[{"type":"thinking","thinking":"Several errors in hf1/hf2:\n\n1. hf1's hder: the `convert h using 1 <;> ring` for hbase — the HasDerivAt for `fun z => -z/2` with derivative `-(1:ℂ)/2`: `(hasDerivAt_id' s₀).neg.div_const 2` — HasDerivAt (fun z => -z/2)? .neg gives HasDerivAt (fun z => -z) (−1); .div_const 2: HasDerivAt (fun z => (−z)/2) ((−1)/2) — the function is `fun z => -z / 2` — same as (−z)/2 ✓ but the DERIVATIVE: −1/2 vs my stated −(1:ℂ)/2 — defeq? −1/2 vs −(1/2): ring-equal ✓ the `convert h using 1 <;> ring` handles the mismatch... The error at 2930 (ring_nf no progress): the `convert h using 1` generated an equality goal that ring can't close — probably the derivative VALUES: −1/2 vs −(1:ℂ)/2 hmm. Actually simpler: state hbase's derivative as (−1)/2 hmm — or avoid: \n   ```\n   have hbase : HasDerivAt (fun z : ℂ => -z / 2) (-(1 : ℂ) / 2) s₀ := by\n     have h := (hasDerivAt_id' s₀).neg.div_const 2\n     -- h : HasDerivAt (fun x => -x / 2) (-1 / 2) s₀\n     convert h using 1\n   ```\n   with the value equality −(1:ℂ)/2 = −1/2 by `ring`?? The error says ring_nf made no progress — so the goal wasn't a ring identity — maybe convert produced a FUNCTION equality: `fun z => -z/2 = fun z => -z/2`? no. Hmm — maybe the h's function form: .div_const on HasDerivAt (fun z => -z): gives HasDerivAt (fun x => -x / 2)?? or (fun x => (-x) / 2)... same. And the value: -1/2 (a division of -1 by 2) vs -(1:ℂ)/2 — ring should prove −(1:ℂ)/2 = −1/2 ✓ it's a ring identity. The error at 2930:30 with TWO ring_nf failures — the `<;> ring` ran on two goals. Maybe convert produced `-(1:ℂ)/2 = -1/2` AND a function equality — the function equality `(fun z => -z / 2) = (fun x => -x / 2)`?? alpha-equiv — convert using 1 should close by rfl... \n\n   Simplify drastically: state the whole hder via a cleaner route: use `HasDerivAt.const_cpow` directly on the identity-ish base:\n   ```\n   have hbase : HasDerivAt (fun z : ℂ => -z / 2) (-(1 : ℂ) / 2) s₀ :=\n     (hasDerivAt_id' s₀).neg.div_const 2\n   ```\n   WITHOUT convert — just check what .neg.div_const 2 produces EXACTLY: (hasDerivAt_id' s₀) : HasDerivAt (fun x => x) (1) s₀; .neg : HasDerivAt (fun x => -x) (-1) s₀; .div_const 2 : HasDerivAt (fun x => -x / 2) (-1 / 2) s₀. So the derivative is −1/2 (as `-1 / 2`) NOT `-(1:ℂ)/2`. Are they the same TERM? −1/2 parses as HDiv.hDiv (−1) 2; −(1:ℂ)/2 = HDiv.hDiv (−(1:ℂ)) 2 — `−1` vs `−(1:ℂ)` — the −1 in −1/2 is also Neg.neg 1 with the OfNat — SAME term actually (both Neg.neg (OfNat.ofNat 1)). So they're identical?? Then hbase := that directly ✓ (the error suggests convert generated something ring couldn't handle — maybe convert itself failed and left ring-goals). Just drop the convert: `exact (hasDerivAt_id' s₀).neg.div_const 2`.