← Session 22 — ii′-1: liCoeff pairing fix + residue machinery
thinking (4313 chars)
The Equiv.tsum_eq rewrite WORKED (the goal moved forward). Now the final per-term cast: the goal at 967: `(1 / ↑↑m) ^ (j + 1) = ↑↑m ^ (-↑j - 1)` — my `rw [← one_div]; rw [← zpow_neg_natCast]` — the first rw [← one_div] — pattern ?a⁻¹ — NOT in the goal (the goal has 1/↑↑m and the zpow — no bare ⁻¹)! — and zpow_neg_natCast — its form: x ^ (↑n : ℤ) = x ^ n — the ← direction: x^n → x^(↑n:ℤ)... hmm — I want (1/↑m)^(j+1) = (↑m)^(−(j+1):ℤ): path: (1/↑m)^(j+1) = (↑m⁻¹)^(j+1) [one_div] = (↑m)^(−(j+1)) [inv_pow? or zpow]. Let me: (1/x)^(j+1) = (x^(j+1))⁻¹ [one_div_pow? `one_div_pow : (1/a)^n = 1/a^n`?] hmm. The clean: (1/x)^(j+1) = x^(−(j+1):ℤ):
- (1/x)^(j+1) = (x⁻¹)^(j+1) [one_div]
- (x⁻¹)^(j+1) = x^(−(j+1)) [inv_pow? — `inv_pow : (a⁻¹)^n = (a^n)⁻¹` ✓ — then (x^(j+1))⁻¹ = x^(−(j+1):ℤ) [zpow_natCast + zpow_neg reversed: ← zpow_natCast gives (x^(j+1)) from x^(↑(j+1):ℤ)... — let me just use zpow directly: (x⁻¹)^(j+1) = x^((−1:ℤ)·(j+1))... — the cleanest single lemma: `one_div_pow`? no. There's `zpow_natCast` and... — let me just compute via: (1/x)^(j+1) = 1/(x^(j+1)) [one_div_pow — if exists] and 1/(x^(j+1)) = x^(−(j+1):ℤ) [zpow_neg_natCast : x^(−(n:ℤ)) = 1/x^n — if that's the form]. So:
```lean
rw [Complex.ofRealCLM_apply]
push_cast
rw [one_div_pow, ← zpow_natCast, ← zpow_neg] -- hmm
```
Let me define the per-term identity cleanly as a standalone to control the forms:
(1/↑m)^(j+1) = (↑m)^(−(j+1):ℤ):
```lean
have hper : ∀ x : ℂ, x ≠ 0 → ∀ p : ℕ, (1 / x) ^ p = x ^ (-(p : ℤ)) := by
intro x hx p
rw [← zpow_natCast] -- x^(−p:ℤ) → x^(−p:ℕ)?? no
...
```
Ugh. The right lemmas: `zpow_neg (x) (n : ℤ) : x ^ (-n) = (x ^ n)⁻¹` and `zpow_natCast (x) (n : ℕ) : x ^ (↑n : ℤ) = x ^ n`. So x^(−(j+1):ℤ) = (x^((j+1:ℕ):ℤ))⁻¹ [zpow_neg] = (x^(j+1))⁻¹ [zpow_natCast] — and (1/x)^(j+1) = 1/(x^(j+1)) [one_div_pow? — `one_div_pow (n) : (1/a)^n = 1/a^n` — exists? — `one_div_pow : (1 / a) ^ n = 1 / a ^ n` ✓ I believe] = (x^(j+1))⁻¹ [one_div]. So per-term: (1/x)^(j+1) = 1/(x^(j+1)) = (x^(j+1))⁻¹ = x^(−(j+1):ℤ) — via:
```lean
intro m
rw [Complex.ofRealCLM_apply]
push_cast
rw [one_div_pow] -- (1/↑↑m)^(j+1) → 1/(↑↑m)^(j+1) — hmm — push_cast might already have normalized (1/↑m)^p to ↑(1/↑m^p)...
