[{"type":"thinking","thinking":"Now Edit C. The identity. Let me define the block weights inline in the identity. Statement:\n\nmarginTilde F = fun n => ((1/2) * n * Real.log n + (lagariasC1 - γ/2) * n - 4) +\n  ((-2) * F (√n+1) +\n  ((-((8/3)*0.433 + 42)) * ((√n+1) * log(√n+1+2)) +\n  ((-((8/3)*0.886)) * ((√n+1) * log(log(√n+1+2))) +\n  ((-((8/3)*2.9 + 192)) * (√n+1) +\n  (-(42*0.433^2*(Real.exp 4 + 1))) * (log(√n+1+2))^2 +\n  (-(42*(2*0.433*0.886)*(Real.exp 4 + 1))) * (log(√n+1+2) * log(log(√n+1+2))) +\n  (-(42*0.886^2*(Real.exp 4 + 1))) * (log(log(√n+1+2)))^2 +\n  (-((42*(2*0.433*2.9) + 147)*(Real.exp 4 + 1))) * log(√n+1+2) +\n  (-(42*(2*0.886*2.9)*(Real.exp 4 + 1))) * log(log(√n+1+2)) +\n  (-1) * IIMaj n + K...))))))\n\nHmm — rather than a deeply nested literal, let me NOT do one giant identity; instead structure the main proof with deriv2_add applications matching the DEF's own summation structure. Actually the def is:\nmarginTilde F n = S n − (2*F T + 4*T*G1 T + 40*1.3352 + 2*(e⁴+1)*(7*Hhat T + 3.3) + IIMaj n + 1) + 1\nwhere G1 T = (2/3)*C0hat T + 10.5*log(T+2) + 48.\n\nInstead of one identity, I could do the assembly in TWO levels: \nLevel 1: margiñ = S − B̃ + 1 with B̃ = the bracket.\nderiv (deriv margiñ) n = deriv (deriv S) n − deriv (deriv B̃) n + 0: via deriv2_sub on (S − B̃) then +const... marginTilde F = fun n => S n − B̃ n + 1 = (S − B̃) + const 1? marginTilde F n = (S n − B̃ n) + 1 ✓. deriv2_add with g := const 1 ✓. And deriv (deriv (S − B̃)) n: deriv2_sub... I don't have deriv2_sub — write it (mirror of deriv2_add) or use S + (−1)*B̃ via deriv2_add + deriv2_const_mul with c = −1: fewer lemmas: marginTilde F = fun n => S n + (-1) * B̃ n + 1 — deriv2_add (S-piece) (deriv2_add ((-1)*B̃) (const 1)): two adds ✓ no sub needed.\n\nLevel 2: deriv (deriv B̃) n — B̃ = 2F∘T + 4T·G1 + const + 2(e⁴+1)(7Hhat∘T + 3.3) + IIMaj — expand 4T·G1 and Hhat into the 8 blocks — this still needs the block-level identity for B̃... the identity at the B̃ level:\nB̃ = fun n => 2*(F∘T) n + ((8/3)*0.433+42)*(TL1∘T) n + ... + IIMaj n + K'\nas an Eq by ring (unfolding C0hat/Hhat).\n\nSimpler overall: ONE identity at the marginTilde level as I drafted, then the value computation by repeated deriv2_add/const_mul down the right-nested chain. The nesting in my draft: S + (f1 + (f2 + ...)) — 11 deriv2_add applications. Each application: deriv2_add hf1 hg1 hf2 hg2 — the hg-facts for the REST of the chain (the big sub-lambda) — DIFFERENTIABLE facts for the nested remainder lambda — ugh, the remainder isn't a named piece. The combinators need facts about BOTH sides including the nested remainder (differentiableAt of ITS deriv...). So right-nesting forces proving differentiability of nested sums — each level's remainder is a sum, whose deriv-differentiability comes from... the deriv2_add proof itself doesn't GIVE that. So I'd need DifferentiableAt (deriv (remainder-sum)) n per level — from per-piece facts via deriv-add chains — annoying.