\n\n2. hf1's hder.const_cpow: `convert hbase.const_cpow ... using 1; ring` — const_cpow's conclusion: HasDerivAt (fun x => c ^ f x) (c ^ f x * Complex.log c * f') x — with f := fun z => −z/2, f' := −1/2: the value: π^(−s₀/2) * Complex.log π * (−1/2) — my stated hder has `(Real.pi : ℂ) ^ (-s₀ / 2) * Complex.log (Real.pi : ℂ) * (-(1 : ℂ) / 2)` — the f' = −1/2 vs −(1:ℂ)/2 — same as above ✓ so const_cpow gives EXACTLY the hder I stated (modulo the f' form) — so drop the convert and just use it directly, with my hder statement adjusted to the const_cpow output form `... * (-1 / 2)`. Simplest: make hder := hbase.const_cpow ... directly (no stated type — use `have hder := hbase.const_cpow (Or.inl ...)`) — then hder.deriv gives the deriv in whatever form. Then the goal after rw [hder.deriv]: `π^(−s₀/2) * log π * (−1/2) / π^(−s₀/2) = −(log π)/2` hmm — wait logDeriv_apply: logDeriv f s₀ = deriv f s₀ / f s₀: so the goal: π^(−s₀/2)*log π*(−1/2) / π^(−s₀/2) = −↑(Real.log π)/2: the `mul_div_assoc, div_self hne1, mul_one` chain: π^… * log π * (−1/2) / π^… — pattern for div_self: `π^(−s₀/2) / π^(−s₀/2)` — the error shows it didn't match because the form is `π^… * log π * (−1/2) / π^…` — parse: ((π^… * log π) * (−1/2)) / π^… — mul_div_assoc: a * b / c = a * (b / c): applied to (π^… * log π) * (−1/2) / π^…: gives π^… * log π * ((−1/2)/π^…)?? mul_div_assoc : a * b / c = a * (b / c) — with a := π^… * log π, b := −1/2, c := π^…: → π^… * log π * ((−1/2) / π^…) — NOT the div_self pattern. The error's goal: `↑Real.pi ^ (-s₀ / 2) * log ↑Real.pi * (-1 / 2 / ↑Real.pi ^ (-s₀ / 2)) = -↑(Real.log Real.pi) / 2` — exactly that wrong split. Fix: field_simp [hne1] then ring:\n   ```\n   rw [hder.deriv]\n   field_simp [hne1]\n   rw [Complex.ofReal_log Real.pi_nonneg]\n   push_cast\n   ring\n   ```\n   field_simp [hne1] clears the denominator: goal becomes π^… * log π * (−1/2) = −↑(Real.log π)/2 * π^…: then ofReal_log + push_cast + ring... the Complex.log π vs ↑(Real.log π): rw [Complex.ofReal_log Real.pi_nonneg] rewrites log ↑Real.pi → ↑(Real.log π) ✓ then `↑(Real.log π) * (−1/2) = −↑(Real.log π)/2` — hmm the LHS after field_simp: π^… * ↑(Real.log π) * (−1/2) = −↑(Real.log π)/2 * π^…: ring ✓ (with ofReal_log rewitten, all casts aligned by push_cast).\n\n   Wait — ofReal_log: Complex.ofReal_log (hx : 0 ≤ x) : ↑(Real.log x) = Complex.log ↑x — so rw [Complex.ofReal_log Real.pi_nonneg] rewrites LHS-to-RHS: occurrences of ↑(Real.log Real.pi) become Complex.log ↑Real.pi. The goal has Complex.log ↑Real.pi (from hder) and ↑(Real.log Real.pi) (in the target −↑(Real.log Real.pi)/2): rw [← Complex.ofReal_log Real.pi_nonneg] rewrites Complex.log ↑π → ↑(Real.log π) ✓ (direction ←). Then push_cast + ring.