[{"type":"thinking","thinking":"Line 106 col 71: `neg_ne_zero.mpr (fun h => hs (by linarith [neg_eq_zero.mp h]))` — col 71 is around `neg_eq_zero.mp h`. The issue: `neg_eq_zero.mp h` — h : −s − 1 = 0 — that's NOT a neg equation! neg_ne_zero : −a ≠ 0 ↔ a ≠ 0 — so from h : −s − 1 = 0 I want s + 1 = 0 then s = −1. neg_eq_zero : −a = 0 ↔ a = 0 — h : −s − 1 = 0 — −s − 1 is not syntactically a negation... My logic chain is wrong anyway. Let me redo: hr : (−s − 1) ≠ 0. From hs : s ≠ −1: −s − 1 = −(s + 1): −s − 1 ≠ 0 ⟺ s + 1 ≠ 0 ⟺ s ≠ −1. Cleanest:\n\n```lean\n  have hr : (-s - 1) ≠ 0 := by\n    intro h\n    apply hs\n    linarith [h]  -- from −s − 1 = 0: s = −1\n```\nh : −s − 1 = 0 — linarith needs s = −1: from h: −s = 1: s = −1 ✓ linear ✓ but linarith on ℂ?? NO — s : ℂ and h : ℂ equation! linarith doesn't work on ℂ. Need: −s − 1 = 0 ⟹ s = −1: `neg_eq_zero`... do: have h1 : -s = 1 := by ... sub_eq_zero: −s − 1 = 0 ⟹ −s = 1 via `sub_eq_zero.mp h : −s = 1`; then s = −1 via neg: `neg_eq_iff_eq_neg`... s = −1: from −s = 1: s = −1: `neg_eq_iff_eq_neg.mp`? −s = 1 ⟺ s = −1: `neg_eq_iff_eq_neg : −a = b ↔ a = −b` ✓: (neg_eq_iff_eq_neg.mp h1) : s = −1 ✓ then exact hs that. All ℂ-safe:\n\n```lean\n  have hr : (-s - 1) ≠ 0 := by\n    intro h\n    exact hs (neg_eq_iff_eq_neg.mp (sub_eq_zero.mp h))\n```\nsub_eq_zero.mp h : -s - 1 = 0 → -s = 1 ✓; neg_eq_iff_eq_neg.mp : (−s = 1) → s = −1 ✓.\n\nSimilarly check my other ℂ-linarith misuses in this file: in hfr proof: `linarith [hx.1]` — hx.1 : k < x (reals ✓ fine). The hfr block: `simp at hx; linarith [hx.1]` — hx : x ∈ Ioo k (k+1): after simp — hmm the `rw [hxeq, hxk] at hx` — hxeq : x = ↑⌊x⌋, hxk : ⌊x⌋ = k: so x = ↑k — wait direction: rw [hxeq, hxk] at hx turns x into ↑k in hx: hx : (↑k : ℝ) ∈ Ioo ↑k (↑k+1)?? then simp: k < k ∧ ... → False ✓ `simp at hx` should close by finding k < k false. Then `linarith [hx.1]` unreachable... the `simp at hx` either closes the goal (False in context) or errors. Let me restructure hfr cleanly:\n\n```lean\n    have hfr : Int.fract x ≠ 0 := by\n      intro h\n      rw [Int.fract] at h  -- h : x − ↑⌊x⌋ = 0\n      have hxeq : x = ⌊x⌋ := sub_eq_zero.mp h\n      have hxk : ⌊x⌋ = (k : ℤ) := Int.floor_eq_on_Ico _ _ ⟨hx.1.le, by exact_mod_cast hx.2⟩\n      rw [hxk] at hxeq  -- hxeq : x = ↑k\n      have : (k : ℝ) < x := hx.1\n      rw [hxeq] at this  -- ↑k < ↑k\n      simp at this  -- or linarith\n```\n`simp at this` on (↑k < ↑k) — norm_num closes (lt_irrefl). Or `exact lt_irrefl _ this`.\n\nAlso in hvv'/hv: `linarith [hx.1]` — hx.1 : k ≤ x or k < x — real ✓ fine.\n\nAlso in huu': `rw [← b1_eq_fract_sub_half (by linarith [hx.1] : (0:ℝ) ≤ x)] at hb` — hb : HasDerivAt (fun y => ↑(b2Fun y)) (↑(Int.fract x − 1/2))... wait ofReal_comp's derivative: HasDerivAt.ofReal_comp (hf : HasDerivAt f u z) : HasDerivAt (fun y => ↑(f y)) u z — the derivative `u` here — looking at the signature again: `HasDerivAt (fun y : ℝ => ↑(f y) : ℝ → ℂ) u z` — hmm u appears uncast — there must be an implicit coercion: HasDerivAt for ℝ → ℂ: the