[{"type":"thinking","thinking":"Now I can see all the errors:\n\n1. **Line 257** (hcontJ/hcomp): `ContinuousAt.comp (hJcont t) (Continuous.continuousAt hpath)` — STILL the HO unification issue: expected `ContinuousAt (HAdd.hAdd ↑(-5/2)) (↑t * I)`. The elaborator decomposed `↑(-5/2) + ↑t * I` as `(HAdd.hAdd ↑(-5/2)) (↑t*I)` with f := `HAdd.hAdd ↑(-5/2)` (partial application!) and x := `↑t * I`. My ascription `ContinuousAt (Jlag n ∘ path) t` — the ∘ form: `Jlag n ∘ path` applied... the problem: ContinuousAt.comp (hg : ContinuousAt g (f x)) (hf : ContinuousAt f x) — from hg := hJcont t : ContinuousAt (Jlag n) (↑(-5/2)+↑t*I): unify `?f ?x` with `↑(-5/2)+↑t*I` — the elaborator picked ?f := HAdd.hAdd ↑(-5/2), ?x := ↑t*I (first-order approximation) instead of ?f := path, ?x := t. My ascription of hcomp's TYPE (ContinuousAt (Jlag n ∘ path) t) doesn't propagate INTO the arguments' unification because `exact hcomp` — wait, I wrote:\n\n```lean\n    have hcomp : ContinuousAt\n        (Jlag n ∘ fun u : ℝ => ((-5 / 2 : ℝ) + (u : ℝ) * Complex.I)) t :=\n      (hJcont t).comp hpath.continuousAt\n```\n\nThe ascription IS there. But the elaborator still assigns ?f ?x from hJcont t's type FIRST (elaborating the term `(hJcont t).comp hpath.continuousAt` against the ascribed type). The expected type gives `ContinuousAt (Jlag n ∘ path) t` — unifying with `ContinuousAt (?g ∘ ?f) ?x` — hmm, ContinuousAt.comp's result type: `ContinuousAt (g ∘ f) x`. Expected: `ContinuousAt (Jlag n ∘ path) t` — so g ∘ f ≡ Jlag n ∘ path, x := t. Elaborator unifies `?g ∘ ?f` with `Jlag n ∘ path` — HO again! It probably can't split. So it then uses hJcont t to determine g := Jlag n and decompose the point... yielding the bad split.\n\nRobust fix: use `ContinuousAt.comp` with EXPLICIT named arguments. Signature: `nonrec theorem ContinuousAt.comp {g : Y → Z} (hg : ContinuousAt g (f x)) (hf : ContinuousAt f x) : ContinuousAt (g ∘ f) x` — the f and x are section variables (implicit). So: `(hJcont t).comp (f := fun u : ℝ => ((-5/2 : ℝ) + (u : ℝ) * Complex.I)) (x := t) hpath.continuousAt`. Wait — with dot notation, hg is the self-arg. Named implicits f, x:\n\n```lean\n    exact ContinuousAt.comp (g := Jlag n)\n      (f := fun u : ℝ => ((-5 / 2 : ℝ) + (u : ℝ) * Complex.I)) (x := t)\n      (hJcont t) hpath.continuousAt\n```\n\nHmm — but are f, x implicit binders named f and x? From the source: `theorem ContinuousAt.comp {g : Y → Z} (hg : ContinuousAt g (f x)) (hf : ContinuousAt f x)` — f and x come from the section: likely `variable {f : X → Y} {x : X} {g : Y → Z}`. Names probably f and x. Risky. \n\nAlternative robust: `Continuous.tendsto`/`Continuous.comp'`... Or use `Continuous.comp`:\n\n```lean\n    exact hcontJ_global... \n```\n\nActually simplest: avoid ContinuousAt entirely — prove hcontJ via Continuous.comp':\n\n```lean\n  have hcontJ : Continuous fun t : ℝ =>\n      Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I) := by\n    apply Continuous.comp' ... \n```\n\nHmm — need `Continuous (Jlag n)` globally — NOT true (Jlag has singularities). So global comp fails; need the ContinuousAt version. \n\nAlternative: `ContinuousAt.comp_of_eq`? Or prove via `ContinuousAt.comp` but give the full type ascription on the TERM:\n\n```lean\n    have hf : ContinuousAt (fun u : ℝ => ((-5 / 2 : ℝ) + (u : ℝ) * Complex.I)) t :=\n      hpath.continuousAt\n    exact (hJcont t).comp hf\n```\n\nNow hf's type is pinned: ContinuousAt path t. Then (hJcont t).comp hf — g := Jlag n, f := path (from hf's type ✓ determined!), x := t ✓. hg := hJcont t : ContinuousAt (Jlag n) (↑(-5/2)+↑t*I) — expected ContinuousAt (Jlag n) (path t) — path t ≡ ↑(-5/2)+↑t*I by beta ✓ defeq ✓. This should work! The earlier failure was hpath.continuousAt with the UNPINNED point ?m.425. Pin via the have.\n\n2. **Line 293:39** \"No goals to be solved\" — line 293 is in hmid: `rwa [hcast] at h` — hmm no. Line 293 col 39 — let me count lines... 293 — probably the `by rw [hre]; norm_num` for hk? \"No goals\" means a tactic ran after the goal was closed: `by rw [hre]; norm_num` — rw [hre] CLOSED the goal (hre : re = -5/2, goal re ≤ 1/2 — rw gives -5/2 ≤ 1/2 — hmm rw doesn't close that)... OR the `by rw [hre]; norm_num` for hσlo in hRtight (goal -5/2 ≤ s.re: rw [hre] → -5/2 ≤ -5/2 → rw's auto-rfl... `-5/2 ≤ -5/2` — is that closed by rfl? `≤` on ℝ: `le_refl` — NOT rfl-defeq... hmm — but norm_num then has no goals if rw closed it? rw closes goals that become `a = a` or `True` via rfl — `-5/2 ≤ -5/2` is not rfl. Hmm.\n\nWait — which line is 293? Let me count from the file. Lines: the hmid block is around 280-300. `hk : ‖kFun n ...