← Session 33 — ii′-3b/3c crux + F-D finding; Lagarias letter package
thinking (7871 chars)
Wait — in the second case (left border, −2n): hb.1 : ((-2·m)).re ∈ {z.re}: mp gives re = z.re: `have h := Set.mem_singleton_iff.mp hb.1 : ((-2*(m:ℂ)):ℂ).re = z.re`; rw [hre0] at h: −2·(m:ℝ) = z.re. Then hre' := h.symm : z.re = −2·(m:ℝ) — hmm, why the .symm? I wrote hre' : z.re = -2·(m:ℝ) := h.symm — h : −2m = z.re → h.symm : z.re = −2m ✓. Then hzr : z.re = −5/2: rw [hzr] at hre' : −5/2 = −2m. hm : m = 5/4: from −5/2 = −2m: 2m = 5/2: m = 5/4 ✓ linarith. Then m = 1 (from m < 2 and hm1 : 1 ≤ m): m = 1: rw [hm5] at hm : (1:ℝ) = 5/4: norm_num → False ✓.
Also the first case: h : z.im = 0?? wait — hb.2 : ((-2·m)).im ∈ {z.im}: mp : im = z.im: rw [rectZ_im, him0] at h: 0 = −T?? hmm: him0 : im = 0; rectZ_im : z.im = −T: h : ((-2·m)).im = z.im → after rw [rectZ_im] at h: im = −T; after rw [him0] at h: 0 = −T. absurd h (0 ≠ −T): ne_of_gt (0 > −T) ✓ (I wrote (0:ℝ) > -T ✓). Wait — h's form: h : ((-2*(m:ℂ)):ℂ).im = z.im — rw [rectZ_im] at h rewrites z.im → −T: h : im = −T; rw [him0] at h rewrites the LHS im → 0: h : 0 = −T ✓. ne_of_gt (0 > −T) : 0 ≠ −T ✓.
Fourth case: h : w.im = ... wait no — fourth case is the RIGHT border: hb.1 : re ∈ {w.re}: mp : re = w.re: rw [hre0] at h : −2m = w.re; h.symm : w.re = −2m; rw [hwr] : 2√n = −2m: nlinarith [hre' : 2√n = −2m, √n ≥ 0, m ≥ 0]: 2√n ≥ 0 and −2m ≤ 0: 2√n = −2m ⟹ both 0 ⟹ √n = 0 ∧ m = 0 — but m ≥ 1: contradiction. nlinarith should find: from hre': 2√n = −2m: −2m ≤ 0 (m ≥ 0) and 2√n ≥ 0 ⟹ √n = 0, m = 0: but m ≥ 1 > 0: contradiction ✓ — need hm1 in context ✓ (m ≥ 1 from obtain). nlinarith [hre', hpos, hm0] — with hm1 : 1 ≤ m in context ✓ works.
Now — the nontrivial border cases (first rcases block): let me re-check those:
- bottom (im = −T): the conj argument: `hTwide (conj s) (isNontrivialZero_conj hnt) (proof √n−1 < (conj s).im) (proof (conj s).im < √n+2)`: (conj s).im = −s.im = T: the bounds: √n−1 < T (from hTn : √n ≤ T — need √n−1 < T: linarith) and T < √n+2 (hTn1 : T ≤ √n+1: linarith ✓). My code:
```
have hconj := hTwide ((starRingEnd ℂ) s) (isNontrivialZero_conj hnt)
(by rw [Complex.conj_im, him]; linarith [hTn])
(by rw [Complex.conj_im, him]; linarith [hTn1])
```
✓ forms ok. Then:
```
have hpos : (0 : ℝ) < 1 / (9 * C0hat (Real.sqrt n + 2)) :=
one_div_pos.mpr (by have := two_le_C0hat (by linarith [hn1sqrt] : (1 : ℝ) ≤ Real.sqrt n + 2); positivity)
```
Hmm — `two_le_C0hat (ht : 1 ≤ t)` — the arg: (by linarith [hn1sqrt] : (1:ℝ) ≤ Real.sqrt n + 2)? hn1sqrt : 1 ≤ √n: √n + 2 ≥ 3 ≥ 1 ✓ linarith ✓. positivity for 0 < 9·C0hat: from two_le_C0hat : 2 ≤ C0hat: 9·C0hat ≥ 18 > 0 ✓ — but positivity can't use the `have :=` — the have is in the proof: `by have := two_le_C0hat ...; positivity` — positivity works on hypotheses ✓ (2 ≤ C0hat in context ⟹ 0 < 9·C0hat ✓ positivity should close... positivity with the hypothesis 2 ≤ C0hat(√n+2): it needs to derive 0 < 9·C0hat(√n+2): linarith would be safer: `one_div_pos.mpr (by have h := two_le_C0hat (by linarith [hn1sqrt] : (1:ℝ) ≤ Real.sqrt n + 2); linarith)`. Let me use linarith for safety.
```
have : |(starRingEnd ℂ) s - T| = 0 := by
rw [Complex.conj_im, him, sub_self, abs_zero]
```
wait — |(conj s).im − T|: conj_im : ((starRingEnd ℂ) s).im = −s.im: rw [Complex.conj_im, him]: −(−T) − T = T − T: sub_self, abs_zero ✓ — but the expression is |(conj s).im − T|: after rw [Complex.conj_im]: |−s.im − T|; rw [him] (him : s.im = −T): |−(−T) − T| = |T − T|: sub_self → |0| → abs_zero → 0 ✓. Hmm — the goal is |(starRingEnd ℂ) s - T| — wait NO: the pigeonhole conclusion is |((starRingEnd ℂ) s).im − T| — the .im! hTwide's conclusion: 1/(9·C0hat) ≤ |ρ.im − T| with ρ := conj s: |(conj s).im − T| ≥ 1/(9C0). My `this` computes |(starRingEnd ℂ) s - T| — that's the norm of the COMPLEX difference, not the im-difference! Bug: should be |((starRingEnd ℂ) s).im − T|. Fix: `have hzero : |(starRingEnd ℂ) s).im - T| = 0 := by rw [Complex.conj_im, him, sub_self, abs_zero]`? sub_self: a − a = 0 — after rw [Complex.conj_im, him]: −(−T) − T = T − T ✓ sub_self → |0| → abs_zero ✓. Then linarith [hconj, hpos, hzero]: 1/(9C0) ≤ 0 < 1/(9C0) contradiction ✓.
