[{"type":"thinking","thinking":"Multiple issues to fix:\n\n1. **hsq field_simp+ring failure** (line 245): the goal after field_simp is a mess with `(s.re^2 + s.im^2)⁻¹` — the issue is my h1/h2 used `s.re / normSq s` but field_simp normalized differently. The remaining goal:\n```\n1 + σ·D⁻¹·2 + σ²·D⁻¹² + T²·(σ²T²·2 + σ⁴ + T⁴)⁻¹ = 1 + σ·D⁻¹·2 + D⁻¹\n```\nHmm — LHS has σ²·(D⁻¹)² + T²·(σ²T²·2+σ⁴+T⁴)⁻¹ — that third term: T²·((σ²+T²)²)⁻¹ should be T²·D⁻² but it's displayed as T²·(σ²T²·2+σ⁴+T⁴)⁻¹ (D² expanded inside the inverse — ring_nf normalized D² = (σ²+T²)² = σ⁴+2σ²T²+T⁴ inside ⁻¹ but didn't simplify σ²·D⁻² + T²·D⁻² = (σ²+T²)·D⁻² = D⁻¹ because the atoms got normalized differently... The problem: `field_simp` didn't fully clear denominators (D⁻¹ remains), then `ring` treats (·)⁻¹ as an atom and the identity σ²·D⁻² + T²·D⁻² = D⁻¹ is NOT a ring identity (needs D·D⁻¹ = 1). \n\nFix: after field_simp, do `ring_nf` first? Or better: multiply out manually. The cleanest: \n```lean\nhave hsq : ... := by\n  rw [Complex.normSq_apply (1 + 1 / s), h1, h2, Complex.normSq_apply s]\n  have hne : (s.re ^ 2 + s.im ^ 2 : ℝ) ≠ 0 := by positivity?no — normSq s ≠ 0: hD.ne'\n  field_simp\n  ring_nf\n```\nHmm, but the error already came from field_simp;ring. The residual goal after field_simp: it cleared SOME denominators but not all? The displayed goal has ⁻¹ everywhere — field_simp seems to have failed to clear because... Actually looking again: the goal shown IS after field_simp (it says unsolved after ring). The LHS: 1 + σD⁻¹·2 + σ²D⁻¹² + T²(σ²T²·2+σ⁴+T⁴)⁻¹ vs RHS 1 + σD⁻¹·2 + D⁻¹. Cancel common: need σ²D⁻² + T²(σ²T²·2+σ⁴+T⁴)⁻¹ = D⁻¹. Note (σ²T²·2+σ⁴+T⁴) = (σ²+T²)² = D². As ring atoms: D⁻¹ = u say; then σ²u² + T²·v⁻¹ where v = σ²T²·2+σ⁴+T⁴ — v is a DIFFERENT atom from D²·... ugh. ring can't since v⁻¹ vs u²: v = D² as ring expression but the ⁻¹ atom differs.\n\nBetter approach: avoid field_simp's inconsistent normalization. Prove hsq by:\n```lean\nhave hsq : Complex.normSq (1 + 1 / s) = 1 + (2 * s.re + 1) / Complex.normSq s := by\n  rw [Complex.normSq_apply (1 + 1 / s), h1, h2, Complex.normSq_apply s]\n  rw [div_add_div_same, ...]  -- manual\n```\nManual algebra: (1 + σ/D)² + (T/D)² = 1 + 2σ/D + σ²/D² + T²/D² and σ²/D² + T²/D² = (σ²+T²)/D² = D/D² = 1/D. Steps:\n```lean\n  have hne : (s.re ^ 2 + s.im ^ 2 : ℝ) ≠ 0 := hD.ne'  -- after rw normSq_apply s, D is replaced... \n```\nHmm — after `rw [Complex.normSq_apply s]`, occurrences of normSq s in the GOAL get replaced by s.re^2 + s.im^2. But hD : 0 < normSq s stays as normSq s — mismatch. Careful: hD.ne' : normSq s ≠ 0 — after rw in goal only, hD unchanged. For field_simp I pass hD.ne' — but the goal now has (s.re^2+s.im^2) not normSq s — field_simp [hD.ne'] won't match! Need the non-neq in the REWRITTEN form: have hne : s.re^2 + s.im^2 ≠ 0 := by rw [← Complex.normSq_apply]; exact hD.ne'. \n\nAlternative cleaner: DON'T rw normSq_apply s in the goal; keep D := normSq s via set BEFORE deriving, so everything stays in D. Let me restructure norm_one_add_one_div_le with `set D := Complex.normSq s with hD_def` at the top, derive h1 h2 hsq in terms of D, and only at the end (hsum) unfold.\n\n```lean\ntheorem norm_one_add_one_div_le {s : ℂ} (hs : s ≠ 0) :\n    ‖1 + 1 / s‖ ≤ 1 + (s.re + 1) / Complex.normSq s := by\n  set D := Complex.normSq s with hD_def\n  have hD : (0:ℝ) < D := Complex.normSq_pos.mpr hs\n  have h1 : (1 + 1 / s).re = 1 + s.re / D := by\n    simp [Complex.add_re, Complex.one_re, Complex.div_re, Complex.one_im, hD_def]\n  have h2 : (1 + 1 / s).im = -s.im / D := by\n    simp [Complex.add_im, Complex.one_im, Complex.div_im, Complex.one_re, hD_def]\n  have hsq : Complex.normSq (1 + 1 / s) = 1 + (2 * s.re + 1) / D := by\n    rw [Complex.normSq_apply (1 + 1 / s), h1, h2]\n    have hDsq : D ^ 2 = (s.re ^ 2 + s.im ^ 2) * D := by\n      rw [hD_def, Complex.normSq_apply s]; ring\n    -- (1 + σ/D)² + (T/D)² = ... : field_simp then ring with hDsq\n    field_simp [hD.ne']\n    -- goal: (D + σ)^2 * D + ... hmm\n    nlinarith [hDsq]  -- or ring_nf then substitute\n  ...