[{"type":"thinking","thinking":"Both `Real.log_two_gt_d9` and `Real.log_two_lt_d9` exist (in Real namespace, ExponentialBounds.lean). Now write the IIMaj piece. This is the biggest remaining chunk — write it in two edits:\n\nEdit 1: defs (iimA, iimB, psiTilde), IIMaj_eq_psiTilde, psiTilde_contDiff, hasDerivAt_psiTilde (D1), deriv2 formula (D2).\nEdit 2: iimaj_deriv2_lt with the factor monotonicity + numerics.\n\nFor Edit 1:\n\n```lean\n/-! ### The IIMaj (B̃₄) term (D-057 §3) -/\n\n/-- Decay rate a = 2 log 2 − 1/2. -/\nprivate def iimA : ℝ := 2 * Real.log 2 - 1 / 2\n\n/-- Decay rate b = 2 log 2. -/\nprivate def iimB : ℝ := 2 * Real.log 2\n\n/-- The IIMaj term rewritten as a function of s = √n:\n4.56·(s+1)·(e^{−as} + e^{−bs}). -/\nprivate def psiTilde (s : ℝ) : ℝ :=\n  4.56 * ((s + 1) * (Real.exp (-iimA * s) + Real.exp (-iimB * s)))\n\ntheorem IIMaj_eq_psiTilde_sqrt (n : ℝ) :\n    IIMaj n = psiTilde (Real.sqrt n) := by\n  unfold IIMaj psiTilde iimA iimB\n  have hrw : (2 : ℝ) ^ (-(2 * Real.sqrt n - 2)) =\n      4 * Real.exp (-(2 * Real.log 2) * Real.sqrt n) := by\n    have h1 : (2 : ℝ) ^ (-(2 * Real.sqrt n - 2)) =\n        (2 : ℝ) ^ (2 - 2 * Real.sqrt n) := by ring_nf\n    rw [h1, Real.rpow_sub (by norm_num), Real.rpow_def_of_pos (by norm_num : (0:ℝ) < 2)]\n    -- 2^2 / 2^(2√n)?? hmm\n    ...\n```\nCareful: 2^(2−2√n) = 2^2 · 2^(−2√n) via Real.rpow_add (hx : 0 < x) : x^(y+z) = x^y · x^z — need form 2 + (−2√n): rw [show (2:ℝ) − 2*Real.sqrt n = 2 + (-(2*Real.sqrt n)) by ring, Real.rpow_add (by norm_num : (0:ℝ) < 2)]. Then 2^(−2√n) = exp(log 2 · (−2√n)) by rpow_def_of_pos ✓ exponent algebra: log 2 * (−2√n) = −(2 log 2)·√n ✓ ring. And 2^(2:ℝ) = 4: Real.rpow_natCast? (2:ℝ)^((2:ℕ):ℝ)... rpow of natural: Real.rpow_two? `Real.rpow_natCast : x ^ (n : ℝ) = x ^ n` ✓ then norm_num.\n\nAnd the first rewrite: -(2√n−2) = 2−2√n: congruence: rw [show (-(2 * Real.sqrt n - 2) : ℝ) = 2 - 2 * Real.sqrt n by ring] ✓ (rewriting inside the exponent — the exponent is a subterm ✓).\n\nThen: (e^{√n/2}+1)·(4·e^{−b√n}) = 4·(e^{√n/2}·e^{−b√n} + e^{−b√n}): and e^{√n/2}·e^{−b√n} = e^{√n/2 − b√n} (−Real.exp_add: exp(x)·exp(y) = exp(x+y) ← Real.exp_add x y : exp(x+y) = exp x * exp y — so rw [← Real.exp_add]) then exponent: √n/2 + (−(2log2)√n) = −(2log2−1/2)√n: have : √n/2 − (2*log2)*√n = -(2*log2−1/2)*√n := by ring — rw inside exp arg.\n\nFull assembly: LHS = 2*(√n+1)*(exp(√n/2)+1)*0.57*(4*exp(−b√n)) — rearrange to 4.56*((√n+1)*(exp(−a√n)+exp(−b√n))): after the exp rewrites, close by ring (atoms: √n, exp(−a√n), exp(−b√n), exp(√n/2) — after ← exp_add, no exp(√n/2) remains ✓; then pure ring in {√n, eA, eB}).\n\nOrder: rw [hrw] — rewrites the 2^(...) to 4*exp(−b√n) ✓. Then: goal: 2*(√n+1)*(exp(√n/2)+1)*0.57*(4*exp(−b√n)) = 4.56*((√n+1)*(exp(−a√n)+exp(−b√n))). Now rewrite exp(−a√n): from exp(√n/2)·exp(−b√n): have hEA : Real.exp (√n/2) * Real.exp (−(2*log2)*√n) = Real.exp (−(2*log2−1/2)*√n) := by rw [← Real.exp_add]; congr 1; ring. Hmm — but −a√n where a unfolds to 2log2−1/2: −iimA*√n — after unfold iimA: −(2*log2−1/2)*√n ✓ matches hEA's RHS ✓.