[{"type":"thinking","thinking":"Several errors:\n\n1. **Line 89** (hΓ1 in §2, and presumably the twin in gFun_one_add_eventuallyEq): the goal after congrArg Complex.re hm is `1 / 2 = (-↑m).re` — the rewrites e3/e2 didn't fire because e's form is `1/2 = (-↑m).re` where the LHS is already `1/2` (not `((1:ℂ)/2).re`) — hm : (1:ℂ)/2 = -(m:ℂ), congrArg Complex.re hm : ((1:ℂ)/2).re = (-(m:ℂ)).re — but it displays as `1 / 2 = (-↑m).re` — Lean already simplified ((1:ℂ)/2).re to 1/2? No wait — e : `1 / 2 = (-↑m).re` — hmm the LHS shows `1 / 2` — that means congrArg gave ((1:ℂ)/2).re but displayed reduced? Actually more likely: `simp` lemmas already normalized during congrArg? No — congrArg Complex.re hm gives (Complex.re ((1:ℂ)/2)) = (Complex.re (-(m:ℂ))) which displays as `1 / 2 = (-↑m).re`?? No — display would be `((1/2)).re`... The error message shows e : 1 / 2 = (-↑m).re — so the LHS got simplified to `1/2` by... hmm, whatever: rw [e3] failed because the pattern ((1:ℂ)/2).re isn't in e (already reduced to 1/2). And rw [e2] failed because... it says \"did not find pattern (-↑↑m).re in 1/2 = (-↑m).re\" — the e2 pattern (-↑↑m).re vs e's (-↑m).re — DOUBLE cast mismatch! e2 : (-↑↑m).re = -↑m where ↑↑m is ((m:ℝ):ℂ)? I wrote ((-((m : ℝ) : ℂ))).re — that's (−((m:ℝ):ℂ)).re displayed as (-↑↑m).re with two arrows (ℝ→ℂ coercion of the ℝ-cast). But e's term is (-↑m).re where ↑m is the direct (m:ℂ) Nat.cast. These differ: -(m:ℂ) vs -((m:ℝ):ℂ). Defeq but not syntactic.\n\nSimplest fix: skip the e2/e3 manual rewriting — use simp on e:\n```lean\n  have hΓ1 : ∀ m : ℕ, (1 : ℂ) / 2 ≠ -(m : ℂ) := by\n    intro m hm\n    have e := congrArg Complex.re hm\n    simp at e  -- e : (1/2 : ℝ) = -(m : ℝ)? then\n    have hm0 : (0 : ℝ) ≤ (m : ℝ) := Nat.cast_nonneg m\n    linarith\n```\n`simp at e` on ((1:ℂ)/2).re = (-(m:ℂ)).re: simp lemmas: Complex.one_re? (1/2 : ℂ).re — is there a simp lemma for (1/2 : ℂ).re = 1/2? `Complex.div_re`? and (-(↑m : ℂ)).re = -(↑m : ℝ) via Complex.neg_re + Complex.ofNat... Let me think: simp should normalize both sides to reals: LHS (1/2 : ℂ).re → 1/2 (via Complex.ofReal... or norm_num). Safer: `simp only [Complex.neg_re] at e` then handle casts... Actually cleanest: norm_num at e? Let me do:\n```lean\n    have e := congrArg Complex.re hm\n    have e' : (1 / 2 : ℝ) = -(m : ℝ) := by\n      have h1 : ((1 : ℂ) / 2).re = 1 / 2 := by norm_num\n      have h2 : (-(m : ℂ)).re = -(m : ℝ) := by simp\n      rw [h1, h2] at e\n      exact e\n    have hm0 : (0 : ℝ) ≤ (m : ℝ) := Nat.cast_nonneg m\n    linarith\n```\nh1 : ((1:ℂ)/2).re = 1/2 by norm_num — hmm does norm_num compute (1/2 : ℂ).re? There are simp lemmas Complex.div_re, Complex.one_re, Complex.ofNat_re... norm_num should handle. h2 : (-(m:ℂ)).re = -(m:ℝ): Complex.neg_re : (-z).re = -z.re; (↑m : ℂ).re = ↑m (Complex.ofNat_re? or by simp). `by simp` should give it (Complex.neg_re, Complex.natCast_re?). But wait — the rw [h1, h2] at e might hit the same display-mismatch: e's LHS is displayed as `1 / 2` — but h1's LHS is ((1:ℂ)/2).re — if e's actual TERM is ((1:ℂ)/2).re (just displayed weirdly), rw works. The error shows the rw DID find... no wait, the error at 89:12 says rw failed: \"Did not find