[{"type":"thinking","thinking":"More errors:\n\n1. **Line 1087 (norm_term_vonMangoldt_X_le_three_pow's hΛ rw)** — \"No goals to be solved\" — hmm, the error shows the context but the actual failing tactic is hidden. 1087:12 — let me look at what's there. Actually 1087 is probably inside the split's `rw [hΛ]` or similar — \"No goals\" means the tactic ran after the goal closed. Let me look at 1080-1090.\n\n2. **Line 1119: unsolved goals** with the context shown ending at hΛ — the error is inside norm_term_vonMangoldt_X_le_three_pow's `rw [hΛ]` region? — actually 1119:68 is in dirichletCX_le's hfin — the `rw [hΛ2]` — no... The context shows n, hn, m, hm, hXre, h2re, hm0, hΛ — that's the norm_term_vonMangoldt_X_le_three_pow context. 1119:68 is probably the `rw [e]` or the final `exact mul_le_mul_of_nonneg_right hle hpos` — hmm.\n\n3. **Line 1165: `ArithmeticFunction.vonMangoldt_prime` unknown** — check the right name for Λ(p) = log p.\n\n4. **Line 1167 (hΛ2's norm_of_nonneg)**: `Complex.norm_of_nonneg Real.log_two_gt_d9.le` — expects `0 ≤ Real.log 2` but got `0.6931... ≤ Real.log 2` — the .le of log_two_gt_d9 gives `0.6931471803 ≤ Real.log 2` — TYPE MISMATCH (it's a ≤ between two specific reals, not 0 ≤). Fix: `Complex.norm_of_nonneg (by positivity : (0 : ℝ) ≤ Real.log 2)` — positivity knows log 2 > 0? — `Real.log_pos (by norm_num : (1:ℝ) < 2)` ✓ or `(Real.log_two_gt_d9.le)` — hmm — norm_of_nonneg needs `0 ≤ Real.log 2`: `Real.log_two_gt_d9 : 0.693... < log 2` ⟹ 0 < log 2 ⟹ 0 ≤ log 2: `by linarith [Real.log_two_gt_d9]` ✓ or positivity (positivity knows log 2 ≥ 0? — it has extensions for log of ≥1: log 2 with 2 ≥ 1: Real.log_nonneg (by norm_num : (1:ℝ) ≤ 2) ✓).\n\n5. **Line 1168**: `‖↑(ArithmeticFunction.vonMangoldt 2)‖` pattern not found in `|ArithmeticFunction.vonMangoldt 2| / 2 ^ (2·√↑n) = Real.log 2 * 2 ^ (-(2·√↑n))` — the target shows `|ArithmeticFunction.vonMangoldt 2|` — the norm displayed as `|·|` (Complex.abs or the norm notation) vs my hΛ2's `‖↑(Λ 2)‖` — the norm_term_eq's output has `‖f n‖` = `‖(fun n => (↑(Λ n)):ℂ) 2‖` = `‖↑(Λ 2)‖` — displayed as `|Λ 2|`?? — hmm, the display shows `|ArithmeticFunction.vonMangoldt 2|` — that's `Complex.abs` or the plain norm — and my rw [hΛ2] pattern is `‖↑(ArithmeticFunction.vonMangoldt 2)‖` — the display `|...|` IS the norm (Lean prints ‖x‖ as |x| sometimes). So why no match? — the goal at that point: `|ArithmeticFunction.vonMangoldt 2| / 2 ^ (2·√↑n) = Real.log 2 * 2 ^ (-(2·√↑n))` — and hΛ2 : `‖↑(Λ 2)‖ = Real.log 2` — should rewrite... — the pattern `‖↑(ArithmeticFunction.vonMangoldt 2)‖` — hmm, maybe the elaborated form is `(‖(ArithmeticFunction.vonMangoldt 2 : ℂ)‖)` with a different cast structure: the norm_term_eq gives `‖(fun n => (↑(Λ n)) : ℂ) 2‖` — beta: `‖(↑(Λ 2) : ℂ)‖` — and my hΛ2's LHS is `‖(ArithmeticFunction.vonMangoldt 2 : ℂ)‖` — `(ArithmeticFunction.vonMangoldt 