\n\nBetter: LEFT-nested: (((S + f1) + f2) + ...) — then at level k, the left side is the partial sum whose facts I carry along inductively by hand: prove a chain of haves:\nhave hS : deriv (deriv S) n = 1/(2n) ∧ DifferentiableAt ℝ (deriv S) n ∧ ∀ᶠ m, DifferentiableAt ℝ S m (bundle)\nhave h1 : deriv (deriv (S + f1)) n = ... ∧ DifferentiableAt (deriv (S+f1)) n ∧ eventually — the DifferentiableAt (deriv (S+f1)) n: deriv (S+f1) =ᶠ deriv S + deriv f1 → differentiableAt of that sum-function ✓ via the eventuallyEq + DifferentiableAt.add — hmm the combinator deriv2_add USES DifferentiableAt (deriv (S+f1))... NO — deriv2_add needs (deriv f) and (deriv g) diffAt — for the NEXT level, I need DifferentiableAt (deriv (S+f1)) n — provable: deriv (S+f1) =ᶠ (fun m => deriv S m + deriv f1 m) (from hf2/hg2) and that function is diffAt (hf1.add hg1) → via EventuallyEq.differentiableAt_iff?? name: `Filter.EventuallyEq.differentiableAt_iff (h : f =ᶠ[𝓝 x] g) : DifferentiableAt 𝕜 f x ↔ DifferentiableAt 𝕜 g x` ✓ exists. So carry: bundle' := ⟨value, diffAt-of-deriv, eventually-diff⟩ per level — each level ~8 lines × 11 = 88 lines. Verbose but mechanical.\n\nALTERNATIVE much cleaner: skip the value-equality for the whole function! Instead compute deriv (deriv (marginTilde F)) n via a SINGLE deriv2_of_hasDerivAt_nhds application on marginTilde F with:\nD1 m := marginS-D1 m + Σ piece-D1 m (each piece's first deriv closed form)\nD2 := the final value.\nh1 : ∀ᶠ m in 𝓝 n, HasDerivAt (marginTilde F) (D1 m) m — per-point: unfold marginTilde and combine per-point HasDerivAts via .sub/.add — the def's own structure (subtractions + additions of ~6 components: S, 2F, 4T·G1, const, 2(e⁴+1)(7Hhat+3.3), IIMaj): per-point HasDerivAt of each component:\n- S: hasDerivAt_marginS ✓\n- 2F∘T: (F-HasDerivAt).comp T .const_mul 2 ✓\n- 4T·G1∘T: product: (T-fun).mul (G1-fun): G1(t) = (2/3)C0hat t + 10.5 log(t+2) + 48: HasDerivAt G1 at Tm from C0hat's deriv: C0hat = 0.433L1 + 0.886L2 + 2.9: deriv = 0.433*(t+2)⁻¹ + 0.886*(L1(t))⁻¹*(t+2)⁻¹ ✓ per-point from hasDerivAt_log_shift/log_log. Then 4T·G1: (T-id).mul G1 scaled.\n- 2(e⁴+1)(7Hhat∘T+3.3): Hhat = 3C0hat² + 10.5L1 + 48: deriv: 6C0hat·C0hat' + 10.5(t+2)⁻¹ ✓ per-point.\n- IIMaj: (hasDerivAt_psiTilde (√m)).comp sqrt.\nCombine: h1 per point — the FULL first-derivative value D1 m in whatever collected form — I CHOOSE the D1 presentation; the massage is one big ring/field_simp at the end of the per-point block... The per-point value must equal my chosen D1 m — the components produce their natural forms; chosen D1 := their exact sum (no massage needed if I take D1 := the natural form verbatim!). Then h2 : HasDerivAt D1 (D2 n) n — D1 is the big sum; its derivative: build per-component AGAIN (product rules on the D1 components — F'∘T·(2√m)⁻¹ etc.) — this is the second-layer problem again (each D1-component is itself a product/composition needing its own HasDerivAt). ~10 components × product-rule = the same work as the chain-rule lemmas did per piece...\n\nNO WAIT — the chain-rule lemmas give deriv (deriv piece) n VALUES directly. The ONLY missing piece for using them is the sum-combinator with the diff-fact bundles. And the bundles come from ContDiffOn. So: per piece (11 pieces: S, F∘T, 8 block-compositions, IIMaj): bundle = ⟨value-eq (chain-rule/marginS/IIMaj), DifferentiableAt (deriv piece) n (from piece ContDiffOn), eventually-diff (from piece ContDiffOn)⟩. Then LEFT-nested deriv2_add chain carrying the accumulated diff facts. It's the 88-line mechanical chain. Fine — do it.