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 | TsumMulIte | [75, 1] | [110, 24] | exact PNat.pos n | α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : AddCommMonoid α
f : ℕ → α
n : ℕ+
h₀ : ↑n ≠ 0
nMul : ℕ → ℕ := fun m => ↑n * m
hnMulInj : Function.Injective fun x => ↑n * x
h₁ : ∑' (k : ℕ), f (↑n * k) = ∑' (k : ℕ), f (nMul k)
h₂ : ∑' (k : ℕ), f (nMul k) = ∑' (a : ↑(Set.range nMul)), f ↑a
k : ℕ
w : ℤ
h... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : AddCommMonoid α
f : ℕ → α
n : ℕ+
h₀ : ↑n ≠ 0
nMul : ℕ → ℕ := fun m => ↑n * m
hnMulInj : Function.Injective fun x => ↑n * x
h₁ : ∑' (k : ℕ), f (↑n * k) = ∑' (k : ℕ), f (nMul k)
h₂ : ∑' (k : ℕ)... |
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | TsumMulIte | [75, 1] | [110, 24] | exact Int.mul_neg_of_pos_of_neg h₃ h₆ | α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : AddCommMonoid α
f : ℕ → α
n : ℕ+
h₀ : ↑n ≠ 0
nMul : ℕ → ℕ := fun m => ↑n * m
hnMulInj : Function.Injective fun x => ↑n * x
h₁ : ∑' (k : ℕ), f (↑n * k) = ∑' (k : ℕ), f (nMul k)
h₂ : ∑' (k : ℕ), f (nMul k) = ∑' (a : ↑(Set.range nMul)), f ↑a
k : ℕ
w : ℤ
h... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : AddCommMonoid α
f : ℕ → α
n : ℕ+
h₀ : ↑n ≠ 0
nMul : ℕ → ℕ := fun m => ↑n * m
hnMulInj : Function.Injective fun x => ↑n * x
h₁ : ∑' (k : ℕ), f (↑n * k) = ∑' (k : ℕ), f (nMul k)
h₂ : ∑' (k : ℕ)... |
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | TsumMulIte | [75, 1] | [110, 24] | intros h₄ | case e_c.mpr
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : AddCommMonoid α
f : ℕ → α
n : ℕ+
h₀ : ↑n ≠ 0
nMul : ℕ → ℕ := fun m => ↑n * m
hnMulInj : Function.Injective fun x => ↑n * x
h₁ : ∑' (k : ℕ), f (↑n * k) = ∑' (k : ℕ), f (nMul k)
h₂ : ∑' (k : ℕ), f (nMul k) = ∑' (a : ↑(Set.range nMul)), f ↑a
... | case e_c.mpr
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : AddCommMonoid α
f : ℕ → α
n : ℕ+
h₀ : ↑n ≠ 0
nMul : ℕ → ℕ := fun m => ↑n * m
hnMulInj : Function.Injective fun x => ↑n * x
h₁ : ∑' (k : ℕ), f (↑n * k) = ∑' (k : ℕ), f (nMul k)
h₂ : ∑' (k : ℕ), f (nMul k) = ∑' (a : ↑(Set.range nMul)), f ↑a
... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_c.mpr
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : AddCommMonoid α
f : ℕ → α
n : ℕ+
h₀ : ↑n ≠ 0
nMul : ℕ → ℕ := fun m => ↑n * m
hnMulInj : Function.Injective fun x => ↑n * x
h₁ : ∑' (k : ℕ), f (↑n * k) = ∑' (k : ℕ), f (nMul k)
h₂... |
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | TsumMulIte | [75, 1] | [110, 24] | rcases h₄ with ⟨w,hw⟩ | case e_c.mpr
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : AddCommMonoid α
f : ℕ → α
n : ℕ+
h₀ : ↑n ≠ 0
nMul : ℕ → ℕ := fun m => ↑n * m
hnMulInj : Function.Injective fun x => ↑n * x
h₁ : ∑' (k : ℕ), f (↑n * k) = ∑' (k : ℕ), f (nMul k)
h₂ : ∑' (k : ℕ), f (nMul k) = ∑' (a : ↑(Set.range nMul)), f ↑a
... | case e_c.mpr.intro
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : AddCommMonoid α
f : ℕ → α
n : ℕ+
h₀ : ↑n ≠ 0
nMul : ℕ → ℕ := fun m => ↑n * m
hnMulInj : Function.Injective fun x => ↑n * x
h₁ : ∑' (k : ℕ), f (↑n * k) = ∑' (k : ℕ), f (nMul k)
h₂ : ∑' (k : ℕ), f (nMul k) = ∑' (a : ↑(Set.range nMul)),... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_c.mpr
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : AddCommMonoid α
f : ℕ → α
n : ℕ+
h₀ : ↑n ≠ 0
nMul : ℕ → ℕ := fun m => ↑n * m
hnMulInj : Function.Injective fun x => ↑n * x
h₁ : ∑' (k : ℕ), f (↑n * k) = ∑' (k : ℕ), f (nMul k)
h₂... |
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | TsumMulIte | [75, 1] | [110, 24] | use w | case e_c.mpr.intro
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : AddCommMonoid α
f : ℕ → α
n : ℕ+
h₀ : ↑n ≠ 0
nMul : ℕ → ℕ := fun m => ↑n * m
hnMulInj : Function.Injective fun x => ↑n * x
h₁ : ∑' (k : ℕ), f (↑n * k) = ∑' (k : ℕ), f (nMul k)
h₂ : ∑' (k : ℕ), f (nMul k) = ∑' (a : ↑(Set.range nMul)),... | case h
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : AddCommMonoid α
f : ℕ → α
n : ℕ+
h₀ : ↑n ≠ 0
nMul : ℕ → ℕ := fun m => ↑n * m
hnMulInj : Function.Injective fun x => ↑n * x
h₁ : ∑' (k : ℕ), f (↑n * k) = ∑' (k : ℕ), f (nMul k)
h₂ : ∑' (k : ℕ), f (nMul k) = ∑' (a : ↑(Set.range nMul)), f ↑a
k w : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_c.mpr.intro
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : AddCommMonoid α
f : ℕ → α
n : ℕ+
h₀ : ↑n ≠ 0
nMul : ℕ → ℕ := fun m => ↑n * m
hnMulInj : Function.Injective fun x => ↑n * x
h₁ : ∑' (k : ℕ), f (↑n * k) = ∑' (k : ℕ), f (nMul... |
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | TsumMulIte | [75, 1] | [110, 24] | simp only [Nat.cast_mul, hw] | case h
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : AddCommMonoid α
f : ℕ → α
n : ℕ+
h₀ : ↑n ≠ 0
nMul : ℕ → ℕ := fun m => ↑n * m
hnMulInj : Function.Injective fun x => ↑n * x
h₁ : ∑' (k : ℕ), f (↑n * k) = ∑' (k : ℕ), f (nMul k)
h₂ : ∑' (k : ℕ), f (nMul k) = ∑' (a : ↑(Set.range nMul)), f ↑a
k w : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : T2Space α
inst✝ : AddCommMonoid α
f : ℕ → α
n : ℕ+
h₀ : ↑n ≠ 0
nMul : ℕ → ℕ := fun m => ↑n * m
hnMulInj : Function.Injective fun x => ↑n * x
h₁ : ∑' (k : ℕ), f (↑n * k) = ∑' (k : ℕ), f (nMul k)
h₂ : ∑' ... |
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | NeedZeroCoeff | [112, 1] | [113, 21] | exact TsumMulIte _ | f : ℕ → ℂ
n : ℕ+
⊢ ∑' (k : ℕ), f (↑n * k) = ∑' (k : ℕ), if ↑↑n ∣ ↑k then f k else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℂ
n : ℕ+
⊢ ∑' (k : ℕ), f (↑n * k) = ∑' (k : ℕ), if ↑↑n ∣ ↑k then f k else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffM0EqualsRues | [115, 1] | [119, 64] | ext1 z | n : ℕ+
⊢ RuesDiff n 0 = Rues n | case h
n : ℕ+
z : ℂ
⊢ RuesDiff n 0 z = Rues n z | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
⊢ RuesDiff n 0 = Rues n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffM0EqualsRues | [115, 1] | [119, 64] | rw [Rues, RuesDiff] | case h
n : ℕ+
z : ℂ
⊢ RuesDiff n 0 z = Rues n z | case h
n : ℕ+
z : ℂ
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + 0 then z ^ k / ↑k.factorial else 0) = ∑' (k : ℕ), z ^ (↑n * k) / ↑(↑n * k).factorial | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ+
z : ℂ
⊢ RuesDiff n 0 z = Rues n z
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffM0EqualsRues | [115, 1] | [119, 64] | simp only [add_zero] | case h
n : ℕ+
z : ℂ
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + 0 then z ^ k / ↑k.factorial else 0) = ∑' (k : ℕ), z ^ (↑n * k) / ↑(↑n * k).factorial | case h
n : ℕ+
z : ℂ
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k then z ^ k / ↑k.factorial else 0) = ∑' (k : ℕ), z ^ (↑n * k) / ↑(↑n * k).factorial | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ+
z : ℂ
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + 0 then z ^ k / ↑k.factorial else 0) = ∑' (k : ℕ), z ^ (↑n * k) / ↑(↑n * k).factorial
