← Session 31 — ii′-5f: tight R-bound 10.5/48 (piece 3 + pinch)
thinking (7163 chars)
`hasDerivAt_iff_tendsto_slope : HasDerivAt f f' x ↔ Tendsto (slope f x) (𝓝[≠] x) (𝓝 f')` — with slope f x y = (y−x)⁻¹ • (f y − f x). For ℂ→ℂ: slope f z x = (x−z)⁻¹ • (f x − f z) = (f x − f z)/(x−z) — smul = division form. So my functions should use slope or the inv-mul form. Simplest: restate my slope functions with slope: (fun x => slope f z x) and (fun x => slope f (conj z) x). slope f z x = (x − z)⁻¹ • (f x − f z) — the smul on ℂ: (x−z)⁻¹ • (f x − f z) = (x−z)⁻¹ * (f x − f z) = (f x − f z)/(x−z) ✓ — defeq-ish? For ℂ the • is * (Module with smul = mul). Let me just use slope directly:
h1 : Tendsto (slope f z) (𝓝[≠] z) (𝓝 (deriv f z)) := hasDerivAt_iff_tendsto_slope.mp hdiff.hasDerivAt
h2's function: conj(slope f z (conj x)) = conj((f(conj x) − f z)/((conj x) − z)) — as a slope: slope f z (conj x) = ((conj x) − z)⁻¹ • (f(conj x) − f z) — the conj of that: conj(((conj x) − z)⁻¹ • (f(conj x) − f z)) — conj_smul: conj(c • w) = conj c • conj w — for the RingHom: map_smul? starRingEnd is a ring hom, map_smul needs module structure — for ℂ self: conj(a·b) = conj a·conj b (map_mul) ✓ the • is just mul.
The EventuallyEq target: slope f (conj z) x = conj(slope f z (conj x)):
slope f (conj z) x = (x − conj z)⁻¹ • (f x − f (conj z)); conj(slope f z (conj x)) = conj(((conj x) − z)⁻¹ • (f(conj x) − f z)).
With hconj: f(conj x) = conj(f x) and hzid: f(conj z) = conj(f z):
conj(slope f z (conj x)) = conj(((conj x − z)⁻¹) • (conj(f x) − f z)) = conj((conj(x − conj z))⁻¹ • conj(f x − conj(f z))) [conj_sub, starRingEnd_self_apply] = (x − conj z)⁻¹ • (f x − conj(f z)) [conj_inv₀, conj_mul...] — messy with smul/mul bridges. Let me avoid slope and just use the explicit quotient form (f x − f z)/(x − z) and prove the tendsto characterization myself — the conversion from the Frechet form to the quotient form: there's `HasDerivAt.tendsto_slope`? Or use `hasDerivAt_iff_tendsto_slope` and bridge slope = quotient: slope f z x = (f x − f z)/(x − z) — `slope_def`? slope f x y = (y−x)⁻¹ • (f y − f x) by def. For ℂ: (y−x)⁻¹ • (f y − f x) = (y−x)⁻¹ * (f y − f x) = (f y − f x)/(y−x) — via smul_eq_mul + div_eq_mul_inv:
```
have hslope : slope f z = fun x : ℂ => (f x - f z) / (x - z) := by
funext x
rw [slope, smul_eq_mul, div_eq_mul_inv]
```
slope's def: `slope (f : 𝕜 → E) (x : 𝕜) : 𝕜 → E := fun y => (y - x)⁻¹ • (f y - f x)` — rw [slope] unfolds? `slope_def`? Let me just: funext x; show ((x − z)⁻¹ • (f x − f z)) = (f x − f z)/(x − z); rw [smul_eq_mul, div_eq_mul_inv]. The `show` needs the slope to unfold — `slope f z x` — delta-unfolds? slope is a def — `show` with the unfolded form should be defeq ✓.
So:
```
have h1 : Filter.Tendsto (fun x : ℂ => (f x - f z) / (x - z)) (𝓝[≠] z)
(𝓝 (deriv f z)) := by
have h := hasDerivAt_iff_tendsto_slope.mp hdiff.hasDerivAt
have hs : (fun x : ℂ => (f x - f z) / (x - z)) = slope f z := by
funext x
show (f x - f z) / (x - z) = (x - z)⁻¹ • (f x - f z)
rw [div_eq_mul_inv, smul_eq_mul]
rw [hs]
exact h
```
Hmm — the show's RHS (x−z)⁻¹ • (f x − f z) — is that slope f z x definitionally? slope f z x = (x − z)⁻¹ • (f x − f z)? slope f x = fun y => (y − x)⁻¹ • (f y − f x): at y = x: (x − z)⁻¹ • (f x − f z) ✓ should be defeq via rfl of the def application. If show fails, use `rw [slope_def]`? or `simp only [slope, ...]`. I'll write `rw [slope, smul_eq_mul, div_eq_mul_inv]` inside funext — `slope` as an equation-lemma: slope f x y = (y−x)⁻¹ • (f y − f x) — there's `slope_def`? Let me hedge: `simp only [slope, smul_eq_mul, div_eq_mul_inv]`? slope might not be simp-unfoldable by name. Try `funext x; rw [slope]; rw [smul_eq_mul, div_eq_mul_inv]` — rw [slope] unfolds the def if slope is an equation? slope is a def: `def slope (f : 𝕜 → E) (x : 𝕜) : 𝕜 → E := fun y => (y - x)⁻¹ • (f y - f x)` — rw [slope] unfolds defs by their equation ✓ usually works.
Similarly the h2/EQ functions with the quotient form — no slope needed there (only h1 uses the slope-characterization; h2's composition and the final HasDerivAt use hasDerivAt_iff_tendsto_slope.mpr with the quotient→slope bridge again).
The final: hasDerivAt_iff_tendsto_slope.mpr — needs Tendsto (slope f (conj z)) (𝓝[≠] (conj z)) (𝓝 (conj (deriv f z))): my h2 gives the quotient form at conj z; bridge: slope f (conj z) = (fun x => (f x − f(conj z))/(x − conj z)) via the same funext; the EventuallyEq h3 connects the quotient forms ✓.
Also the nhdsWithin lemma names: `nhdsWithin_le_nhds` is root-namespace (from the Calculus usages ✓ no Filter prefix).