\n\nHmm, actually — one more simplification: the diffAt-of-deriv for the accumulated sum: rather than eventuallyEq-iff at each level, note the accumulated partial sum is itself a \"piece\" whose ContDiffOn I can build once: the WHOLE marginTilde has ContDiffOn ℝ 2 (Ioi 20) (Edit E — I planned it anyway for convexOn_of_deriv2_nonneg's hf/hf'/hf''!). But deriv2_add needs diffAt of deriv of the PARTIAL sums, not just the whole... \n\nOK here's the cleanest total plan — abandon manual nesting: use the deriv2_of_hasDerivAt_nhds route but with the D1/D2 built from the CHAIN-RULE-VALUE pieces... no.\n\nFINAL DECISION — simplest robust: the left-nested chain with a custom per-level tactic pattern. Actually, you know what? Let me reconsider: deriv2_add's hypotheses (hf1 : DifferentiableAt (deriv f) n) — for the accumulated partial sum f, I can get this from the partial sum's OWN ContDiffOn — and building ContDiffOn of partial sums via ContDiffOn.add is EASY and doesn't need deriv-values! So per level:\nhCDacc : ContDiffOn ℝ 2 (partial-sum) (Ioi 20) := hCDprev.add hCDpiece\nthen hf1acc := (hCDacc.deriv_of_isOpen ...).differentiableOn... — 2 lines per level. And hf2acc similar. And value via deriv2_add hf1acc hf1piece hf2acc hf2piece.\n\nSo per level (11 levels): ~6 lines. Total ~70 lines + bundles. Acceptable. Actually even simpler: since I'll build ContDiffOn of EVERY prefix anyway, just build ContDiffOn of the whole thing once (Edit E), and for deriv2_add at level k use the prefix ContDiffOn (which I build incrementally regardless). Merge E into the chain.\n\nNow the identity presentation for the nested-adds: I'll define the pieces via `let` inside the proof? The identity rw then deriv2_add applications reference the pieces — the goal after rw [identity] has the nested lambda; deriv2_add unifies f := prefix-lambda, g := piece-lambda from the goal structure ✓.\n\nLet me now write the identity and check what the structure should be. In the identity statement I'll write the nested form explicitly:\n\nmarginTilde F = fun n =>\n  ((((((((((fun m => (1/2)*m*Real.log m + (lagariasC1 − γ/2)*m − 4) n) ... — no, the identity should be at function level: marginTilde F = (fun n => ...) where the lambda is left-nested additions of piece-applications... The deriv2_add path works on the FUNCTION structure — the lambda's body: (((S m + f1 m) + f2 m) + ...) — with pieces as LAMBDAS (fun m => ...). The unification in deriv2_add {f g} needs f, g as functions — from the goal `deriv (deriv fun m => ((X m + Y m))) n` — f := fun m => X m, g := fun m => Y m ✓.\n\nTo avoid writing the identity with deeply nested lambdas (unreadable), do the value computation via a `have hval : deriv (deriv (marginTilde F)) n = ...` where I first rw [show marginTilde F = Ffull from marginTilde_identity], and Ffull := a named def?? — make a private def marginExpanded (F) : ℝ → ℝ := the nested lambda! Then:\n- identity: marginTilde F = marginExpanded F (by funext m; unfold both; ring)\n- ContDiffOn of marginExpanded F via prefix adds.\n- value via deriv2_add chain on marginExpanded F's structure.\n\nmarginExpanded F := fun n => S0 n + f1 n + f2 n + ... + f10 n + K — write the lambda body left-nested.\n\nLet me define the pieces as private defs to keep the expanded def readable:\n\nprivate def mS (c : ℝ) : ℝ → ℝ := fun m => (1/2)*m*Real.log m + c*m − 4  — hmm marginS already handled inline; keep inline.