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffM0EqualsRues | [115, 1] | [119, 64] | rw [NeedZeroCoeff (λ (k : ℕ) => z ^ k / (Nat.factorial k)) n] | case h
n : ℕ+
z : ℂ
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k then z ^ k / ↑k.factorial else 0) = ∑' (k : ℕ), z ^ (↑n * k) / ↑(↑n * k).factorial | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ+
z : ℂ
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k then z ^ k / ↑k.factorial else 0) = ∑' (k : ℕ), z ^ (↑n * k) / ↑(↑n * k).factorial
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | simp_rw [RuesDiff, ←tsum_mul_left] | n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
⊢ RuesDiff n m (z * rou) = rou ^ (-m) * RuesDiff n m z | n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) =
∑' (x : ℕ), rou ^ (-m) * if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
⊢ RuesDiff n m (z * rou) = rou ^ (-m) * RuesDiff n m z
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | congr | n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) =
∑' (x : ℕ), rou ^ (-m) * if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0 | case e_f
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
⊢ (fun k => if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) = fun x =>
rou ^ (-m) * if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) =
∑' (x : ℕ), rou ^ (-m) * if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | ext1 k | case e_f
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
⊢ (fun k => if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) = fun x =>
rou ^ (-m) * if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0 | case e_f.h
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
⊢ (if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) =
rou ^ (-m) * if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
⊢ (fun k => if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) = fun x =>
rou ^ (-m) * if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | simp only [zpow_neg, mul_ite, mul_zero] | case e_f.h
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
⊢ (if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) =
rou ^ (-m) * if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0 | case e_f.h
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
⊢ (if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) =
if ↑↑n ∣ ↑k + m then (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial) else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
⊢ (if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) =
rou ^ (-m) * if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | have h₀ := Classical.em (↑↑n ∣ ↑k + m) | case e_f.h
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
⊢ (if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) =
if ↑↑n ∣ ↑k + m then (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial) else 0 | case e_f.h
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀ : ↑↑n ∣ ↑k + m ∨ ¬↑↑n ∣ ↑k + m
⊢ (if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) =
if ↑↑n ∣ ↑k + m then (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial) else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
⊢ (if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) =
if ↑↑n ∣ ↑k + m then (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial) else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | rcases h₀ with h₀a | h₀b | case e_f.h
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀ : ↑↑n ∣ ↑k + m ∨ ¬↑↑n ∣ ↑k + m
⊢ (if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) =
if ↑↑n ∣ ↑k + m then (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial) else 0 | case e_f.h.inl
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀a : ↑↑n ∣ ↑k + m
⊢ (if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) =
if ↑↑n ∣ ↑k + m then (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial) else 0
case e_f.h.inr
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀b : ¬↑↑n ∣ ↑k + m
⊢ (if ↑↑n ∣ ↑k + m then (z... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀ : ↑↑n ∣ ↑k + m ∨ ¬↑↑n ∣ ↑k + m
⊢ (if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) =
if ↑↑n ∣ ↑k + m then (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial) else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | simp_rw [if_pos h₀a] | case e_f.h.inl
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀a : ↑↑n ∣ ↑k + m
⊢ (if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) =
if ↑↑n ∣ ↑k + m then (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial) else 0 | case e_f.h.inl
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀a : ↑↑n ∣ ↑k + m
⊢ (z * rou) ^ k / ↑k.factorial = (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.inl
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀a : ↑↑n ∣ ↑k + m
⊢ (if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) =
if ↑↑n ∣ ↑k + m then (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial) else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | rw [mul_pow z rou k] | case e_f.h.inl
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀a : ↑↑n ∣ ↑k + m
⊢ (z * rou) ^ k / ↑k.factorial = (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial) | case e_f.h.inl
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀a : ↑↑n ∣ ↑k + m
⊢ z ^ k * rou ^ k / ↑k.factorial = (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.inl
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀a : ↑↑n ∣ ↑k + m
⊢ (z * rou) ^ k / ↑k.factorial = (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | have h₁ : rou ^ k = (rou ^ m)⁻¹ := by
obtain ⟨k₂, kmDiv⟩ := h₀a
have h₂ : rou ^ (↑k + m) = 1 := by
rw [kmDiv, zpow_mul]
simp only [zpow_natCast, h, one_zpow]
have h₃ := congrArg (λ (z₀ : ℂ) => z₀ * (rou ^ m)⁻¹) h₂
simp only [one_mul, ne_eq, inv_eq_zero] at h₃
have h₄ := RouNot0 n rou h
rw [zpow_add₀... | case e_f.h.inl
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀a : ↑↑n ∣ ↑k + m
⊢ z ^ k * rou ^ k / ↑k.factorial = (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial) | case e_f.h.inl
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀a : ↑↑n ∣ ↑k + m
h₁ : rou ^ k = (rou ^ m)⁻¹
⊢ z ^ k * rou ^ k / ↑k.factorial = (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.inl
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀a : ↑↑n ∣ ↑k + m
⊢ z ^ k * rou ^ k / ↑k.factorial = (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | rw [h₁] | case e_f.h.inl
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀a : ↑↑n ∣ ↑k + m
h₁ : rou ^ k = (rou ^ m)⁻¹
⊢ z ^ k * rou ^ k / ↑k.factorial = (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial) | case e_f.h.inl
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀a : ↑↑n ∣ ↑k + m
h₁ : rou ^ k = (rou ^ m)⁻¹
