[{"type":"thinking","thinking":"Errors:\n1. `congr 1; field_simp` in exp_le_inv_sub_sixteen leaves `x * 16 = x * ↑16` — a CAST issue: field_simp turned the goal into x * 16 = x * ↑16 where one side has the ℕ-cast 16 and the other... interesting: `↑16 * (x/16) = x` became `x * 16 = x * ↑16` — hmm, field_simp normalized to x*16 vs x*↑16 — the cast: `↑16` is ((16:ℕ):ℝ) and plain `16` is OfNat. They're the same value but syntactically different. Fix: after field_simp, `norm_num` or `ring` should close x*16 = x*↑16? push_cast: ↑16 = 16. Use: `congr 1; field_simp; norm_num` — or better `push_cast; ring`? The goal x * 16 = x * ↑16: `norm_num` should unify casts. Let me use: congr 1; field_simp [h16]; norm_num? Hmm — actually cleaner: avoid field_simp: goal after congr 1 is `x = ↑16 * (x / 16)` or `↑16 * (x/16) = x`. Just: `rw [mul_div_assoc, mul_div_cancel_left₀]`? mul_div_cancel_left₀ : a ≠ 0 → b * a / b = b... Form: ↑16 * (x/16): rw [mul_div_assoc'] : a*(b/c) = a*b/c... then 16*x/16 = x: `div_self`? 16*x/16 = x needs mul_div... Let me think: ↑16 * (x / 16) = (↑16 * x)/16 = x·(↑16)/↑16... Simplest robust: `have : ((16:ℕ):ℝ) * (x/16) = x := by field_simp [h16]` — didn't work. Try: `by rw [mul_div_assoc, div_self h16, mul_one]`? mul_div_assoc : a * b / c = a * (b / c)? mul_div_assoc : a * (b / c) = a * b / c (one direction). (↑16 * x)/↑16 = x: that's mul_div_cancel_right₀? `mul_div_cancel_right₀ (h : b ≠ 0) : a * b / b = a`. So: rw [mul_div_assoc, mul_div_cancel_right₀ x h16]?? mul_div_cancel_right₀ signature: (a : G) (h : b ≠ 0) : a * b / b = a — as rw [mul_div_cancel_right₀ x h16]?? It rewrites x * ↑16 / ↑16 → x. Path: ↑16 * (x/16) →(mul_div_assoc) ↑16 * x / 16 — hmm ↑16 * x vs x * ↑16 — mul_div_assoc : a * (b / c) = (a * b) / c: gives (↑16 * x)/↑16; mul_div_cancel_right₀ matches a * b / b with a := x, b := ↑16 — but we have ↑16 * x / ↑16 — commuted. rw [mul_comm ↑16 x] first. Ugh. \n\n   Cleanest: skip the intermediate h4-rewrite approach; prove h4 by:\n   ```\n   have h4 : Real.exp x = (Real.exp (x / 16)) ^ 16 := by\n     have h16' : (16 : ℝ) ≠ 0 := by norm_num\n     rw [← Real.exp_nat_mul (x / 16) 16]\n     congr 1\n     rw [show ((16:ℕ):ℝ) * (x / 16) = x by field_simp]\n   ```\n   field_simp failing similarly... The root issue: (↑(16:ℕ) : ℝ) * (x/16) = x. field_simp normalizes divisions: ↑16 * (x/16) → ↑16 * x / 16 → cross... it produced x*16 = x*↑16 which is cast-mismatch noise. `norm_num` after field_simp should close x*16 = x*↑16 (norm_num handles casts: ↑16 evaluates to 16). So: `congr 1; field_simp; norm_num`? But field_simp left \"unsolved goals\" as an ERROR at 46:53 — the whole `field_simp` call is at fault for leaving the goal. Adding `; norm_num` after should close. Actually — even simpler: use `push_cast` then `ring`:\n   goal `↑16 * (x / 16) = x` (or reversed): push_cast turns ↑16 into 16 (OfNat): then 16*(x/16) = x by ring. So: `congr 1; push_cast; ring`. But wait — does congr 1 give ↑16 * (x/16) = x or x = ↑16*(x/16)? Either way ring handles. But is the goal exactly that? After rw [← Real.exp_nat_mul (x/16) 16]: goal Real.exp x = Real.exp (↑16 * (x/16)) — congr 1: x = ↑16 * (x/16) ✓. push_cast; ring ✓. Same fix for exp_eleven/five/four: goal after rw: Real.exp 11 = Real.exp (↑11 * 1): congr 1: 11 = ↑11 * 1: push_cast; ring or norm_num. The earlier compile — did exp_eleven fail? Errors listed only 46:53 for the field_simp — exp_eleven at line ~62 used `congr 1; norm_num` and NO error was reported for it! So norm_num closes 11 = ↑11 * 1 fine (norm_num handles the cast). So for h4: `congr 1; norm_num`?? goal x = ↑16*(x/16) — norm_num can't (has free x and division). Use field_simp [h16] then norm_num: field_simp leaves x*16 = x*↑16, norm_num closes. Hmm — but actually maybe `ring_nf` closes x = ↑16*(x/16) directly? ring_nf normalizes casts and divisions-as-multiplications: x vs 16·x·(1/16) — ring_nf should close it (fields handled by ring_nf? ring_nf handles division by constants ✓). Try `congr 1; ring_nf`... but the cast ↑16: ring_nf might not push casts. `push_cast; ring_nf`. I'll use `congr 1; push_cast; ring` — ring after push_cast: 16*(x/16) = x — ring handles division by nonzero constant ✓ (ring works in fields, treating /16 as *16⁻¹; 16*16⁻¹ = 1 needs field... ring CAN do this: 16*(x*16⁻¹) = x·(16·16⁻¹) — ring normalizes 16*16⁻¹ to 1 in fields? Yes, ring handles inverse of literals).\n\n2. `pow_le_pow_left` unknown identifier — in this Mathlib version it must be named differently! But exp_one_pow_le compiled?? No wait — errors at 51, 56, 59 are exp_one_pow_le, exp_eleven... no: 51:8, 56:2, 59:2 — line 51 is the end of exp_one_pow_le's proof (pow_le_pow_left line). So pow_le_pow_left is UNKNOWN. In recent Mathlib it was renamed: `pow_le_pow_left` → `pow_le_pow_left₀`? or `pow_le_pow_left` has different arg order (ha : 0 ≤ a) (hab : a ≤ b) (n)... In Mathlib4 current: `pow_le_pow_left₀ (ha : 0 ≤ a) (hab : a ≤ b) (n : ℕ) : a ^ n ≤ b ^ n`? Hmm, and there's `pow_le_pow_left (hn : 0 ≤ x)?? ` — let me grep.\n\n3. `div_le_div_right` unknown — renamed: `div_le_div_right` → `div_le_div_right₀`? or the iff form `div_le_div_iff_right`? Let me grep both.","signature":""},{"type":"text","text":"Renamed lemmas in this Mathlib version. Finding the current names."},{"type":"tool_use","id":"Bash_142","name":"Bash","input":{"command":"cd D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"theorem pow_le_pow_left\" Mathlib/Algebra/Order/ | head -5; grep -rn \"theorem div_le_div_right\\b\\|theorem div_le_div_right₀\" Mathlib/Algebra/Order/ | head -5","description":"Find current pow_le_pow_left / div_le_div_right names"},"extra_content":null}]