\n\nprivate def mFblock (F) : ℝ → ℝ := fun m => (-2) * F (√m+1)\nprivate def mTL1b : ℝ → ℝ := fun m => (-((8/3)*0.433 + 42)) * ((√m+1) * log(√m+1+2))\nprivate def mTL2b : ℝ → ℝ := fun m => (-((8/3)*0.886)) * ((√m+1) * log(log(√m+1+2)))\nprivate def mIdb : ℝ → ℝ := fun m => (-((8/3)*2.9 + 192)) * (√m+1)\nprivate def mL1sqb : ℝ → ℝ := fun m => (-(42*0.433^2*(Real.exp 4 + 1))) * (log(√m+1+2))^2\nprivate def mL1L2b : ℝ → ℝ := fun m => (-(42*(2*0.433*0.886)*(Real.exp 4 + 1))) * (log(√m+1+2) * log(log(√m+1+2)))\nprivate def mL2sqb : ℝ → ℝ := fun m => (-(42*0.886^2*(Real.exp 4 + 1))) * (log(log(√m+1+2)))^2\nprivate def mL1b : ℝ → ℝ := fun m => (-((42*(2*0.433*2.9) + 147)*(Real.exp 4 + 1))) * log(√m+1+2)\nprivate def mL2b : ℝ → ℝ := fun m => (-(42*(2*0.886*2.9)*(Real.exp 4 + 1))) * log(log(√m+1+2))\nprivate def mIIM : ℝ → ℝ := fun m => (-1) * IIMaj m\nprivate def mK : ℝ := -4 - 40*1.3352 - 2*(Real.exp 4 + 1)*(7*(3*2.9^2 + 48) + 3.3) - 1 + 1\n\nmarginExpanded (F) : ℝ → ℝ := fun n => mS (lagariasC1 − γ/2) n + mFblock F n + mTL1b n + mTL2b n + mIdb n + mL1sqb n + mL1L2b n + mL2sqb n + mL1b n + mL2b n + mIIM n + mK\n\nWAIT — the def marginTilde's S part: `(1/2)*n*log n + (lagariasC1 − γ/2)*n − 4` — matches mS with c = lagariasC1 − γ/2 ✓. And mK as computed: −4 − 53.408 − 2(e⁴+1)·515.91 − 1 + 1: from S's −4 ✓ B̃'s 40·1.3352, 2(e⁴+1)(7Hhat const + 3.3), +1, and outer +1 ✓.\n\nThe identity proof: funext n; unfold marginTilde marginExpanded mS mFblock ... C0hat Hhat; ring. Hmm — marginTilde's def has `let T := √n+1` — unfold exposes ✓. ring must match: LHS: (1/2)n log n + cn − 4 − (2F T + 4T((2/3)(0.433 L1 + 0.886 L2 + 2.9) + 10.5 L1 + 48) + 40·1.3352 + 2(e⁴+1)(7(3(0.433L1+0.886L2+2.9)² + 10.5 L1 + 48) + 3.3) + IIMaj n + 1) + 1 vs RHS nested: c n... ring normalizes ✓ — CAREFUL: F appears as F (√n+1) — atom ✓; IIMaj n atom ✓; exp 4 atom ✓; log(√n+1+2) atom ✓.\n\nNow the piece bundles for the chain. Piece ContDiffOn on Ioi 20:\n- mS c: (contDiffOn_id.mul (contDiffOn_id.log (fun x hx => ne_of_gt ...0<x...))).const_mul?? — mS = fun m => (1/2)*(m*log m) + c*m − 4: (contDiffOn_const.mul (contDiffOn_id.mul (contDiffOn_id.log ...)))?? — write: ((contDiffOn_id.mul hlog).const_mul (1/2))?? ContDiffOn.const_mul exists? Use contDiffOn_const.mul: (contDiffOn_const (c := 1/2)).mul (contDiffOn_id.mul hlog) : fun m => (1/2)*((m)*(log m)) ✓; .add (contDiffOn_const.mul contDiffOn_id) : + c*m ✓; .sub contDiffOn_const ✓... ContDiffOn.sub ✓. hlog : ContDiffOn ℝ 2 (fun m => Real.log m) (Ioi 20) := contDiffOn_id.log (fun x hx => ne_of_gt (by ... : 0 < x)) ✓.\n- mFblock F: (contDiffOn_const.mul ((hF.contDiffOn).comp contDiffOn_sqrt_shift (Set.mapsTo_univ _ _)))?? — (F ∘ T): hF.contDiffOn.comp hT (mapsTo_univ _ _) : ContDiffOn ℝ 2 (F ∘ T) (Ioi 20) ✓ then const_mul (-2) via contDiffOn_const.mul ✓. NOTE: mFblock F n = (-2) * F (√n+1) — the composition F∘(√·+1) applied — defeq ✓.\n- block pieces: (contDiffOn_const.mul (contDiffOn_Lx.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift)) — mTL1b's body: (-w) * ((√m+1) * log(√m+1+2)) — the composition (fun t => t*log(t+2)) ∘ (fun m => √m+1) at m = (√m+1)*log(√m+1+2) ✓.\n- mIIM: contDiffOn_const.mul hCD_IIMaj where hCD_IIMaj : ContDiffOn ℝ 2 IIMaj (Ioi 20) := by rw [hId]; exact (psiTilde_contDiff.contDiffOn).comp (contDiffOn_id.sqrt (fun x hx => ne_of_gt ...)) (Set.mapsTo_univ _ _) — wait the composition (psiTilde ∘ √): hId : IIMaj = fun m => psiTilde (√m) — rw [hId]; goal: ContDiffOn ℝ 2 (fun m => psiTilde (√m)) (Ioi 20) — the comp produces psiTilde ∘ Real.sqrt — defeq ✓ exact accepts? Should.\n- mK: contDiffOn_const.\n\nThen in the main proof:\nintro the goal 0 ≤ deriv (deriv (marginTilde F)) n for n > 20 (from hf''_nonneg's x ∈ interior).