\n  exact abs_sub_abs_le_abs_sub _ _\n```\nabs_sub_abs_le_abs_sub (a b : α) : |a| − |b| ≤ |a − b|: with a := s.re/2+m, b := s.re/2+R/2: |s.re/2+m| − |s.re/2+R/2| ≤ |(s.re/2+m) − (s.re/2+R/2)| = |m−R/2| hmm — direction: I want |m−R/2| − (s.re/2+R/2) ≤ |s.re/2+m|: from |a|−|b| ≤ |a−b| with a−b = m−R/2: |a|−|b| ≤ |m−R/2| — wrong direction. Use: |m−R/2| = |(s.re/2+m) − (s.re/2+R/2)| ≥ |s.re/2+m| − |s.re/2+R/2| (abs_sub_abs_le_abs_sub (s.re/2+m) (s.re/2+R/2)) ⟹ |s.re/2+m| ≥ |m−R/2| − |s.re/2+R/2| = |m−R/2| − (s.re/2+R/2) (s.re/2+R/2 ≥ 0 ✓ abs_of_nonneg). So:\n```\nhave h4 : |(m : ℝ) - R / 2| - (s.re / 2 + R / 2) ≤ |s.re / 2 + (m : ℝ)| := by\n  have h5 : |(s.re / 2 + (m : ℝ)) - (s.re / 2 + R / 2)| = |(m : ℝ) - R / 2| := by\n    have h6 : (s.re / 2 + (m : ℝ)) - (s.re / 2 + R / 2) = (m : ℝ) - R / 2 := by ring\n    rw [h6]\n  have h7 := abs_sub_abs_le_abs_sub (s.re / 2 + (m : ℝ)) (s.re / 2 + R / 2)\n  rw [h5] at h7\n  have h8 : |s.re / 2 + R / 2| = s.re / 2 + R / 2 := abs_of_nonneg hge\n  linarith [h7, h8]\n```\n✓. Then final: ‖s/2+m‖ ≥ |s.re/2+m| ≥ |m−R/2| − 50/R ≥ 1/4 − 50/R ≥ 1/8: linarith [h1, h4, hgap', h9, hlt, h50]: h4 : |m−R/2| − (s.re/2+R/2) ≤ |s.re/2+m|; hgap' : 1/4 ≤ |R/2−m| = |m−R/2| (h9); hlt : s.re/2+R/2 < 50/R; h50 : 50/R ≤ 1/8. Chain: |s.re/2+m| ≥ 1/4 − 50/R ≥ 1/4 − 1/8 = 1/8 ✓ linarith ✓.\n\n8. Real.pi_gt_three exists (used in norm_logPi_div_two_le) ✓ I think. Let me check the pi_lt name — grep Mathlib for pi_lt.","signature":""},{"type":"text","text":"Multiple fixes needed. First check the π bounds available in this Mathlib."},{"type":"tool_use","id":"Bash_180","name":"Bash","input":{"command":"cd \"D:/Python/Quantum/research/lean/NguyenRH1\" && grep -rn \"pi_lt\\|pi_gt\" ../../../zeta-23-lean/.lake/packages/mathlib/Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean 2>/dev/null | head -8; grep -rn \"theorem pi_lt_315\\|pi_lt_315\" ../../../zeta-23-lean/.lake/packages/mathlib/Mathlib/ 2>/dev/null | head -3","description":"Find pi bounds"},"extra_content":null}]