2 : ℂ)` vs `((fun n => (↑(Λ n)) : ℂ) 2)` — beta-redex not reduced → pattern mismatch! Fix: `show` the beta-reduced form first or `simp only` with beta. The hfin block: after `rw [hΛ2]`... — actually simpler: state hΛ2 as `‖(fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) 2‖ = Real.log 2`? — or add `show` before rw. Let me restructure hfin's tail with a show:\n```\n    have hΛ2 : ‖(ArithmeticFunction.vonMangoldt 2 : ℂ)‖ = Real.log 2 := ...\n    rw [show ‖(fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) 2‖ =\n        ‖(ArithmeticFunction.vonMangoldt 2 : ℂ)‖ from rfl, hΛ2]\n```\nor `simp only [hΛ2]`?? — hΛ2 is an equation, simp only [hΛ2] rewrites ✓ and simp handles beta-redexes ✓ better than rw here. Use simp only for the norm rewrites in hfin.\n\nActually — the norm_term_eq's LHS has `‖LSeries.term ...‖` — after `simp only [LSeries.norm_term_eq, hXre]` and the if-rewrites, the remaining term is `‖(fun n => (↑(Λ n)) : ℂ) 2‖ / 2 ^ (2·√↑n)` — need to rewrite to `Real.log 2 * 2^{−2√n}`: \n```\n    have hΛ2 : ‖(fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) 2‖ = Real.log 2 := by\n      show ‖(ArithmeticFunction.vonMangoldt 2 : ℂ)‖ = Real.log 2\n      rw [show ArithmeticFunction.vonMangoldt 2 = Real.log 2 from\n        ArithmeticFunction.vonMangoldt_apply_prime Nat.prime_two]  -- name TBD\n      exact Complex.norm_of_nonneg (by linarith [Real.log_two_gt_d9] : (0 : ℝ) ≤ Real.log 2)\n    rw [hΛ2]\n```\nHmm — the `show ‖(ArithmeticFunction.vonMangoldt 2 : ℂ)‖ = Real.log 2` — the goal is `‖(fun n => (↑(Λ n)):ℂ) 2‖` — beta-defeq ✓ show passes ✓ then rw inside ✓.\n\n6. **Line 1232 (log_nine_quarters_ge's hprod mul_le_mul)**: hb : 0 ≤ Real.exp 0.8 expected but got 0 ≤ 2.2254 — the mul_le_mul's hb : 0 ≤ b with b := Real.exp 0.8?? — mul_le_mul (hab : a ≤ b) (hcd : c ≤ d) (hc : 0 ≤ c) (hb : 0 ≤ b) : a·c ≤ b·d — with a := 2.2254, b := exp 0.8, c := 1.01106, d := exp 0.011: hc : 0 ≤ 1.01106 ✓ norm_num; hb : 0 ≤ exp 0.8 — I passed (by norm_num : 0 ≤ 2.2254) — WRONG content for hb's slot. The slots: (hc : 0 ≤ c := 1.01106) (hb : 0 ≤ b := exp 0.8)?? — wait the error says expected `0 ≤ Real.exp 0.8` got `0 ≤ 2.2254` — so hb is `0 ≤ exp 0.8`: fix with `(Real.exp_pos _).le`. And hc at 1236 similar? — 1236:46 — another mul_le_mul arg issue in the same region — the `mul_le_mul h08 h011` — h08 : 2.2254 ≤ exp 0.8 ✓ hab ✓; h011 : 1.01106 ≤ exp 0.011 ✓ hcd ✓; then hc : 0 ≤ 1.01106 ✓ norm_num ✓; hb : 0 ≤ exp 0.8 ✓ (Real.exp_pos _).le ✓.\n\nWait — the 1236 error: \"argument hprod has type 9/4 ≤ exp 0.811 but expected Real.exp ?m ≤ ?m\" — that's `Real.log_le_log (Real.exp_pos _) hprod` — log_le_log signature: `Real.log_le_log (hx : 0 < x) (h : x ≤ y) : log x ≤ log y` — hmm — `Real.log_le_log (h1 : 0 < x) (h2 : x ≤ y)`? — the error says hprod expected to be `Real.exp ?m ≤ ?m` — log_le_log's SECOND arg should be x ≤ y with x := 9/4, y := exp 0.811 — hprod : 9/4 ≤ exp 0.811 ✓ matches — but the error says expected `Real.exp ?m ≤ ?m` — meaning the FIRST arg's type drove the unification: `Real.log_le_log (Real.exp_pos _)` — the first arg is `0 < exp ?