[{"type":"thinking","thinking":"Errors:\n\n1. **Line 522 (halg step 3)**: the last calc step `_ = 4·exp(−cσ) + 4·(2:ℝ)^(−σ) := by rw [h2b]` — rw [h2b] rewrites `2^(−σ)` → `exp(−log2 σ)` — the goal at that point: `4·exp(−cσ) + 4·2^(−σ) = 4·exp(−cσ) + 4·exp(−log2 σ)` — rw [h2b] rewrites the LHS's 2^(−σ) → exp form: making both sides identical ✓ — the error shows the goal AFTER the failed... wait no, the error shows goal `4·exp(−cσ) + 4·2^(−σ) = 4·exp(−cσ) + 4·exp(−log2 σ)` UNSOLVED — rw [h2b] should rewrite LHS's 2^(−σ)... hmm — h2b : (2:ℝ)^(−σ) = exp(−log2·σ): rw [h2b] rewrites occurrences of `2^(−σ)` → `exp(−log2 σ)`: the LHS has `4 * 2 ^ (-σ)` — contains 2^(−σ) ✓ should rewrite... The displayed goal has LHS unchanged (still 2^(−σ)) and RHS in exp form — wait, the ERROR is at 522:71 — unsolved goals with that exact form — meaning rw [h2b] did NOT fire?? — hmm, or it fired on the WRONG side: it rewrote the RHS's `exp(−log2 σ)`?? no, h2b's LHS pattern is `2 ^ (-σ)` — RHS of the goal has exp, not 2^(−σ)... \n\nOH WAIT — I see: this is the calc's last step where after the previous step the goal is `4·exp(−cσ) + 4·2^(−σ) = 4·exp(−cσ) + 4·exp(−log2 σ)` — the calc step's expected RHS (from halg's statement) is `4·exp(−cσ) + 4·2^(−σ)` and I'm proving the step `X = 4·exp(−cσ)+4·2^(−σ)` where X = 4·exp(−cσ)+4·exp(−log2 σ) — so the goal is `4·exp(−cσ)+4·exp(−log2 σ) = 4·exp(−cσ)+4·2^(−σ)` and rw [h2b] should rewrite 2^(−σ)→exp on the RHS making them equal ✓. The displayed goal shows `LHS(exp) = RHS(2^)`?? — the error shows: `⊢ 4·exp(−cσ) + 4·2^(−σ) = 4·exp(−cσ) + 4·exp(−log2 σ)` — LEFT has 2^(−σ), RIGHT has exp. So rw [h2b] rewrote the LHS?! It rewrote the RIGHT side's... no wait — if rw [h2b] fired on the RIGHT's 2^(−σ)... the right side in the error IS exp form — and left is 2^ — so the rw [h2b] flipped the WRONG side: it rewrote occurrences of 2^(−σ) — there was ONE occurrence (RHS in the original orientation) — after rewriting, the goal became exp = exp?? and should close by rfl. But the error shows them STILL different (2^ vs exp). So rw [h2b] did NOT rewrite, and the goal is unsolved because... the calc step mismatch: the calc expected `4·exp(−cσ)+4·(2:ℝ)^(−σ)` as the FINAL RHS (halg's statement) — the step goal is `4·exp(−cσ)+4·exp(−log2σ) = 4·exp(−cσ)+4·2^(−σ)` — and rw [h2b] — hmm — h2b is an equation (2:ℝ)^(−σ) = exp(−(log 2)·σ) — wait, h2b's statement is `(2 : ℝ) ^ (-σ) = Real.exp (-(Real.log 2) * σ)` — with exponent `-(Real.log 2) * σ` — but the goal's exp term is `Real.exp (-Real.log 2 * σ)` — NOTE: `-(Real.log 2) * σ` vs `-Real.log 2 * σ` — same term (−log2·σ) ✓. And 2^(−σ) — the goal has `2 ^ (-σ)` ✓ matches h2b's LHS ✓. rw [h2b] should rewrite it...