\n\nThen LHS has exp(√n/2) — write: (exp(√n/2)+1)*(4*exp(−b√n)) = 4*(exp(√n/2)*exp(−b√n) + exp(−b√n)) by ring; rw [hEA]?? hEA rewrites the product → then ring. But rw [hEA] needs the exact product subterm `Real.exp (Real.sqrt n / 2) * Real.exp (-(2 * Real.log 2) * Real.sqrt n)` to appear — after `ring`-normalization?? NO — ring for REWRITING goal is bad (goal is an equation — use a calc or conv). Cleaner: transform goal into the factored form via `linear_combination`?? Overkill. Use:\n\n  have h1 : (Real.exp (√n/2) + 1) * (4 * Real.exp (−b√n)) =\n      4 * (Real.exp (−a√n) + Real.exp (−b√n)) := by\n    rw [show (Real.exp (√n/2) + 1) * (4 * Real.exp (−b√n)) =\n        4 * (Real.exp (√n/2) * Real.exp (−b√n) + Real.exp (−b√n)) by ring, hEA]\n  -- then goal: 2*(√n+1)*[that]... hmm the goal's structure: 2 * (√n+1) * (exp(√n/2)+1) * 0.57 * (4*exp(−b√n)) — the subterm (exp(√n/2)+1)*(4*exp(−b√n)) is NOT contiguous (0.57 in between). Commutative ring — rw can't. \n\nAlternative: prove the whole identity by: rw [hrw]; then `have hEA`; then `linear_combination 0.228 * ...`? No. Best: conv/ring_nf to normalize both sides into a polynomial in atoms {√n, exp(√n/2), exp(−a√n), exp(−b√n)} — then rewrite exp(√n/2)·exp(−b√n) via hEA inside... ring_nf doesn't do that rewrite.\n\nTrick: substitute eA := exp(−a√n), eB := exp(−b√n), eC := exp(√n/2) and hCE : eC * eB = eA (hEA) — then goal is linear in eC: both sides: LHS = 2*0.57*4*(√n+1)*(eC+1)*eB = 4.56*(√n+1)*(eC*eB + eB); RHS = 4.56*(√n+1)*(eA + eB). rw [show eC*eB = eA from hEA]?? — the goal after unfolding has (exp(√n/2)+1) and exp(−b√n) multiplied somewhere — the product eC*eB appears only after expansion: (eC+1)*eB = eC*eB + eB — distributivity. So: rw [mul_add_one?? hmm: (exp(√n/2)+1)*X... the goal term is (exp(√n/2)+1) * (4*exp(−b√n)) — not expanded. \n\nDo: rw [hrw]; rw [show ∀ X Y Z : ℝ, (X + 1) * (4 * Y) = 4 * (X * Y + Y) from fun _ _ _ => by ring]?? — rewriting with a ∀-equation: rw [show (Real.exp (√n/2) + 1) * (4 * Real.exp (−b√n)) = 4 * (Real.exp (√n/2) * Real.exp (−b√n) + Real.exp (−b√n)) by ring] — a `show` with the SPECIFIC instance — that rewrites that exact subterm ✓ then rw [hEA] ✓ then ring ✓. OK that works (verbose but robust).\n\npsiTilde_contDiff:\n```lean\ntheorem psiTilde_contDiff : ContDiff ℝ 2 psiTilde := by\n  unfold psiTilde iimA iimB\n  apply ContDiff.const_mul\n  apply ContDiff.mul\n  · exact contDiff_id.add contDiff_const\n  · apply ContDiff.add\n    · exact (contDiff_const.mul contDiff_id).exp\n    · exact (contDiff_const.mul contDiff_id).exp\n```\nContDiff.const_mul : (c : ℝ) → ContDiff ℝ n f → ContDiff ℝ n (fun x => c * f x)? signature: `ContDiff.const_mul (h : ContDiff 𝕜 n f) (c : ℝ)?? order` — check: `theorem ContDiff.const_mul (hf : ContDiff 𝕜 i f) (c : 𝕜) : ContDiff 𝕜 i fun x => c * f x` — hmm c might come first. Use `contDiff_const.mul ...` instead: (contDiff_const (c := 4.56)).mul h — contDiff_const : ContDiff 𝕜 n fun _ => c ✓ .mul ✓ function fun x => 4.56 * (...) ✓ but psiTilde's body is `4.56 * (...)