‖ ≤ 2 := kFun_norm_le_two (by rw [hre]; norm_num)` — col 39... \"No goals to be solved\" — if rw [hre] rewrites goal `(term).re ≤ 1/2` — wait, hre : (term).re = -5/2. rw [hre] at goal: replaces (term).re with -5/2 → goal `-5/2 ≤ 1/2` — NOT closed by rfl — then norm_num closes ✓. So that's not it.\n\nHmm, line 293 col 39. Let me look at the file around line 293. Actually — the `by rw [hre]; norm_num` appears TWICE in the hRtight line (hσlo and hσhi) and once in hmid (hk) and once in htail (hk). For hσlo : `-5/2 ≤ s.re`: rw [hre] : hre : s.re = -5/2 — rewrites `s.re` → `-5/2`: goal `-5/2 ≤ -5/2` — rw tries rfl — `le_refl (-5/2)`... is `a ≤ a` closed by `rfl`? In Lean 4 Mathlib, `≤` on ℝ is `Real.le` — defined how? Real.instLE: le := fun a b => ... Actually for REAL, `a ≤ a` — `Real.le` is defined as `LE.le` via... In Lean core, Real.le a b := ∃ ... no. Modern Mathlib: `Real.le` is defined via the order on Cauchy sequences?? No — Real is a quotient; but the `≤` instance: `instLEReal : LE ℝ` with le := Real.le, and `Real.le (x y : ℝ) : Prop := x < y ∨ x = y`?? Hmm — I recall `Real.le` IS defined as `x < y ∨ x = y` in Mathlib (Real.lt as... actually `Real.le_def : x ≤ y ↔ x < y ∨ x = y`). If Real.le x y is DEFINITIONALLY x < y ∨ x = y, then `a ≤ a` := `a < a ∨ a = a` — proved by `Or.inr rfl` — is that defeq to `rfl`? The rfl-closer in rw uses `Eq.refl`?? rw closes goals with `rfl` tactic — which tries `Eq.refl`, `Iff.refl`, `HEq.refl`... NOT Or.inr rfl. Hmm, but the `rfl` tactic also does `refl` for registered refl-lemmas? The `rfl` tactic: tries `Eq.refl` then `refl` attribute theorems (like `le_refl`!). The `refl` tactic extension finds `le_refl` tagged with @[refl] ✓. So rw's post-rw rfl closes `-5/2 ≤ -5/2` via le_refl ✓✓. THAT's the \"No goals to be solved\": rw [hre] CLOSED the hσlo goal, leaving `norm_num` with no goals!\n\nLine 293 col 39 — the first `(by rw [hre]; norm_num)` in the hRtight line (hσlo). Fix: use `(by rw [hre])` alone for hσlo (rw closes it), keep norm_num only where needed (hσhi: -5/2 ≤ 3 — rw [hre] → `-5/2 ≤ 3` — NOT closed by rw's rfl — norm_num needed ✓). So:\n- hσlo: `by rw [hre]` \n- hσhi: `by rw [hre]; norm_num`\n\nAnd in hmid/htail hk: goal `(term).re ≤ 1/2`: rw [hre] → `-5/2 ≤ 1/2` — needs norm_num ✓ keep.\n\nHmm wait — but actually is `-5/2 ≤ -5/2` closed by rw-rfl? The error says \"No goals to be solved\" at 293:39 — confirming yes for one of the sites. But which line is 293? Let me count precisely from my edit. The hRtight line:\n\n```\n    have hRtight := logDeriv_riemannZeta_sub_nearZeros_le_tight (t := t) (s := s)\n      hZC hZC2 hNLZ ht10 (by rw [hre]; norm_num) (by rw [hre]; norm_num) him\n      (zeta_ne_zero_left_vertical t)\n```\n\nIf 293:39 is the hσlo by-block here — then hmid's hk by-block is around line 285... hmm — but if rw [hre]; norm_num failed at hσlo with \"no goals\", it means rw closed it. But then the SAME pattern at hmid's hk (goal (term).re ≤ 1/2) — rw [hre] gives -5/2 ≤ 1/2 — NOT closed — norm_num fine ✓ no error there. And htail's hk — same ✓. So only the hσlo site errors ✓ consistent with \"293:39\" being a single location. \n\nBut wait — \"No goals to be solved\" at col 39 — the hσlo `by rw [hre]; norm_num` — col 39 could be where `norm_num` starts... roughly. OK.\n\nActually hold on — could 293 be in hmid instead? Let me count lines from my written file. I'll count: the §2 block starts around line 208. Z2hat_nonneg ~10 lines → vIntegral starts ~222. hmid around 283-300. hRtight around 315. Line 293 falls in hmid: `hk : ‖kFun n ((-5/2 : ℝ) + (t:ℝ)*Complex.I)‖ ≤ 2 := kFun_norm_le_two (by rw [hre]; norm_num)` — col 39 would be mid-line... that by-block: goal `(term).re ≤ 1/2` — rw [hre] rewrites to `-5/2 ≤ 1/2` — hmm rw's rfl tries le_refl — `-5/2 ≤ 1/2` is NOT refl — so norm_num runs ✓ no \"no goals\" error. Contradiction. So 293 must be the hσlo site. I'll fix by making hσlo `by rw [hre]` — wait but if rw leaves `-5/2 ≤ -5/2` and closes via rfl ✓ done — `by rw [hre]` suffices. But SAFER (avoid relying on rw-rfl-le_refl): `by rw [hre]` — if rw doesn't close it, error \"unsolved goals\" — then I'd see it. Alternatively write `(ge_of_eq hre.symm)`... wait direction: ge_of_eq (h : a = b) : a ≥ b. hre : s.re = -5/2 — hre.symm : -5/2 = s.re — ge_of_eq hre.symm : -5/2 ≥ s.re — WRONG. `le_of_eq hre.symm` : le_of_eq (h : a = b) : a ≤ b — hre.symm : -5/2 = s.re gives -5/2 ≤ s.re ✓! So hσlo := `le_of_eq hre.symm`. Clean, no rw. \n\nHmm — actually simplest that definitely works: `hre.ge` — `Eq.ge : a = b → a ≥ b` — hre.ge : s.re ≥ -5/2 i.e. -5/2 ≤ s.re ✓. Or `hre.symm.le`? Eq.le : a = b → a ≤ b — hre.symm.le : -5/2 ≤ s.re ✓. I'll use `hre.symm.le`.\n\nWait no — hre : s.re = -5/2. hre.symm : -5/2 = s.re. hre.symm.le : -5/2 ≤ s.re ✓ YES.\n\nAnd hσhi : s.re ≤ 3 := by rw [hre]; norm_num — after rw goal -5/2 ≤ 3 — norm_num ✓ keep. But wait — \"No goals\" issue here? rw [hre] → `-5/2 ≤ 3` — not refl — norm_num runs ✓ fine.