Hmm wait — the goal of `this` is `|(starRingEnd ℂ) s - T| = 0` — with (starRingEnd ℂ) s being ℂ — the pigeonhole's hconj : 1/(9·C0hat(√n+2)) ≤ |((starRingEnd ℂ) s).im − T| — the .im is there. So I need hzero about |((starRingEnd ℂ) s).im − T|. Fix that.
- top: same fix: `have : |s.im - T| = 0 := by rw [him, sub_self, abs_zero]` ✓ that one was right.
- left (re = −5/2): `rw [hre'] at hnt; exact (by norm_num : ¬ ((0:ℝ) < -5/2 ∧ -5/2 < 1)) ⟨hnt.2.1, hnt.2.2⟩` — hnt : IsNontrivialZero s: hnt.2.1 : 0 < s.re; hnt.2.2 : s.re < 1: after rw [hre'] at hnt: 0 < −5/2 ∧ −5/2 < 1 — hre' : s.re = −5/2: rw at hnt rewrites s.re → −5/2 in hnt?? rw at a hypothesis rewrites occurrences of s.re — hnt.2.1's statement is 0 < s.re ✓ becomes 0 < −5/2 ✓. But can rw rewrite inside a structure (hnt.2.1)? rw [hre'] at hnt rewrites s.re everywhere in hnt's TYPE (IsNontrivialZero s = ζ s = 0 ∧ 0 < s.re ∧ s.re < 1) ✓ becomes ζ s = 0 ∧ 0 < −5/2 ∧ −5/2 < 1 ✓. Then ⟨hnt.2.1, hnt.2.2⟩ : 0 < −5/2 ∧ −5/2 < 1 ✓ contradiction by norm_num ✓.
Wait — hre' : s.re = -5/2 — from hb.1 : s.re ∈ {z.re}: Set.mem_singleton_iff.mp hb.1 : s.re = z.re: need rw [rectZ_re]: my code: `have hre' : s.re = -5 / 2 := by have := hb.1; rw [rectZ_re] at this; exact Set.mem_singleton_iff.mp this` — hb.1 : s.re ∈ {z.re}: rw [rectZ_re] at this: s.re ∈ {−5/2}: mp : s.re = −5/2 ✓. In my written code:
```
have hre' : s.re = -5 / 2 := by
have := hb.1
rw [rectZ_re] at this
exact Set.mem_singleton_iff.mp this
```
✓ good.
- right: `exact riemannZeta_ne_zero_of_one_le_re (by rw [hre']; linarith [Real.one_le_sqrt.mpr (by exact_mod_cast hn : (1:ℝ) ≤ n)]) hζ` — hre' : s.re = 2·√n: 1 ≤ s.re: rw [hre'] : 1 ≤ 2·√n: linarith with √n ≥ 1 ✓.
Also — the hre' in the right case: `have hre' : s.re = 2 * Real.sqrt n := by have := hb.1; rw [rectW_re] at this; exact Set.mem_singleton_iff.mp this` ✓.
Now the backward hSz direction's hr for nontrivial: `⟨by linarith [hnt.2.1], by linarith [hnt.2.2, Real.one_le_sqrt.mpr ...]⟩` — s.re ∈ Icc [−5/2, 2√n]: −5/2 ≤ s.re (from s.re > 0 ✓) and s.re ≤ 2√n (from s.re < 1 and 1 ≤ √n ⟹ 2√n ≥ 2 > 1 > s.re... s.re < 1 ≤ √n ≤ 2√n ✓ linarith ✓).
The `hi` for nontrivial: `exact abs_le.mp himT` — himT : |s.im| ≤ T: abs_le.mp : −T ≤ s.im ∧ s.im ≤ T ✓ matches Icc [−T, T] ✓.
The h1mem/h1notb: h2n : 1 < 2·√n: from √n ≥ 1: 2√n ≥ 2 > 1 ✓ linarith [Real.one_le_sqrt.mpr (by exact_mod_cast hn : (1:ℝ) ≤ n)] — Real.one_le_sqrt.mpr needs (1:ℝ) ≤ n ✓ exact_mod_cast hn : (1:ℝ) ≤ (n:ℝ)?? hn : 1 ≤ n (ℕ): exact_mod_cast hn : (1:ℝ) ≤ (n:ℝ) ✓.
Hmm — in h1notb's first case: `exact absurd (Set.mem_singleton_iff.mp hb.2) (by rw [rectZ_im] ; exact ne_of_gt (by linarith : (0 : ℝ) < T))` — hb.2 : (1:ℂ).im ∈ {z.im}: mp : im = z.im: the by-block's goal: ¬((1:ℂ).im = z.im): rw [rectZ_im]: ¬(im = −T): (1:ℂ).im = 0: the rw [rectZ_im] rewrites z.im → −T ✓; then ne_of_gt (0 < T) : 0 ≠ T — but the equation is im = −T with im = 0: 0 = −T?? — ne_of_gt gives 0 ≠ T; the goal is (1:ℂ).im ≠ −T — after rectZ_im: (1:ℂ).im ≠ −T: is (1:ℂ).im displayed as 0? Complex.one_im : (1:ℂ).im = 0 — simp? The goal after rw [rectZ_im]: ¬ ((1:ℂ).im = -T): exact needs (1:ℂ).im ≠ −T: (1:ℂ).im = 0 definitionally? Complex.one_im is rfl? (1:ℂ).im — 1 = One.toOfNat1 — its im: OfNat.ofNat 1 .im — computes to 0 definitionally? In practice `(1 : ℂ).im` reduces to 0 by `Complex.one_re/im` simp lemmas, but definitionally OfNat's im... `Complex.ofNat_im`? Hmm — safer: rw [rectZ_im, Complex.one_im]: goal 0 ≠ −T: `exact ne_of_gt (by linarith [hT1] : (0:ℝ) > -T)`. My code lacks the Complex.one_im rewrite and has the wrong sign (0 < T giving 0 ≠ T vs needed 0 ≠ −T). Fix both places (bottom: 0 ≠ −T; top: 0 ≠ T ✓ was right).
Let me fix these remaining issues.
