[{"type":"thinking","thinking":"1. **572 linarith** (hdisj): h1 : 2^jX − 1 < z.im; h2 : z.im ≤ −(2^jX) — and h2jX : 0 < 2^jX — linarith failed?! The context shows h2jX : 0 < 2^j*X is present. z.im > 2^jX − 1 and z.im ≤ −2^jX ⟹ 2^jX − 1 < −2^jX ⟹ 2·2^jX < 1 ⟹ 2^jX < 1/2 — but 2^jX ≥ 200·... — the contradiction needs 2^jX ≥ something > 1/2: h2jX1 : 1 ≤ 2^jX is defined AFTER (inside the next calc step) — not in scope here! linarith [h2jX] alone: 2^jX > 0 doesn't contradict 2^jX < 1/2. Need 2^jX ≥ 1/2: from X ≥ 200: 2^jX ≥ 200: but that fact (hpowX is defined after too). Give linarith the fact: `linarith [hX, one_le...]`— the needed: 1/2 ≤ 2^jX: from X ≥ 200 and 2^j ≥ 1: 2^jX ≥ 200 ✓: linarith [hX, h1le] where h1le : (1:ℝ) ≤ (2:ℝ)^j := by exact_mod_cast Nat.one_le_two_pow — then 2^jX ≥ 1·X ≥ 200 — nlinarith (product!) — linarith can't multiply. Provide the product explicitly:\n```lean\n    have hge : (1 : ℝ) ≤ (2 : ℝ) ^ j * X := by\n      have h1le : (1 : ℝ) ≤ (2 : ℝ) ^ j := by exact_mod_cast Nat.one_le_two_pow\n      nlinarith [h1le, hX]\n    linarith\n```\nAdd that before linarith in hdisj.\n\n2. **613**: `rw [abs_neg, abs_of_nonneg h2j1X.le]` — the target `|-2^(j+1)*X| ≤ 2^(j+1)*X`: abs_neg's pattern `|(-?a)|` should match |-2^(j+1)*X|... the error says pattern |−?a| not found in `|-2 ^ (j + 1) * X|` — hmm — the term is |−(2^(j+1)*X)| with the neg OUTSIDE: `-(2 ^ (j + 1) * X)` displays as `-2 ^ (j + 1) * X` — abs_neg's pattern `|-?a|` — the expression is `|Neg.neg (2^(j+1)*X)|` ✓ should match... unless the actual term is different: the max's first arg is `|-(2:ℝ)^(j+1) * X|` — parses as |(-(2^(j+1)))·X|?? NO: -(2:ℝ)^(j+1) * X = (-(2^(j+1))) * X?? Operator precedence: unary minus binds tighter than *: `-(2 ^ (j + 1)) * X` = (−2^{j+1})·X. And abs_neg : |−a| = |a| with a := 2^{j+1}: gives |2^{j+1}·X|... wait the term is |(-2^{j+1})·X| — abs_neg needs the WHOLE thing as −a: (−2^{j+1})·X = −(2^{j+1}·X)? — as TERMS these differ: Neg.mul vs mul-neg... `neg_mul : -(a*b) = (-a)*b`? — hmm the parsed term: `|-(2:ℝ)^(j+1) * X|` — the unary minus on 2^(j+1) times X — the abs argument is `(-(2^(j+1))) * X` (HMul (Neg.neg (2^(j+1))) X) — abs_neg's pattern Neg.neg ?a matches the WHOLE abs argument if it's a Neg.neg — but the argument is HMul (Neg.neg ...) X — NOT a Neg.neg at the top ⟹ abs_neg doesn't fire. Fix: use `neg_mul` first to normalize: `rw [neg_mul, abs_neg, abs_of_nonneg h2j1X.le]`?? neg_mul : −a * b = −(a * b)? or the reverse: neg_mul : -(a * b) = -a * b... I want (−2^{j+1})·X = −(2^{j+1}·X): `neg_mul (a b) : -(a*b) = (-a)*b`?? the standard: `neg_mul : -a * b = -(a * b)` ✓ — rw [neg_mul] turns (−2^{j+1})·X into −(2^{j+1}·X) ✓ then abs_neg ✓ then abs_of_nonneg ✓. Wait — which direction: neg_mul : (-a) * b = -(a * b)? — Mathlib: `neg_mul (a b : α) : -a * b = -(a * b)`? or `neg_mul : -(a * b) = -a * b`? In Mathlib `neg_mul : -a * b = -(a * b)` ✓ (left-neg to outer). Hmm