\n\n3. hf2's hder: `rw [h.deriv]` — h : HasDerivAt (Gamma ∘ fun x => x/2)?? the comp: `(differentiableAt_Gamma _ hs).hasDerivAt.comp s₀ (hasDerivAt_id' s₀ |>.div_const 2)` — HasDerivAt.comp (hg : HasDerivAt g g' (f x)) (hf : HasDerivAt f f' x) : HasDerivAt (g ∘ f) (g' * f') x — the composed function is `Gamma ∘ fun x => x / 2` — my goal mentions `deriv (fun z => Gamma (z / 2)) s₀` — NOT syntactically `Gamma ∘ (fun x => x/2)` — the rewrite failed since h.deriv is about `Gamma ∘ fun x => x/2`. Fix: change hf2's deriv-equality to use the ∘ form, or state h differently: HasDerivAt (fun z => Gamma (z/2)) — via `(hg).comp x hf` gives the ∘ form; to get the lambda form use `HasDerivAt.comp_of_eq`? or just: `((Complex.differentiableAt_Gamma _ hs).hasDerivAt).comp s₀ ...` produces HasDerivAt (Gamma ∘ fun x => x/2) — and deriv (Gamma ∘ ...) — then the subsequent rw [hder] needs the goal's `deriv (fun z => Gamma (z / 2)) s₀` to match. Are `Gamma ∘ fun x => x / 2` and `fun z => Gamma (z / 2)` defeq? Yes (eta) but not syntactically — rw fails. Fix: state hder as the deriv-equality directly via `deriv_comp`? There's `deriv.comp`?? In Mathlib: `deriv_comp (hg : DifferentiableAt 𝕜 g (f x)) (hf : DifferentiableAt 𝕜 f x) : deriv (g ∘ f) x = deriv g (f x) * deriv f x`?? hmm — for ℂ→ℂ (1-dim): `deriv.comp : deriv (h₂ ∘ h₁) x = deriv h₂ (h₁ x) * deriv h₁ x` — `deriv_comp (hh₂ : DifferentiableAt 𝕜 h₂ (h₁ x)) (hh₁ : DifferentiableAt 𝕜 h₁ x)` ✓ exists. But still the ∘-vs-lambda mismatch in the GOAL: goal `deriv (fun z => Gamma (z/2)) s₀ = ...` — rewrite the goal's lambda to ∘: `show deriv (Complex.Gamma ∘ fun z : ℂ => z / 2) s₀ = _` — is that accepted? (fun z => Gamma (z/2)) vs Gamma ∘ (fun z => z/2): defeq (eta) — `show` with defeq type ✓ works. Then rw [deriv_comp (Complex.differentiableAt_Gamma _ hs) differentiableAt_id.div_const 2's differentiableAt]: \n   ```\n   have hder : deriv (fun z : ℂ => Complex.Gamma (z / 2)) s₀\n       = deriv Complex.Gamma (s₀ / 2) * (1 / 2) := by\n     have h1 : deriv (Complex.Gamma ∘ fun z : ℂ => z / 2) s₀\n         = deriv Complex.Gamma (s₀ / 2) * deriv (fun z : ℂ => z / 2) s₀ :=\n       deriv_comp _ (Complex.differentiableAt_Gamma _ hs) (differentiableAt_id.div_const 2)\n     have h2 : deriv (fun z : ℂ => z / 2) s₀ = 1 / 2 := by\n       rw [deriv_div_const, deriv_id'']\n     rw [h1, h2]\n   ```\n   Hmm — wait deriv_comp signature: `deriv_comp (hh₂ : DifferentiableAt 𝕜 h₂ (h₁ x)) (hh₁ : DifferentiableAt 𝕜 h₁ x) : deriv (h₂ ∘ h₁) x = deriv h₂ (h₁ x) * deriv h₁ x`? Let me double check the arg order and the form (g'*(f') vs (f')*(g') — in 1-dim it's commutative anyway). And `deriv (fun z => z/2) s₀ = 1/2`: deriv_div_const : deriv (fun x => f x / c) x = deriv f x / c; deriv_id'' : deriv id x = 1 — the lambda (fun z => z/2) — is it syntactically `fun z => z / 2` vs `fun z => z / (2:ℂ)` ✓ same. deriv_div_const: `deriv_div_const (c : 𝕜) : deriv (fun x => f x / c) x = deriv f x / c` ✓ with f := fun z => z hmm — (fun z => z) vs id: deriv_id'' : deriv id x = 1 — the lambda might not match id syntactically: `fun z => z` IS id by eta — rw [deriv_id''] expects `deriv id s₀`: use `deriv_id'`? or just: (hasDerivAt_id' s₀).div_const 2 |>.deriv : deriv (fun x => x / 2) s₀ = 1/2 ✓ cleaner:\n   ```\n   have h2 : deriv (fun z : ℂ => z / 2) s₀ = 1 / 2 := ((hasDerivAt_id' s₀).div_const 2).deriv\n   ```\n   HasDerivAt.div_const: (hf : HasDerivAt f f' x) : HasDerivAt (fun x => f x / c) (f' / c) x: with f := id-lambda, f' = 1: deriv (fun x => x/2) s₀ = 1/2 ✓ .deriv ✓.\n\n   And the `show` bridge: the goal after rw [logDeriv_apply] in hf2: `deriv (fun z => Gamma (z / 2)) s₀ / Gamma (s₀ / 2) = (1/2) * digamma (s₀/2)`. My hder is about deriv (fun z => Gamma (z/2)) s₀ — but I prove it via the ∘ form with the show... Let me restructure hf2 cleanly:\n   ```\n   have hf2 : logDeriv (fun z : ℂ => Complex.Gamma (z / 2)) s₀\n       = (1 / 2 : ℂ) * Complex.digamma (s₀ / 2) := by\n     rw [logDeriv_apply]\n     have hder : deriv (fun z : ℂ => Complex.Gamma (z / 2)) s₀\n         = deriv Complex.Gamma (s₀ / 2) * (1 / 2) := by\n       have h1 : deriv (Complex.Gamma ∘ fun z : ℂ => z / 2) s₀\n           = deriv Complex.Gamma (s₀ / 2) * deriv (fun z : ℂ => z / 2) s₀ :=\n         deriv_comp _ (Complex.differentiableAt_Gamma _ hs) (differentiableAt_id.div_const 2)\n       have h2 : deriv (fun z : ℂ => z / 2) s₀ = 1 / 2 :=\n         ((hasDerivAt_id' s₀).div_const 2).deriv\n       have h3 : deriv (fun z : ℂ => Complex.Gamma (z / 2)) s₀\n           = deriv (Complex.Gamma ∘ fun z : ℂ => z / 2) s₀ := rfl\n       rw [h3, h1, h2]\n     rw [hder]\n     have hdig : deriv Complex.Gamma (s₀ / 2)\n         = Complex.digamma (s₀ / 2) * Complex.Gamma (s₀ / 2) := by\n       rw [Complex.digamma_def, logDeriv_apply]\n       field_simp [hne2]\n     rw [hdig]\n     field_simp [hne2]\n     ring\n   ```\n   h3 by rfl: deriv (fun z => Gamma (z/2)) s₀ vs deriv (Gamma ∘ fun z => z/2) s₀ — defeq, rfl should work (eta).\n   Wait — does `deriv_comp` exist by that exact name? In Mathlib: `deriv_comp (hh₂ : DifferentiableAt 𝕜 h₂ (h₁ x)) (hh₁ : DifferentiableAt 𝕜 h₁ x) : deriv (h₂ ∘ h₁) x = deriv h₂ (h₁ x) * deriv h₁ x` — I believe yes (FDeriv/Basic or Deriv/Basic). There's also `deriv.comp` hmm. Let me just try; fallback: the HasDerivAt.comp route with the show-bridge on the hder's LHS.\n\nAlso hf1's hbase: drop convert, exact directly.\n\nLet me apply these fixes.","signature":""},{"type":"text","text":"Fixing hf1/hf2: drop the fragile `convert`s, use `deriv_comp`, and `field_simp` for the division."