Fixing the remaining sign/im issues in the identity proof.
tool: Edit ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiAssembly.lean",
"old_string": " rcases hb with hb | hb | hb | hb\n \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.2) (by rw [rectZ_im] ; exact ne_of_gt (by linarith : (0 : \u211d) < T))\n \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.1) (by rw [rectZ_re]; norm_num)\n \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.2) (by rw [rectW_im]; exact ne_of_gt (by linarith : (0 : \u211d) < T))\n \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.1) (by rw [rectW_re]; exact ne_of_gt h2n)",
"new_string": " rcases hb with hb | hb | hb | hb\n \u00b7 have h := Set.mem_singleton_iff.mp hb.2\n rw [rectZ_im, Complex.one_im] at h\n exact absurd h (ne_of_gt (by linarith [hT1] : (0 : \u211d) > -T))\n \u00b7 have h := Set.mem_singleton_iff.mp hb.1\n rw [rectZ_re, Complex.one_re] at h\n norm_num at h\n \u00b7 have h := Set.mem_singleton_iff.mp hb.2\n rw [rectW_im, Complex.one_im] at h\n exact absurd h (ne_of_gt (by linarith [hT1] : (0 : \u211d) < T))\n \u00b7 have h := Set.mem_singleton_iff.mp hb.1\n rw [rectW_re, Complex.one_re] at h\n exact absurd h (ne_of_gt h2n)"
}
result 2026-08-25 05:55:36Z → 2026-08-25 05:55:36Z
{
"filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiAssembly.lean",
"oldString": " rcases hb with hb | hb | hb | hb\n \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.2) (by rw [rectZ_im] ; exact ne_of_gt (by linarith : (0 : \u211d) < T))\n \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.1) (by rw [rectZ_re]; norm_num)\n \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.2) (by rw [rectW_im]; exact ne_of_gt (by linarith : (0 : \u211d) < T))\n \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.1) (by rw [rectW_re]; exact ne_of_gt h2n)",
"newString": " rcases hb with hb | hb | hb | hb\n \u00b7 have h := Set.mem_singleton_iff.mp hb.2\n rw [rectZ_im, Complex.one_im] at h\n exact absurd h (ne_of_gt (by linarith [hT1] : (0 : \u211d) > -T))\n \u00b7 have h := Set.mem_singleton_iff.mp hb.1\n rw [rectZ_re, Complex.one_re] at h\n norm_num at h\n \u00b7 have h := Set.mem_singleton_iff.mp hb.2\n rw [rectW_im, Complex.one_im] at h\n exact absurd h (ne_of_gt (by linarith [hT1] : (0 : \u211d) < T))\n \u00b7 have h := Set.mem_singleton_iff.mp hb.1\n rw [rectW_re, Complex.one_re] at h\n exact absurd h (ne_of_gt h2n)",
"originalFile": "/-\nLiAssembly.lean \u2014 tier ii\u2032-6-b: the sides identity and the bridge\n(D-071 \u00a74.1\u20134.3).\n\n\u00a71 (`sf_abs_le_rectInt_add_zeroSum`): the rectangle residue theorem\n`rectInt'_Jlag` (ii\u2032-1b) on the D-036 variant-B rectangle\n(z = \u22125/2 \u2212 iT, w = 2\u221an + iT, T zero-avoiding-wide from ii\u2032-6's L3-wide)\nplus the crux `sf_eq_resW1` (ii\u2032-3c):\n |Sf n| = |(ResW1 n).re| \u2264 \u2016RectInt\u2016 + \u2016\u03a3_{Sz} mult\u00b7(1\u2212(\u03c1/(\u03c1\u22121))\u207f)\u2016.\n\n\u00a72 (`zeroSum_norm_le_four_FAlt_add_one`): the bridge \u2014 under RHUpTo\n3\u00d710\u00b9\u00b2 every zero in Sz is on the line (|\u03c1/(\u03c1\u22121)| = 1), so each term\n\u2264 2\u00b7mult, and the both-signs mult-count is 2\u00d7 the one-sign count\n(conj symmetry + `zeroMult_conj` + the perimeter `ZeroCountGlobalProp`)\n\u27f9 \u2016\u03a3\u2016 \u2264 4\u00b7FAlt T + 1 (the +1 is the trivial-zero term at \u22122).\n\nPerimeter used: ZeroCountProp (the crux), RHUpTo, ZeroCountGlobalProp,\nNoLowZeros.\n-/\nimport Mathlib\nimport NguyenRH1.LiSides\nimport NguyenRH1.LiCrux\nimport NguyenRH1.LiContour\nimport NguyenRH1.LiTightR\nimport NguyenRH1.LiFEntire\nimport NguyenRH1.LiCircleBound\n\nopen Zeta23 Complex Filter Topology\nopen scoped Nat\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 0. The both-signs ordinate window -/\n\n/-- The nontrivial zeros with |\u03b3| \u2264 T (the rectangle's zero content,\nminus the trivial zero \u22122 which is handled separately). -/\ndef zeroBothWindow (T : \u211d) : Set \u2102 :=\n {\u03c1 | IsNontrivialZero \u03c1 \u2227 |\u03c1.im| \u2264 T}\n\n/-- Finiteness of the both-signs window (via Zeta23 `zerosIn_finite`). -/\ntheorem zeroBothWindow_finite (T : \u211d) : (zeroBothWindow T).Finite := by\n apply Set.Finite.subset (zerosIn_finite (-T) T)\n intro \u03c1 h\u03c1\n refine \u27e8h\u03c1.1, ?_, ?