an occurrence of the pattern (-↑↑m).re in the target expression 1 / 2 = (-↑m).re\" — so e's term contains (-↑m).re (single cast) and my e2 was about (-↑↑m).re (double). So e's LHS `1 / 2` — hmm, is it possible e's LHS is literally (1/2 : ℝ)?? If congrArg Complex.re hm produced ((1:ℂ)/2).re = ... — no way to get plain 1/2 without simp. UNLESS hm itself is (1:ℂ)/2 = −↑m where (1:ℂ)/2 has already been elaborated... and Complex.re ((1:ℂ)/2) — Lean might display `((1 : ℂ) / 2).re` as... The message `1 / 2 = (-↑m).re` is in the GOAL display after `rw [e3]`?? No — the error is at the rw [e3, e2] at e line (89:12). It reports the pattern (-↑↑m).re (from e2) not found in `1 / 2 = (-↑m).re` — so e's current form IS `1 / 2 = (-↑m).re`. e3's rewrite must have succeeded first ((1:ℂ)/2).re → 1/2? e3 : ((1 : ℂ) / 2).re = 1 / 2 — rw [e3] at e replaced ((1:ℂ)/2).re by 1/2 ✓ worked. Then e2's pattern (-↑↑m).re — the actual term is (-↑m).re with ↑m = Nat.cast to ℂ — my e2 was built with ((-((m : ℝ) : ℂ))) — the ofReal-of-natcast double arrow. Mismatch.\n\nFix: make e2 about (-(m : ℂ)).re:\n```lean\n    have e2 : (-(m : ℂ)).re = -(m : ℝ) := by simp\n```\n`(-(m : ℂ)).re` — displays as (-↑m).re ✓ matches. simp proves: Complex.neg_re → -(↑m).re; (↑m : ℂ).re = ↑m via Complex.ofNat_re / Complex.natCast_re — simp set should include these by default (Complex.natCast_re is simp? `Complex.natCast_re : (↑n : ℂ).re = n` — likely simp). Try by simp.\n\n2. **Line 124** (`ring_nf` suggestion; \"Not a comparison: s ∈ {0}\"): this is in hs1' — `apply hs0; linear_combination h` where hs0 : s ∈ {0}ᶜ — apply hs0 turns goal into s ∈ {0}?? hs0 : s ∈ {0}ᶜ = (s ∈ {0} → False). `apply hs0` — goal after intro h: False. apply hs0 : s ∈ {0}ᶜ → False... hs0's type: s ∈ {0}ᶜ which is s ∉ {0} = (s ∈ {0} → False). apply hs0 to goal False gives goal s ∈ {0}. Then linear_combination can't prove s ∈ {0} (not a comparison). Fix: hs0 is used as s ≠ 0 — in LiMoments they passed h0 from self_mem_nhdsWithin directly to gFun_eq_momentB's (h0 : s ≠ 0) — defeq works for APPLICATION but `apply` exposes the set-membership. Fix: \n```lean\n  have hs0' : s ≠ 0 := hs0\n```\nand use hs0'... or `exact hs0 (by linear_combination h : s = 0)`? Membership s ∈ {0} is defeq s = 0: `exact hs0 (show s = 0 by linear_combination h)`? The show's target s = 0 vs needed s ∈ {0}: defeq ✓ should work via exact. Hmm — show s = 0 then exact hs0 that: hs0 expects s ∈ {0}; s = 0 is defeq to s ∈ ({0} : Set ℂ)? s ∈ {a} is Set.mem_singleton_iff-reducible... it's defeq (singleton defined via setOf). `exact hs0 (show s = 0 by linear_combination h)` — likely works. Safer: `exact hs0 (Set.mem_singleton_iff.mpr (by linear_combination h))`. Hmm Set.mem_singleton_iff : a ∈ {b} ↔ a = b. .mpr (h : s = 0) : s ∈ {0} ✓ robust. Actually wait — in the error the goal became s ∈ {0} and linear_combination errored. Replace:\n```lean\n  have hs1' : (1 : ℂ) + s ≠ 1 := by\n    intro h\n    apply hs0\n    have e : s = 0 := by linear_combination h\n    exact Set.mem_singleton_iff.mpr e\n```\nHmm — apply hs0 then exact Set.mem_singleton_iff.mpr e ✓.