},{"type":"tool_use","id":"Edit_169","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean","old_string":"  have hf1 : logDeriv (fun z : ℂ => (Real.pi : ℂ) ^ (-z / 2)) s₀\n      = -(Real.log Real.pi : ℂ) / 2 := by\n    rw [logDeriv_apply]\n    have hder : HasDerivAt (fun z : ℂ => (Real.pi : ℂ) ^ (-z / 2))\n        ((Real.pi : ℂ) ^ (-s₀ / 2) * Complex.log (Real.pi : ℂ) * (-(1 : ℂ) / 2)) s₀ := by\n      have hbase : HasDerivAt (fun z : ℂ => -z / 2) (-(1 : ℂ) / 2) s₀ := by\n        have h := (hasDerivAt_id' s₀).neg.div_const 2\n        convert h using 1 <;> ring\n      convert hbase.const_cpow (Or.inl (by exact_mod_cast Real.pi_ne_zero :\n        (Real.pi : ℂ) ≠ 0)) using 1\n      ring\n    rw [hder.deriv]\n    rw [mul_div_assoc, div_self hne1, mul_one]\n    rw [Complex.ofReal_log Real.pi_nonneg]\n    push_cast\n    ring\n  have hf2 : logDeriv (fun z : ℂ => Complex.Gamma (z / 2)) s₀\n      = (1 / 2 : ℂ) * Complex.digamma (s₀ / 2) := by\n    rw [logDeriv_apply]\n    have hder : deriv (fun z : ℂ => Complex.Gamma (z / 2)) s₀\n        = deriv Complex.Gamma (s₀ / 2) * (1 / 2) := by\n      have h := ((Complex.differentiableAt_Gamma _ hs).hasDerivAt).comp s₀\n        (hasDerivAt_id' s₀ |>.div_const 2)\n      rw [h.deriv]\n    rw [hder]\n    have hdig : deriv Complex.Gamma (s₀ / 2)\n        = Complex.digamma (s₀ / 2) * Complex.Gamma (s₀ / 2) := by\n      rw [Complex.digamma_def, logDeriv_apply]\n      field_simp [hne2]\n    rw [hdig]\n    field_simp [hne2]\n    ring\n  rw [hf1, hf2]","new_string":"  have hf1 : logDeriv (fun z : ℂ => (Real.pi : ℂ) ^ (-z / 2)) s₀\n      = -(Real.log Real.pi : ℂ) / 2 := by\n    rw [logDeriv_apply]\n    have hbase : HasDerivAt (fun z : ℂ => -z / 2) (-(1 : ℂ) / 2) s₀ :=\n      (hasDerivAt_id' s₀).neg.div_const 2\n    have hder := hbase.const_cpow (Or.inl (by exact_mod_cast Real.pi_ne_zero :\n      (Real.pi : ℂ) ≠ 0))\n    rw [hder.deriv]\n    field_simp [hne1]\n    rw [← Complex.ofReal_log Real.pi_nonneg]\n    push_cast\n    ring\n  have hf2 : logDeriv (fun z : ℂ => Complex.Gamma (z / 2)) s₀\n      = (1 / 2 : ℂ) * Complex.digamma (s₀ / 2) := by\n    rw [logDeriv_apply]\n    have hder : deriv (fun z : ℂ => Complex.Gamma (z / 2)) s₀\n        = deriv Complex.Gamma (s₀ / 2) * (1 / 2) := by\n      have h1 : deriv (Complex.Gamma ∘ fun z : ℂ => z / 2) s₀\n          = deriv Complex.Gamma (s₀ / 2) * deriv (fun z : ℂ => z / 2) s₀ :=\n        deriv_comp _ (Complex.differentiableAt_Gamma _ hs) (differentiableAt_id.div_const 2)\n      have h2 : deriv (fun z : ℂ => z / 2) s₀ = 1 / 2 :=\n        ((hasDerivAt_id' s₀).div_const 2).deriv\n      have h3 : deriv (fun z : ℂ => Complex.Gamma (z / 2)) s₀\n          = deriv (Complex.Gamma ∘ fun z : ℂ => z / 2) s₀ := rfl\n      rw [h3, h1, h2]\n    rw [hder]\n    have hdig : deriv Complex.Gamma (s₀ / 2)\n        = Complex.digamma (s₀ / 2) * Complex.Gamma (s₀ / 2) := by\n      rw [Complex.digamma_def, logDeriv_apply]\n      field_simp [hne2]\n    rw [hdig]\n    field_simp [hne2]\n    ring\n  rw [hf1, hf2]"},"extra_content":null}]