[{"type":"thinking","thinking":"Errors:\n\n1. Line 2646: push_neg deprecated warning (just a warning — but banked gotcha says push_neg is deprecated; leave it or change to `push Not`. It's a warning not error — the file had push_neg uses before? The warning appeared — fix it: use `push Not at hbad`.\n\n2. Line 2664: `not_lt.mp` — not_lt is an iff: ¬ a < b ↔ b ≤ a — (not_lt).mp gives b ≤ a from ¬(a < b) ✓ — but the application structure: `(not_lt).mp (by ...)` where the by-block proves ¬(|x| < 1)?? — my code: `(not_lt).mp (by have h' := mem_nearZerosSub.mp h; linarith [h', hx]) hx` — I applied the mp to the by-block AND to hx — the by-block proves ¬(|x| < 1)?? no wait — hfar's statement: ρ ∉ nearZerosSub t — the proof: fun h => (not_lt).mp (by ...) hx — the (not_lt).mp produces 1 ≤ |x| from ¬(|x| < 1) — and then applied to hx?? — the structure is wrong. I want: hfar : ρ ∉ nearZerosSub t = ¬(ρ ∈ nearZerosSub t) = ¬(|x| < 1) (via mem_nearZerosSub). So: hfar := fun hmem => by have h' := mem_nearZerosSub.mp hmem; linarith [h', hx] — this proves False from hmem — i.e. ρ ∉ nearZerosSub t ✓ directly:\n```\n  have hfar : ρ ∉ nearZerosSub t := fun hmem => by\n    have h' := mem_nearZerosSub.mp hmem\n    linarith [h', hx]\n```\n— a term-mode lambda with a tactic block — `fun hmem => by ...` — allowed ✓.\n\n3. Line 2686: the show-rw for the division form: my show : 4.5·m/(x·√(1+x²)) = (4.5·m)·(1/(x·√(1+x²))) — but the goal has `4.5 * ↑(zeroMult ↑ρ) / (x * √(1 + ((↑ρ).im - t) ^ 2))` — the x is a set-variable: `set x : ℝ := |ρ.1.im - t|` — the goal's sqrt arg is (↑ρ).im − t, NOT x!! Because farG's body uses |ρ.1.im − t| and set x := that — but the farG was already rw'd BEFORE the set?? Order: rw [farG, if_neg hfar, farW] THEN set k, set x — the farG's body expression |ρ.1.im − t| appears — set x replaces occurrences ✓ should work... The show-rw pattern has √(1 + x ^ 2) but the goal has √(1 + ((↑ρ).im - t) ^ 2) — the set x := |...| should have folded it. UNLESS set happened but the farG-rw introduced the term AFTER set? Order in my code: rw [farG, if_neg hfar, farW]; set k := ...; set x := ... — the farG-rw is BEFORE set ✓ so set should fold... Hmm — but farW's rw introduced ANOTHER |ρ.1.im−t|?? no — farW is about k. The goal shows the LHS with √((↑ρ).im−t)² — unfolded — and √(1+x²) in my show — mismatch means set x didn't fold (maybe because the term is |ρ.1.im − t| with the abs — set x := |ρ.1.im − t| — set folds occurrences of the TERM |ρ.1.im − t| — the goal's √(1+((↑ρ).im−t)²) contains (↑ρ).im − t WITHOUT the abs bars!! farG's def has Real.sqrt (1 + (ρ.1.im - t) ^ 2) — WITHOUT abs! and |ρ.1.im − t| WITH abs in the other factor. set x := |ρ.1.im − t| only folds the abs-ed term. The sqrt-arg (ρ.1.im − t) is a different term (no abs). So my show-rw must use the un-abs-ed form. Fix: don't use set for the division rewrites; write them with the explicit