` — and it unified the y-side expecting `Real.exp ?m ≤ ?m` — because Real.log_le_log's form might be `(hx : 0 < x) (hxy : x ≤ y)` with x from the FIRST arg — if I pass (Real.exp_pos 0.811) : 0 < exp 0.811 — then x := exp 0.811 and the second arg should be `exp 0.811 ≤ ?` — but hprod is `9/4 ≤ exp 0.811` — MISMATCH in order!! The goal is log(9/4) ≤ log(exp 0.811) = 0.811 — so I need log_le_log (h : 0 < 9/4) (hprod : 9/4 ≤ exp 0.811): `Real.log_le_log (by norm_num : (0:ℝ) < 9/4) hprod` ✓ then rw [Real.log_exp] at the result ✓. Fix the arg order.\n\n7. **Line 1263 (four_ninths_rpow_sqrt_le's h2a)**: unsolved goals ⊢ False — the `mul_le_mul_of_nonpos_left hx hlogneg.le` — signature: mul_le_mul_of_nonpos_left (h : a ≤ b) (hc : c ≤ 0) : b * c ≤ a * c — with a := 2.236, b := x, c := log(4/9): gives `x·log(4/9) ≤ 2.236·log(4/9)` — but h2a is stated as `Real.log (4 / 9) * x ≤ Real.log (4 / 9) * 2.236` (the log FIRST) — so I need mul_le_mul_of_nonpos_right?? — no: mul_le_mul_of_nonpos_right (h : a ≤ b) (hc : 0 ≤ c) : a·c ≤ b·c — positive multiplier. For the NEGATIVE multiplier on the RIGHT with log-first: log(4/9)·x ≤ log(4/9)·2.236 ⟸ x ≥ 2.236, log(4/9) ≤ 0: the multiplier log(4/9) is on the LEFT: mul_le_mul_of_nonpos_left (h : x ≥ 2.236) (hc : log(4/9) ≤ 0) : x·log(4/9) ≤ 2.236·log(4/9) — gives x·log ≤ 2.236·log — but I want log·x ≤ log·2.236 — COMMUTED — the lemma's output is `x·c ≤ 2.236·c`, my h2a states `c·x ≤ c·2.236` — mul comm: rewrite h2a's statement to `x * Real.log (4/9) ≤ 2.236 * Real.log (4/9)`?? — but the goal's h1 has `Real.log (4/9) * x` (log-first from rpow_def_of_pos : exp (log a * x)) — so the exponent is log(4/9)·x — and mul_le_mul_of_nonpos_left hx hlogneg.le gives `x·log(4/9) ≤ 2.236·log(4/9)` — the h1's exponent form is `log(4/9)·x` — COMMUTED — so I should state h2a as `Real.log (4 / 9) * x ≤ Real.log (4 / 9) * 2.236` and prove via `mul_le_mul_of_nonneg_left`?? — NO — the multiplier log(4/9) is NEGATIVE and on the LEFT: log(4/9)·x vs log(4/9)·2.236 with x ≥ 2.236: negative·bigger ≤ negative·smaller ✓: this is mul_le_mul_of_nonpos_left?? — mul_le_mul_of_nonpos_left (h : a ≤ b) (hc : c ≤ 0) : b * c ≤ a * c — c on the RIGHT. For c on the LEFT with c ≤ 0: there's `mul_le_mul_of_nonpos_right`?? no that's (h : a ≤ b) (hc : 0 ≤ c) — positive. The LEFT-negative form: `mul_le_mul_of_nonpos_left`?? hmm — let me think: `mul_le_mul_of_nonpos_left (h : a ≤ b) (hc : c ≤ 0) : b * c ≤ a * c` — b·c with c ≤ 0 — c is on the RIGHT of b·c. My case: c·x with c = log(4/9) ≤ 0 on the LEFT: rewrite as x·c? — state h2a as `x * Real.log (4 / 9) ≤ 2.236 * Real.log (4 / 9) := mul_le_mul_of_nonpos_left hx hlogneg.le` ✓ — but the exponent in h1 is `Real.log (4 / 9) * x` — commuted — handle by rewriting the exponent order: h1's exp (Real.log (4/9) * x) — and h2's form should be `Real.log (4/9) * x ≤ -1.8135` — with h2a in the x-first form: `x * Real.log (4/9) ≤ 2.236 * Real.log (4/9)` — then h2b : `2.236 * Real.log (4/9) ≤ 2.236 * (-0.811)` via mul_le_mul_of_nonneg_left hlog49 (0 ≤ 2.236) ✓ — and the calc connects log(4/9)·x ≤ 2.236·log(4/9) ≤ 2.236·(−0.811) — the first link needs x·log ≤ 2.236·log vs h1's log·x — commuted — the calc starts at `Real.log (4 / 9) * x` — so h2a should be stated log-first but PROVED via the mul-comm... — simplest: make h2b: `Real.log (4 / 9) * x ≤ Real.log (4 / 9) * 2.236 := by rw [mul_comm (Real.log (4 / 9)) x, mul_comm (Real.log (4 / 9)) 2.236]; exact mul_le_mul_of_nonpos_left hx hlogneg.le`?? — or just avoid: rewrite h1's exponent: `rw [h1, mul_comm (Real.log (4 / 9)) x]`?? — no wait — the goal after rw [h1] is `Real.exp (Real.log (4/9) * x) ≤ 0.17631` — h2's form `Real.log (4/9) * x ≤ -1.8135` — prove `Real.log (4 / 9) * x ≤ 2.236 * (-0.811)` directly:\n```\n    have h2 : Real.log (4 / 9) * x ≤ (-1.8135 : ℝ) := by\n      have h2a : Real.log (4 / 9) * x ≤ Real.log (4 / 9) * 2.236 := by\n        apply mul_le_mul_of_nonneg_left _ (by norm_num : (0 : ℝ) ≤ 2.236)\n        · sorry\n```\nhmm no — mul_le_mul_of_nonneg_left : a ≤ b → 0 ≤ c → c·a ≤ c·b — with c := log(4/9) < 0 — WRONG sign. The correct lemma for left-negative-multiplier: `mul_le_mul_of_nonpos_left (h : a ≤ b) (hc : c ≤ 0) : b * c ≤ a * c` — c on the RIGHT. For the LEFT: there might be `mul_le_mul_of_nonpos_right (h : b ≤ a) (hc : c ≤ 0) : a·c ≤ b·c`?? — hmm — \"nonpos_right\": (h : b ≤ a) (hc : c ≤ 0) : a * c ≤ b * c?? no... Let me think again: mul_le_mul_of_nonpos_right (h : a ≤ b) (hc : 0 ≤ c) : a·c ≤ b·c — that's nonneg_right. For c ≤ 0 on the right: mul_le_mul_of_nonpos_left — confusingly named (the ≤ 0 multiplier is ON the LEFT?? or RIGHT?). From the earlier usage in this session: `mul_le_mul_of_nonpos_right hx hlogneg.le` at h2a (my ORIGINAL draft) — I wrote mul_le_mul_of_nonpos_right hx hlogneg.le for `x·log(4/9) ≤ 2.236·log(4/9)` — mul_le_mul_of_nonpos_right (h : a ≤ b) (hc : c ≤ 0) : b * c ≤ a * c — YES that exists?? — the error at 1263 was \"unsolved goals ⊢ False\" — maybe the lemma name is wrong or the arg order. Mathlib: `mul_le_mul_of_nonpos_left (h : a ≤ b) (hc : c ≤ 0) : b * c ≤ a * c` and `mul_le_mul_of_nonpos_right (h : b ≤ a) (hc : c ≤ 0) : a * c ≤ b * c`?? — hmm no. Let me recall the actual Mathlib signatures:\n- mul_le_mul_of_nonneg_left : a ≤ b → 0 ≤ c → c * a ≤ c * b\n- mul_le_mul_of_nonneg_right : a ≤ b → 0 ≤ c → a * c ≤ b * c\n- mul_le_mul_of_nonpos_left : b ≤ a → c ≤ 0 → a * c ≤ b * c  (b ≤ a with c ≤ 0 flips to a·c ≤ b·c)\n- mul_le_mul_of_nonpos_right : b ≤ a → c ≤ 0 → c * a ≤ c * b  (c on the left)\n\nSo for `Real.log (4/9) * x ≤ Real.log (4/9) * 2.236` with 2.236 ≤ x and log(4/9) ≤ 0: mul_le_mul_of_nonpos_right (h : x ≥ 2.236 i.e. 2.236 ≤ x) (hc : log(4/9) ≤ 0) : log(4/9)·x ≤ log(4/9)·2.236 ✓ — signature `mul_le_mul_of_nonpos_right (h : b ≤ a) (hc : c ≤ 0) : c * a ≤ c * b` — with a := x, b := 2.236, c := log(4/9): needs `b ≤ a` := `2.236 ≤ x` = hx ✓ and hc := hlogneg.le ✓ — output `log(4/9)·x ≤ log(4/9)·2.236` ✓✓ — so my ORIGINAL draft's `mul_le_mul_of_nonpos_right hx hlogneg.le` for the x·log form was WRONG (that gives a·c ≤ b·c with c on the RIGHT — same statement here coincidentally?). And my EDITED version used mul_le_mul_of_nonpos_left hx hlogneg.le — mul_le_mul_of_nonpos_left (h : b ≤ a) (hc : c ≤ 0) : a * c ≤ b * c — with a := x, b := 2.236: needs 2.236 ≤ x ✓ gives `x·log(4/9) ≤ 2.236·log(4/9)` — the output has x FIRST — but h2a states log-first `Real.log (4/9) * x ≤ Real.log (4/9) * 2.236` — mismatch ✗. \n\nUgh — the two lemmas: mul_le_mul_of_nonpos_left : (b ≤ a) → (c ≤ 0) → a·c ≤ b·c (c RIGHT, flips). mul_le_mul_of_nonpos_right : (b ≤ a) → (c ≤ 0) → c·a ≤ c·b (c LEFT, flips). So for the log-LEFT form `log(4/9)·x ≤ log(4/9)·2.236`: mul_le_mul_of_nonpos_right (h : 2.236 ≤ x) (hc : log(4/9) ≤ 0) ✓ — wait that gives c·a ≤ c·b with a := x, b := 2.236: log·x ≤ log·2.236 ✓✓. So h2a := mul_le_mul_of_nonpos_right hx hlogneg.le ✓ (my ORIGINAL draft was right; my edit broke it to nonpos_left). And h2b : `Real.log (4/9) * 2.236 ≤ (-0.811) * 2.236` : mul_le_mul_of_nonneg_right hlog49 (0 ≤ 2.236) ✓ (a ≤ b → 0 ≤ c → a·c ≤ b·c ✓ log(4/9) ≤ −0.811 with 2.236 ≥ 0 ✓).\n\nHmm wait — but actually is mul_le_mul_of_nonpos_right's signature (h : b ≤ a) (hc : c ≤ 0) : c·a ≤ c·b? — hmm — I believe: `mul_le_mul_of_nonpos_right (h : b ≤ a) (hc : c ≤ 0) : a * c ≤ b * c`?? no... Let me just check with grep to be sure.\n\n8. **Line 1275 (calc step form)**: `Real.exp (x * Real.log (4 / 9))` vs expected `Real.exp (Real.log (4 / 9) * x)` — the calc's first step I wrote `Real.exp (x * Real.log (4 / 9)) ≤ Real.exp (-1.8135)` but h1 gave `Real.exp (Real.log (4/9) * x)` — fix the calc to start at `Real.exp (Real.log (4 / 9) * x)`.\n\nLet me check mul_le_mul_of_nonpos signatures and then fix everything.","signature":""},{"type":"text","text":"Checking the exact `mul_le_mul_of_nonpos_*` signatures and the vonMangoldt prime-lemma name."},{"type":"tool_use","id":"Bash_431","name":"Bash","input":{"command":"cd D:/Python/Quantum/zeta-23-lean && grep -rn \"theorem mul_le_mul_of_nonpos_left\\|theorem mul_le_mul_of_nonpos_right\" .lake/packages/mathlib/Mathlib/Algebra/Order/ 2>/dev/null | head -4; grep -rn \"vonMangoldt_prime\\b\\|vonMangoldt_apply_prime\" .lake/packages/mathlib/Mathlib/NumberTheory/ArithmeticFunction/VonMangoldt.lean 2>/dev/null | head -4","description":"Check mul_le_mul_of_nonpos signatures and vonMangoldt prime lemma"},"extra_content":null}]