← Session 22 — ii′-1: liCoeff pairing fix + residue machinery
thinking (3370 chars)
The file now ends with the sorry placeholder and no `end NguyenRH1`. I need to complete the ending of iteratedDeriv_DinfC_half properly. The goal at that point (after congr 2): the remaining goal is the series identity:
`((1/2:ℂ))^(−(k:ℤ)−1) + ∑' m, (1/2 + ↑m + 1)^(−(k:ℤ)−1) = ∑' m, (1/2 + ↑m)^(−(k:ℤ)−1)`
and I did `rw [tsum_eq_zero_add hsum]` — which rewrites the RHS: ∑' m, f m = f 0 + ∑' m, f (m+1) — wait — tsum_eq_zero_add : ∑' n, f n = f 0 + ∑' n, f (n + 1) — rw [tsum_eq_zero_add hsum] rewrites ∑' m, (1/2+↑m)^(−(k+1)) → (1/2+↑0)^(−(k+1)) + ∑' m, (1/2+↑(m+1))^(−(k+1)) — so the goal becomes:
(1/2)^(−(k+1)) + Σ_{m}(1/2+↑m+1)^{...} = (1/2+↑0)^{...} + Σ_{m}(1/2+↑(m+1))^{...}
— the first terms: (1/2)^(−(k+1)) vs (1/2+↑0)^(−(k+1)) — ↑(0:ℕ) = 0, 1/2+0 = 1/2 ✓ equal modulo cast — and the tail sums: Σ_m (1/2+↑m+1)^{...} vs Σ_m (1/2+↑(m+1))^{...} — per-term: ↑(m+1) = ↑m+1 ✓ — so close by tsum_congr + per-term push_cast/ring:
```lean
have hsum := summable_zpow_half_add_nat k hk
rw [tsum_eq_zero_add hsum]
have e0 : ((1 / 2 : ℂ) + ((0 : ℕ) : ℂ)) ^ (-(k : ℤ) - 1) = ((1 : ℂ) / 2) ^ (-(k : ℤ) - 1) := by simp
rw [e0]
apply add_left_cancel_iff.mpr -- hmm the forms
apply tsum_congr
intro m
push_cast
ring
```
Hmm — after rw [e0]: goal (1/2)^(−(k+1)) + Σ_{m}(1/2+↑m+1)^(−(k+1)) = (1/2)^(−(k+1)) + Σ_{m}(1/2+↑(m+1))^(−(k+1)) — `apply add_left_cancel_iff.mpr` — add_left_cancel_iff : a + b = a + c ↔ b = c ✓ — leaves Σ (1/2+↑m+1)^... = Σ (1/2+↑(m+1))^... — tsum_congr per-term: (1/2 + ↑m + 1)^(−(k+1)) = (1/2 + ↑(m+1))^(−(k+1)) — congr 1 on the zpow base: 1/2 + ↑m + 1 = 1/2 + ↑(m+1) — push_cast + ring ✓. Let me write:
```lean
have hsum := summable_zpow_half_add_nat k hk
rw [tsum_eq_zero_add hsum]
have e0 : ((1 / 2 : ℂ) + ((0 : ℕ) : ℂ)) ^ (-(k : ℤ) - 1) = ((1 : ℂ) / 2) ^ (-(k : ℤ) - 1) := by
simp
rw [e0]
apply add_left_cancel_iff.mpr
apply tsum_congr
intro m
push_cast
ring
```
Hmm — wait — tsum_eq_zero_add rewrites the GOAL's tsum — the goal's RHS is ∑' m, ((1/2:ℂ)+↑m)^(−(k:ℤ)−1) — after rw: → f 0 + ∑' m, f (m+1) = ((1/2)+↑0)^... + ∑' m, ((1/2)+↑(m+1))^... ✓. But — hmm — does the tsum_eq_zero_add rewrite fire on the goal's RHS tsum? The goal at that point: (1/2)^z + Σ(1/2+↑m+1)^z = Σ(1/2+↑m)^z — rw [tsum_eq_zero_add hsum] — the pattern ∑' n, ?f n — matches BOTH sums (the LHS's Σ(1/2+↑m+1)^z is also a tsum)! — the FIRST match gets rewritten — the traversal order — the goal is Eq LHS RHS — LHS has the Σ(1/2+↑m+1)^z tsum and the (1/2)^z... — hmm the first tsum in the goal is the LHS's Σ(1/2+↑m+1)^z!! I want to rewrite the RHS's. Damn. Use `conv_rhs => rw [tsum_eq_zero_add hsum]`? — conv_rhs navigation ✓:
```lean
conv_rhs => rw [tsum_eq_zero_add hsum]
```
✓ deterministic.
