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 | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | ring_nf | z : ℂ
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * ((1 - I) / 2)) * cexp (z * ((1 + I) / 2)) * 4⁻¹ +
cexp (z * ((1 - I) / 2)) * cexp (-(z * ((1 + I) / 2))) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (z * ((1 + I) / 2)) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (-(z * ((1 + I) / 2))) * 4⁻¹ | z : ℂ
⊢ cexp (-(z * I)) * (1 / 4) + cexp (-z) * (1 / 4) + cexp (z * I) * (1 / 4) + cexp z * (1 / 4) =
cexp (z * (1 / 2) + z * I * (-1 / 2)) * cexp (z * (1 / 2) + z * I * (1 / 2)) * (1 / 4) +
cexp (z * (1 / 2) + z * I * (-1 / 2)) * cexp (z * (-1 / 2) + z * I * (-1 / 2)) * (1 / 4) +
cexp (z * (1 / 2) + z * I * (1 / 2)) * cexp (z * (-1 / 2) + z * I * (1 / 2)) * (1 / 4) +
cexp (z * (-1 / 2) + z * I * (-1 / 2)) * cexp (z * (-1 / 2) + z * I * (1 / 2)) * (1 / 4) | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ cexp (z * -I) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * ((1 - I) / 2)) * cexp (z * ((1 + I) / 2)) * 4⁻¹ +
cexp (z * ((1 - I) / 2)) * cexp (-(z * ((1 + I) / 2))) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (z * ((1 + I) / 2)) * 4⁻¹ +
cexp (-(z * ((1 - I) / 2))) * cexp (-(z * ((1 + I) / 2))) * 4⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | simp only [Int.ofNat_eq_coe, Nat.cast_one, Int.cast_one, Nat.cast_ofNat, one_div,
Int.cast_negOfNat, mul_neg, mul_one, neg_mul] | z : ℂ
⊢ cexp (-(z * I)) * (1 / 4) + cexp (-z) * (1 / 4) + cexp (z * I) * (1 / 4) + cexp z * (1 / 4) =
cexp (z * (1 / 2) + z * I * (-1 / 2)) * cexp (z * (1 / 2) + z * I * (1 / 2)) * (1 / 4) +
cexp (z * (1 / 2) + z * I * (-1 / 2)) * cexp (z * (-1 / 2) + z * I * (-1 / 2)) * (1 / 4) +
cexp (z * (1 / 2) + z * I * (1 / 2)) * cexp (z * (-1 / 2) + z * I * (1 / 2)) * (1 / 4) +
cexp (z * (-1 / 2) + z * I * (-1 / 2)) * cexp (z * (-1 / 2) + z * I * (1 / 2)) * (1 / 4) | z : ℂ
⊢ cexp (-(z * I)) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * 2⁻¹ + z * I * (-1 / 2)) * cexp (z * 2⁻¹ + z * I * 2⁻¹) * 4⁻¹ +
cexp (z * 2⁻¹ + z * I * (-1 / 2)) * cexp (z * (-1 / 2) + z * I * (-1 / 2)) * 4⁻¹ +
cexp (z * 2⁻¹ + z * I * 2⁻¹) * cexp (z * (-1 / 2) + z * I * 2⁻¹) * 4⁻¹ +
cexp (z * (-1 / 2) + z * I * (-1 / 2)) * cexp (z * (-1 / 2) + z * I * 2⁻¹) * 4⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ cexp (-(z * I)) * (1 / 4) + cexp (-z) * (1 / 4) + cexp (z * I) * (1 / 4) + cexp z * (1 / 4) =
cexp (z * (1 / 2) + z * I * (-1 / 2)) * cexp (z * (1 / 2) + z * I * (1 / 2)) * (1 / 4) +
cexp (z * (1 / 2) + z * I * (-1 / 2)) * cexp (z * (-1 / 2) + z * I * (-1 / 2)) * (1 / 4) +
cexp (z * (1 / 2) + z * I * (1 / 2)) * cexp (z * (-1 / 2) + z * I * (1 / 2)) * (1 / 4) +
cexp (z * (-1 / 2) + z * I * (-1 / 2)) * cexp (z * (-1 / 2) + z * I * (1 / 2)) * (1 / 4)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | simp_rw [Complex.exp_add] | z : ℂ
⊢ cexp (-(z * I)) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * 2⁻¹ + z * I * (-1 / 2)) * cexp (z * 2⁻¹ + z * I * 2⁻¹) * 4⁻¹ +
cexp (z * 2⁻¹ + z * I * (-1 / 2)) * cexp (z * (-1 / 2) + z * I * (-1 / 2)) * 4⁻¹ +
cexp (z * 2⁻¹ + z * I * 2⁻¹) * cexp (z * (-1 / 2) + z * I * 2⁻¹) * 4⁻¹ +
cexp (z * (-1 / 2) + z * I * (-1 / 2)) * cexp (z * (-1 / 2) + z * I * 2⁻¹) * 4⁻¹ | z : ℂ
⊢ cexp (-(z * I)) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * 2⁻¹) * cexp (z * I * (-1 / 2)) * (cexp (z * 2⁻¹) * cexp (z * I * 2⁻¹)) * 4⁻¹ +
cexp (z * 2⁻¹) * cexp (z * I * (-1 / 2)) * (cexp (z * (-1 / 2)) * cexp (z * I * (-1 / 2))) * 4⁻¹ +
cexp (z * 2⁻¹) * cexp (z * I * 2⁻¹) * (cexp (z * (-1 / 2)) * cexp (z * I * 2⁻¹)) * 4⁻¹ +
cexp (z * (-1 / 2)) * cexp (z * I * (-1 / 2)) * (cexp (z * (-1 / 2)) * cexp (z * I * 2⁻¹)) * 4⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ cexp (-(z * I)) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * 2⁻¹ + z * I * (-1 / 2)) * cexp (z * 2⁻¹ + z * I * 2⁻¹) * 4⁻¹ +
cexp (z * 2⁻¹ + z * I * (-1 / 2)) * cexp (z * (-1 / 2) + z * I * (-1 / 2)) * 4⁻¹ +
cexp (z * 2⁻¹ + z * I * 2⁻¹) * cexp (z * (-1 / 2) + z * I * 2⁻¹) * 4⁻¹ +
cexp (z * (-1 / 2) + z * I * (-1 / 2)) * cexp (z * (-1 / 2) + z * I * 2⁻¹) * 4⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | ring_nf | z : ℂ
⊢ cexp (-(z * I)) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * 2⁻¹) * cexp (z * I * (-1 / 2)) * (cexp (z * 2⁻¹) * cexp (z * I * 2⁻¹)) * 4⁻¹ +
cexp (z * 2⁻¹) * cexp (z * I * (-1 / 2)) * (cexp (z * (-1 / 2)) * cexp (z * I * (-1 / 2))) * 4⁻¹ +
cexp (z * 2⁻¹) * cexp (z * I * 2⁻¹) * (cexp (z * (-1 / 2)) * cexp (z * I * 2⁻¹)) * 4⁻¹ +
cexp (z * (-1 / 2)) * cexp (z * I * (-1 / 2)) * (cexp (z * (-1 / 2)) * cexp (z * I * 2⁻¹)) * 4⁻¹ | z : ℂ
⊢ cexp (-(z * I)) * (1 / 4) + cexp (-z) * (1 / 4) + cexp (z * I) * (1 / 4) + cexp z * (1 / 4) =