forms:\n```\n  rw [show (4.5 : ℝ) * (1 / ((k : ℝ) * Real.sqrt (1 + (k : ℝ) ^ 2)) * (zeroMult ρ.1 : ℝ))\n      = (4.5 * (zeroMult ρ.1 : ℝ)) * (1 / ((k : ℝ) * Real.sqrt (1 + (k : ℝ) ^ 2))) from by ring]\n  rw [show (4.5 : ℝ) * (zeroMult ρ.1 : ℝ) / (|ρ.1.im - t| * Real.sqrt (1 + (ρ.1.im - t) ^ 2))\n      = (4.5 * (zeroMult ρ.1 : ℝ)) * (1 / (|ρ.1.im - t| * Real.sqrt (1 + (ρ.1.im - t) ^ 2))) from by\n        rw [div_eq_mul_inv]]\n```\nand hmono/hpos with the explicit forms: hmono : (k:ℝ)·√(1+k²) ≤ |x|·√(1+(ρ.1.im−t)²)... hmm — but hmono is stated with x: (k:ℝ)·√(1+k²) ≤ x·√(1+x²) — where x = |ρ.1.im−t| (the abs). The goal's sqrt arg (ρ.1.im−t) vs x = |ρ.1.im−t| — NOT the same term. The farG denominator: |ρ.1.im − t| * Real.sqrt (1 + (ρ.1.im − t) ^ 2) — the FIRST factor has abs, the sqrt-arg has NO abs. So hmono must relate (k)·√(1+k²) ≤ |x|·√(1+(ρ.1.im−t)²): the sqrt-arg (ρ.1.im−t)² = |ρ.1.im−t|² = x² (sq_abs): so √(1+(ρ.1.im−t)²) = √(1+x²) (via sq_abs: (ρ.1.im−t)² = |ρ.1.im−t|²? — sq_abs (a) : |a|² = a² — so (ρ.1.im−t)² = |ρ.1.im−t|² = x² via (sq_abs _).symm). Let me restructure farG_le_farW_mul to avoid set and handle the abs forms explicitly:\n```\nlemma farG_le_farW_mul {t : ℝ} (ρ : {ρ : ℂ // IsNontrivialZero ρ})\n    (hx : 1 ≤ |ρ.1.im - t|) :\n    farG t ρ ≤ 4.5 * (farW (Nat.floor |ρ.1.im - t|) * (zeroMult ρ.1 : ℝ)) := by\n  classical\n  have hfar : ρ ∉ nearZerosSub t := fun hmem => by\n    have h' := mem_nearZerosSub.mp hmem\n    linarith [h', hx]\n  rw [farG, if_neg hfar, farW]\n  have hk1 : (1 : ℝ) ≤ ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ) := by\n    have h1 := Nat.floor_mono hx\n    rw [Nat.floor_one] at h1\n    exact_mod_cast h1\n  have hkx : ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ) ≤ |ρ.1.im - t| :=\n    Nat.floor_le (abs_nonneg _)\n  have hsq : (ρ.1.im - t) ^ 2 = |ρ.1.im - t| ^ 2 := by rw [sq_abs]\n  have hmono : ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ)\n        * Real.sqrt (1 + ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ) ^ 2)\n      ≤ |ρ.1.im - t| * Real.sqrt (1 + (ρ.1.im - t) ^ 2) := by\n    rw [hsq]\n    apply mul_le_mul hkx _ (Real.sqrt_nonneg _) (by positivity)\n    apply Real.sqrt_le_sqrt\n    have h1 : ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ) ≤ |ρ.1.im - t| := hkx\n    nlinarith [h1, abs_nonneg (ρ.1.im - t)]\n  ...\n```\nwait — hmono's sqrt comparison: √(1+k²) ≤ √(1+|x|²): k² ≤ |x|² (hkx + nonneg): nlinarith [hkx, abs_nonneg, sq_nonneg...] ✓.\nThe mul_le_mul (h₁ : k ≤ |x|) (h₂ : √ ≤ √) (0 ≤ a₂ = √) (0 ≤ b₁ = |x|): (Real.sqrt_nonneg _) (abs_nonneg _) ✓.