← Session 22 — ii′-1: liCoeff pairing fix + residue machinery
thinking (3750 chars)
Issues:
1. **Line 206**: `have hne2 : ¬(i + 2 = 1) := by omega` — omega fails with "No usable constraints"! On `¬(i + 2 = 1)` with i : ℕ?! That's bizarre. Unless... the `i` in scope is NOT ℕ? In hid's rcases third case: `rcases i with _ | _ | i` — the third case's new i : ℕ ✓. Hmm — "no usable constraints" for ¬(i+2=1)... Wait — maybe the goal at line 206 is NOT ¬(i+2=1) — line 206 is `have hne2 : ¬(i + 2 = 1) := by omega`? — the error is INSIDE the by omega — if the stated type has the if-condition as I wrote, omega should handle ¬(i+2=1) easily. Unless the `i` is shadowed and of a different type... OR the issue: `omega` sees `i + 2 = 1` where `2` and `1` are... hmm no. Let me look at the exact line. Actually — wait, maybe line 206 is the hne1 line? hne1 := Nat.succ_ne_zero i — no omega there. Let me view the file around 200-210.
2. **Line 248**: `tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within` — the exact hc.tendsto — the expected type is STILL the full nhdsWithin form — so the `apply` didn't decompose — the lemma name might be wrong (unknown identifier would error differently...) — "expected type Tendsto ... (𝓝[≠] 0) (𝓝 1)" for the first apply-goal — meaning `apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within` left goal 1 = the FULL target — so the apply matched WRONG (the lemma's conclusion unified with the target directly?? that would close it... no). Actually if the lemma doesn't exist, there'd be "unknown identifier" — the error is that the FIRST subgoal after apply is the full Tendsto (𝓝[≠] 0) (𝓝 1) — so apply found the lemma but its conclusion is the nhdsWithin-form and one of its PREMISES is also nhdsWithin form?? The lemma: `tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within (h : Tendsto f (𝓝 a) (𝓝 b)) (h' : ∀ᶠ x in 𝓝[s] a, f x ∈ t) : Tendsto f (𝓝[s] a) (𝓝[t] b)` — apply should give goals: Tendsto f (𝓝 0) (𝓝 1)?? hmm — wait — the target is Tendsto (1+·) (𝓝[≠] 0) (𝓝[≠] 1) — apply unifies conclusion Tendsto ?f (𝓝[?s] ?a) (𝓝[?t] ?b) — ?s := {0}ᶜ etc. — first goal: Tendsto (1+·) (𝓝 0) (𝓝 1)... the error says the first goal IS Tendsto (fun s => 1+s) (𝓝[≠] 0) (𝓝 1) — that's NOT the lemma's first premise form (𝓝 0 not 𝓝[≠] 0)... So maybe the lemma name resolved to something else with a different premise order — OR apply unified differently. The error display: hc.tendsto has type Tendsto ... (𝓝 0) (𝓝 (1 + 0)), expected Tendsto ... (𝓝[≠] 0) (𝓝 1) — TWO mismatches: 𝓝 0 vs 𝓝[≠] 0 AND 𝓝 (1+0) vs 𝓝 1. So the goal after apply was (𝓝[≠] 0) (𝓝 1)-form — meaning the apply's first premise unified as the nhdsWithin one — so the lemma I applied probably has premises in a different order or my lemma name maps to another statement. Let me just look up the exact lemma name.
3. **Line 256**: add_eq_left.mp — the error shows `add_eq_right.mp h` in the message — wait no: "in the application add_eq_right.mp h" — hmm my edit used add_eq_left.mp — but the error says add_eq_right?? — OH — the error output might be from a STALE compile? No — it recompiled. The error: "add_eq_right.mp h" — hmm, but I wrote add_eq_left.mp hcontra — unless the error message's lemma display is confusingly showing... Actually looking carefully: "in the application add_eq_right.mp h" — there's no h in my current code at 256 (I used hcontra). So this error output IS stale?? No wait — the error is at 256:30 — and says add_eq_right.mp h — my current file has `add_eq_left.mp hcontra` at that area. Confusing — OH WAIT. I see — the compile output IS from the current file; the error text mentions `add_eq_right.mp` — maybe Lean's `add_eq_left` doesn't exist and it fell back?? No — unknown identifier would say so. Hmm — let me look at the current file content around those lines to be sure of what's there.
Let me look at the current file state around the errors.
