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 | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | ring | n k : ℕ+
m : ℤ
x i j : ℕ
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j)
h₁ : ↑↑n ≠ 0
y : ℤ
h₃ : ↑i - ↑j = ↑↑k * y
h₄ : ↑↑k * y < ↑↑k
h₅ : -↑↑k < ↑↑k * y
h₆ : ↑↑k > 0
h₇ : y < 1
⊢ -↑↑k = ↑↑k * -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
x i j : ℕ
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j)
h₁ : ↑↑n ≠ 0
y : ℤ
h₃ : ↑i - ↑j = ↑↑k * y
h₄ : ↑↑k * y < ↑↑k
h₅ : -↑↑k < ↑↑k * y
h₆ : ↑↑k > 0
h₇ : y < 1
⊢ -↑↑k = ↑↑k * -1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | exact (mul_lt_mul_left h₆).mp h₅ | n k : ℕ+
m : ℤ
x i j : ℕ
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j)
h₁ : ↑↑n ≠ 0
y : ℤ
h₃ : ↑i - ↑j = ↑↑k * y
h₄ : ↑↑k * y < ↑↑k
h₅ : ↑↑k * -1 < ↑↑k * y
h₆ : ↑↑k > 0
h₇ : y < 1
⊢ -1 < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
x i j : ℕ
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j)
h₁ : ↑↑n ≠ 0
y : ℤ
h₃ : ↑i - ↑j = ↑↑k * y
h₄ : ↑↑k * y < ↑↑k
h₅ : ↑↑k * -1 < ↑↑k * y
h₆ : ↑↑k > 0
h₇ : y < 1
⊢ -1 < y
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | linarith | n k : ℕ+
m : ℤ
x i j : ℕ
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j)
h₁ : ↑↑n ≠ 0
y : ℤ
h₃ : ↑i - ↑j = ↑↑k * y
h₄ : ↑↑k * y < ↑↑k
h₅ : ↑↑k * -1 < ↑↑k * y
h₆ : ↑↑k > 0
h₇ : y < 1
h₈ : -1 < y
⊢ y = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
x i j : ℕ
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j)
h₁ : ↑↑n ≠ 0
y : ℤ
h₃ : ↑i - ↑j = ↑↑k * y
h₄ : ↑↑k * y < ↑↑k
h₅ : ↑↑k * -1 < ↑↑k * y
h₆ : ↑↑k > 0
h₇ : y < 1
h₈ : -1 < y
⊢ y = 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | intros h₀ | case e_f.h.e_c.mp
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
⊢ ↑↑n ∣ ↑x + m → ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) | case e_f.h.e_c.mp
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
h₀ : ↑↑n ∣ ↑x + m
⊢ ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_c.mp
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
⊢ ↑↑n ∣ ↑x + m → ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | obtain ⟨w, h₁⟩ := h₀ | case e_f.h.e_c.mp
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
h₀ : ↑↑n ∣ ↑x + m
⊢ ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) | case e_f.h.e_c.mp.intro
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
⊢ ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_c.mp
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
h₀ : ↑↑n ∣ ↑x + m
⊢ ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + (↑↑n * ↑i) := by
intros i
ring_nf | case e_f.h.e_c.mp.intro
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
⊢ ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) | case e_f.h.e_c.mp.intro
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_c.mp.intro
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
⊢ ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | simp_rw [h₂, h₁] | case e_f.h.e_c.mp.intro
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) | case e_f.h.e_c.mp.intro
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑i | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_c.mp.intro
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | use ((-w) % k).toNat | case e_f.h.e_c.mp.intro
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑i | case h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ (-w % ↑↑k).toNat < ↑k ∧ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑(-w % ↑↑k).toNat | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_c.mp.intro
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑i
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | constructor | case h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ (-w % ↑↑k).toNat < ↑k ∧ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑(-w % ↑↑k).toNat | case h.left
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ (-w % ↑↑k).toNat < ↑k
case h.right
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑(-w % ↑↑k).toNat | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ (-w % ↑↑k).toNat < ↑k ∧ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑(-w % ↑↑k).toNat
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | intros i | n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
⊢ ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i | n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
i : ℕ
⊢ ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
⊢ ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | ring_nf | n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
i : ℕ
⊢ ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
i : ℕ
⊢ ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | refine (Int.toNat_lt' ?h.left.hn).mpr ?h.left.a | case h.left
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ (-w % ↑↑k).toNat < ↑k | case h.left.hn
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ ↑k ≠ 0
case h.left.a