⊢ z ^ k * (rou ^ m)⁻¹ / ↑k.factorial = (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial) | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.inl
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀a : ↑↑n ∣ ↑k + m
h₁ : rou ^ k = (rou ^ m)⁻¹
⊢ z ^ k * rou ^ k / ↑k.factorial = (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | ring | case e_f.h.inl
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀a : ↑↑n ∣ ↑k + m
h₁ : rou ^ k = (rou ^ m)⁻¹
⊢ z ^ k * (rou ^ m)⁻¹ / ↑k.factorial = (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.inl
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀a : ↑↑n ∣ ↑k + m
h₁ : rou ^ k = (rou ^ m)⁻¹
⊢ z ^ k * (rou ^ m)⁻¹ / ↑k.factorial = (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | obtain ⟨k₂, kmDiv⟩ := h₀a | n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀a : ↑↑n ∣ ↑k + m
⊢ rou ^ k = (rou ^ m)⁻¹ | case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
⊢ rou ^ k = (rou ^ m)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀a : ↑↑n ∣ ↑k + m
⊢ rou ^ k = (rou ^ m)⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | have h₂ : rou ^ (↑k + m) = 1 := by
rw [kmDiv, zpow_mul]
simp only [zpow_natCast, h, one_zpow] | case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
⊢ rou ^ k = (rou ^ m)⁻¹ | case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
⊢ rou ^ k = (rou ^ m)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
⊢ rou ^ k = (rou ^ m)⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | have h₃ := congrArg (λ (z₀ : ℂ) => z₀ * (rou ^ m)⁻¹) h₂ | case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
⊢ rou ^ k = (rou ^ m)⁻¹ | case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : (fun z₀ => z₀ * (rou ^ m)⁻¹) (rou ^ (↑k + m)) = (fun z₀ => z₀ * (rou ^ m)⁻¹) 1
⊢ rou ^ k = (rou ^ m)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
⊢ rou ^ k = (rou ^ m)⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | simp only [one_mul, ne_eq, inv_eq_zero] at h₃ | case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : (fun z₀ => z₀ * (rou ^ m)⁻¹) (rou ^ (↑k + m)) = (fun z₀ => z₀ * (rou ^ m)⁻¹) 1
⊢ rou ^ k = (rou ^ m)⁻¹ | case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : rou ^ (↑k + m) * (rou ^ m)⁻¹ = (rou ^ m)⁻¹
⊢ rou ^ k = (rou ^ m)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : (fun z₀ => z₀ * (rou ^ m)⁻¹) (rou ^ (↑k + m)) = (fun z₀ => z₀ * (rou ^ m)⁻¹) 1
⊢ rou ^ k = (rou ^ m)⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | have h₄ := RouNot0 n rou h | case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : rou ^ (↑k + m) * (rou ^ m)⁻¹ = (rou ^ m)⁻¹
⊢ rou ^ k = (rou ^ m)⁻¹ | case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : rou ^ (↑k + m) * (rou ^ m)⁻¹ = (rou ^ m)⁻¹
h₄ : rou ≠ 0
⊢ rou ^ k = (rou ^ m)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : rou ^ (↑k + m) * (rou ^ m)⁻¹ = (rou ^ m)⁻¹
⊢ rou ^ k = (rou ^ m)⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | rw [zpow_add₀ h₄ ↑k m] at h₃ | case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : rou ^ (↑k + m) * (rou ^ m)⁻¹ = (rou ^ m)⁻¹
h₄ : rou ≠ 0
⊢ rou ^ k = (rou ^ m)⁻¹ | case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : rou ^ ↑k * rou ^ m * (rou ^ m)⁻¹ = (rou ^ m)⁻¹
h₄ : rou ≠ 0
⊢ rou ^ k = (rou ^ m)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : rou ^ (↑k + m) * (rou ^ m)⁻¹ = (rou ^ m)⁻¹
h₄ : rou ≠ 0
⊢ rou ^ k = (rou ^ m)⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | rw [←h₃] | case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : rou ^ ↑k * rou ^ m * (rou ^ m)⁻¹ = (rou ^ m)⁻¹
h₄ : rou ≠ 0
⊢ rou ^ k = (rou ^ m)⁻¹ | case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : rou ^ ↑k * rou ^ m * (rou ^ m)⁻¹ = (rou ^ m)⁻¹
h₄ : rou ≠ 0
⊢ rou ^ k = rou ^ ↑k * rou ^ m * (rou ^ m)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : rou ^ ↑k * rou ^ m * (rou ^ m)⁻¹ = (rou ^ m)⁻¹
h₄ : rou ≠ 0
⊢ rou ^ k = (rou ^ m)⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | have h₅ : rou ^ m ≠ 0 := by
exact zpow_ne_zero m h₄ | case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : rou ^ ↑k * rou ^ m * (rou ^ m)⁻¹ = (rou ^ m)⁻¹
h₄ : rou ≠ 0
⊢ rou ^ k = rou ^ ↑k * rou ^ m * (rou ^ m)⁻¹ | case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : rou ^ ↑k * rou ^ m * (rou ^ m)⁻¹ = (rou ^ m)⁻¹
h₄ : rou ≠ 0
h₅ : rou ^ m ≠ 0
⊢ rou ^ k = rou ^ ↑k * rou ^ m * (rou ^ m)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : rou ^ ↑k * rou ^ m * (rou ^ m)⁻¹ = (rou ^ m)⁻¹
h₄ : rou ≠ 0
⊢ rou ^ k = rou ^ ↑k * rou ^ m * (rou ^ m)⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | field_simp | case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : rou ^ ↑k * rou ^ m * (rou ^ m)⁻¹ = (rou ^ m)⁻¹
h₄ : rou ≠ 0
h₅ : rou ^ m ≠ 0
⊢ rou ^ k = rou ^ ↑k * rou ^ m * (rou ^ m)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : rou ^ ↑k * rou ^ m * (rou ^ m)⁻¹ = (rou ^ m)⁻¹
h₄ : rou ≠ 0
h₅ : rou ^ m ≠ 0
⊢ rou ^ k = rou ^ ↑k * rou ^ m * (rou ^ m)⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | rw [kmDiv, zpow_mul] | n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
⊢ rou ^ (↑k + m) = 1 | n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
⊢ (rou ^ ↑↑n) ^ k₂ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
⊢ rou ^ (↑k + m) = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | simp only [zpow_natCast, h, one_zpow] | n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
⊢ (rou ^ ↑↑n) ^ k₂ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
⊢ (rou ^ ↑↑n) ^ k₂ = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | exact zpow_ne_zero m h₄ | n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : rou ^ ↑k * rou ^ m * (rou ^ m)⁻¹ = (rou ^ m)⁻¹
h₄ : rou ≠ 0
⊢ rou ^ m ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
k₂ : ℤ
kmDiv : ↑k + m = ↑↑n * k₂
h₂ : rou ^ (↑k + m) = 1
h₃ : rou ^ ↑k * rou ^ m * (rou ^ m)⁻¹ = (rou ^ m)⁻¹
h₄ : rou ≠ 0
⊢ rou ^ m ≠ 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffRotationallySymmetric | [121, 1] | [145, 25] | simp_rw [if_neg h₀b] | case e_f.h.inr
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀b : ¬↑↑n ∣ ↑k + m
⊢ (if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) =
if ↑↑n ∣ ↑k + m then (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial) else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.inr
n : ℕ+
m : ℤ
z rou : ℂ
h : rou ^ ↑n = 1
k : ℕ
h₀b : ¬↑↑n ∣ ↑k + m
⊢ (if ↑↑n ∣ ↑k + m then (z * rou) ^ k / ↑k.factorial else 0) =
if ↑↑n ∣ ↑k + m then (rou ^ m)⁻¹ * (z ^ k / ↑k.factorial) else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | Dvd.dvd.addMultiple | [147, 1] | [154, 38] | have h₁ : n ∣ (k * n) := by
exact Int.dvd_mul_left k n | n m k : ℤ
⊢ n ∣ m ↔ n ∣ m + k * n | n m k : ℤ
h₁ : n ∣ k * n
⊢ n ∣ m ↔ n ∣ m + k * n | Please generate a tactic in lean4 to solve the state.