` — ✓ match. Then apply ContDiff.mul (contDiff_const) (rest). Write with `refine contDiff_const.mul ?_` etc. ContDiff.mul : ContDiff 𝕜 n f → ContDiff 𝕜 n g → ContDiff 𝕜 n (f * g)? or (fun x => f x * g x) — it's the latter (Pi.mul) — psiTilde body `4.56 * (X)` elaborates as HMul.hMul 4.56 X vs (fun x => 4.56) * X — after unfold psiTilde the goal shows 4.56 * ((s+1)*(...)) — is that Pi.mul application? No — it's pointwise HMul on the real value — hmm: psiTilde s = 4.56 * (...) with s the lambda var: the FUNCTION is fun s => 4.56 * (body s) — ContDiff.mul gives ContDiff fun x => f x * g x = pointwise ✓ matches. `apply ContDiff.mul` unifies f := fun _ => 4.56, g := body ✓.\n\nh1 (per-point HasDerivAt psiTilde (D1 x)):\n```lean\ntheorem hasDerivAt_psiTilde (x : ℝ) :\n    HasDerivAt psiTilde\n      (4.56 * ((Real.exp (-iimA * x) + Real.exp (-iimB * x)) +\n        (x + 1) * (-iimA * Real.exp (-iimA * x) - iimB * Real.exp (-iimB * x)))) x := by\n  unfold psiTilde iimA iimB\n  have hu : HasDerivAt (fun t => t + 1) 1 x := (hasDerivAt_id' x).add_const 1\n  have hva : HasDerivAt (fun t => Real.exp (-(2 * Real.log 2 - 1 / 2) * t))\n      (Real.exp (-(2 * Real.log 2 - 1 / 2) * x) * (-(2 * Real.log 2 - 1 / 2))) x :=\n    (((hasDerivAt_id' x).const_mul (-(2 * Real.log 2 - 1 / 2)))).exp\n  have hvb : HasDerivAt (fun t => Real.exp (-(2 * Real.log 2) * t))\n      (Real.exp (-(2 * Real.log 2) * x) * (-(2 * Real.log 2))) x :=\n    (((hasDerivAt_id' x).const_mul (-(2 * Real.log 2)))).exp\n  have hv := hva.add hvb\n  have huv := hu.mul hv\n  have h := huv.const_mul 4.56\n  -- massage derivative value\n  convert h using 2 <;> ring\n```\nIssues: (1) `.exp` on HasDerivAt (fun t => c*t) — gives exp(c*t)*(c)?? HasDerivAt.exp : HasDerivAt f f' x → HasDerivAt (fun x => Real.exp (f x)) (Real.exp (f x) * f') x ✓. (2) huv's function: fun t => (t+1)*(exp(-a t)+exp(-b t)) ✓; const_mul 4.56 → fun t => 4.56 * (...) ✓ matches unfolded psiTilde? psiTilde's body: 4.56 * ((s+1)*(exp + exp)) ✓ (fun t => 4.56 * ((t+1)*(...))) ✓. (3) derivative value: 4.56 * (1 * (eA+eB) + (x+1) * (eA*(−a) + eB*(−b))) vs target 4.56 * ((eA+eB) + (x+1)*(−a eA − b eB)): convert using 2 → goal: 4.56*(1*(eA+eB) + (x+1)*(eA*−a + eB*−b)) = 4.56*((eA+eB) + (x+1)*(−a*eA − b*eB)): ring ✓. But wait — the a/b in hva/hvb are the UNFOLDED expressions (2log2−1/2 etc.) while target has iimA/iimB — I unfolded them at the start ✓ both unfolded ✓... but the STATEMENT of hasDerivAt_psiTilde uses iimA/iimB (folded) — after `unfold ...` the GOAL is unfolded ✓ and h's value is in unfolded terms ✓ convert matches.\n\nAlso const_mul signature: HasDerivAt.const_mul (c : ℝ) (hf : HasDerivAt f f' x) : HasDerivAt (fun x => c * f x) (c * f') x — c first? In Mathlib: `theorem HasDerivAt.const_mul (c : 𝕜) (hf : HasDerivAt f f' x) : HasDerivAt (fun x => c * f x) (c * f') x` — hmm dot notation huv.const_mul 4.56 → const_mul huv 4.56?? dot notation: huv fills first explicit arg of TYPE HasDerivAt — if signature is (c) (hf), then huv.const_mul 4.56 = HasDerivAt.const_mul 4.56 huv? NO — dot notation inserts huv at the first explicit argument of matching type (HasDerivAt) — which is hf (second position) — giving const_mul 4.56 huv ✓ works either way.