url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | have h₁ : ∀ x ∈ range ↑n, Summable (λ (x_1 : ℕ) => (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑(Nat.factorial x_1) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) := by
intros k _
exact Summable.smul_const (ExpTaylorSeriesSummable (z * cexp (2 * ↑π * (↑k / ↑↑n) * I))) _ | n : ℕ+
m : ℤ
z : ℂ
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n,
∑' (x_1 : ℕ),
(z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n | n : ℕ+
m : ℤ
z : ℂ
h₁ :
∀ x ∈ range ↑n,
Summable fun x_1 =>
(z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n,
∑' (x_1 : ℕ),
(z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n,
∑' (x_1 : ℕ),
(z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | have h₂ := (tsum_sum h₁).symm | n : ℕ+
m : ℤ
z : ℂ
h₁ :
∀ x ∈ range ↑n,
Summable fun x_1 =>
(z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n,
∑' (x_1 : ℕ),
(z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n | n : ℕ+
m : ℤ
z : ℂ
h₁ :
∀ x ∈ range ↑n,
Summable fun x_1 =>
(z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)
h₂ :
∑ i ∈ range ↑n,
∑' (b : ℕ), (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I) =
∑' (b : ℕ),
∑ i ∈ range ↑n, (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I)
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n,
∑' (x_1 : ℕ),
(z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
h₁ :
∀ x ∈ range ↑n,
Summable fun x_1 =>
(z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n,
∑' (x_1 : ℕ),
(z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | clear h₁ | n : ℕ+
m : ℤ
z : ℂ
h₁ :
∀ x ∈ range ↑n,
Summable fun x_1 =>
(z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)
h₂ :
∑ i ∈ range ↑n,
∑' (b : ℕ), (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I) =
∑' (b : ℕ),
∑ i ∈ range ↑n, (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I)
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n,
∑' (x_1 : ℕ),
(z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n | n : ℕ+
m : ℤ
z : ℂ
h₂ :
∑ i ∈ range ↑n,
∑' (b : ℕ), (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I) =
∑' (b : ℕ),
∑ i ∈ range ↑n, (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I)
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n,
∑' (x_1 : ℕ),
(z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
h₁ :
∀ x ∈ range ↑n,
Summable fun x_1 =>
(z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)
h₂ :
∑ i ∈ range ↑n,
∑' (b : ℕ), (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I) =
∑' (b : ℕ),
∑ i ∈ range ↑n, (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I)
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n,
∑' (x_1 : ℕ),
(z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | simp_rw [h₂] | n : ℕ+
m : ℤ
z : ℂ
h₂ :
∑ i ∈ range ↑n,
∑' (b : ℕ), (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I) =
∑' (b : ℕ),
∑ i ∈ range ↑n, (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I)
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n,
∑' (x_1 : ℕ),
(z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n | n : ℕ+
m : ℤ
z : ℂ
h₂ :
∑ i ∈ range ↑n,
∑' (b : ℕ), (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I) =
∑' (b : ℕ),
∑ i ∈ range ↑n, (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I)
⊢ RuesDiff n m z =
(∑' (b : ℕ),
∑ i ∈ range ↑n, (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I)) /
↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
h₂ :
∑ i ∈ range ↑n,
∑' (b : ℕ), (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I) =
∑' (b : ℕ),
∑ i ∈ range ↑n, (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I)
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n,
∑' (x_1 : ℕ),
(z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | clear h₂ | n : ℕ+
m : ℤ
z : ℂ
h₂ :
∑ i ∈ range ↑n,
∑' (b : ℕ), (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I) =
∑' (b : ℕ),
∑ i ∈ range ↑n, (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I)
⊢ RuesDiff n m z =
(∑' (b : ℕ),
∑ i ∈ range ↑n, (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I)) /
↑↑n | n : ℕ+
m : ℤ
z : ℂ
⊢ RuesDiff n m z =
(∑' (b : ℕ),
∑ i ∈ range ↑n, (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I)) /
↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
h₂ :
∑ i ∈ range ↑n,
∑' (b : ℕ), (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I) =
∑' (b : ℕ),
∑ i ∈ range ↑n, (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I)
⊢ RuesDiff n m z =
(∑' (b : ℕ),
∑ i ∈ range ↑n, (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I)) /
↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | simp_rw [mul_pow, ←Complex.exp_nat_mul] | n : ℕ+
m : ℤ
z : ℂ
⊢ RuesDiff n m z =
(∑' (b : ℕ),
∑ i ∈ range ↑n, (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I)) /
↑↑n | n : ℕ+
m : ℤ
z : ℂ
⊢ RuesDiff n m z =
(∑' (b : ℕ),
∑ x ∈ range ↑n,
z ^ b * cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
⊢ RuesDiff n m z =
(∑' (b : ℕ),
∑ i ∈ range ↑n, (z * cexp (2 * ↑π * (↑i / ↑↑n) * I)) ^ b / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑i / ↑↑n) * I)) /
↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | have h₃ : ∀ (b x : ℕ), z ^ b * cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) / ↑(Nat.factorial b) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I) =
(z ^ b / ↑(Nat.factorial b)) * (cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) := by
intros b x
ring_nf | n : ℕ+
m : ℤ
z : ℂ
⊢ RuesDiff n m z =
(∑' (b : ℕ),
∑ x ∈ range ↑n,
z ^ b * cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n | n : ℕ+
m : ℤ
z : ℂ
h₃ :
∀ (b x : ℕ),
z ^ b * cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I) =
z ^ b / ↑b.factorial * (cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I))
⊢ RuesDiff n m z =
(∑' (b : ℕ),
∑ x ∈ range ↑n,
z ^ b * cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
⊢ RuesDiff n m z =
(∑' (b : ℕ),
∑ x ∈ range ↑n,
z ^ b * cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | simp_rw [h₃, ←Finset.mul_sum, ←Complex.exp_add, ←tsum_div_const, RuesDiff] | n : ℕ+
m : ℤ
z : ℂ
h₃ :
∀ (b x : ℕ),
z ^ b * cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I) =
z ^ b / ↑b.factorial * (cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I))
⊢ RuesDiff n m z =
(∑' (b : ℕ),
∑ x ∈ range ↑n,
z ^ b * cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n | n : ℕ+
m : ℤ
z : ℂ
h₃ :
∀ (b x : ℕ),
z ^ b * cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I) =
z ^ b / ↑b.factorial * (cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I))
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑' (x : ℕ),
(z ^ x / ↑x.factorial *
∑ x_1 ∈ range ↑n, cexp (↑x * (2 * ↑π * (↑x_1 / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x_1 / ↑↑n) * I)) /
↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
h₃ :
∀ (b x : ℕ),
z ^ b * cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I) =
z ^ b / ↑b.factorial * (cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I))
⊢ RuesDiff n m z =
(∑' (b : ℕ),
∑ x ∈ range ↑n,
z ^ b * cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | clear h₃ | n : ℕ+
m : ℤ
z : ℂ
h₃ :
∀ (b x : ℕ),
z ^ b * cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I) =
z ^ b / ↑b.factorial * (cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I))
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑' (x : ℕ),
(z ^ x / ↑x.factorial *
∑ x_1 ∈ range ↑n, cexp (↑x * (2 * ↑π * (↑x_1 / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x_1 / ↑↑n) * I)) /
↑↑n | n : ℕ+
m : ℤ
z : ℂ
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑' (x : ℕ),
(z ^ x / ↑x.factorial *
∑ x_1 ∈ range ↑n, cexp (↑x * (2 * ↑π * (↑x_1 / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x_1 / ↑↑n) * I)) /
↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
h₃ :
∀ (b x : ℕ),
z ^ b * cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I) =
z ^ b / ↑b.factorial * (cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I))
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑' (x : ℕ),
(z ^ x / ↑x.factorial *