STATE:
n m k : ℤ
⊢ n ∣ m ↔ n ∣ m + k * n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | Dvd.dvd.addMultiple | [147, 1] | [154, 38] | constructor | n m k : ℤ
h₁ : n ∣ k * n
⊢ n ∣ m ↔ n ∣ m + k * n | case mp
n m k : ℤ
h₁ : n ∣ k * n
⊢ n ∣ m → n ∣ m + k * n
case mpr
n m k : ℤ
h₁ : n ∣ k * n
⊢ n ∣ m + k * n → n ∣ m | Please generate a tactic in lean4 to solve the state.
STATE:
n m k : ℤ
h₁ : n ∣ k * n
⊢ n ∣ m ↔ n ∣ m + k * n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | Dvd.dvd.addMultiple | [147, 1] | [154, 38] | exact Int.dvd_mul_left k n | n m k : ℤ
⊢ n ∣ k * n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n m k : ℤ
⊢ n ∣ k * n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | Dvd.dvd.addMultiple | [147, 1] | [154, 38] | intros h₀ | case mp
n m k : ℤ
h₁ : n ∣ k * n
⊢ n ∣ m → n ∣ m + k * n | case mp
n m k : ℤ
h₁ : n ∣ k * n
h₀ : n ∣ m
⊢ n ∣ m + k * n | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
n m k : ℤ
h₁ : n ∣ k * n
⊢ n ∣ m → n ∣ m + k * n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | Dvd.dvd.addMultiple | [147, 1] | [154, 38] | exact Dvd.dvd.add h₀ h₁ | case mp
n m k : ℤ
h₁ : n ∣ k * n
h₀ : n ∣ m
⊢ n ∣ m + k * n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
n m k : ℤ
h₁ : n ∣ k * n
h₀ : n ∣ m
⊢ n ∣ m + k * n
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | Dvd.dvd.addMultiple | [147, 1] | [154, 38] | intros h₂ | case mpr
n m k : ℤ
h₁ : n ∣ k * n
⊢ n ∣ m + k * n → n ∣ m | case mpr
n m k : ℤ
h₁ : n ∣ k * n
h₂ : n ∣ m + k * n
⊢ n ∣ m | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
n m k : ℤ
h₁ : n ∣ k * n
⊢ n ∣ m + k * n → n ∣ m
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | Dvd.dvd.addMultiple | [147, 1] | [154, 38] | exact (Int.dvd_add_left h₁).mp h₂ | case mpr
n m k : ℤ
h₁ : n ∣ k * n
h₂ : n ∣ m + k * n
⊢ n ∣ m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
n m k : ℤ
h₁ : n ∣ k * n
h₂ : n ∣ m + k * n
⊢ n ∣ m
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffMPeriodic | [156, 1] | [163, 10] | ext1 z | n : ℕ+
m k : ℤ
⊢ RuesDiff n m = RuesDiff n (m + k * ↑↑n) | case h
n : ℕ+
m k : ℤ
z : ℂ
⊢ RuesDiff n m z = RuesDiff n (m + k * ↑↑n) z | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m k : ℤ
⊢ RuesDiff n m = RuesDiff n (m + k * ↑↑n)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffMPeriodic | [156, 1] | [163, 10] | simp_rw [RuesDiff] | case h
n : ℕ+
m k : ℤ
z : ℂ
⊢ RuesDiff n m z = RuesDiff n (m + k * ↑↑n) z | case h
n : ℕ+
m k : ℤ
z : ℂ
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑' (k_1 : ℕ), if ↑↑n ∣ ↑k_1 + (m + k * ↑↑n) then z ^ k_1 / ↑k_1.factorial else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ+
m k : ℤ
z : ℂ
⊢ RuesDiff n m z = RuesDiff n (m + k * ↑↑n) z
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffMPeriodic | [156, 1] | [163, 10] | congr | case h
n : ℕ+
m k : ℤ
z : ℂ
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑' (k_1 : ℕ), if ↑↑n ∣ ↑k_1 + (m + k * ↑↑n) then z ^ k_1 / ↑k_1.factorial else 0 | case h.e_f
n : ℕ+
m k : ℤ
z : ℂ
⊢ (fun k => if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = fun k_1 =>
if ↑↑n ∣ ↑k_1 + (m + k * ↑↑n) then z ^ k_1 / ↑k_1.factorial else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ+
m k : ℤ
z : ℂ
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑' (k_1 : ℕ), if ↑↑n ∣ ↑k_1 + (m + k * ↑↑n) then z ^ k_1 / ↑k_1.factorial else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffMPeriodic | [156, 1] | [163, 10] | ext1 K | case h.e_f
n : ℕ+
m k : ℤ
z : ℂ
⊢ (fun k => if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = fun k_1 =>
if ↑↑n ∣ ↑k_1 + (m + k * ↑↑n) then z ^ k_1 / ↑k_1.factorial else 0 | case h.e_f.h
n : ℕ+
m k : ℤ
z : ℂ
K : ℕ
⊢ (if ↑↑n ∣ ↑K + m then z ^ K / ↑K.factorial else 0) = if ↑↑n ∣ ↑K + (m + k * ↑↑n) then z ^ K / ↑K.factorial else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e_f
n : ℕ+
m k : ℤ
z : ℂ
⊢ (fun k => if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = fun k_1 =>
if ↑↑n ∣ ↑k_1 + (m + k * ↑↑n) then z ^ k_1 / ↑k_1.factorial else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffMPeriodic | [156, 1] | [163, 10] | congr 1 | case h.e_f.h
n : ℕ+
m k : ℤ
z : ℂ
K : ℕ
⊢ (if ↑↑n ∣ ↑K + m then z ^ K / ↑K.factorial else 0) = if ↑↑n ∣ ↑K + (m + k * ↑↑n) then z ^ K / ↑K.factorial else 0 | case h.e_f.h.e_c
n : ℕ+
m k : ℤ
z : ℂ
K : ℕ
⊢ (↑↑n ∣ ↑K + m) = (↑↑n ∣ ↑K + (m + k * ↑↑n)) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e_f.h
n : ℕ+
m k : ℤ
z : ℂ
K : ℕ
⊢ (if ↑↑n ∣ ↑K + m then z ^ K / ↑K.factorial else 0) = if ↑↑n ∣ ↑K + (m + k * ↑↑n) then z ^ K / ↑K.factorial else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffMPeriodic | [156, 1] | [163, 10] | rw [Dvd.dvd.addMultiple (↑↑n) (↑K + m) k] | case h.e_f.h.e_c
n : ℕ+
m k : ℤ
z : ℂ
K : ℕ
⊢ (↑↑n ∣ ↑K + m) = (↑↑n ∣ ↑K + (m + k * ↑↑n)) | case h.e_f.h.e_c
n : ℕ+
m k : ℤ
z : ℂ
K : ℕ
⊢ (↑↑n ∣ ↑K + m + k * ↑↑n) = (↑↑n ∣ ↑K + (m + k * ↑↑n)) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e_f.h.e_c
n : ℕ+
m k : ℤ
z : ℂ
K : ℕ
⊢ (↑↑n ∣ ↑K + m) = (↑↑n ∣ ↑K + (m + k * ↑↑n))
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffMPeriodic | [156, 1] | [163, 10] | ring_nf | case h.e_f.h.e_c
n : ℕ+
m k : ℤ
z : ℂ
K : ℕ
⊢ (↑↑n ∣ ↑K + m + k * ↑↑n) = (↑↑n ∣ ↑K + (m + k * ↑↑n)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e_f.h.e_c
n : ℕ+
m k : ℤ
z : ℂ
K : ℕ
⊢ (↑↑n ∣ ↑K + m + k * ↑↑n) = (↑↑n ∣ ↑K + (m + k * ↑↑n))
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | simp_rw [RuesDiff] | n k : ℕ+
m : ℤ
z : ℂ
⊢ RuesDiff n m z = ∑ k₀ ∈ range ↑k, RuesDiff (n * k) (↑↑n * ↑k₀ + m) z | n k : ℕ+
m : ℤ
z : ℂ
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑ x ∈ range ↑k, ∑' (k_1 : ℕ), if ↑↑(n * k) ∣ ↑k_1 + (↑↑n * ↑x + m) then z ^ k_1 / ↑k_1.factorial else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
z : ℂ
⊢ RuesDiff n m z = ∑ k₀ ∈ range ↑k, RuesDiff (n * k) (↑↑n * ↑k₀ + m) z
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₀ : ∀ x ∈ range k, Summable (λ (k_1 : ℕ) => if ↑↑(n * k) ∣ ↑k_1 + (↑↑n * ↑x + m) then z ^ k_1 / ↑k_1.factorial else 0) := by
intros x _
exact RuesDiffSummable (n * k) _ z | n k : ℕ+
m : ℤ
z : ℂ
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑ x ∈ range ↑k, ∑' (k_1 : ℕ), if ↑↑(n * k) ∣ ↑k_1 + (↑↑n * ↑x + m) then z ^ k_1 / ↑k_1.factorial else 0 | n k : ℕ+
m : ℤ
z : ℂ
h₀ : ∀ x ∈ range ↑k, Summable fun k_1 => if ↑↑(n * k) ∣ ↑k_1 + (↑↑n * ↑x + m) then z ^ k_1 / ↑k_1.factorial else 0