\n\nSimilarly .const_mul on (hasDerivAt_id' x): (hasDerivAt_id' x).const_mul c : HasDerivAt (fun t => c*t) (c*1)?? — HasDerivAt.const_mul gives (fun x => c * f x) (c * f') = (fun t => c*t) (c*1) — derivative c*1 vs needed c: massage — hmm hva's stated derivative: exp(−a x) * (−a) — from .exp of const_mul result: f' = c*1?? The const_mul result: HasDerivAt (fun t => −a*t) (−a*1) x — then .exp: exp(−a x) * (−a*1) — my stated hva derivative is exp(−a x) * (−a) — mismatch c*1 vs c — add mul_one massage: rwa [mul_one] at h?? (−a)*1: rw [mul_one] turns into −a ✓. Or state with *(−a * 1)... Let me instead use `.const_mul` result derivative form: after .exp the derivative is Real.exp (−a*x) * (−a * 1). Massage: convert using 2; ring at the end of the whole h — the final convert h using 2 <;> ring handles the FULL derivative expression including the *1's? convert using 2: goals: function eq (should be rfl-ish — convert might auto-close or leave for ring) and value eq — ring closes 4.56*(1*(eA+eB) + (x+1)*(eA*(−a*1) + eB*(−b*1))) = target ✓ ring handles.\n\nActually wait — will `convert h using 2 <;> ring` leave the function-equality goal to ring? ring can't prove function equalities with exp atoms? The functions: goal `fun t => 4.56 * ((t+1)*(exp(−a t)+exp(−b t)))` vs h's `fun t => 4.56 * ((t+1)*(exp(−a t)+exp(−b t)))` — identical ✓ closed by rfl automatically in convert. The derivative value goal → ring ✓. But HasDerivAt psiTilde ... — psiTilde unfolded in goal — h's function must match the unfolded form ✓.\n\nHmm — one concern: `unfold psiTilde iimA iimB` — unfold at goal ✓.\n\nD2 formula:\n```lean\ntheorem deriv2_psiTilde (x : ℝ) :\n    deriv (deriv psiTilde) x =\n      4.56 * (2 * (-iimA * Real.exp (-iimA * x) - iimB * Real.exp (-iimB * x)) +\n        (x + 1) * (iimA ^ 2 * Real.exp (-iimA * x) + iimB ^ 2 * Real.exp (-iimB * x))) := by\n  have h1 : ∀ᶠ y in 𝓝 x, HasDerivAt psiTilde (D1-expr y) y :=\n    Filter.eventually_of_forall?? — unknown earlier! used filter_upwards [] pattern instead:\n    by filter_upwards [] with y; exact hasDerivAt_psiTilde y\n  have h2 : HasDerivAt (D1-function) (D2-value) x := ...\n  exact deriv2_of_hasDerivAt_nhds h1 h2\n```\nwhere D1-function := fun t => 4.56 * ((exp(−a t)+exp(−b t)) + (t+1)*(−a e − b e)).\n\nh2: D1 = const_mul 4.56 of (w + z) with w = fun t => eA + eB (w' = eA*(−a) + eB*(−b) = sum of hva/hvb derivatives), z = fun t => (t+1)*(−a eA − b eB): z = u.mul g where g = fun t => −a*eA − b*eB: g' = −a*(eA*(−a)) − b*(eB*(−b)) = a²eA + b²eB: g = (const_mul (−a) eA-fun).sub?? g = fun t => (−a*eA t) − (b*eB t): ((hasDerivAt eA-fun).const_mul (−a)).sub ((...).const_mul b)?? — HasDerivAt.sub ✓. g' = (−a)*(eA x*(−a)) − b*(eB x*(−b)) — massage to a²eA + b²eB by ring (atoms eA, eB, a, b, x) ✓ (−a)*(−a) = a² — ring normalizes ✓ but a is the unfolded (2log2−1/2) EXPRESSION — ring treats it as atom ✓ (−A)*(−A) = A² ✓ and A^2 in target ✓ match.