\n\nHmm — unless the issue is that after step 2, the calc goal is `4·exp(−cσ) + 4·exp(−log2 σ) = 4·exp(−cσ) + 4·2^(−σ)` — and rw [h2b] rewrites BOTH... no, only one occurrence of 2^(−σ) (the RHS). After rewrite: exp = exp ✓ rfl-close ✓. The error \"unsolved goals ⊢ 4·exp(−cσ) + 4·2^(−σ) = 4·exp(−cσ) + 4·exp(−log2 σ)\" — the sides are SWAPPED from what I said — LHS is 2^, RHS is exp — so the calc step is: `X = 4·exp(−cσ)+4·2^(−σ)` where X = `4·exp(−cσ)+4·exp(−log2 σ)` — NO WAIT, the error's goal shows LHS = 2^ form and RHS = exp form: `4·exp(−cσ) + 4·2^(−σ) = 4·exp(−cσ) + 4·exp(−log2 σ)`. That's the step AFTER rw [h2b] fired — it rewrote the RHS's `4·2^(−σ)`... no — if h2b fired on the ORIGINAL goal `4·exp(−cσ)+4·exp(−log2σ) = 4·exp(−cσ)+4·2^(−σ)`, it rewrites the RHS's 2^(−σ) → exp(−log2 σ): giving `4·exp(−cσ)+4·exp(−log2 σ) = 4·exp(−cσ)+4·exp(−log2 σ)` ✓ rfl. The error shows `2^` on the LEFT and exp on the RIGHT — meaning the goal at the error is the ORIGINAL halg statement's RHS after my calc's steps misaligned: step 2's result was `4·exp(−cσ)+4·exp(−log2 σ)` (call it Y) and the calc's declared final is `4·exp(−cσ)+4·2^(−σ)` (call it Z): the step-3 goal is Y = Z: `4·exp(−cσ)+4·exp(−log2 σ) = 4·exp(−cσ)+4·2^(−σ)`. rw [h2b] rewrites 2^(−σ) → exp(−log2 σ): in Y=Z, the 2^(−σ) appears in Z (RHS) ✓ → Z becomes exp form: Y = Z' with both sides identical ✓ rfl ✓. But the error's goal has 2^ on the LEFT... — so the step-3 goal was `Z = Y`?? That means my calc's LAST step's declared form got flipped: I wrote `_ = 4·exp(−cσ) + 4·(2:ℝ)^(−σ) := by rw [h2b]` — the calc chain: step1 proved LHS = A; step2 proved A = Y; step3 must prove Y = Z (halg's RHS). The displayed goal `4·exp(−cσ) + 4·2^(−σ) = 4·exp(−cσ) + 4·exp(−log2 σ)` = `Z = Y`?! — that's step-3 with sides flipped — meaning the calc parsed my step-3 as proving Z = Y instead of Y = Z?? That happens when the PREVIOUS step produced Z = Y by mistake...\n\nHmm, this is getting too deep. The cleanest fix: close the whole thing with `rw [h2b]` replaced by `rw [← h2b]`?? no. Simplest: finish with `congr 1; rw [h2b]`?? Or restructure halg's last step to just use h2b at the STATEMENT level: after step 2 the result has exp form; halg's statement needs 2^(−σ): rw [h2b] at the START of halg (I did that at the beginning: `rw [h2a, h2b]` — h2b rewrote the statement's 2^(−σ) to exp form — so halg's goal is fully in exp-form from the start! Then the calc proves the exp-form goal and the final `_ = ... + 4·(2:ℝ)^(−σ)` step is WRONG — the goal after rw [h2a, h2b] is `(exp(4/9σ)+1)·(4·exp(−log2 σ)) = 4·exp(−cσ) + 4·exp(−log2 σ)` — the calc should END at the exp-form RHS, NOT the 2^(−σ) form. My step-3 declared `_ = ... + 4·(2:ℝ)^(−σ)` which FLIPPED back — creating the Z = Y mess. FIX: drop step 3 (the goal is already closed at step 2 in exp form):\n```\n    calc (Real.exp ((4 / 9) * σ) + 1) * (4 * Real.exp (-(Real.log 2) * σ))\n        = 4 * (Real.exp ((4 / 9) * σ) * Real.exp (-(Real.log 2) * σ)) +\n          4 * Real.exp (-(Real.log 2) * σ) := by ring\n      _ = 4 * Real.exp (-(Real.log 2 - 4 / 9) * σ) + 4 * Real.exp (-(Real.log 2) * σ) := by\n        rw [hexp_alg]\n```\nAfter step 2: `4·exp(−cσ)+4·exp(−log2 σ)` = the goal's RHS exactly (both in exp form after the initial rw [h2a, h2b]) ✓ rfl-close ✓. And halg's STATEMENT keeps the 2^(−σ) form (the initial rw [h2a, h2b] transformed the GOAL, not the statement — halg's TYPE stays `(exp+1)·2^(−(σ−2)) = 4·exp(−cσ)+4·2^(−σ)` ✓ which is what the envelope use needs ✓). So just delete step 3.