[{"type":"thinking","thinking":"Several errors to fix:\n\n1. `summand_star`: `map_sub` didn't apply — `star (a - b)` — star here is `starRingEnd ℂ`? The `map_sub` expects the function to be a ring hom applied. `star (x - y)` — the simp lemma is `star_sub` or `map_sub (starRingEnd ℂ)`. The error: \"Did not find pattern ?f (?a − ?b)\" — meaning star is NOT displayed as an application of a ring hom; it's the `star` operation (typeclass). So use `star_sub`, `star_one`, `star_pow`, `star_div₀`? In Mathlib: `star_sub : star (a - b) = star a - star b`; `star_div' : star (a / b) = star a / star b` (with a prime maybe); `star_pow : star (a ^ n) = (star a) ^ n`; `star_one : star 1 = 1`. Let me rewrite with those.\n\n2. `liSummand_eq`: `Complex.add_conj` pattern mismatch: `z + star z` — the target has `1 - ... + star (1 - ...)` i.e. `f + star f` but the pattern is `?z + (starRingEnd ℂ) ?z` — so `star` in my term is not syntactically `starRingEnd ℂ` application — it IS `star` function (the class). `Complex.add_conj : z + star z = 2 * (z.re : ℂ)` — hmm, if the lemma states `z + star z` with star the class function, the pattern should match... The error shows the pattern as `(starRingEnd ℂ) ?z` — meaning Complex.add_conj uses `starRingEnd ℂ z`, while my term has `star (...)`. These ARE different: `star z` vs `(starRingEnd ℂ) z`. They're defeq but not syntactically equal. Fix: rewrite `star` to `starRingEnd ℂ` first, or use `Complex.conj` — in Mathlib, `star z = conj z` for ℂ (`Complex.conj_eq_star`? there's `Complex.star_def : star z = conj z`). And `Complex.add_conj : z + conj z = 2 * (z.re : ℂ)`? Actually I recall `Complex.add_conj (z : ℂ) : z + conj z = 2 * (z.re : ℂ)`. So: convert star to conj: `Complex.star_def : star z = conj z`. Hmm wait — in my def I wrote `star (ρ : ℂ)`. Let me check: maybe simpler to rewrite summand_star using conj from the start: in liSummand use `star` (fine for the def) but in the proof convert.\n\n3. `pow_one_add_inv_two_mul_le` line 42: \"No goals\" — the `gcongr` closed more than expected. Restructure.\n\n4. Line 51: after `rw [← Real.exp_nat_mul]` — the rewrite of `(exp (1/2))^2` — `Real.exp_nat_mul : Real.exp (n * x) = (Real.exp x) ^ n`? The issue: `← Real.exp_nat_mul` rewrites `(Real.exp x)^n` to `Real.exp (n*x)` — pattern with n : ℕ. The target `Real.exp (1/2 : ℝ) ^ 2` — with 2 : ℕ? After `rw [← Real.exp_nat_mul]` the goal became `Real.exp (↑2 * (1/2)) ≤ 2^2` — the rewrite worked but then `this : Real.exp (2 * (1/2)) = Real.exp 1` has `2` as ℝ literal vs `↑2` — mismatch. Fix: use `show Real.exp ((2 : ℝ) * (1/2)) = Real.exp 1` and congr + norm_num, or use `Real.exp_one_lt_d9` differently: Real.exp (1/2) ≤ 2: maybe