[{"type":"thinking","thinking":"Several mismatches:\n\n1. hcap 5c: the capstone gives ≤ 100000·(log|t|)² but hcap's stated bound is 200000 — need `.trans` with 100000 ≤ 200000: `exact h.trans (by ...)`: (mul_le_mul_of_nonneg_left (by norm_num : (100000:ℝ) ≤ 200000) ...) hmm: 100000·X ≤ 200000·X with X = (log|t|)² ≥ 0: mul_le_mul_of_nonneg_right (by norm_num : (100000:ℝ) ≤ 200000) (pow_two_nonneg _) — wait the forms: 100000 * (Real.log |s.im|)^2 ≤ 200000 * (Real.log |s.im|)^2: mul_le_mul_of_nonneg_right (by norm_num) (pow_two_nonneg _) ✓. So: `exact (logDeriv_riemannZeta_sub_nearZeros_le (t := s.im) ht10 rfl hhalf (by linarith [hreg1] : s.re ≤ 7 / 2) hζs).trans (mul_le_mul_of_nonneg_right (by norm_num : (100000:ℝ) ≤ 200000) (pow_two_nonneg _))`.\n\n2. h1's expected type: `32 * C0hat(...) * (3 * A0count * Real.log(...))` — the h1 in the final calc step of region (ii): its statement was written with (3 * A0count * Real.log(...)) but h2.trans h4 produces (32·Ĉ)·((3·A₀)·(2L)) form... The h1's statement: `32 * C0hat (...) * (3 * A0count * Real.log (3 * max ...)) ≤ 192·A₀·Ĉ·L²` — and h2.trans h4 : `32 * C0hat(...) * (3 * A0count) * Real.log(...) ≤ 192·...` — associativity AGAIN: (32·Ĉ·(3·A₀))·log vs (32·Ĉ)·((3·A₀)·log): h1's statement should be in the produced form: `32 * C0hat (4 * ↑k + 9 / 2) * (3 * A0count) * Real.log (3 * max |s.im - 1| |s.im + 1| + 5) ≤ ...` — i.e. left-assoc (32·Ĉ·(3·A₀))·log. Fix h1's statement: change `(3 * A0count * Real.log (3 * max |s.im - 1| |s.im + 1| + 5))` → `(3 * A0count) * Real.log (3 * max |s.im - 1| |s.im + 1| + 5)`. And hnear's stated bound: `32 * C0hat (...) * (3 * A0count * Real.log (...))` — the calc's mul_le_mul_of_nonneg_left nearZeros_count_le produces (32Ĉ)·(3A₀·log) — as ((32Ĉ))·((3A₀)·log)?? mul_le_mul_of_nonneg_left (h : b ≤ a) (hc : 0 ≤ c) : c * b ≤ c * a: with c := 32Ĉ: (32Ĉ)·(3A₀·log) — the hnear's final stated form: `32 * C0hat (...) * (3 * A0count * Real.log (...))` = (32Ĉ)·((3A₀)·log) ✓ that MATCHED (the calc closed ✓ since no error reported for hnear). Then h1 converts hnear's form to 192...: h1's statement must be EXACTLY hnear's RHS: `32 * C0hat (...) * (3 * A0count * Real.log (3 * max |s.im - 1| |s.im + 1| + 5))` — which is what I wrote... but the error says h2.trans h4 has type `32 * C0hat(...) * (3 * A0count) * Real.log(...)` (left-assoc: ((32Ĉ)·(3A₀))·log) vs h1's expected `32 * C0hat(...) * (3 * A0count * Real.log(...))` ((32Ĉ)·((3A₀)·log)) — DIFFERENT associativity between h2's output and h1's statement. h2 was built via mul_le_mul_of_nonneg_left hlogmax (hc : 0 ≤ (32Ĉ)·(3A₀)) : ((32Ĉ)·(3A₀))·log(...) ≤ ((32Ĉ)·(3A₀))·(2L) — LEFT-assoc ((32Ĉ)·(3A₀))·X. So h2's LHS form: `32 * C0hat (...) * (3 * A0count) * Real.log(...)` (left-assoc). But hnear's RHS (from its calc): the mul_le_mul_of_nonneg_left nearZeros_count_le (by positivity : 0 ≤ 32·Ĉ): (32Ĉ)·(3A₀·log) — RIGHT-nested (32Ĉ)·((3A₀)·log). So hnear's type is `32 * C0hat (...) * (3 * A0count * Real.log (...))