← Session 35 — ii′-6-R: Lagarias 6.1 right-tail re-proof (E-032/D-073)
thinking (1080 chars)
The bridge pattern (LiTightR:175–197) is exactly what I need to extract as a standalone lemma:
```
∑ ρ ∈ nearZerosSub t, (zeroMult ρ.1 : ℝ) ≤ Z2hat t
```
via: sum_image (Subtype.ext) + subset into (zeroAbsWindow_finite t).toFinset + sum_le_sum_of_subset_of_nonneg + hZC2 t ht. And Z2hat t is DEFINITIONALLY the closed form (0.6·log(t+2) + 0.886·loglog + 2.7) — `rfl` on the last step? The last two steps: `_ = (((zeroAbsWindow_finite t).toFinset.sum fun ρ => zeroMult ρ) : ℝ) := rfl` then `≤ [closed form] := hZC2 t ht`. So the standalone lemma:
```lean
theorem nearZerosSub_mult_sum_le (hZC2 : ZeroCountProp2) {t : ℝ} (ht : 10 ≤ t) :
∑ ρ ∈ nearZerosSub t, (zeroMult ρ.1 : ℝ) ≤ Z2hat t := ...
```
with `Z2hat t` unfolding to the closed form (Z2hat is a def; the hZC2 conclusion is the closed form — need `show` or `unfold Z2hat` / `rw [Z2hat]`).
Now check Mathlib names: Real.arcsinh, hasDerivAt_arcsinh, integral_comp_neg, integral_eq_sub_of_hasDerivAt, integral_finset_sum, norm_integral_le_integral_norm, integral_add_adjacent_intervals, ContinuousOn.intervalIntegrable.
The bridge pattern is exactly clonable (LiTightR:175–197). Checking the Mathlib API names I'll need for the scaffolding.
tool: Bash
input
{
"command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && cat > Scratch.lean <<'EOF'\nimport Mathlib\n\nopen intervalIntegral in\nexample : True := by\n -- probe names\n have h1 := @intervalIntegral.integral_comp_neg\n have h2 := @intervalIntegral.integral_eq_sub_of_hasDerivAt\n have h3 := @intervalIntegral.integral_finset_sum\n have h4 := @intervalIntegral.norm_integral_le_integral_norm\n have h5 := @intervalIntegral.integral_add_adjacent_intervals\n have h6 := @ContinuousOn.intervalIntegrable\n have h7 := @Real.hasDerivAt_arcsinh\n have h8 := @intervalIntegral.integral_mono_on\n have h9 := @intervalIntegral.integral_add\n trivial\n\n#print Real.arcsinh\n#print Real.hasDerivAt_arcsinh\n#check @intervalIntegral.integral_comp_neg\n#check @intervalIntegral.integral_eq_sub_of_hasDerivAt\n#check @intervalIntegral.integral_finset_sum\n#check @intervalIntegral.norm_integral_le_integral_norm\n#check @intervalIntegral.integral_add_adjacent_intervals\n#check @ContinuousOn.intervalIntegrable\n#check @intervalIntegral.integral_mono_on\n#check @intervalIntegral.integral_add\nEOF\n~/.elan/bin/lake env lean Scratch.lean 2>&1 | head -60",
"description": "Probe Mathlib interval-integral and arcsinh API signatures",
"timeout": 240000
}
raw content_json (database row)
[{"type":"thinking","thinking":"The bridge pattern (LiTightR:175–197) is exactly what I need to extract as a standalone lemma:\n\n```\n∑ ρ ∈ nearZerosSub t, (zeroMult ρ.1 : ℝ) ≤ Z2hat t\n```\nvia: sum_image (Subtype.ext) + subset into (zeroAbsWindow_finite t).toFinset + sum_le_sum_of_subset_of_nonneg + hZC2 t ht. And Z2hat t is DEFINITIONALLY the closed form (0.6·log(t+2) + 0.886·loglog + 2.7) — `rfl` on the last step? The last two steps: `_ = (((zeroAbsWindow_finite t).toFinset.sum fun ρ => zeroMult ρ) : ℝ) := rfl` then `≤ [closed form] := hZC2 t ht`. So the standalone lemma:\n\n```lean\ntheorem nearZerosSub_mult_sum_le (hZC2 : ZeroCountProp2) {t : ℝ} (ht : 10 ≤ t) :\n ∑ ρ ∈ nearZerosSub t, (zeroMult ρ.1 : ℝ) ≤ Z2hat t := ...\n```\nwith `Z2hat t` unfolding to the closed form (Z2hat is a def; the hZC2 conclusion is the closed form — need `show` or `unfold Z2hat` / `rw [Z2hat]`).\n\nNow check Mathlib names: Real.arcsinh, hasDerivAt_arcsinh, integral_comp_neg, integral_eq_sub_of_hasDerivAt, integral_finset_sum, norm_integral_le_integral_norm, integral_add_adjacent_intervals, ContinuousOn.intervalIntegrable.","signature":""},{"type":"text","text":"The bridge pattern is exactly clonable (LiTightR:175–197). Checking the Mathlib API names I'll need for the scaffolding."},{"type":"tool_use","id":"Bash_41","name":"Bash","input":{"command":"cd /d/Python/Quantum/research/lean/NguyenRH1 && cat > Scratch.lean <<'EOF'\nimport Mathlib\n\nopen intervalIntegral in\nexample : True := by\n -- probe names\n have h1 := @intervalIntegral.integral_comp_neg\n have h2 := @intervalIntegral.integral_eq_sub_of_hasDerivAt\n have h3 := @intervalIntegral.integral_finset_sum\n have h4 := @intervalIntegral.norm_integral_le_integral_norm\n have h5 := @intervalIntegral.integral_add_adjacent_intervals\n have h6 := @ContinuousOn.intervalIntegrable\n have h7 := @Real.hasDerivAt_arcsinh\n have h8 := @intervalIntegral.integral_mono_on\n have h9 := @intervalIntegral.integral_add\n trivial\n\n#print Real.arcsinh\n#print Real.hasDerivAt_arcsinh\n#check @intervalIntegral.integral_comp_neg\n#check @intervalIntegral.integral_eq_sub_of_hasDerivAt\n#check @intervalIntegral.integral_finset_sum\n#check @intervalIntegral.norm_integral_le_integral_norm\n#check @intervalIntegral.integral_add_adjacent_intervals\n#check @ContinuousOn.intervalIntegrable\n#check @intervalIntegral.integral_mono_on\n#check @intervalIntegral.integral_add\nEOF\n~/.elan/bin/lake env lean Scratch.lean 2>&1 | head -60","description":"Probe Mathlib interval-integral and arcsinh API signatures","timeout":240000},"extra_content":null}]