tool: Read ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiZetaCoeff.lean",
"offset": 196,
"limit": 75
}
result 2026-08-22 05:37:55Z → 2026-08-24 04:25:59Z
{
"type": "text",
"file": {
"filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean",
"content": " linarith [h2]\n rcases mul_eq_zero.mp h5 with h8 | h8\n \u00b7 have h9 : s.re = R := by linarith [h8]\n linarith [h9, hreg2]\n \u00b7 linarith [h8]\n apply Complex.ext_iff.mpr\n rw [h4, him]\n simp\n rw [hsR] at hm\n have hm2 : (m : \u211d) = R / 2 := by\n have h2 : (-(R : \u211d) : \u2102) / 2 = ((-R / 2 : \u211d) : \u2102) := by push_cast; ring\n have h3 : (-R / 2 : \u211d) = -(m : \u211d) := by\n have h1 : (-(R : \u211d) : \u2102) / 2 = -(m : \u2102) := hm\n rw [h2] at h1\n have h4 : ((-R / 2 : \u211d) : \u2102) = -(m : \u2102) := h1\n have h5 : (-R / 2 : \u211d) = -(m : \u211d) := by exact_mod_cast h4\n exact h5\n linarith [h3]\n have hgap' := hgapInt (m : \u2124)\n have hcast : ((m : \u2124) : \u211d) = (m : \u211d) := by push_cast; ring\n rw [hcast, \u2190 hm2] at hgap'\n rw [show R / 2 - R / 2 = (0 : \u211d) by ring, abs_zero] at hgap'\n norm_num at hgap'\n \u00b7 exact hm (by\n have h1 : (s / 2).im \u2260 0 := by\n have h2 : (s / 2).im = s.im / 2 := by simp\n rw [h2]\n exact div_ne_zero him (two_ne_zero : (2 : \u211d) \u2260 0)\n have h3 : (-(m : \u2102)).im = 0 := by\n rw [Complex.neg_im, Complex.natCast_im]\n have h4 : (s / 2).im = (-(m : \u2102)).im := congrArg Complex.im hm\n rw [h3] at h4\n exact h1 h4)\n -- the FE identity\n have hG : Complex.Gamma\u211d s \u2260 0 := by\n rw [ne_eq, Complex.Gamma\u211d_eq_zero_iff]\n rintro \u27e8m, hm\u27e9\n have h1 : s / 2 = -(m : \u2102) := by\n rw [hm]\n ring",
"numLines": 40,
"startLine": 3570,
"totalLines": 3940
}
}
raw content_json (database row)
[{"type":"thinking","thinking":"Issues:\n\n1. **Line 206**: `have hne2 : ¬(i + 2 = 1) := by omega` — omega fails with \"No usable constraints\"! On `¬(i + 2 = 1)` with i : ℕ?! That's bizarre. Unless... the `i` in scope is NOT ℕ? In hid's rcases third case: `rcases i with _ | _ | i` — the third case's new i : ℕ ✓. Hmm — \"no usable constraints\" for ¬(i+2=1)... Wait — maybe the goal at line 206 is NOT ¬(i+2=1) — line 206 is `have hne2 : ¬(i + 2 = 1) := by omega`? — the error is INSIDE the by omega — if the stated type has the if-condition as I wrote, omega should handle ¬(i+2=1) easily. Unless the `i` is shadowed and of a different type... OR the issue: `omega` sees `i + 2 = 1` where `2` and `1` are... hmm no. Let me look at the exact line. Actually — wait, maybe line 206 is the hne1 line? hne1 := Nat.succ_ne_zero i — no omega there. Let me view the file around 200-210.\n\n2. **Line 248**: `tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within` — the exact hc.tendsto — the expected type is STILL the full nhdsWithin form — so the `apply` didn't decompose — the lemma name might be wrong (unknown identifier would error differently...) — \"expected type Tendsto ... (𝓝[≠] 0) (𝓝 1)\" for the first apply-goal — meaning `apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within` left goal 1 = the FULL target — so the apply matched WRONG (the lemma's conclusion unified with the target directly?? that would close it... no). Actually if the lemma doesn't exist, there'd be \"unknown identifier\" — the error is that the FIRST subgoal after apply is the full Tendsto (𝓝[≠] 0) (𝓝 1) — so apply found the lemma but its conclusion is the nhdsWithin-form and one of its PREMISES is also nhdsWithin form?? The lemma: `tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within (h : Tendsto f (𝓝 a) (𝓝 b)) (h' : ∀ᶠ x in 𝓝[s] a, f x ∈ t) : Tendsto f (𝓝[s] a) (𝓝[t] b)` — apply should give goals: Tendsto f (𝓝 0) (𝓝 1)?? hmm — wait — the target is Tendsto (1+·) (𝓝[≠] 0) (𝓝[≠] 1) — apply unifies conclusion Tendsto ?f (𝓝[?s] ?a) (𝓝[?t] ?b) — ?s := {0}ᶜ etc. — first goal: Tendsto (1+·) (𝓝 0) (𝓝 1)... the error says the first goal IS Tendsto (fun s => 1+s) (𝓝[≠] 0) (𝓝 1) — that's NOT the lemma's first premise form (𝓝 0 not 𝓝[≠] 0)... So maybe the lemma name resolved to something else with a different premise order — OR apply unified differently. The error display: hc.tendsto has type Tendsto ... (𝓝 0) (𝓝 (1 + 0)), expected Tendsto ... (𝓝[≠] 0) (𝓝 1) — TWO mismatches: 𝓝 0 vs 𝓝[≠] 0 AND 𝓝 (1+0) vs 𝓝 1. So the goal after apply was (𝓝[≠] 0) (𝓝 1)-form — meaning the apply's first premise unified as the nhdsWithin one — so the lemma I applied probably has premises in a different order or my lemma name maps to another statement. Let me just look up the exact lemma name.\n\n3. **Line 256**: add_eq_left.mp — the error shows `add_eq_right.mp h` in the message — wait no: \"in the application add_eq_right.mp h\" — hmm my edit used add_eq_left.mp — but the error says add_eq_right?? — OH — the error output might be from a STALE compile? No — it recompiled. The error: \"add_eq_right.mp h\" — hmm, but I wrote add_eq_left.mp hcontra — unless the error message's lemma display is confusingly showing... Actually looking carefully: \"in the application add_eq_right.mp h\" — there's no h in my current code at 256 (I used hcontra). So this error output IS stale?? No wait — the error is at 256:30 — and says add_eq_right.mp h — my current file has `add_eq_left.mp hcontra` at that area. Confusing — OH WAIT. I see — the compile output IS from the current file; the error text mentions `add_eq_right.mp` — maybe Lean's `add_eq_left` doesn't exist and it fell back?? No — unknown identifier would say so. Hmm — let me look at the current file content around those lines to be sure of what's there.","signature":""},{"type":"text","text":"Let me look at the current file state around the errors."},{"type":"tool_use","id":"Read_211","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiZetaCoeff.lean","offset":196,"limit":75},"extra_content":null}]