_\u27e9\n \u00b7 linarith [abs_le.mp h\u03c1.2 |>.1]\n \u00b7 exact abs_le.mp h\u03c1.2 |>.2\n\n/-- The negative ordinate window: nontrivial zeros with \u2212T \u2264 \u03b3 < 0. -/\ndef zeroGlobalWindowNeg (T : \u211d) : Set \u2102 :=\n {\u03c1 | IsNontrivialZero \u03c1 \u2227 -T \u2264 \u03c1.im \u2227 \u03c1.im < 0}\n\n/-- Finiteness of the negative window. -/\ntheorem zeroGlobalWindowNeg_finite (T : \u211d) : (zeroGlobalWindowNeg T).Finite := by\n apply Set.Finite.subset (zerosIn_finite (-T) 0)\n intro \u03c1 h\u03c1\n refine \u27e8h\u03c1.1, h\u03c1.2.1, by linarith [h\u03c1.2.2]\u27e9\n\n/-! ## 1. The sides identity -/\n\n/-- The rectangle for the L0 assembly (D-036 variant B): z = \u22125/2 \u2212 iT,\nw = 2\u221an + iT. -/\nabbrev rectZ (T : \u211d) : \u2102 := ((-5 / 2 : \u211d) : \u2102) - (T : \u211d) * Complex.I\nabbrev rectW (n : \u2115) (T : \u211d) : \u2102 := ((2 * Real.sqrt n : \u211d) : \u2102) + (T : \u211d) * Complex.I\n\nlemma rectZ_re (T : \u211d) : (rectZ T).re = -5 / 2 := by simp [rectZ]\nlemma rectZ_im (T : \u211d) : (rectZ T).im = -T := by simp [rectZ]\nlemma rectW_re (n : \u2115) (T : \u211d) : (rectW n T).re = 2 * Real.sqrt n := by simp [rectW]\nlemma rectW_im (n : \u2115) (T : \u211d) : (rectW n T).im = T := by simp [rectW]\n\n/-- **The sides identity** (D-071 \u00a74.1): on the variant-B rectangle with\nthe wide zero-avoiding ordinate,\n|Sf n| \u2264 \u2016RectangleIntegral' (Jlag n) z w\u2016 + \u2016\u03a3_{Sz} mult\u00b7(1\u2212(\u03c1/(\u03c1\u22121))\u207f)\u2016,\nSz = {nontrivial zeros, |\u03b3| \u2264 T} \u222a {\u22122}. -/\ntheorem sf_abs_le_rectInt_add_zeroSum (hZC : ZeroCountProp) {n : \u2115} (hn : 1 \u2264 n)\n {T : \u211d} (hT1 : 1 \u2264 T) (hTn : Real.sqrt n \u2264 T) (hTn1 : T \u2264 Real.sqrt n + 1)\n (hTwide : \u2200 \u03c1 : \u2102, IsNontrivialZero \u03c1 \u2192 Real.sqrt n - 1 < \u03c1.im \u2192\n \u03c1.im < Real.sqrt n + 2 \u2192 1 / (9 * C0hat (Real.sqrt n + 2)) \u2264 |\u03c1.im - T|) :\n |Sf n| \u2264 \u2016RectangleIntegral' (Jlag n) (rectZ T) (rectW n T)\u2016 +\n \u2016\u2211 \u03c1 \u2208 ((zeroBothWindow_finite T).toFinset \u222a {(-2 : \u2102)}),\n (zeroMult \u03c1 : \u2102) * (1 - (\u03c1 / (\u03c1 - 1)) ^ n)\u2016 := by\n set z := rectZ T\n set w := rectW n T\n have hn1sqrt : (1 : \u211d) \u2264 Real.sqrt n := Real.one_le_sqrt.mpr (by exact_mod_cast hn)\n have hre : z.re \u2264 w.re := by\n rw [rectZ_re, rectW_re]\n have h := Real.sqrt_nonneg (n : \u211d)\n linarith\n have him : z.im \u2264 w.im := by\n rw [rectZ_im, rectW_im]\n linarith\n -- 1 is interior\n have h1mem : (1 : \u2102) \u2208 Rectangle z w := by\n have hr : (1 : \u2102).re \u2208 Set.uIcc z.re w.re := by\n rw [Set.uIcc_of_le hre, rectZ_re, rectW_re, Complex.one_re, Set.mem_Icc]\n constructor <;> [linarith; linarith [Real.sqrt_nonneg (n : \u211d)]]\n have hi : (1 : \u2102).im \u2208 Set.uIcc z.im w.im := by\n rw [Set.uIcc_of_le him, rectZ_im, rectW_im, Complex.one_im, Set.mem_Icc]\n constructor <;> linarith\n exact \u27e8hr, hi\u27e9\n have h1notb : (1 : \u2102) \u2209 RectangleBorder z w := by\n intro hb\n have h1re : (1 : \u2102).re = 1 := Complex.one_re\n have h1im : (1 : \u2102).im = 0 := Complex.one_im\n have h2n : (1 : \u211d) < 2 * Real.sqrt n := by linarith [Real.one_le_sqrt.mpr (by exact_mod_cast hn : (1 : \u211d) \u2264 n)]\n rcases hb with hb | hb | hb | hb\n \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.2) (by rw [rectZ_im] ; exact ne_of_gt (by linarith : (0 : \u211d) < T))\n \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.1) (by rw [rectZ_re]; norm_num)\n \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.2) (by rw [rectW_im]; exact ne_of_gt (by linarith : (0 : \u211d) < T))\n \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.1) (by rw [rectW_re]; exact ne_of_gt h2n)\n have h1int : Rectangle z w \u2208 \ud835\udcdd (1 : \u2102) :=\n rectangle_mem_nhds_of_mem_not_border hre him h1mem h1notb\n -- the border is zero-free\n have hborder : \u2200 s \u2208 RectangleBorder z w, riemannZeta s \u2260 0 := by\n intro s hs h\u03b6\n rcases zeta_zero_classification h\u03b6 with hnt | h0 | hn2\n \u00b7 -- nontrivial: re \u2208 (0,1); the only border pieces with re in (0,1) are top/bottom\n rcases hs with hb | hb | hb | hb\n \u00b7 -- bottom: im = \u2212T; conj \u03c1 has im = T in the window \u2014 pigeonhole contradiction\n have him : s.im = -T := by\n have := hb.2\n rw [rectZ_im] at this\n exact Set.mem_singleton_iff.mp this\n have hconj := hTwide ((starRingEnd \u2102) s) (isNontrivialZero_conj hnt)\n (by rw [Complex.conj_im, him]; linarith [hTn])\n (by rw [Complex.conj_im, him]; linarith [hTn1])\n have hpos : (0 : \u211d) < 1 / (9 * C0hat (Real.sqrt n + 2)) :=\n one_div_pos.mpr (by have := two_le_C0hat (by linarith [hn1sqrt] : (1 : \u211d) \u2264 Real.sqrt n + 2); positivity)\n have : |(starRingEnd \u2102) s - T| = 0 := by\n rw [Complex.conj_im, him, sub_self, abs_zero]\n linarith [hconj]\n \u00b7 -- left: re = \u22125/2, not in (0,1)\n have hre' : s.re = -5 / 2 := by\n have := hb.1\n rw [rectZ_re] at this\n exact Set.mem_singleton_iff.mp this\n rw [hre'] at hnt\n exact (by norm_num : \u00ac ((0 : \u211d) < -5 / 2 \u2227 -5 / 2 < 1)) \u27e8hnt.2.1, hnt.2.2\u27e9\n \u00b7 -- top: im = T; pigeonhole contradiction directly\n have him : s.im = T := by\n have := hb.2\n rw [rectW_im] at this\n exact Set.mem_singleton_iff.mp this\n have hdir := hTwide s hnt (by rw [him]; exact hTn) (by rw [him]; linarith [hTn1])\n have hpos : (0 : \u211d) < 1 / (9 * C0hat (Real.sqrt n + 2)) :=\n one_div_pos.mpr (by have := two_le_C0hat (by linarith [hn1sqrt] : (1 : \u211d) \u2264 Real.sqrt n + 2); positivity)\n have : |s.im - T| = 0 := by rw [him, sub_self, abs_zero]\n linarith [hdir]\n \u00b7 -- right: re = 2\u221an \u2265 2 > 1\n have hre' : s.re = 2 * Real.sqrt n := by\n have := hb.1\n rw [rectW_re] at this\n exact Set.mem_singleton_iff.mp this\n exact riemannZeta_ne_zero_of_one_le_re (by rw [hre']; linarith [Real.one_le_sqrt.mpr (by exact_mod_cast hn : (1 : \u211d) \u2264 n)]) h\u03b6\n \u00b7 -- s = 0: not on the border (im 0 = \u00b1T needs T = 0; re 0 = \u22125/2 / 2\u221an false)\n rw [h0] at hs\n rcases hs with hb | hb | hb | hb\n \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.2) (by rw [rectZ_im, Complex.zero_im]; exact ne_of_gt (by linarith : (0 : \u211d) < T))\n \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.1) (by rw [rectZ_re, Complex.zero_re]; norm_num)\n \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.2) (by rw [rectW_im, Complex.zero_im]; exact ne_of_gt (by linarith : (0 : \u211d) < T))\n \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.1) (by rw [rectW_re, Complex.zero_re]; norm_num [Real.sqrt_nonneg (n : \u211d)])\n \u00b7 -- s = \u22122n, n \u2265 1: re = \u22122n; border re-pieces are \u22125/2 and 2\u221an; im = 0 \u2260 \u00b1T\n obtain \u27e8m, hm1, rfl\u27e9 := hn2\n have him0 : ((-2 * (m : \u2102)) : \u2102).im = 0 := by\n simp [Complex.neg_re, Complex.neg_im, Complex.mul_re, Complex.mul_im]\n have hre0 : ((-2 * (m : \u2102)) : \u2102).re = -2 * (m : \u211d) := by\n simp [Complex.mul_re, Complex.mul_im]\n rcases hs with hb | hb | hb | hb\n \u00b7 have h := Set.mem_singleton_iff.mp hb.2\n rw [rectZ_im, him0] at h\n exact absurd h (ne_of_gt (by linarith [hT1] : (0 : \u211d) > -T))\n \u00b7 have hre' : z.re = -2 * (m : \u211d) := by\n have h := Set.mem_singleton_iff.mp hb.1\n rw [hre0] at h\n exact h.symm\n have hzr : z.re = -5 / 2 := rectZ_re T\n rw [hzr] at hre'\n have hm : (m : \u211d) = 5 / 4 := by linarith\n have hm5 : m = 1 := by\n have h6 : (m : \u211d) < 2 := by linarith\n have h7 : m < 2 := by exact_mod_cast h6\n exact le_antisymm (Nat.lt_succ_iff.mp h7) hm1\n rw [hm5] at hm\n norm_num at hm\n \u00b7 have h := Set.mem_singleton_iff.mp hb.2\n rw [rectW_im, him0] at h\n exact absurd h (ne_of_gt (by linarith [hT1] : (0 : \u211d) < T))\n \u00b7 have hre' : w.re = -2 * (m : \u211d) := by\n have h := Set.mem_singleton_iff.mp hb.1\n rw [hre0] at h\n exact h.symm\n have hwr : w.re = 2 * Real.sqrt n := rectW_re n T\n rw [hwr] at hre'\n have hpos : (0 : \u211d) \u2264 Real.sqrt n := Real.sqrt_nonneg _\n have hm0 : (0 : \u211d) \u2264 (m : \u211d) := Nat.cast_nonneg _\n nlinarith [hre', hpos, hm0]\n -- Sz and its characterization\n set Sz : Finset \u2102 := (zeroBothWindow_finite T).toFinset \u222a {(-2 : \u2102)} with hSz_def\n have hSz : \u2200 s : \u2102, s \u2208 Sz \u2194 s \u2208 Rectangle z w \u2227 zetaA s = 0 := by\n intro s\n constructor\n \u00b7 intro hs\n rw [hSz_def, Finset.mem_union, Finset.mem_singleton] at hs\n rcases hs with hs | rfl\n \u00b7 have hmem : s \u2208 zeroBothWindow T := (zeroBothWindow_finite T).mem_toFinset.mp hs\n obtain \u27e8hnt, himT\u27e9 := hmem\n have hs1 : s \u2260 1 := by\n intro h\n rw [h, Complex.one_re] at hnt\n exact lt_irrefl 1 hnt.2.2\n refine \u27e8\u27e8?_, ?_\u27e9, ?_\u27e9\n \u00b7 have hr : s.re \u2208 Set.uIcc z.re w.re := by\n rw [Set.uIcc_of_le hre, rectZ_re, rectW_re, Set.mem_Icc]\n exact \u27e8by linarith [hnt.2.1], by linarith [hnt.2.2, Real.one_le_sqrt.mpr (by exact_mod_cast hn : (1 : \u211d) \u2264 n)]\u27e9\n exact hr\n \u00b7 have hi : s.im \u2208 Set.uIcc z.im w.im := by\n rw [Set.uIcc_of_le him, rectZ_im, rectW_im, Set.mem_Icc]\n exact abs_le.mp himT\n exact hi\n \u00b7 rw [zetaA_apply hs1, hnt.1, mul_zero]\n \u00b7 refine \u27e8\u27e8?