\n\n2. **Line 562 (e in integral_exp_neg_le)**: `have e : -Real.exp (-c * 3) / c = Real.exp (-3 * c) / c := by congr 1; congr 1; ring` — the congr 1 gave `−Real.exp(−(c·3)) = Real.exp(−(c·3))` — the first congr 1 reduced to the exp args: `-Real.exp(-c*3)` vs `Real.exp(-3*c)` — the exponents: `-(c*3)` vs `-(3*c)` — my `ring` after congr 1 was supposed to prove `-(c*3) = -(3*c)` ✓ but the error shows goal `−Real.exp(−(c·3)) = Real.exp(−(c·3))` — hmm the two congr 1's over-reduced: `−X = −Y` from congr 1 → X = Y (the negation congr) then congr 1 again → exp-args... The displayed leftover: `-Real.exp (-(c * 3)) = Real.exp (-(c * 3))` — WAIT that says −E = E with the SAME exponent?! That would need E = 0 — nonsense — OH I see: `Real.exp (-c * 3)` parses as `Real.exp ((-c) * 3)` — and `-3 * c` as written in e's statement `Real.exp (-3 * c)` parses as `Real.exp ((-3) * c)` — different groupings ((−c)·3 vs (−3)·c) — equal by comm but the congr 1's left a negation mismatch. This whole `e` is over-engineered: simplify by writing the exponent as `-3 * c` everywhere and proving by `ring_nf` or direct:\n```\n  have e : -Real.exp (-c * 3) / c = Real.exp (-3 * c) / c := by\n    congr 1\n    congr 1\n    ring\n```\nThe error was AFTER those tactics left `−Real.exp (−(c·3)) = Real.exp (−(c·3))`... hmm wait no — the error at 562:61 is `unsolved goals ⊢ -Real.exp (-(c * 3)) = Real.exp (-(c * 3))` — the SAME exponent −(c·3) on both sides, one with outer negation. So after `congr 1; congr 1`, the goal became −E = E?! That means the FIRST congr 1 turned `-Real.exp(-c*3)/c = Real.exp(-3*c)/c` into `-Real.exp(-c*3) = Real.exp(-3*c)` (dropped /c ✓), the SECOND congr 1 turned it into `Real.exp(-c*3) = -Real.exp(-3*c)`?? or the arg-level: `-(c*3) = -(3*c)`... the displayed leftover is `−E(−(c·3)) = E(−(c·3))` — hmm, identical exponent. I think the second congr 1 peeled the outer negation WRONG (congr on negation: −X = −Y ⟸ X = Y ✓ but it went the other way producing −X = Y??). Whatever — replace e's proof with a direct approach:\n```\n  have e : -Real.exp (-c * 3) / c = Real.exp (-3 * c) / c := by\n    rw [show (-3 : ℝ) * c = -(c * 3) by ring]\n```\n`(-3)*c = -(c*3)` ✓ by ring — then Real.exp(-(c·3)) matches the LHS's Real.exp(-c*3) — wait LHS has `-Real.exp (-c * 3) / c` — with −(c·3) as the exponent: `-c * 3` = (−c)·3. And `show (-3:ℝ)*c = -(c*3)`: rw turns Real.exp((-3)*c) into Real.exp(−(c·3)) ✓ both sides: `-Real.exp(-(c·3))/c = Real.exp(-(c·3))/c` ✓ rfl ✓. \n\nActually — hold on, why do I even have `e` in this form? Looking at integral_exp_neg_le's tail:\n```\n  rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hF hint]\n  have h1 : -Real.exp (-c * XR) / c ≤ 0 := ...\n  have e : -Real.exp (-c * 3) / c = Real.exp (-3 * c) / c := ...