cexp (z * (1 / 2)) * cexp (z * I * (-1 / 2)) ^ 2 * cexp (z * (-1 / 2)) * (1 / 4) +
cexp (z * (1 / 2)) * cexp (z * I * (1 / 2)) ^ 2 * cexp (z * (-1 / 2)) * (1 / 4) +
cexp (z * (1 / 2)) ^ 2 * cexp (z * I * (-1 / 2)) * cexp (z * I * (1 / 2)) * (1 / 4) +
cexp (z * I * (-1 / 2)) * cexp (z * I * (1 / 2)) * cexp (z * (-1 / 2)) ^ 2 * (1 / 4) | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ cexp (-(z * I)) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * 2⁻¹) * cexp (z * I * (-1 / 2)) * (cexp (z * 2⁻¹) * cexp (z * I * 2⁻¹)) * 4⁻¹ +
cexp (z * 2⁻¹) * cexp (z * I * (-1 / 2)) * (cexp (z * (-1 / 2)) * cexp (z * I * (-1 / 2))) * 4⁻¹ +
cexp (z * 2⁻¹) * cexp (z * I * 2⁻¹) * (cexp (z * (-1 / 2)) * cexp (z * I * 2⁻¹)) * 4⁻¹ +
cexp (z * (-1 / 2)) * cexp (z * I * (-1 / 2)) * (cexp (z * (-1 / 2)) * cexp (z * I * 2⁻¹)) * 4⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | simp only [Int.ofNat_eq_coe, Nat.cast_one, Int.cast_one, Nat.cast_ofNat, one_div,
Int.cast_negOfNat, mul_neg, mul_one, neg_mul] | z : ℂ
⊢ cexp (-(z * I)) * (1 / 4) + cexp (-z) * (1 / 4) + cexp (z * I) * (1 / 4) + cexp z * (1 / 4) =
cexp (z * (1 / 2)) * cexp (z * I * (-1 / 2)) ^ 2 * cexp (z * (-1 / 2)) * (1 / 4) +
cexp (z * (1 / 2)) * cexp (z * I * (1 / 2)) ^ 2 * cexp (z * (-1 / 2)) * (1 / 4) +
cexp (z * (1 / 2)) ^ 2 * cexp (z * I * (-1 / 2)) * cexp (z * I * (1 / 2)) * (1 / 4) +
cexp (z * I * (-1 / 2)) * cexp (z * I * (1 / 2)) * cexp (z * (-1 / 2)) ^ 2 * (1 / 4) | z : ℂ
⊢ cexp (-(z * I)) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * 2⁻¹) * cexp (z * I * (-1 / 2)) ^ 2 * cexp (z * (-1 / 2)) * 4⁻¹ +
cexp (z * 2⁻¹) * cexp (z * I * 2⁻¹) ^ 2 * cexp (z * (-1 / 2)) * 4⁻¹ +
cexp (z * 2⁻¹) ^ 2 * cexp (z * I * (-1 / 2)) * cexp (z * I * 2⁻¹) * 4⁻¹ +
cexp (z * I * (-1 / 2)) * cexp (z * I * 2⁻¹) * cexp (z * (-1 / 2)) ^ 2 * 4⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ cexp (-(z * I)) * (1 / 4) + cexp (-z) * (1 / 4) + cexp (z * I) * (1 / 4) + cexp z * (1 / 4) =
cexp (z * (1 / 2)) * cexp (z * I * (-1 / 2)) ^ 2 * cexp (z * (-1 / 2)) * (1 / 4) +
cexp (z * (1 / 2)) * cexp (z * I * (1 / 2)) ^ 2 * cexp (z * (-1 / 2)) * (1 / 4) +
cexp (z * (1 / 2)) ^ 2 * cexp (z * I * (-1 / 2)) * cexp (z * I * (1 / 2)) * (1 / 4) +
cexp (z * I * (-1 / 2)) * cexp (z * I * (1 / 2)) * cexp (z * (-1 / 2)) ^ 2 * (1 / 4)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | simp_rw [←Complex.exp_nat_mul, ←Complex.exp_add] | z : ℂ
⊢ cexp (-(z * I)) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * 2⁻¹) * cexp (z * I * (-1 / 2)) ^ 2 * cexp (z * (-1 / 2)) * 4⁻¹ +
cexp (z * 2⁻¹) * cexp (z * I * 2⁻¹) ^ 2 * cexp (z * (-1 / 2)) * 4⁻¹ +
cexp (z * 2⁻¹) ^ 2 * cexp (z * I * (-1 / 2)) * cexp (z * I * 2⁻¹) * 4⁻¹ +
cexp (z * I * (-1 / 2)) * cexp (z * I * 2⁻¹) * cexp (z * (-1 / 2)) ^ 2 * 4⁻¹ | z : ℂ
⊢ cexp (-(z * I)) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * 2⁻¹ + ↑2 * (z * I * (-1 / 2)) + z * (-1 / 2)) * 4⁻¹ +
cexp (z * 2⁻¹ + ↑2 * (z * I * 2⁻¹) + z * (-1 / 2)) * 4⁻¹ +
cexp (↑2 * (z * 2⁻¹) + z * I * (-1 / 2) + z * I * 2⁻¹) * 4⁻¹ +
cexp (z * I * (-1 / 2) + z * I * 2⁻¹ + ↑2 * (z * (-1 / 2))) * 4⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ cexp (-(z * I)) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * 2⁻¹) * cexp (z * I * (-1 / 2)) ^ 2 * cexp (z * (-1 / 2)) * 4⁻¹ +
cexp (z * 2⁻¹) * cexp (z * I * 2⁻¹) ^ 2 * cexp (z * (-1 / 2)) * 4⁻¹ +
cexp (z * 2⁻¹) ^ 2 * cexp (z * I * (-1 / 2)) * cexp (z * I * 2⁻¹) * 4⁻¹ +
cexp (z * I * (-1 / 2)) * cexp (z * I * 2⁻¹) * cexp (z * (-1 / 2)) ^ 2 * 4⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | ring_nf | z : ℂ
⊢ cexp (-(z * I)) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * 2⁻¹ + ↑2 * (z * I * (-1 / 2)) + z * (-1 / 2)) * 4⁻¹ +
cexp (z * 2⁻¹ + ↑2 * (z * I * 2⁻¹) + z * (-1 / 2)) * 4⁻¹ +
cexp (↑2 * (z * 2⁻¹) + z * I * (-1 / 2) + z * I * 2⁻¹) * 4⁻¹ +
cexp (z * I * (-1 / 2) + z * I * 2⁻¹ + ↑2 * (z * (-1 / 2))) * 4⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ cexp (-(z * I)) * 4⁻¹ + cexp (-z) * 4⁻¹ + cexp (z * I) * 4⁻¹ + cexp z * 4⁻¹ =
cexp (z * 2⁻¹ + ↑2 * (z * I * (-1 / 2)) + z * (-1 / 2)) * 4⁻¹ +
cexp (z * 2⁻¹ + ↑2 * (z * I * 2⁻¹) + z * (-1 / 2)) * 4⁻¹ +
cexp (↑2 * (z * 2⁻¹) + z * I * (-1 / 2) + z * I * 2⁻¹) * 4⁻¹ +
cexp (z * I * (-1 / 2) + z * I * 2⁻¹ + ↑2 * (z * (-1 / 2))) * 4⁻¹
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | rfl | z : ℂ
⊢ ↑4 = 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ ↑4 = 4
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | have h₁b := ExpToNatPowersOfI 3 | z : ℂ
⊢ cexp (↑π * I * (3 / 2)) = -I | z : ℂ
h₁b : cexp (↑π * I * ↑3 / 2) = I ^ 3
⊢ cexp (↑π * I * (3 / 2)) = -I | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ cexp (↑π * I * (3 / 2)) = -I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | simp only [Nat.cast_ofNat] at h₁b | z : ℂ