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑ x ∈ range ↑k, ∑' (k_1 : ℕ), if ↑↑(n * k) ∣ ↑k_1 + (↑↑n * ↑x + m) then z ^ k_1 / ↑k_1.factorial else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
z : ℂ
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑ x ∈ range ↑k, ∑' (k_1 : ℕ), if ↑↑(n * k) ∣ ↑k_1 + (↑↑n * ↑x + m) then z ^ k_1 / ↑k_1.factorial else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | rw [← tsum_sum h₀] | n k : ℕ+
m : ℤ
z : ℂ
h₀ : ∀ x ∈ range ↑k, Summable fun k_1 => if ↑↑(n * k) ∣ ↑k_1 + (↑↑n * ↑x + m) then z ^ k_1 / ↑k_1.factorial else 0
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑ x ∈ range ↑k, ∑' (k_1 : ℕ), if ↑↑(n * k) ∣ ↑k_1 + (↑↑n * ↑x + m) then z ^ k_1 / ↑k_1.factorial else 0 | n k : ℕ+
m : ℤ
z : ℂ
h₀ : ∀ x ∈ range ↑k, Summable fun k_1 => if ↑↑(n * k) ∣ ↑k_1 + (↑↑n * ↑x + m) then z ^ k_1 / ↑k_1.factorial else 0
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑' (b : ℕ), ∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑b + (↑↑n * ↑i + m) then z ^ b / ↑b.factorial else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
z : ℂ
h₀ : ∀ x ∈ range ↑k, Summable fun k_1 => if ↑↑(n * k) ∣ ↑k_1 + (↑↑n * ↑x + m) then z ^ k_1 / ↑k_1.factorial else 0
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑ x ∈ range ↑k, ∑' (k_1 : ℕ), if ↑↑(n * k) ∣ ↑k_1 + ... |
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | clear h₀ | n k : ℕ+
m : ℤ
z : ℂ
h₀ : ∀ x ∈ range ↑k, Summable fun k_1 => if ↑↑(n * k) ∣ ↑k_1 + (↑↑n * ↑x + m) then z ^ k_1 / ↑k_1.factorial else 0
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑' (b : ℕ), ∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑b + (↑↑n * ↑i + m) then z ^ b / ↑b.factorial else 0 | n k : ℕ+
m : ℤ
z : ℂ
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑' (b : ℕ), ∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑b + (↑↑n * ↑i + m) then z ^ b / ↑b.factorial else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
z : ℂ
h₀ : ∀ x ∈ range ↑k, Summable fun k_1 => if ↑↑(n * k) ∣ ↑k_1 + (↑↑n * ↑x + m) then z ^ k_1 / ↑k_1.factorial else 0
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑' (b : ℕ), ∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑b + (↑↑n... |
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | congr | n k : ℕ+
m : ℤ
z : ℂ
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑' (b : ℕ), ∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑b + (↑↑n * ↑i + m) then z ^ b / ↑b.factorial else 0 | case e_f
n k : ℕ+
m : ℤ
z : ℂ
⊢ (fun k => if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = fun b =>
∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑b + (↑↑n * ↑i + m) then z ^ b / ↑b.factorial else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
z : ℂ
⊢ (∑' (k : ℕ), if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) =
∑' (b : ℕ), ∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑b + (↑↑n * ↑i + m) then z ^ b / ↑b.factorial else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | ext1 x | case e_f
n k : ℕ+
m : ℤ
z : ℂ
⊢ (fun k => if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = fun b =>
∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑b + (↑↑n * ↑i + m) then z ^ b / ↑b.factorial else 0 | case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0) =
∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m) then z ^ x / ↑x.factorial else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f
n k : ℕ+
m : ℤ
z : ℂ
⊢ (fun k => if ↑↑n ∣ ↑k + m then z ^ k / ↑k.factorial else 0) = fun b =>
∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑b + (↑↑n * ↑i + m) then z ^ b / ↑b.factorial else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | let f₀ : ℕ → Prop := (λ (i : ℕ) => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)) | case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0) =
∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m) then z ^ x / ↑x.factorial else 0 | case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0) =
∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m) then z ^ x / ↑x.factorial else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0) =
∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m) then z ^ x / ↑x.factorial else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₁ : ∀ i ∈ range ↑k, ∀ j ∈ range ↑k, f₀ i → f₀ j → i = j := by
intros i hir j hjr hi hj
simp [f₀] at hi hj
simp only [mem_range] at hir hjr
clear f₀ z
rw [←Int.modEq_zero_iff_dvd] at hi hj
have h₀ := Int.ModEq.sub hi hj
simp only [add_sub_add_left_eq_sub, add_sub_add_right_eq_sub, sub_self] at h₀
c... | case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0) =
∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m) then z ^ x / ↑x.factorial else 0 | case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
h₁ : ∀ i ∈ range ↑k, ∀ j ∈ range ↑k, f₀ i → f₀ j → i = j
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0) =
∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m) then z ^ x / ↑x.factorial else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0) =
∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m) then z ^ x / ↑x.factorial else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₂ := Finset.sum_ite_zero (range ↑k) f₀ h₁ (z ^ x / ↑x.factorial) | case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
h₁ : ∀ i ∈ range ↑k, ∀ j ∈ range ↑k, f₀ i → f₀ j → i = j
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0) =
∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m) then z ^ x / ↑x.factorial else 0 | case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
h₁ : ∀ i ∈ range ↑k, ∀ j ∈ range ↑k, f₀ i → f₀ j → i = j
h₂ :
(∑ i ∈ range ↑k, if f₀ i then z ^ x / ↑x.factorial else 0) = if ∃ i ∈ range ↑k, f₀ i then z ^ x / ↑x.factorial else 0
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factori... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
h₁ : ∀ i ∈ range ↑k, ∀ j ∈ range ↑k, f₀ i → f₀ j → i = j
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0) =
∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i +... |
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | clear h₁ | case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
h₁ : ∀ i ∈ range ↑k, ∀ j ∈ range ↑k, f₀ i → f₀ j → i = j
h₂ :
(∑ i ∈ range ↑k, if f₀ i then z ^ x / ↑x.factorial else 0) = if ∃ i ∈ range ↑k, f₀ i then z ^ x / ↑x.factorial else 0
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factori... | case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