_, ?_\u27e9, ?_\u27e9\n \u00b7 have hr : (-2 : \u2102).re \u2208 Set.uIcc z.re w.re := by\n rw [Set.uIcc_of_le hre, rectZ_re, rectW_re, Set.mem_Icc]\n have h2re : (-2 : \u2102).re = -2 := by simp\n rw [h2re]\n constructor\n \u00b7 norm_num\n \u00b7 have h0' : (0 : \u211d) \u2264 Real.sqrt n := Real.sqrt_nonneg _\n linarith\n exact hr\n \u00b7 have hi : (-2 : \u2102).im \u2208 Set.uIcc z.im w.im := by\n rw [Set.uIcc_of_le him, rectZ_im, rectW_im, Set.mem_Icc]\n have h2im : (-2 : \u2102).im = 0 := by simp\n rw [h2im]\n constructor <;> linarith [hT1]\n exact hi\n \u00b7 have h2ne : (-2 : \u2102) \u2260 1 := by norm_num\n rw [zetaA_apply h2ne]\n have h\u03b62 : riemannZeta (-2 : \u2102) = 0 := by\n have h := riemannZeta_neg_two_mul (n := 1) le_rfl\n rw [show (-2 : \u2102) * ((1 : \u2115) : \u2102) = -2 by push_cast; ring] at h\n exact h\n rw [h\u03b62, mul_zero]\n \u00b7 intro \u27e8hmem, hzA\u27e9\n have hs1 : s \u2260 1 := by\n intro h\n rw [h, zetaA_one] at hzA\n exact one_ne_zero hzA\n have h\u03b6 : riemannZeta s = 0 := by\n have e := zetaA_apply hs1\n rw [e, mul_eq_zero] at hzA\n rcases hzA with h | h\n \u00b7 exact absurd (sub_eq_zero.mp h) hs1\n \u00b7 exact h\n rcases zeta_zero_classification h\u03b6 with hnt | h0 | hn2\n \u00b7 rw [hSz_def, Finset.mem_union]\n left\n apply (zeroBothWindow_finite T).mem_toFinset.mpr\n refine \u27e8hnt, ?_\u27e9\n have hi := hmem.2\n rw [Set.uIcc_of_le him, rectZ_im, rectW_im, Set.mem_Icc] at hi\n exact abs_le.mpr hi\n \u00b7 have h00 : zetaA 0 = 0 := by rwa [h0] at hzA\n rw [zetaA_apply (by norm_num : (0 : \u2102) \u2260 1), riemannZeta_zero] at h00\n norm_num at h00\n \u00b7 obtain \u27e8m, hm1, rfl\u27e9 := hn2\n rw [hSz_def, Finset.mem_union]\n have hr := hmem.1\n rw [Set.uIcc_of_le hre, rectZ_re, rectW_re, Set.mem_Icc] at hr\n have hre2 : ((-2 * (m : \u2102)) : \u2102).re = -2 * (m : \u211d) := by\n simp [Complex.mul_re, Complex.neg_re]\n rw [hre2] at hr\n have hm1' : m = 1 := by\n have h1 : (m : \u211d) \u2264 5 / 4 := by\n have := hr.1\n have hm0 : (0 : \u211d) \u2264 (m : \u211d) := Nat.cast_nonneg _\n nlinarith [this, hm0]\n have h2 : (m : \u211d) \u2265 1 := by exact_mod_cast hm1\n have hm2 : m \u2264 1 := by\n have h6 : (m : \u211d) < 2 := by linarith\n exact_mod_cast Nat.lt_succ_iff.mp (by exact_mod_cast h6 : m < 2)\n exact le_antisymm hm2 hm1\n rw [hm1']\n right\n simp\n -- the identity\n have hid := rectInt'_Jlag n hre him h1int hborder Sz hSz\n have hcrux := sf_eq_resW1 hZC hn\n rw [\u2190 hcrux]\n refine le_trans (abs_re_le_norm _) ?_\n have e : ResW1 n = RectangleIntegral' (Jlag n) z w -\n \u2211 \u03c1 \u2208 Sz, (zeroMult \u03c1 : \u2102) * (1 - (\u03c1 / (\u03c1 - 1)) ^ n) := by\n linear_combination hid\n rw [e]\n exact norm_sub_le _ _\n\nend NguyenRH1\n",
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" have h1im : (1 : \u2102).im = 0 := Complex.one_im",
" have h2n : (1 : \u211d) < 2 * Real.sqrt n := by linarith [Real.one_le_sqrt.mpr (by exact_mod_cast hn : (1 : \u211d) \u2264 n)]",
" rcases hb with hb | hb | hb | hb",
"- \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.2) (by rw [rectZ_im] ; exact ne_of_gt (by linarith : (0 : \u211d) < T))",
"- \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.1) (by rw [rectZ_re]; norm_num)",
"- \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.2) (by rw [rectW_im]; exact ne_of_gt (by linarith : (0 : \u211d) < T))",
"- \u00b7 exact absurd (Set.mem_singleton_iff.mp hb.1) (by rw [rectW_re]; exact ne_of_gt h2n)",
"+ \u00b7 have h := Set.mem_singleton_iff.mp hb.2",
"+ rw [rectZ_im, Complex.one_im] at h",
"+ exact absurd h (ne_of_gt (by linarith [hT1] : (0 : \u211d) > -T))",
"+ \u00b7 have h := Set.mem_singleton_iff.mp hb.1",
"+ rw [rectZ_re, Complex.one_re] at h",
"+ norm_num at h",
"+ \u00b7 have h := Set.mem_singleton_iff.mp hb.2",
"+ rw [rectW_im, Complex.one_im] at h",
"+ exact absurd h (ne_of_gt (by linarith [hT1] : (0 : \u211d) < T))",
"+ \u00b7 have h := Set.mem_singleton_iff.mp hb.1",
"+ rw [rectW_re, Complex.one_re] at h",
"+ exact absurd h (ne_of_gt h2n)",
" have h1int : Rectangle z w \u2208 \ud835\udcdd (1 : \u2102) :=",
" rectangle_mem_nhds_of_mem_not_border hre him h1mem h1notb",
" -- the border is zero-free"