h₁b : cexp (↑π * I * ↑3 / 2) = I ^ 3
⊢ cexp (↑π * I * (3 / 2)) = -I | z : ℂ
h₁b : cexp (↑π * I * 3 / 2) = I ^ 3
⊢ cexp (↑π * I * (3 / 2)) = -I | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₁b : cexp (↑π * I * ↑3 / 2) = I ^ 3
⊢ cexp (↑π * I * (3 / 2)) = -I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | have h₁b₁ : ↑π * I * 3 / 2 = ↑π * I * (3 / 2) := by
ring | z : ℂ
h₁b : cexp (↑π * I * 3 / 2) = I ^ 3
⊢ cexp (↑π * I * (3 / 2)) = -I | z : ℂ
h₁b : cexp (↑π * I * 3 / 2) = I ^ 3
h₁b₁ : ↑π * I * 3 / 2 = ↑π * I * (3 / 2)
⊢ cexp (↑π * I * (3 / 2)) = -I | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₁b : cexp (↑π * I * 3 / 2) = I ^ 3
⊢ cexp (↑π * I * (3 / 2)) = -I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | rw [h₁b₁] at h₁b | z : ℂ
h₁b : cexp (↑π * I * 3 / 2) = I ^ 3
h₁b₁ : ↑π * I * 3 / 2 = ↑π * I * (3 / 2)
⊢ cexp (↑π * I * (3 / 2)) = -I | z : ℂ
h₁b : cexp (↑π * I * (3 / 2)) = I ^ 3
h₁b₁ : ↑π * I * 3 / 2 = ↑π * I * (3 / 2)
⊢ cexp (↑π * I * (3 / 2)) = -I | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₁b : cexp (↑π * I * 3 / 2) = I ^ 3
h₁b₁ : ↑π * I * 3 / 2 = ↑π * I * (3 / 2)
⊢ cexp (↑π * I * (3 / 2)) = -I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | rw [h₁b] | z : ℂ
h₁b : cexp (↑π * I * (3 / 2)) = I ^ 3
h₁b₁ : ↑π * I * 3 / 2 = ↑π * I * (3 / 2)
⊢ cexp (↑π * I * (3 / 2)) = -I | z : ℂ
h₁b : cexp (↑π * I * (3 / 2)) = I ^ 3
h₁b₁ : ↑π * I * 3 / 2 = ↑π * I * (3 / 2)
⊢ I ^ 3 = -I | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₁b : cexp (↑π * I * (3 / 2)) = I ^ 3
h₁b₁ : ↑π * I * 3 / 2 = ↑π * I * (3 / 2)
⊢ cexp (↑π * I * (3 / 2)) = -I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | clear h₁b h₁b₁ | z : ℂ
h₁b : cexp (↑π * I * (3 / 2)) = I ^ 3
h₁b₁ : ↑π * I * 3 / 2 = ↑π * I * (3 / 2)
⊢ I ^ 3 = -I | z : ℂ
⊢ I ^ 3 = -I | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₁b : cexp (↑π * I * (3 / 2)) = I ^ 3
h₁b₁ : ↑π * I * 3 / 2 = ↑π * I * (3 / 2)
⊢ I ^ 3 = -I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | have h₅ : I ^ (3 : ℕ) = I ^ (3 : ℤ) := by
exact rfl | z : ℂ
⊢ I ^ 3 = -I | z : ℂ
h₅ : I ^ 3 = I ^ 3
⊢ I ^ 3 = -I | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ I ^ 3 = -I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | clear h₅ | z : ℂ
h₅ : I ^ 3 = I ^ 3
⊢ I ^ 3 = -I | z : ℂ
⊢ I ^ 3 = -I | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₅ : I ^ 3 = I ^ 3
⊢ I ^ 3 = -I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | have h₆ : (3 : ℤ) = 2 + 1 := by
exact rfl | z : ℂ
⊢ I ^ 3 = -I | z : ℂ
h₆ : 3 = 2 + 1
⊢ I ^ 3 = -I | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ I ^ 3 = -I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | rw [h₆] | z : ℂ
h₆ : 3 = 2 + 1
⊢ I ^ 3 = -I | z : ℂ
h₆ : 3 = 2 + 1
⊢ I ^ (2 + 1) = -I | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₆ : 3 = 2 + 1
⊢ I ^ 3 = -I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | clear h₆ | z : ℂ
h₆ : 3 = 2 + 1
⊢ I ^ (2 + 1) = -I | z : ℂ
⊢ I ^ (2 + 1) = -I | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₆ : 3 = 2 + 1
⊢ I ^ (2 + 1) = -I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | rw [zpow_add₀ I_ne_zero] | z : ℂ
⊢ I ^ (2 + 1) = -I | z : ℂ
⊢ I ^ 2 * I ^ 1 = -I | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ I ^ (2 + 1) = -I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | have h₇ : (2 : ℤ) = 1 + 1 := by
exact rfl | z : ℂ
⊢ I ^ 2 * I ^ 1 = -I | z : ℂ
h₇ : 2 = 1 + 1
⊢ I ^ 2 * I ^ 1 = -I | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ I ^ 2 * I ^ 1 = -I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | rw [h₇] | z : ℂ
h₇ : 2 = 1 + 1
⊢ I ^ 2 * I ^ 1 = -I | z : ℂ
h₇ : 2 = 1 + 1
⊢ I ^ (1 + 1) * I ^ 1 = -I | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₇ : 2 = 1 + 1
⊢ I ^ 2 * I ^ 1 = -I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | clear h₇ | z : ℂ
h₇ : 2 = 1 + 1
⊢ I ^ (1 + 1) * I ^ 1 = -I | z : ℂ
⊢ I ^ (1 + 1) * I ^ 1 = -I | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₇ : 2 = 1 + 1
⊢ I ^ (1 + 1) * I ^ 1 = -I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | rw [zpow_add₀ I_ne_zero] | z : ℂ
⊢ I ^ (1 + 1) * I ^ 1 = -I | z : ℂ
⊢ I ^ 1 * I ^ 1 * I ^ 1 = -I | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ I ^ (1 + 1) * I ^ 1 = -I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | simp only [zpow_one, I_mul_I, neg_mul, one_mul] | z : ℂ
⊢ I ^ 1 * I ^ 1 * I ^ 1 = -I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ I ^ 1 * I ^ 1 * I ^ 1 = -I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | ring | z : ℂ
h₁b : cexp (↑π * I * 3 / 2) = I ^ 3
⊢ ↑π * I * 3 / 2 = ↑π * I * (3 / 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