\n```\nAfter field_simp [hD.ne'] on goal `(1 + s.re/D)^2 + (-s.im/D)^2 = 1 + (2*s.re+1)/D`: field_simp multiplies through by D² (it clears all denominators): goal becomes like `(D + s.re)^2 * D + s.im^2 * D = (1*D + (2*s.re+1)) * D^2`? — both sides ×D². LHS: (1+σ/D)²·D² = (D+σ)²·... hmm field_simp's exact normal form varies. The identity to prove after clearing: (D+σ)²·? ... Let me just think: (1+σ/D)² + (T/D)² = 1+(2σ+1)/D. Multiply by D²: (D+σ)² + T² = D² + (2σ+1)D. With D = σ²+T² (hD_def + normSq_apply). So after field_simp: goal `(D + s.re)^2 + s.im^2 = D^2 + (2*s.re + 1)*D` (or some ring-normal equivalent). This is NOT a ring identity (needs D = σ²+T²). So `nlinarith [hD_rel]` where hD_rel : D = s.re^2 + s.im^2 — expand: (D+σ)² + T² = D²+2σD+σ²+T²; RHS D²+2σD+D: need σ²+T² = D ✓ exactly hD_rel. So `nlinarith [hD_rel]`? The difference: LHS−RHS = σ²+T²−D = 0 given hD_rel — linear_combination hD_rel! `linear_combination hD_rel` should work: LHS − RHS = (σ² + T²) − D ✓ as ring identity modulo hD_rel. Let me use: \n```lean\n    field_simp [hD.ne']\n    have hD_rel : D = s.re ^ 2 + s.im ^ 2 := by rw [hD_def, Complex.normSq_apply s]\n    linear_combination hD_rel\n```\nHmm — linear_combination: goal LHS = RHS; checks (LHS − RHS) − (coeff·(h_lhs − h_rhs)) = 0 by ring. (LHS−RHS) = (D+σ)²+T² − (D²+(2σ+1)D) = σ²+T²−D (ring). hD_rel: D − (σ²+T²). So (LHS−RHS) = −(D − (σ²+T²)): linear_combination (-hD_rel)? or linear_combination (hD_rel) with sign handled: linear_combination hD_rel means subtract 1*(D − (σ²+T²))... it verifies (LHS−RHS) − 1·(D − (σ²+T²)) = 0 by ring: (σ²+T²−D) − (D−σ²−T²) = 2(σ²+T²) − 2D ≠ 0 as ring identity ✗. So use `linear_combination (-hD_rel)`? Hmm — linear_combination exp uses exp as the coefficient difference: goal: LHS−RHS = exp's (lhs−rhs)?? The tactic: `linear_combination e` proves LHS = RHS by checking LHS − RHS − (e.lhs − e.rhs) = 0 via ring. e = hD_rel : D = σ²+T². So check: [(D+σ)²+T² − (D²+(2σ+1)D)] − [D − (σ²+T²)] = (σ²+T²−D) − (D−σ²−T²) = 2σ²+2T²−2D — not 0 by ring ✗. So coefficient should be... the difference LHS−RHS equals σ²+T²−D = −(D−σ²−T²). So linear_combination (-hD_rel)? then check: (LHS−RHS) − (−(D − (σ²+T²))) = (σ²+T²−D) + (D−σ²−T²) = 0 ✓. Hmm wait but actually I'm not sure of field_simp's exact output form. Might be multiplied differently (e.g. by D² vs D⁴). To be safe: `ring_nf` after field_simp then `linear_combination`? Or simplest robust: nlinarith [hD_rel]: nlinarith should manage since after expanding the products it reduces to σ²+T² = D... nlinarith can square hypotheses etc. But the goal after field_simp is an EQUALITY — nlinarith proves equalities by splitting ≤ both ways? nlinarith can close equalities? It can if it reduces to linarith on both directions — nlinarith does handle `=` goals by splitting into two ≤. OK: `nlinarith [hD_rel]` — but the products: (D+σ)² — hypothesis products considered: hD_rel is linear in atoms D, σ²?? no — σ² is σ*σ... nlinarith treats monomials: goal has D², σD, σ², T². hD_rel : D = σ²+T². The needed inference: substitute D... nlinarith multiplies pairs of hypotheses — it can multiply hD_rel by D? (D = σ²+T²) × ... it considers products of hypotheses with each other and with the negated goal. For equality splitting: two ≤ goals. Goal ≤ direction: (D+σ)²+T² ≤ D²+(2σ+1)D ⟺ σ²+T² ≤ D. From hD_rel directly (linear in atoms σ²,T²,D after expansion? σ² is a monomial atom ✓ linarith over monomials). nlinarith's preprocessing expands powers? It uses ring_nf on hypotheses/goal first I believe. Should work. If not, iterate.