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[{"type":"thinking","thinking":"Wait — in the second case (left border, −2n): hb.1 : ((-2·m)).re ∈ {z.re}: mp gives re = z.re: `have h := Set.mem_singleton_iff.mp hb.1 : ((-2*(m:ℂ)):ℂ).re = z.re`; rw [hre0] at h: −2·(m:ℝ) = z.re. Then hre' := h.symm : z.re = −2·(m:ℝ) — hmm, why the .symm? I wrote hre' : z.re = -2·(m:ℝ) := h.symm — h : −2m = z.re → h.symm : z.re = −2m ✓. Then hzr : z.re = −5/2: rw [hzr] at hre' : −5/2 = −2m. hm : m = 5/4: from −5/2 = −2m: 2m = 5/2: m = 5/4 ✓ linarith. Then m = 1 (from m < 2 and hm1 : 1 ≤ m): m = 1: rw [hm5] at hm : (1:ℝ) = 5/4: norm_num → False ✓.\n\nAlso the first case: h : z.im = 0?? wait — hb.2 : ((-2·m)).im ∈ {z.im}: mp : im = z.im: rw [rectZ_im, him0] at h: 0 = −T?? hmm: him0 : im = 0; rectZ_im : z.im = −T: h : ((-2·m)).im = z.im → after rw [rectZ_im] at h: im = −T; after rw [him0] at h: 0 = −T. absurd h (0 ≠ −T): ne_of_gt (0 > −T) ✓ (I wrote (0:ℝ) > -T ✓). Wait — h's form: h : ((-2*(m:ℂ)):ℂ).im = z.im — rw [rectZ_im] at h rewrites z.im → −T: h : im = −T; rw [him0] at h rewrites the LHS im → 0: h : 0 = −T ✓. ne_of_gt (0 > −T) : 0 ≠ −T ✓.\n\nFourth case: h : w.im = ... wait no — fourth case is the RIGHT border: hb.1 : re ∈ {w.re}: mp : re = w.re: rw [hre0] at h : −2m = w.re; h.symm : w.re = −2m; rw [hwr] : 2√n = −2m: nlinarith [hre' : 2√n = −2m, √n ≥ 0, m ≥ 0]: 2√n ≥ 0 and −2m ≤ 0: 2√n = −2m ⟹ both 0 ⟹ √n = 0 ∧ m = 0 — but m ≥ 1: contradiction. nlinarith should find: from hre': 2√n = −2m: −2m ≤ 0 (m ≥ 0) and 2√n ≥ 0 ⟹ √n = 0, m = 0: but m ≥ 1 > 0: contradiction ✓ — need hm1 in context ✓ (m ≥ 1 from obtain). nlinarith [hre', hpos, hm0] — with hm1 : 1 ≤ m in context ✓ works.\n\nNow — the nontrivial border cases (first rcases block): let me re-check those:\n- bottom (im = −T): the conj argument: `hTwide (conj s) (isNontrivialZero_conj hnt) (proof √n−1 < (conj s).im) (proof (conj s).im < √n+2)`: (conj s).im = −s.im = T: the bounds: √n−1 < T (from hTn : √n ≤ T — need √n−1 < T: linarith) and T < √n+2 (hTn1 : T ≤ √n+1: linarith ✓). My code:\n```\n have hconj := hTwide ((starRingEnd ℂ) s) (isNontrivialZero_conj hnt)\n (by rw [Complex.conj_im, him]; linarith [hTn])\n (by rw [Complex.conj_im, him]; linarith [hTn1])\n```\n✓ forms ok. Then:\n```\n have hpos : (0 : ℝ) < 1 / (9 * C0hat (Real.sqrt n + 2)) :=\n one_div_pos.mpr (by have := two_le_C0hat (by linarith [hn1sqrt] : (1 : ℝ) ≤ Real.sqrt n + 2); positivity)\n```\nHmm — `two_le_C0hat (ht : 1 ≤ t)` — the arg: (by linarith [hn1sqrt] : (1:ℝ) ≤ Real.sqrt n + 2)? hn1sqrt : 1 ≤ √n: √n + 2 ≥ 3 ≥ 1 ✓ linarith ✓. positivity for 0 < 9·C0hat: from two_le_C0hat : 2 ≤ C0hat: 9·C0hat ≥ 18 > 0 ✓ — but positivity can't use the `have :=` — the have is in the proof: `by have := two_le_C0hat ...; positivity` — positivity works on hypotheses ✓ (2 ≤ C0hat in context ⟹ 0 < 9·C0hat ✓ positivity should close... positivity with the hypothesis 2 ≤ C0hat(√n+2): it needs to derive 0 < 9·C0hat(√n+2): linarith would be safer: `one_div_pos.mpr (by have h := two_le_C0hat (by linarith [hn1sqrt] : (1:ℝ) ≤ Real.sqrt n + 2); linarith)`. Let me use linarith for safety.\n```\n have : |(starRingEnd ℂ) s - T| = 0 := by\n rw [Complex.conj_im, him, sub_self, abs_zero]\n```\nwait — |(conj s).im − T|: conj_im : ((starRingEnd ℂ) s).im = −s.im: rw [Complex.conj_im, him]: −(−T) − T = T − T: sub_self, abs_zero ✓ — but the expression is |(conj s).im − T|: after rw [Complex.conj_im]: |−s.im − T|; rw [him] (him : s.im = −T): |−(−T) − T| = |T − T|: sub_self → |0| → abs_zero → 0 ✓. Hmm — the goal is |(starRingEnd ℂ) s - T| — wait NO: the pigeonhole conclusion is |((starRingEnd ℂ) s).im − T| — the .im! hTwide's conclusion: 1/(9·C0hat) ≤ |ρ.im − T| with ρ := conj s: |(conj s).im − T| ≥ 1/(9C0). My `this` computes |(starRingEnd ℂ) s - T| — that's the norm of the COMPLEX difference, not the im-difference! Bug: should be |((starRingEnd ℂ) s).im − T|. Fix: `have hzero : |(starRingEnd ℂ) s).im - T| = 0 := by rw [Complex.conj_im, him, sub_self, abs_zero]`? sub_self: a − a = 0 — after rw [Complex.conj_im, him]: −(−T) − T = T − T ✓ sub_self → |0| → abs_zero ✓. Then linarith [hconj, hpos, hzero]: 1/(9C0) ≤ 0 < 1/(9C0) contradiction ✓.\n\nHmm wait — the goal of `this` is `|(starRingEnd ℂ) s - T| = 0` — with (starRingEnd ℂ) s being ℂ — the pigeonhole's hconj : 1/(9·C0hat(√n+2)) ≤ |((starRingEnd ℂ) s).im − T| — the .im is there. So I need hzero about |((starRingEnd ℂ) s).im − T|. Fix that.\n\n- top: same fix: `have : |s.im - T| = 0 := by rw [him, sub_self, abs_zero]` ✓ that one was right.