` and h2's LHS is `32 * C0hat (...) * (3 * A0count) * Real.log (...)` — DIFFERENT. So h1's statement must start from h2's form (left-assoc): `32 * C0hat (...) * (3 * A0count) * Real.log (...) ≤ ...` — but then the calc step `_ ≤ ...300...` uses h1 : ... and the PREVIOUS calc step gives hnear's right-nested form — mismatch at the calc's `_ ≤ 200000·L|t|² + [right-nested]` — the add_le_add hcap hnear produces `200000·|t|² + (32Ĉ)·((3A₀)·log)` — and the next step's LHS stated as `200000·R² + 300...` — the transition: `_ ≤ dirichletC2 + 200000·L² + 300·A₀ĈL² := by ...nlinarith [hlogt, h1, h5, ...]` — h1 must connect hnear's nested form to 192·A₀ĈL². Simplest: make h1's LHS match h2's form (left-assoc) AND bridge hnear's nested form: change h1's statement to the left-assoc form and in the calc, the add_le_add step... ugh — actually the cleanest: change hnear's FINAL form to the left-assoc too: hnear's last calc step used mul_le_mul_of_nonneg_left nearZeros_count_le (by positivity : (0:ℝ) ≤ 32 * C0hat (...)): (32Ĉ)·(3A₀·log) — nested. If instead I apply it as `mul_le_mul_of_nonneg_left nearZeros_count_le (by positivity : (0:ℝ) ≤ 32 * C0hat (...) * (3 * A0count))` with h'logmax?? no — the count bound is 3A₀·log ≤ 3A₀·2L: TWO steps: first mul_le_mul_of_nonneg_left nearZeros_count_le (0 ≤ 32Ĉ): (32Ĉ)·(3A₀·log) nested; then hlogmax's: mul_le_mul_of_nonneg_left hlogmax (0 ≤ 32Ĉ·3A₀): ((32Ĉ)·(3A₀))·log ≤ ((32Ĉ)·(3A₀))·2L. The two forms differ in nesting of the FIRST two factors. To keep ONE consistent form, make h1's statement match h2's left-assoc form AND adjust the calc's expectation: the calc step `_ ≤ 200000·(log|t|)² + (32Ĉ·(3A₀·log))` (add_le_add hcap hnear) — then the final nlinarith combines with h1 : (32Ĉ)·(3A₀)·log ≤ 192A₀ĈL² — DIFFERENT terms for the near-sum: nlinarith treats them as atoms — the two nestings are DIFFERENT atoms!! linarith/nlinarith won't match them. So: make hnear's final form LEFT-assoc from the start: in hnear's last calc step, instead of `mul_le_mul_of_nonneg_left nearZeros_count_le (by positivity)`, first reshape: `... = (32 * C0hat (...)) * (3 * A0count) * Real.log (...) := by ring` — no wait: (32Ĉ)·(3A₀·log) = ((32Ĉ)·(3A₀))·log by ring (mul_assoc) — add a calc step `_ = 32 * C0hat (...) * (3 * A0count) * Real.log (...) := by ring`? Hmm — mul_assoc: a·(b·c) = (a·b)·c ✓ ring closes. But then hnear's TYPE (the lemma statement) must also be left-assoc: change hnear's statement to `... ≤ 32 * C0hat (...) * (3 * A0count) * Real.log (...)` and internally: after the mul_le_mul step (nested), add `... = (left-assoc form) := by ring` — but ring proves EQUALITY — the calc chain: `... ≤ (32Ĉ)·(3A₀·log) := mul_le_mul... _ = (32Ĉ)·(3A₀)·log := by ring` ✓ — the calc's final form is left-assoc ✓ matching the lemma statement.