\n\n3. **Line 333** (hlog's norm_le_insert): TYPE MISMATCH — `norm_le_insert ?a ?b : ‖?b‖ ≤ ‖?a‖ + ‖?a - ?b‖` — I had the ARGUMENT ORDER WRONG! `norm_le_insert (a b) : ‖a‖ ≤ ‖b‖ + ‖a - b‖`?? From the alias: `norm_le_insert := norm_le_norm_add_norm_sub`. And the error shows: `norm_le_insert ?m.1567 ?m.1568 has type ‖?m.1568‖ ≤ ‖?m.1567‖ + ‖?m.1567 - ?m.1568‖`. So `norm_le_insert a b : ‖b‖ ≤ ‖a‖ + ‖a - b‖`. I want: ‖logDeriv‖ ≤ ‖Σ‖ + ‖logDeriv − Σ‖ — hmm: ‖logDeriv‖ = ‖Σ + (logDeriv − Σ)‖... The lemma with a := Σ, b := logDeriv: ‖logDeriv‖ ≤ ‖Σ‖ + ‖Σ − logDeriv‖ — gives ‖Σ − logDeriv‖ not ‖logDeriv − Σ‖. Mismatch: my stated h1 has `‖logDeriv riemannZeta s - Σ‖` as the second term — the lemma gives ‖Σ − logDeriv‖. Fix: state h1 with the sum SUBTRACTED the other way: `‖logDeriv‖ ≤ ‖Σ‖ + ‖Σ - logDeriv‖`?? Then I can't directly use hRtight (‖logDeriv − Σ‖ ≤ ...). Need norm_sub_rev: ‖a − b‖ = ‖b − a‖. OR use norm_le_insert' — the OTHER alias: `norm_le_insert' := norm_le_norm_add_norm_sub'` : ‖?a‖ ≤ ‖?b‖ + ‖?a - ?b‖? From MStructure usage: `norm_le_insert' (R • x) (R • P • R • x)` proving `‖v 0‖ ≤ ‖h‖ + ‖v 0 - h‖` (ControlledClosure line 87: `‖v 0‖ ≤ ‖h‖ + ‖v 0 - h‖ := norm_le_insert' _ _`). So `norm_le_insert' (a b) : ‖a‖ ≤ ‖b‖ + ‖a - b‖` — EXACTLY what I need: a := logDeriv ζ s, b := Σ: ‖logDeriv‖ ≤ ‖Σ‖ + ‖logDeriv − Σ‖ ✓.\n\nFix: `norm_le_insert' _ _`.\n\n4. **Line 366** (hleft's integral_comp_neg): the `_` placeholder couldn't be inferred: `intervalIntegral.integral_comp_neg ?m.2050` — the HO unification `?f (-x) ≡ ‖Jlag n (↑(-5/2) + ↑(-t) * I)‖` failed — \"has type @intervalIntegral ... (fun x => ?f (-x)) ... = ∫ x in -?b..-?a, ?f x but expected ...\" — the metavariable ?f couldn't be resolved by unification (the fun x => ?f (-x) vs fun t => ‖Jlag n (↑(-5/2)+↑(-t)*I)‖ — pattern ?f (-x) — should be Miller... but maybe the coercion ↑(-t) makes it not a Miller pattern? ?f applied to `-x` (a bound-var-headed term) — `↑(-x)`? Actually the integrand is `‖Jlag n (↑(-5/2) + ↑(-x) * I)‖` — ?f's argument is `-x` — Miller pattern (?f applied to distinct bound vars/function of bound var) — `?f (-x)` is a Miller pattern with argument `-x` — solvable ?f := fun u => ‖Jlag n (↑(-5/2)+↑u*I)‖... but Lean's unifier may have failed because... the expected type has `@intervalIntegral ℝ Real.normedAddCommGroup RCLike.toInner...` — wait, look at the mismatch: `?m.2046` (the NormedAddCommGroup) vs `Real.normedAddCommGroup` — the F's codomain got unified with ℝ ✓ fine — the actual failure: the unifier couldn't assign ?m.2050 (?f). Because I passed `_` — Lean tried to elaborate `integral_comp_neg _` THEN unify with the ascription — the `_` stays a metavariable; the unification `fun x => ?f (-x) =?= fun t => ‖Jlag n (↑(-5/2) + ↑(-t)*I)‖` — HO — Lean might fail this particular HO (the argument -x is fine Miller, but maybe the issue is intervalIntegral's first args ?m.2045/2046/2047 also metavariables, blocking). Solution: pass f EXPLICITLY (it's an explicit arg!):\n\n```lean\n      _ = ∫ t in -(-10 : ℝ)..-(-T),\n            ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ :=\n          intervalIntegral.integral_comp_neg\n            (fun t : ℝ => ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖)\n```\n\nWith f pinned, the LHS `∫ x in ?a..?b, (fun t => ‖Jlag n (↑(-5/2)+↑t*I)‖) (-x)` — beta ≡ `∫ x in ?a..?b, ‖Jlag n (↑(-5/2)+↑(-x)*I)‖` — unify with the ascription's LHS ✓ ?a := -T, ?b := -10 ✓.\n\n5. **Line 375:80 \"unsolved goals\"** — the hsplit block: after the two rewrites, goal `A_x + (B_x + C_x) = A_t + (B_t + C_t)` — NOT closed by rw's rfl?? The context shows hsplit's proof state with that goal remaining. Why didn't rfl close it? Alpha-equivalence IS handled by rfl... Hmm — wait, look at the goal: LHS `A + (B + C)` (right-nested ✓), RHS — the hsplit STATEMENT's RHS — displayed `A_t + B_t + C_t` — hmm the statement I wrote was `A + (B + C)`. But maybe the elaborator re-associated? No. Look at error 375's goal RHS: `∫ -T..-10, F + ∫ -10..10, F + ∫ 10..T, F` — could be either nesting (parens elided in display). If it's `A + (B + C)` ✓ same as LHS modulo alpha — rfl should close...\n\nUNLESS the RHS is actually `(A + B) + C` (left-nested) — did I write left-nested? My hsplit statement:\n\n```\n      ∫ t in (-T)..(-10 : ℝ), F... +\n        (∫ t in (-10 : ℝ)..(10 : ℝ), F... +\n          ∫ t in (10 : ℝ)..T, F...)\n```\n\nI explicitly parenthesized (B + C) ✓ right-nested. Hmm — but wait, the `+` at line end: `A + \\n (B + C)` — parses as A + (B+C) ✓.