h₁b : cexp (↑π * I * 3 / 2) = I ^ 3
⊢ ↑π * I * 3 / 2 = ↑π * I * (3 / 2)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | exact rfl | z : ℂ
⊢ I ^ 3 = I ^ 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ I ^ 3 = I ^ 3
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | exact rfl | z : ℂ
⊢ 3 = 2 + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ 3 = 2 + 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | exact rfl | z : ℂ
⊢ 2 = 1 + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ 2 = 1 + 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | nth_rw 2 [←ExpPiMulIHalf] | z : ℂ
⊢ cexp (↑π * I * 2⁻¹) = I | z : ℂ
⊢ cexp (↑π * I * 2⁻¹) = cexp (↑(π / 2) * I) | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ cexp (↑π * I * 2⁻¹) = I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | congr 1 | z : ℂ
⊢ cexp (↑π * I * 2⁻¹) = cexp (↑(π / 2) * I) | case e_z
z : ℂ
⊢ ↑π * I * 2⁻¹ = ↑(π / 2) * I | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ cexp (↑π * I * 2⁻¹) = cexp (↑(π / 2) * I)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | simp only [ofReal_div, ofReal_ofNat] | case e_z
z : ℂ
⊢ ↑π * I * 2⁻¹ = ↑(π / 2) * I | case e_z
z : ℂ
⊢ ↑π * I * 2⁻¹ = ↑π / 2 * I | Please generate a tactic in lean4 to solve the state.
STATE:
case e_z
z : ℂ
⊢ ↑π * I * 2⁻¹ = ↑(π / 2) * I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | ring_nf | case e_z
z : ℂ
⊢ ↑π * I * 2⁻¹ = ↑π / 2 * I | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_z
z : ℂ
⊢ ↑π * I * 2⁻¹ = ↑π / 2 * I
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | rw [Inv.inv, Complex.instInv, normSq] | z : ℂ
⊢ (1 + I)⁻¹ = (1 - I) / 2 | z : ℂ
⊢ {
inv := fun z =>
(starRingEnd ℂ) z *
↑({ toFun := fun z => z.re * z.re + z.im * z.im, map_zero' := normSq.proof_1, map_one' := normSq.proof_2,
map_mul' := normSq.proof_3 }
z)⁻¹ }.1
(1 + I) =
(1 - I) / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ (1 + I)⁻¹ = (1 - I) / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | simp only [MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, ofReal_inv, ofReal_add, ofReal_mul,
map_add, map_one, conj_I, add_re, one_re, I_re, add_zero, ofReal_one, mul_one, add_im, one_im,
I_im, zero_add] | z : ℂ
⊢ {
inv := fun z =>
(starRingEnd ℂ) z *
↑({ toFun := fun z => z.re * z.re + z.im * z.im, map_zero' := normSq.proof_1, map_one' := normSq.proof_2,
map_mul' := normSq.proof_3 }
z)⁻¹ }.1
(1 + I) =
(1 - I) / 2 | z : ℂ
⊢ (1 + -I) * (1 + 1)⁻¹ = (1 - I) / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ {
inv := fun z =>
(starRingEnd ℂ) z *
↑({ toFun := fun z => z.re * z.re + z.im * z.im, map_zero' := normSq.proof_1, map_one' := normSq.proof_2,
map_mul' := normSq.proof_3 }
z)⁻¹ }.1
(1 + I) =
(1 - I) / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | ring_nf | z : ℂ
⊢ (1 + -I) * (1 + 1)⁻¹ = (1 - I) / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ (1 + -I) * (1 + 1)⁻¹ = (1 - I) / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | rw [Inv.inv, Complex.instInv, normSq] | z : ℂ
⊢ (1 - I)⁻¹ = (1 + I) / 2 | z : ℂ
⊢ {
inv := fun z =>
(starRingEnd ℂ) z *
↑({ toFun := fun z => z.re * z.re + z.im * z.im, map_zero' := normSq.proof_1, map_one' := normSq.proof_2,
map_mul' := normSq.proof_3 }
z)⁻¹ }.1
(1 - I) =
(1 + I) / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ (1 - I)⁻¹ = (1 + I) / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | simp only [MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, ofReal_inv, ofReal_add, ofReal_mul,
map_sub, map_one, conj_I, sub_neg_eq_add, sub_re, one_re, I_re, sub_zero, ofReal_one, mul_one,
sub_im, one_im, I_im, zero_sub, ofReal_neg, mul_neg, neg_neg] | z : ℂ
⊢ {
inv := fun z =>
(starRingEnd ℂ) z *
↑({ toFun := fun z => z.re * z.re + z.im * z.im, map_zero' := normSq.proof_1, map_one' := normSq.proof_2,
map_mul' := normSq.proof_3 }
z)⁻¹ }.1
(1 - I) =
(1 + I) / 2 | z : ℂ
⊢ (1 + I) * (1 + 1)⁻¹ = (1 + I) / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ {
inv := fun z =>
(starRingEnd ℂ) z *
↑({ toFun := fun z => z.re * z.re + z.im * z.im, map_zero' := normSq.proof_1, map_one' := normSq.proof_2,
map_mul' := normSq.proof_3 }
z)⁻¹ }.1
(1 - I) =
(1 + I) / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesN4EqualsCoshCosh | [420, 1] | [488, 10] | ring_nf | z : ℂ
⊢ (1 + I) * (1 + 1)⁻¹ = (1 + I) / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ (1 + I) * (1 + 1)⁻¹ = (1 + I) / 2
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | ExpSumOfRuesDiff | [490, 1] | [494, 13] | rw [←RuesN1EqualsExp, ←RuesDiffM0EqualsRues] | k : ℕ+
z : ℂ
⊢ cexp z = ∑ k₀ ∈ range ↑k, RuesDiff k (↑k₀) z | k : ℕ+
z : ℂ
⊢ RuesDiff 1 0 z = ∑ k₀ ∈ range ↑k, RuesDiff k (↑k₀) z | Please generate a tactic in lean4 to solve the state.