\nThen the assemble:\n```\n  have hm : (0 : ℝ) ≤ (zeroMult ρ.1 : ℝ) := Nat.cast_nonneg _\n  have hpos : (0 : ℝ) < ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ)\n      * Real.sqrt (1 + ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ) ^ 2) := by\n    have h1 : (0 : ℝ) < ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ) := by linarith [hk1]\n    positivity\n  rw [show (4.5 : ℝ) * (1 / (((Nat.floor |ρ.1.im - t| : ℕ) : ℝ)\n        * Real.sqrt (1 + ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ) ^ 2)) * (zeroMult ρ.1 : ℝ))\n      = (4.5 * (zeroMult ρ.1 : ℝ)) * (1 / (((Nat.floor |ρ.1.im - t| : ℕ) : ℝ)\n        * Real.sqrt (1 + ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ) ^ 2))) from by ring]\n  rw [show (4.5 : ℝ) * (zeroMult ρ.1 : ℝ) / (|ρ.1.im - t| * Real.sqrt (1 + (ρ.1.im - t) ^ 2))\n      = (4.5 * (zeroMult ρ.1 : ℝ)) * (1 / (|ρ.1.im - t| * Real.sqrt (1 + (ρ.1.im - t) ^ 2))) from by\n        rw [div_eq_mul_inv]]\n  exact mul_le_mul_of_nonneg_left (one_div_le_one_div_of_le hpos hmono)\n    (mul_nonneg (by norm_num : (0 : ℝ) ≤ 4.5) hm)\n```\nWait — the goal after rw [farG, if_neg hfar, farW]: `4.5 * (zeroMult ρ.1 : ℝ) / (|ρ.1.im - t| * Real.sqrt (1 + (ρ.1.im - t) ^ 2)) ≤ 4.5 * (farW ⌊|ρ.1.im - t|⌋₊ * (zeroMult ρ.1 : ℝ))` — farW unfolds to 1/(k·√(1+k²)): `4.5 * (1/(k·√(1+k²)) · m)`. My first show-rw rewrites the RHS: 4.5·(1/(k√)·m) → (4.5·m)·(1/(k√)) ✓. Second show-rw rewrites the LHS: 4.5·m/(x·√(1+y²)) → (4.5·m)·(1/(x·√(1+y²))) ✓. Then the goal: (4.5·m)·(1/(x√)) ≤ (4.5·m)·(1/(k√)) — one_div_le_one_div_of_le (0 < k√) (k√ ≤ x√) gives 1/(x√) ≤ 1/(k√) ✓ then mul_le_mul_of_nonneg_left ✓.\n\n4. Line 2702: `exact_mod_cast hk2` in hk2R — `(Nat.floor_le ...).trans (by exact_mod_cast hk2)` — the trans order wrong: I want (k:ℝ) ≤ t: from hk2 : k ≤ ⌊t⌋₊ and ⌊t⌋₊ ≤ t (Nat.floor_le): (k:ℝ) ≤ (⌊t⌋₊ : ℝ) ≤ t: exact_mod_cast hk2 gives (k:ℝ) ≤ (Nat.floor t : ℝ) ✓ then .trans (Nat.floor_le ...): so: `have hk2R : (k : ℝ) ≤ t := (by exact_mod_cast hk2 : (k : ℝ) ≤ (Nat.floor t : ℝ)).trans (Nat.floor_le (by linarith : (0:ℝ) ≤ t))`.\n\n5. Line 2823: the hterm arg type mismatch — unit_family_le's hterm : ∀ ρ ∈ FS, g ρ ≤ W (j ρ) * (zeroMult (ρ:ℂ):ℝ) — my farG_le_farW_mul gives farG t ρ ≤ 4.5·(farW ⌊x⌋₊·m) — the form 4.5·(farW k·m) vs W(j ρ)·m with W k = 4.5·farW k: (4.5·farW k)·m vs 4.5·(farW k·m) — ring-equal but not syntactically. Fix: give the lambda a form matching or bridge: (fun ρ hρ => by have h := farG_le_farW_mul ρ (...); rw [mul_comm (4.5:ℝ) ...]... simplest: change farG_le_farW_mul's conclusion to `farG t ρ ≤ (4.5 * farW (Nat.floor |ρ.1.im - t|)) * (zeroMult ρ.1 : ℝ)` — then it matches W(j ρ)·m with W := fun k => 4.5·farW k ✓. Edit the lemma's statement: `farG t ρ ≤ 4.5 * farW (Nat.floor |ρ.1.im - t|) * (zeroMult ρ.1 : ℝ)` — parsed as (4.5·farW)·m ✓. And its proof's final: the goal after my two show-rws: (4.5·m)·(1/(x√)) ≤ (4.5·farW k)·m — hmm — the RHS: (4.5·(1/(k√)))·m vs my show-rw's (4.5·m)·(1/(k√)) — different association. Fix the first show-rw to produce the (4.5/(k√))·m form:\n```\n  rw [show (4.5 : ℝ) * (1 / (...) * (zeroMult ρ.1 : ℝ))\n      = (4.5 * (1 / (...))) * (zeroMult ρ.1 : ℝ) from by ring]\n```\nwait — farW