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ -w % ↑↑k < ↑↑k | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ (-w % ↑↑k).toNat < ↑k
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | exact PNat.ne_zero k | case h.left.hn
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ ↑k ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.hn
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ ↑k ≠ 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | refine Int.emod_lt_of_pos (-w) ?h.left.a.H | case h.left.a
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ -w % ↑↑k < ↑↑k | case h.left.a.H
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ 0 < ↑↑k | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.a
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ -w % ↑↑k < ↑↑k
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | refine Int.ofNat_pos.mpr ?h.left.a.H.a | case h.left.a.H
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ 0 < ↑↑k | case h.left.a.H.a
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ 0 < ↑k | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.a.H
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ 0 < ↑↑k
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | exact PNat.pos k | case h.left.a.H.a
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ 0 < ↑k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.a.H.a
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ 0 < ↑k
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₃ : ↑(-w % ↑↑k).toNat = (-w % ↑↑k) := by
refine Int.toNat_of_nonneg ?_
refine Int.emod_nonneg (-w) ?_
exact Ne.symm (NeZero.ne' (k : ℤ)) | case h.right
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑(-w % ↑↑k).toNat | case h.right
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
h₃ : ↑(-w % ↑↑k).toNat = -w % ↑↑k
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑(-w % ↑↑k).toNat | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑(-w % ↑↑k).toNat
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | rw [h₃] | case h.right
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
h₃ : ↑(-w % ↑↑k).toNat = -w % ↑↑k
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑(-w % ↑↑k).toNat | case h.right
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
h₃ : ↑(-w % ↑↑k).toNat = -w % ↑↑k
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
h₃ : ↑(-w % ↑↑k).toNat = -w % ↑↑k
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * ↑(-w % ↑↑k).toNat
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | clear h₁ h₂ h₃ m z x | case h.right
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
h₃ : ↑(-w % ↑↑k).toNat = -w % ↑↑k
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k) | case h.right
n k : ℕ+
w : ℤ
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
h₃ : ↑(-w % ↑↑k).toNat = -w % ↑↑k
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | suffices h₀ : ↑↑k ∣ w + (-w % ↑↑k) | case h.right
n k : ℕ+
w : ℤ
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k) | case h.right
n k : ℕ+
w : ℤ
h₀ : ↑↑k ∣ w + -w % ↑↑k
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k)
case h₀
n k : ℕ+
w : ℤ
⊢ ↑↑k ∣ w + -w % ↑↑k | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
n k : ℕ+
w : ℤ
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | refine Int.toNat_of_nonneg ?_ | n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ ↑(-w % ↑↑k).toNat = -w % ↑↑k | n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ 0 ≤ -w % ↑↑k | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ ↑(-w % ↑↑k).toNat = -w % ↑↑k
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | refine Int.emod_nonneg (-w) ?_ | n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ 0 ≤ -w % ↑↑k | n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ ↑↑k ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ 0 ≤ -w % ↑↑k
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | exact Ne.symm (NeZero.ne' (k : ℤ)) | n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ ↑↑k ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
w : ℤ
h₁ : ↑x + m = ↑↑n * w
h₂ : ∀ (i : ℕ), ↑x + (↑↑n * ↑i + m) = ↑x + m + ↑↑n * ↑i
⊢ ↑↑k ≠ 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₁ := mul_dvd_mul_left (n : ℤ) h₀ | case h.right
n k : ℕ+
w : ℤ
h₀ : ↑↑k ∣ w + -w % ↑↑k
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k) | case h.right
n k : ℕ+
w : ℤ
h₀ : ↑↑k ∣ w + -w % ↑↑k
h₁ : ↑↑n * ↑↑k ∣ ↑↑n * (w + -w % ↑↑k)
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
n k : ℕ+
w : ℤ
h₀ : ↑↑k ∣ w + -w % ↑↑k
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | ring_nf at * | case h.right
n k : ℕ+
w : ℤ
h₀ : ↑↑k ∣ w + -w % ↑↑k
h₁ : ↑↑n * ↑↑k ∣ ↑↑n * (w + -w % ↑↑k)
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k) | case h.right
n k : ℕ+
w : ℤ
h₀ : ↑↑k ∣ w + -w % ↑↑k
h₁ : ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k)
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
n k : ℕ+
w : ℤ
h₀ : ↑↑k ∣ w + -w % ↑↑k
h₁ : ↑↑n * ↑↑k ∣ ↑↑n * (w + -w % ↑↑k)
