[{"type":"thinking","thinking":"Progress — the re bullet now reduces to `2 - (↑t * (1/2)).im + s.re * (1/2) = 2 + s.re * (1/2)` — needs (↑t·(1/2)).im = 0. My simp only set didn't include the mul_im evaluation. The goal shows `(↑t * (1/2)).im` — wait, that's (↑t * (1/2)).im — the (↑t/2 · I) got... hmm interesting: the form is `↑t * (1/2)` — so the (t/2)·I was rewritten... the goal `2 - (↑t * (1/2)).im + s.re * (1/2) = 2 + s.re * (1/2)` — needs (↑t * (1/2)).im = 0: ((↑t)·(1/2:ℂ)).im = t·(1/2).im = 0: `(1/2 : ℂ).im = 0` — via OfNat/division simp: (1/2:ℂ).im — simp only [Complex.div_im]? or since (1/2:ℂ) is real-valued... Let me add `Complex.mul_im` and the OfNat-im evaluation. Actually — simpler: the whole issue is my simp-only set is fighting the (↑t/2)·I form. Let me just use a bigger simp set: `simp [Complex.add_re, Complex.mul_re, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im]` and then `ring` if anything remains. Plain simp should fully evaluate all the .re/.im of literal-ish terms. Earlier `by simp` alone didn't fully close (left the arithmetic) — so `simp [...]; ring` hmm but simp might close. The pattern that works per the error: the goal after my simp only was partially evaluated. Let me use: `simp only [Complex.add_re, Complex.mul_re, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im]` + handle the `(↑t * (1/2)).im`: (↑t * (1/2:ℂ)).im — via mul_im: (↑t).re·(1/2).im + (↑t).im·(1/2).re = t·0 + 0·(1/2) = 0: needs (1/2:ℂ).im = 0 and .re = 1/2: those are OfNat-division — simp only doesn't evaluate (1/2:ℂ).im without a lemma. There's `Complex.div_im`? (a/b).im = (a.im*b.re − a.re*b.im)/(...) — for b = 2: (1·2 − 0)/(4) hmm. Ugh. The clean fix: avoid the (t/2)·I form — write w₀ := 1 + ((t/2 : ℝ) : ℂ)·I: the cast-first form: ((t/2:ℝ):ℂ)·I: mul_im: ((t/2:ℝ):ℂ).re·1 + ((t/2:ℝ):ℂ).im·0·... = (t/2)·1 − 0·0... (a·I).im = a.re: via mul_im: (a.re·I.re... I.re = 0, I.im = 1: (a·I).im = a.re·1 + a.im·0 = a.re = t/2 ✓. So w₀ := 1 + ((t/2 : ℝ):ℂ) * Complex.I. Then (w₀).re = 1 + (t/2), .im = t/2 directly via mul_re/mul_im + ofReal_re/ofReal_im + I_re/I_im ✓ clean. Let me redefine w₀ that way and check hw₀eq (w₀ = (2 + t·I)/2 — the bridge: 1 + ((t/2:ℝ):ℂ)·I = (2 + t·I)/2: by Complex.ext: re: 1 + t/2 = (2+t·I...).re/... hmm — ext: re: (1 + ↑((t/2))·I).re = 1 + t/2; ((2 + t·I)/2).re = (2 + t·I).re/2 = 2/2 = 1?? — wait ((2+t·I)/2).re = (2+t·I).re/2 = 2/2 = 1? No: (2+t·I).re = 2, so /2 gives 1 ✓ and im: ((2+tI)/2).im = t/2 ✓. So ext: re: 1 + t/2 vs 1 — NO WAIT: (1 + ↑(t/2)·I).re = 1 + 0 = 1 (the ↑(t/2)·I has re 0). And ((2+tI)/2).re = 1. So re: 1 = 1 ✓; im: t/2 = t/2 ✓. So hw₀eq : (2 + t*I)/2 = 1 + ↑(t/2)*I: by rw [Complex.ext_iff]; constructor <;> simp [Complex.div_re, ...]