← Session 24 — ii′-2…4: swaps, (4.10) design, contour + website roadmap fixes
thinking (1735 chars)
`mul_div_cancel_left₀ (b : M₀) {a : M₀} (ha : a ≠ 0) : a * b / a = b` — so `mul_div_cancel_left₀ (riemannZeta s) hΓR : Gammaℝ s * riemannZeta s / Gammaℝ s = riemannZeta s` — the implicit a is the DENOMINATOR. So for hz: goal `riemannZeta s = ΓR s * ζ s / ΓR s` — (mul_div_cancel_left₀ (riemannZeta s) hΓR).symm ✓... wait — hz goal after rw [hΛ]: `riemannZeta s = Gammaℝ s * riemannZeta s / Gammaℝ s` and mul_div_cancel_left₀ gives `ΓR·ζ/ΓR = ζ` — so exact `(mul_div_cancel_left₀ _ hΓR).symm` ✓ — but the implicit {a} must unify a := Gammaℝ s ✓ from hΓR's type. My code: `rw [hΛ, mul_div_cancel_left₀ _ hΓR]` — rw with the equation ΓR·ζ/ΓR = ζ: rewrites LHS pattern in goal: goal `ζ s = ΓR s * ζ s / ΓR s` — rw [mul_div_cancel_left₀ _ hΓR] rewrites `ΓR s * ζ s / ΓR s` → `ζ s`: goal ζ s = ζ s ✓ closed by rfl ✓. But `_` for b: mul_div_cancel_left₀ (b) {a} (ha : a ≠ 0) — explicit b then implicit a then ha: `mul_div_cancel_left₀ _ hΓR` — _ = b (the ζ value), hΓR : Gammaℝ s ≠ 0 — a implicit from ha ✓. OK ✓.
For hG: goal Γ(s/2) = ((s/2)(s/2+1)·Γ(s/2)) / ((s/2)(s/2+1)): `(mul_div_cancel_left₀ (Complex.Gamma (s/2)) (mul_ne_zero h2ne h21ne)).symm` ✓.
Now let me apply all fixes:
Fix A (ball typing):
```lean
have hball : (-2 : ℂ) ∈ Metric.ball (-2 : ℂ) (1 : ℝ) :=
Metric.mem_ball_self (zero_lt_one : (0:ℝ) < 1)
exact (hd.analyticOnNhd Metric.isOpen_ball) _ hball
```
and make sure the DifferentiableOn ball is ascribed: `Metric.ball ((-2 : ℂ)) (1 : ℝ)`.
Fix B (hcpi): use congr approach.
Fix C (h1 sign): the negated form + norm_neg.
Fix D (comp): show-ascribe the inner function.
Fix E (Hfun_neg_two): field_simp; ring.
Fix F (hG): mul_div_cancel_left₀ route.
Also check `Complex.cpow_def_of_ne_zero` name: grep.
Applying the fix set. First check `cpow_def_of_ne_zero` exact name.
tool: Bash
input
{
"command": "cd D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"theorem cpow_def_of_ne_zero\\|lemma cpow_def_of_ne_zero\" Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | head -2",
"description": "Verify cpow_def_of_ne_zero"
}
raw content_json (database row)
[{"type":"thinking","thinking":"`mul_div_cancel_left₀ (b : M₀) {a : M₀} (ha : a ≠ 0) : a * b / a = b` — so `mul_div_cancel_left₀ (riemannZeta s) hΓR : Gammaℝ s * riemannZeta s / Gammaℝ s = riemannZeta s` — the implicit a is the DENOMINATOR. So for hz: goal `riemannZeta s = ΓR s * ζ s / ΓR s` — (mul_div_cancel_left₀ (riemannZeta s) hΓR).symm ✓... wait — hz goal after rw [hΛ]: `riemannZeta s = Gammaℝ s * riemannZeta s / Gammaℝ s` and mul_div_cancel_left₀ gives `ΓR·ζ/ΓR = ζ` — so exact `(mul_div_cancel_left₀ _ hΓR).symm` ✓ — but the implicit {a} must unify a := Gammaℝ s ✓ from hΓR's type. My code: `rw [hΛ, mul_div_cancel_left₀ _ hΓR]` — rw with the equation ΓR·ζ/ΓR = ζ: rewrites LHS pattern in goal: goal `ζ s = ΓR s * ζ s / ΓR s` — rw [mul_div_cancel_left₀ _ hΓR] rewrites `ΓR s * ζ s / ΓR s` → `ζ s`: goal ζ s = ζ s ✓ closed by rfl ✓. But `_` for b: mul_div_cancel_left₀ (b) {a} (ha : a ≠ 0) — explicit b then implicit a then ha: `mul_div_cancel_left₀ _ hΓR` — _ = b (the ζ value), hΓR : Gammaℝ s ≠ 0 — a implicit from ha ✓. OK ✓.\n\nFor hG: goal Γ(s/2) = ((s/2)(s/2+1)·Γ(s/2)) / ((s/2)(s/2+1)): `(mul_div_cancel_left₀ (Complex.Gamma (s/2)) (mul_ne_zero h2ne h21ne)).symm` ✓.\n\nNow let me apply all fixes:\n\nFix A (ball typing): \n```lean\n have hball : (-2 : ℂ) ∈ Metric.ball (-2 : ℂ) (1 : ℝ) :=\n Metric.mem_ball_self (zero_lt_one : (0:ℝ) < 1)\n exact (hd.analyticOnNhd Metric.isOpen_ball) _ hball\n```\nand make sure the DifferentiableOn ball is ascribed: `Metric.ball ((-2 : ℂ)) (1 : ℝ)`.\n\nFix B (hcpi): use congr approach.\n\nFix C (h1 sign): the negated form + norm_neg.\n\nFix D (comp): show-ascribe the inner function.\n\nFix E (Hfun_neg_two): field_simp; ring.\n\nFix F (hG): mul_div_cancel_left₀ route.\n\nAlso check `Complex.cpow_def_of_ne_zero` name: grep.","signature":""},{"type":"text","text":"Applying the fix set. First check `cpow_def_of_ne_zero` exact name."},{"type":"tool_use","id":"Bash_113","name":"Bash","input":{"command":"cd D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"theorem cpow_def_of_ne_zero\\|lemma cpow_def_of_ne_zero\" Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | head -2","description":"Verify cpow_def_of_ne_zero"},"extra_content":null}]