And `Filter.tendsto_nhdsWithin_iff` — didn't find; use: Tendsto f (𝓝[s] a) (𝓝 b) = Tendsto f (𝓝 a) (𝓝 b) ⊓ principal... The punctured case: 𝓝[≠] a = 𝓝 a ⊓ principal {a}ᶜ: `tendsto_nhdsWithin_of_tendsto`? — there's `ContinuousAt.tendsto_nhdsWithin`? or build via `Filter.Tendsto.inf`:
hmap : Tendsto conj (𝓝[≠] (conj z)) (𝓝[≠] z): 𝓝[≠] x = 𝓝[{x}ᶜ] x: Tendsto conj (𝓝[{conj z}ᶜ] (conj z)) (𝓝[{z}ᶜ] z) — via `tendsto_nhdsWithin_of_tendsto_nhds_of_eventually`? Hmm — simplest:
```
have hmap : Filter.Tendsto (starRingEnd ℂ) (𝓝[≠] ((starRingEnd ℂ) z)) (𝓝[≠] z) := by
have hcont : Continuous ((starRingEnd ℂ) : ℂ → ℂ) := Complex.continuous_conj
apply (hcont.tendsto ((starRingEnd ℂ) z)).inf
rw [Filter.tendsto_principal_principal]
exact fun x hx => by
show (starRingEnd ℂ) x ∈ {z}ᶜ
rw [Set.mem_compl_singleton_iff] at hx ⊢
intro hbad
apply hx
have h3 := congrArg (starRingEnd ℂ) hbad
rwa [starRingEnd_self_apply] at h3
```
`Filter.Tendsto.inf (h : Tendsto f l (𝓝 a)) : Tendsto f l (principal s)`?? Tendsto.inf : Tendsto f l (𝓝 a) → Tendsto f l (𝓟 s) → Tendsto f l (𝓝 a ⊓ 𝓟 s) — and 𝓝[≠] z = 𝓝 z ⊓ 𝓟 {z}ᶜ — the nhdsWithin def: 𝓝[s] a = 𝓝 a ⊓ 𝓟 s ✓ and 𝓝[≠] a = 𝓝[{a}ᶜ] a ✓. Tendsto f l₁ (l₂ ⊓ l₃) from components: `tendsto_inf`? — `Filter.Tendsto.inf (h₁ : Tendsto f l₁ l₂) (h₂ : Tendsto f l₁ l₃) : Tendsto f l₁ (l₂ ⊓ l₃)` ✓ exists. And Tendsto f l (𝓟 s) ⟺ ∀ x ∈ l-ish, f x ∈ s eventually: `tendsto_principal_principal : Tendsto f (𝓟 s) (𝓟 t) ↔ ∀ x ∈ s, f x ∈ t` ✓. The nhdsWithin destructure: 𝓝[≠] a is notation for 𝓝[{a}ᶜ] a — and 𝓝[s] a = 𝓝 a ⊓ 𝓟 s via `nhdsWithin` def — the Tendsto target 𝓝[≠] z: rewrite via `nhdsWithin` def: `show Filter.Tendsto _ _ (𝓝 z ⊓ 𝓟 {z}ᶜ)`? The notation 𝓝[≠] z — there's `nhdsWithin` and the def `nhdsWithin a s = 𝓝 a ⊓ 𝓟 s` — unfold via `nhdsWithin` — `rw [nhdsWithin]`? The def: `def nhdsWithin (a : α) (s : Set α) : Filter α := 𝓝 a ⊓ 𝓟 s` — rw [nhdsWithin] should unfold.
Hmm — also `Continuous.tendsto (x) : Tendsto f (𝓝 x) (𝓝 (f x))` ✓.
Let me also fix `tendsto_nhdsWithin_iff` — replace with the inf approach above.
Now the Filter.EventuallyEq.filter_mono: signature `EventuallyEq.filter_mono (hl : f =ᶠ[l₁] g) (h : l₂ ≤ l₁) : f =ᶠ[l₂] g`? — from the grep the name exists (EventuallyEq.filter_mono). Use h4.filter_mono nhdsWithin_le_nhds with nhdsWithin_le_nhds root ✓.
Also `isOpen_compl_singleton.eventually_nhds` — signature: `IsOpen.eventually_nhds (hs : IsOpen s) (hx : x ∈ s) : ∀ᶠ y in 𝓝 x, y ∈ s` ✓ — isOpen_compl_singleton : IsOpen {a}ᶜ ✓ and membership conj z ∈ {1}ᶜ = conj z ≠ 1: Set.mem_compl_singleton_iff.mpr h1 ✓.
Let me apply all these fixes.
Rewriting the tendsto parts of §14 (slope characterization, nhdsWithin via inf, mono structure) plus the §13 fixes.
tool: Write ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\patch_s1314.py",
"content": "p = 'NguyenRH1/LiTightR.lean'\nsrc = open(p, encoding='utf-8').read()\n\n# log_add_le: robust nonnegs + the 1+c/a-1 bridge\nold = \"\"\"/-- log(a + c) \u2264 log a + c/a (the log-increment machine). -/\nlemma log_add_le (a c : \u211d) (ha : 0 < a) (hc : 0 \u2264 c) :\n Real.log (a + c) \u2264 Real.log a + c / a := by\n have h1 : a + c = a * (1 + c / a) := by field_simp [ha.ne']; ring\n rw [h1, Real.log_mul ha.ne' (by positivity : (1 : \u211d) + c / a \u2260 0)]\n have h2 : Real.log (1 + c / a) \u2264 c / a :=\n Real.log_le_sub_one_of_pos (by positivity : (0 : \u211d) < 1 + c / a)\n linarith [h2]\"\"\"\nnew = \"\"\"/-- log(a + c) \u2264 log a + c/a (the log-increment machine). -/\nlemma log_add_le (a c : \u211d) (ha : 0 < a) (hc : 0 \u2264 c) :\n Real.log (a + c) \u2264 Real.log a + c / a := by\n have hca : (0 : \u211d) \u2264 c / a := div_nonneg hc ha.le\n have h1ca : (0 : \u211d) < 1 + c / a := by linarith\n have h1 : a + c = a * (1 + c / a) := by field_simp [ha.ne']; ring\n rw [h1, Real.log_mul ha.ne' h1ca.ne']\n have h2 : Real.log (1 + c / a) \u2264 c / a := by\n have h3 := Real.log_le_sub_one_of_pos (x := 1 + c / a) h1ca\n rwa [show (1 : \u211d) + c / a - 1 = c / a from by ring] at h3\n linarith [h2]\"\"\"\nassert old in src\nsrc = src.replace(old, new)\n\n# C0hat_two_mul_add_one_le: log_add_le call-site nonneg args fine (log_add_le fixed)\n# div_le_div unknown -> div_le_div_iff\u2080 bridge in C0hat_two_t_add_one_le\nold = \"\"\" have hlog2le : Real.log 2 \u2264 0.6932 := Real.log_two_lt_d9.le.trans (by norm_num)\n have h10 : Real.log 2 / Real.log (t + 3) \u2264 0.6932 / 2.5 := by\n apply div_le_div hlog2le hlog13 (by norm_num : (0 : \u211d) \u2264 2.5) hlogpos\n linarith [h6, h7, h10]\"\"\"\nnew = \"\"\" have hlog2le : Real.log 2 \u2264 0.6932 := Real.log_two_lt_d9.le.trans (by norm_num)\n have h10 : Real.log 2 / Real.log (t + 3) \u2264 0.6932 / 2.5 := by\n rw [div_le_div_iff\u2080 hlogpos (by norm_num : (0 : \u211d) < 2.5)]\n have h11 : (0.6932 : \u211d) * 2.5 \u2264 0.6932 * Real.log (t + 3) :=\n mul_le_mul_of_nonneg_left hlog13 (by norm_num)\n have h12 : Real.log 2 * 2.5 \u2264 0.6932 * 2.5 :=\n mul_le_mul_of_nonneg_right hlog2le (by norm_num)\n linarith [h11, h12]\n linarith [h6, h7, h10]\"\"\"\nassert old in src\nsrc = src.replace(old, new)\n\n# hasDerivAt_conj_of_conj_eq: full rewrite of the tendsto parts\nold = \"\"\" have h1 : Filter.Tendsto (fun x : \u2102 => (f x - f z) / (x - z)) (\ud835\udcdd[\u2260] z)\n (\ud835\udcdd (deriv f z)) :=\n hasDerivAt_iff_tendsto.mp hdiff.hasDerivAt\n have hmap : Filter.Tendsto (starRingEnd \u2102) (\ud835\udcdd[\u2260] ((starRingEnd \u2102) z)) (\ud835\udcdd[\u2260] z) := by\n have hcont : Continuous ((starRingEnd \u2102) : \u2102 \u2192 \u2102) := Complex.continuous_conj\n have hT := hcont.tendsto ((starRingEnd \u2102) z)\n rw [Filter.tendsto_nhdsWithin_iff]\n refine \u27e8hT.mono_left Filter.nhdsWithin_le_nhds, ?_\u27e9\n rw [Filter.eventually_nhdsWithin_iff]\n exact Filter.Eventually.of_forall (fun x (hx : x \u2260 (starRingEnd \u2102) z) => by\n show (starRingEnd \u2102) x \u2260 z\n intro hbad\n apply hx\n have h3 := congrArg (starRingEnd \u2102) hbad\n rwa [starRingEnd_self_apply] at h3)\n have h2 : Filter.Tendsto (fun x : \u2102 =>\n (starRingEnd \u2102) ((f ((starRingEnd \u2102) x) - f z) / (((starRingEnd \u2102) x) - z)))\n (\ud835\udcdd[\u2260] ((starRingEnd \u2102) z)) (\ud835\udcdd ((starRingEnd \u2102) (deriv f z))) := by\n have hcont : Continuous ((starRingEnd \u2102) : \u2102 \u2192 \u2102) := Complex.continuous_conj\n exact (hcont.tendsto _).comp (h1.comp hmap)\n have h3 : (fun x : \u2102 => (f x - f ((starRingEnd \u2102) z)) / (x - (starRingEnd \u2102) z))\n =\u1da0[\ud835\udcdd[\u2260] ((starRingEnd \u2102) z)] fun x : \u2102 =>\n (starRingEnd \u2102) ((f ((starRingEnd \u2102) x) - f z) / (((starRingEnd \u2102) x) - z)) := by\n apply Filter.EventuallyEq.filter_mono ?_ Filter.nhdsWithin_le_nhds\n exact hconj.mono (fun s hs => by\n rw [hs, hzid]\n simp only [map_sub, map_div\u2080, starRingEnd_self_apply])\n exact hasDerivAt_iff_tendsto.mpr (Filter.Tendsto.congr' h3 h2)\"\"\"\nnew = \"\"\" have hslope : (fun x : \u2102 => (f x - f z) / (x - z)) = slope f z := by\n funext x\n rw [slope, smul_eq_mul, div_eq_mul_inv]\n have h1 : Filter.Tendsto (fun x : \u2102 => (f x - f z) / (x - z)) (\ud835\udcdd[\u2260] z)\n (\ud835\udcdd (deriv f z)) := by\n rw [hslope]\n exact hasDerivAt_iff_tendsto_slope.mp hdiff.hasDerivAt\n have hmap : Filter.Tendsto (starRingEnd \u2102) (\ud835\udcdd[\u2260] ((starRingEnd \u2102) z)) (\ud835\udcdd[\u2260] z) := by\n have hcont : Continuous ((starRingEnd \u2102) : \u2102 \u2192 \u2102) := Complex.continuous_conj\n have hT := hcont.tendsto ((starRingEnd \u2102) z)\n have hp : Filter.Tendsto (starRingEnd \u2102) (\ud835\udcdd[\u2260] ((starRingEnd \u2102) z)) (Filter.principal {z}\u1d9c) := by\n rw [Filter.tendsto_principal_principal]\n intro x hx\n rw [Set.mem_compl_singleton_iff] at hx\n rw [Set.mem_compl_singleton_iff]\n intro hbad\n apply hx\n have h3 := congrArg (starRingEnd \u2102) hbad\n rwa [starRingEnd_self_apply] at h3\n have h2 := hT.inf hp\n have hunfold : (\ud835\udcdd[\u2260] z) = \ud835\udcdd z \u2293 Filter.principal {z}\u1d9c := rfl\n rw [hunfold]\n exact h2\n have h2 : Filter.Tendsto (fun x : \u2102 =>\n (starRingEnd \u2102) ((f ((starRingEnd \u2102) x) - f z) / (((starRingEnd \u2102) x) - z)))\n (\ud835\udcdd[\u2260] ((starRingEnd \u2102) z)) (\ud835\udcdd ((starRingEnd \u2102) (deriv f z))) := by\n have hcont : Continuous ((starRingEnd \u2102) : \u2102 \u2192 \u2102) := Complex.continuous_conj\n exact (hcont.tendsto _).comp (h1.comp hmap)\n have h4 : (fun x : \u2102 => (f x - f ((starRingEnd \u2102) z)) / (x - (starRingEnd \u2102) z))\n =\u1da0[\ud835\udcdd ((starRingEnd \u2102) z)] fun x : \u2102 =>\n (starRingEnd \u2102) ((f ((starRingEnd \u2102) x) - f z) / (((starRingEnd \u2102) x) - z)) := by\n exact hconj.mono (fun s hs => by\n rw [hs, hzid]\n simp only [map_sub, map_div\u2080, starRingEnd_self_apply])\n have h3 : (fun x : \u2102 => (f x - f ((starRingEnd \u2102) z)) / (x - (starRingEnd \u2102) z))\n =\u1da0[\ud835\udcdd[\u2260] ((starRingEnd \u2102) z)] fun x : \u2102 =>\n (starRingEnd \u2102) ((f ((starRingEnd \u2102) x) - f z) / (((starRingEnd \u2102) x) - z)) :=\n h4.filter_mono nhdsWithin_le_nhds\n have hslope2 : (fun x : \u2102 => (f x - f ((starRingEnd \u2102) z)) / (x - (starRingEnd \u2102) z))\n = slope f ((starRingEnd \u2102) z) := by\n funext x\n rw [slope, smul_eq_mul, div_eq_mul_inv]\n rw [hslope2] at h3\n exact hasDerivAt_iff_tendsto_slope.mpr (Filter.Tendsto.congr' h3 h2)\"\"\"\nassert old in src\nsrc = src.replace(old, new)\n\nopen(p, 'w', encoding='utf-8').write(src)\nprint(\"done\")\n"
}
result 2026-08-24 18:37:19Z → 2026-08-24 18:37:19Z
{
"type": "create",
"filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\patch_s1314.py",
"content": "p = 'NguyenRH1/LiTightR.lean'\nsrc = open(p, encoding='utf-8').read()\n\n# log_add_le: robust nonnegs + the 1+c/a-1 bridge\nold = \"\"\"/-- log(a + c) \u2264 log a + c/a (the log-increment machine). -/\nlemma log_add_le (a c : \u211d) (ha : 0 < a) (hc : 0 \u2264 c) :\n Real.log (a + c) \u2264 Real.log a + c / a := by\n have h1 : a + c = a * (1 + c / a) := by field_simp [ha.ne']; ring\n rw [h1, Real.log_mul ha.ne' (by positivity : (1 : \u211d) + c / a \u2260 0)]\n have h2 : Real.log (1 + c / a) \u2264 c / a :=\n Real.log_le_sub_one_of_pos (by positivity : (0 : \u211d) < 1 + c / a)\n linarith [h2]\"\"\"\nnew = \"\"\"/-- log(a + c) \u2264 log a + c/a (the log-increment machine). -/\nlemma log_add_le (a c : \u211d) (ha : 0 < a) (hc : 0 \u2264 c) :\n Real.log (a + c) \u2264 Real.log a + c / a := by\n have hca : (0 : \u211d) \u2264 c / a := div_nonneg hc ha.le\n have h1ca : (0 : \u211d) < 1 + c / a := by linarith\n have h1 : a + c = a * (1 + c / a) := by field_simp [ha.ne']; ring\n rw [h1, Real.log_mul ha.ne' h1ca.ne']\n have h2 : Real.log (1 + c / a) \u2264 c / a := by\n have h3 := Real.log_le_sub_one_of_pos (x := 1 + c / a) h1ca\n rwa [show (1 : \u211d) + c / a - 1 = c / a from by ring] at h3\n linarith [h2]\"\"\"\nassert old in src\nsrc = src.replace(old, new)\n\n# C0hat_two_mul_add_one_le: log_add_le call-site nonneg args fine (log_add_le fixed)\n# div_le_div unknown -> div_le_div_iff\u2080 bridge in C0hat_two_t_add_one_le\nold = \"\"\" have hlog2le : Real.log 2 \u2264 0.6932 := Real.log_two_lt_d9.le.trans (by norm_num)\n have h10 : Real.log 2 / Real.log (t + 3) \u2264 0.6932 / 2.5 := by\n apply div_le_div hlog2le hlog13 (by norm_num : (0 : \u211d) \u2264 2.5) hlogpos\n linarith [h6, h7, h10]\"\"\"\nnew = \"\"\" have hlog2le : Real.log 2 \u2264 0.6932 := Real.log_two_lt_d9.le.trans (by norm_num)\n have h10 : Real.log 2 / Real.log (t + 3) \u2264 0.6932 / 2.5 := by\n rw [div_le_div_iff\u2080 hlogpos (by norm_num : (0 : \u211d) < 2.5)]\n have h11 : (0.6932 : \u211d) * 2.5 \u2264 0.6932 * Real.log (t + 3) :=\n mul_le_mul_of_nonneg_left hlog13 (by norm_num)\n have h12 : Real.log 2 * 2.5 \u2264 0.6932 * 2.5 :=\n mul_le_mul_of_nonneg_right hlog2le (by norm_num)\n linarith [h11, h12]\n linarith [h6, h7, h10]\"\"\"\nassert old in src\nsrc = src.replace(old, new)\n\n# hasDerivAt_conj_of_conj_eq: full rewrite of the tendsto parts\nold = \"\"\" have h1 : Filter.Tendsto (fun x : \u2102 => (f x - f z) / (x - z)) (\ud835\udcdd[\u2260] z)\n (\ud835\udcdd (deriv f z)) :=\n hasDerivAt_iff_tendsto.mp hdiff.hasDerivAt\n have hmap : Filter.Tendsto (starRingEnd \u2102) (\ud835\udcdd[\u2260] ((starRingEnd \u2102) z)) (\ud835\udcdd[\u2260] z) := by\n have hcont : Continuous ((starRingEnd \u2102) : \u2102 \u2192 \u2102) := Complex.continuous_conj\n have hT := hcont.tendsto ((starRingEnd \u2102) z)\n rw [Filter.tendsto_nhdsWithin_iff]\n refine \u27e8hT.mono_left Filter.nhdsWithin_le_nhds, ?