rw [← zpow_natCast, ← zpow_neg] -- 1/(↑↑m)^(j+1) → ((↑↑m)^(j+1))⁻¹ → ↑↑m^(−(j+1):ℤ) — hmm the forms
```
The push_cast on ↑( (1/↑m)^(j+1) ) — push_cast turns it into (1/↑m)^(j+1) with ℂ casts — the goal after push_cast: (1/↑↑m)^(j+1) = ↑↑m^(−↑j−1) — then one_div_pow : (1/a)^n = 1/a^n: LHS → 1/(↑↑m)^(j+1) — then need 1/(↑↑m)^(j+1) = ↑↑m^(−↑j−1): ← zpow_natCast : (↑↑m)^((j+1:ℕ):ℤ) = (↑↑m)^(j+1) — hmm direction: zpow_natCast : x^(↑n:ℤ) = x^n — the RHS of my goal is x^(−↑j−1:ℤ) = x^(−((j+1:ℕ):ℤ))... — the exponent display `-↑j - 1` — an INTEGER expression −(j:ℤ)−1 — and the npow exponent (j+1:ℕ) — the bridge: `−↑j − 1 = −((j+1:ℕ):ℤ)` via push_cast/ring ✓ (I've used this show-pattern ✓). So:
```lean
intro m
rw [Complex.ofRealCLM_apply]
push_cast
have hexp : (-(j : ℤ) - 1 : ℤ) = -(((j + 1 : ℕ) : ℤ)) := by push_cast; ring
rw [hexp, one_div_pow, zpow_neg_natCast'?...]
```
`zpow_neg_natCast` — does it exist? — there might be `zpow_neg_natCast : x ^ (-(n:ℤ)) = 1 / x ^ n`? — or I combine zpow_neg + zpow_natCast: x^(−(n:ℤ)) = (x^(n:ℤ))⁻¹ [zpow_neg] = (x^n)⁻¹ [zpow_natCast] — so the target: 1/(x^(j+1)) = x^(−(j+1):ℤ): rw [show x^(−((j+1:ℕ):ℤ)) = (x^(j+1))⁻¹ from by rw [zpow_neg, zpow_natCast]] then one_div: (x^(j+1))⁻¹ = 1/(x^(j+1)) — `inv_eq_one_div`? — `inv_eq_one_div : a⁻¹ = 1/a` ✓. So:
```lean
rw [hexp]
rw [show (↑↑m : ℂ) ^ (-(((j + 1 : ℕ) : ℤ))) = (((↑↑m : ℂ) ^ (j + 1))⁻¹) from by
rw [zpow_neg, zpow_natCast]]
rw [one_div_pow, inv_eq_one_div]
```
Hmm — the show's rewrite direction: the goal after push_cast+hexp: (1/↑↑m)^(j+1) = ↑↑m^(−((j+1:ℕ):ℤ)) — the show rewrites the RHS's zpow → ((↑↑m)^(j+1))⁻¹ ✓ then one_div_pow on the LHS (1/↑↑m)^(j+1) → 1/(↑↑m)^(j+1) ✓ then inv_eq_one_div: ((↑↑m)^(j+1))⁻¹ → 1/(↑↑m)^(j+1) ✓ — both sides 1/(↑↑m)^(j+1) ✓ rfl by rw ✓.
Let me rewrite the per-term block in hcast with this. Also the error at 970 — cascade from hcast's failure (the rw [hcast] failed because hcast's proof failed) — will resolve once hcast is fixed.
Let me look at the current hcast block and rewrite it.
Fixing the per-term zpow/div bridge in `hcast`.