\n\nSo why \"unsolved goals\"?? OH WAIT — I see it — line 375:80 — col 80 — that's not the hsplit rw failing to close; it's the SECOND rw line? Let me look at my hsplit proof:\n\n```\n    rw [← intervalIntegral.integral_add_adjacent_intervals (hI (-T) (-10))\n      (hI (-10) T),\n      ← intervalIntegral.integral_add_adjacent_intervals (hI (-10) 10) (hI 10 T)]\n```\n\nLine 375 col 80 — hmm, `(hI 10 T)` is around col 80. \"unsolved goals\" with the full context displayed — this error is at the END of the rw (after both rewrites, the rfl didn't close). So the remaining goal `A_x+(B_x+C_x) = A_t+(B_t+C_t)` was left. rw's auto-rfl: uses `Eq.refl` — for a goal `X = Y` with X, Y alpha-equivalent — `Eq.refl` proves `X = X` — needs defeq X ≡ Y — alpha IS defeq ✓ should close!! Unless the rewrites produced something subtly different — like the second rewrite ALSO rewriting the newly-created subterm in an unexpected spot, or the endpoints differing: `(hI (-10) T)` — the b-point: first rewrite: a=-T, b=-10, c=T ✓. Second: a=-10, b=10, c=T ✓.\n\nHmm wait — maybe the problem: after the FIRST rewrite, the goal's LHS became `(∫ x in -T..-10, F x) + ∫ x in -10..T, F x`. The SECOND rewrite pattern: `∫ x in -10..T, F x` — but ALSO — hold on, could the second rewrite have rewritten the WRONG occurrence or... there's only one.\n\nLet me think about the ACTUAL displayed unsolved goal once more:\n\nLHS: `(∫ (x : ℝ) in -T..-10, F) + ((∫ (x : ℝ) in -10..10, F) + ∫ (x : ℝ) in 10..T, F)`\nRHS: `∫ (t : ℝ) in -T..-10, F + ∫ (t : ℝ) in -10..10, F + ∫ (t : ℝ) in 10..T, F`\n\nIf RHS is `(A + B) + C` (left-nested) then goal = `A+(B+C) = (A+B)+C` — NOT rfl! Then the error is my hsplit STATEMENT — maybe the elaborator parsed `A + \\n (B + C)` differently... OR the calc step that consumed hsplit re-associated? The unsolved goal is inside hsplit's proof (before the calc). Hmm — actually WAIT. Maybe I mis-transcribed and the statement's RHS parsed as left-nested because the parens... let me look at my edit text again:\n\n```\n  have hsplit : ∫ t in (-T)..T, ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ =\n      ∫ t in (-T)..(-10 : ℝ), ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ +\n        (∫ t in (-10 : ℝ)..(10 : ℝ),\n          ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ +\n          ∫ t in (10 : ℝ)..T, ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖) := by\n```\n\nRight-nested ✓ definitely. So the goal after rewrites is `A_x + (B_x + C_x) = A_t + (B_t + C_t)` — defeq (alpha) — rw's rfl should close!!! Why not?\n\nHmm — is it possible the rewrites did NOT fire and the \"unsolved goals\" is the rw COMPLAINING...? No — \"unsolved goals\" at 375:80 with the displayed post-rewrite goal means the rewrites FIRED and left the goal open. rw auto-runs rfl only at the END of a rw sequence — it should have closed. Unless... rfl failed due to the integrals being non-rfl-defeq for a DEEPER reason: the first integral's integrand uses bound var x, RHS uses t — alpha ✓ defeq. Hmm, unless `integral_add_adjacent_intervals`'s instance produced DIFFERENT endpoints: `-10` from `(hI (-10) T)` vs `-10` in the goal ✓ same.\n\nOH WAIT. I bet I know — the `↑(-t)` in... no. Hmm. Actually — let me reconsider: maybe rw DID close the goal and \"unsolved goals\" refers to something else at col 80: `(hI 10 T)` — hmm `(hI 10 T)` — the `10` here — hI's args are a b : ℝ — `10` elaborates as ℝ ✓. \n\nHmm, actually — let me reconsider whether the error is \"unsolved goals\" from the `have hsplit` as a whole: after `rw [...]` the tactic block ended with goals remaining (the rfl didn't close). Lean reports \"unsolved goals\" at the END of the `by` block — location pointing to the last tactic. So yes — the two rewrites happened, rfl didn't close.\n\nWhy wouldn't rfl close `A_x+(B_x+C_x) = A_t+(B_t+C_t)`?? Alpha-equivalence should make them defeq... UNLESS they're not alpha-equivalent because the ENDPOINTS differ: LHS second endpoint `-10` from hI (-10)... in the second rewrite: `(hI (-10) 10)` and `(hI 10 T)`: lemma instance b = 10: `∫ x in -10..10, F x` ✓ `∫ x in 10..T, F x` ✓. RHS `∫ t in -10..10, F` `∫ t in 10..T, F` ✓ match.\n\nFirst rewrite: `∫ x in -T..-10, F x + ∫ x in -10..T, F x` — the endpoints `-T`, `-10`, `T` ✓.\n\nHmm hmm. So both sides ARE alpha-equal. Why no rfl?? \n\nOH. I think I finally see it. rw's auto-rfl: `rw` tries to close the goal with `rfl` ONLY when the rewrite produces a goal of the form `a = a` or `True`... no — rw always attempts `rfl` (with the `Eq.refl`/`Iff.refl` + `refl` lemmas) after rewriting. For `X = Y` with X ≡alpha Y — `Eq.refl` : X = X — unify X =?= Y — defeq ✓. This MUST close. Unless the goal is not `=` but something else... The displayed goal IS `=`. \n\nHmm, wait — maybe the issue is that the rewrites left metavariables? The hI instances are concrete...