∑ x_1 ∈ range ↑n, cexp (↑x * (2 * ↑π * (↑x_1 / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x_1 / ↑↑n) * I)) /
↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | congr | n : ℕ+
m : ℤ
z : ℂ
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑' (x : ℕ),
(z ^ x / ↑x.factorial *
∑ x_1 ∈ range ↑n, cexp (↑x * (2 * ↑π * (↑x_1 / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x_1 / ↑↑n) * I)) /
↑↑n | case e_f
n : ℕ+
m : ℤ
z : ℂ
⊢ (fun k => if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = fun x =>
(z ^ x / ↑x.factorial *
∑ x_1 ∈ range ↑n, cexp (↑x * (2 * ↑π * (↑x_1 / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x_1 / ↑↑n) * I)) /
↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑' (x : ℕ),
(z ^ x / ↑x.factorial *
∑ x_1 ∈ range ↑n, cexp (↑x * (2 * ↑π * (↑x_1 / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x_1 / ↑↑n) * I)) /
↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | ext1 k | case e_f
n : ℕ+
m : ℤ
z : ℂ
⊢ (fun k => if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = fun x =>
(z ^ x / ↑x.factorial *
∑ x_1 ∈ range ↑n, cexp (↑x * (2 * ↑π * (↑x_1 / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x_1 / ↑↑n) * I)) /
↑↑n | case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
(z ^ k / ↑k.factorial * ∑ x ∈ range ↑n, cexp (↑k * (2 * ↑π * (↑x / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f
n : ℕ+
m : ℤ
z : ℂ
⊢ (fun k => if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = fun x =>
(z ^ x / ↑x.factorial *
∑ x_1 ∈ range ↑n, cexp (↑x * (2 * ↑π * (↑x_1 / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x_1 / ↑↑n) * I)) /
↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | have h₄ : ∀ (x : ℕ), ↑k * (2 * ↑π * (↑x / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x / ↑↑n) * I = (2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I) := by
intros x
ring_nf | case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
(z ^ k / ↑k.factorial * ∑ x ∈ range ↑n, cexp (↑k * (2 * ↑π * (↑x / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n | case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₄ : ∀ (x : ℕ), ↑k * (2 * ↑π * (↑x / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x / ↑↑n) * I = 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
(z ^ k / ↑k.factorial * ∑ x ∈ range ↑n, cexp (↑k * (2 * ↑π * (↑x / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
(z ^ k / ↑k.factorial * ∑ x ∈ range ↑n, cexp (↑k * (2 * ↑π * (↑x / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | simp_rw [h₄] | case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₄ : ∀ (x : ℕ), ↑k * (2 * ↑π * (↑x / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x / ↑↑n) * I = 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
(z ^ k / ↑k.factorial * ∑ x ∈ range ↑n, cexp (↑k * (2 * ↑π * (↑x / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n | case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₄ : ∀ (x : ℕ), ↑k * (2 * ↑π * (↑x / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x / ↑↑n) * I = 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
(z ^ k / ↑k.factorial * ∑ x ∈ range ↑n, cexp (2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I)) / ↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₄ : ∀ (x : ℕ), ↑k * (2 * ↑π * (↑x / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x / ↑↑n) * I = 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
(z ^ k / ↑k.factorial * ∑ x ∈ range ↑n, cexp (↑k * (2 * ↑π * (↑x / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | clear h₄ | case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₄ : ∀ (x : ℕ), ↑k * (2 * ↑π * (↑x / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x / ↑↑n) * I = 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
(z ^ k / ↑k.factorial * ∑ x ∈ range ↑n, cexp (2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I)) / ↑↑n | case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
(z ^ k / ↑k.factorial * ∑ x ∈ range ↑n, cexp (2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I)) / ↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₄ : ∀ (x : ℕ), ↑k * (2 * ↑π * (↑x / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x / ↑↑n) * I = 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
(z ^ k / ↑k.factorial * ∑ x ∈ range ↑n, cexp (2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I)) / ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | have h₅ := RouGeometricSumEqIte n (↑k + m) | case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
(z ^ k / ↑k.factorial * ∑ x ∈ range ↑n, cexp (2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I)) / ↑↑n | case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
(z ^ k / ↑k.factorial * ∑ x ∈ range ↑n, cexp (2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I)) / ↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
(z ^ k / ↑k.factorial * ∑ x ∈ range ↑n, cexp (2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I)) / ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | have h₆ : ∀ (x : ℕ), (2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I) = (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) := by
intros x
simp only [Int.cast_add, Int.cast_natCast]
ring_nf | case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
(z ^ k / ↑k.factorial * ∑ x ∈ range ↑n, cexp (2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I)) / ↑↑n | case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
(z ^ k / ↑k.factorial * ∑ x ∈ range ↑n, cexp (2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I)) / ↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
(z ^ k / ↑k.factorial * ∑ x ∈ range ↑n, cexp (2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I)) / ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | simp_rw [h₆, h₅] | case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
(z ^ k / ↑k.factorial * ∑ x ∈ range ↑n, cexp (2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I)) / ↑↑n | case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = (z ^ k / ↑k.factorial * if ↑↑n ∣ ↑k + m then ↑↑n else 0) / ↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
(z ^ k / ↑k.factorial * ∑ x ∈ range ↑n, cexp (2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I)) / ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | simp only [mul_ite, mul_zero] | case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = (z ^ k / ↑k.factorial * if ↑↑n ∣ ↑k + m then ↑↑n else 0) / ↑↑n | case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial * ↑↑n else 0) / ↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = (z ^ k / ↑k.factorial * if ↑↑n ∣ ↑k + m then ↑↑n else 0) / ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | have hem := Classical.em (↑↑n ∣ ↑k + m) | case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial * ↑↑n else 0) / ↑↑n | case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
hem : ↑↑n ∣ ↑k + m ∨ ¬↑↑n ∣ ↑k + m
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial * ↑↑n else 0) / ↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial * ↑↑n else 0) / ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | rcases hem with hemt | hemf | case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
hem : ↑↑n ∣ ↑k + m ∨ ¬↑↑n ∣ ↑k + m
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial * ↑↑n else 0) / ↑↑n | case e_f.h.inl
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
hemt : ↑↑n ∣ ↑k + m
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial * ↑↑n else 0) / ↑↑n
case e_f.h.inr
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
hemf : ¬↑↑n ∣ ↑k + m
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial * ↑↑n else 0) / ↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
hem : ↑↑n ∣ ↑k + m ∨ ¬↑↑n ∣ ↑k + m
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial * ↑↑n else 0) / ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | intros k | n : ℕ+
m : ℤ
z : ℂ
⊢ ∀ (k : ℕ),
cexp (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) =
∑' (k_1 : ℕ), (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial | n : ℕ+
m : ℤ
z : ℂ
k : ℕ
⊢ cexp (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) = ∑' (k_1 : ℕ), (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
⊢ ∀ (k : ℕ),
cexp (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) =
∑' (k_1 : ℕ), (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | exact ExpTsumForm (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) | n : ℕ+
m : ℤ
z : ℂ
k : ℕ
⊢ cexp (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) = ∑' (k_1 : ℕ), (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
⊢ cexp (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) = ∑' (k_1 : ℕ), (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | intros k _ | n : ℕ+
m : ℤ
z : ℂ
⊢ ∀ x ∈ range ↑n,
Summable fun x_1 =>
(z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I) | n : ℕ+
m : ℤ
z : ℂ
k : ℕ
a✝ : k ∈ range ↑n
⊢ Summable fun x_1 => (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑k / ↑↑n) * I) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
⊢ ∀ x ∈ range ↑n,
Summable fun x_1 =>
(z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | exact Summable.smul_const (ExpTaylorSeriesSummable (z * cexp (2 * ↑π * (↑k / ↑↑n) * I))) _ | n : ℕ+
m : ℤ
z : ℂ
k : ℕ
a✝ : k ∈ range ↑n
⊢ Summable fun x_1 => (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑k / ↑↑n) * I) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
a✝ : k ∈ range ↑n
⊢ Summable fun x_1 => (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) ^ x_1 / ↑x_1.factorial * cexp (↑m * 2 * ↑π * (↑k / ↑↑n) * I)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | intros b x | n : ℕ+
m : ℤ
z : ℂ
⊢ ∀ (b x : ℕ),
z ^ b * cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I) =
z ^ b / ↑b.factorial * (cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) | n : ℕ+
m : ℤ
z : ℂ
b x : ℕ
⊢ z ^ b * cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I) =
z ^ b / ↑b.factorial * (cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
⊢ ∀ (b x : ℕ),
z ^ b * cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I) =
z ^ b / ↑b.factorial * (cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I))
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | ring_nf | n : ℕ+
m : ℤ
z : ℂ
b x : ℕ
⊢ z ^ b * cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I) =
z ^ b / ↑b.factorial * (cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
b x : ℕ
⊢ z ^ b * cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) / ↑b.factorial * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I) =
z ^ b / ↑b.factorial * (cexp (↑b * (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I))
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | intros x | n : ℕ+
m : ℤ
z : ℂ
k : ℕ
⊢ ∀ (x : ℕ), ↑k * (2 * ↑π * (↑x / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x / ↑↑n) * I = 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I | n : ℕ+
m : ℤ
z : ℂ
k x : ℕ
⊢ ↑k * (2 * ↑π * (↑x / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x / ↑↑n) * I = 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
⊢ ∀ (x : ℕ), ↑k * (2 * ↑π * (↑x / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x / ↑↑n) * I = 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | ring_nf | n : ℕ+
m : ℤ
z : ℂ
k x : ℕ
⊢ ↑k * (2 * ↑π * (↑x / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x / ↑↑n) * I = 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
k x : ℕ
⊢ ↑k * (2 * ↑π * (↑x / ↑↑n) * I) + ↑m * 2 * ↑π * (↑x / ↑↑n) * I = 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | intros x | n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
⊢ ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I) | n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
x : ℕ
⊢ 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
⊢ ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | simp only [Int.cast_add, Int.cast_natCast] | n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
x : ℕ
⊢ 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I) | n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
x : ℕ
⊢ 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n * I) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
x : ℕ
⊢ 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | ring_nf | n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
x : ℕ
⊢ 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n * I) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
x : ℕ
⊢ 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n * I)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | simp_rw [if_pos hemt] | case e_f.h.inl
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
hemt : ↑↑n ∣ ↑k + m
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial * ↑↑n else 0) / ↑↑n | case e_f.h.inl
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
hemt : ↑↑n ∣ ↑k + m
⊢ z ^ k / ↑k.factorial = z ^ k / ↑k.factorial * ↑↑n / ↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.inl
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
hemt : ↑↑n ∣ ↑k + m
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial * ↑↑n else 0) / ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | ring_nf | case e_f.h.inl
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
hemt : ↑↑n ∣ ↑k + m
⊢ z ^ k / ↑k.factorial = z ^ k / ↑k.factorial * ↑↑n / ↑↑n | case e_f.h.inl
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
hemt : ↑↑n ∣ ↑k + m
⊢ z ^ k * (↑k.factorial)⁻¹ = z ^ k * (↑k.factorial)⁻¹ * ↑↑n * (↑↑n)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.inl
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
hemt : ↑↑n ∣ ↑k + m
⊢ z ^ k / ↑k.factorial = z ^ k / ↑k.factorial * ↑↑n / ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | simp only [ne_eq, Nat.cast_eq_zero, PNat.ne_zero, not_false_eq_true, mul_inv_cancel_right₀] | case e_f.h.inl
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
hemt : ↑↑n ∣ ↑k + m
⊢ z ^ k * (↑k.factorial)⁻¹ = z ^ k * (↑k.factorial)⁻¹ * ↑↑n * (↑↑n)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.inl
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
hemt : ↑↑n ∣ ↑k + m
⊢ z ^ k * (↑k.factorial)⁻¹ = z ^ k * (↑k.factorial)⁻¹ * ↑↑n * (↑↑n)⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | simp_rw [if_neg hemf] | case e_f.h.inr
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
hemf : ¬↑↑n ∣ ↑k + m
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial * ↑↑n else 0) / ↑↑n | case e_f.h.inr
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
hemf : ¬↑↑n ∣ ↑k + m
⊢ 0 = 0 / ↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.inr
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
hemf : ¬↑↑n ∣ ↑k + m
⊢ (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = (if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial * ↑↑n else 0) / ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | simp only [zero_div] | case e_f.h.inr
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
hemf : ¬↑↑n ∣ ↑k + m
⊢ 0 = 0 / ↑↑n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.inr
n : ℕ+
m : ℤ
z : ℂ
k : ℕ
h₅ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)) = if ↑↑n ∣ ↑k + m then ↑↑n else 0
h₆ : ∀ (x : ℕ), 2 * ↑π * ((↑k + ↑m) * ↑x / ↑↑n) * I = 2 * ↑π * (↑(↑k + m) * ↑x / ↑↑n * I)
hemf : ¬↑↑n ∣ ↑k + m
⊢ 0 = 0 / ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesNMthIteratedDeriv | [354, 1] | [356, 23] | rw [←RuesDiffM0EqualsRues, RuesDiffIteratedDeriv] | n m : ℕ+
⊢ iteratedDeriv (↑m) (Rues n) = RuesDiff n ↑↑m | n m : ℕ+
⊢ RuesDiff n (↑↑m + 0) = RuesDiff n ↑↑m | Please generate a tactic in lean4 to solve the state.
STATE:
n m : ℕ+
⊢ iteratedDeriv (↑m) (Rues n) = RuesDiff n ↑↑m
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesNMthIteratedDeriv | [354, 1] | [356, 23] | simp only [add_zero] | n m : ℕ+
⊢ RuesDiff n (↑↑m + 0) = RuesDiff n ↑↑m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n m : ℕ+
⊢ RuesDiff n (↑↑m + 0) = RuesDiff n ↑↑m
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffMod | [358, 1] | [363, 10] | rw [RuesDiffMPeriodic n (m % n) (m / n)] | n : ℕ+
m : ℤ
⊢ RuesDiff n m = RuesDiff n (m % ↑↑n) | n : ℕ+
m : ℤ
⊢ RuesDiff n m = RuesDiff n (m % ↑↑n + m / ↑↑n * ↑↑n) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
⊢ RuesDiff n m = RuesDiff n (m % ↑↑n)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffMod | [358, 1] | [363, 10] | nth_rw 1 [←Int.ediv_add_emod' m n] | n : ℕ+
m : ℤ
⊢ RuesDiff n m = RuesDiff n (m % ↑↑n + m / ↑↑n * ↑↑n) | n : ℕ+
m : ℤ
⊢ RuesDiff n (m / ↑↑n * ↑↑n + m % ↑↑n) = RuesDiff n (m % ↑↑n + m / ↑↑n * ↑↑n) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
⊢ RuesDiff n m = RuesDiff n (m % ↑↑n + m / ↑↑n * ↑↑n)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffMod | [358, 1] | [363, 10] | suffices h₀ : m / ↑↑n * ↑↑n + m % ↑↑n = m % ↑↑n + m / ↑↑n * ↑↑n | n : ℕ+
m : ℤ
⊢ RuesDiff n (m / ↑↑n * ↑↑n + m % ↑↑n) = RuesDiff n (m % ↑↑n + m / ↑↑n * ↑↑n) | n : ℕ+
m : ℤ
h₀ : m / ↑↑n * ↑↑n + m % ↑↑n = m % ↑↑n + m / ↑↑n * ↑↑n
⊢ RuesDiff n (m / ↑↑n * ↑↑n + m % ↑↑n) = RuesDiff n (m % ↑↑n + m / ↑↑n * ↑↑n)
case h₀
n : ℕ+
m : ℤ
⊢ m / ↑↑n * ↑↑n + m % ↑↑n = m % ↑↑n + m / ↑↑n * ↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
⊢ RuesDiff n (m / ↑↑n * ↑↑n + m % ↑↑n) = RuesDiff n (m % ↑↑n + m / ↑↑n * ↑↑n)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffMod | [358, 1] | [363, 10] | exact congrArg (RuesDiff n) h₀ | n : ℕ+
m : ℤ
h₀ : m / ↑↑n * ↑↑n + m % ↑↑n = m % ↑↑n + m / ↑↑n * ↑↑n
⊢ RuesDiff n (m / ↑↑n * ↑↑n + m % ↑↑n) = RuesDiff n (m % ↑↑n + m / ↑↑n * ↑↑n)
case h₀
n : ℕ+
m : ℤ
⊢ m / ↑↑n * ↑↑n + m % ↑↑n = m % ↑↑n + m / ↑↑n * ↑↑n | case h₀
n : ℕ+
m : ℤ
⊢ m / ↑↑n * ↑↑n + m % ↑↑n = m % ↑↑n + m / ↑↑n * ↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
h₀ : m / ↑↑n * ↑↑n + m % ↑↑n = m % ↑↑n + m / ↑↑n * ↑↑n
⊢ RuesDiff n (m / ↑↑n * ↑↑n + m % ↑↑n) = RuesDiff n (m % ↑↑n + m / ↑↑n * ↑↑n)
case h₀
n : ℕ+
m : ℤ
⊢ m / ↑↑n * ↑↑n + m % ↑↑n = m % ↑↑n + m / ↑↑n * ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffMod | [358, 1] | [363, 10] | ring_nf | case h₀
n : ℕ+
m : ℤ
⊢ m / ↑↑n * ↑↑n + m % ↑↑n = m % ↑↑n + m / ↑↑n * ↑↑n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀
n : ℕ+
m : ℤ
⊢ m / ↑↑n * ↑↑n + m % ↑↑n = m % ↑↑n + m / ↑↑n * ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | ExpPiMulIHalf | [365, 1] | [368, 14] | rw [exp_mul_I] | ⊢ cexp (↑(π / 2) * I) = I | ⊢ (↑(π / 2)).cos + (↑(π / 2)).sin * I = I | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ cexp (↑(π / 2) * I) = I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | ExpPiMulIHalf | [365, 1] | [368, 14] | simp only [ofReal_div, ofReal_ofNat, Complex.cos_pi_div_two, Complex.sin_pi_div_two, one_mul,
zero_add] | ⊢ (↑(π / 2)).cos + (↑(π / 2)).sin * I = I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ (↑(π / 2)).cos + (↑(π / 2)).sin * I = I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | ExpToNatPowersOfI | [370, 1] | [383, 14] | induction' k with K Kih | k : ℕ
⊢ cexp (↑π * I * ↑k / 2) = I ^ k | case zero
⊢ cexp (↑π * I * ↑0 / 2) = I ^ 0
case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
⊢ cexp (↑π * I * ↑(K + 1) / 2) = I ^ (K + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
k : ℕ
⊢ cexp (↑π * I * ↑k / 2) = I ^ k
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | ExpToNatPowersOfI | [370, 1] | [383, 14] | simp only [Nat.zero_eq, CharP.cast_eq_zero, mul_zero, zero_div, Complex.exp_zero, pow_zero] | case zero
⊢ cexp (↑π * I * ↑0 / 2) = I ^ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
⊢ cexp (↑π * I * ↑0 / 2) = I ^ 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | ExpToNatPowersOfI | [370, 1] | [383, 14] | simp_rw [Nat.cast_succ] | case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
⊢ cexp (↑π * I * ↑(K + 1) / 2) = I ^ (K + 1) | case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
⊢ cexp (↑π * I * (↑K + 1) / 2) = I ^ (K + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
⊢ cexp (↑π * I * ↑(K + 1) / 2) = I ^ (K + 1)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | ExpToNatPowersOfI | [370, 1] | [383, 14] | have h₀ : ↑π * I * (↑K + 1) / 2 = ↑π * I * ↑K / 2 + ↑(π / 2) * I := by
simp only [ofReal_div, ofReal_ofNat]
ring_nf | case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
⊢ cexp (↑π * I * (↑K + 1) / 2) = I ^ (K + 1) | case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
h₀ : ↑π * I * (↑K + 1) / 2 = ↑π * I * ↑K / 2 + ↑(π / 2) * I
⊢ cexp (↑π * I * (↑K + 1) / 2) = I ^ (K + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
⊢ cexp (↑π * I * (↑K + 1) / 2) = I ^ (K + 1)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | ExpToNatPowersOfI | [370, 1] | [383, 14] | rw [h₀] | case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
h₀ : ↑π * I * (↑K + 1) / 2 = ↑π * I * ↑K / 2 + ↑(π / 2) * I
⊢ cexp (↑π * I * (↑K + 1) / 2) = I ^ (K + 1) | case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
h₀ : ↑π * I * (↑K + 1) / 2 = ↑π * I * ↑K / 2 + ↑(π / 2) * I
⊢ cexp (↑π * I * ↑K / 2 + ↑(π / 2) * I) = I ^ (K + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
h₀ : ↑π * I * (↑K + 1) / 2 = ↑π * I * ↑K / 2 + ↑(π / 2) * I
⊢ cexp (↑π * I * (↑K + 1) / 2) = I ^ (K + 1)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | ExpToNatPowersOfI | [370, 1] | [383, 14] | clear h₀ | case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
h₀ : ↑π * I * (↑K + 1) / 2 = ↑π * I * ↑K / 2 + ↑(π / 2) * I
⊢ cexp (↑π * I * ↑K / 2 + ↑(π / 2) * I) = I ^ (K + 1) | case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
⊢ cexp (↑π * I * ↑K / 2 + ↑(π / 2) * I) = I ^ (K + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
h₀ : ↑π * I * (↑K + 1) / 2 = ↑π * I * ↑K / 2 + ↑(π / 2) * I
⊢ cexp (↑π * I * ↑K / 2 + ↑(π / 2) * I) = I ^ (K + 1)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | ExpToNatPowersOfI | [370, 1] | [383, 14] | rw [Complex.exp_add, Kih, ExpPiMulIHalf] | case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
⊢ cexp (↑π * I * ↑K / 2 + ↑(π / 2) * I) = I ^ (K + 1) | case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
⊢ I ^ K * I = I ^ (K + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
⊢ cexp (↑π * I * ↑K / 2 + ↑(π / 2) * I) = I ^ (K + 1)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | ExpToNatPowersOfI | [370, 1] | [383, 14] | have h₂ := zpow_add₀ I_ne_zero K 1 | case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
⊢ I ^ K * I = I ^ (K + 1) | case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
h₂ : I ^ (↑K + 1) = I ^ ↑K * I ^ 1
⊢ I ^ K * I = I ^ (K + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
⊢ I ^ K * I = I ^ (K + 1)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | ExpToNatPowersOfI | [370, 1] | [383, 14] | simp only [zpow_natCast, zpow_one] at h₂ | case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
h₂ : I ^ (↑K + 1) = I ^ ↑K * I ^ 1
⊢ I ^ K * I = I ^ (K + 1) | case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
h₂ : I ^ (↑K + 1) = I ^ K * I
⊢ I ^ K * I = I ^ (K + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
h₂ : I ^ (↑K + 1) = I ^ ↑K * I ^ 1
⊢ I ^ K * I = I ^ (K + 1)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | ExpToNatPowersOfI | [370, 1] | [383, 14] | rw [←h₂] | case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
h₂ : I ^ (↑K + 1) = I ^ K * I
⊢ I ^ K * I = I ^ (K + 1) | case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
h₂ : I ^ (↑K + 1) = I ^ K * I
⊢ I ^ (↑K + 1) = I ^ (K + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
h₂ : I ^ (↑K + 1) = I ^ K * I
⊢ I ^ K * I = I ^ (K + 1)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | ExpToNatPowersOfI | [370, 1] | [383, 14] | exact rfl | case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
h₂ : I ^ (↑K + 1) = I ^ K * I
⊢ I ^ (↑K + 1) = I ^ (K + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
h₂ : I ^ (↑K + 1) = I ^ K * I
⊢ I ^ (↑K + 1) = I ^ (K + 1)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | ExpToNatPowersOfI | [370, 1] | [383, 14] | simp only [ofReal_div, ofReal_ofNat] | K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
⊢ ↑π * I * (↑K + 1) / 2 = ↑π * I * ↑K / 2 + ↑(π / 2) * I | K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
⊢ ↑π * I * (↑K + 1) / 2 = ↑π * I * ↑K / 2 + ↑π / 2 * I | Please generate a tactic in lean4 to solve the state.
STATE:
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
⊢ ↑π * I * (↑K + 1) / 2 = ↑π * I * ↑K / 2 + ↑(π / 2) * I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | ExpToNatPowersOfI | [370, 1] | [383, 14] | ring_nf | K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
⊢ ↑π * I * (↑K + 1) / 2 = ↑π * I * ↑K / 2 + ↑π / 2 * I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
K : ℕ
Kih : cexp (↑π * I * ↑K / 2) = I ^ K
⊢ ↑π * I * (↑K + 1) / 2 = ↑π * I * ↑K / 2 + ↑π / 2 * I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesNEqualsExpSum | [385, 1] | [389, 48] | rw [←RuesDiffM0EqualsRues, RuesDiffEqualsExpSum] | n : ℕ+
z : ℂ
⊢ Rues n z = (∑ m ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑m / ↑↑n) * I))) / ↑↑n | n : ℕ+
z : ℂ
⊢ (∑ k₀ ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑k₀ / ↑↑n) * I) + ↑0 * 2 * ↑π * (↑k₀ / ↑↑n) * I)) / ↑↑n =
(∑ m ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑m / ↑↑n) * I))) / ↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
z : ℂ
⊢ Rues n z = (∑ m ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑m / ↑↑n) * I))) / ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesNEqualsExpSum | [385, 1] | [389, 48] | congr | n : ℕ+
z : ℂ
⊢ (∑ k₀ ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑k₀ / ↑↑n) * I) + ↑0 * 2 * ↑π * (↑k₀ / ↑↑n) * I)) / ↑↑n =
(∑ m ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑m / ↑↑n) * I))) / ↑↑n | case e_a.e_f
n : ℕ+
z : ℂ
⊢ (fun k₀ => cexp (z * cexp (2 * ↑π * (↑k₀ / ↑↑n) * I) + ↑0 * 2 * ↑π * (↑k₀ / ↑↑n) * I)) = fun m =>
cexp (z * cexp (2 * ↑π * (↑m / ↑↑n) * I)) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
z : ℂ
⊢ (∑ k₀ ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑k₀ / ↑↑n) * I) + ↑0 * 2 * ↑π * (↑k₀ / ↑↑n) * I)) / ↑↑n =
(∑ m ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑m / ↑↑n) * I))) / ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesNEqualsExpSum | [385, 1] | [389, 48] | ext1 k | case e_a.e_f
n : ℕ+
z : ℂ
⊢ (fun k₀ => cexp (z * cexp (2 * ↑π * (↑k₀ / ↑↑n) * I) + ↑0 * 2 * ↑π * (↑k₀ / ↑↑n) * I)) = fun m =>
cexp (z * cexp (2 * ↑π * (↑m / ↑↑n) * I)) | case e_a.e_f.h
n : ℕ+
z : ℂ
k : ℕ
⊢ cexp (z * cexp (2 * ↑π * (↑k / ↑↑n) * I) + ↑0 * 2 * ↑π * (↑k / ↑↑n) * I) = cexp (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f
n : ℕ+
z : ℂ
⊢ (fun k₀ => cexp (z * cexp (2 * ↑π * (↑k₀ / ↑↑n) * I) + ↑0 * 2 * ↑π * (↑k₀ / ↑↑n) * I)) = fun m =>
cexp (z * cexp (2 * ↑π * (↑m / ↑↑n) * I))
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesNEqualsExpSum | [385, 1] | [389, 48] | simp only [Int.cast_zero, zero_mul, add_zero] | case e_a.e_f.h
n : ℕ+
z : ℂ
k : ℕ
⊢ cexp (z * cexp (2 * ↑π * (↑k / ↑↑n) * I) + ↑0 * 2 * ↑π * (↑k / ↑↑n) * I) = cexp (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_f.h
n : ℕ+
z : ℂ
k : ℕ
⊢ cexp (z * cexp (2 * ↑π * (↑k / ↑↑n) * I) + ↑0 * 2 * ↑π * (↑k / ↑↑n) * I) = cexp (z * cexp (2 * ↑π * (↑k / ↑↑n) * I))
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesZ0Equals1 | [391, 1] | [394, 65] | rw [RuesNEqualsExpSum] | n : ℕ+
⊢ Rues n 0 = 1 | n : ℕ+
⊢ (∑ m ∈ range ↑n, cexp (0 * cexp (2 * ↑π * (↑m / ↑↑n) * I))) / ↑↑n = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
⊢ Rues n 0 = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesZ0Equals1 | [391, 1] | [394, 65] | simp only [zero_mul, Complex.exp_zero, sum_const, card_range, nsmul_eq_mul, mul_one, ne_eq,
Nat.cast_eq_zero, PNat.ne_zero, not_false_eq_true, div_self] | n : ℕ+
⊢ (∑ m ∈ range ↑n, cexp (0 * cexp (2 * ↑π * (↑m / ↑↑n) * I))) / ↑↑n = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
⊢ (∑ m ∈ range ↑n, cexp (0 * cexp (2 * ↑π * (↑m / ↑↑n) * I))) / ↑↑n = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN1EqualsExp | [396, 1] | [399, 36] | ext1 z | ⊢ Rues 1 = cexp | case h
z : ℂ
⊢ Rues 1 z = cexp z | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Rues 1 = cexp
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN1EqualsExp | [396, 1] | [399, 36] | rw [Rues, ExpTsumForm] | case h
z : ℂ
⊢ Rues 1 z = cexp z | case h
z : ℂ
⊢ ∑' (k : ℕ), z ^ (↑1 * k) / ↑(↑1 * k).factorial = ∑' (k : ℕ), z ^ k / ↑k.factorial | Please generate a tactic in lean4 to solve the state.
STATE:
case h
z : ℂ
⊢ Rues 1 z = cexp z
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN1EqualsExp | [396, 1] | [399, 36] | simp only [PNat.one_coe, one_mul] | case h
z : ℂ
⊢ ∑' (k : ℕ), z ^ (↑1 * k) / ↑(↑1 * k).factorial = ∑' (k : ℕ), z ^ k / ↑k.factorial | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
z : ℂ
⊢ ∑' (k : ℕ), z ^ (↑1 * k) / ↑(↑1 * k).factorial = ∑' (k : ℕ), z ^ k / ↑k.factorial
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN2EqualsCosh | [401, 1] | [418, 8] | ext1 z | ⊢ Rues 2 = Complex.cosh | case h
z : ℂ
⊢ Rues 2 z = z.cosh | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Rues 2 = Complex.cosh
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN2EqualsCosh | [401, 1] | [418, 8] | rw [RuesNEqualsExpSum, Complex.cosh] | case h
z : ℂ
⊢ Rues 2 z = z.cosh | case h
z : ℂ
⊢ (∑ m ∈ range ↑2, cexp (z * cexp (2 * ↑π * (↑m / ↑↑2) * I))) / ↑↑2 = (cexp z + cexp (-z)) / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
z : ℂ
⊢ Rues 2 z = z.cosh
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN2EqualsCosh | [401, 1] | [418, 8] | have h₀ : range (2 : ℕ+) = {0, 1} := by
rfl | case h
z : ℂ
⊢ (∑ m ∈ range ↑2, cexp (z * cexp (2 * ↑π * (↑m / ↑↑2) * I))) / ↑↑2 = (cexp z + cexp (-z)) / 2 | case h
z : ℂ
h₀ : range ↑2 = {0, 1}
⊢ (∑ m ∈ range ↑2, cexp (z * cexp (2 * ↑π * (↑m / ↑↑2) * I))) / ↑↑2 = (cexp z + cexp (-z)) / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
z : ℂ
⊢ (∑ m ∈ range ↑2, cexp (z * cexp (2 * ↑π * (↑m / ↑↑2) * I))) / ↑↑2 = (cexp z + cexp (-z)) / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN2EqualsCosh | [401, 1] | [418, 8] | simp_rw [h₀, Finset.sum] | case h
z : ℂ
h₀ : range ↑2 = {0, 1}
⊢ (∑ m ∈ range ↑2, cexp (z * cexp (2 * ↑π * (↑m / ↑↑2) * I))) / ↑↑2 = (cexp z + cexp (-z)) / 2 | case h
z : ℂ
h₀ : range ↑2 = {0, 1}
⊢ (Multiset.map (fun x => cexp (z * cexp (2 * ↑π * (↑x / ↑↑2) * I))) {0, 1}.val).sum / ↑↑2 = (cexp z + cexp (-z)) / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
z : ℂ
h₀ : range ↑2 = {0, 1}
⊢ (∑ m ∈ range ↑2, cexp (z * cexp (2 * ↑π * (↑m / ↑↑2) * I))) / ↑↑2 = (cexp z + cexp (-z)) / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN2EqualsCosh | [401, 1] | [418, 8] | simp only [mem_singleton, insert_val, singleton_val, Multiset.mem_singleton, not_false_eq_true,
Multiset.ndinsert_of_not_mem, Multiset.map_cons, CharP.cast_eq_zero, zero_div, mul_zero,
zero_mul, Complex.exp_zero, mul_one, Multiset.map_singleton, Nat.cast_one, one_div,
Multiset.sum_cons, Multiset.sum_singleton] | case h
z : ℂ
h₀ : range ↑2 = {0, 1}
⊢ (Multiset.map (fun x => cexp (z * cexp (2 * ↑π * (↑x / ↑↑2) * I))) {0, 1}.val).sum / ↑↑2 = (cexp z + cexp (-z)) / 2 | case h
z : ℂ
h₀ : range ↑2 = {0, 1}
⊢ (cexp z + cexp (z * cexp (2 * ↑π * (↑↑2)⁻¹ * I))) / ↑↑2 = (cexp z + cexp (-z)) / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
z : ℂ
h₀ : range ↑2 = {0, 1}
⊢ (Multiset.map (fun x => cexp (z * cexp (2 * ↑π * (↑x / ↑↑2) * I))) {0, 1}.val).sum / ↑↑2 = (cexp z + cexp (-z)) / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN2EqualsCosh | [401, 1] | [418, 8] | have h₁ : cexp (2 * ↑π * (↑↑(2 : ℕ+))⁻¹ * I) = -1 := by
have h₂ : 2 * (π : ℂ) * (↑↑(2 : ℕ+))⁻¹ = π := by
field_simp
rw [h₂]
simp only [exp_pi_mul_I] | case h
z : ℂ
h₀ : range ↑2 = {0, 1}
⊢ (cexp z + cexp (z * cexp (2 * ↑π * (↑↑2)⁻¹ * I))) / ↑↑2 = (cexp z + cexp (-z)) / 2 | case h
z : ℂ
h₀ : range ↑2 = {0, 1}
h₁ : cexp (2 * ↑π * (↑↑2)⁻¹ * I) = -1
⊢ (cexp z + cexp (z * cexp (2 * ↑π * (↑↑2)⁻¹ * I))) / ↑↑2 = (cexp z + cexp (-z)) / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
z : ℂ
h₀ : range ↑2 = {0, 1}
⊢ (cexp z + cexp (z * cexp (2 * ↑π * (↑↑2)⁻¹ * I))) / ↑↑2 = (cexp z + cexp (-z)) / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN2EqualsCosh | [401, 1] | [418, 8] | simp_rw [h₁] | case h
z : ℂ
h₀ : range ↑2 = {0, 1}
h₁ : cexp (2 * ↑π * (↑↑2)⁻¹ * I) = -1
⊢ (cexp z + cexp (z * cexp (2 * ↑π * (↑↑2)⁻¹ * I))) / ↑↑2 = (cexp z + cexp (-z)) / 2 | case h
z : ℂ
h₀ : range ↑2 = {0, 1}
h₁ : cexp (2 * ↑π * (↑↑2)⁻¹ * I) = -1
⊢ (cexp z + cexp (z * -1)) / ↑↑2 = (cexp z + cexp (-z)) / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
z : ℂ
h₀ : range ↑2 = {0, 1}
h₁ : cexp (2 * ↑π * (↑↑2)⁻¹ * I) = -1
⊢ (cexp z + cexp (z * cexp (2 * ↑π * (↑↑2)⁻¹ * I))) / ↑↑2 = (cexp z + cexp (-z)) / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN2EqualsCosh | [401, 1] | [418, 8] | simp only [mul_neg, mul_one] | case h
z : ℂ
h₀ : range ↑2 = {0, 1}
h₁ : cexp (2 * ↑π * (↑↑2)⁻¹ * I) = -1
⊢ (cexp z + cexp (z * -1)) / ↑↑2 = (cexp z + cexp (-z)) / 2 | case h
z : ℂ
h₀ : range ↑2 = {0, 1}
h₁ : cexp (2 * ↑π * (↑↑2)⁻¹ * I) = -1
⊢ (cexp z + cexp (-z)) / ↑↑2 = (cexp z + cexp (-z)) / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
z : ℂ
h₀ : range ↑2 = {0, 1}
h₁ : cexp (2 * ↑π * (↑↑2)⁻¹ * I) = -1
⊢ (cexp z + cexp (z * -1)) / ↑↑2 = (cexp z + cexp (-z)) / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN2EqualsCosh | [401, 1] | [418, 8] | congr | case h
z : ℂ
h₀ : range ↑2 = {0, 1}
h₁ : cexp (2 * ↑π * (↑↑2)⁻¹ * I) = -1
⊢ (cexp z + cexp (-z)) / ↑↑2 = (cexp z + cexp (-z)) / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
z : ℂ
h₀ : range ↑2 = {0, 1}
h₁ : cexp (2 * ↑π * (↑↑2)⁻¹ * I) = -1
⊢ (cexp z + cexp (-z)) / ↑↑2 = (cexp z + cexp (-z)) / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN2EqualsCosh | [401, 1] | [418, 8] | rfl | z : ℂ
⊢ range ↑2 = {0, 1} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ range ↑2 = {0, 1}
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN2EqualsCosh | [401, 1] | [418, 8] | have h₂ : 2 * (π : ℂ) * (↑↑(2 : ℕ+))⁻¹ = π := by
field_simp | z : ℂ
h₀ : range ↑2 = {0, 1}
⊢ cexp (2 * ↑π * (↑↑2)⁻¹ * I) = -1 | z : ℂ
h₀ : range ↑2 = {0, 1}
h₂ : 2 * ↑π * (↑↑2)⁻¹ = ↑π
⊢ cexp (2 * ↑π * (↑↑2)⁻¹ * I) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₀ : range ↑2 = {0, 1}
⊢ cexp (2 * ↑π * (↑↑2)⁻¹ * I) = -1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN2EqualsCosh | [401, 1] | [418, 8] | rw [h₂] | z : ℂ
h₀ : range ↑2 = {0, 1}
h₂ : 2 * ↑π * (↑↑2)⁻¹ = ↑π
⊢ cexp (2 * ↑π * (↑↑2)⁻¹ * I) = -1 | z : ℂ
h₀ : range ↑2 = {0, 1}
h₂ : 2 * ↑π * (↑↑2)⁻¹ = ↑π
⊢ cexp (↑π * I) = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₀ : range ↑2 = {0, 1}
h₂ : 2 * ↑π * (↑↑2)⁻¹ = ↑π
⊢ cexp (2 * ↑π * (↑↑2)⁻¹ * I) = -1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN2EqualsCosh | [401, 1] | [418, 8] | simp only [exp_pi_mul_I] | z : ℂ
h₀ : range ↑2 = {0, 1}
h₂ : 2 * ↑π * (↑↑2)⁻¹ = ↑π
⊢ cexp (↑π * I) = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₀ : range ↑2 = {0, 1}
h₂ : 2 * ↑π * (↑↑2)⁻¹ = ↑π
⊢ cexp (↑π * I) = -1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN2EqualsCosh | [401, 1] | [418, 8] | field_simp | z : ℂ
h₀ : range ↑2 = {0, 1}
⊢ 2 * ↑π * (↑↑2)⁻¹ = ↑π | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₀ : range ↑2 = {0, 1}
⊢ 2 * ↑π * (↑↑2)⁻¹ = ↑π
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | rw [RuesNEqualsExpSum, Complex.cosh, Complex.cosh] | z : ℂ
⊢ Rues 4 z = (z / (1 + I)).cosh * (z / (1 - I)).cosh | z : ℂ
⊢ (∑ m ∈ range ↑4, cexp (z * cexp (2 * ↑π * (↑m / ↑↑4) * I))) / ↑↑4 =
(cexp (z / (1 + I)) + cexp (-(z / (1 + I)))) / 2 * ((cexp (z / (1 - I)) + cexp (-(z / (1 - I)))) / 2) | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ Rues 4 z = (z / (1 + I)).cosh * (z / (1 - I)).cosh
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | have h₀ : (4 : ℕ+) = (4 : ℕ) := by
rfl | z : ℂ
⊢ (∑ m ∈ range ↑4, cexp (z * cexp (2 * ↑π * (↑m / ↑↑4) * I))) / ↑↑4 =
(cexp (z / (1 + I)) + cexp (-(z / (1 + I)))) / 2 * ((cexp (z / (1 - I)) + cexp (-(z / (1 - I)))) / 2) | z : ℂ
h₀ : ↑4 = 4
⊢ (∑ m ∈ range ↑4, cexp (z * cexp (2 * ↑π * (↑m / ↑↑4) * I))) / ↑↑4 =
(cexp (z / (1 + I)) + cexp (-(z / (1 + I)))) / 2 * ((cexp (z / (1 - I)) + cexp (-(z / (1 - I)))) / 2) | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ (∑ m ∈ range ↑4, cexp (z * cexp (2 * ↑π * (↑m / ↑↑4) * I))) / ↑↑4 =
(cexp (z / (1 + I)) + cexp (-(z / (1 + I)))) / 2 * ((cexp (z / (1 - I)) + cexp (-(z / (1 - I)))) / 2)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | simp_rw [h₀, Finset.sum] | z : ℂ
h₀ : ↑4 = 4
⊢ (∑ m ∈ range ↑4, cexp (z * cexp (2 * ↑π * (↑m / ↑↑4) * I))) / ↑↑4 =
(cexp (z / (1 + I)) + cexp (-(z / (1 + I)))) / 2 * ((cexp (z / (1 - I)) + cexp (-(z / (1 - I)))) / 2) | z : ℂ
h₀ : ↑4 = 4
⊢ (Multiset.map (fun x => cexp (z * cexp (2 * ↑π * (↑x / ↑4) * I))) (range 4).val).sum / ↑4 =
(cexp (z / (1 + I)) + cexp (-(z / (1 + I)))) / 2 * ((cexp (z / (1 - I)) + cexp (-(z / (1 - I)))) / 2) | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₀ : ↑4 = 4
⊢ (∑ m ∈ range ↑4, cexp (z * cexp (2 * ↑π * (↑m / ↑↑4) * I))) / ↑↑4 =
(cexp (z / (1 + I)) + cexp (-(z / (1 + I)))) / 2 * ((cexp (z / (1 - I)) + cexp (-(z / (1 - I)))) / 2)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | clear h₀ | z : ℂ
h₀ : ↑4 = 4
⊢ (Multiset.map (fun x => cexp (z * cexp (2 * ↑π * (↑x / ↑4) * I))) (range 4).val).sum / ↑4 =
(cexp (z / (1 + I)) + cexp (-(z / (1 + I)))) / 2 * ((cexp (z / (1 - I)) + cexp (-(z / (1 - I)))) / 2) | z : ℂ
⊢ (Multiset.map (fun x => cexp (z * cexp (2 * ↑π * (↑x / ↑4) * I))) (range 4).val).sum / ↑4 =
(cexp (z / (1 + I)) + cexp (-(z / (1 + I)))) / 2 * ((cexp (z / (1 - I)) + cexp (-(z / (1 - I)))) / 2) | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₀ : ↑4 = 4
⊢ (Multiset.map (fun x => cexp (z * cexp (2 * ↑π * (↑x / ↑4) * I))) (range 4).val).sum / ↑4 =
(cexp (z / (1 + I)) + cexp (-(z / (1 + I)))) / 2 * ((cexp (z / (1 - I)) + cexp (-(z / (1 - I)))) / 2)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | simp only [range_val, Multiset.range_succ, Multiset.range_zero, Multiset.cons_zero,
Nat.cast_ofNat, Multiset.map_cons, Nat.cast_one, one_div, Multiset.map_singleton,
CharP.cast_eq_zero, zero_div, mul_zero, zero_mul, Complex.exp_zero, mul_one, Multiset.sum_cons,
Multiset.sum_singleton] | z : ℂ
⊢ (Multiset.map (fun x => cexp (z * cexp (2 * ↑π * (↑x / ↑4) * I))) (range 4).val).sum / ↑4 =
(cexp (z / (1 + I)) + cexp (-(z / (1 + I)))) / 2 * ((cexp (z / (1 - I)) + cexp (-(z / (1 - I)))) / 2) | z : ℂ
⊢ (cexp (z * cexp (2 * ↑π * (3 / 4) * I)) +
(cexp (z * cexp (2 * ↑π * (2 / 4) * I)) + (cexp (z * cexp (2 * ↑π * 4⁻¹ * I)) + cexp z))) /
4 =
(cexp (z / (1 + I)) + cexp (-(z / (1 + I)))) / 2 * ((cexp (z / (1 - I)) + cexp (-(z / (1 - I)))) / 2) | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ (Multiset.map (fun x => cexp (z * cexp (2 * ↑π * (↑x / ↑4) * I))) (range 4).val).sum / ↑4 =
(cexp (z / (1 + I)) + cexp (-(z / (1 + I)))) / 2 * ((cexp (z / (1 - I)) + cexp (-(z / (1 - I)))) / 2)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | ring_nf | z : ℂ
⊢ (cexp (z * cexp (2 * ↑π * (3 / 4) * I)) +
(cexp (z * cexp (2 * ↑π * (2 / 4) * I)) + (cexp (z * cexp (2 * ↑π * 4⁻¹ * I)) + cexp z))) /
4 =
(cexp (z / (1 + I)) + cexp (-(z / (1 + I)))) / 2 * ((cexp (z / (1 - I)) + cexp (-(z / (1 - I)))) / 2) | z : ℂ
⊢ cexp (z * cexp (↑π * I * (3 / 2))) * (1 / 4) + cexp (z * cexp (↑π * I)) * (1 / 4) +
cexp (z * cexp (↑π * I * (1 / 2))) * (1 / 4) +
cexp z * (1 / 4) =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * (1 / 4) + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * (1 / 4) +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * (1 / 4) +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * (1 / 4) | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ (cexp (z * cexp (2 * ↑π * (3 / 4) * I)) +
(cexp (z * cexp (2 * ↑π * (2 / 4) * I)) + (cexp (z * cexp (2 * ↑π * 4⁻¹ * I)) + cexp z))) /
4 =
(cexp (z / (1 + I)) + cexp (-(z / (1 + I)))) / 2 * ((cexp (z / (1 - I)) + cexp (-(z / (1 - I)))) / 2)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | simp only [one_div, exp_pi_mul_I, mul_neg, mul_one] | z : ℂ
⊢ cexp (z * cexp (↑π * I * (3 / 2))) * (1 / 4) + cexp (z * cexp (↑π * I)) * (1 / 4) +
cexp (z * cexp (↑π * I * (1 / 2))) * (1 / 4) +
cexp z * (1 / 4) =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * (1 / 4) + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * (1 / 4) +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * (1 / 4) +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * (1 / 4) | z : ℂ
⊢ cexp (z * cexp (↑π * I * (3 / 2))) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * cexp (↑π * I * 2⁻¹)) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ cexp (z * cexp (↑π * I * (3 / 2))) * (1 / 4) + cexp (z * cexp (↑π * I)) * (1 / 4) +
cexp (z * cexp (↑π * I * (1 / 2))) * (1 / 4) +
cexp z * (1 / 4) =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * (1 / 4) + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * (1 / 4) +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * (1 / 4) +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * (1 / 4)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | have h₁ : cexp (↑π * I * (3 / 2)) = -I := by
have h₁b := ExpToNatPowersOfI 3
simp only [Nat.cast_ofNat] at h₁b
have h₁b₁ : ↑π * I * 3 / 2 = ↑π * I * (3 / 2) := by
ring
rw [h₁b₁] at h₁b
rw [h₁b]
clear h₁b h₁b₁
have h₅ : I ^ (3 : ℕ) = I ^ (3 : ℤ) := by
exact rfl
rw [h₅]
clear h₅
have h₆ : (3 : ℤ) = 2 + 1 := by
exact rfl
rw [h₆]
clear h₆
rw [zpow_add₀ I_ne_zero]
have h₇ : (2 : ℤ) = 1 + 1 := by
exact rfl
rw [h₇]
clear h₇
rw [zpow_add₀ I_ne_zero]
simp only [zpow_one, I_mul_I, neg_mul, one_mul] | z : ℂ
⊢ cexp (z * cexp (↑π * I * (3 / 2))) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * cexp (↑π * I * 2⁻¹)) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | z : ℂ
h₁ : cexp (↑π * I * (3 / 2)) = -I
⊢ cexp (z * cexp (↑π * I * (3 / 2))) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * cexp (↑π * I * 2⁻¹)) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ cexp (z * cexp (↑π * I * (3 / 2))) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * cexp (↑π * I * 2⁻¹)) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | rw [h₁] | z : ℂ
h₁ : cexp (↑π * I * (3 / 2)) = -I
⊢ cexp (z * cexp (↑π * I * (3 / 2))) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * cexp (↑π * I * 2⁻¹)) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | z : ℂ
h₁ : cexp (↑π * I * (3 / 2)) = -I
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * cexp (↑π * I * 2⁻¹)) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₁ : cexp (↑π * I * (3 / 2)) = -I
⊢ cexp (z * cexp (↑π * I * (3 / 2))) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * cexp (↑π * I * 2⁻¹)) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | clear h₁ | z : ℂ
h₁ : cexp (↑π * I * (3 / 2)) = -I
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * cexp (↑π * I * 2⁻¹)) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | z : ℂ
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * cexp (↑π * I * 2⁻¹)) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₁ : cexp (↑π * I * (3 / 2)) = -I
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * cexp (↑π * I * 2⁻¹)) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | have h₂ : cexp (↑π * I * 2⁻¹) = I := by
nth_rw 2 [←ExpPiMulIHalf]
congr 1
simp only [ofReal_div, ofReal_ofNat]
ring_nf | z : ℂ
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * cexp (↑π * I * 2⁻¹)) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | z : ℂ
h₂ : cexp (↑π * I * 2⁻¹) = I
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * cexp (↑π * I * 2⁻¹)) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * cexp (↑π * I * 2⁻¹)) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | rw [h₂] | z : ℂ
h₂ : cexp (↑π * I * 2⁻¹) = I
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * cexp (↑π * I * 2⁻¹)) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | z : ℂ
h₂ : cexp (↑π * I * 2⁻¹) = I
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₂ : cexp (↑π * I * 2⁻¹) = I
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * cexp (↑π * I * 2⁻¹)) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | clear h₂ | z : ℂ
h₂ : cexp (↑π * I * 2⁻¹) = I
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | z : ℂ
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₂ : cexp (↑π * I * 2⁻¹) = I
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | have h₃ : (1 + I)⁻¹ = (1 - I) / 2 := by
rw [Inv.inv, Complex.instInv, normSq]
simp only [MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, ofReal_inv, ofReal_add, ofReal_mul,
map_add, map_one, conj_I, add_re, one_re, I_re, add_zero, ofReal_one, mul_one, add_im, one_im,
I_im, zero_add]
ring_nf | z : ℂ
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | z : ℂ
h₃ : (1 + I)⁻¹ = (1 - I) / 2
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | rw [h₃] | z : ℂ
h₃ : (1 + I)⁻¹ = (1 - I) / 2
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | z : ℂ
h₃ : (1 + I)⁻¹ = (1 - I) / 2
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * ((1 - I) / 2)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * ((1 - I) / 2)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₃ : (1 + I)⁻¹ = (1 - I) / 2
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * (1 + I)⁻¹) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * (1 + I)⁻¹) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * (1 + I)⁻¹)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | clear h₃ | z : ℂ
h₃ : (1 + I)⁻¹ = (1 - I) / 2
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * ((1 - I) / 2)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * ((1 - I) / 2)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | z : ℂ
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * ((1 - I) / 2)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * ((1 - I) / 2)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₃ : (1 + I)⁻¹ = (1 - I) / 2
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * ((1 - I) / 2)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * ((1 - I) / 2)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | have h₄ : (1 - I)⁻¹ = (1 + I) / 2 := by
rw [Inv.inv, Complex.instInv, normSq]
simp only [MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, ofReal_inv, ofReal_add, ofReal_mul,
map_sub, map_one, conj_I, sub_neg_eq_add, sub_re, one_re, I_re, sub_zero, ofReal_one, mul_one,
sub_im, one_im, I_im, zero_sub, ofReal_neg, mul_neg, neg_neg]
ring_nf | z : ℂ
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * ((1 - I) / 2)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * ((1 - I) / 2)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | z : ℂ
h₄ : (1 - I)⁻¹ = (1 + I) / 2
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * ((1 - I) / 2)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * ((1 - I) / 2)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * ((1 - I) / 2)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * ((1 - I) / 2)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | rw [h₄] | z : ℂ
h₄ : (1 - I)⁻¹ = (1 + I) / 2
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * ((1 - I) / 2)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * ((1 - I) / 2)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ | z : ℂ
h₄ : (1 - I)⁻¹ = (1 + I) / 2
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * ((1 - I) / 2)) * cexp (z * ((1 + I) / 2)) * 4⁻¹ +
cexp (z * ((1 - I) / 2)) * cexp (-(z * ((1 + I) / 2))) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (z * ((1 + I) / 2)) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (-(z * ((1 + I) / 2))) * 4⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₄ : (1 - I)⁻¹ = (1 + I) / 2
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * ((1 - I) / 2)) * cexp (z * (1 - I)⁻¹) * 4⁻¹ + cexp (z * ((1 - I) / 2)) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (z * (1 - I)⁻¹) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (-(z * (1 - I)⁻¹)) * 4⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | clear h₄ | z : ℂ
h₄ : (1 - I)⁻¹ = (1 + I) / 2
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * ((1 - I) / 2)) * cexp (z * ((1 + I) / 2)) * 4⁻¹ +
cexp (z * ((1 - I) / 2)) * cexp (-(z * ((1 + I) / 2))) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (z * ((1 + I) / 2)) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (-(z * ((1 + I) / 2))) * 4⁻¹ | z : ℂ
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * ((1 - I) / 2)) * cexp (z * ((1 + I) / 2)) * 4⁻¹ +
cexp (z * ((1 - I) / 2)) * cexp (-(z * ((1 + I) / 2))) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (z * ((1 + I) / 2)) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (-(z * ((1 + I) / 2))) * 4⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₄ : (1 - I)⁻¹ = (1 + I) / 2
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * ((1 - I) / 2)) * cexp (z * ((1 + I) / 2)) * 4⁻¹ +
cexp (z * ((1 - I) / 2)) * cexp (-(z * ((1 + I) / 2))) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (z * ((1 + I) / 2)) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (-(z * ((1 + I) / 2))) * 4⁻¹
TACTIC:
|
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