derivative type is ℂ, so u must be ℂ-coerced — the lemma's u : ℝ and the result derivative (↑u : ℂ)? The displayed signature `... ) u z` with u : ℝ — that can't typecheck unless HasDerivAt's derivative arg is ℝ... For functions ℝ → ℂ, HasDerivAt f f' x has f' : ℂ. So the lemma must coerce: probably the actual statement has `(u : ℂ)` coercion implicit via ↑. The displayed text from grep might just render the coercion invisibly. Fine — hb : HasDerivAt (fun y => ↑(b2Fun y)) (↑(fract x − 1/2)) x presumably. Then rw [← b1_eq_fract_sub_half ...] at hb: rewrites occurrences of `Int.fract x - 1/2` (real) inside — the casted ↑(fract x − 1/2) — the pattern Int.fract x − 1/2 appears under the cast ✓ rw finds it (rw works under coercions ✓) giving ↑(B1 x) ✓.\n\nAlso hfr's `by exact_mod_cast hx.2`: hx.2 : x < (k:ℝ)+1: need x < ⌊x⌋+1 form for the Ico: Int.floor_eq_on_Ico's hypothesis: x ∈ Ico (↑n) (↑n+1): ⟨⌊x⌋ ≤ x (from hx.1.le after cast: (k:ℝ) ≤ x and ⌊x⌋ = k... wait — circular: I'm PROVING hxk via the Ico with n = k: need (k:ℝ) ≤ x (have hx.1.le ✓ after cast from k to ℤ... the Ico membership: ((k:ℤ):ℝ) ≤ x — from hx.1 : (k:ℝ) < x — but (k:ℤ):ℝ vs (k:ℕ):ℝ — same value: push_cast/▸. Let me write:\n```lean\n      have hxk : ⌊x⌋ = (k : ℤ) := by\n        apply Int.floor_eq_on_Ico\n        constructor\n        · have : ((k : ℤ) : ℝ) ≤ x := by push_cast; linarith [hx.1.le]\n          exact this\n        · have : x < ((k : ℤ) : ℝ) + 1 := by push_cast; linarith [hx.2]\n          exact this\n```\n`Int.floor_eq_on_Ico (n : ℤ) (x : α) (h : x ∈ Set.Ico ↑n (↑n + 1)) : ⌊x⌋ = n` ✓. Membership via constructor ⟨le, lt⟩ ✓ (Ico membership is a conjunction... Set.mem_Ico: a ≤ x ∧ x < b ✓ `⟨·, ·⟩` after `show`... `apply Int.floor_eq_on_Ico` leaves the Ico-membership goal: `refine ⟨?_, ?_⟩`? Ico mem is And ✓.\n\nThis is getting intricate — let me rewrite the whole intervalIntegral_b1_cpow proof block cleanly with all these fixes.","signature":""},{"type":"text","text":"The `linarith`-on-ℂ misuse at line 106 plus the `hfr` block need restructuring. Rewriting the unit-IBP proof cleanly."},{"type":"tool_use","id":"Edit_210","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiRBound.lean","old_string":"  have hk0 : (0:ℝ) < (k : ℝ) := by exact_mod_cast hk\n  have hr : (-s - 1) ≠ 0 := neg_ne_zero.mpr (fun h => hs (by linarith [neg_eq_zero.mp h]))\n  have hu : ContinuousOn (fun x : ℝ => (b2Fun x : ℂ)) (Set.Icc (k : ℝ) (k + 1 : ℝ)) :=\n    (continuous_ofReal.comp continuous_b2Fun).continuousOn\n  have hv : ContinuousOn (fun x : ℝ => (x : ℂ) ^ (-s - 1)) (Set.Icc (k : ℝ) (k + 1 : ℝ)) := by\n    intro x hx\n    have hx0 : x ≠ 0 := by linarith [hx.1]\n    exact (hasDerivAt_ofReal_cpow_const hx0 hr).continuousAt.continuousWithinAt\n  have huu' : ∀ x ∈ Set.Ioo (min (k : ℝ) (k + 1 : ℝ)) (max (k : ℝ) (k + 1 : ℝ)),\n      HasDerivAt (fun x : ℝ => (b2Fun x : ℂ)) ((B1 x : ℂ)) x := by\n    intro x hx\n    rw [min_eq_left (by linarith : (0:ℝ) ≤ 1), max_eq_right (by linarith : (0:ℝ) ≤ 1)] at hx\n    have hfr : Int.fract x ≠ 