h₂ :
(∑ i ∈ range ↑k, if f₀ i then z ^ x / ↑x.factorial else 0) = if ∃ i ∈ range ↑k, f₀ i then z ^ x / ↑x.factorial else 0
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0) =
∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑x + (↑↑n... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
h₁ : ∀ i ∈ range ↑k, ∀ j ∈ range ↑k, f₀ i → f₀ j → i = j
h₂ :
(∑ i ∈ range ↑k, if f₀ i then z ^ x / ↑x.factorial else 0) = if ∃ i ∈ range ↑k, f₀ i then z ^ x / ↑... |
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | simp only [PNat.mul_coe, Nat.cast_mul, mem_range, f₀] at h₂ ⊢ | case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
h₂ :
(∑ i ∈ range ↑k, if f₀ i then z ^ x / ↑x.factorial else 0) = if ∃ i ∈ range ↑k, f₀ i then z ^ x / ↑x.factorial else 0
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0) =
∑ i ∈ range ↑k, if ↑↑(n * k) ∣ ↑x + (↑↑n... | case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
h₂ :
(∑ x_1 ∈ range ↑k, if ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑x_1 + m) then z ^ x / ↑x.factorial else 0) =
if ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) then z ^ x / ↑x.factorial else 0
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factoria... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
h₂ :
(∑ i ∈ range ↑k, if f₀ i then z ^ x / ↑x.factorial else 0) = if ∃ i ∈ range ↑k, f₀ i then z ^ x / ↑x.factorial else 0
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.fac... |
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | rw [h₂] | case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
h₂ :
(∑ x_1 ∈ range ↑k, if ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑x_1 + m) then z ^ x / ↑x.factorial else 0) =
if ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) then z ^ x / ↑x.factorial else 0
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factoria... | case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
h₂ :
(∑ x_1 ∈ range ↑k, if ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑x_1 + m) then z ^ x / ↑x.factorial else 0) =
if ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) then z ^ x / ↑x.factorial else 0
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factoria... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
h₂ :
(∑ x_1 ∈ range ↑k, if ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑x_1 + m) then z ^ x / ↑x.factorial else 0) =
if ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) then z ^ x / ↑x... |
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | clear h₂ f₀ | case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
h₂ :
(∑ x_1 ∈ range ↑k, if ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑x_1 + m) then z ^ x / ↑x.factorial else 0) =
if ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) then z ^ x / ↑x.factorial else 0
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factoria... | case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0) =
if ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) then z ^ x / ↑x.factorial else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
h₂ :
(∑ x_1 ∈ range ↑k, if ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑x_1 + m) then z ^ x / ↑x.factorial else 0) =
if ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) then z ^ x / ↑x... |
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | congr | case e_f.h
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0) =
if ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) then z ^ x / ↑x.factorial else 0 | case e_f.h.e_c
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
⊢ (↑↑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
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
⊢ (if ↑↑n ∣ ↑x + m then z ^ x / ↑x.factorial else 0) =
if ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m) then z ^ x / ↑x.factorial else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | rw [←iff_eq_eq] | case e_f.h.e_c
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
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
⊢ ↑↑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
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] | constructor | case e_f.h.e_c
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
⊢ ↑↑n ∣ ↑x + m → ∃ i < ↑k, ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
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 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_c
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] | intros x _ | n k : ℕ+
m : ℤ
z : ℂ
⊢ ∀ x ∈ range ↑k, Summable fun k_1 => if ↑↑(n * k) ∣ ↑k_1 + (↑↑n * ↑x + m) then z ^ k_1 / ↑k_1.factorial else 0 | n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
a✝ : x ∈ range ↑k
⊢ Summable fun k_1 => if ↑↑(n * k) ∣ ↑k_1 + (↑↑n * ↑x + m) then z ^ k_1 / ↑k_1.factorial else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
z : ℂ
⊢ ∀ x ∈ range ↑k, Summable fun k_1 => if ↑↑(n * k) ∣ ↑k_1 + (↑↑n * ↑x + m) then z ^ k_1 / ↑k_1.factorial else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | exact RuesDiffSummable (n * k) _ z | n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
a✝ : x ∈ range ↑k
⊢ Summable fun k_1 => if ↑↑(n * k) ∣ ↑k_1 + (↑↑n * ↑x + m) then z ^ k_1 / ↑k_1.factorial else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
a✝ : x ∈ range ↑k
⊢ Summable fun k_1 => if ↑↑(n * k) ∣ ↑k_1 + (↑↑n * ↑x + m) then z ^ k_1 / ↑k_1.factorial else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | intros i hir j hjr hi hj | n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
⊢ ∀ i ∈ range ↑k, ∀ j ∈ range ↑k, f₀ i → f₀ j → i = j | n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
i : ℕ
hir : i ∈ range ↑k
j : ℕ
hjr : j ∈ range ↑k
hi : f₀ i
hj : f₀ j
⊢ i = j | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
⊢ ∀ i ∈ range ↑k, ∀ j ∈ range ↑k, f₀ i → f₀ j → i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | simp [f₀] at hi hj | n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
i : ℕ
hir : i ∈ range ↑k
j : ℕ
hjr : j ∈ range ↑k
hi : f₀ i
hj : f₀ j
⊢ i = j | n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
i : ℕ
hir : i ∈ range ↑k
j : ℕ
hjr : j ∈ range ↑k
hi : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
hj : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑j + m)
⊢ i = j | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
i : ℕ
hir : i ∈ range ↑k
j : ℕ
hjr : j ∈ range ↑k