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | exact h₁ | case h.right
n k : ℕ+
w : ℤ
h₀ : ↑↑k ∣ w + -w % ↑↑k
h₁ : ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k)
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
n k : ℕ+
w : ℤ
h₀ : ↑↑k ∣ w + -w % ↑↑k
h₁ : ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k)
⊢ ↑↑n * ↑↑k ∣ ↑↑n * w + ↑↑n * (-w % ↑↑k)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | refine Int.dvd_of_emod_eq_zero ?h₀.H | case h₀
n k : ℕ+
w : ℤ
⊢ ↑↑k ∣ w + -w % ↑↑k | case h₀.H
n k : ℕ+
w : ℤ
⊢ (w + -w % ↑↑k) % ↑↑k = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀
n k : ℕ+
w : ℤ
⊢ ↑↑k ∣ w + -w % ↑↑k
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₀ : (0 : ℤ) = 0 % k := by
exact rfl | case h₀.H
n k : ℕ+
w : ℤ
⊢ (w + -w % ↑↑k) % ↑↑k = 0 | case h₀.H
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
⊢ (w + -w % ↑↑k) % ↑↑k = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀.H
n k : ℕ+
w : ℤ
⊢ (w + -w % ↑↑k) % ↑↑k = 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | rw [h₀] | case h₀.H
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
⊢ (w + -w % ↑↑k) % ↑↑k = 0 | case h₀.H
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
⊢ (w + -w % ↑↑k) % ↑↑k = 0 % ↑↑k | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀.H
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
⊢ (w + -w % ↑↑k) % ↑↑k = 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | refine Eq.symm (Int.ModEq.eq ?h₀.H.a) | case h₀.H
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
⊢ (w + -w % ↑↑k) % ↑↑k = 0 % ↑↑k | case h₀.H.a
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
⊢ 0 ≡ w + -w % ↑↑k [ZMOD ↑↑k] | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀.H
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
⊢ (w + -w % ↑↑k) % ↑↑k = 0 % ↑↑k
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₁ : -w % ↑↑k ≡ -w [ZMOD ↑↑k] := by
exact Int.mod_modEq (-w) ↑↑k | case h₀.H.a
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
⊢ 0 ≡ w + -w % ↑↑k [ZMOD ↑↑k] | case h₀.H.a
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
h₁ : -w % ↑↑k ≡ -w [ZMOD ↑↑k]
⊢ 0 ≡ w + -w % ↑↑k [ZMOD ↑↑k] | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀.H.a
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
⊢ 0 ≡ w + -w % ↑↑k [ZMOD ↑↑k]
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₂ : w ≡ w [ZMOD ↑↑k] := by exact rfl | case h₀.H.a
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
h₁ : -w % ↑↑k ≡ -w [ZMOD ↑↑k]
⊢ 0 ≡ w + -w % ↑↑k [ZMOD ↑↑k] | case h₀.H.a
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
h₁ : -w % ↑↑k ≡ -w [ZMOD ↑↑k]
h₂ : w ≡ w [ZMOD ↑↑k]
⊢ 0 ≡ w + -w % ↑↑k [ZMOD ↑↑k] | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀.H.a
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
h₁ : -w % ↑↑k ≡ -w [ZMOD ↑↑k]
⊢ 0 ≡ w + -w % ↑↑k [ZMOD ↑↑k]
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₃ := Int.ModEq.add h₂ h₁ | case h₀.H.a
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
h₁ : -w % ↑↑k ≡ -w [ZMOD ↑↑k]
h₂ : w ≡ w [ZMOD ↑↑k]
⊢ 0 ≡ w + -w % ↑↑k [ZMOD ↑↑k] | case h₀.H.a
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
h₁ : -w % ↑↑k ≡ -w [ZMOD ↑↑k]
h₂ : w ≡ w [ZMOD ↑↑k]
h₃ : w + -w % ↑↑k ≡ w + -w [ZMOD ↑↑k]
⊢ 0 ≡ w + -w % ↑↑k [ZMOD ↑↑k] | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀.H.a
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
h₁ : -w % ↑↑k ≡ -w [ZMOD ↑↑k]
h₂ : w ≡ w [ZMOD ↑↑k]
⊢ 0 ≡ w + -w % ↑↑k [ZMOD ↑↑k]
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | simp only [add_right_neg] at h₃ | case h₀.H.a
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
h₁ : -w % ↑↑k ≡ -w [ZMOD ↑↑k]
h₂ : w ≡ w [ZMOD ↑↑k]
h₃ : w + -w % ↑↑k ≡ w + -w [ZMOD ↑↑k]
⊢ 0 ≡ w + -w % ↑↑k [ZMOD ↑↑k] | case h₀.H.a
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
h₁ : -w % ↑↑k ≡ -w [ZMOD ↑↑k]
h₂ : w ≡ w [ZMOD ↑↑k]
h₃ : w + -w % ↑↑k ≡ 0 [ZMOD ↑↑k]
⊢ 0 ≡ w + -w % ↑↑k [ZMOD ↑↑k] | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀.H.a
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
h₁ : -w % ↑↑k ≡ -w [ZMOD ↑↑k]
h₂ : w ≡ w [ZMOD ↑↑k]
h₃ : w + -w % ↑↑k ≡ w + -w [ZMOD ↑↑k]
⊢ 0 ≡ w + -w % ↑↑k [ZMOD ↑↑k]
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | exact h₃.symm | case h₀.H.a
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
h₁ : -w % ↑↑k ≡ -w [ZMOD ↑↑k]
h₂ : w ≡ w [ZMOD ↑↑k]
h₃ : w + -w % ↑↑k ≡ 0 [ZMOD ↑↑k]
⊢ 0 ≡ w + -w % ↑↑k [ZMOD ↑↑k] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₀.H.a
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
h₁ : -w % ↑↑k ≡ -w [ZMOD ↑↑k]
h₂ : w ≡ w [ZMOD ↑↑k]
h₃ : w + -w % ↑↑k ≡ 0 [ZMOD ↑↑k]
⊢ 0 ≡ w + -w % ↑↑k [ZMOD ↑↑k]
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | exact rfl | n k : ℕ+
w : ℤ
⊢ 0 = 0 % ↑↑k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
w : ℤ
⊢ 0 = 0 % ↑↑k
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | exact Int.mod_modEq (-w) ↑↑k | n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
⊢ -w % ↑↑k ≡ -w [ZMOD ↑↑k] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