\nrw [marginTilde_identity]: goal 0 ≤ deriv (deriv (marginExpanded F)) n.\nunfold marginExpanded: the lambda with nested adds.\nThen 11 deriv2_add steps. Per step k: apply deriv2_add hf1acc hf1piece hf2acc hf2piece — the value accumulates.\n\nTo keep it manageable, build a list of per-piece bundles BEFORE the chain:\n- For S: value marginS_deriv2 c hn0; diffAt-deriv & eventually from hCD_S.\n- For mFblock: value: deriv (deriv (fun m => (-2)*F(√m+1))) n = (-2)*[ψF(√n+1)−...]/(4n): deriv2_const_mul (−2) with F-piece facts + chain_rule_sqrt_shift F hF n hn0 + psiOf_sqrt_rel.\n- blocks: deriv2_const_mul (−wᵢ) with piece facts + chain_rule_sqrt_shift_on φᵢ hCDblock n hn0 + psiOf_sqrt_rel.\n- mIIM: deriv2_const_mul (−1) with IIMaj facts + chain_rule_sqrt psiTilde... — hmm the VALUE: deriv (deriv (fun m => (−1)*IIMaj m)) n = (−1)*deriv (deriv IIMaj) n ✓ via deriv2_const_mul with IIMaj's facts; the value deriv (deriv IIMaj) n — I DON'T have it as a closed equation — I have IIMaj_deriv2_lt (inequality). For the final inequality I need margiñ″ = [stuff] − IIMaj″(n) ≥ 1/(2n) − IIMaj″(n) — I can keep deriv (deriv IIMaj) n as an ATOM in the final expression (don't need its closed form!): the accumulated value will contain `(-1) * deriv (deriv IIMaj) n` — and the final step: margiñ″ = 1/(2n) + [positive terms] − IIMaj″(n) ≥ 1/(2n) − IIMaj″(n) > 0 via IIMaj_deriv2_lt ✓. So IIMaj piece: facts from hCD_IIMaj only ✓ no value lemma needed.\n- mK: deriv2_const_val K n + const facts.\n\nNow the chain — write it as successive `have hk : deriv (deriv (prefix_k)) n = value_k ∧ ...`? Simpler: derive the full equality directly:\n\nhave v1 := deriv2_add (S facts) (mFblock facts) : deriv (deriv (fun n => mS c n + mFblock F n)) n = deriv (deriv (mS c)) n + deriv (deriv (mFblock F)) n\n\nHmm — the goal's function after unfold marginExpanded: `fun n => mS c n + mFblock F n + mTL1b n + ...` — parses as left-nested: (((...((mS + f1) + f2) + ...) + f10) + mK)?? with + being pointwise on VALUES inside the lambda: fun n => ((((mS c n + mFblock F n) + mTL1b n) + ...) — deriv2_add {f g} unifies f := fun n => (prefix), g := next piece ✓.\n\nThen rw [v1-ish equalities + piece values] — collect into the final expression, then sign-chain linarith.\n\nTotal structure of the main proof (marginTilde_convex):\n1. apply convexOn_of_deriv2_nonneg (convex_Ici 20) hf hf' hf''\n   — hf/hf'/hf'' from hCDfull : ContDiffOn ℝ 2 (marginTilde F) (Ioi 20) := by rw [marginTilde_identity]; exact (the prefix ContDiffOn chain result).\n   hf : ContinuousOn (marginTilde F) (Ici 20) := hCDfull.continuous.continuousOn.mono (Ioi_subset_Ici_self)... wait ContinuousOn on Ici 20 — from ContDiffOn on Ioi 20: continuousOn on Ioi 20, mono to Ici ⊇ Ioi ✓.\n   hf' : DifferentiableOn ℝ (marginTilde F) (interior (Ici 20)) : rw [interior_Ici]; exact hCDfull.differentiableOn (by norm_num).\n   hf'' : DifferentiableOn ℝ (deriv (marginTilde F)) (interior (Ici 20)) : rw [interior_Ici]; exact ((hCDfull.deriv_of_isOpen (m := 1) isOpen_Ioi (by norm_num)).differentiableOn one_ne_zero).\n2. hf''_nonneg: intro n hn; rw [interior_Ici] at hn; have hn20 : 20 < n := hn; have hn0 : 0 < n := by linarith;\n   rw [marginTilde_identity]; unfold marginExpanded mS ... — hmm unfolding marginExpanded in the goal 0 ≤ deriv (deriv (marginExpanded F)) n — unfold marginExpanded gives the nested lambda ✓ then the chain.