Then the goal: (1/2)^z + Σ_m(1/2+↑m+1)^z = (1/2+↑0)^z + Σ_m(1/2+↑(m+1))^z. Then e0 turns (1/2+↑0)^z → (1/2)^z... wait — after conv_rhs, the RHS's f 0 = (1/2 + ↑(0))^z — hmm — f 0 with f m := (1/2+↑m)^z: f 0 = (1/2 + ↑0)^z ✓ — e0 matches ✓ then the two (1/2)^z cancel via add_left_cancel_iff.mpr → the tail sums equal via tsum_congr per-term push_cast/ring ✓.
Let me also double check `tsum_eq_zero_add`'s exact form: `tsum_eq_zero_add (hf : Summable f) : ∑' n, f n = f 0 + ∑' n, f (n + 1)`? — hmm — it might be `tsum_eq_zero_add : ∑' (n : ℕ), f n = f 0 + ∑' (n : ℕ), f (n + 1)` ✓ I'm fairly confident.
Let me write the fixed ending.
Completing the ending properly (conv_rhs navigation + the per-term cast algebra).
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigamma.lean",
"old_string": " congr 2\n have hsum := summable_zpow_half_add_nat k hk\n rw [tsum_eq_zero_add hsum]\n apply Finset.sum_congr rfl |>.trans' |>.symm -- placeholder, will fix\n sorry",
"new_string": " congr 2\n have hsum := summable_zpow_half_add_nat k hk\n conv_rhs => rw [tsum_eq_zero_add hsum]\n have e0 : ((1 / 2 : \u2102) + ((0 : \u2115) : \u2102)) ^ (-(k : \u2124) - 1) = ((1 : \u2102) / 2) ^ (-(k : \u2124) - 1) := by\n simp\n rw [e0]\n apply add_left_cancel_iff.mpr\n apply tsum_congr\n intro m\n push_cast\n ring"
}
raw content_json (database row)
[{"type":"thinking","thinking":"The file now ends with the sorry placeholder and no `end NguyenRH1`. I need to complete the ending of iteratedDeriv_DinfC_half properly. The goal at that point (after congr 2): the remaining goal is the series identity:\n`((1/2:ℂ))^(−(k:ℤ)−1) + ∑' m, (1/2 + ↑m + 1)^(−(k:ℤ)−1) = ∑' m, (1/2 + ↑m)^(−(k:ℤ)−1)` \n\nand I did `rw [tsum_eq_zero_add hsum]` — which rewrites the RHS: ∑' m, f m = f 0 + ∑' m, f (m+1) — wait — tsum_eq_zero_add : ∑' n, f n = f 0 + ∑' n, f (n + 1) — rw [tsum_eq_zero_add hsum] rewrites ∑' m, (1/2+↑m)^(−(k+1)) → (1/2+↑0)^(−(k+1)) + ∑' m, (1/2+↑(m+1))^(−(k+1)) — so the goal becomes:\n(1/2)^(−(k+1)) + Σ_{m}(1/2+↑m+1)^{...} = (1/2+↑0)^{...} + Σ_{m}(1/2+↑(m+1))^{...}\n— the first terms: (1/2)^(−(k+1)) vs (1/2+↑0)^(−(k+1)) — ↑(0:ℕ) = 0, 1/2+0 = 1/2 ✓ equal modulo cast — and the tail sums: Σ_m (1/2+↑m+1)^{...} vs Σ_m (1/2+↑(m+1))^{...} — per-term: ↑(m+1) = ↑m+1 ✓ — so close by tsum_congr + per-term push_cast/ring:\n```lean\n have hsum := summable_zpow_half_add_nat k hk\n rw [tsum_eq_zero_add hsum]\n have e0 : ((1 / 2 : ℂ) + ((0 : ℕ) : ℂ)) ^ (-(k : ℤ) - 1) = ((1 : ℂ) / 2) ^ (-(k : ℤ) - 1) := by simp\n rw [e0]\n apply add_left_cancel_iff.mpr -- hmm the forms\n apply tsum_congr\n intro m\n push_cast\n ring\n```\nHmm — after rw [e0]: goal (1/2)^(−(k+1)) + Σ_{m}(1/2+↑m+1)^(−(k+1)) = (1/2)^(−(k+1)) + Σ_{m}(1/2+↑(m+1))^(−(k+1)) — `apply