⊢ -w % ↑↑k ≡ -w [ZMOD ↑↑k]
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | exact rfl | n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
h₁ : -w % ↑↑k ≡ -w [ZMOD ↑↑k]
⊢ w ≡ w [ZMOD ↑↑k] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
w : ℤ
h₀ : 0 = 0 % ↑↑k
h₁ : -w % ↑↑k ≡ -w [ZMOD ↑↑k]
⊢ w ≡ w [ZMOD ↑↑k]
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | intros h₀ | case e_f.h.e_c.mpr
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
⊢ (∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)) → ↑↑n ∣ ↑x + m | case e_f.h.e_c.mpr
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
h₀ : ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
⊢ ↑↑n ∣ ↑x + m | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_c.mpr
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
⊢ (∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)) → ↑↑n ∣ ↑x + m
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | obtain ⟨w, _, h₂⟩ := h₀ | case e_f.h.e_c.mpr
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
h₀ : ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
⊢ ↑↑n ∣ ↑x + m | case e_f.h.e_c.mpr.intro.intro
n k : ℕ+
m : ℤ
z : ℂ
x w : ℕ
left✝ : w < ↑k
h₂ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m)
⊢ ↑↑n ∣ ↑x + m | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_c.mpr
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
h₀ : ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
⊢ ↑↑n ∣ ↑x + m
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₃ := dvd_of_mul_right_dvd h₂ | case e_f.h.e_c.mpr.intro.intro
n k : ℕ+
m : ℤ
z : ℂ
x w : ℕ
left✝ : w < ↑k
h₂ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m)
⊢ ↑↑n ∣ ↑x + m | case e_f.h.e_c.mpr.intro.intro
n k : ℕ+
m : ℤ
z : ℂ
x w : ℕ
left✝ : w < ↑k
h₂ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m)
h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m)
⊢ ↑↑n ∣ ↑x + m | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_c.mpr.intro.intro
n k : ℕ+
m : ℤ
z : ℂ
x w : ℕ
left✝ : w < ↑k
h₂ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m)
⊢ ↑↑n ∣ ↑x + m
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₄ : (n : ℤ) ∣ ↑↑n * ↑w := by exact Int.dvd_mul_right (↑n) w | case e_f.h.e_c.mpr.intro.intro
n k : ℕ+
m : ℤ
z : ℂ
x w : ℕ
left✝ : w < ↑k
h₂ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m)
h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m)
⊢ ↑↑n ∣ ↑x + m | case e_f.h.e_c.mpr.intro.intro
n k : ℕ+
m : ℤ
z : ℂ
x w : ℕ
left✝ : w < ↑k
h₂ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m)
h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m)
h₄ : ↑↑n ∣ ↑↑n * ↑w
⊢ ↑↑n ∣ ↑x + m | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_c.mpr.intro.intro
n k : ℕ+
m : ℤ
z : ℂ
x w : ℕ
left✝ : w < ↑k
h₂ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m)
h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m)
⊢ ↑↑n ∣ ↑x + m
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | rw [(show ↑x + (↑↑n * ↑w + m) = ↑↑n * ↑w + ↑(x + m) by ring_nf)] at h₃ | case e_f.h.e_c.mpr.intro.intro
n k : ℕ+
m : ℤ
z : ℂ
x w : ℕ
left✝ : w < ↑k
h₂ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m)
h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m)
h₄ : ↑↑n ∣ ↑↑n * ↑w
⊢ ↑↑n ∣ ↑x + m | case e_f.h.e_c.mpr.intro.intro
n k : ℕ+
m : ℤ
z : ℂ
x w : ℕ
left✝ : w < ↑k
h₂ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m)
h₃ : ↑↑n ∣ ↑↑n * ↑w + (↑x + m)
h₄ : ↑↑n ∣ ↑↑n * ↑w
⊢ ↑↑n ∣ ↑x + m | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_c.mpr.intro.intro
n k : ℕ+
m : ℤ
z : ℂ
x w : ℕ
left✝ : w < ↑k
h₂ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m)
h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m)
h₄ : ↑↑n ∣ ↑↑n * ↑w
⊢ ↑↑n ∣ ↑x + m
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | exact (Int.dvd_iff_dvd_of_dvd_add h₃).mp h₄ | case e_f.h.e_c.mpr.intro.intro
n k : ℕ+
m : ℤ
z : ℂ
x w : ℕ
left✝ : w < ↑k
h₂ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m)
h₃ : ↑↑n ∣ ↑↑n * ↑w + (↑x + m)
h₄ : ↑↑n ∣ ↑↑n * ↑w
⊢ ↑↑n ∣ ↑x + m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_c.mpr.intro.intro
n k : ℕ+
m : ℤ
z : ℂ
x w : ℕ
left✝ : w < ↑k
h₂ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m)
h₃ : ↑↑n ∣ ↑↑n * ↑w + (↑x + m)
h₄ : ↑↑n ∣ ↑↑n * ↑w
⊢ ↑↑n ∣ ↑x + m
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | exact Int.dvd_mul_right (↑n) w | n k : ℕ+
m : ℤ
z : ℂ
x w : ℕ
left✝ : w < ↑k
h₂ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m)
h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m)
⊢ ↑↑n ∣ ↑↑n * ↑w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
z : ℂ
x w : ℕ
left✝ : w < ↑k
h₂ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m)
h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m)
⊢ ↑↑n ∣ ↑↑n * ↑w
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | ring_nf | n k : ℕ+
m : ℤ
z : ℂ
x w : ℕ
left✝ : w < ↑k
h₂ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m)
h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m)