\n   Hmm wait — the calc currently ends `... := mul_le_mul_of_nonneg_left nearZeros_count_le (by positivity)`. Append a step `... _ = 32 * C0hat (...) * (3 * A0count) * Real.log (...) := by ring`. And change hnear's statement to left-assoc. And h1's statement: starts from the left-assoc: `32 * C0hat (...) * (3 * A0count) * Real.log (3 * max ...) ≤ 192·A₀·Ĉ·L²` — h2's form ✓.\n\n   Hmm wait — actually h1's statement I wrote: `32 * C0hat (4 * (k:ℝ) + 9 / 2) * (3 * A0count * Real.log (3 * max |s.im - 1| |s.im + 1| + 5))` — nested. Change to left-assoc `(3 * A0count) * Real.log (...)`... wait NO — the error showed h1's EXPECTED type as `32 * C0hat (...) * (3 * A0count * Real.log (...))` (from MY statement) and h2.trans h4's type as `32 * C0hat (...) * (3 * A0count) * Real.log (...)` (left-assoc) — so h2.trans h4 produces LEFT-assoc and my h1 statement was NESTED. Unify everything to LEFT-assoc.\n\n3. Line 3582 (hsR's hm2): `↑(-R) / 2 = -↑m` vs `-↑R / 2 = -↑m` — the hm : s/2 = −↑m with s = ↑(-R): after rw [hsR] at hm: hm : (↑(-R))/2 = −↑m — my h2 : (↑(-R))/2 = ↑(-R/2) — but the elaborator's hm is `↑(-R) / 2` (cast of −R divided by 2) vs my pattern `-(↑R) / 2` — hmm the forms: `(-(R:ℝ) : ℂ) / 2` vs `-((R:ℝ) : ℂ) / 2`?? The hsR : s = (-R : ℝ) — the coercion (↑(-R) : ℂ). hm : (↑(-R))/2 = −↑m. My h2 : (↑(-R) : ℂ)/2 = ((-R/2 : ℝ) : ℂ): push_cast; ring — should prove: (↑(−R))/2 vs ↑(−R/2): push_cast normalizes ↑(−R/2) to (↑(−R))/2?? The error shows h2's produced form mismatched the target `↑(-R) / 2 = -↑m` vs `-↑R / 2 = -↑m` — so after rw [h2] at hm the hm became `↑(-R/2) = -↑m`?? no — the error is in hm2's proof: `rw [h2] at h1` where h1 : (↑(-R))/2 = −↑m — h2's LHS pattern (↑(-R))/2 — is that what h2 says? h2 : (-(R : ℝ) : ℂ) / 2 = ((-R / 2 : ℝ) : ℂ) — the LHS `(-(R:ℝ) : ℂ)/2` vs hm's `↑(-R)/2`: `(-(R:ℝ) : ℂ)` vs `↑(-R)` — SAME? (−R : ℝ) coerced to ℂ = ↑(-R) and (-(R:ℝ) : ℂ) = -↑R — these are DIFFERENT terms (cast-of-neg vs neg-of-cast)! The error message: `↑(-R) / 2 = -↑m` (hm's actual form) vs my h2's `-↑R / 2` — so hm has ↑(-R) (cast first) and my h2 has -↑R. Fix: push_cast at hm FIRST to normalize: or state h2's LHS as (↑(-R) : ℂ)/2. Simplest: in hm2: \n   ```\n   have hm2 : (m : ℝ) = R / 2 := by\n     have h1 : (-(R : ℝ) : ℂ) / 2 = -(m : ℂ) := hm\n     have h2 : (-R / 2 : ℝ) = -(m : ℝ) := by\n       have h3 : ((-R / 2 : ℝ) : ℂ) = (-(R : ℝ) : ℂ) / 2 := by push_cast; ring\n       rw [h3] at h1?? — hmm direction: I want to convert h1's LHS to a cast form for exact_mod_cast:\n       have h4 : ((-(m : ℝ) : ℝ) : ℂ) = -(m : ℂ) := by push_cast; ring\n       rw [← h3, ← h4] at h1  -- h1 : ((-R/2 : ℝ) : ℂ) = ((-(m:ℝ)) : ℂ)\n       exact_mod_cast h1\n     linarith [h2]\n   ```\n   Wait — `(-(m : ℝ) : ℝ)` is just -(m:ℝ) : ℝ; ((-(m:ℝ)) : ℂ) = -(↑m) by push_cast ✓. And ((-R/2 : ℝ):ℂ) = (↑(-R))/2 by push_cast — the push_cast normal form of ↑(−R/2): −(↑R)/2 or ↑(−R)/2? push_cast pushes casts OUT: ↑(−R/2) = (↑(−R))/2 = −(↑R)/2·... cast_neg: ↑(−x) = −↑x: ↑(−R/2) = −(↑(R/2)) = −((↑R)/2) hmm — depends on simp order. The robust way: use congrArg Complex.re on hm: hm : s/2 = −↑m: with s = ↑(−R): (s/2).re = (−↑m).re: (s/2).re = s.re/2 (hre-like: simp) = (↑(-R)).re/2 = −R/2: (−↑m).re = −↑m: so −R/2 = −↑m in ℝ ✓ then m = R/2:\n   ```\n   have hm2 : (m : ℝ) = R / 2 := by\n     have h1 := congrArg Complex.re hm\n     have h2 : ((-(R : ℝ) : ℂ) / 2).re = -(m : ℝ) := by\n       rw [Complex.neg_re, Complex.natCast_re] at h1 ⊢\n       ... \n   ```\n   getting messy. Cleanest robust: avoid congrArg: \n   ```\n   have hm2 : (m : ℝ) = R / 2 := by\n     have h1 : (↑(-R) : ℂ) / 2 = -(m : ℂ) := hm\n     have h2 : (↑(-R) : ℂ) / 2 = ((-(R / 2) : ℝ) : ℂ) := by\n       rw [show ((-(R / 2) : ℝ) : ℂ) = ((-(R / 2) : ℝ) : ℂ) from rfl]\n       push_cast\n       ring\n     rw [h2] at h1\n     have h3 : -(R / 2) = -(m : ℝ) := by exact_mod_cast h1\n     linarith [h3]\n   ```\n   h2 : (↑(-R))/2 = ↑(-(R/2)): push_cast normalizes ↑(-(R/2)) → ??? push_cast with the `ring` fallback — the goal (↑(-R))/2 = ↑(-(R/2)): push_cast should rewrite ↑(-(R/2)) to -(↑R)/2 and (↑(-R))/2 to -(↑R)/2 hmm — actually `Complex.ofReal_div_ofNat`?? I'm not sure push_cast handles ↑(-R)/2 vs ↑(-(R/2)) — they're EQUAL by ofReal_div + ofReal_neg: ↑(-(R/2)) = -↑(R/2) = -(↑R/2); (↑(-R))/2 = (↑(-R))/2 = (-↑R)/2 = -(↑R/2) ✓ same normal form. push_cast normalizes casts: ofReal_neg, ofReal_div (if exists)... `ofReal_div : (↑(x / y) : ℂ) = ↑x / ↑y` ✓ simp. So push_cast turns ↑(-(R/2)) into -↑R/2 hmm: ↑(−(R/2)): ofReal_neg → −↑(R/2); ↑(R/2): ofReal_div → ↑R/↑2 = ↑R/2: so −(↑R/2). And (↑(−R))/2: −↑R/2 ✓ same ✓ then ring (or rfl-close).\n\n   Alternatively MUCH simpler: skip hsR's complex detour — hm : s/2 = −↑m and hsR : s = ↑(-R): rw [hsR] at hm gives hm : (↑(-R))/2 = −↑m: apply congrArg Complex.re: ((↑(-R))/2).re = (−↑m).re: LHS: ((↑(-R))/2).re: simp: ((↑(-R)).re)/2 = (−R)/2: ((−R:ℂ)).re = −R (Complex.ofReal_re) — via simp [Complex.ofReal_re] hmm: (↑(-R) : ℂ) = ((−R : ℝ) : ℂ): ofReal(−R): re = −R ✓ ((−R:ℂ)/2).re = −R/2: simp should do: `have h1 := congrArg Complex.re hm; simp [Complex.ofReal_re] at h1` — after simp, h1 : −R/2 = −↑m (in ℝ with casts) — then exact_mod_cast/linarith. The RHS (−↑m).re = −↑m (neg_re + natCast_re) ✓ simp handles. Let me do that:\n   ```\n   have hm2 : (m : ℝ) = R / 2 := by\n     have h1 := congrArg Complex.re hm\n     simp only [Complex.neg_re, Complex.natCast_re] at h1\n     have h2 : ((-(R : ℝ) : ℂ) / 2).re = -R / 2 := by simp\n     rw [hsR] at hm?? \n   ```\n   OK let me look at the actual current code around 3582 to write the precise fix. The current:\n   ```\n   rw [hsR] at hm\n   have hm2 : (m : ℝ) = R / 2 := by\n     have h2 : (-(R : ℝ) : ℂ) / 2 = ((-R / 2 : ℝ) : ℂ) := by push_cast; ring\n     have h3 : (-R / 2 : ℝ) = -(m : ℝ) := by\n       have h1 : (-(R : ℝ) : ℂ) / 2 = -(m : ℂ) := hm\n       rw [h2] at h1\n       have h4 : ((-R / 2 : ℝ) : ℂ) = -(m : ℂ) := h1\n       have h5 : (-R / 2 : ℝ) = -(m : ℝ) := by exact_mod_cast h4\n       exact h5\n     linarith [h3]\n   ```\n   Error: after rw [h2] at h1: h1 : ((-R/2 : ℝ) : ℂ) = −(m:ℂ)?? — the error said \"has type ↑(-R) / 2 = -↑m but expected -↑R / 2 = -↑m\" — so hm's actual form after rw [hsR] is `↑(-R) / 2 = -↑m` and my h1's stated form `(-(R : ℝ) : ℂ) / 2 = -(m : ℂ)` MISMATCHES (the have h1 : ... := hm ascription failed). Fix: change h1's form to `((-R : ℝ) : ℂ) / 2 = -(m : ℂ)` — matching hm's `↑(-R) / 2 = -↑m`: `(-R : ℝ)` vs `-(R : ℝ)` — hsR : s = (-R : ℝ) — the term I wrote in hsR's statement: `s = (-R : ℝ)` — parsed as `s = ↑(-R)`?? `(-R : ℝ)` is −R : ℝ then coerced — `↑(-R)`. My h1 wrote `(-(R : ℝ) : ℂ)` — neg-of-cast — mismatch. Unify: change hsR's statement to `s = ((-R : ℝ) : ℂ)` and hm2's h1 to the same form, with h2 : (((-R : ℝ):ℂ))/2 = ((-R/2 : ℝ):ℂ) via push_cast; ring — hmm push_cast on `↑(-R)/2 = ↑(-R/2)`: RHS ↑(-R/2) → -↑R/2 (ofReal_neg + ofReal_div + ofReal... ); LHS ↑(-R)/2 → (-↑R)/2 → -(↑R/2)?? ofReal_neg then div: (-↑R)/2 = -(↑R/2)? ofReal_div on LHS: ↑(-R)/↑2 hmm — actually ↑(-R)/2 : the /2 is complex division: ((↑(-R)) : ℂ)/((2:ℕ):ℂ)?? the 2 is OfNat — ↑(-R)/2: HDiv ↑(-R) 2: push_cast: no cast on the whole (it's already a complex division of a cast): hmm — push_cast pushes the INNER cast: ↑(-R) → -↑R (ofReal_neg): LHS becomes (-↑R)/2 = -(↑R/2)? that's ring-level. RHS ↑(-(R/2)): ofReal_neg → -↑(R/2); ofReal_div → -(↑R/↑2) hmm ↑(R/2) → ↑R/2 (ofReal_div — is ofReal_div a simp/push_cast lemma? `ofReal_div : (↑(x / y)) = ↑x / ↑y` — yes simp). So both sides -(↑R/2)... LHS: (-↑R)/2 — ring normalizes to -(↑R/2)? then `ring` closes ✓. So:\n   ```\n   have hm2 : (m : ℝ) = R / 2 := by\n     have h1 : ((-R : ℝ) : ℂ) / 2 = -(m : ℂ) := hm\n     have h2 : ((-R : ℝ) : ℂ) / 2 = ((-R / 2 : ℝ) : ℂ) := by push_cast; ring\n     rw [h2] at h1\n     have h4 : ((-R / 2 : ℝ) : ℂ) = -(m : ℂ) := h1\n     have h5 : (-R / 2 : ℝ) = -(m : ℝ) := by exact_mod_cast h4\n     linarith [h5]\n   ```\n   And hsR's statement: `s = ((-R : ℝ) : ℂ)` — but hsR is PROVED via Complex.ext_iff.mpr with `rw [h4, him]; simp` — h4 : s.re = -R: ext: s.re = ((-R:ℂ)).re and s.im = ((-R:ℂ)).im: simp proves ((−R:ℂ)).re = −R and im = 0 ✓ fine.\n   Hmm — and hm : s/2 = −↑m: rw [hsR] at hm: hm : ((−R:ℂ))/2 = −↑m ✓ matches h1 ✓.