\n\nActually simpler: `intro h; exact hs0 (by linear_combination h : s = 0)` — ascription (term : type): `exact hs0 (show s = 0 by linear_combination h)` — does exact accept the defeq s = 0 vs s ∈ {0}? exact uses isDefEq — s ∈ {0} unfolds to s = 0? Set.instSingletonSet: {a} = Set.singleton a := {x | x = a}; s ∈ {x | x = a} is defeq (s = a) ✓ yes.\n\n3. **Line 167** (hPlog): after field_simp, leftover goal `s * s⁻¹ * 2 + s⁻¹ = 1 + s * s⁻¹ + s⁻¹`. Hmm — the field_simp normalized but the goal isn't closed: `s * s⁻¹ * 2 + s⁻¹ = 1 + s * s⁻¹ + s⁻¹` — that's `2·(s·s⁻¹) + s⁻¹ = 1 + s·s⁻¹ + s⁻¹` — needs s·s⁻¹ = 1: field_simp should have used hs0... it cleared denominators but the resulting goal needs `ring` with s·s⁻¹=1 — but s·s⁻¹ is NOT a ring identity (needs field). My trailing `ring` failed because s·s⁻¹ isn't 1 in a general ring. Hmm — wait actually the goal displayed IS the remaining goal after `field_simp [hs0', hs0]; ring`?? The error says \"unsolved goals ... ⊢ s * s⁻¹ * 2 + s⁻¹ = 1 + s * s⁻¹ + s⁻¹\" — so field_simp produced this and ring couldn't close (s*s⁻¹ atom). Something's off: what was the goal before field_simp? After rw [hderiv, show ((1:ℂ)+s)−1 = s]: goal: ((1+s) − 1/2) / ((1+s)*s/2) = 1/(1+s) + 1/s. field_simp [hs0', hs0] — cross-multiplies: ((1+s) − 1/2)·(1+s)·s·2... hmm the leftover `s * s⁻¹ * 2 + s⁻¹ = 1 + s * s⁻¹ + s⁻¹` looks like field_simp didn't fully clear (s⁻¹ present). Maybe field_simp got confused by the (1+s)⁻¹ vs 1/(1+s) forms. \n\nLet me instead prove hPlog more directly: ((1+s) − 1/2) / ((1+s)*s/2) = 1/(1+s) + 1/s. Multiply out: LHS = (2(1+s) − 1)/((1+s)s) = (1 + 2s)/((1+s)s). RHS = (s + (1+s))/((1+s)s) = (1+2s)/((1+s)s) ✓. field_simp then ring should work if field_simp fully clears. The problem might be that field_simp treats `1 / (1 + s)` fine... The leftover suggests field_simp rewrote partially. Alternative robust: \n```lean\n    rw [hderiv, show ((1 : ℂ) + s) - 1 = s by ring]\n    have hne1 : (1 + s) * s ≠ 0 := mul_ne_zero hs0' hs0\n    field_simp [hs0', hs0]\n    ring_nf\n    rw [mul_inv_cancel hs0]  -- s * s⁻¹ = 1\n    ring? \n```\nHmm messy. Let me think again: maybe cleaner to compute logDeriv directly as 1/(1+s) + 1/s by deriving it as a SUM of logDerivs: logDeriv (t(t−1)/2) — actually t(t−1)/2 = (1/2)·t·(t−1): logDeriv (const · t · (t−1)) = logDeriv t + logDeriv (t−1) (const factor cancels: logDeriv_mul with const ≠ 0, then logDeriv_const? hmm logDeriv (fun t => c * f t) — there's logDeriv_const_mul? Might not exist. Skip.)\n\nAlternative: use the identity 1/(1+s) + 1/s = (1 + 2s)/(s(1+s)) and prove equality of fractions via `div_eq_div_iff`:\n```lean\n    rw [hderiv, show ((1 : ℂ) + s) - 1 = s by ring]\n    rw [div_add_div _ _ hs0' hs0]  -- 1/(1+s) + 1/s = (1·s + (1+s)·1)/((1+s)·s)\n    -- goal: ((1+s) − 1/2)/((1+s)*s/2) = (1*s + (1+s)*1)/((1+s)*s)\n    rw [div_eq_div_iff (by ... : (1+s)*s/2 ≠ 0) (by ... : (1+s)*s ≠ 0)]\n    ring\n```\ndiv_add_div : a/b + c/d = (a·d + c·b)/(b·d)? Signature: div_add_div (a c : α) {b d : α} (hb : b ≠ 0) (hd : d ≠ 0) : a / b + c / d = (a * d + c * b) / (b * d) ✓. rw [div_add_div 1 1 hs0' hs0] — hmm the explicit args (a c): `div_add_div 1 1 hs0' hs0`? a := 1, c := 1 ✓. Gives 1/(1+s) + 1/s = (1*s + 1*(1+s))/((1+s)*s). \ndiv_eq_div_iff (hb : b ≠ 0) (hd : d ≠ 0) : a/b = c/d ↔ a*d = c*b ✓.