0 := by\n      intro h\n      have hxk : ⌊x⌋ = (k : ℤ) := Int.floor_eq_on_Ico _ _\n        ⟨hx.1.le, by exact_mod_cast hx.2⟩\n      have hxeq : x = ⌊x⌋ := by\n        have := sub_eq_zero.mp h\n        rw [Int.fract] at this\n        exact this.symm\n      rw [hxeq, hxk] at hx\n      simp at hx\n      linarith [hx.1]\n    have hb := (hasDerivAt_b2Fun hfr).ofReal_comp\n    rw [← b1_eq_fract_sub_half (by linarith [hx.1] : (0:ℝ) ≤ x)] at hb\n    exact hb\n  have hvv' : ∀ x ∈ Set.Ioo (min (k : ℝ) (k + 1 : ℝ)) (max (k : ℝ) (k + 1 : ℝ)),\n      HasDerivAt (fun x : ℝ => (x : ℂ) ^ (-s - 1)) ((-(s + 1)) * (x : ℂ) ^ (-s - 2)) x := by\n    intro x hx\n    rw [min_eq_left (by linarith : (0:ℝ) ≤ 1), max_eq_right (by linarith : (0:ℝ) ≤ 1)] at hx\n    have hx0 : x ≠ 0 := by linarith [hx.1]\n    have h := hasDerivAt_ofReal_cpow_const hx0 hr\n    refine h.congr_deriv ?_\n    rw [show -s - 1 - 1 = -s - 2 by ring]\n    ring","new_string":"  have hk0 : (0:ℝ) < (k : ℝ) := by exact_mod_cast hk\n  have hr : (-s - 1) ≠ 0 := by\n    intro h\n    exact hs (neg_eq_iff_eq_neg.mp (sub_eq_zero.mp h))\n  have hu : ContinuousOn (fun x : ℝ => (b2Fun x : ℂ)) (Set.Icc (k : ℝ) (k + 1 : ℝ)) :=\n    (continuous_ofReal.comp continuous_b2Fun).continuousOn\n  have hv : ContinuousOn (fun x : ℝ => (x : ℂ) ^ (-s - 1)) (Set.Icc (k : ℝ) (k + 1 : ℝ)) := by\n    intro x hx\n    have hx0 : x ≠ 0 := by linarith [hx.1]\n    exact (hasDerivAt_ofReal_cpow_const hx0 hr).continuousAt.continuousWithinAt\n  have hIco_x : ∀ x ∈ Set.Ioo (k : ℝ) (k + 1 : ℝ), x ∈ Set.Ico (⌊x⌋ : ℝ) (⌊x⌋ + 1) :=\n    fun x _ => ⟨Int.floor_le x, Int.lt_floor_add_one x⟩\n  have huu' : ∀ x ∈ Set.Ioo (min (k : ℝ) (k + 1 : ℝ)) (max (k : ℝ) (k + 1 : ℝ)),\n      HasDerivAt (fun x : ℝ => (b2Fun x : ℂ)) ((B1 x : ℂ)) x := by\n    intro x hx\n    rw [min_eq_left (by linarith : (0:ℝ) ≤ 1), max_eq_right (by linarith : (0:ℝ) ≤ 1)] at hx\n    have hfr : Int.fract x ≠ 0 := by\n      intro h\n      rw [Int.fract] at h\n      have hxeq : x = ⌊x⌋ := sub_eq_zero.mp h\n      have hxk : ⌊x⌋ = (k : ℤ) := by\n        apply Int.floor_eq_on_Ico\n        refine ⟨?_, ?_⟩\n        · have h1 : ((k : ℤ) : ℝ) ≤ x := by push_cast; linarith [hx.1.le]\n          exact h1\n        · have h2 : x < ((k : ℤ) : ℝ) + 1 := by push_cast; linarith [hx.2]\n          exact h2\n      rw [hxk] at hxeq\n      have hlt : (k : ℝ) < x := hx.1\n      rw [hxeq] at hlt\n      norm_num at hlt\n    have hb := (hasDerivAt_b2Fun hfr).ofReal_comp\n    rw [← b1_eq_fract_sub_half (by linarith [hx.1] : (0:ℝ) ≤ x)] at hb\n    exact hb\n  have hvv' : ∀ x ∈ Set.Ioo (min (k : ℝ) (k + 1 : ℝ)) (max (k : ℝ) (k + 1 : ℝ)),\n      HasDerivAt (fun x : ℝ => (x : ℂ) ^ (-s - 1)) ((-(s + 1)) * (x : ℂ) ^ (-s - 2)) x := by\n    intro x hx\n    rw [min_eq_left (by linarith : (0:ℝ) ≤ 1), max_eq_right (by linarith : (0:ℝ) ≤ 1)] at hx\n    have hx0 : x ≠ 0 := by linarith [hx.1]\n    have h := hasDerivAt_ofReal_cpow_const hx0 hr\n    refine h.congr_deriv ?_\n    rw [show -s - 1 - 1 = -s - 2 by ring]\n    ring"},"extra_content":null}]