STATE:
k : ℕ+
z : ℂ
⊢ cexp z = ∑ k₀ ∈ range ↑k, RuesDiff k (↑k₀) z
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | ExpSumOfRuesDiff | [490, 1] | [494, 13] | have h₀ := RuesDiffSumOfRuesDiff 1 k 0 z | k : ℕ+
z : ℂ
⊢ RuesDiff 1 0 z = ∑ k₀ ∈ range ↑k, RuesDiff k (↑k₀) z | k : ℕ+
z : ℂ
h₀ : RuesDiff 1 0 z = ∑ k₀ ∈ range ↑k, RuesDiff (1 * k) (↑↑1 * ↑k₀ + 0) z
⊢ RuesDiff 1 0 z = ∑ k₀ ∈ range ↑k, RuesDiff k (↑k₀) z | Please generate a tactic in lean4 to solve the state.
STATE:
k : ℕ+
z : ℂ
⊢ RuesDiff 1 0 z = ∑ k₀ ∈ range ↑k, RuesDiff k (↑k₀) z
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | ExpSumOfRuesDiff | [490, 1] | [494, 13] | simp only [one_mul, PNat.val_ofNat, Nat.cast_one, add_zero] at h₀ | k : ℕ+
z : ℂ
h₀ : RuesDiff 1 0 z = ∑ k₀ ∈ range ↑k, RuesDiff (1 * k) (↑↑1 * ↑k₀ + 0) z
⊢ RuesDiff 1 0 z = ∑ k₀ ∈ range ↑k, RuesDiff k (↑k₀) z | k : ℕ+
z : ℂ
h₀ : RuesDiff 1 0 z = ∑ x ∈ range ↑k, RuesDiff k (↑x) z
⊢ RuesDiff 1 0 z = ∑ k₀ ∈ range ↑k, RuesDiff k (↑k₀) z | Please generate a tactic in lean4 to solve the state.
STATE:
k : ℕ+
z : ℂ
h₀ : RuesDiff 1 0 z = ∑ k₀ ∈ range ↑k, RuesDiff (1 * k) (↑↑1 * ↑k₀ + 0) z
⊢ RuesDiff 1 0 z = ∑ k₀ ∈ range ↑k, RuesDiff k (↑k₀) z
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | ExpSumOfRuesDiff | [490, 1] | [494, 13] | assumption | k : ℕ+
z : ℂ
h₀ : RuesDiff 1 0 z = ∑ x ∈ range ↑k, RuesDiff k (↑x) z
⊢ RuesDiff 1 0 z = ∑ k₀ ∈ range ↑k, RuesDiff k (↑k₀) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k : ℕ+
z : ℂ
h₀ : RuesDiff 1 0 z = ∑ x ∈ range ↑k, RuesDiff k (↑x) z
⊢ RuesDiff 1 0 z = ∑ k₀ ∈ range ↑k, RuesDiff k (↑k₀) z
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouForm | [496, 1] | [500, 10] | rw [(Complex.exp_nat_mul _ n).symm, Complex.exp_eq_one_iff] | n : ℕ+
x : ℕ
⊢ cexp (2 * ↑π * (↑x / ↑↑n) * I) ^ ↑n = 1 | n : ℕ+
x : ℕ
⊢ ∃ n_1, ↑↑n * (2 * ↑π * (↑x / ↑↑n) * I) = ↑n_1 * (2 * ↑π * I) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
x : ℕ
⊢ cexp (2 * ↑π * (↑x / ↑↑n) * I) ^ ↑n = 1
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouForm | [496, 1] | [500, 10] | use x | n : ℕ+
x : ℕ
⊢ ∃ n_1, ↑↑n * (2 * ↑π * (↑x / ↑↑n) * I) = ↑n_1 * (2 * ↑π * I) | case h
n : ℕ+
x : ℕ
⊢ ↑↑n * (2 * ↑π * (↑x / ↑↑n) * I) = ↑↑x * (2 * ↑π * I) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
x : ℕ
⊢ ∃ n_1, ↑↑n * (2 * ↑π * (↑x / ↑↑n) * I) = ↑n_1 * (2 * ↑π * I)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouForm | [496, 1] | [500, 10] | field_simp | case h
n : ℕ+
x : ℕ
⊢ ↑↑n * (2 * ↑π * (↑x / ↑↑n) * I) = ↑↑x * (2 * ↑π * I) | case h
n : ℕ+
x : ℕ
⊢ 2 * ↑π * ↑x * I = ↑x * (2 * ↑π * I) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ+
x : ℕ
⊢ ↑↑n * (2 * ↑π * (↑x / ↑↑n) * I) = ↑↑x * (2 * ↑π * I)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RouForm | [496, 1] | [500, 10] | ring_nf | case h
n : ℕ+
x : ℕ
⊢ 2 * ↑π * ↑x * I = ↑x * (2 * ↑π * I) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ+
x : ℕ
⊢ 2 * ↑π * ↑x * I = ↑x * (2 * ↑π * I)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | Sum3Cycle | [502, 1] | [505, 28] | rw [sum_comm] | M : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : AddCommMonoid M
s : Finset α
t : Finset β
u : Finset γ
f : α → β → γ → M
⊢ ∑ a ∈ s, ∑ b ∈ t, ∑ c ∈ u, f a b c = ∑ b ∈ t, ∑ c ∈ u, ∑ a ∈ s, f a b c | M : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : AddCommMonoid M
s : Finset α
t : Finset β
u : Finset γ
f : α → β → γ → M
⊢ ∑ y ∈ t, ∑ x ∈ s, ∑ c ∈ u, f x y c = ∑ b ∈ t, ∑ c ∈ u, ∑ a ∈ s, f a b c | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : AddCommMonoid M
s : Finset α
t : Finset β
u : Finset γ
f : α → β → γ → M
⊢ ∑ a ∈ s, ∑ b ∈ t, ∑ c ∈ u, f a b c = ∑ b ∈ t, ∑ c ∈ u, ∑ a ∈ s, f a b c
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | Sum3Cycle | [502, 1] | [505, 28] | simp_rw [@sum_comm _ _ γ] | M : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : AddCommMonoid M
s : Finset α
t : Finset β
u : Finset γ
f : α → β → γ → M
⊢ ∑ y ∈ t, ∑ x ∈ s, ∑ c ∈ u, f x y c = ∑ b ∈ t, ∑ c ∈ u, ∑ a ∈ s, f a b c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : AddCommMonoid M
s : Finset α
t : Finset β
u : Finset γ
f : α → β → γ → M
⊢ ∑ y ∈ t, ∑ x ∈ s, ∑ c ∈ u, f x y c = ∑ b ∈ t, ∑ c ∈ u, ∑ a ∈ s, f a b c
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | rw [RuesDiffEqualsExpSum] | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ RuesDiff n m (z₀ + z₁) = ∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ k₀ ∈ range ↑n, cexp ((z₀ + z₁) * cexp (2 * ↑π * (↑k₀ / ↑↑n) * I) + ↑m * 2 * ↑π * (↑k₀ / ↑↑n) * I)) / ↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ RuesDiff n m (z₀ + z₁) = ∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | simp_rw [Complex.exp_add, RightDistribClass.right_distrib, Complex.exp_add, ExpSumOfRuesDiff n (z₀ * _), ExpSumOfRuesDiff n (z₁ * _)] | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ k₀ ∈ range ↑n, cexp ((z₀ + z₁) * cexp (2 * ↑π * (↑k₀ / ↑↑n) * I) + ↑m * 2 * ↑π * (↑k₀ / ↑↑n) * I)) / ↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ x ∈ range ↑n,
((∑ k₀ ∈ range ↑n, RuesDiff n (↑k₀) (z₀ * cexp (2 * ↑π * (↑x / ↑↑n) * I))) *
∑ k₀ ∈ range ↑n, RuesDiff n (↑k₀) (z₁ * cexp (2 * ↑π * (↑x / ↑↑n) * I))) *
cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ k₀ ∈ range ↑n, cexp ((z₀ + z₁) * cexp (2 * ↑π * (↑k₀ / ↑↑n) * I) + ↑m * 2 * ↑π * (↑k₀ / ↑↑n) * I)) / ↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | simp_rw [RuesDiffRotationallySymmetric n _ _ _ (RouForm n _), Finset.sum_mul, Finset.mul_sum, Finset.sum_mul, ←Complex.exp_int_mul] | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ x ∈ range ↑n,
((∑ k₀ ∈ range ↑n, RuesDiff n (↑k₀) (z₀ * cexp (2 * ↑π * (↑x / ↑↑n) * I))) *
∑ k₀ ∈ range ↑n, RuesDiff n (↑k₀) (z₁ * cexp (2 * ↑π * (↑x / ↑↑n) * I))) *
cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
∑ x_2 ∈ range ↑n,
cexp (↑(-↑x_1) * (2 * ↑π * (↑x / ↑↑n) * I)) * RuesDiff n (↑x_1) z₀ *
(cexp (↑(-↑x_2) * (2 * ↑π * (↑x / ↑↑n) * I)) * RuesDiff n (↑x_2) z₁) *
cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ x ∈ range ↑n,
((∑ k₀ ∈ range ↑n, RuesDiff n (↑k₀) (z₀ * cexp (2 * ↑π * (↑x / ↑↑n) * I))) *
∑ k₀ ∈ range ↑n, RuesDiff n (↑k₀) (z₁ * cexp (2 * ↑π * (↑x / ↑↑n) * I))) *
cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | rw [Sum3Cycle] | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
∑ x_2 ∈ range ↑n,
cexp (↑(-↑x_1) * (2 * ↑π * (↑x / ↑↑n) * I)) * RuesDiff n (↑x_1) z₀ *
(cexp (↑(-↑x_2) * (2 * ↑π * (↑x / ↑↑n) * I)) * RuesDiff n (↑x_2) z₁) *
cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ b ∈ range ↑n,
∑ c ∈ range ↑n,
∑ a ∈ range ↑n,
cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑b) z₀ *
(cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑c) z₁) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
∑ x_2 ∈ range ↑n,
cexp (↑(-↑x_1) * (2 * ↑π * (↑x / ↑↑n) * I)) * RuesDiff n (↑x_1) z₀ *
(cexp (↑(-↑x_2) * (2 * ↑π * (↑x / ↑↑n) * I)) * RuesDiff n (↑x_2) z₁) *
cexp (↑m * 2 * ↑π * (↑x / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | have h₀ : ∀ (a b c : ℕ), cexp (↑(-(b : ℤ)) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑b) z₀ * (cexp (↑(-(c : ℤ)) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑c) z₁) * cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I) =
RuesDiff n (↑b) z₀ * RuesDiff n (↑c) z₁ * (cexp (↑(-(b : ℤ)) * (2 * ↑π * (↑a / ↑↑n) * I)) * (cexp (↑(-(c : ℤ)) * (2 * ↑π * (↑a / ↑↑n) * I))) * cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I)) := by
intros a b c
ring_nf | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ b ∈ range ↑n,
∑ c ∈ range ↑n,
∑ a ∈ range ↑n,
cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑b) z₀ *
(cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑c) z₁) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
h₀ :
∀ (a b c : ℕ),
cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑b) z₀ *
(cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑c) z₁) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I) =
RuesDiff n (↑b) z₀ * RuesDiff n (↑c) z₁ *
(cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I))
⊢ (∑ b ∈ range ↑n,
∑ c ∈ range ↑n,
∑ a ∈ range ↑n,
cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑b) z₀ *
(cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑c) z₁) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ b ∈ range ↑n,
∑ c ∈ range ↑n,
∑ a ∈ range ↑n,
cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑b) z₀ *
(cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑c) z₁) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | simp_rw [h₀, ←Complex.exp_add, ←Finset.mul_sum] | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
h₀ :
∀ (a b c : ℕ),
cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑b) z₀ *
(cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑c) z₁) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I) =
RuesDiff n (↑b) z₀ * RuesDiff n (↑c) z₁ *
(cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I))
⊢ (∑ b ∈ range ↑n,
∑ c ∈ range ↑n,
∑ a ∈ range ↑n,
cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑b) z₀ *
(cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑c) z₁) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
h₀ :
∀ (a b c : ℕ),
cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑b) z₀ *
(cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑c) z₁) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I) =
RuesDiff n (↑b) z₀ * RuesDiff n (↑c) z₁ *
(cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I))
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ i ∈ range ↑n,
cexp
(↑(-↑x) * (2 * ↑π * (↑i / ↑↑n) * I) + ↑(-↑x_1) * (2 * ↑π * (↑i / ↑↑n) * I) +
↑m * 2 * ↑π * (↑i / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
h₀ :
∀ (a b c : ℕ),
cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑b) z₀ *
(cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑c) z₁) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I) =
RuesDiff n (↑b) z₀ * RuesDiff n (↑c) z₁ *
(cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I))
⊢ (∑ b ∈ range ↑n,
∑ c ∈ range ↑n,
∑ a ∈ range ↑n,
cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑b) z₀ *
(cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑c) z₁) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | clear h₀ | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
h₀ :
∀ (a b c : ℕ),
cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑b) z₀ *
(cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑c) z₁) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I) =
RuesDiff n (↑b) z₀ * RuesDiff n (↑c) z₁ *
(cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I))
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ i ∈ range ↑n,
cexp
(↑(-↑x) * (2 * ↑π * (↑i / ↑↑n) * I) + ↑(-↑x_1) * (2 * ↑π * (↑i / ↑↑n) * I) +
↑m * 2 * ↑π * (↑i / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ i ∈ range ↑n,
cexp
(↑(-↑x) * (2 * ↑π * (↑i / ↑↑n) * I) + ↑(-↑x_1) * (2 * ↑π * (↑i / ↑↑n) * I) +
↑m * 2 * ↑π * (↑i / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
h₀ :
∀ (a b c : ℕ),
cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑b) z₀ *
(cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑c) z₁) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I) =
RuesDiff n (↑b) z₀ * RuesDiff n (↑c) z₁ *
(cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I))
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ i ∈ range ↑n,
cexp
(↑(-↑x) * (2 * ↑π * (↑i / ↑↑n) * I) + ↑(-↑x_1) * (2 * ↑π * (↑i / ↑↑n) * I) +
↑m * 2 * ↑π * (↑i / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | simp only [Int.cast_neg, Int.cast_natCast, neg_mul] | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ i ∈ range ↑n,
cexp
(↑(-↑x) * (2 * ↑π * (↑i / ↑↑n) * I) + ↑(-↑x_1) * (2 * ↑π * (↑i / ↑↑n) * I) +
↑m * 2 * ↑π * (↑i / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ x_2 ∈ range ↑n,
cexp
(-(↑x * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + -(↑x_1 * (2 * ↑π * (↑x_2 / ↑↑n) * I)) +
↑m * 2 * ↑π * (↑x_2 / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ i ∈ range ↑n,
cexp
(↑(-↑x) * (2 * ↑π * (↑i / ↑↑n) * I) + ↑(-↑x_1) * (2 * ↑π * (↑i / ↑↑n) * I) +
↑m * 2 * ↑π * (↑i / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | have h₁ : ∀ (x x_1 x_2 : ℕ), -(↑x * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + -(↑x_1 * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + ↑m * 2 * ↑π * (↑x_2 / ↑↑n) * I =