h₄ : ↑↑n ∣ ↑↑n * ↑w
⊢ ↑x + (↑↑n * ↑w + m) = ↑↑n * ↑w + (↑x + m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
z : ℂ
x w : ℕ
left✝ : w < ↑k
h₂ : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑w + m)
h₃ : ↑↑n ∣ ↑x + (↑↑n * ↑w + m)
h₄ : ↑↑n ∣ ↑↑n * ↑w
⊢ ↑x + (↑↑n * ↑w + m) = ↑↑n * ↑w + (↑x + m)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffNthIteratedDeriv | [258, 1] | [261, 10] | rw [RuesDiffIteratedDeriv, RuesDiffMPeriodic n m 1] | n : ℕ+
m : ℤ
⊢ iteratedDeriv (↑n) (RuesDiff n m) = RuesDiff n m | n : ℕ+
m : ℤ
⊢ RuesDiff n (↑↑n + m) = RuesDiff n (m + 1 * ↑↑n) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
⊢ iteratedDeriv (↑n) (RuesDiff n m) = RuesDiff n m
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffNthIteratedDeriv | [258, 1] | [261, 10] | simp only [one_mul] | n : ℕ+
m : ℤ
⊢ RuesDiff n (↑↑n + m) = RuesDiff n (m + 1 * ↑↑n) | n : ℕ+
m : ℤ
⊢ RuesDiff n (↑↑n + m) = RuesDiff n (m + ↑↑n) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
⊢ RuesDiff n (↑↑n + m) = RuesDiff n (m + 1 * ↑↑n)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffNthIteratedDeriv | [258, 1] | [261, 10] | ring_nf | n : ℕ+
m : ℤ
⊢ RuesDiff n (↑↑n + m) = RuesDiff n (m + ↑↑n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
⊢ RuesDiff n (↑↑n + m) = RuesDiff n (m + ↑↑n)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | have h₀ : ∀ (x : ℕ), (2 * ↑π * (↑k * ↑x / ↑↑n * I)) = ↑x * (2 * ↑π * (↑k / ↑↑n * I)) := by
intros x
ring_nf | n : ℕ+
k : ℤ
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k * ↑x / ↑↑n * I)) = if ↑↑n ∣ k then ↑↑n else 0 | n : ℕ+
k : ℤ
h₀ : ∀ (x : ℕ), 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I))
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k * ↑x / ↑↑n * I)) = if ↑↑n ∣ k then ↑↑n else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
k : ℤ
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k * ↑x / ↑↑n * I)) = if ↑↑n ∣ k then ↑↑n else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | simp_rw [h₀, Complex.exp_nat_mul] | n : ℕ+
k : ℤ
h₀ : ∀ (x : ℕ), 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I))
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k * ↑x / ↑↑n * I)) = if ↑↑n ∣ k then ↑↑n else 0 | n : ℕ+
k : ℤ
h₀ : ∀ (x : ℕ), 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I))
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
k : ℤ
h₀ : ∀ (x : ℕ), 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I))
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k * ↑x / ↑↑n * I)) = if ↑↑n ∣ k then ↑↑n else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | clear h₀ | n : ℕ+
k : ℤ
h₀ : ∀ (x : ℕ), 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I))
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0 | n : ℕ+
k : ℤ
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
k : ℤ
h₀ : ∀ (x : ℕ), 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I))
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | have hem := Classical.em (↑↑n ∣ k) | n : ℕ+
k : ℤ
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0 | n : ℕ+
k : ℤ
hem : ↑↑n ∣ k ∨ ¬↑↑n ∣ k
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
k : ℤ
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | have h₂ : (n : ℂ) ≠ 0 := by exact Ne.symm (NeZero.ne' (n : ℂ)) | n : ℕ+
k : ℤ
hem : ↑↑n ∣ k ∨ ¬↑↑n ∣ k
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0 | n : ℕ+
k : ℤ
hem : ↑↑n ∣ k ∨ ¬↑↑n ∣ k
h₂ : ↑↑n ≠ 0
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
k : ℤ
hem : ↑↑n ∣ k ∨ ¬↑↑n ∣ k
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | rcases hem with hemt | hemf | n : ℕ+
k : ℤ
hem : ↑↑n ∣ k ∨ ¬↑↑n ∣ k
h₂ : ↑↑n ≠ 0
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0 | case inl
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemt : ↑↑n ∣ k
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0
case inr
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
k : ℤ
hem : ↑↑n ∣ k ∨ ¬↑↑n ∣ k
h₂ : ↑↑n ≠ 0
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | intros x | n : ℕ+
k : ℤ
⊢ ∀ (x : ℕ), 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I)) | n : ℕ+
k : ℤ
x : ℕ
⊢ 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I)) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
k : ℤ
⊢ ∀ (x : ℕ), 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I))
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | ring_nf | n : ℕ+
k : ℤ
x : ℕ
⊢ 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
k : ℤ
x : ℕ
⊢ 2 * ↑π * (↑k * ↑x / ↑↑n * I) = ↑x * (2 * ↑π * (↑k / ↑↑n * I))
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | exact Ne.symm (NeZero.ne' (n : ℂ)) | n : ℕ+
k : ℤ
hem : ↑↑n ∣ k ∨ ¬↑↑n ∣ k
⊢ ↑↑n ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
k : ℤ
hem : ↑↑n ∣ k ∨ ¬↑↑n ∣ k
⊢ ↑↑n ≠ 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | rw [h₁, if_pos hemt] | case inl
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemt : ↑↑n ∣ k
h₁ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = ∑ x ∈ range ↑n, 1