hi : f₀ i
hj : f₀ j
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | simp only [mem_range] at hir hjr | n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
i : ℕ
hir : i ∈ range ↑k
j : ℕ
hjr : j ∈ range ↑k
hi : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
hj : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑j + m)
⊢ i = j | n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
i j : ℕ
hi : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
hj : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑j + m)
hir : i < ↑k
hjr : j < ↑k
⊢ i = j | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
i : ℕ
hir : i ∈ range ↑k
j : ℕ
hjr : j ∈ range ↑k
hi : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
hj : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑j + m)
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | clear f₀ z | n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
i j : ℕ
hi : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
hj : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑j + m)
hir : i < ↑k
hjr : j < ↑k
⊢ i = j | n k : ℕ+
m : ℤ
x i j : ℕ
hi : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
hj : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑j + m)
hir : i < ↑k
hjr : j < ↑k
⊢ i = j | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
z : ℂ
x : ℕ
f₀ : ℕ → Prop := fun i => ↑↑(n * k) ∣ ↑x + (↑↑n * ↑i + m)
i j : ℕ
hi : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
hj : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑j + m)
hir : i < ↑k
hjr : j < ↑k
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | rw [←Int.modEq_zero_iff_dvd] at hi hj | n k : ℕ+
m : ℤ
x i j : ℕ
hi : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
hj : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑j + m)
hir : i < ↑k
hjr : j < ↑k
⊢ i = j | n k : ℕ+
m : ℤ
x i j : ℕ
hi : ↑x + (↑↑n * ↑i + m) ≡ 0 [ZMOD ↑↑n * ↑↑k]
hj : ↑x + (↑↑n * ↑j + m) ≡ 0 [ZMOD ↑↑n * ↑↑k]
hir : i < ↑k
hjr : j < ↑k
⊢ i = j | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
x i j : ℕ
hi : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑i + m)
hj : ↑↑n * ↑↑k ∣ ↑x + (↑↑n * ↑j + m)
hir : i < ↑k
hjr : j < ↑k
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₀ := Int.ModEq.sub hi hj | n k : ℕ+
m : ℤ
x i j : ℕ
hi : ↑x + (↑↑n * ↑i + m) ≡ 0 [ZMOD ↑↑n * ↑↑k]
hj : ↑x + (↑↑n * ↑j + m) ≡ 0 [ZMOD ↑↑n * ↑↑k]
hir : i < ↑k
hjr : j < ↑k
⊢ i = j | n k : ℕ+
m : ℤ
x i j : ℕ
hi : ↑x + (↑↑n * ↑i + m) ≡ 0 [ZMOD ↑↑n * ↑↑k]
hj : ↑x + (↑↑n * ↑j + m) ≡ 0 [ZMOD ↑↑n * ↑↑k]
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑x + (↑↑n * ↑i + m) - (↑x + (↑↑n * ↑j + m)) ≡ 0 - 0 [ZMOD ↑↑n * ↑↑k]
⊢ i = j | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
x i j : ℕ
hi : ↑x + (↑↑n * ↑i + m) ≡ 0 [ZMOD ↑↑n * ↑↑k]
hj : ↑x + (↑↑n * ↑j + m) ≡ 0 [ZMOD ↑↑n * ↑↑k]
hir : i < ↑k
hjr : j < ↑k
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | simp only [add_sub_add_left_eq_sub, add_sub_add_right_eq_sub, sub_self] at h₀ | n k : ℕ+
m : ℤ
x i j : ℕ
hi : ↑x + (↑↑n * ↑i + m) ≡ 0 [ZMOD ↑↑n * ↑↑k]
hj : ↑x + (↑↑n * ↑j + m) ≡ 0 [ZMOD ↑↑n * ↑↑k]
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑x + (↑↑n * ↑i + m) - (↑x + (↑↑n * ↑j + m)) ≡ 0 - 0 [ZMOD ↑↑n * ↑↑k]
⊢ i = j | n k : ℕ+
m : ℤ
x i j : ℕ
hi : ↑x + (↑↑n * ↑i + m) ≡ 0 [ZMOD ↑↑n * ↑↑k]
hj : ↑x + (↑↑n * ↑j + m) ≡ 0 [ZMOD ↑↑n * ↑↑k]
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑↑n * ↑i - ↑↑n * ↑j ≡ 0 [ZMOD ↑↑n * ↑↑k]
⊢ i = j | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
x i j : ℕ
hi : ↑x + (↑↑n * ↑i + m) ≡ 0 [ZMOD ↑↑n * ↑↑k]
hj : ↑x + (↑↑n * ↑j + m) ≡ 0 [ZMOD ↑↑n * ↑↑k]
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑x + (↑↑n * ↑i + m) - (↑x + (↑↑n * ↑j + m)) ≡ 0 - 0 [ZMOD ↑↑n * ↑↑k]
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | clear hi hj | n k : ℕ+
m : ℤ
x i j : ℕ
hi : ↑x + (↑↑n * ↑i + m) ≡ 0 [ZMOD ↑↑n * ↑↑k]
hj : ↑x + (↑↑n * ↑j + m) ≡ 0 [ZMOD ↑↑n * ↑↑k]
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑↑n * ↑i - ↑↑n * ↑j ≡ 0 [ZMOD ↑↑n * ↑↑k]
⊢ i = j | n k : ℕ+
m : ℤ
x i j : ℕ
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑↑n * ↑i - ↑↑n * ↑j ≡ 0 [ZMOD ↑↑n * ↑↑k]
⊢ i = j | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ+
m : ℤ
x i j : ℕ
hi : ↑x + (↑↑n * ↑i + m) ≡ 0 [ZMOD ↑↑n * ↑↑k]
hj : ↑x + (↑↑n * ↑j + m) ≡ 0 [ZMOD ↑↑n * ↑↑k]
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑↑n * ↑i - ↑↑n * ↑j ≡ 0 [ZMOD ↑↑n * ↑↑k]
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | rw [Int.modEq_zero_iff_dvd, (show (↑↑n * ↑i - ↑↑n * ↑j : ℤ) = ↑↑n * (↑i - ↑j) by ring)] at h₀ | n k : ℕ+
m : ℤ
x i j : ℕ
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑↑n * ↑i - ↑↑n * ↑j ≡ 0 [ZMOD ↑↑n * ↑↑k]
⊢ i = j | n k : ℕ+
m : ℤ
x i j : ℕ
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j)
⊢ i = j | 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 * ↑i - ↑↑n * ↑j ≡ 0 [ZMOD ↑↑n * ↑↑k]
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₁ : (n : ℤ) ≠ 0 := by
exact Ne.symm (NeZero.ne' (n : ℤ)) | n k : ℕ+
m : ℤ
x i j : ℕ
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j)
⊢ i = j | n k : ℕ+
m : ℤ
x i j : ℕ
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j)
h₁ : ↑↑n ≠ 0
⊢ i = j | 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)
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₂ : (k : ℤ) ∣ ↑i - ↑j := by exact (mul_dvd_mul_iff_left h₁).mp h₀ | n k : ℕ+
m : ℤ
x i j : ℕ
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j)
h₁ : ↑↑n ≠ 0
⊢ i = j | n k : ℕ+
m : ℤ
x i j : ℕ
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j)
h₁ : ↑↑n ≠ 0
h₂ : ↑↑k ∣ ↑i - ↑j
⊢ i = j | 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
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | obtain ⟨y, h₃⟩ := h₂ | n k : ℕ+
m : ℤ
x i j : ℕ
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j)
h₁ : ↑↑n ≠ 0
h₂ : ↑↑k ∣ ↑i - ↑j
⊢ i = j | case intro
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
⊢ i = j | 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
h₂ : ↑↑k ∣ ↑i - ↑j
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₄ : k * y < k := by
linarith | case intro
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
⊢ i = j | case intro
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
⊢ i = j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
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
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₅ : -k < k * y := by
linarith | case intro