\n((1+s)*s/2) ≠ 0: div_ne_zero (mul_ne_zero hs0' hs0) two_ne_zero; (1+s)*s ≠ 0: mul_ne_zero hs0' hs0.\nThen ring: ((1+s) − 1/2)·((1+s)·s) = (1·s + 1·(1+s))·((1+s)·s/2): ring ✓ (no inverses).\n\n4. **Line 194** — \"unsolved goals\" in hΛeq's field_simp [htΓ]? The error shows unsolved goals after some point in the hE/hPlog region — cascade. Let me fix the three identified issues and rebuild to see what remains.\n\nAlso check hX's field_simp [hs0]: (1 − (1 − s·X))/s = X — LHS = (s·X)/s = X: field_simp [hs0] should close outright (banked: field_simp closes goals outright; if it leaves `s * (X) = X * s`-style goals ring needed). The error at 194:47 might be hX — \"unsolved goals\" with context shown. 194:47 — let me look at line 194 in the written file to identify exactly. Rather than guess: fix the 3 known issues and rerun.\n\nFor the hΓ1 fix (two places — §2 and gFun_one_add_eventuallyEq):\n```lean\n  have hΓ1 : ∀ m : ℕ, (1 : ℂ) / 2 ≠ -(m : ℂ) := by\n    intro m hm\n    have e := congrArg Complex.re hm\n    have e2 : (-(m : ℂ)).re = -(m : ℝ) := by simp\n    have e3 : ((1 : ℂ) / 2).re = 1 / 2 := by norm_num\n    rw [e3, e2] at e\n    have hm0 : (0 : ℝ) ≤ (m : ℝ) := Nat.cast_nonneg m\n    linarith\n```\nWait — but in the FIRST hΓ1 the error was only e2 (e3 worked). So fix e2's form: (-(m : ℂ)).re — display (-↑m).re ✓. simp should prove (-(m:ℂ)).re = -(m:ℝ): Complex.neg_re gives -(↑m : ℂ).re; then (↑m : ℂ).re = ↑m : ℝ — simp lemma Complex.natCast_re? hmm not sure of name — `Complex.ofNat_re`? Let me just compute: (-(m : ℂ)).re: `by simp [Complex.neg_re]` — Complex.neg_re : (-z).re = -z.re, then (↑m).re — simp knows Complex.ofNat_re? There's `Complex.natCast_re : (↑n : ℂ).re = ↑n`? I think `Complex.natCast_re` or `Complex.ofNat_re` exists as simp. by simp should close.\n\nFor hs1' fix:\n```lean\n  have hs1' : (1 : ℂ) + s ≠ 1 := by\n    intro h\n    exact hs0 (show s = 0 by linear_combination h)\n```\n\nFor hPlog fix: the div_add_div + div_eq_div_iff route.\n\nLet me apply these fixes.","signature":""},{"type":"text","text":"Four fixes needed: the re-cast mismatch in hΓ1, the set-membership in hs1′, and the hPlog fraction algebra. Applying."},{"type":"tool_use","id":"Edit_105","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCrux.lean","old_string":"  have hΓ1 : ∀ m : ℕ, (1 : ℂ) / 2 ≠ -(m : ℂ) := by\n    intro m hm\n    have e := congrArg Complex.re hm\n    have e2 : ((-((m : ℝ) : ℂ))).re = -(m : ℝ) := by simp\n    have e3 : ((1 : ℂ) / 2).re = 1 / 2 := by simp\n    rw [e3, e2] at e\n    have hm0 : (0 : ℝ) ≤ (m : ℝ) := Nat.cast_nonneg m\n    linarith\n  have hΓcont : ContinuousAt Complex.Gammaℝ 1 :=\n    (differentiableAt_Gammaℝ_of_not_neg_nat hΓ1).continuousAt","new_string":"  have hΓ1 : ∀ m : ℕ, (1 : ℂ) / 2 ≠ -(m : ℂ) := by\n    intro m hm\n    have e := congrArg Complex.re hm\n    have e2 : (-(m : ℂ)).re = -(m : ℝ) := by simp\n    have e3 : ((1 : ℂ) / 2).re = 1 / 2 := by norm_num\n    rw [e3, e2] at e\n    have hm0 : (0 : ℝ) ≤ (m : ℝ) := Nat.cast_nonneg m\n    linarith\n  have hΓcont : ContinuousAt Complex.Gammaℝ 1 :=\n    (differentiableAt_Gammaℝ_of_not_neg_nat hΓ1).continuousAt"},"extra_content":null}]