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0 | case inl
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemt : ↑↑n ∣ k
h₁ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = ∑ x ∈ range ↑n, 1
⊢ ∑ x ∈ range ↑n, 1 = ↑↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemt : ↑↑n ∣ k
h₁ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = ∑ x ∈ range ↑n, 1
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | simp only [sum_const, card_range, nsmul_eq_mul, mul_one] | case inl
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemt : ↑↑n ∣ k
h₁ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = ∑ x ∈ range ↑n, 1
⊢ ∑ x ∈ range ↑n, 1 = ↑↑n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemt : ↑↑n ∣ k
h₁ : ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = ∑ x ∈ range ↑n, 1
⊢ ∑ x ∈ range ↑n, 1 = ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | congr | n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemt : ↑↑n ∣ k
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = ∑ x ∈ range ↑n, 1 | case e_f
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemt : ↑↑n ∣ k
⊢ (fun x => cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x) = fun x => 1 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemt : ↑↑n ∣ k
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = ∑ x ∈ range ↑n, 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | ext1 x | case e_f
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemt : ↑↑n ∣ k
⊢ (fun x => cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x) = fun x => 1 | case e_f.h
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemt : ↑↑n ∣ k
x : ℕ
⊢ cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemt : ↑↑n ∣ k
⊢ (fun x => cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x) = fun x => 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | obtain ⟨k₂, kDiv⟩ := hemt | case e_f.h
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemt : ↑↑n ∣ k
x : ℕ
⊢ cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 1 | case e_f.h.intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
⊢ cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemt : ↑↑n ∣ k
x : ℕ
⊢ cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | rw [kDiv] | case e_f.h.intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
⊢ cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 1 | case e_f.h.intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
⊢ cexp (2 * ↑π * (↑(↑↑n * k₂) / ↑↑n * I)) ^ x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
⊢ cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | field_simp | case e_f.h.intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
⊢ cexp (2 * ↑π * (↑(↑↑n * k₂) / ↑↑n * I)) ^ x = 1 | case e_f.h.intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
⊢ cexp (2 * ↑π * (↑k₂ * I)) ^ x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
⊢ cexp (2 * ↑π * (↑(↑↑n * k₂) / ↑↑n * I)) ^ x = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | suffices h₃ : cexp (2 * ↑π * (↑k₂ * I)) = 1 | case e_f.h.intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
⊢ cexp (2 * ↑π * (↑k₂ * I)) ^ x = 1 | case e_f.h.intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
h₃ : cexp (2 * ↑π * (↑k₂ * I)) = 1
⊢ cexp (2 * ↑π * (↑k₂ * I)) ^ x = 1
case h₃
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
⊢ cexp (2 * ↑π * (↑k₂ * I)) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
⊢ cexp (2 * ↑π * (↑k₂ * I)) ^ x = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | rw [h₃] | case e_f.h.intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
h₃ : cexp (2 * ↑π * (↑k₂ * I)) = 1
⊢ cexp (2 * ↑π * (↑k₂ * I)) ^ x = 1 | case e_f.h.intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
h₃ : cexp (2 * ↑π * (↑k₂ * I)) = 1
⊢ 1 ^ x = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
h₃ : cexp (2 * ↑π * (↑k₂ * I)) = 1
⊢ cexp (2 * ↑π * (↑k₂ * I)) ^ x = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | simp only [one_pow] | case e_f.h.intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
h₃ : cexp (2 * ↑π * (↑k₂ * I)) = 1
⊢ 1 ^ x = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
h₃ : cexp (2 * ↑π * (↑k₂ * I)) = 1
⊢ 1 ^ x = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | refine Complex.exp_eq_one_iff.mpr ?h₃.a | case h₃
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
⊢ cexp (2 * ↑π * (↑k₂ * I)) = 1 | case h₃.a
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
⊢ ∃ n, 2 * ↑π * (↑k₂ * I) = ↑n * (2 * ↑π * I) | Please generate a tactic in lean4 to solve the state.
STATE:
case h₃
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
⊢ cexp (2 * ↑π * (↑k₂ * I)) = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | use k₂ | case h₃.a
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
⊢ ∃ n, 2 * ↑π * (↑k₂ * I) = ↑n * (2 * ↑π * I) | case h
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
⊢ 2 * ↑π * (↑k₂ * I) = ↑k₂ * (2 * ↑π * I) | Please generate a tactic in lean4 to solve the state.