simpler: Real.exp (1/2) < Real.exp 1 (strict monotone) and Real.exp 1 < 2.72 < ... but exp(1/2) ≤ 2 needs the numeric. Alternative: `Real.exp_lt` ... simplest robust: have h : Real.exp (1/2:ℝ) ≤ 2 := by\n   rw [show (2:ℝ) = Real.exp (Real.log 2) by rw [Real.exp_log (by norm_num)]]\n   apply Real.exp_le_exp.mpr; apply Real.log... hmm exp(1/2) ≤ exp(log 2) ⟺ 1/2 ≤ log 2 ⟺ ... log 2 ≥ 1/2: `Real.log_two_gt_d9 : 0.6931471803 < Real.log 2` ✓ linarith. Clean!\n\n5. `geom_sum_mul w n` : `(∑) * (w−1) = w^n − 1` — the order is flipped: it's `∑ * (x−1)` not `(x−1) * ∑`. Use `mul_geom_sum`? or `geom_sum_mul` gives `(∑ i ∈ range n, x^i) * (x−1) = x^n − 1` ✓ as shown. So use that form: S * (w−1) = wⁿ − 1. Adjust: I need (w−1)·S: just commute: (w−1)·S = S·(w−1) = wⁿ−1.\n\nLet me fix all errors with a rewrite of the file. Also check `Complex.inv_def` usage later — let me simplify hre1's proof: `((1:ℂ)/z).re = z.re/‖z‖^2`: (1/z) = z⁻¹: re(z⁻¹) = z.re/‖z‖². There's `Complex.inv_re : (z⁻¹).re = z.re / (Complex.normSq z)`? Let me not guess: do it via conj: 1/z = star z / ‖z‖² (Complex.inv_def : z⁻¹ = star z / ‖z‖ ^ 2 — hmm ‖z‖² as (‖z‖^2 : ℝ) cast to ℂ). I recall `Complex.inv_def : z⁻¹ = (star z) / (Complex.normSq z)` with normSq cast. Then re: ((star z)/normSq z).re = (star z).re/normSq z = z.re/normSq z (div by real: `Complex.div_re`?). And normSq z = ‖z‖²: `Complex.normSq_eq_norm_sq`? Hmm — there is `Complex.normSq_eq_abs : normSq z = (Complex.abs z)^2` and `Complex.norm_eq_abs : ‖z‖ = Complex.abs z`. So normSq z = ‖z‖^2 ✓ via those.\n\nRather than guessing lemma names inline, I'll write the hinv proof with explicit simp lemmas and iterate.\n\nActually, let me restructure hre1: instead of going through inv_re, note:\n(n : ℂ)·(1/z) = n·(star z)/normSq z (Complex.inv_def); re of that = n·z.re/normSq z. And I want n·z.re/‖z‖^2 — equal by normSq_eq_abs + norm_eq_abs.\n\nLet me rewrite the whole scratch with fixes:\n\nAlso potential issue: `abs_pow` — |a^n| = |a|^n — in hA I use abs_div, abs_mul, abs_of_nonneg, then div_le_div... Let me simplify hA:\n|n·z.re/‖z‖²| = n·|z.re|/‖z‖² (n>0, ‖z‖²>0) ≤ n·1/‖z‖² = n/‖z‖². Write:\nhave hA : |n * z.re / ‖z‖ ^ 2| ≤ n / ‖z‖ ^ 2 := by\n  rw [abs_div, abs_mul]\n  have e1 : |(n : ℝ)| = n := abs_of_nonneg hn0.le\n  have e2 : |‖z‖ ^ 2| = ‖z‖ ^ 2 := abs_of_nonneg (by positivity)\n  rw [e1, e2]\n  apply div_le_div_of_nonneg_right? — div_le_div_of_nonneg_right? Hmm: to show n·|z.re|/‖z‖² ≤ n/‖z‖² with ‖z‖² > 0: `div_le_div_right hzn2` : a/c ≤ b/c ⟺ a ≤ b... lemma name `div_le_div_right (hc : 0 < c) : a / c ≤ b / c ↔ a ≤ b`. So:\n  apply (div_le_div_right (by positivity)).mpr\n  exact mul_le_mul_of_nonneg_left (abs_le.mpr ⟨by linarith, hzre1⟩) hn0.le\n  — wait: mul_le_mul_of_nonneg_left : b ≤ c → 0 ≤ a → a*b ≤ a*c: here n·|z.re| ≤ n·1 ✓.