... hmm the (2+tI)/2's re: via `Complex.div_re`? or since 2 = ((2:ℝ):ℂ): (2 + tI)/2 = (2+tI)·((1/2:ℝ):ℂ)?? Let me just: hw₀eq : (2 + t * Complex.I) / 2 = 1 + ((t / 2 : ℝ) : ℂ) * Complex.I := by rw [Complex.ext_iff]; constructor <;> [simp; simp] — the re: ((2+tI)/2).re = 2/2?? — (a/2).re with 2 : ℂ: div_re: (a.re·2 + a.im·0)/(4+0) = a.re/2 = 1 ✓ simp should evaluate. im: (a/2).im = a.im/2 = t/2 ✓. And the RHS: (1 + ↑(t/2)·I).re = 1 ✓ .im = t/2 ✓. So `constructor <;> simp` might work with the right lemmas. Risky — let me use the safest known-good: `by ring` on the complex identity directly: (2 + t·I)/2 = 1 + ↑(t/2)·I: is this a ring identity? LHS: (2 + t·I)/2; RHS: 1 + ↑(t/2)·I — the cast ↑(t/2) = ((t/2:ℝ):ℂ) — as a ℂ term: (t/2)·I where t/2 is real-division — the identity (2+t·I)/2 = 1 + (t/2)·I holds in ℂ ✓ and ring normalizes casts? ↑(t/2) is a coe-application — ring treats ↑(t/2) as an atom — the identity (2 + t·I)/2 = 1 + ↑(t/2)·I with atoms t·I... hmm — the LHS has t (real) coerced: (2 + ↑t·I)/2 — atoms: ↑t, I, ↑(t/2): the identity (2 + ↑t·I)/2 = 1 + ↑(t/2)·I requires ↑(t/2) = ↑t/2 — a CAST identity, not ring. So ring fails. Use: rw [show ((t/2 : ℝ):ℂ) = (t:ℂ)/2 by push_cast; ring] then ring: (2 + t·I)/2 = 1 + (t/2)·I ✓ ring over ℂ with t as atom ✓. So:\nhw₀eq : (2 + t * Complex.I) / 2 = 1 + ((t / 2 : ℝ) : ℂ) * Complex.I := by\n  rw [show ((t / 2 : ℝ) : ℂ) = (t : ℂ) / 2 by push_cast; ring]\n  ring\n\nThen hw₀im : w₀.im = t/2 with w₀ := 1 + ↑(t/2)·I: by rw [hw₀]; simp [Complex.add_im, Complex.mul_im, Complex.ofReal_re] — (↑(t/2)·I).im = (↑(t/2)).re = t/2: mul_im: (a·I).im = a.re·I.im + a.im·I.re = a.re·1 + a.im·0 = a.re: ofReal_re: t/2 ✓. So simp [Complex.add_im, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im] ✓.\n\nLet me rewrite the w₀-related parts with this cast-first form:\n- log_diff_le's statement: `Complex.log (1 + (t / 2) * Complex.I)` — change to `Complex.log (1 + ((t / 2 : ℝ) : ℂ) * Complex.I)`? The statement's w₀ form propagates to digamma_diff_le's hw₀eq. Let me keep the STATEMENTS with `(t / 2) * Complex.I` (the un-cast form — Lean elaborates (t/2) : ℝ then cast: ((t/2:ℝ):ℂ)·I automatically? `(t / 2) * Complex.I` with t : ℝ: t/2 : ℝ, then ·I needs ℂ: coercion ↑ — so the elaborated form IS ((t/2:ℝ):ℂ)·I ✓ automatically! Good — so w₀ := 1 + (t/2)·I already elaborates to the cast-first form. The earlier issue was my simp-only sets not handling the (↑t/2)·I... wait — the error showed `(↑t * (1/2)).im` — so the elaboration produced ↑t * (1/2:ℂ)?? That means (t/2) was elaborated as ↑t·(1/2:ℂ) — division INSIDE ℂ (the /2 happened after coercion: ↑t / 2 in ℂ = ↑t · (1/2:ℂ)). Hmm — so `(t/2 : ℝ) * I` elaborated as (↑t/2 : ℂ)·I with the division in ℂ. To force the real-division-first: write `((t / 2 : ℝ) : ℂ) * Complex.I` explicitly. Let me check the current w₀ def: `set w₀ : ℂ := 1 + (t / 2) * Complex.I with hw₀` — the (t/2) elaborated at ℂ (since ·I forces ℂ): (↑t)/(2:ℂ). That's the issue. Change all w₀ forms to the explicit cast: `1 + ((t / 2 : ℝ) : ℂ) * Complex.I`.