\nz' = 1*g x + (x+1)*g' = g x + (x+1)(a²eA + b²eB).\nD2-value = 4.56 * (w' + z') = 4.56 * ((eA*(−a) + eB*(−b)) + (g x + (x+1)*(a²eA+b²eB))) = 4.56 * (2*(−a eA − b eB) + (x+1)(a²eA+b²eB)) — since g x = −a eA − b eB ✓ convert+ring.\n\nThen also deriv_psiTilde: deriv psiTilde x = D1 x from hasDerivAt_psiTilde x |>.deriv ✓.\n\nThis is a lot of code. Write Edit 1 now (defs, identity, contDiff, hasDerivAt_psiTilde, deriv_psiTilde, deriv2_psiTilde).","signature":""},{"type":"text","text":"Both bounds exist. Writing the IIMaj machinery (defs, identity, smoothness, derivatives)."},{"type":"tool_use","id":"Edit_155","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":"/-- [SKELETON] Margin convexity (D-057, draft §3.1 Step 4): for any","new_string":"/-! ### The IIMaj (B̃₄) term (D-057 §3) -/\n\n/-- Decay rate a = 2 log 2 − 1/2 ≈ 0.8863. -/\nprivate def iimA : ℝ := 2 * Real.log 2 - 1 / 2\n\n/-- Decay rate b = 2 log 2 ≈ 1.3863. -/\nprivate def iimB : ℝ := 2 * Real.log 2\n\n/-- The IIMaj term as a function of s = √n: 4.56·(s+1)·(e^{−as} + e^{−bs}). -/\nprivate def psiTilde (s : ℝ) : ℝ :=\n  4.56 * ((s + 1) * (Real.exp (-iimA * s) + Real.exp (-iimB * s)))\n\n/-- [PROVED session 011] IIMaj n = ψ̃(√n): the rpow/exp bookkeeping. -/\ntheorem IIMaj_eq_psiTilde_sqrt (n : ℝ) :\n    IIMaj n = psiTilde (Real.sqrt n) := by\n  have hrw : (2 : ℝ) ^ (-(2 * Real.sqrt n - 2)) =\n      4 * Real.exp (-(2 * Real.log 2) * Real.sqrt n) := by\n    rw [show (-(2 * Real.sqrt n - 2) : ℝ) = 2 + -(2 * Real.sqrt n) by ring,\n      Real.rpow_add (by norm_num : (0 : ℝ) < 2),\n      Real.rpow_def_of_pos (by norm_num : (0 : ℝ) < 2)]\n    have e1 : (2 : ℝ) ^ (2 : ℝ) = 4 := by\n      rw [Real.rpow_natCast]; norm_num\n    rw [e1]\n    ring_nf\n  have hEA : Real.exp (Real.sqrt n / 2) * Real.exp (-(2 * Real.log 2) * Real.sqrt n) =\n      Real.exp (-(2 * Real.log 2 - 1 / 2) * Real.sqrt n) := by\n    rw [← Real.exp_add]\n    congr 1\n    ring\n  unfold IIMaj psiTilde iimA iimB\n  rw [hrw,\n    show (Real.exp (Real.sqrt n / 2) + 1) * (4 * Real.exp (-(2 * Real.log 2) * Real.sqrt n)) =\n      4 * (Real.exp (Real.sqrt n / 2) * Real.exp (-(2 * Real.log 2) * Real.sqrt n) +\n        Real.exp (-(2 * Real.log 2) * Real.sqrt n)) by ring,\n    hEA]\n  ring\n\n/-- [PROVED session 011] ψ̃ is globally C² (polynomial × exponentials). -/\ntheorem psiTilde_contDiff : ContDiff ℝ 2 psiTilde := by\n  unfold psiTilde iimA iimB\n  refine contDiff_const.mul ?_\n  refine ContDiff.mul ?_ ?