_\u27e9\n rw [Filter.eventually_nhdsWithin_iff]\n exact Filter.Eventually.of_forall (fun x (hx : x \u2260 (starRingEnd \u2102) z) => by\n show (starRingEnd \u2102) x \u2260 z\n intro hbad\n apply hx\n have h3 := congrArg (starRingEnd \u2102) hbad\n rwa [starRingEnd_self_apply] at h3)\n have h2 : Filter.Tendsto (fun x : \u2102 =>\n (starRingEnd \u2102) ((f ((starRingEnd \u2102) x) - f z) / (((starRingEnd \u2102) x) - z)))\n (\ud835\udcdd[\u2260] ((starRingEnd \u2102) z)) (\ud835\udcdd ((starRingEnd \u2102) (deriv f z))) := by\n have hcont : Continuous ((starRingEnd \u2102) : \u2102 \u2192 \u2102) := Complex.continuous_conj\n exact (hcont.tendsto _).comp (h1.comp hmap)\n have h3 : (fun x : \u2102 => (f x - f ((starRingEnd \u2102) z)) / (x - (starRingEnd \u2102) z))\n =\u1da0[\ud835\udcdd[\u2260] ((starRingEnd \u2102) z)] fun x : \u2102 =>\n (starRingEnd \u2102) ((f ((starRingEnd \u2102) x) - f z) / (((starRingEnd \u2102) x) - z)) := by\n apply Filter.EventuallyEq.filter_mono ?_ Filter.nhdsWithin_le_nhds\n exact hconj.mono (fun s hs => by\n rw [hs, hzid]\n simp only [map_sub, map_div\u2080, starRingEnd_self_apply])\n exact hasDerivAt_iff_tendsto.mpr (Filter.Tendsto.congr' h3 h2)\"\"\"\nnew = \"\"\" have hslope : (fun x : \u2102 => (f x - f z) / (x - z)) = slope f z := by\n funext x\n rw [slope, smul_eq_mul, div_eq_mul_inv]\n have h1 : Filter.Tendsto (fun x : \u2102 => (f x - f z) / (x - z)) (\ud835\udcdd[\u2260] z)\n (\ud835\udcdd (deriv f z)) := by\n rw [hslope]\n exact hasDerivAt_iff_tendsto_slope.mp hdiff.hasDerivAt\n have hmap : Filter.Tendsto (starRingEnd \u2102) (\ud835\udcdd[\u2260] ((starRingEnd \u2102) z)) (\ud835\udcdd[\u2260] z) := by\n have hcont : Continuous ((starRingEnd \u2102) : \u2102 \u2192 \u2102) := Complex.continuous_conj\n have hT := hcont.tendsto ((starRingEnd \u2102) z)\n have hp : Filter.Tendsto (starRingEnd \u2102) (\ud835\udcdd[\u2260] ((starRingEnd \u2102) z)) (Filter.principal {z}\u1d9c) := by\n rw [Filter.tendsto_principal_principal]\n intro x hx\n rw [Set.mem_compl_singleton_iff] at hx\n rw [Set.mem_compl_singleton_iff]\n intro hbad\n apply hx\n have h3 := congrArg (starRingEnd \u2102) hbad\n rwa [starRingEnd_self_apply] at h3\n have h2 := hT.inf hp\n have hunfold : (\ud835\udcdd[\u2260] z) = \ud835\udcdd z \u2293 Filter.principal {z}\u1d9c := rfl\n rw [hunfold]\n exact h2\n have h2 : Filter.Tendsto (fun x : \u2102 =>\n (starRingEnd \u2102) ((f ((starRingEnd \u2102) x) - f z) / (((starRingEnd \u2102) x) - z)))\n (\ud835\udcdd[\u2260] ((starRingEnd \u2102) z)) (\ud835\udcdd ((starRingEnd \u2102) (deriv f z))) := by\n have hcont : Continuous ((starRingEnd \u2102) : \u2102 \u2192 \u2102) := Complex.continuous_conj\n exact (hcont.tendsto _).comp (h1.comp hmap)\n have h4 : (fun x : \u2102 => (f x - f ((starRingEnd \u2102) z)) / (x - (starRingEnd \u2102) z))\n =\u1da0[\ud835\udcdd ((starRingEnd \u2102) z)] fun x : \u2102 =>\n (starRingEnd \u2102) ((f ((starRingEnd \u2102) x) - f z) / (((starRingEnd \u2102) x) - z)) := by\n exact hconj.mono (fun s hs => by\n rw [hs, hzid]\n simp only [map_sub, map_div\u2080, starRingEnd_self_apply])\n have h3 : (fun x : \u2102 => (f x - f ((starRingEnd \u2102) z)) / (x - (starRingEnd \u2102) z))\n =\u1da0[\ud835\udcdd[\u2260] ((starRingEnd \u2102) z)] fun x : \u2102 =>\n (starRingEnd \u2102) ((f ((starRingEnd \u2102) x) - f z) / (((starRingEnd \u2102) x) - z)) :=\n h4.filter_mono nhdsWithin_le_nhds\n have hslope2 : (fun x : \u2102 => (f x - f ((starRingEnd \u2102) z)) / (x - (starRingEnd \u2102) z))\n = slope f ((starRingEnd \u2102) z) := by\n funext x\n rw [slope, smul_eq_mul, div_eq_mul_inv]\n rw [hslope2] at h3\n exact hasDerivAt_iff_tendsto_slope.mpr (Filter.Tendsto.congr' h3 h2)\"\"\"\nassert old in src\nsrc = src.replace(old, new)\n\nopen(p, 'w', encoding='utf-8').write(src)\nprint(\"done\")\n",
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[{"type":"thinking","thinking":"`hasDerivAt_iff_tendsto_slope : HasDerivAt f f' x ↔ Tendsto (slope f x) (𝓝[≠] x) (𝓝 f')` — with slope f x y = (y−x)⁻¹ • (f y − f x). For ℂ→ℂ: slope f z x = (x−z)⁻¹ • (f x − f z) = (f x − f z)/(x−z) — smul = division form. So my functions should use slope or the inv-mul form. Simplest: restate my slope functions with slope: (fun x => slope f z x) and (fun x => slope f (conj z) x). slope f z x = (x − z)⁻¹ • (f x − f z) — the smul on ℂ: (x−z)⁻¹ • (f x − f z) = (x−z)⁻¹ * (f x − f z) = (f x − f z)/(x−z) ✓ — defeq-ish? For ℂ the • is * (Module with smul = mul). Let me just use slope directly:\n\nh1 : Tendsto (slope f z) (𝓝[≠] z) (𝓝 (deriv f z)) := hasDerivAt_iff_tendsto_slope.mp hdiff.hasDerivAt\nh2's function: conj(slope f z (conj x)) = conj((f(conj x) − f z)/((conj x) − z)) — as a slope: slope f z (conj x) = ((conj x) − z)⁻¹ • (f(conj x) − f z) — the conj of that: conj(((conj x) − z)⁻¹ • (f(conj x) − f z)) — conj_smul: conj(c • w) = conj c • conj w — for the RingHom: map_smul? starRingEnd is a ring hom, map_smul needs module structure — for ℂ self: conj(a·b) = conj a·conj b (map_mul) ✓ the • is just mul.