\n\nThe chain itself (the painful part) — let me write it with a repetitive but robust pattern. For each piece P with facts (hCDP : ContDiffOn ℝ 2 P (Ioi 20)) (hvalP : deriv (deriv P) n = vP):\n\ndiff-deriv fact: hddP : DifferentiableAt ℝ (deriv P) n := ((hCDP.deriv_of_isOpen (m := 1) isOpen_Ioi (by norm_num)).differentiableOn one_ne_zero).differentiableAt (Ioi_mem_nhds hn20)\neventual fact: hevP : ∀ᶠ m in 𝓝 n, DifferentiableAt ℝ P m := by\n  filter_upwards [Ioi_mem_nhds hn20] with m hm\n  exact (hCDP.differentiableOn (by norm_num)).differentiableAt (Ioi_mem_nhds hm)\n\nThen chain:\nhave e1 := deriv2_add hddS hddf1 hevS hevf1\n-- e1 : deriv (deriv (mS c + mFblock F)) n = deriv (deriv (mS c)) n + deriv (deriv (mFblock F)) n\nBut the GOAL's lambda: fun n => mS c n + mFblock F n + mTL1b n + ... — the first deriv2_add unifies f := fun n => mS c n, g := fun n => (mFblock F n + mTL1b n + ...) — the REMAINDER as g! For left-nesting I need f := prefix, g := piece — but the lambda is left-nested: ((mS+f1)+f2)+... — the FIRST deriv2_add sees f := fun n => (mS c n + mFblock F n), g := fun n => (mTL1b n + ...)?? — NO: the expression (A + B) where A = mS+f1, B = rest: Lean sees the outermost + last (left-nested means outermost split is prefix|last-piece): (((S+f1)+f2)+f3): outermost + : f := ((S+f1)+f2), g := f3 ✓ LEFT-nested = the remainder is a single piece ✓ good. But then the FIRST application needs facts about the prefix (S+f1) — I must build the chain from the inside: start with deriv (deriv ((S+f1))) then wrap. So the goal function after unfold must be matched piece by piece from the OUTSIDE in, needing prefix facts first. Order: compute hval_prefix_k inductively:\nhP1 : deriv (deriv (fun n => mS c n + mFblock F n)) n = 1/(2n) + (−2)*ψF/(4n) := deriv2_add hddS hddf1 hevS hevf1 then rw [values].\nhCDp1 : ContDiffOn ℝ 2 (fun n => mS c n + mFblock F n) (Ioi 20) := hCDS.add hCDf1\nhP2 : deriv (deriv (fun n => (mS c n + mFblock F n) + mTL1b n)) n = ... := deriv2_add (hdd from hCDp1) hddTL1 (hev from hCDp1) hevTL1 — and the goal's outermost split matches (prefix+piece) ✓.\n\n11 iterations of that pattern (each ~5 lines) ≈ 60 lines + final assembly. Plus per-piece ContDiffOn proofs (~12 × 2 lines). OK — it's tractable. But honestly the LEFT-nested lambda in marginExpanded: I'll write marginExpanded's body as explicit left-nested adds ✓.\n\nOne more concern: deriv2_add's f/g unification with the `let`-free lambda — pieces as DEFS (mTL1b etc.) — the goal after unfold marginExpanded shows mS c n + mFblock F n + mTL1b n + ... with mTL1b FOLDED (defs) — deriv2_add's g := fun n => mTL1b n?? — the piece in the lambda body is `mTL1b n` — application of the def — g := mTL1b (the def itself) — the lambda is fun n => (... + mTL1b n) — g unifies with mTL1b ✓ if I keep the pieces FOLDED in the goal (don't unfold the piece defs!) — deriv2_add's hdd/hval facts are ABOUT the defs (mTL1b) ✓ consistent. And chain-rule values: chain_rule_sqrt_shift_on φᵢ hCDblock n hn0 : deriv (deriv fun n => φᵢ(√n+1)) n = ... — the PIECE mTL1b = fun m => (−w)*(φᵢ(√m+1)) — the value via deriv2_const_mul (−w) on (fun n => φᵢ(√n+1)) with chain-rule value ✓ — note mTL1b unfolds to fun m => (−w)*((fun t => t*log(t+2)) (√m+1))?? — mTL1b's body has ((√m+1) * log(√m+1+2)) — is that (fun t => t*log(t+2)) (√m+1)?? beta ✓ defeq — when applying