\n\nFor hB: |((S−n)/z).re| ≤ ‖(S−n)/z‖ = ‖S−n‖/‖z‖ ≤ (n²/‖z‖)/‖z‖ = n²/‖z‖² ✓ (avoid div_pow issues):\nhave hB : |((S - n) / z).re| ≤ n ^ 2 / ‖z‖ ^ 2 := by\n  have h1 : |((S - n) / z).re| ≤ ‖(S - n) / z‖ := Complex.abs_re_le_norm _\n  have h2 : ‖(S - n) / z‖ ≤ (n ^ 2 / ‖z‖) / ‖z‖ := by\n    rw [norm_div]\n    exact div_le_div_of_nonneg_right hSn hzn -- hmm name: div_le_div_of_nonneg_right? For a/b ≤ c/b: `div_le_div_of_nonneg_right : a ≤ b → 0 ≤ c → a / c ≤ b / c`? Hmm — actually the lemma for dividing by the same positive: `div_le_div_of_nonneg_right`?? I think the modern name is `div_le_div_right` (iff) or `div_le_div_of_le_left`... The monotone one: `div_le_div_of_nonneg_right (h : a ≤ b) (hc : 0 ≤ c) : a / c ≤ b / c` — hmm I don't remember exactly; there's `div_le_div_right (hc : 0 < c) : a / c ≤ b / c ↔ a ≤ b`. Use the iff: (div_le_div_right hzn).mpr hSn.\n  calc |((S - n) / z).re| ≤ ‖(S - n) / z‖ := h1\n    _ ≤ (n ^ 2 / ‖z‖) / ‖z‖ := h2\n    _ = n ^ 2 / ‖z‖ ^ 2 := by ring\n  — the last ring: (n²/‖z‖)/‖z‖ = n²/‖z‖²: `div_div` = n²/(‖z‖·‖z‖) = n²/‖z‖²: ring should handle with pow_two: maybe `by rw [div_div, ← pow_two]`.\n\nhSn's inner calc: I wrote a weird structure with `apply mul_le_mul_of_nonneg_left _ ... ; apply ... ; · have ...` — mismatched bullets. Rewrite cleanly:\nh3: (1/‖z‖) * M * (n*(n−1)/2) ≤ (1/‖z‖) * 2 * (n²/2):\n  = (1/‖z‖) · [M · (n(n−1)/2)] ≤ (1/‖z‖)·[2·(n²/2)]: two mul_le_mul_of_nonneg_left applications:\n  have hb1 : (n:ℝ)*(n-1)/2 ≤ n^2/2 := by nlinarith [hn0]\n  have hb2 : M * (n*(n-1)/2) ≤ 2 * (n^2/2) := mul_le_mul_of_nonneg hpow hb1 (by positivity) (by positivity)\n  -- mul_le_mul_of_nonneg : a ≤ b → c ≤ d → 0 ≤ c → 0 ≤ b → a*c ≤ b*d\n  then (1/‖z‖)·[...] ≤ (1/‖z‖)·[...]: mul_le_mul_of_nonneg_left hb2 (by positivity).\n\nAlso `Finset.mul_sum` usage: ∑ (c·(j·M)) = c·M·∑ j — I wrote `(1/‖z‖) * (j * M)` — factor c·M: rewrite each term: (1/‖z‖)*(j*M) = (1/‖z‖*M)*j ✓ then Finset.mul_sum.\n\nThe hpow: (1+1/(2n))^{n-1} ≤ (1+1/(2n))^n: `pow_le_pow_right₀ (ha : 0 ≤ a) (hb : 1 ≤ b)... ` hmm — pow_le_pow_right₀ signature: `pow_le_pow_right₀ (ha : 0 ≤ a) (hb : 1 ≤ b) : a ^ m ≤ a ^ n ↔ m ≤ n`? No wait — for exponent comparison with base ≥ 1: `pow_le_pow_right₀ (hb : 1 ≤ b) (h : m ≤ n) : b ^ m ≤ b ^ n`? In recent Mathlib: `pow_le_pow_right₀ (ha : 1 ≤ a) (hmn : m ≤ n) : a ^ m ≤ a ^ n` ✓ I think. Use that.\n\ngeom_sum_mul direction fix: hgeom : ∑ j ∈ range n, w^j) * (w - 1) = w ^ n - 1 := geom_sum_mul w n. Then 1 − wⁿ = −(wⁿ−1) = −S·(w−1) = −(w−1)·S — fine by comm.\n\nhw1 : w − 1 = -(1/z): then 1 − wⁿ = −(w−1)·S = (1/z)·S ✓.