\nThe EventuallyEq target: slope f (conj z) x = conj(slope f z (conj x)):\nslope f (conj z) x = (x − conj z)⁻¹ • (f x − f (conj z)); conj(slope f z (conj x)) = conj(((conj x) − z)⁻¹ • (f(conj x) − f z)).\nWith hconj: f(conj x) = conj(f x) and hzid: f(conj z) = conj(f z):\nconj(slope f z (conj x)) = conj(((conj x − z)⁻¹) • (conj(f x) − f z)) = conj((conj(x − conj z))⁻¹ • conj(f x − conj(f z))) [conj_sub, starRingEnd_self_apply] = (x − conj z)⁻¹ • (f x − conj(f z)) [conj_inv₀, conj_mul...] — messy with smul/mul bridges. Let me avoid slope and just use the explicit quotient form (f x − f z)/(x − z) and prove the tendsto characterization myself — the conversion from the Frechet form to the quotient form: there's `HasDerivAt.tendsto_slope`? Or use `hasDerivAt_iff_tendsto_slope` and bridge slope = quotient: slope f z x = (f x − f z)/(x − z) — `slope_def`? slope f x y = (y−x)⁻¹ • (f y − f x) by def. For ℂ: (y−x)⁻¹ • (f y − f x) = (y−x)⁻¹ * (f y − f x) = (f y − f x)/(y−x) — via smul_eq_mul + div_eq_mul_inv: \n```\nhave hslope : slope f z = fun x : ℂ => (f x - f z) / (x - z) := by\n funext x\n rw [slope, smul_eq_mul, div_eq_mul_inv]\n```\nslope's def: `slope (f : 𝕜 → E) (x : 𝕜) : 𝕜 → E := fun y => (y - x)⁻¹ • (f y - f x)` — rw [slope] unfolds? `slope_def`? Let me just: funext x; show ((x − z)⁻¹ • (f x − f z)) = (f x − f z)/(x − z); rw [smul_eq_mul, div_eq_mul_inv]. The `show` needs the slope to unfold — `slope f z x` — delta-unfolds? slope is a def — `show` with the unfolded form should be defeq ✓.\n\nSo:\n```\n have h1 : Filter.Tendsto (fun x : ℂ => (f x - f z) / (x - z)) (𝓝[≠] z)\n (𝓝 (deriv f z)) := by\n have h := hasDerivAt_iff_tendsto_slope.mp hdiff.hasDerivAt\n have hs : (fun x : ℂ => (f x - f z) / (x - z)) = slope f z := by\n funext x\n show (f x - f z) / (x - z) = (x - z)⁻¹ • (f x - f z)\n rw [div_eq_mul_inv, smul_eq_mul]\n rw [hs]\n exact h\n```\nHmm — the show's RHS (x−z)⁻¹ • (f x − f z) — is that slope f z x definitionally? slope f z x = (x − z)⁻¹ • (f x − f z)? slope f x = fun y => (y − x)⁻¹ • (f y − f x): at y = x: (x − z)⁻¹ • (f x − f z) ✓ should be defeq via rfl of the def application. If show fails, use `rw [slope_def]`? or `simp only [slope, ...]`. I'll write `rw [slope, smul_eq_mul, div_eq_mul_inv]` inside funext — `slope` as an equation-lemma: slope f x y = (y−x)⁻¹ • (f y − f x) — there's `slope_def`? Let me hedge: `simp only [slope, smul_eq_mul, div_eq_mul_inv]`? slope might not be simp-unfoldable by name. Try `funext x; rw [slope]; rw [smul_eq_mul, div_eq_mul_inv]` — rw [slope] unfolds the def if slope is an equation? slope is a def: `def slope (f : 𝕜 → E) (x : 𝕜) : 𝕜 → E := fun y => (y - x)⁻¹ • (f y - f x)` — rw [slope] unfolds defs by their equation ✓ usually works.\n\nSimilarly the h2/EQ functions with the quotient form — no slope needed there (only h1 uses the slope-characterization; h2's composition and the final HasDerivAt use hasDerivAt_iff_tendsto_slope.mpr with the quotient→slope bridge again).\n\nThe final: hasDerivAt_iff_tendsto_slope.mpr — needs Tendsto (slope f (conj z)) (𝓝[≠] (conj z)) (𝓝 (conj (deriv f z))): my h2 gives the quotient form at conj z; bridge: slope f (conj z) = (fun x => (f x − f(conj z))/(x − conj z)) via the same funext; the EventuallyEq h3 connects the quotient forms ✓.\n\nAlso the nhdsWithin lemma names: `nhdsWithin_le_nhds` is root-namespace (from the Calculus usages ✓ no Filter prefix).