tool: Read ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigamma.lean",
"offset": 955,
"limit": 22
}
result 2026-08-22 10:26:08Z → 2026-08-22 10:26:08Z
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"content": " rw [lagariasA, if_neg hj1]\n push_cast\n congr 1\n have hcast : (\u2191(\u2211' m : {m : \u2115 // Odd m \u2227 3 \u2264 m}, (1 / (m : \u211d)) ^ (j + 1)) : \u2102) =\n \u2211' m : {m : \u2115 // Odd m \u2227 3 \u2264 m}, (((m : \u2115) : \u2102)) ^ (-(j : \u2124) - 1) := by\n have h2 := Complex.ofRealCLM.map_tsum (oddSlice_summable (j + 1) (by omega))\n rw [Complex.ofRealCLM_apply] at h2\n rw [h2]\n apply tsum_congr\n intro m\n rw [Complex.ofRealCLM_apply]\n push_cast\n rw [\u2190 one_div]\n rw [\u2190 zpow_neg_natCast]\n ring_nf\n rw [hcast]",
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"totalLines": 970
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raw content_json (database row)
[{"type":"thinking","thinking":"The Equiv.tsum_eq rewrite WORKED (the goal moved forward). Now the final per-term cast: the goal at 967: `(1 / ↑↑m) ^ (j + 1) = ↑↑m ^ (-↑j - 1)` — my `rw [← one_div]; rw [← zpow_neg_natCast]` — the first rw [← one_div] — pattern ?a⁻¹ — NOT in the goal (the goal has 1/↑↑m and the zpow — no bare ⁻¹)! — and zpow_neg_natCast — its form: x ^ (↑n : ℤ) = x ^ n — the ← direction: x^n → x^(↑n:ℤ)... hmm — I want (1/↑m)^(j+1) = (↑m)^(−(j+1):ℤ): path: (1/↑m)^(j+1) = (↑m⁻¹)^(j+1) [one_div] = (↑m)^(−(j+1)) [inv_pow? or zpow]. Let me: (1/x)^(j+1) = (x^(j+1))⁻¹ [one_div_pow? `one_div_pow : (1/a)^n = 1/a^n`?] hmm. The clean: (1/x)^(j+1) = x^(−(j+1):ℤ):\n- (1/x)^(j+1) = (x⁻¹)^(j+1) [one_div]\n- (x⁻¹)^(j+1) = x^(−(j+1)) [inv_pow? — `inv_pow : (a⁻¹)^n = (a^n)⁻¹` ✓ — then (x^(j+1))⁻¹ = x^(−(j+1):ℤ) [zpow_natCast + zpow_neg reversed: ← zpow_natCast gives (x^(j+1)) from x^(↑(j+1):ℤ)... — let me just use zpow directly: (x⁻¹)^(j+1) = x^((−1:ℤ)·(j+1))... — the cleanest single lemma: `one_div_pow`? no. There's `zpow_natCast` and... — let me just compute via: (1/x)^(j+1) = 1/(x^(j+1)) [one_div_pow — if exists] and 1/(x^(j+1)) = x^(−(j+1):ℤ) [zpow_neg_natCast : x^(−(n:ℤ)) = 1/x^n — if that's the form]. So:\n```lean\n rw [Complex.ofRealCLM_apply]\n push_cast\n rw [one_div_pow, ← zpow_natCast, ← zpow_neg] -- hmm\n```\nLet me define the per-term identity cleanly as a standalone to control the forms:\n(1/↑m)^(j+1) = (↑m)^(−(j+1):ℤ):\n```lean\n have hper : ∀ x : ℂ, x ≠ 0 → ∀ p : ℕ, (1 / x) ^ p = x ^ (-(p : ℤ)) := by\n intro x hx p\n rw [← zpow_natCast] -- x^(−p:ℤ) → x^(−p:ℕ)?? no\n ...