\n\nAnd the im/re computations with the explicit cast form are clean.\n\nLet me find/replace the w₀ forms:\n- log_diff_le statement: `Complex.log (1 + (t / 2) * Complex.I)` → `Complex.log (1 + ((t / 2 : ℝ) : ℂ) * Complex.I)`.\n- `set w₀ : ℂ := 1 + (t / 2) * Complex.I with hw₀` → `set w₀ : ℂ := 1 + ((t / 2 : ℝ) : ℂ) * Complex.I with hw₀`.\n- hzim's simp: (z y).im where z y = w₀ + ↑y·↑c: (w₀).im + (↑y·↑c).im = t/2 + y·0: simp [Complex.add_im, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im, Complex.I_im, Complex.I_re] ✓.\n- htarget bullets: with the explicit-cast w₀: (w₀ + ↑c).re = (1 + ↑(t/2)·I + ↑(s.re/2+1)).re = 1 + 0 + (s.re/2+1): simp only [Complex.add_re, Complex.mul_re, Complex.ofReal_re, Complex.I_re, Complex.I_im, Complex.ofReal_im, mul_zero, mul_one, sub_zero] hmm (↑(t/2)·I).re = (↑(t/2)).re·I.re − (↑(t/2)).im·I.im = (t/2)·0 − 0·1 = 0 ✓. So the re bullet: simp only [...] gives 1 + (0 − 0) + (s.re/2+1) = s.re/2+2 — then ring.\n  \n  And h1 : (s/2 + 2).re = s.re/2 + 2 := by simp — did that work? The error at 564 showed the goal reduced to `2 - (↑t * (1/2)).im + s.re * (1/2) = 2 + s.re * (1/2)` — meaning h1's form `s.re/2 + 2` got matched, but the w₀-side evaluation left (↑t·(1/2)).im — wait no: the goal shows `2 - (↑t * (1/2)).im + s.re * (1/2)` — this came from the RHS (s/2+2).re evaluation: (s/2+2).re with s/2 = ↑s·(1/2:ℂ)?: (s/2).re = (s.re·(1/2) + s.im·0)/... — the form `s.re * (1/2)` ✓ and `2 - (↑t*(1/2)).im` — hmm confusing. Whatever — with the explicit-cast w₀ and proper simp sets it should normalize.\n\nLet me also double check `h1 : (s / 2 + 2).re = s.re / 2 + 2 := by simp` — the (s/2).re: (s/2) = s·(1/2:ℂ)? or s/(2:ℂ): div_re: (s/2).re = (s.re·2 + s.im·0)/(4) hmm = s.re·2/4 = s.re/2 — simp should do it via Complex.div_re + norm_num... The error display suggests simp DID handle it (the `s.re * (1/2)` form). OK.\n\nLet me make the w₀ form fixes.","signature":""},{"type":"text","text":"The `(t/2)·I` elaborated with the division inside ℂ. Forcing the real-division-first cast form."},{"type":"tool_use","id":"Bash_233","name":"Bash","input":{"command":"cd D:/Python/Quantum/research/lean/NguyenRH1/NguyenRH1 && grep -n \"(t / 2) \\* Complex.I\" LiTightR.lean","description":"Find all (t/2)·I occurrences"},"extra_content":null}]