\nAnd `Filter.tendsto_nhdsWithin_iff` — didn't find; use: Tendsto f (𝓝[s] a) (𝓝 b) = Tendsto f (𝓝 a) (𝓝 b) ⊓ principal... The punctured case: 𝓝[≠] a = 𝓝 a ⊓ principal {a}ᶜ: `tendsto_nhdsWithin_of_tendsto`? — there's `ContinuousAt.tendsto_nhdsWithin`? or build via `Filter.Tendsto.inf`:\nhmap : Tendsto conj (𝓝[≠] (conj z)) (𝓝[≠] z): 𝓝[≠] x = 𝓝[{x}ᶜ] x: Tendsto conj (𝓝[{conj z}ᶜ] (conj z)) (𝓝[{z}ᶜ] z) — via `tendsto_nhdsWithin_of_tendsto_nhds_of_eventually`? Hmm — simplest: \n```\n have hmap : Filter.Tendsto (starRingEnd ℂ) (𝓝[≠] ((starRingEnd ℂ) z)) (𝓝[≠] z) := by\n have hcont : Continuous ((starRingEnd ℂ) : ℂ → ℂ) := Complex.continuous_conj\n apply (hcont.tendsto ((starRingEnd ℂ) z)).inf\n rw [Filter.tendsto_principal_principal]\n exact fun x hx => by\n show (starRingEnd ℂ) x ∈ {z}ᶜ\n rw [Set.mem_compl_singleton_iff] at hx ⊢\n intro hbad\n apply hx\n have h3 := congrArg (starRingEnd ℂ) hbad\n rwa [starRingEnd_self_apply] at h3\n```\n`Filter.Tendsto.inf (h : Tendsto f l (𝓝 a)) : Tendsto f l (principal s)`?? Tendsto.inf : Tendsto f l (𝓝 a) → Tendsto f l (𝓟 s) → Tendsto f l (𝓝 a ⊓ 𝓟 s) — and 𝓝[≠] z = 𝓝 z ⊓ 𝓟 {z}ᶜ — the nhdsWithin def: 𝓝[s] a = 𝓝 a ⊓ 𝓟 s ✓ and 𝓝[≠] a = 𝓝[{a}ᶜ] a ✓. Tendsto f l₁ (l₂ ⊓ l₃) from components: `tendsto_inf`? — `Filter.Tendsto.inf (h₁ : Tendsto f l₁ l₂) (h₂ : Tendsto f l₁ l₃) : Tendsto f l₁ (l₂ ⊓ l₃)` ✓ exists. And Tendsto f l (𝓟 s) ⟺ ∀ x ∈ l-ish, f x ∈ s eventually: `tendsto_principal_principal : Tendsto f (𝓟 s) (𝓟 t) ↔ ∀ x ∈ s, f x ∈ t` ✓. The nhdsWithin destructure: 𝓝[≠] a is notation for 𝓝[{a}ᶜ] a — and 𝓝[s] a = 𝓝 a ⊓ 𝓟 s via `nhdsWithin` def — the Tendsto target 𝓝[≠] z: rewrite via `nhdsWithin` def: `show Filter.Tendsto _ _ (𝓝 z ⊓ 𝓟 {z}ᶜ)`? The notation 𝓝[≠] z — there's `nhdsWithin` and the def `nhdsWithin a s = 𝓝 a ⊓ 𝓟 s` — unfold via `nhdsWithin` — `rw [nhdsWithin]`? The def: `def nhdsWithin (a : α) (s : Set α) : Filter α := 𝓝 a ⊓ 𝓟 s` — rw [nhdsWithin] should unfold.\n\nHmm — also `Continuous.tendsto (x) : Tendsto f (𝓝 x) (𝓝 (f x))` ✓.\n\nLet me also fix `tendsto_nhdsWithin_iff` — replace with the inf approach above.\n\nNow the Filter.EventuallyEq.filter_mono: signature `EventuallyEq.filter_mono (hl : f =ᶠ[l₁] g) (h : l₂ ≤ l₁) : f =ᶠ[l₂] g`? — from the grep the name exists (EventuallyEq.filter_mono). Use h4.filter_mono nhdsWithin_le_nhds with nhdsWithin_le_nhds root ✓.\n\nAlso `isOpen_compl_singleton.eventually_nhds` — signature: `IsOpen.eventually_nhds (hs : IsOpen s) (hx : x ∈ s) : ∀ᶠ y in 𝓝 x, y ∈ s` ✓ — isOpen_compl_singleton : IsOpen {a}ᶜ ✓ and membership conj z ∈ {1}ᶜ = conj z ≠ 1: Set.mem_compl_singleton_iff.mpr h1 ✓.\n\nLet me apply all these fixes.","signature":""},{"type":"text","text":"Rewriting the tendsto parts of §14 (slope characterization, nhdsWithin via inf, mono structure) plus the §13 fixes."},{"type":"tool_use","id":"Write_136","name":"Write","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\patch_s1314.py","content":"p = 'NguyenRH1/LiTightR.lean'\nsrc = open(p, encoding='utf-8').read()\n\n# log_add_le: robust nonnegs + the 1+c/a-1 bridge\nold = \"\"\"/-- log(a + c) ≤ log a + c/a (the log-increment machine). -/\nlemma log_add_le (a c : ℝ) (ha : 0 < a) (hc : 0 ≤ c) :\n Real.log (a + c) ≤ Real.log a + c / a := by\n have h1 : a + c = a * (1 + c / a) := by field_simp [ha.ne']; ring\n rw [h1, Real.log_mul ha.ne' (by positivity : (1 : ℝ) + c / a ≠ 0)]\n have h2 : Real.log (1 + c / a) ≤ c / a :=\n Real.log_le_sub_one_of_pos (by positivity : (0 : ℝ) < 1 + c / a)\n linarith [h2]\"\"\"\nnew = \"\"\"/-- log(a + c) ≤ log a + c/a (the log-increment machine). -/\nlemma log_add_le (a c : ℝ) (ha : 0 < a) (hc : 0 ≤ c) :\n Real.log (a + c) ≤ Real.log a + c / a := by\n have hca : (0 : ℝ) ≤ c / a := div_nonneg hc ha.le\n have h1ca : (0 : ℝ) < 1 + c / a := by linarith\n have h1 : a + c = a * (1 + c / a) := by field_simp [ha.ne']; ring\n rw [h1, Real.log_mul ha.ne' h1ca.ne']\n have h2 : Real.log (1 + c / a) ≤ c / a := by\n have h3 := Real.log_le_sub_one_of_pos (x := 1 + c / a) h1ca\n rwa [show (1 : ℝ) + c / a - 1 = c / a from by ring] at h3\n linarith [h2]\"\"\"\nassert old in src\nsrc = src.replace(old, new)\n\n# C0hat_two_mul_add_one_le: log_add_le call-site nonneg args fine (log_add_le fixed)\n# div_le_div unknown -> div_le_div_iff₀ bridge in C0hat_two_t_add_one_le\nold = \"\"\" have hlog2le : Real.log 2 ≤ 0.6932 := Real.log_two_lt_d9.le.trans (by norm_num)\n have h10 : Real.log 2 / Real.log (t + 3) ≤ 0.6932 / 2.5 := by\n apply div_le_div hlog2le hlog13 (by norm_num : (0 : ℝ) ≤ 2.5) hlogpos\n linarith [h6, h7, h10]\"\"\"\nnew = \"\"\" have hlog2le : Real.log 2 ≤ 0.6932 := Real.log_two_lt_d9.le.trans (by norm_num)\n have h10 : Real.log 2 / Real.log (t + 3) ≤ 0.6932 / 2.5 := by\n rw [div_le_div_iff₀ hlogpos (by norm_num : (0 : ℝ) < 2.5)]\n have h11 : (0.6932 : ℝ) * 2.5 ≤ 0.6932 * Real.log (t + 3) :=\n mul_le_mul_of_nonneg_left hlog13 (by norm_num)\n have h12 : Real.log 2 * 2.5 ≤ 0.6932 * 2.5 :=\n mul_le_mul_of_nonneg_right hlog2le (by norm_num)\n linarith [h11, h12]\n linarith [h6, h7, h10]\"\"\"\nassert old in src\nsrc = src.replace(old, new)\n\n# hasDerivAt_conj_of_conj_eq: full rewrite of the tendsto parts\nold = \"\"\" have h1 : Filter.Tendsto (fun x : ℂ => (f x - f z) / (x - z)) (𝓝[≠] z)\n (𝓝 (deriv f z)) :=\n hasDerivAt_iff_tendsto.mp hdiff.hasDerivAt\n have hmap : Filter.Tendsto (starRingEnd ℂ) (𝓝[≠] ((starRingEnd ℂ) z)) (𝓝[≠] z) := by\n have hcont : Continuous ((starRingEnd ℂ) : ℂ → ℂ) := Complex.continuous_conj\n have hT := hcont.tendsto ((starRingEnd ℂ) z)\n rw [Filter.tendsto_nhdsWithin_iff]\n refine ⟨hT.mono_left Filter.nhdsWithin_le_nhds, ?_⟩\n rw [Filter.eventually_nhdsWithin_iff]\n exact Filter.Eventually.of_forall (fun x (hx : x ≠ (starRingEnd ℂ) z) => by\n show (starRingEnd ℂ) x ≠ z\n intro hbad\n apply hx\n have h3 := congrArg (starRingEnd ℂ) hbad\n rwa [starRingEnd_self_apply] at h3)\n have h2 : Filter.Tendsto (fun x : ℂ =>\n (starRingEnd ℂ) ((f ((starRingEnd ℂ) x) - f z) / (((starRingEnd ℂ) x) - z)))\n (𝓝[≠] ((starRingEnd ℂ) z)) (𝓝 ((starRingEnd ℂ) (deriv f z))) := by\n have hcont : Continuous ((starRingEnd ℂ) : ℂ → ℂ) := Complex.continuous_conj\n exact (hcont.tendsto _).comp (h1.comp hmap)\n have h3 : (fun x : ℂ => (f x - f ((starRingEnd ℂ) z)) / (x - (starRingEnd ℂ) z))\n =ᶠ[𝓝[≠] ((starRingEnd ℂ) z)] fun x : ℂ =>\n (starRingEnd ℂ) ((f ((starRingEnd ℂ) x) - f z) / (((starRingEnd ℂ) x) - z)) := by\n apply Filter.EventuallyEq.filter_mono ?_ Filter.nhdsWithin_le_nhds\n exact hconj.mono (fun s hs => by\n rw [hs, hzid]\n simp only [map_sub, map_div₀, starRingEnd_self_apply])\n exact hasDerivAt_iff_tendsto.mpr (Filter.Tendsto.congr' h3 h2)\"\"\"\nnew = \"\"\" have hslope : (fun x : ℂ => (f x - f z) / (x - z)) = slope f z := by\n funext x\n rw [slope, smul_eq_mul, div_eq_mul_inv]\n have h1 : Filter.Tendsto (fun x : ℂ => (f x - f z) / (x - z)) (𝓝[≠] z)\n (𝓝 (deriv f z)) := by\n rw [hslope]\n exact hasDerivAt_iff_tendsto_slope.mp hdiff.hasDerivAt\n have hmap : Filter.Tendsto (starRingEnd ℂ) (𝓝[≠] ((starRingEnd ℂ) z)) (𝓝[≠] z) := by\n have hcont : Continuous ((starRingEnd ℂ) : ℂ → ℂ) := Complex.continuous_conj\n have hT := hcont.tendsto ((starRingEnd ℂ) z)\n have hp : Filter.Tendsto (starRingEnd ℂ) (𝓝[≠] ((starRingEnd ℂ) z)) (Filter.principal {z}ᶜ) := by\n rw [Filter.tendsto_principal_principal]\n intro x hx\n rw [Set.mem_compl_singleton_iff] at hx\n rw [Set.mem_compl_singleton_iff]\n intro hbad\n apply hx\n have h3 := congrArg (starRingEnd ℂ) hbad\n rwa [starRingEnd_self_apply] at h3\n have h2 := hT.inf hp\n have hunfold : (𝓝[≠] z) = 𝓝 z ⊓ Filter.principal {z}ᶜ := rfl\n rw [hunfold]\n exact h2\n have h2 : Filter.Tendsto (fun x : ℂ =>\n (starRingEnd ℂ) ((f ((starRingEnd ℂ) x) - f z) / (((starRingEnd ℂ) x) - z)))\n (𝓝[≠] ((starRingEnd ℂ) z)) (𝓝 ((starRingEnd ℂ) (deriv f z))) := by\n have hcont : Continuous ((starRingEnd ℂ) : ℂ → ℂ) := Complex.continuous_conj\n exact (hcont.tendsto _).comp (h1.comp hmap)\n have h4 : (fun x : ℂ => (f x - f ((starRingEnd ℂ) z)) / (x - (starRingEnd ℂ) z))\n =ᶠ[𝓝 ((starRingEnd ℂ) z)] fun x : ℂ =>\n (starRingEnd ℂ) ((f ((starRingEnd ℂ) x) - f z) / (((starRingEnd ℂ) x) - z)) := by\n exact hconj.mono (fun s hs => by\n rw [hs, hzid]\n simp only [map_sub, map_div₀, starRingEnd_self_apply])\n have h3 : (fun x : ℂ => (f x - f ((starRingEnd ℂ) z)) / (x - (starRingEnd ℂ) z))\n =ᶠ[𝓝[≠] ((starRingEnd ℂ) z)] fun x : ℂ =>\n (starRingEnd ℂ) ((f ((starRingEnd ℂ) x) - f z) / (((starRingEnd ℂ) x) - z)) :=\n h4.filter_mono nhdsWithin_le_nhds\n have hslope2 : (fun x : ℂ => (f x - f ((starRingEnd ℂ) z)) / (x - (starRingEnd ℂ) z))\n = slope f ((starRingEnd ℂ) z) := by\n funext x\n rw [slope, smul_eq_mul, div_eq_mul_inv]\n rw [hslope2] at h3\n exact hasDerivAt_iff_tendsto_slope.mpr (Filter.Tendsto.congr' h3 h2)\"\"\"\nassert old in src\nsrc = src.replace(old, new)\n\nopen(p, 'w', encoding='utf-8').write(src)\nprint(\"done\")\n"},"extra_content":null}]