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
⊢ i = j | case intro
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
⊢ i = j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
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
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₆ : (k : ℤ) > 0 := by
linarith | case intro
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
⊢ i = j | case intro
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
⊢ i = j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
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
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₇ : y < 1 := by
exact (mul_lt_iff_lt_one_right h₆).mp h₄ | case intro
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
⊢ i = j | case intro
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
⊢ i = j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
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
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | nth_rw 1 [(show -(k : ℤ) = ↑↑k * -1 by ring)] at h₅ | case intro
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
⊢ i = j | case intro
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
⊢ i = j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
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
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₈ : -1 < y := by
exact (mul_lt_mul_left h₆).mp h₅ | case intro
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
⊢ i = j | case intro
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
⊢ i = j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
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
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₉ : y = 0 := by
linarith | case intro
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
⊢ i = j | case intro
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
h₉ : y = 0
⊢ i = j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
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
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | rw [h₉] at h₃ | case intro
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
h₉ : y = 0
⊢ i = j | case intro
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 * 0
h₄ : ↑↑k * y < ↑↑k
h₅ : ↑↑k * -1 < ↑↑k * y
h₆ : ↑↑k > 0
h₇ : y < 1
h₈ : -1 < y
h₉ : y = 0
⊢ i = j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
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
h₉ : y = 0
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | simp only [mul_zero] at h₃ | case intro
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 * 0
h₄ : ↑↑k * y < ↑↑k
h₅ : ↑↑k * -1 < ↑↑k * y
h₆ : ↑↑k > 0
h₇ : y < 1
h₈ : -1 < y
h₉ : y = 0
⊢ i = j | case intro
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
h₄ : ↑↑k * y < ↑↑k
h₅ : ↑↑k * -1 < ↑↑k * y
h₆ : ↑↑k > 0
h₇ : y < 1
h₈ : -1 < y
h₉ : y = 0
h₃ : ↑i - ↑j = 0
⊢ i = j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
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 * 0
h₄ : ↑↑k * y < ↑↑k
h₅ : ↑↑k * -1 < ↑↑k * y
h₆ : ↑↑k > 0
h₇ : y < 1
h₈ : -1 < y
h₉ : y = 0
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | clear n hir hjr m h₀ h₁ h₄ h₅ h₆ h₇ h₈ h₉ y x k | case intro
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
h₄ : ↑↑k * y < ↑↑k
h₅ : ↑↑k * -1 < ↑↑k * y
h₆ : ↑↑k > 0
h₇ : y < 1
h₈ : -1 < y
h₉ : y = 0
h₃ : ↑i - ↑j = 0
⊢ i = j | case intro
i j : ℕ
h₃ : ↑i - ↑j = 0
⊢ i = j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
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
h₄ : ↑↑k * y < ↑↑k
h₅ : ↑↑k * -1 < ↑↑k * y
h₆ : ↑↑k > 0
h₇ : y < 1
h₈ : -1 < y
h₉ : y = 0
h₃ : ↑i - ↑j = 0
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | refine Int.ofNat_inj.mp ?intro.a | case intro
i j : ℕ
h₃ : ↑i - ↑j = 0
⊢ i = j | case intro.a
i j : ℕ
h₃ : ↑i - ↑j = 0
⊢ ↑i = ↑j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
i j : ℕ
h₃ : ↑i - ↑j = 0
⊢ i = j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | have h₀ := congrArg (λ (k : ℤ) => k + j) h₃ | case intro.a
i j : ℕ
h₃ : ↑i - ↑j = 0
⊢ ↑i = ↑j | case intro.a
i j : ℕ
h₃ : ↑i - ↑j = 0
h₀ : (fun k => k + ↑j) (↑i - ↑j) = (fun k => k + ↑j) 0
⊢ ↑i = ↑j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.a
i j : ℕ
h₃ : ↑i - ↑j = 0
⊢ ↑i = ↑j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | simp only [sub_add_cancel, zero_add] at h₀ | case intro.a
i j : ℕ
h₃ : ↑i - ↑j = 0
h₀ : (fun k => k + ↑j) (↑i - ↑j) = (fun k => k + ↑j) 0
⊢ ↑i = ↑j | case intro.a
i j : ℕ
h₃ : ↑i - ↑j = 0
h₀ : ↑i = ↑j
⊢ ↑i = ↑j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.a
i j : ℕ
h₃ : ↑i - ↑j = 0
h₀ : (fun k => k + ↑j) (↑i - ↑j) = (fun k => k + ↑j) 0
⊢ ↑i = ↑j
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | exact h₀ | case intro.a
i j : ℕ
h₃ : ↑i - ↑j = 0
h₀ : ↑i = ↑j
⊢ ↑i = ↑j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.a
i j : ℕ
h₃ : ↑i - ↑j = 0
h₀ : ↑i = ↑j
⊢ ↑i = ↑j
TACTIC:
|
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 - ↑↑n * ↑j
⊢ ↑↑n * ↑i - ↑↑n * ↑j = ↑↑n * (↑i - ↑j) | 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 - ↑↑n * ↑j
⊢ ↑↑n * ↑i - ↑↑n * ↑j = ↑↑n * (↑i - ↑j)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | exact Ne.symm (NeZero.ne' (n : ℤ)) | n k : ℕ+
m : ℤ
x i j : ℕ
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j)
⊢ ↑↑n ≠ 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)
⊢ ↑↑n ≠ 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | exact (mul_dvd_mul_iff_left h₁).mp h₀ | n k : ℕ+
m : ℤ
x i j : ℕ
hir : i < ↑k
hjr : j < ↑k
h₀ : ↑↑n * ↑↑k ∣ ↑↑n * (↑i - ↑j)
h₁ : ↑↑n ≠ 0
⊢ ↑↑k ∣ ↑i - ↑j | 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
⊢ ↑↑k ∣ ↑i - ↑j
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
⊢ ↑↑k * y < ↑↑k | 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
⊢ ↑↑k * y < ↑↑k
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
⊢ -↑↑k < ↑↑k * 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
⊢ -↑↑k < ↑↑k * 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 < ↑↑k * y
⊢ ↑↑k > 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 < ↑↑k * y
⊢ ↑↑k > 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffSumOfRuesDiff | [165, 1] | [256, 48] | exact (mul_lt_iff_lt_one_right h₆).mp h₄ | 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
⊢ y < 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
⊢ y < 1
TACTIC:
|
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