\n  rw [e]\n  linarith [h1]\n```\nThe integral = F(XR) − F(3) = (−exp(−c·XR)/c) − (−exp(−c·3)/c) = −exp(−c·XR)/c + exp(−c·3)/c. The `e` rewrites exp(−c·3)/c's EXPONENT to −3c... actually the issue: the FTC gives `-Real.exp (-c * XR) / c - -Real.exp (-c * 3) / c` — and I want to bound by exp(−3c)/c. The `e` as written normalizes the second term's exponent. The error chain at 567: \"linarith failed\" — cascade of e failing. Fix e as above.\n\nHmm wait — also: `Real.exp (-c * 3)` vs `Real.exp (-(c * 3))`: `-c * 3` = `HMul.hMul (-c) 3` and `-(c * 3)` = `Neg.neg (c*3)` — DIFFERENT terms ((−c)·3 vs −(c·3)) — defeq but not syntactically equal. The FTC's F(3) = −exp(−c·3)/c uses −c·3 = (−c)·3 — my e's target is exp(−3·c) = exp((−3)·c) — third form! Ugh — all three are defeq (−c·3 = −(c·3) = (−3)·c) but rw needs syntactic match. The robust `e`:\n```\n  have e : -Real.exp (-c * 3) / c = Real.exp (-3 * c) / c := by\n    congr 1\n    congr 1\n    ring\n```\nfailed. New approach — prove the whole tail without e:\n```\n  rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hF hint]\n  have h1 : -Real.exp (-c * XR) / c ≤ 0 := by\n    apply div_nonpos_of_nonpos_of_nonneg _ hc.le\n    linarith [Real.exp_pos (-c * XR)]\n  have e : Real.exp (-c * 3) = Real.exp (-3 * c) := by\n    congr 1\n    ring\n  rw [e]\n  -- goal: -exp(-c·XR)/c - -exp(-3c)/c ≤ exp(-3c)/c\n  linarith [h1]\n```\ne : Real.exp(−c·3) = Real.exp(−3·c): congr 1 → exponent identity −c·3 = −3·c ✓ ring ✓. After rw [e] the goal: `−exp(−c·XR)/c − −exp(−3c)/c ≤ exp(−3c)/c` — i.e. −exp(−c·XR)/c + exp(−3c)/c ≤ exp(−3c)/c ⟺ −exp(−c·XR)/c ≤ 0 ✓ h1 ✓ linarith [h1] closes ✓.\n\n3. Line 567's linarith failure — cascade of the e issue; fixed by the new e.","signature":""},{"type":"text","text":"Two fixes: drop the misaligned last calc step in `halg` (the initial `rw [h2a, h2b]` already put the goal in exp form), and repair the exponent-normalization `e` in `integral_exp_neg_le`."},{"type":"tool_use","id":"Edit_266","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","old_string":"      _ = 4 * Real.exp (-(Real.log 2 - 4 / 9) * σ) + 4 * Real.exp (-(Real.log 2) * σ) := by\n        rw [hexp_alg]\n      _ = 4 * Real.exp (-(Real.log 2 - 4 / 9) * σ) + 4 * (2 : ℝ) ^ (-σ) := by\n        rw [h2b]","new_string":"      _ = 4 * Real.exp (-(Real.log 2 - 4 / 9) * σ) + 4 * Real.exp (-(Real.log 2) * σ) := by\n        rw [hexp_alg]"},"extra_content":null},{"type":"tool_use","id":"Edit_267","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","old_string":"  rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hF hint]\n  have h1 : -Real.exp (-c * XR) / c ≤ 0 := by\n    apply div_nonpos_of_nonpos_of_nonneg _ hc.le\n    linarith [Real.exp_pos (-c * XR)]\n  have e : -Real.exp (-c * 3) / c = Real.exp (-3 * c) / c := by\n    congr 1\n    congr 1\n    ring\n  rw [e]\n  linarith [h1]","new_string":"  rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hF hint]\n  have h1 : -Real.exp (-c * XR) / c ≤ 0 := by\n    apply div_nonpos_of_nonpos_of_nonneg _ hc.le\n    linarith [Real.exp_pos (-c * XR)]\n  have e : Real.exp (-c * 3) = Real.exp (-3 * c) := by\n    congr 1\n    ring\n  rw [e]\n  linarith [h1]"},"extra_content":null}]