\n\n2. **h1/h2 simp warnings + possibly wrong form**: simp [Complex.div_re] unused arg — because simp's built-in `Complex.div_re`? The warning says div_re unused in h1 — meaning simp closed h1 without needing div_re? h1 : (1 + 1/s).re = 1 + s.re/normSq s. simp [add_re, one_re] reduced LHS to 1 + (1/s).re and then? To finish it needs (1/s).re = s.re/normSq s — maybe simp found `Complex.div_re`... the warning says unused — maybe simp used `Complex.div_re` is marked simp already and my explicit arg is redundant? Whatever — warnings only; but the h1 FORM: it succeeded, giving `1 + s.re / normSq s`? h1's displayed in the error context: `h1 : (1 + 1 / s).re = 1 + s.re / normSq s` ✓ good. Fix warnings by trimming args: h1: simp [Complex.add_re, Complex.one_re]; h2: simp [Complex.add_im, Complex.one_im]. Hmm but will they still close? The warning's \"Try this\" suggests exactly those reduced arg lists. Use them.\n\n3. **n = 0 case in exp_four**: \n```\ncase inl ⊢ 0 ≤ Real.exp 4 + 1\n```\nMy `simp [this]` didn't close. Goal after simp only [kFun, pow_zero]: ‖kFun 0 (s+1)‖ — simp only [kFun, pow_zero] turns ‖(…)^0 − 1‖ into ‖1 − 1‖... it left `0 ≤ Real.exp 4 + 1`? The context shows goal `0 ≤ Real.exp 4 + 1` — so simp reduced ‖1−1‖ = ‖0‖ = 0 ✓ leaving `0 ≤ exp 4 + 1` — just `positivity` closes. Replace the whole branch:\n```lean\n  · subst hn\n    simp only [kFun, pow_zero, sub_self, norm_zero]\n    positivity\n```\nkFun 0 (s+1) = ((s+1)/(s+1−1))^0 − 1 = 1 − 1 = 0: simp only [kFun, pow_zero, sub_self] gives ‖0‖ ≤ ... then norm_zero: 0 ≤ exp 4 + 1: positivity ✓.\n\n4. **`le_or_lt` unknown**: the name is `le_or_lt : a ≤ b ∨ b < a`? Hmm — it exists as `le_or_lt`? Maybe renamed `le_or_lt` → `le_or_lt`... The error says unknown identifier. Current Mathlib: `le_or_lt (a b : α) : a ≤ b ∨ b < a` — hmm maybe it's `le_or_lt` in Order namespace or `_root_.le_or_lt`... It might be `lt_or_ge`/`le_or_lt` — let me use `rcases le_or_lt 0 (s.re + 1)` → if unknown, use `by_cases hσp : 0 ≤ s.re + 1` with the negative case via `not_le.mp`. Use by_cases (always works).\n\n5. **`pow_le_pow_left` unknown**: renamed! Current Mathlib: `pow_le_pow_left (ha : 0 ≤ a) (hab : a ≤ b) (n : ℕ)` might now be `pow_le_pow_left₀`? The grep earlier showed `pow_le_one₀` with ₀ suffix — so pow_le_pow_left is probably `pow_le_pow_left₀`. Let me check.\n\n6. **`div_le_div_left` unknown**: also renamed? Maybe `div_le_div_iff_left`? or `div_le_div_left` exists with different args. Check.\n\nLet me grep for the current names.","signature":""},{"type":"text","text":"Several renames to hunt down (`pow_le_pow_left`, `div_le_div_left`, `le_or_lt`) plus a field_simp normalization issue. Checking current names."},{"type":"tool_use","id":"Bash_83","name":"Bash","input":{"command":"cd D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"theorem pow_le_pow_left\\|lemma pow_le_pow_left\" Mathlib/Algebra/Order/ 2>/dev/null -r | head -6; grep -rn \"theorem div_le_div_left\\|lemma div_le_div_left\\|theorem div_le_div_right\\|lemma div_le_div_right\\|theorem div_le_div_of\" Mathlib/Algebra/Order/ -r 2>/dev/null | head -8; grep -rn \"theorem le_or_lt\\|lemma le_or_lt\" Mathlib/Order/ -r 2>/dev/null | head -4","description":"Find current names for pow/div comparison lemmas"},"extra_content":null}]