\n\nLet me look at col 375:80 more carefully. Line 375 might be the line with `← intervalIntegral.integral_add_adjacent_intervals (hI (-10) 10) (hI 10 T)]`. Col 80 ≈ `(hI 10 T)`. \"unsolved goals\" — hmm — this could be an error from ELABORATING the rw rule: the second rewrite rule instance `(hI (-10) 10) (hI 10 T)` — the lemma `integral_add_adjacent_intervals hab hbc : A + B = C` — the REVERSED rewrite `←`: pattern C = `∫ x in -10..T, F x`. In the goal (after first rewrite), occurrences of `∫ x in -10..T, F x`: ONE (in LHS). Rewrite to `∫ -10..10 + ∫ 10..T` ✓. Then rfl: `A + ((B + C)) = A + (B + C)` alpha ✓...\n\nOK you know what — I'll stop theorizing. The pragmatic fix: end hsplit's proof with `; ring` or explicit: since the goal after rewrites is X = Y with alpha-equivalent sides, close with `rfl` explicitly... if plain rfl fails, something deeper is wrong and I'll see its error. Actually — safer: restructure hsplit to avoid the issue entirely:\n\n```lean\n  have hsplit : ... := by\n    have h1 := intervalIntegral.integral_add_adjacent_intervals (hI (-T) (-10)) (hI (-10) T)\n    have h2 := intervalIntegral.integral_add_adjacent_intervals (hI (-10) 10) (hI 10 T)\n    rw [← h1, ← h2]\n```\n\nSame thing. Alternatively — maybe the problem is genuinely alpha — and rw's rfl DOES handle alpha... I'm quite sure it does.\n\nHmm wait — actually, let me re-read the displayed unsolved goal ONE more time, character by character:\n\n```\n⊢ (∫ (x : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑x * I)‖) +\n      ((∫ (x : ℝ) in -10..10, ‖Jlag n (↑(-5 / 2) + ↑x * I)‖) + ∫ (x : ℝ) in 10..T, ‖Jlag n (↑(-5 / 2) + ↑x * I)‖) =\n    ∫ (t : ℝ) in -T..-10,\n      ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ +\n        ∫ (t : ℝ) in -10..10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ + ∫ (t : ℝ) in 10..T, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖\n```\n\nThe LEFT side's OUTER parens: `(∫ -T..-10) + ((∫ -10..10) + ∫ 10..T)`. The RIGHT side: `∫ -T..-10 + ∫ -10..10 + ∫ 10..T` — the pretty printer with elided parens... For right-nested `A + (B + C)` the printer shows `A + (B + C)` — with parens? Lean's pretty printer ELIDES parens based on precedence — `+` is right-associative in Lean 4!! `a + b + c` PARSES as `a + (b + c)` (right-assoc, precedence 65). And prints WITHOUT parens when right-nested: `a + b + c`. Left-nested `(a + b) + c` prints WITH parens: `(a + b) + c`.\n\nSo: the LHS displays `(A) + ((B) + (C))` — the explicit parens on each integral are just the printer around ∫-terms. `(A) + ((B) + (C))` = A + (B + C) right-nested — printer shows parens only where needed for clarity... it showed `((∫...) + (∫...))` — hmm the outer parens around the integrals themselves (∫ terms are atoms, printer may parenthesize them under + for clarity).\n\nThe RHS `A + B + C` — right-nested, no parens needed. So RHS IS right-nested `A + (B + C)` ✓ SAME as LHS modulo alpha. rfl should close!!!\n\nArgh. OK — unless the error is NOT at the rfl stage but the SECOND rw failed to find its pattern and the \"unsolved goals\" is a CASCADE: if the second rewrite failed with \"pattern not found\", we'd see that error instead... \n\nActually — WAIT. Read the error list order: 375:80 \"unsolved goals\" — then 392:12 rw hleft pattern-not-found. hsplit (375) FAILED with unsolved goals — BUT hsplit still appears in the context at 392! When a `have` fails, Lean inserts a sorry and CONTINUES — hsplit : ... exists as a sorry-backed hypothesis. So 392's rw [hleft] — hleft also appears ✓ (its proof had error 366, sorry-backed). The 392 error: rw [hleft] pattern not found in the calc goal `A + (B + C) ≤ RHS`... the pattern `∫ t in -T..-10, F` IS the A — should be found! Unless... OH WAIT. The calc step:\n\n```\n    _ = ∫ t in (-T)..(-10 : ℝ), F + (∫ -10..10 + ∫ 10..T) := hsplit\n    _ ≤ 4 * T * ... := by rw [hleft]; ...\n```\n\nThe `_ ≤ RHS` step's starting point is the previous calc line's RHS: `∫ -T..-10, F + (∫ -10..10, F + ∫ 10..T, F)`. rw [hleft] : pattern `∫ t in -T..-10, F` — present ✓✓✓. But the error says NOT FOUND?!?!\n\nUnless the calc's previous line didn't elaborate (because hsplit is sorry'd?) — no, the goal DISPLAYED at 392 shows exactly `A + (B + C) ≤ RHS` with A = `∫ t in -T..-10, F`.\n\nPattern: `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖`.\nA: `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖`.\n\nIDENTICAL. Why would rw not find it?? \n\nOHHHH. I see it now!!! Look at hleft's statement — its LHS integral is `∫ (t : ℝ) in -T..-10, ...` — and in hleft's CONTEXT (error 375's context): `hleft : ∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ = ∫ (t : ℝ) in 10..T, ...` ✓. And the pattern printed in error 392: `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖` ✓ same. Target A ✓ same...\n\nHmm hold on, maybe the issue is that hleft is SORRY-BACKED (error 366 killed its proof) — no, that doesn't affect rw pattern matching (hleft : LHS = RHS as a term exists regardless).\n\nWAIT — maybe error 392's target is NOT what I think. Re-read:\n\n```\nNguyenRH1/LiSideBounds.lean:392:12: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern\n  ∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖\nin the target expression\n  ∫ (t : ℝ) in -T..-10,\n      ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ +\n        ∫ (t : ℝ) in -10..10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ + ∫ (t : ℝ) in 10..T, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ ≤\n    4 * T * (2 / 3 * Z2hat T + 10.5 * Real.log (T + 2) + 48) + 40 * 1.3352\n```\n\nThe target displayed IS `A + B + C ≤ RHS`. And the pattern is A. NOT FOUND?! \n\nUnless... the target's A and the pattern differ in the DISPLAYED-BUT-INVISIBLE part: the integrand! Pattern integrand: `‖Jlag n (↑(-5 / 2) + ↑t * I)‖`. Target integrand: same...\n\nHmm — could there be TWO different `I`s or norms? No.\n\nCould `-T` differ: pattern `-T` vs target `-T` — T is the same local.\n\nCould the difference be `-10`: pattern `..-10` — from hleft's statement `(-10 : ℝ)` — elaborates `Neg.neg (10 : ℝ)` where `10` is `OfNat.ofNat 10`. Target's `-10` — from MY calc statement `(-10 : ℝ)` — same elaboration ✓.\n\nI'm clearly missing something. Let me think about 375's \"unsolved goals\" differently: if rfl failed to close `X = Y` (alpha-equal), maybe they're NOT alpha-equal — e.g., the second rewrite's `10` endpoints: `(hI (-10) 10)` gives b := 10 — `∫ x in -10..10, F x`. RHS's B: `∫ t in (-10 : ℝ)..(10 : ℝ), F` ✓ same.\n\nHmm — what if the rewrites fired in a DIFFERENT place: the FIRST reversed rewrite `∫ x in -T..T, F x` — does this pattern occur anywhere ELSE in the goal? The RHS of hsplit's statement — no ∫-T..T there ✓. \n\nWhat if `integral_add_adjacent_intervals`'s μ unification picked a DIFFERENT measure? hI uses MeasureTheory.volume ✓.\n\nOK — radical idea: maybe the \"unsolved goals\" error at 375:80 is NOT about rfl at all — col 80 is inside the rw RULE expression: `← intervalIntegral.integral_add_adjacent_intervals (hI (-10) 10) (hI 10 T)]` — elaborating `hI 10 T`: `hI : ∀ a b : ℝ, IntervalIntegrable F volume a b` — `hI 10 T` — WAIT. `10` and `T` — FINE. Hmm — but what about `hI (-10) 10` — fine too.\n\nActually — \"unsolved goals\" errors display the remaining goals at the END of a tactic block. The position 375:80 — the position of the whole `by` block or the last tactic. The displayed goal is the post-rw goal. So rfl DID fail. Since alpha-equivalence can't fail...\n\nLet me question my assumption: are the two sides REALLY alpha-equal? The LHS integrals have bound var `x` (from the lemma), RHS `t`. In Lean 4 kernel, `∀`/`∫` binders with different names are alpha-EQUIVALENT and the kernel's defeq DOES handle alpha. `rfl` via Eq.refl — kernel defeq — alpha ✓. IT MUST CLOSE.\n\n...unless the sides differ in some OTHER invisible way. What could differ between the rewrite-produced integrals and my stated ones? THE INTEGRAND: lemma-produced: `(fun t => ‖Jlag n (↑(-5/2) + ↑t * I)‖) x` — a BETA-REDEX if not reduced during elaboration! If the rewrite produced `∫ x in -T..-10, (fun t => ‖Jlag n (↑(-5/2)+↑t*I)‖) x` (redex), the printer would still show `‖Jlag n (↑(-5/2)+↑x*I)‖` (printer beta-reduces for display!). And `rfl` — kernel defeq handles beta ✓ closes anyway.\n\nI give up theorizing — pragmatic: replace the rw-proof of hsplit with a calc-style proof that closes each step explicitly, and for the final assembly replace `rw [hleft]` with `conv` or a `calc`-rewriting approach. OR simply: after the two rewrites, add explicit `rfl`-attempt and see... Can't iterate blindly — each lake run is ~1 min. Let me make the robust changes:\n\nFor hsplit — avoid the reversed-lemma rewrite; instead prove the equation forward:\n\n```lean\n  have hsplit : ∫ t in (-T)..T, F = A + (B + C) := by\n    rw [← intervalIntegral.integral_add_adjacent_intervals (hI (-10) 10) (hI 10 T),\n      ← intervalIntegral.integral_add_adjacent_intervals (hI (-T) (-10)) (hI (-10) T)]\n```\n\nOrder swapped: first rewrite RHS's `(B + C)`-part? The goal: `∫ -T..T, F = A + (B + C)`. rw [← lemma2] : pattern `∫ x in -10..T, F` — occurs in... NOWHERE in the goal! The goal has ∫(-T..T) and A(∫-T..-10) and B(∫-10..10) and C(∫10..T). No ∫(-10..T). So that order fails.\n\nThe original order: first `← lemma1` (pattern ∫(-T..T)) — occurs once (LHS) ✓ → LHS becomes ∫(-T..-10) + ∫(-10..T). Then `← lemma2` (pattern ∫(-10..T)) ✓ → ∫(-10..10)+∫(10..T). Final: LHS = A+(B+C) syntactically (with x-binders). Goal: A+(B+C) = A+(B+C) — rfl. This SHOULD work...\n\nYou know — maybe the real culprit: after the FIRST rewrite, does the goal's RHS also contain a match for the FIRST pattern? No (RHS has no ∫-T..T). Hmm.\n\nLet me try yet another lens: maybe `rw` failed NOT at rfl but during the second rewrite — the pattern `∫ x in -10..T, (fun t => ‖Jlag n (↑(-5/2)+↑t*I)‖) x` vs the goal's `∫ x in -10..T, ‖Jlag n (↑(-5/2)+↑x*I)‖` — the rewrite rule (from hI-typed instance): the integrand in the rule is `F' x` where F' := the lambda from hI — elaborated as redex or reduced? If the rule's integrand is `(fun t => ...) x` (redex) and the goal's is the REDUCED `‖Jlag n (↑(-5/2)+↑x*I)‖` — rw matching: `rw` matches patterns up to \"reducible\" defeq? rw uses `kabstract` — keyed matching at REDUCIBLE transparency — beta-redexes... kabstract should handle beta. Hmm — but the FIRST rewrite worked (the error only complains at the END) — actually do we KNOW the first rewrite worked? The displayed post-failure goal shows LHS fully split — both rewrites FIRED ✓.\n\nSo genuinely: after both rewrites, goal `LHS' = RHS` with LHS' ≡alpha RHS, and rw's rfl DIDN'T close. This contradicts my understanding of rw... UNLESS the two sides are NOT alpha-equal due to different `10`/`-10` ELABORATION: in `(hI (-10) 10)`, the `10 : ℝ` — OfNat 10. In my statement, `(10 : ℝ)` — OfNat 10 ✓ same. `-10`: `Neg.neg (OfNat 10)` both ✓.\n\nWAIT WAIT WAIT. Look at the displayed goal once more — the SECOND integral on the LHS: `∫ (x : ℝ) in -10..10` and on the RHS: `∫ (t : ℝ) in -10..10`. The FIRST integral: LHS `-T..-10` RHS `-T..-10`. THIRD: `10..T` both. Integrands ✓. Nesting: LHS `(A) + ((B) + (C))`, RHS `A + B + C`.\n\nHmm — if `+` were LEFT-assoc, RHS `A + B + C` = `(A+B)+C` ≠ LHS `A+(B+C)` — NOT defeq, rfl fails, \"unsolved goals\" ✓✓✓. Is `+` right or left associative in Lean 4?? HAdd.hAdd notation `+` : `infixl:65 \"+\" => HAdd.hAdd` — **infixl** — LEFT-ASSOCIATIVE!!! \n\nTHAT'S IT!!! `a + b + c` parses as `(a + b) + c`. My hsplit RHS: I wrote `A + \\n (B + C)` — with EXPLICIT parens around B + C — so it parsed as `A + (B + C)` — RIGHT-nested by explicit parens. The rewrites produced `A + (B + C)` too. Both right-nested... then they'd match...\n\nBUT — the printer: for right-nested `A + (B + C)` with left-assoc `+`, the printer MUST print `A + (B + C)` (parens required, else misread). The displayed RHS `A + B + C` (no parens) = LEFT-nested `(A + B) + C`!!! So my statement's RHS... did the parens get LOST? I wrote:\n\n```\n      ∫ t in (-T)..(-10 : ℝ), ‖...‖ +\n        (∫ t in (-10 : ℝ)..(10 : ℝ), ‖...‖ +\n          ∫ t in (10 : ℝ)..T, ‖...‖)\n```\n\nParens present ✓. Hmm — but the DISPLAY at 375 shows RHS without parens... contradiction? Unless the elaborator... no, elaboration preserves structure. The printer at 375's RHS: `∫ -T..-10, F + ∫ -10..10, F + ∫ 10..T, F` — if this were right-nested, the printer would show `A + (B + C)`... \n\nHmm — actually maybe Lean's printer for `HAdd.hAdd A (HAdd.hAdd B C)` prints `A + (B + C)` WITH parens ✓, and for `(A+B)+C` prints `A + B + C`. The display shows NO parens ⟹ RHS is LEFT-nested `(A+B)+C`. So my written `A + (B + C)` became left-nested?!? IMPOSSIBLE unless... the parens I wrote were consumed differently: `(∫ -10..10, F + ∫ 10..T, F)` — wait, maybe I DIDN'T write the parens correctly — the open-paren was BEFORE `∫ -10..10` ✓ and close after the ∫ 10..T term ✓. That gives A + (B + C)...\n\nHold on, let me look at 392's target again:\n\n```\n∫ (t : ℝ) in -T..-10, F +\n    ∫ (t : ℝ) in -10..10, F + ∫ (t : ℝ) in 10..T, F ≤\n```\n\nno parens — left-nested. And the LHS at 375: `(∫ -T..-10, F) + ((∫ -10..10, F) + ∫ 10..T, F)` — WITH parens — right-nested (explicit). So LHS (rewrites) = A + (B + C), RHS (my statement) = (A + B) + C. MY STATEMENT'S RHS IS LEFT-NESTED! But I wrote parens!!!\n\nUnless — OH NO. I see it. In my hsplit STATEMENT (as written in the Edit), let me re-examine:\n\n```\n      ∫ t in (-T)..(-10 : ℝ), ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ +\n        (∫ t in (-10 : ℝ)..(10 : ℝ),\n          ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ +\n          ∫ t in (10 : ℝ)..T, ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖) := by\n```\n\nThis is `A + (B + C)` — explicit parens. The display shows `(A+B)+C`. CONTRADICTION. Unless the file got written differently than I think — let me READ the actual file around hsplit to see what's really there! Maybe my Edit string didn't have the parens (I should check what I actually sent). Looking at my edit payload above... In the edit I wrote:\n\n```\n  have hsplit : ∫ t in (-T)..T, ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ =\n      ∫ t in (-T)..(-10 : ℝ), ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ +\n        (∫ t in (-10 : ℝ)..(10 : ℝ),\n          ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ +\n          ∫ t in (10 : ℝ)..T, ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖) := by\n```\n\nParens there ✓. Hmm — but ALSO the calc's `=`-step:\n\n```\n    _ = ∫ t in (-T)..(-10 : ℝ), ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ +\n          (∫ t in (-10 : ℝ)..(10 : ℝ),\n            ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ +\n            ∫ t in (10 : ℝ)..T,\n              ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖) := hsplit\n```\n\nAlso right-nested ✓.\n\nSo both should be right-nested. The 392 target `A + B + C ≤ ...` (no parens, left-nested) — CONTRADICTS my written calc. UNLESS the printer elides... Ugh.\n\nOK actually — hmm, maybe I have the associativity backwards: if `+` is `infixl` (left-assoc), then `A + B + C` = `(A+B)+C`, and printing right-nested `A + (B + C)` REQUIRES parens. The 375-LHS display `(A) + ((B) + (C))` — hmm those parens could be the printer's grouping for clarity around ∫ terms, NOT structural. If the printer puts parens around each ∫-term for legibility: `(∫..) + ((∫..) + (∫..))` — the inner `((∫) + (∫))` has parens = structural right-nesting ✓. So LHS = right-nested ✓ as expected. RHS `∫ + ∫ + ∫` no parens = left-nested. So RHS ≠ LHS. CONCLUSION: my hsplit statement's RHS elaborated LEFT-nested — the parens I wrote... \n\n...OR the printer DOESN'T add parens for right-nested + and prints `A + B + C` for BOTH nestings (trusting the reader's infixl assumption to parse it back as left-nested — that would be a printer BUG producing non-roundtrip terms). Lean's printer IS supposed to roundtrip. Hmm — actually for atoms A B C, `A + (B + C)` prints as `A + (B + C)` — I'm fairly confident parens appear.\n\nFINE. Whatever the nesting, the robust fix that eliminates the question: make hsplit's RHS EXACTLY what the rewrites produce — and make the calc consume it via hsplit as a rewrite, not an equation-match. Simplest bulletproof approach: RESTRUCTURE. Don't state hsplit with a hand-written RHS at all. Instead:\n\n```lean\n  have h1 := intervalIntegral.integral_add_adjacent_intervals (hI (-T) (-10)) (hI (-10) T)\n  have h2 := intervalIntegral.integral_add_adjacent_intervals (hI (-10) 10) (hI 10 T)\n```\n\nh1 : ∫(-T..-10) + ∫(-10..T) = ∫(-T..T); h2 : ∫(-10..10) + ∫(10..T) = ∫(-10..T). Then h1 ▸ h2: `∫(-T..T) = ∫(-T..-10) + (∫(-10..10) + ∫(10..T))`:\n\n```lean\n  have hsplit : ∫ t in (-T)..T, F = ∫(-T..-10) + (∫(-10..10) + ∫(10..T)) := by\n    rw [← h1, ← h2]\n```\n\nSame as before. The problem persists conceptually... \n\nNEW ANGLE: use `calc` with explicit transitivity through h1/h2:\n\n```lean\n  have hsplit : ... := by\n    calc ∫ t in (-T)..T, F \n        = ∫ t in (-T)..(-10:ℝ), F + ∫ t in (-10:ℝ)..T, F :=\n          (intervalIntegral.integral_add_adjacent_intervals (hI _ _) (hI _ _)).symm\n      _ = ∫ t in (-T)..(-10:ℝ), F + (∫ t in (-10:ℝ)..(10:ℝ), F + ∫ t in (10:ℝ)..T, F) := by\n          rw [← intervalIntegral.integral_add_adjacent_intervals (hI (-10) 10) (hI 10 T)]\n```\n\nFirst calc step: RHS is `A + D` where D = ∫(-10..T) — I write that form; the lemma instance proves `A + D = ∫(-T..T)` so .symm ✓ — but wait, the ascription's LHS `∫(-T..T)` and RHS `A + D` — the instance `(lemma hI hI).symm : ∫(-T..T) = A + D` — its LHS `∫ x in -T..T, F x` matches the calc LHS ✓ its RHS `A + D` matches my written ✓ (modulo binder alpha — calc steps check defeq ✓ alpha OK — this is the KEY difference: calc-step typechecking uses full defeq, unlike rw's syntactic-ish matching... rw should too though).\n\nSecond step: goal `A + D = A + (B + C)` — rw [← lemma2] rewrites D → B + C ✓ then rfl (alpha) ✓.\n\nHmm — but this relies on the same rfl-at-end. If alpha-rfl was the failure, it fails again. But we ESTABLISHED the sides are alpha-equal-right-nested... unless the printer mislead me and my statement is really LEFT-nested (parse issue!). If my written parens somehow didn't take effect — e.g., if the `+` at line end followed by `(∫...)` on the next line — the parser might have parsed `A + (B + C)` correctly ✓ I'm confident parens work.\n\nYou know what, let me just LOOK at the actual file content to see what's really written there, then test hypotheses empirically. And also check: maybe the hsplit statement in the file lost its parens due to my edit formatting. Let me read the file section.","signature":""},{"type":"text","text":"Several distinct issues. Let me look at the actual file content around the failing spots first."},{"type":"tool_use","id":"Read_44","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean","offset":245,"limit":160},"extra_content":null}]