(2 * ↑π * (((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n) * I)) := by
intros x x_1 x_2
ring_nf | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ x_2 ∈ range ↑n,
cexp
(-(↑x * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + -(↑x_1 * (2 * ↑π * (↑x_2 / ↑↑n) * I)) +
↑m * 2 * ↑π * (↑x_2 / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
h₁ :
∀ (x x_1 x_2 : ℕ),
-(↑x * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + -(↑x_1 * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + ↑m * 2 * ↑π * (↑x_2 / ↑↑n) * I =
2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I)
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ x_2 ∈ range ↑n,
cexp
(-(↑x * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + -(↑x_1 * (2 * ↑π * (↑x_2 / ↑↑n) * I)) +
↑m * 2 * ↑π * (↑x_2 / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ x_2 ∈ range ↑n,
cexp
(-(↑x * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + -(↑x_1 * (2 * ↑π * (↑x_2 / ↑↑n) * I)) +
↑m * 2 * ↑π * (↑x_2 / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | simp_rw [h₁] | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
h₁ :
∀ (x x_1 x_2 : ℕ),
-(↑x * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + -(↑x_1 * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + ↑m * 2 * ↑π * (↑x_2 / ↑↑n) * I =
2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I)
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ x_2 ∈ range ↑n,
cexp
(-(↑x * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + -(↑x_1 * (2 * ↑π * (↑x_2 / ↑↑n) * I)) +
↑m * 2 * ↑π * (↑x_2 / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
h₁ :
∀ (x x_1 x_2 : ℕ),
-(↑x * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + -(↑x_1 * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + ↑m * 2 * ↑π * (↑x_2 / ↑↑n) * I =
2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I)
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ x_2 ∈ range ↑n, cexp (2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I))) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
h₁ :
∀ (x x_1 x_2 : ℕ),
-(↑x * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + -(↑x_1 * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + ↑m * 2 * ↑π * (↑x_2 / ↑↑n) * I =
2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I)
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ x_2 ∈ range ↑n,
cexp
(-(↑x * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + -(↑x_1 * (2 * ↑π * (↑x_2 / ↑↑n) * I)) +
↑m * 2 * ↑π * (↑x_2 / ↑↑n) * I)) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | clear h₁ | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
h₁ :
∀ (x x_1 x_2 : ℕ),
-(↑x * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + -(↑x_1 * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + ↑m * 2 * ↑π * (↑x_2 / ↑↑n) * I =
2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I)
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ x_2 ∈ range ↑n, cexp (2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I))) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ x_2 ∈ range ↑n, cexp (2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I))) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
h₁ :
∀ (x x_1 x_2 : ℕ),
-(↑x * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + -(↑x_1 * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + ↑m * 2 * ↑π * (↑x_2 / ↑↑n) * I =
2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I)
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ x_2 ∈ range ↑n, cexp (2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I))) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | have h₂ : ∀ (x x_1 : ℕ), (m : ℂ) - (x : ℂ) - (x_1 : ℂ) = @Int.cast ℂ Ring.toIntCast (m - (x : ℤ) - (x_1 : ℤ)) := by
intros x x_1
norm_cast | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ x_2 ∈ range ↑n, cexp (2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I))) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
h₂ : ∀ (x x_1 : ℕ), ↑m - ↑x - ↑x_1 = ↑(m - ↑x - ↑x_1)
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ x_2 ∈ range ↑n, cexp (2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I))) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ x_2 ∈ range ↑n, cexp (2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I))) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | simp_rw [h₂, RouGeometricSumEqIte] | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
h₂ : ∀ (x x_1 : ℕ), ↑m - ↑x - ↑x_1 = ↑(m - ↑x - ↑x_1)
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ x_2 ∈ range ↑n, cexp (2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I))) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
h₂ : ∀ (x x_1 : ℕ), ↑m - ↑x - ↑x_1 = ↑(m - ↑x - ↑x_1)
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n, RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ * if ↑↑n ∣ m - ↑x - ↑x_1 then ↑↑n else 0) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
h₂ : ∀ (x x_1 : ℕ), ↑m - ↑x - ↑x_1 = ↑(m - ↑x - ↑x_1)
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n,
RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ *
∑ x_2 ∈ range ↑n, cexp (2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I))) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | clear h₂ | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
h₂ : ∀ (x x_1 : ℕ), ↑m - ↑x - ↑x_1 = ↑(m - ↑x - ↑x_1)
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n, RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ * if ↑↑n ∣ m - ↑x - ↑x_1 then ↑↑n else 0) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n, RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ * if ↑↑n ∣ m - ↑x - ↑x_1 then ↑↑n else 0) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
h₂ : ∀ (x x_1 : ℕ), ↑m - ↑x - ↑x_1 = ↑(m - ↑x - ↑x_1)
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n, RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ * if ↑↑n ∣ m - ↑x - ↑x_1 then ↑↑n else 0) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | simp only [mul_ite, mul_zero, sum_range] | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n, RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ * if ↑↑n ∣ m - ↑x - ↑x_1 then ↑↑n else 0) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁ | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ i : Fin ↑n,