STATE:
case h₃.a
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
⊢ ∃ n, 2 * ↑π * (↑k₂ * I) = ↑n * (2 * ↑π * I)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | ring_nf | case h
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
⊢ 2 * ↑π * (↑k₂ * I) = ↑k₂ * (2 * ↑π * I) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
x : ℕ
k₂ : ℤ
kDiv : k = ↑↑n * k₂
⊢ 2 * ↑π * (↑k₂ * I) = ↑k₂ * (2 * ↑π * I)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | rw [if_neg hemf] | case inr
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0 | case inr
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = if ↑↑n ∣ k then ↑↑n else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | have h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1 := by
by_contra h
rw [Complex.exp_eq_one_iff] at h
obtain ⟨m, h⟩ := h
rw [(show 2 * ↑π * (↑k / ↑↑n * I) = (↑k / ↑↑n) * (2 * ↑π * I) by ring)] at h
have h₃ := mul_right_cancel₀ Complex.two_pi_I_ne_zero h
field_simp at h₃
have h₄ : k = m * n := by
exact mod_c... | case inr
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 0 | case inr
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | rw [geom_sum_eq h₀] | case inr
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 0 | case inr
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ (cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n - 1) / (cexp (2 * ↑π * (↑k / ↑↑n * I)) - 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ ∑ x ∈ range ↑n, cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ x = 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | suffices h₁ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ (n : ℕ) = 1 | case inr
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ (cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n - 1) / (cexp (2 * ↑π * (↑k / ↑↑n * I)) - 1) = 0 | case inr
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
h₁ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n = 1
⊢ (cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n - 1) / (cexp (2 * ↑π * (↑k / ↑↑n * I)) - 1) = 0
case h₁
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ cexp... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ (cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n - 1) / (cexp (2 * ↑π * (↑k / ↑↑n * I)) - 1) = 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | by_contra h | n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
⊢ cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1 | n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h : cexp (2 * ↑π * (↑k / ↑↑n * I)) = 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
⊢ cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | rw [Complex.exp_eq_one_iff] at h | n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h : cexp (2 * ↑π * (↑k / ↑↑n * I)) = 1
⊢ False | n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h : ∃ n_1, 2 * ↑π * (↑k / ↑↑n * I) = ↑n_1 * (2 * ↑π * I)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h : cexp (2 * ↑π * (↑k / ↑↑n * I)) = 1
⊢ False
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | obtain ⟨m, h⟩ := h | n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h : ∃ n_1, 2 * ↑π * (↑k / ↑↑n * I) = ↑n_1 * (2 * ↑π * I)
⊢ False | case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : 2 * ↑π * (↑k / ↑↑n * I) = ↑m * (2 * ↑π * I)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h : ∃ n_1, 2 * ↑π * (↑k / ↑↑n * I) = ↑n_1 * (2 * ↑π * I)
⊢ False
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | rw [(show 2 * ↑π * (↑k / ↑↑n * I) = (↑k / ↑↑n) * (2 * ↑π * I) by ring)] at h | case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : 2 * ↑π * (↑k / ↑↑n * I) = ↑m * (2 * ↑π * I)
⊢ False | case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : 2 * ↑π * (↑k / ↑↑n * I) = ↑m * (2 * ↑π * I)
⊢ False
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | have h₃ := mul_right_cancel₀ Complex.two_pi_I_ne_zero h | case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
⊢ False | case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
h₃ : ↑k / ↑↑n = ↑m
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
⊢ False
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | field_simp at h₃ | case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
h₃ : ↑k / ↑↑n = ↑m
⊢ False | case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
h₃ : ↑k = ↑m * ↑↑n
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
h₃ : ↑k / ↑↑n = ↑m
⊢ False
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | have h₄ : k = m * n := by
exact mod_cast h₃ | case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
h₃ : ↑k = ↑m * ↑↑n
⊢ False | case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
h₃ : ↑k = ↑m * ↑↑n
h₄ : k = m * ↑↑n
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
h₃ : ↑k = ↑m * ↑↑n
⊢ False
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | have h₅ : (n : ℤ) ∣ k := by
exact Dvd.intro_left m (id (Eq.symm h₄)) | case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
h₃ : ↑k = ↑m * ↑↑n
h₄ : k = m * ↑↑n
⊢ False | case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
h₃ : ↑k = ↑m * ↑↑n
h₄ : k = m * ↑↑n
h₅ : ↑↑n ∣ k
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
h₃ : ↑k = ↑m * ↑↑n
h₄ : k = m * ↑↑n
⊢ False
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | apply hemf | case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
h₃ : ↑k = ↑m * ↑↑n
h₄ : k = m * ↑↑n
h₅ : ↑↑n ∣ k
⊢ False | case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
h₃ : ↑k = ↑m * ↑↑n
h₄ : k = m * ↑↑n
h₅ : ↑↑n ∣ k
⊢ ↑↑n ∣ k | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
h₃ : ↑k = ↑m * ↑↑n
h₄ : k = m * ↑↑n
h₅ : ↑↑n ∣ k
⊢ False
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | exact h₅ | case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
h₃ : ↑k = ↑m * ↑↑n
h₄ : k = m * ↑↑n
h₅ : ↑↑n ∣ k
⊢ ↑↑n ∣ k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
h₃ : ↑k = ↑m * ↑↑n
h₄ : k = m * ↑↑n
h₅ : ↑↑n ∣ k
⊢ ↑↑n ∣ k
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | ring | n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : 2 * ↑π * (↑k / ↑↑n * I) = ↑m * (2 * ↑π * I)
⊢ 2 * ↑π * (↑k / ↑↑n * I) = ↑k / ↑↑n * (2 * ↑π * I) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : 2 * ↑π * (↑k / ↑↑n * I) = ↑m * (2 * ↑π * I)
⊢ 2 * ↑π * (↑k / ↑↑n * I) = ↑k / ↑↑n * (2 * ↑π * I)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | exact mod_cast h₃ | n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
h₃ : ↑k = ↑m * ↑↑n
⊢ k = m * ↑↑n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
h₃ : ↑k = ↑m * ↑↑n
⊢ k = m * ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | exact Dvd.intro_left m (id (Eq.symm h₄)) | n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
h₃ : ↑k = ↑m * ↑↑n
h₄ : k = m * ↑↑n
⊢ ↑↑n ∣ k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
m : ℤ
h : ↑k / ↑↑n * (2 * ↑π * I) = ↑m * (2 * ↑π * I)
h₃ : ↑k = ↑m * ↑↑n
h₄ : k = m * ↑↑n
⊢ ↑↑n ∣ k
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | rw [h₁] | case inr
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
h₁ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n = 1
⊢ (cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n - 1) / (cexp (2 * ↑π * (↑k / ↑↑n * I)) - 1) = 0 | case inr
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
h₁ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n = 1
⊢ (1 - 1) / (cexp (2 * ↑π * (↑k / ↑↑n * I)) - 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
h₁ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n = 1
⊢ (cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n - 1) / (cexp (2 * ↑π * (↑k / ↑↑n * I)) - 1) = 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | simp only [sub_self, zero_div] | case inr
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
h₁ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n = 1
⊢ (1 - 1) / (cexp (2 * ↑π * (↑k / ↑↑n * I)) - 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
h₁ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n = 1
⊢ (1 - 1) / (cexp (2 * ↑π * (↑k / ↑↑n * I)) - 1) = 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | rw [(Complex.exp_nat_mul _ n).symm] | case h₁
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n = 1 | case h₁
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ cexp (↑↑n * (2 * ↑π * (↑k / ↑↑n * I))) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ cexp (2 * ↑π * (↑k / ↑↑n * I)) ^ ↑n = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | refine Complex.exp_eq_one_iff.mpr ?h₁.a | case h₁
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ cexp (↑↑n * (2 * ↑π * (↑k / ↑↑n * I))) = 1 | case h₁.a
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ ∃ n_1, ↑↑n * (2 * ↑π * (↑k / ↑↑n * I)) = ↑n_1 * (2 * ↑π * I) | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ cexp (↑↑n * (2 * ↑π * (↑k / ↑↑n * I))) = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | use k | case h₁.a
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ ∃ n_1, ↑↑n * (2 * ↑π * (↑k / ↑↑n * I)) = ↑n_1 * (2 * ↑π * I) | case h
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ ↑↑n * (2 * ↑π * (↑k / ↑↑n * I)) = ↑k * (2 * ↑π * I) | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁.a
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ ∃ n_1, ↑↑n * (2 * ↑π * (↑k / ↑↑n * I)) = ↑n_1 * (2 * ↑π * I)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | field_simp | case h
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ ↑↑n * (2 * ↑π * (↑k / ↑↑n * I)) = ↑k * (2 * ↑π * I) | case h
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ 2 * ↑π * (↑k * I) = ↑k * (2 * ↑π * I) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ ↑↑n * (2 * ↑π * (↑k / ↑↑n * I)) = ↑k * (2 * ↑π * I)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouGeometricSumEqIte | [263, 1] | [308, 14] | ring_nf | case h
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ 2 * ↑π * (↑k * I) = ↑k * (2 * ↑π * I) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ+
k : ℤ
h₂ : ↑↑n ≠ 0
hemf : ¬↑↑n ∣ k
h₀ : cexp (2 * ↑π * (↑k / ↑↑n * I)) ≠ 1
⊢ 2 * ↑π * (↑k * I) = ↑k * (2 * ↑π * I)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | simp_rw [Complex.exp_add] | n : ℕ+
m : ℤ
z : ℂ
⊢ RuesDiff n m z = (∑ k₀ ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑k₀ / ↑↑n) * I) + ↑m * 2 * ↑π * (↑k₀ / ↑↑n) * I)) / ↑↑n | n : ℕ+
m : ℤ
z : ℂ
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) * 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 = (∑ k₀ ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑k₀ / ↑↑n) * I) + ↑m * 2 * ↑π * (↑k₀ / ↑↑n) * I)) / ↑↑n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | have h₀ : ∀ (k : ℕ), cexp (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) = ∑' (k_1 : ℕ), (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) ^ k_1 / ↑(Nat.factorial k_1) := by
intros k
exact ExpTsumForm (z * cexp (2 * ↑π * (↑k / ↑↑n) * I)) | n : ℕ+
m : ℤ
z : ℂ
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n | n : ℕ+
m : ℤ
z : ℂ
h₀ :
∀ (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
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) * 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, cexp (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) * 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 [h₀] | n : ℕ+
m : ℤ
z : ℂ
h₀ :
∀ (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
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) / ↑↑n | n : ℕ+
m : ℤ
z : ℂ
h₀ :
∀ (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
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n,
(∑' (k_1 : ℕ), (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial) *
cexp (↑m * 2 * ↑π *... | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
h₀ :
∀ (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
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n, cexp (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) * cexp (↑m *... |
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | clear h₀ | n : ℕ+
m : ℤ
z : ℂ
h₀ :
∀ (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
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n,
(∑' (k_1 : ℕ), (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial) *
cexp (↑m * 2 * ↑π *... | n : ℕ+
m : ℤ
z : ℂ
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n,
(∑' (k_1 : ℕ), (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ k_1 / ↑k_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
h₀ :
∀ (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
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n,
(∑' (k_1 : ℕ), (z * cexp (2 * ↑π * (↑x / ↑↑n) ... |
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffEqualsExpSum | [310, 1] | [352, 25] | simp_rw [←tsum_mul_right] | n : ℕ+
m : ℤ
z : ℂ
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n,
(∑' (k_1 : ℕ), (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial) *
cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n | 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 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z : ℂ
⊢ RuesDiff n m z =
(∑ x ∈ range ↑n,
(∑' (k_1 : ℕ), (z * cexp (2 * ↑π * (↑x / ↑↑n) * I)) ^ k_1 / ↑k_1.factorial) *
cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n
TACTIC:
|
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