add_left_cancel_iff.mpr` — add_left_cancel_iff : a + b = a + c ↔ b = c ✓ — leaves Σ (1/2+↑m+1)^... = Σ (1/2+↑(m+1))^... — tsum_congr per-term: (1/2 + ↑m + 1)^(−(k+1)) = (1/2 + ↑(m+1))^(−(k+1)) — congr 1 on the zpow base: 1/2 + ↑m + 1 = 1/2 + ↑(m+1) — push_cast + ring ✓. Let me write:\n```lean\n have hsum := summable_zpow_half_add_nat k hk\n rw [tsum_eq_zero_add hsum]\n have e0 : ((1 / 2 : ℂ) + ((0 : ℕ) : ℂ)) ^ (-(k : ℤ) - 1) = ((1 : ℂ) / 2) ^ (-(k : ℤ) - 1) := by\n simp\n rw [e0]\n apply add_left_cancel_iff.mpr\n apply tsum_congr\n intro m\n push_cast\n ring\n```\nHmm — wait — tsum_eq_zero_add rewrites the GOAL's tsum — the goal's RHS is ∑' m, ((1/2:ℂ)+↑m)^(−(k:ℤ)−1) — after rw: → f 0 + ∑' m, f (m+1) = ((1/2)+↑0)^... + ∑' m, ((1/2)+↑(m+1))^... ✓. But — hmm — does the tsum_eq_zero_add rewrite fire on the goal's RHS tsum? The goal at that point: (1/2)^z + Σ(1/2+↑m+1)^z = Σ(1/2+↑m)^z — rw [tsum_eq_zero_add hsum] — the pattern ∑' n, ?f n — matches BOTH sums (the LHS's Σ(1/2+↑m+1)^z is also a tsum)! — the FIRST match gets rewritten — the traversal order — the goal is Eq LHS RHS — LHS has the Σ(1/2+↑m+1)^z tsum and the (1/2)^z... — hmm the first tsum in the goal is the LHS's Σ(1/2+↑m+1)^z!! I want to rewrite the RHS's. Damn. Use `conv_rhs => rw [tsum_eq_zero_add hsum]`? — conv_rhs navigation ✓:\n```lean\n conv_rhs => rw [tsum_eq_zero_add hsum]\n```\n✓ deterministic.\n\nThen the goal: (1/2)^z + Σ_m(1/2+↑m+1)^z = (1/2+↑0)^z + Σ_m(1/2+↑(m+1))^z. Then e0 turns (1/2+↑0)^z → (1/2)^z... wait — after conv_rhs, the RHS's f 0 = (1/2 + ↑(0))^z — hmm — f 0 with f m := (1/2+↑m)^z: f 0 = (1/2 + ↑0)^z ✓ — e0 matches ✓ then the two (1/2)^z cancel via add_left_cancel_iff.mpr → the tail sums equal via tsum_congr per-term push_cast/ring ✓.\n\nLet me also double check `tsum_eq_zero_add`'s exact form: `tsum_eq_zero_add (hf : Summable f) : ∑' n, f n = f 0 + ∑' n, f (n + 1)`? — hmm — it might be `tsum_eq_zero_add : ∑' (n : ℕ), f n = f 0 + ∑' (n : ℕ), f (n + 1)` ✓ I'm fairly confident.\n\nLet me write the fixed ending.","signature":""},{"type":"text","text":"Completing the ending properly (conv_rhs navigation + the per-term cast algebra)."},{"type":"tool_use","id":"Edit_473","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigamma.lean","old_string":" congr 2\n have hsum := summable_zpow_half_add_nat k hk\n rw [tsum_eq_zero_add hsum]\n apply Finset.sum_congr rfl |>.trans' |>.symm -- placeholder, will fix\n sorry","new_string":" congr 2\n have hsum := summable_zpow_half_add_nat k hk\n conv_rhs => rw [tsum_eq_zero_add hsum]\n have e0 : ((1 / 2 : ℂ) + ((0 : ℕ) : ℂ)) ^ (-(k : ℤ) - 1) = ((1 : ℂ) / 2) ^ (-(k : ℤ) - 1) := by\n simp\n rw [e0]\n apply add_left_cancel_iff.mpr\n apply tsum_congr\n intro m\n push_cast\n ring"},"extra_content":null}]