unfolds to 1/(k√) so the RHS of the goal is 4.5·(1/(k√)·m): show-rw to (4.5·(1/(k√)))·m? Hmm — then LHS: 4.5·m/(x√) → (4.5·m)/(x√): the inequality: (4.5·m)/(x√) ≤ (4.5/(k√))·m = (4.5/(k√))·m: rewrite LHS: 4.5·m/(x√) = (4.5/(x√))·m (ring): then mul_le_mul_of_nonneg_right (1/(x√) ≤ 1/(k√)) hm... cleaner:\n```\n  rw [show (4.5 : ℝ) * (zeroMult ρ.1 : ℝ) / (|ρ.1.im - t| * Real.sqrt (1 + (ρ.1.im - t) ^ 2))\n      = (4.5 / (|ρ.1.im - t| * Real.sqrt (1 + (ρ.1.im - t) ^ 2))) * (zeroMult ρ.1 : ℝ) from by ring]\n  rw [show (4.5 : ℝ) * (farW (Nat.floor |ρ.1.im - t|) * (zeroMult ρ.1 : ℝ))\n      = (4.5 * farW (Nat.floor |ρ.1.im - t|)) * (zeroMult ρ.1 : ℝ) from by ring]\n```\nhmm — but the goal's RHS after rw [farW] is 4.5·(farW k·m) with farW REDUCED to 1/(k√)? I did rw [farG, if_neg hfar, farW] — farW unfolds → 1/(k√). So the goal: 4.5·m/(x√(1+y²)) ≤ 4.5·(1/(k√(1+k²))·m). Restructure the whole ending:\n```\n  rw [show (4.5 : ℝ) * (zeroMult ρ.1 : ℝ) / (|ρ.1.im - t| * Real.sqrt (1 + (ρ.1.im - t) ^ 2))\n      = (4.5 * (zeroMult ρ.1 : ℝ)) * (1 / (|ρ.1.im - t| * Real.sqrt (1 + (ρ.1.im - t) ^ 2))) from by\n        rw [div_eq_mul_inv]]\n  rw [show (4.5 : ℝ) * (1 / ((⌊|ρ.1.im - t|⌋₊ : ℝ) * Real.sqrt (1 + (⌊|ρ.1.im - t|⌋₊ : ℝ) ^ 2))\n        * (zeroMult ρ.1 : ℝ))\n      = (4.5 * (zeroMult ρ.1 : ℝ)) * (1 / ((⌊|ρ.1.im - t|⌋₊ : ℝ)\n        * Real.sqrt (1 + (⌊|ρ.1.im - t|⌋₊ : ℝ) ^ 2))) from by ring]\n  exact mul_le_mul_of_nonneg_left (one_div_le_one_div_of_le hpos hmono)\n    (mul_nonneg (by norm_num : (0 : ℝ) ≤ 4.5) hm)\n```\n— with the lemma statement in the form farG t ρ ≤ 4.5 * farW ⌊x⌋₊ * (zeroMult ρ.1 : ℝ) — the goal's RHS after rw [farW]: 4.5·(1/(k√)·m): the second show-rw pattern must match `4.5 * (1/(k√(1+k²)) * ↑(zeroMult ↑ρ))` — and turns it into (4.5·m)·(1/(k√)) ✓ then mul_le_mul_of_nonneg_left (1/(x√) ≤ 1/(k√)) (0 ≤ 4.5·m) ✓.\n\nThe Nat.floor notation ⌊·⌋₊ — my hkx/hk1/hmono/hpos use (Nat.floor |...| : ℕ) — let me just use the ⌊⌋₊ notation consistently: `Nat.floor |ρ.1.im - t|` — write out Nat.floor.\n\nAlso fix the hfar structure (fun hmem => by ...).\n\nLet me rewrite farG_le_farW_mul wholesale with an Edit.","signature":""},{"type":"text","text":"Several form mismatches. Rewriting `farG_le_farW_mul` wholesale and fixing the smaller issues."},{"type":"tool_use","id":"Write_167","name":"Write","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\patch_s16.py","content":"p = 'NguyenRH1/LiTightR.lean'\nsrc = open(p, encoding='utf-8').read()\n\n# rewrite farG_le_farW_mul wholesale\nstart = src.index(\"/-- farG is bounded by 4.5·farW(⌊x⌋₊)·m (the monotone weight chain). -/\")\nend = src.index(\"/-- **The main bucket count**\")\nnew_lemma = \"\"\"/-- farG is bounded by 4.5·farW(⌊x⌋₊)·m (the monotone weight chain). -/\nlemma farG_le_farW_mul {t : ℝ} (ρ : {ρ : ℂ // IsNontrivialZero ρ})\n    (hx : 1 ≤ |ρ.1.im - t|) :\n    farG t ρ ≤ 4.5 * farW (Nat.floor |ρ.1.im - t|) * (zeroMult ρ.1 : ℝ) := by\n  classical\n  have hfar : ρ ∉ nearZerosSub t := fun hmem => by\n    have h' := mem_nearZerosSub.mp hmem\n    linarith [h', hx]\n  rw [farG, if_neg hfar, farW]\n  have hk1 : (1 : ℝ) ≤ ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ) := by\n    have h1 := Nat.floor_mono hx\n    rw [Nat.floor_one] at h1\n    exact_mod_cast h1\n  have hkx : ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ) ≤ |ρ.1.im - t| :=\n    Nat.floor_le (abs_nonneg _)\n  have hsq : (ρ.1.im - t) ^ 2 = |ρ.1.im - t| ^ 2 := by rw [sq_abs]\n  have hmono : ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ)\n        * Real.sqrt (1 + ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ) ^ 2)\n      ≤ |ρ.1.im - t| * Real.sqrt (1 + (ρ.1.im - t) ^ 2) := by\n    rw [hsq]\n    apply mul_le_mul hkx _ (Real.sqrt_nonneg _) (abs_nonneg _)\n    apply Real.sqrt_le_sqrt\n    have h1 : (0 : ℝ) ≤ ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ) := by positivity\n    nlinarith [hkx, h1, abs_nonneg (ρ.1.im - t)]\n  have hm : (0 : ℝ) ≤ (zeroMult ρ.1 : ℝ) := Nat.cast_nonneg _\n  have hpos : (0 : ℝ) < ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ)\n      * Real.sqrt (1 + ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ) ^ 2) := by\n    have h1 : (0 : ℝ) < ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ) := by linarith [hk1]\n    positivity\n  rw [show (4.5 : ℝ) * (zeroMult ρ.1 : ℝ) / (|ρ.1.im - t| * Real.sqrt (1 + (ρ.1.im - t) ^ 2))\n      = (4.5 * (zeroMult ρ.1 : ℝ)) * (1 / (|ρ.1.im - t| * Real.sqrt (1 + (ρ.1.im - t) ^ 2))) from by\n        rw [div_eq_mul_inv]]\n  rw [show (4.5 : ℝ) * (1 / (((Nat.floor |ρ.1.im - t| : ℕ) : ℝ)\n        * Real.sqrt (1 + ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ) ^ 2)) * (zeroMult ρ.1 : ℝ))\n      = (4.5 * (zeroMult ρ.1 : ℝ)) * (1 / (((Nat.floor |ρ.1.im - t| : ℕ) : ℝ)\n        * Real.sqrt (1 + ((Nat.floor |ρ.1.im - t| : ℕ) : ℝ) ^ 2))) from by ring]\n  exact mul_le_mul_of_nonneg_left (one_div_le_one_div_of_le hpos hmono)\n    (mul_nonneg (by norm_num : (0 : ℝ) ≤ 4.5) hm)\n\n\"\"\"\nsrc = src[:start] + new_lemma + src[end:]\n\n# hk2R: fix the trans order\nold = \"\"\"  have hk2R : (k : ℝ) ≤ t := (Nat.floor_le (by linarith : (0 : ℝ) ≤ t)).trans\n    (by exact_mod_cast hk2)\"\"\"\nnew = \"\"\"  have hk2R : (k : ℝ) ≤ t :=\n    (by exact_mod_cast hk2 : (k : ℝ) ≤ (Nat.floor t : ℝ)).trans\n      (Nat.floor_le (by linarith : (0 : ℝ) ≤ t))\"\"\"\nassert old in src\nsrc = src.replace(old, new)\n\n# push_neg deprecation at mainFinset_bucket_range\nsrc = src.replace(\"\"\"    by_contra hbad\n    push_neg at hbad\n    have h2 : Nat.floor t + 1 ≤ Nat.floor |ρ.1.im - t| := hbad\"\"\", \"\"\"    by_contra hbad\n    push Not at hbad\n    have h2 : Nat.floor t + 1 ≤ Nat.floor |ρ.1.im - t| := hbad\"\"\")\n\nopen(p, 'w', encoding='utf-8').write(src)\nprint(\"done\")\n"},"extra_content":null}]