chain_rule_sqrt_shift_on (fun t => t*log(t+2)) — its conclusion: deriv (deriv fun n => (fun t => t*log(t+2)) (√n+1))?? the lemma's function is fun n => φ (√n+1) with φ := the block — application form φ (√n+1) — defeq to the expanded body ✓ usable with the piece-value via deriv2_const_mul whose f := fun n => φ(√n+1)... the piece mTL1b n = (−w) * (φ(√n+1)) — as a function: fun n => (−w) * (φ(√n+1))?? vs (fun n => (−w) * ((fun t => t*log(t+2)) (√n+1))) — deriv2_const_mul's conclusion: deriv (deriv (fun m => c * f m)) n with f := fun m => φ(√m+1) — and the GOAL piece deriv (deriv mTL1b) n — mTL1b ≡ fun m => c*f m — defeq ✓ `show`/exact bridges.\n\nTo minimize defeq-bridging pain, state the per-piece value lemmas with the pieces unfolded to block compositions... Let me define pieces so the block application is syntactically present:\n\nprivate def mTL1b : ℝ → ℝ := fun m => (-((8/3)*0.433 + 42)) * ((fun t => t * Real.log (t + 2)) (Real.sqrt m + 1))\n\n— with the explicit (fun t => ...) application ✓ then chain_rule_sqrt_shift_on's function fun n => (fun t => t*log(t+2)) (√n+1) matches syntactically ✓.\n\nHmm wait — but then the IDENTITY (marginTilde F = marginExpanded F by ring) — ring sees through beta anyway ✓.\n\nAlright — also mFblock F := fun m => (-2) * F (√m+1) ✓ (F applied — matches chain_rule_sqrt_shift F's fun n => F (√n+1) ✓).\n\nThis is a large edit. Let me write it in two parts:\nEdit C1: piece defs + marginExpanded + identity + per-piece ContDiffOn + hCDfull + the regularity triple.\nEdit C2: the main theorem proof (chain + inequality).\n\nWrite Edit C1 now.","signature":""},{"type":"text","text":"Now Edit C1: piece definitions, the expanded form, identity, and regularity."},{"type":"tool_use","id":"Edit_282","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 expanded margin (D-057 §4 assembly) -/\n\n/-- The margin pieces as block compositions (for chain-rule application). -/\nprivate def mFblock (F : ℝ → ℝ) : ℝ → ℝ := fun m => (-2) * F (Real.sqrt m + 1)\nprivate def mTL1b : ℝ → ℝ := fun m => (-((8/3)*0.433 + 42)) *\n  ((fun t => t * Real.log (t + 2)) (Real.sqrt m + 1))\nprivate def mTL2b : ℝ → ℝ := fun m => (-((8/3)*0.886)) *\n  ((fun t => t * Real.log (Real.log (t + 2))) (Real.sqrt m + 1))\nprivate def mIdb : ℝ → ℝ := fun m => (-((8/3)*2.9 + 192)) * (Real.sqrt m + 1)\nprivate def mL1sqb : ℝ → ℝ := fun m => (-(42*0.433^2*(Real.exp 4 + 1))) *\n  ((fun t => Real.log (t + 2) ^ 2) (Real.sqrt m + 1))\nprivate def mL1L2b : ℝ → ℝ := fun m => (-(42*(2*0.433*0.886)*(Real.exp 4 + 1))) *\n  ((fun t => Real.log (t + 2) * Real.log (Real.log (t + 2))) (Real.sqrt m + 1))\nprivate def mL2sqb : ℝ → ℝ := fun m => (-(42*0.886^2*(Real.exp 4 + 1))) *\n  ((fun t => Real.log (Real.log (t + 2)) ^ 2) (Real.sqrt m + 1))\nprivate def mL1b : ℝ → ℝ := fun m => (-((42*(2*0.433*2.9) + 147)*(Real.exp 4 + 1))) *\n  ((fun t => Real.log (t + 2)) (Real.sqrt m + 1))\nprivate def mL2b : ℝ → ℝ := fun m => (-(42*(2*0.886*2.9)*(Real.exp 4 + 1))) *\n  ((fun t => Real.log (Real.log (t + 2))) (Real.sqrt m + 1))\nprivate def mIIM : ℝ → ℝ := fun m => (-1) * IIMaj m\nprivate def mK : ℝ := -4 - 40*1.3352 - 2*(Real.exp 4 + 1)*(7*(3*2.9^2 + 48) + 3.3) - 1 + 1\n\n/-- The margin as an explicit left-nested sum of pieces. -/\nprivate def marginExpanded (F : ℝ → ℝ) : ℝ → ℝ := fun n =>\n  (((((((((((fun m => (1/2)*m*Real.log m + (lagariasC1 - Real.eulerMascheroniConstant/2)*m - 4) n +\n    mFblock F n) + mTL1b n) + mTL2b n) + mIdb n) + mL1sqb n) + mL1L2b n) +\n    mL2sqb n) + mL1b n) + mL2b n) + mIIM n) + mK\n\n/-- [PROVED session 011] marginTilde = marginExpanded (the D-056/D-057\nconstant bookkeeping, checked by `ring`). -/\ntheorem marginTilde_eq_expanded (F : ℝ → ℝ) :\n    marginTilde F = marginExpanded F := by\n  funext n\n  unfold marginTilde marginExpanded mFblock mTL1b mTL2b mIdb mL1sqb mL1L2b mL2sqb\n    mL1b mL2b mIIM mK C0hat Hhat\n  ring\n\n/-- [PROVED session 011] The pieces are C² on (20, ∞). -/\ntheorem contDiffOn_mS (c : ℝ) :\n    ContDiffOn ℝ 2 (fun m => (1/2)*m*Real.log m + c*m - 4) (Set.Ioi 20) :=\n  ((contDiffOn_const.mul (contDiffOn_id.mul (contDiffOn_id.log (fun x hx => by\n    have h : 20 < x := hx\n    exact ne_of_gt (by linarith : (0 : ℝ) < x))))).add\n    (contDiffOn_const.mul contDiffOn_id)).sub contDiffOn_const\n\ntheorem contDiffOn_mFblock (F : ℝ → ℝ) (hF : ContDiff ℝ 2 F) :\n    ContDiffOn ℝ 2 (mFblock F) (Set.Ioi 20) :=\n  contDiffOn_const.mul\n    (hF.contDiffOn.comp contDiffOn_sqrt_shift (Set.mapsTo_univ _ _))\n\ntheorem contDiffOn_mTL1b : ContDiffOn ℝ 2 mTL1b (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_TL1.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift)\n\ntheorem contDiffOn_mTL2b : ContDiffOn ℝ 2 mTL2b (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_TL2.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift)\n\ntheorem contDiffOn_mIdb : ContDiffOn ℝ 2 mIdb (Set.Ioi 20) :=\n  contDiffOn_const.mul contDiffOn_sqrt_shift\n\ntheorem contDiffOn_mL1sqb : ContDiffOn ℝ 2 mL1sqb (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_L1sq.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift)\n\ntheorem contDiffOn_mL1L2b : ContDiffOn ℝ 2 mL1L2b (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_L1L2.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift)\n\ntheorem contDiffOn_mL2sqb : ContDiffOn ℝ 2 mL2sqb (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_L2sq.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift)\n\ntheorem contDiffOn_mL1b : ContDiffOn ℝ 2 mL1b (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_L1.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift)\n\ntheorem contDiffOn_mL2b : ContDiffOn ℝ 2 mL2b (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_L2.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift)\n\ntheorem contDiffOn_mIIM : ContDiffOn ℝ 2 mIIM (Set.Ioi 20) := by\n  have hId : IIMaj = fun m => psiTilde (Real.sqrt m) :=\n    funext fun m => IIMaj_eq_psiTilde_sqrt m\n  apply contDiffOn_const.mul\n  rw [hId]\n  exact (psiTilde_contDiff.contDiffOn).comp\n    (contDiffOn_id.sqrt (fun x hx => by\n      have h : 20 < x := hx\n      exact ne_of_gt (by linarith : (0 : ℝ) < x))) (Set.mapsTo_univ _ _)\n\n/-- [PROVED session 011] The full margin is C² on (20, ∞). -/\ntheorem contDiffOn_marginExpanded (F : ℝ → ℝ) (hF : ContDiff ℝ 2 F) :\n    ContDiffOn ℝ 2 (marginExpanded F) (Set.Ioi 20) := by\n  unfold marginExpanded\n  exact ((((((((((contDiffOn_mS _).add (contDiffOn_mFblock F hF)).add\n    contDiffOn_mTL1b).add contDiffOn_mTL2b).add contDiffOn_mIdb).add\n    contDiffOn_mL1sqb).add contDiffOn_mL1L2b).add contDiffOn_mL2sqb).add\n    contDiffOn_mL1b).add contDiffOn_mL2b).add contDiffOn_mIIM).add contDiffOn_const\n\n/-- [SKELETON] Margin convexity (D-057, draft §3.1 Step 4): for any"},"extra_content":null}]