\n\n- left (re = −5/2): `rw [hre'] at hnt; exact (by norm_num : ¬ ((0:ℝ) < -5/2 ∧ -5/2 < 1)) ⟨hnt.2.1, hnt.2.2⟩` — hnt : IsNontrivialZero s: hnt.2.1 : 0 < s.re; hnt.2.2 : s.re < 1: after rw [hre'] at hnt: 0 < −5/2 ∧ −5/2 < 1 — hre' : s.re = −5/2: rw at hnt rewrites s.re → −5/2 in hnt?? rw at a hypothesis rewrites occurrences of s.re — hnt.2.1's statement is 0 < s.re ✓ becomes 0 < −5/2 ✓. But can rw rewrite inside a structure (hnt.2.1)? rw [hre'] at hnt rewrites s.re everywhere in hnt's TYPE (IsNontrivialZero s = ζ s = 0 ∧ 0 < s.re ∧ s.re < 1) ✓ becomes ζ s = 0 ∧ 0 < −5/2 ∧ −5/2 < 1 ✓. Then ⟨hnt.2.1, hnt.2.2⟩ : 0 < −5/2 ∧ −5/2 < 1 ✓ contradiction by norm_num ✓.\n\nWait — hre' : s.re = -5/2 — from hb.1 : s.re ∈ {z.re}: Set.mem_singleton_iff.mp hb.1 : s.re = z.re: need rw [rectZ_re]: my code: `have hre' : s.re = -5 / 2 := by have := hb.1; rw [rectZ_re] at this; exact Set.mem_singleton_iff.mp this` — hb.1 : s.re ∈ {z.re}: rw [rectZ_re] at this: s.re ∈ {−5/2}: mp : s.re = −5/2 ✓. In my written code:\n```\n have hre' : s.re = -5 / 2 := by\n have := hb.1\n rw [rectZ_re] at this\n exact Set.mem_singleton_iff.mp this\n```\n✓ good.\n\n- right: `exact riemannZeta_ne_zero_of_one_le_re (by rw [hre']; linarith [Real.one_le_sqrt.mpr (by exact_mod_cast hn : (1:ℝ) ≤ n)]) hζ` — hre' : s.re = 2·√n: 1 ≤ s.re: rw [hre'] : 1 ≤ 2·√n: linarith with √n ≥ 1 ✓.\n\nAlso — the hre' in the right case: `have hre' : s.re = 2 * Real.sqrt n := by have := hb.1; rw [rectW_re] at this; exact Set.mem_singleton_iff.mp this` ✓.\n\nNow the backward hSz direction's hr for nontrivial: `⟨by linarith [hnt.2.1], by linarith [hnt.2.2, Real.one_le_sqrt.mpr ...]⟩` — s.re ∈ Icc [−5/2, 2√n]: −5/2 ≤ s.re (from s.re > 0 ✓) and s.re ≤ 2√n (from s.re < 1 and 1 ≤ √n ⟹ 2√n ≥ 2 > 1 > s.re... s.re < 1 ≤ √n ≤ 2√n ✓ linarith ✓).\n\nThe `hi` for nontrivial: `exact abs_le.mp himT` — himT : |s.im| ≤ T: abs_le.mp : −T ≤ s.im ∧ s.im ≤ T ✓ matches Icc [−T, T] ✓.\n\nThe h1mem/h1notb: h2n : 1 < 2·√n: from √n ≥ 1: 2√n ≥ 2 > 1 ✓ linarith [Real.one_le_sqrt.mpr (by exact_mod_cast hn : (1:ℝ) ≤ n)] — Real.one_le_sqrt.mpr needs (1:ℝ) ≤ n ✓ exact_mod_cast hn : (1:ℝ) ≤ (n:ℝ)?? hn : 1 ≤ n (ℕ): exact_mod_cast hn : (1:ℝ) ≤ (n:ℝ) ✓.\n\nHmm — in h1notb's first case: `exact absurd (Set.mem_singleton_iff.mp hb.2) (by rw [rectZ_im] ; exact ne_of_gt (by linarith : (0 : ℝ) < T))` — hb.2 : (1:ℂ).im ∈ {z.im}: mp : im = z.im: the by-block's goal: ¬((1:ℂ).im = z.im): rw [rectZ_im]: ¬(im = −T): (1:ℂ).im = 0: the rw [rectZ_im] rewrites z.im → −T ✓; then ne_of_gt (0 < T) : 0 ≠ T — but the equation is im = −T with im = 0: 0 = −T?? — ne_of_gt gives 0 ≠ T; the goal is (1:ℂ).im ≠ −T — after rectZ_im: (1:ℂ).im ≠ −T: is (1:ℂ).im displayed as 0? Complex.one_im : (1:ℂ).im = 0 — simp? The goal after rw [rectZ_im]: ¬ ((1:ℂ).im = -T): exact needs (1:ℂ).im ≠ −T: (1:ℂ).im = 0 definitionally? Complex.one_im is rfl? (1:ℂ).im — 1 = One.toOfNat1 — its im: OfNat.ofNat 1 .im — computes to 0 definitionally? In practice `(1 : ℂ).im` reduces to 0 by `Complex.one_re/im` simp lemmas, but definitionally OfNat's im... `Complex.ofNat_im`? Hmm — safer: rw [rectZ_im, Complex.one_im]: goal 0 ≠ −T: `exact ne_of_gt (by linarith [hT1] : (0:ℝ) > -T)`. My code lacks the Complex.one_im rewrite and has the wrong sign (0 < T giving 0 ≠ T vs needed 0 ≠ −T). Fix both places (bottom: 0 ≠ −T; top: 0 ≠ T ✓ was right).\n\nLet me fix these remaining issues.","signature":""},{"type":"text","text":"Fixing the remaining sign/im issues in the identity proof."},{"type":"tool_use","id":"Edit_268","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiAssembly.lean","old_string":" rcases hb with hb | hb | hb | hb\n · exact absurd (Set.mem_singleton_iff.mp hb.2) (by rw [rectZ_im] ; exact ne_of_gt (by linarith : (0 : ℝ) < T))\n · exact absurd (Set.mem_singleton_iff.mp hb.1) (by rw [rectZ_re]; norm_num)\n · exact absurd (Set.mem_singleton_iff.mp hb.2) (by rw [rectW_im]; exact ne_of_gt (by linarith : (0 : ℝ) < T))\n · exact absurd (Set.mem_singleton_iff.mp hb.1) (by rw [rectW_re]; exact ne_of_gt h2n)","new_string":" rcases hb with hb | hb | hb | hb\n · have h := Set.mem_singleton_iff.mp hb.2\n rw [rectZ_im, Complex.one_im] at h\n exact absurd h (ne_of_gt (by linarith [hT1] : (0 : ℝ) > -T))\n · have h := Set.mem_singleton_iff.mp hb.1\n rw [rectZ_re, Complex.one_re] at h\n norm_num at h\n · have h := Set.mem_singleton_iff.mp hb.2\n rw [rectW_im, Complex.one_im] at h\n exact absurd h (ne_of_gt (by linarith [hT1] : (0 : ℝ) < T))\n · have h := Set.mem_singleton_iff.mp hb.1\n rw [rectW_re, Complex.one_re] at h\n exact absurd h (ne_of_gt h2n)"},"extra_content":null}]