\n\n4. hG's `exact hpoles m h1` — hpoles takes (m : ℕ) and the proof s/2 ≠ −(m:ℂ) as an IMPLICIT?? hpoles : ∀ m : ℕ, s / 2 ≠ -(m : ℂ) — so `hpoles m h1 : False` ✓ then exact that — but the error shows \"this term has type s / 2 = -↑m / Note: Expected a function because this term is being applied\" — the `exact hpoles m h1` — hmm hpoles m h1 : False ✓ exact False-elim... `exact hpoles m h1` — hpoles m h1 is a proof of False: exact (hpoles m h1) should work via False.elim... Actually `exact h` with h : False works (False.elim is defeq-inferred). The error mentions \"Expected a function\" — because hpoles's TYPE: ∀ m : ℕ, s/2 ≠ −↑m — after h1 : s/2 = −↑m: hpoles m h1 : False ✓. Hmm — the error at 3591: `rw [show R / 2 - R / 2 = (0 : ℝ) by ring, abs_zero] at hgap'` — the hgap' from hgapInt (m : ℤ) with hcast + hm2: rw [hcast, ← hm2] at hgap': hgap' : 1/4 ≤ |R/2 − (m:ℝ)|: rw [← hm2] : hm2 : (m:ℝ) = R/2: ← hm2 rewrites R/2 → (m:ℝ)?? — rw [← hm2] at hgap' rewrites occurrences of hm2's RHS (R/2) by its LHS ((m:ℝ)) — hgap' has |R/2 − ↑m| — rw [← hm2] turns R/2 into ↑m: |↑m − ↑m|?? — NO: I want to turn ↑m into R/2: rw [hm2] (forward): ↑m → R/2: |R/2 − R/2| ✓. My code: `rw [hcast, ← hm2] at hgap'` — WRONG direction: should be `rw [hcast, hm2] at hgap'` — hcast : ((m:ℤ):ℝ) = (m:ℝ): hgap' : 1/4 ≤ |R/2 − ((m:ℤ):ℝ)|: rw [hcast] : |R/2 − (m:ℝ)|: rw [hm2] : |R/2 − R/2| ✓ then the show. Fix direction.\n\nLet me also double check hG's hpoles error — 3591's error is the rw direction one. The \"Expected a function\" note after — cascade. Fix the rw direction and the h1 forms.","signature":""},{"type":"text","text":"Multiple form-mismatches. Applying fixes: 100000→200000 trans, left-assoc unification for the near-sum forms, hm2's cast forms, and the rw direction at hgap'."},{"type":"tool_use","id":"Edit_210","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean","old_string":"          rw [hsR] at hm\n          have hm2 : (m : ℝ) = R / 2 := by\n            have h2 : (-(R : ℝ) : ℂ) / 2 = ((-R / 2 : ℝ) : ℂ) := by push_cast; ring\n            have h3 : (-R / 2 : ℝ) = -(m : ℝ) := by\n              have h1 : (-(R : ℝ) : ℂ) / 2 = -(m : ℂ) := hm\n              rw [h2] at h1\n              have h4 : ((-R / 2 : ℝ) : ℂ) = -(m : ℂ) := h1\n              have h5 : (-R / 2 : ℝ) = -(m : ℝ) := by exact_mod_cast h4\n              exact h5\n            linarith [h3]\n          have hgap' := hgapInt (m : ℤ)\n          have hcast : ((m : ℤ) : ℝ) = (m : ℝ) := by push_cast; ring\n          rw [hcast, ← hm2] at hgap'\n          rw [show R / 2 - R / 2 = (0 : ℝ) by ring, abs_zero] at hgap'\n          norm_num at hgap'","new_string":"          rw [hsR] at hm\n          have hm2 : (m : ℝ) = R / 2 := by\n            have h1 : ((-R : ℝ) : ℂ) / 2 = -(m : ℂ) := hm\n            have h2 : ((-R : ℝ) : ℂ) / 2 = ((-R / 2 : ℝ) : ℂ) := by push_cast; ring\n            rw [h2] at h1\n            have h4 : ((-R / 2 : ℝ) : ℂ) = -(m : ℂ) := h1\n            have h5 : (-R / 2 : ℝ) = -(m : ℝ) := by exact_mod_cast h4\n            linarith [h5]\n          have hgap' := hgapInt (m : ℤ)\n          have hcast : ((m : ℤ) : ℝ) = (m : ℝ) := by push_cast; ring\n          rw [hcast, hm2] at hgap'\n          rw [show R / 2 - R / 2 = (0 : ℝ) by ring, abs_zero] at hgap'\n          norm_num at hgap'"},"extra_content":null}]