\n\nFor hgeomj: (w^j − 1) = (∑ i ∈ range j, w^i)·(w−1) := geom_sum_mul w j ✓.\n\nsummand_star with star lemmas: star_sub, star_one, star_pow, star_div₀? — for division with star: `star_div₀ : star (a / b) = star a / star b`? There's `star_div'` in Mathlib (star (a / b) = star a / star b) for GroupWithZero + Star. Name candidates: `star_div₀`, `star_div'`. I'll write `simp only [star_sub, star_one, star_pow, star_div₀, star_natCast]` and if star_div₀ doesn't exist try `star_div'`. Actually safer: use `map_` versions with starRingEnd: first convert star to starRingEnd ℂ: `Complex.star_def : star z = conj z`?? Hmm — actually in current Mathlib, `star (z : ℂ) = conj z` where conj = starRingEnd ℂ: the lemma `Complex.star_def : star z = conj z` — no wait, I think conj is DEFINED via starRingEnd: `Complex.conj : ℂ →* ℂ` is starRingEnd ℂ. And `star z` unfolds to `starRingEnd ℂ z` — defeq. The rw failure earlier: map_sub pattern `?f (?a − ?b)` — star (a−b) should match with ?f = star?? The error suggests `star` in goal is not an application — because `star` is `star (R:=ℂ) x` with instance arguments, still an application though... The actual issue might be that map_sub applies to RingHomClass functions, and `star` for ℂ is `starRingEnd` but the `star x` notation doesn't unfold to an application of a bundled hom. Use `star_sub` etc. instead:\n\nsummand_star (n : ℕ) (z : ℂ) :\n    (1 : ℂ) - (1 - 1 / (star z)) ^ n = star ((1 - (1 - 1 / z) ^ n)) := by\n  rw [star_sub, star_one, star_pow, star_sub, star_one, star_div₀, star_one]\nIf star_div₀ doesn't exist, try star_div'. Let me check quickly which exists in this Mathlib via grep.\n\nAlso liSummand_eq's use of Complex.add_conj: after summand_star rewrite, goal has `star (1 - (1 - 1/ρ)^n)` with star the class-fn. Complex.add_conj : z + star z = 2 * (z.re : ℂ)? or with conj? Check via grep too.\n\nLet me grep Mathlib for these names.","signature":""},{"type":"text","text":"Fixable errors — let me check the exact lemma names in this Mathlib version first."},{"type":"tool_use","id":"Bash_85","name":"Bash","input":{"command":"cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"theorem star_div₀\\|theorem star_div'\\|lemma star_div₀\\|lemma star_div'\" Mathlib/Algebra/Star/ | head -4; grep -rn \"theorem Complex.add_conj\\|lemma Complex.add_conj\\|add_conj \" Mathlib/Data/Complex/Basic.lean Mathlib/Analysis/SpecialFunctions/*.lean 2>/dev/null | head -4; grep -rn \"theorem inv_def\\|inv_re \" Mathlib/Data/Complex/Basic.lean | head -4; grep -rn \"theorem geom_sum_mul \\|lemma geom_sum_mul \" Mathlib/Algebra/GeomSum.lean | head -3","description":"Check exact lemma names in vendored Mathlib"},"extra_content":null}]