_\n  · exact contDiff_id.add contDiff_const\n  · exact ((contDiff_const.mul contDiff_id).exp).add (contDiff_const.mul contDiff_id).exp\n\n/-- [PROVED session 011] ψ̃′ in closed form (per-point, global). -/\ntheorem hasDerivAt_psiTilde (x : ℝ) :\n    HasDerivAt psiTilde\n      (4.56 * ((Real.exp (-iimA * x) + Real.exp (-iimB * x)) +\n        (x + 1) * (-iimA * Real.exp (-iimA * x) - iimB * Real.exp (-iimB * x)))) x := by\n  unfold psiTilde iimA iimB\n  have hu : HasDerivAt (fun t => t + 1) 1 x := (hasDerivAt_id' x).add_const 1\n  have hva : HasDerivAt (fun t => Real.exp (-(2 * Real.log 2 - 1 / 2) * t))\n      (Real.exp (-(2 * Real.log 2 - 1 / 2) * x) * (-(2 * Real.log 2 - 1 / 2))) x := by\n    have h := ((hasDerivAt_id' x).const_mul (-(2 * Real.log 2 - 1 / 2))).exp\n    rwa [mul_one] at h\n  have hvb : HasDerivAt (fun t => Real.exp (-(2 * Real.log 2) * t))\n      (Real.exp (-(2 * Real.log 2) * x) * (-(2 * Real.log 2))) x := by\n    have h := ((hasDerivAt_id' x).const_mul (-(2 * Real.log 2))).exp\n    rwa [mul_one] at h\n  have h := (hu.mul (hva.add hvb)).const_mul 4.56\n  convert h using 2 <;> ring\n\n/-- [PROVED session 011] deriv ψ̃ = the closed D1 form. -/\ntheorem deriv_psiTilde (x : ℝ) :\n    deriv psiTilde x =\n      4.56 * ((Real.exp (-iimA * x) + Real.exp (-iimB * x)) +\n        (x + 1) * (-iimA * Real.exp (-iimA * x) - iimB * Real.exp (-iimB * x))) :=\n  (hasDerivAt_psiTilde x).deriv\n\n/-- [PROVED session 011] ψ̃″ in closed form. -/\ntheorem deriv2_psiTilde (x : ℝ) :\n    deriv (deriv psiTilde) x =\n      4.56 * (2 * (-iimA * Real.exp (-iimA * x) - iimB * Real.exp (-iimB * x)) +\n        (x + 1) * (iimA ^ 2 * Real.exp (-iimA * x) +\n          iimB ^ 2 * Real.exp (-iimB * x))) := by\n  have h1 : ∀ᶠ y in 𝓝 x, HasDerivAt psiTilde\n      (4.56 * ((Real.exp (-iimA * y) + Real.exp (-iimB * y)) +\n        (y + 1) * (-iimA * Real.exp (-iimA * y) - iimB * Real.exp (-iimB * y)))) y := by\n    filter_upwards [] with y\n    exact hasDerivAt_psiTilde y\n  have h2 : HasDerivAt\n      (fun t => 4.56 * ((Real.exp (-iimA * t) + Real.exp (-iimB * t)) +\n        (t + 1) * (-iimA * Real.exp (-iimA * t) - iimB * Real.exp (-iimB * t))))\n      (4.56 * (2 * (-iimA * Real.exp (-iimA * x) - iimB * Real.exp (-iimB * x)) +\n        (x + 1) * (iimA ^ 2 * Real.exp (-iimA * x) +\n          iimB ^ 2 * Real.exp (-iimB * x)))) x := by\n    unfold iimA iimB\n    have hu : HasDerivAt (fun t => t + 1) 1 x := (hasDerivAt_id' x).add_const 1\n    have hva : HasDerivAt (fun t => Real.exp (-(2 * Real.log 2 - 1 / 2) * t))\n        (Real.exp (-(2 * Real.log 2 - 1 / 2) * x) * (-(2 * Real.log 2 - 1 / 2))) x := by\n      have h := ((hasDerivAt_id' x).const_mul (-(2 * Real.log 2 - 1 / 2))).exp\n      rwa [mul_one] at h\n    have hvb : HasDerivAt (fun t => Real.exp (-(2 * Real.log 2) * t))\n        (Real.exp (-(2 * Real.log 2) * x) * (-(2 * Real.log 2))) x := by\n      have h := ((hasDerivAt_id' x).const_mul (-(2 * Real.log 2))).exp\n      rwa [mul_one] at h\n    have hw := hva.add hvb\n    have hg := (hva.const_mul (-(2 * Real.log 2 - 1 / 2))).sub\n      (hvb.const_mul (2 * Real.log 2))\n    have hz := hu.mul hg\n    have h := (hw.add hz).const_mul 4.56\n    convert h using 2 <;> ring\n  exact deriv2_of_hasDerivAt_nhds h1 h2\n\n/-- [SKELETON] Margin convexity (D-057, draft §3.1 Step 4): for any"},"extra_content":null}]