∑ i_1 : Fin ↑n, if ↑↑n ∣ m - ↑↑i - ↑↑i_1 then RuesDiff n (↑↑i) z₀ * RuesDiff n (↑↑i_1) z₁ * ↑↑n else 0) /
↑↑n =
∑ i : Fin ↑n, RuesDiff n (↑↑i) z₀ * RuesDiff n (m - ↑↑i) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ x ∈ range ↑n,
∑ x_1 ∈ range ↑n, RuesDiff n (↑x) z₀ * RuesDiff n (↑x_1) z₁ * if ↑↑n ∣ m - ↑x - ↑x_1 then ↑↑n else 0) /
↑↑n =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (m - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | sorry | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ i : Fin ↑n,
∑ i_1 : Fin ↑n, if ↑↑n ∣ m - ↑↑i - ↑↑i_1 then RuesDiff n (↑↑i) z₀ * RuesDiff n (↑↑i_1) z₁ * ↑↑n else 0) /
↑↑n =
∑ i : Fin ↑n, RuesDiff n (↑↑i) z₀ * RuesDiff n (m - ↑↑i) z₁ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ (∑ i : Fin ↑n,
∑ i_1 : Fin ↑n, if ↑↑n ∣ m - ↑↑i - ↑↑i_1 then RuesDiff n (↑↑i) z₀ * RuesDiff n (↑↑i_1) z₁ * ↑↑n else 0) /
↑↑n =
∑ i : Fin ↑n, RuesDiff n (↑↑i) z₀ * RuesDiff n (m - ↑↑i) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | intros a b c | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ ∀ (a b c : ℕ),
cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑b) z₀ *
(cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑c) z₁) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I) =
RuesDiff n (↑b) z₀ * RuesDiff n (↑c) z₁ *
(cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I)) | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
a b c : ℕ
⊢ cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑b) z₀ *
(cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑c) z₁) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I) =
RuesDiff n (↑b) z₀ * RuesDiff n (↑c) z₁ *
(cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I)) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ ∀ (a b c : ℕ),
cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑b) z₀ *
(cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑c) z₁) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I) =
RuesDiff n (↑b) z₀ * RuesDiff n (↑c) z₁ *
(cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I))
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | ring_nf | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
a b c : ℕ
⊢ cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑b) z₀ *
(cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑c) z₁) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I) =
RuesDiff n (↑b) z₀ * RuesDiff n (↑c) z₁ *
(cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
a b c : ℕ
⊢ cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑b) z₀ *
(cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) * RuesDiff n (↑c) z₁) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I) =
RuesDiff n (↑b) z₀ * RuesDiff n (↑c) z₁ *
(cexp (↑(-↑b) * (2 * ↑π * (↑a / ↑↑n) * I)) * cexp (↑(-↑c) * (2 * ↑π * (↑a / ↑↑n) * I)) *
cexp (↑m * 2 * ↑π * (↑a / ↑↑n) * I))
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | intros x x_1 x_2 | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ ∀ (x x_1 x_2 : ℕ),
-(↑x * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + -(↑x_1 * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + ↑m * 2 * ↑π * (↑x_2 / ↑↑n) * I =
2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I) | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
x x_1 x_2 : ℕ
⊢ -(↑x * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + -(↑x_1 * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + ↑m * 2 * ↑π * (↑x_2 / ↑↑n) * I =
2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ ∀ (x x_1 x_2 : ℕ),
-(↑x * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + -(↑x_1 * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + ↑m * 2 * ↑π * (↑x_2 / ↑↑n) * I =
2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | ring_nf | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
x x_1 x_2 : ℕ
⊢ -(↑x * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + -(↑x_1 * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + ↑m * 2 * ↑π * (↑x_2 / ↑↑n) * I =
2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
x x_1 x_2 : ℕ
⊢ -(↑x * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + -(↑x_1 * (2 * ↑π * (↑x_2 / ↑↑n) * I)) + ↑m * 2 * ↑π * (↑x_2 / ↑↑n) * I =
2 * ↑π * ((↑m - ↑x - ↑x_1) * ↑x_2 / ↑↑n * I)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | intros x x_1 | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ ∀ (x x_1 : ℕ), ↑m - ↑x - ↑x_1 = ↑(m - ↑x - ↑x_1) | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
x x_1 : ℕ
⊢ ↑m - ↑x - ↑x_1 = ↑(m - ↑x - ↑x_1) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
⊢ ∀ (x x_1 : ℕ), ↑m - ↑x - ↑x_1 = ↑(m - ↑x - ↑x_1)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffArgumentSumRule | [507, 1] | [531, 8] | norm_cast | n : ℕ+
m : ℤ
z₀ z₁ : ℂ
x x_1 : ℕ
⊢ ↑m - ↑x - ↑x_1 = ↑(m - ↑x - ↑x_1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
z₀ z₁ : ℂ
x x_1 : ℕ
⊢ ↑m - ↑x - ↑x_1 = ↑(m - ↑x - ↑x_1)
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesArgumentSumRule | [533, 1] | [540, 10] | rw [←RuesDiffM0EqualsRues, RuesDiffArgumentSumRule] | n : ℕ+
z₀ z₁ : ℂ
⊢ Rues n (z₀ + z₁) = ∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (↑↑n - ↑k) z₁ | n : ℕ+
z₀ z₁ : ℂ
⊢ ∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (0 - ↑k) z₁ =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (↑↑n - ↑k) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
z₀ z₁ : ℂ
⊢ Rues n (z₀ + z₁) = ∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (↑↑n - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesArgumentSumRule | [533, 1] | [540, 10] | congr | n : ℕ+
z₀ z₁ : ℂ
⊢ ∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (0 - ↑k) z₁ =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (↑↑n - ↑k) z₁ | case e_f
n : ℕ+
z₀ z₁ : ℂ
⊢ (fun k => RuesDiff n (↑k) z₀ * RuesDiff n (0 - ↑k) z₁) = fun k => RuesDiff n (↑k) z₀ * RuesDiff n (↑↑n - ↑k) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
z₀ z₁ : ℂ
⊢ ∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (0 - ↑k) z₁ =
∑ k ∈ range ