[{"type":"thinking","thinking":"Great progress — only TWO errors remain, both about the same thing:\n\n**Error 1 (hsplit)**: The rewrites succeeded but left the goal as `LHS' = RHS` where LHS' = `∫(-T..-10) + (∫(-10..10) + ∫(10..T))` and RHS is the stated form — but note the ASSOCIATIVITY difference: LHS' shows `∫(-T..-10) + (∫(-10..10) + ∫(10..T))` (pretty-printed with parens) vs RHS `∫(-T..-10) + ∫(-10..10) + ∫(10..T)` (left-assoc). Wait, look carefully:\n\nGoal after rw:\n```\n⊢ (∫ -T..-10) + ((∫ -10..10) + ∫ 10..T) =\n    ∫ -T..-10 + ∫ -10..10 + ∫ 10..T\n```\n\nHmm — actually looking at the error message for line 392 (that's the SECOND error — the rw [hleft] one). The first error context shows the goal:\n\n```\n⊢ (∫ (x : ℝ) in -T..-10, ...) +\n      ((∫ (x : ℝ) in -10..10, ...) + ∫ (x : ℝ) in 10..T, ...) =\n    ∫ (t : ℝ) in -T..-10, ... + ∫ (t : ℝ) in -10..10, ... + ∫ (t : ℝ) in 10..T, ...\n```\n\nThe LHS has `(A) + ((B) + (C))` (right-nested from my rewrites) and the RHS has `A + B + C` (left-nested from the hsplit statement I wrote). Wait — I wrote hsplit's RHS as `A + (B + C)`:\n\n```\n      ∫ t in (-T)..(-10 : ℝ), ... +\n        (∫ t in (-10 : ℝ)..(10 : ℝ), ... +\n          ∫ t in (10 : ℝ)..T, ...) := by\n```\n\nThat's `A + (B + C)` — right-nested ✓ same as LHS'. Hmm, but the error at 392 says rw [hleft] \"did not find the pattern ∫ -T..-10, F in the target\" — and displays the target as containing `∫ -T..-10, F + ...`. Wait the target DOES contain it!\n\nLook at error 2 (line 392, the rw [hleft]): \"Did not find an occurrence of the pattern ∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5/2) + ↑t * I)‖ in the target expression\" — but the target displayed literally begins with `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ + ...`.\n\nThe pattern vs the occurrence differ how? hleft's LHS is `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5/2) + ↑t * I)‖`. The target contains `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5/2) + ↑t * I)‖`. These LOOK identical... but the bound variable NAME differs (t vs t — same). Hmm.\n\nOH WAIT. I see it. The pretty-printed target at error 2 shows the goal AFTER the first error's context... Let me re-read. Error 1's context (top of output) shows goal:\n\n```\n⊢ (∫ (x : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑x * I)‖) +\n      ((∫ (x : ℝ) in -10..10, ...) + ∫ (x : ℝ) in 10..T, ...) =\n    ∫ (t : ℝ) in -T..-10,\n      ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ +\n        ∫ (t : ℝ) in -10..10, ... + ∫ (t : ℝ) in 10..T, ...\n```\n\nBound var `x` on the LHS (from the integral_add_adjacent_intervals rewrites — the lemma's bound var is x) vs bound var `t` on the RHS (my hsplit statement uses t). So after my two rw's, the goal is `A_x + (B_x + C_x) = A_t + (B_t + C_t)` — alpha-equivalent but rw's final rfl-closure requires SYNTACTIC/rfl-defeq equality — alpha-equivalence IS defeq and rw closes with rfl which handles alpha... Hmm, actually rw closes the goal with `rfl` automatically after rewriting — the rfl check uses defeq (with alpha ✓). But the error suggests the goal was NOT closed — meaning the rw succeeded but the result wasn't closed by rfl?\n\nWait — the first error message's actual error text is cut off (the output starts mid-context). The visible first block ends with `⊢ ... = ...` and then the SECOND error starts at 392. So the first error is at some line < 392 — likely the hsplit rw (line ~385). The error message itself was truncated. What could it be? Possibly \"motive is not type correct\" or the rw failed to close. Hmm.\n\nActually — maybe the issue: the two rewrites produced a goal that's NOT syntactically the RHS because of the NESTING: my first rewrite turned `∫(-T..T)` into `∫(-T..-10) + ∫(-10..T)`; the second turned `∫(-10..T)` into `∫(-10..10) + ∫(10..T)`. Result: `∫(-T..-10) + (∫(-10..10) + ∫(10..T))`. The hsplit RHS (as elaborated): I wrote `A + (B + C)` ✓ same. So the goal after rewrites is `A_x + (B_x + C_x) = A_t + (B_t + C_t)` — rw tries rfl: alpha-equiv — rfl via defeq — alpha-renaming IS defeq ✓ should close.\n\nHmm wait, maybe the FIRST error is elsewhere. Let me count: the output shows the tail. The errors: one at 392:12 (rw [hleft] failure). The first error — its message text got cut. Its context shows the hsplit-shaped goal. Then error 2 at 392.\n\nFor error 2: rw [hleft] failed to find the pattern in the goal. The goal (from hsplit, now in context as hsplit : ...) — wait NO. Error 2's context shows hsplit as a HYPOTHESIS (successfully proved!) and the GOAL is:\n\n```\n⊢ ∫ (t : ℝ) in -T..-10, F + ∫ (t : ℝ) in -10..10, F + ∫ (t : ℝ) in 10..T, F ≤ 4*T*(...) + 40*1.3352\n```\n\nHmm interesting — so the goal at error 2 has `∫ -T..-10 + ∫ -10..10 + ∫ 10..T` (left-nested!). But hsplit's RHS was right-nested `A + (B + C)`... The calc step `_ = A + (B + C) := hsplit` — wait, my calc:\n\n```\n    _ = ∫ t in (-T)..(-10 : ℝ), F +\n          (∫ t in (-10 : ℝ)..(10 : ℝ), F +\n            ∫ t in (10 : ℝ)..T, F) := hsplit\n```\n\nand then \n\n```\n    _ ≤ 4 * T * ... := by rw [hleft]; ...\n```\n\nThe goal at error 2 shows `A + B + C ≤ ...` — LEFT-nested. But I wrote right-nested `A + (B + C)`. Hmm — unless the elaborator re-associated? No. Look again at error 2's target display:\n\n```\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 pretty-printer indents under the first +... The indentation suggests `A + (B + C)` printed with the (B+C) indented — \"∫ -10..10, F + ∫ 10..T, F\" on one line at deeper indent — yes! This IS right-nested: `A + (B + C)` — the pretty printer elides parens. So the target DOES contain `∫ -T..-10, F` as a subterm (the A).\n\nAnd the pattern (from hleft's LHS): `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5/2) + ↑t * I)‖`. The target's A: `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5/2) + ↑t * I)‖`. IDENTICAL?!\n\nUnless... the `-T` in hleft vs the `-T` in the goal differ? Both `-T`. Hmm, or `(-10 : ℝ)` vs `-10`: hleft's LHS I wrote `(-T)..(-10 : ℝ)` — elaborates `Neg.neg (10 : ℝ)` vs the goal's `-10` — same.\n\nWAIT. Maybe the issue: `rw [hleft]` — hleft's LHS has bound var name `t`, the target's integral also `t` — irrelevant for matching.\n\nHmm, let me look at error 1 more carefully — maybe error 1 is the hsplit rw FAILING to fully close, leaving a weird goal, and hsplit got a different shape. But hsplit's displayed type in error-2 context: \n\n```\nhsplit :\n  ∫ (t : ℝ) in -T..T, F =\n    ∫ (t : ℝ) in -T..-10, F + ∫ (t : ℝ) in -10..10, F + ∫ (t : ℝ) in 10..T, F\n```\n\nHmm — displayed left-nested-looking but probably right-nested with parens elided.\n\nOK here's another thought — the error at 392: rw [hleft] — the pattern is hleft's LHS: `∫ t in -T..-10, F`. The target: `A + (B + C) ≤ RHS` where A = `∫ t in -T..-10, F`. These match... UNLESS the -T in A vs hleft differ in some hidden way — e.g., in hleft, `-T` came from `(-T)` in MY statement ✓ same T. \n\nHold on — actually, look VERY carefully at error 2's pattern vs target:\n\nPattern: `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖`\nTarget A: `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖`\n\nIdentical as printed. But maybe the difference is in the BOUND VARIABLE being t vs x — no, both t.\n\nHmm, hmm. What about `↑(-t)` — not relevant.\n\nLet me reconsider: maybe error 1 (truncated) is at line ~385 (hsplit's rw) — and it's the SAME \"did not find pattern\" error for the SECOND adjacent-intervals rewrite: pattern `∫ x in -10..T, F` — after the first rewrite, the goal contains `∫ x in -10..T, F x`? The first rewrite replaced `∫ x in -T..T, F` with `(∫ x in -T..-10, F) + ∫ x in -10..T, F` ✓ contains it. Hmm.\n\nActually wait — I see it now! Look at the FIRST error context goal:\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```\n\nThis goal is an `=` goal — this is hsplit's proof state after the two rewrites. The error message (cut off) is probably \"tactic 'rewrite' failed, equality or iff proof expected\"? No... Or maybe the rw SUCCEEDED and left this goal which rw's auto-rfl DIDN'T close because... the LHS has bound var x, RHS bound var t — alpha — rfl should handle. UNLESS the RHS is genuinely different: RHS = hsplit's stated RHS = `A_t + (B_t + C_t)` where my statement's integrals use `(t : ℝ)` as binder. LHS (post-rw) uses x. Alpha-equivalent ✓.\n\nHmm, so what's the error? Maybe the error IS that the goal remains and `rw` reported failure to close? rw automatically tries rfl after rewriting — if it fails, no error, the goal just remains (rw leaves it as a new goal — not an error!). An error only if NO goals get closed... no — rw never errors for leaving goals; it errors when the pattern isn't found.\n\nSo error 1 must be a pattern-not-found too! Pattern: `∫ x in -T..T, F` for the first rewrite — the goal at hsplit-proof-start is `∫ t in -T..T, F = RHS`. The pattern from the reversed lemma: `∫ x in -T..T, F x` — with bound var x vs goal's t — rw matches up to alpha ✓ fine.\n\nHmm — the lemma instance: `intervalIntegral.integral_add_adjacent_intervals (hI (-T) (-10)) (hI (-10) T)` — hI a b : IntervalIntegrable F' volume a b where F' = `(fun t => ‖Jlag n (↑(-5/2)+↑t*I)‖)`. The lemma: `(∫ x in a..b, ?f x) + ∫ x in b..c, ?f x = ∫ x in a..c, ?f x` — with ?f unified from hI's type: ?f := F'. Instance: `(∫ x in -T..-10, F' x) + ∫ x in -10..T, F' x = ∫ x in -T..T, F' x`. Note: `F' x` is a BETA-REDEX: `(fun t => ‖Jlag n (↑(-5/2)+↑t*I)‖) x`. When elaborating the lemma instance, does Lean beta-reduce? The lemma statement has `f x` with f a variable — instantiated f := F' gives `(fun t => ...) x` — Lean's elaborator INSTANTIATES but typically beta-reduces metavariable assignments... the resulting rewrite rule's RHS: `∫ x in -T..T, (fun t => ‖...‖) x` — vs the goal's `∫ t in -T..T, ‖...‖` — pattern matching for rw: the rewrite pattern (lemma RHS) vs goal subterm — rw uses keyed matching + defeq — should handle beta ✓.\n\nI'm going in circles. Let me look at the FULL error output, not just tail -50. Re-run capturing more, or better — since elaboration is incremental, let me view errors from the top. Actually simpler: the errors are only 2 (the output showed 2 error blocks + contexts). Error 1's message got cut off the top. Let me re-run with head instead.\n\nActually wait — I realize both errors might be the SAME root cause and there's a simpler explanation. Look at error 2 again:\n\n\"Did not find an occurrence of the pattern `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖`\"\n\nTarget: contains `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖` as the first summand...\n\nHmm — BUT WAIT: in the target, the first summand's integral might have bound variable `x` not `t`! The display shows:\n\n```\n⊢ ∫ (t : ℝ) in -T..-10,\n      ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ +\n```\n\nbound var t ✓. Hmm no.\n\nOK — different idea: maybe the issue is the `-10` in hleft's statement vs `-10` in hsplit's — hleft: `∫ t in (-T)..(-10 : ℝ)` — the `-10` here: `Neg.neg (10 : ℝ)`. hsplit: same ✓.\n\nHmm, what about `-T` — in hleft the endpoint is `-T` where T is the section... in error 2's target the endpoints are `-T..-10` ✓.\n\nYou know what — let me just look at the FULL first error. Run again with head -60.","signature":""},{"type":"text","text":"Two errors, the first one's message got cut off. Let me see the full output from the top."},{"type":"tool_use","id":"Bash_43","name":"Bash","input":{"command":"export PATH=\"$HOME/.elan/bin:$PATH\" && cd \"D:/Python/Quantum/research/lean/NguyenRH1\" && lake env lean NguyenRH1/LiSideBounds.lean 2>&1 | head -60","description":"Show first errors of LiSideBounds elaboration","timeout":600000},"extra_content":null}]