actually I recall `neg_mul : -(a * b) = -a * b` (outer to left)... Either way rw normalizes toward a canonical side. Let me instead use `abs_neg` after converting via `show`: the abs goal: |−2^{j+1}·X| ≤ 2^{j+1}X: rewrite the inside to −(2^{j+1}X): `rw [show -(2:ℝ)^(j+1) * X = -((2:ℝ)^(j+1) * X) by ring]`?? the parse of `-(2:ℝ)^(j+1) * X` — is it (−(2^{j+1}))·X or −(2^{j+1}·X)? Precedence: unary minus vs *: in Lean `−a * b` parses as `(-a) * b`? or `−(a*b)`? I believe unary minus binds TIGHTER than * in Lean: `-a * b` = `(-a) * b`... Actually no — in Lean 4 Mathlib, `-a * b` parses as `-(a * b)`!! (negation has LOWER precedence than *). Hmm — the journal-011 gotcha: \"-a*b vs -(a*b) breaks Real.exp_neg rewrites\" — suggesting the two forms differ — so `-2^(j+1) * X` = −(2^{j+1}·X)?? If unary minus binds LOOSER than *, then -2^(j+1) * X = -(2^(j+1) * X) ✓ and abs_neg SHOULD match... but the error shows it didn't. So the parse gave (−2^{j+1})·X?? — but that contradicts... The error's target display: `|-2 ^ (j + 1) * X|` — if it were −(2^{j+1}·X) the display would be the same. The pattern |−?a| didn't match ⟹ the term inside is NOT a top-level Neg.neg — so it's (Neg.neg (2^(j+1))) * X. OK so unary minus bound TIGHTER. Then: `rw [show (-(2:ℝ)^(j+1)) * X = -((2:ℝ)^(j+1) * X) from by ring]`?? the show's LHS `(-(2:ℝ)^(j+1)) * X` — hmm the parse of that... ugh. Alternative: use `abs_of_nonneg` after establishing the argument is a neg of nonneg... simplest: `rw [abs_of_nonpos (by linarith [h2j1X])]`?? abs_of_nonpos : a ≤ 0 → |a| = −a: for (−2^{j+1})·X ≤ 0: from 2^{j+1}·X ≥ 0: nlinarith... the term (−2^{j+1})·X vs −(2^{j+1}·X) — argh the parse ambiguity.\n\nCleanest: avoid the abs forms: `have hmax : max |-(2:ℝ)^(j+1)*X| ... ≤ ...` — replace the whole max_le/abs dance by: |anything| ≥ 0 and the max ≤ ... via: for the negative-side: |−2^{j+1}X| = 2^{j+1}X: `abs_of_nonneg`?? no. Use `norm_num`/simp with abs_neg: simp [abs_neg]: simp should handle: the goal max |−2^{j+1}X| |−2^jX| ≤ 2^{j+1}X: simp [abs_neg, abs_of_nonneg h2j1X.le...]... \n\nHonestly the robust one-liner: prove |−y| = |y| issues away: `have e1 : |-(2:ℝ)^(j+1)*X| = (2:ℝ)^(j+1)*X := by rw [abs_neg, abs_of_nonneg h2j1X.le]` — abs_neg : |−a| = |a| — the term |−(2^{j+1}·X)|?? — the parse... OK let me just TEST the parse in a probe to settle it.","signature":""},{"type":"text","text":"Let me settle the unary-minus parse question with a probe."},{"type":"tool_use","id":"Bash_394","name":"Bash","input":{"command":"cd D:/Python/Quantum/research/lean/NguyenRH1 && cat > Probe.lean << 'EOF'\nimport Mathlib\nopen Real\nexample (X : ℝ) (j : ℕ) (h : 0 ≤ (2:ℝ)^(j+1) * X) : |-(2:ℝ)^(j+1) * X| = (2:ℝ)^(j+1) * X := by\n  rw [abs_neg, abs_of_nonneg h]\nEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean Probe.lean 2>&1 | head -8","description":"Probe abs_neg on -2^(j+1)*X","timeout":300000},"extra_content":null}]