\n```\nUgh. The right lemmas: `zpow_neg (x) (n : ℤ) : x ^ (-n) = (x ^ n)⁻¹` and `zpow_natCast (x) (n : ℕ) : x ^ (↑n : ℤ) = x ^ n`. So x^(−(j+1):ℤ) = (x^((j+1:ℕ):ℤ))⁻¹ [zpow_neg] = (x^(j+1))⁻¹ [zpow_natCast] — and (1/x)^(j+1) = 1/(x^(j+1)) [one_div_pow? — `one_div_pow (n) : (1/a)^n = 1/a^n` — exists? — `one_div_pow : (1 / a) ^ n = 1 / a ^ n` ✓ I believe] = (x^(j+1))⁻¹ [one_div]. So per-term: (1/x)^(j+1) = 1/(x^(j+1)) = (x^(j+1))⁻¹ = x^(−(j+1):ℤ) — via:\n```lean\n intro m\n rw [Complex.ofRealCLM_apply]\n push_cast\n rw [one_div_pow] -- (1/↑↑m)^(j+1) → 1/(↑↑m)^(j+1) — hmm — push_cast might already have normalized (1/↑m)^p to ↑(1/↑m^p)... \n rw [← zpow_natCast, ← zpow_neg] -- 1/(↑↑m)^(j+1) → ((↑↑m)^(j+1))⁻¹ → ↑↑m^(−(j+1):ℤ) — hmm the forms\n```\nThe push_cast on ↑( (1/↑m)^(j+1) ) — push_cast turns it into (1/↑m)^(j+1) with ℂ casts — the goal after push_cast: (1/↑↑m)^(j+1) = ↑↑m^(−↑j−1) — then one_div_pow : (1/a)^n = 1/a^n: LHS → 1/(↑↑m)^(j+1) — then need 1/(↑↑m)^(j+1) = ↑↑m^(−↑j−1): ← zpow_natCast : (↑↑m)^((j+1:ℕ):ℤ) = (↑↑m)^(j+1) — hmm direction: zpow_natCast : x^(↑n:ℤ) = x^n — the RHS of my goal is x^(−↑j−1:ℤ) = x^(−((j+1:ℕ):ℤ))... — the exponent display `-↑j - 1` — an INTEGER expression −(j:ℤ)−1 — and the npow exponent (j+1:ℕ) — the bridge: `−↑j − 1 = −((j+1:ℕ):ℤ)` via push_cast/ring ✓ (I've used this show-pattern ✓). So:\n```lean\n intro m\n rw [Complex.ofRealCLM_apply]\n push_cast\n have hexp : (-(j : ℤ) - 1 : ℤ) = -(((j + 1 : ℕ) : ℤ)) := by push_cast; ring\n rw [hexp, one_div_pow, zpow_neg_natCast'?...]\n```\n`zpow_neg_natCast` — does it exist? — there might be `zpow_neg_natCast : x ^ (-(n:ℤ)) = 1 / x ^ n`? — or I combine zpow_neg + zpow_natCast: x^(−(n:ℤ)) = (x^(n:ℤ))⁻¹ [zpow_neg] = (x^n)⁻¹ [zpow_natCast] — so the target: 1/(x^(j+1)) = x^(−(j+1):ℤ): rw [show x^(−((j+1:ℕ):ℤ)) = (x^(j+1))⁻¹ from by rw [zpow_neg, zpow_natCast]] then one_div: (x^(j+1))⁻¹ = 1/(x^(j+1)) — `inv_eq_one_div`? — `inv_eq_one_div : a⁻¹ = 1/a` ✓. So:\n```lean\n rw [hexp]\n rw [show (↑↑m : ℂ) ^ (-(((j + 1 : ℕ) : ℤ))) = (((↑↑m : ℂ) ^ (j + 1))⁻¹) from by\n rw [zpow_neg, zpow_natCast]]\n rw [one_div_pow, inv_eq_one_div]\n```\nHmm — the show's rewrite direction: the goal after push_cast+hexp: (1/↑↑m)^(j+1) = ↑↑m^(−((j+1:ℕ):ℤ)) — the show rewrites the RHS's zpow → ((↑↑m)^(j+1))⁻¹ ✓ then one_div_pow on the LHS (1/↑↑m)^(j+1) → 1/(↑↑m)^(j+1) ✓ then inv_eq_one_div: ((↑↑m)^(j+1))⁻¹ → 1/(↑↑m)^(j+1) ✓ — both sides 1/(↑↑m)^(j+1) ✓ rfl by rw ✓.\n\nLet me rewrite the per-term block in hcast with this. Also the error at 970 — cascade from hcast's failure (the rw [hcast] failed because hcast's proof failed) — will resolve once hcast is fixed.\n\nLet me look at the current hcast block and rewrite it.","signature":""},{"type":"text","text":"Fixing the per-term zpow/div bridge in `hcast`."},{"type":"tool_use","id":"Read_521","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigamma.lean","offset":955,"limit":22},"extra_content":null}]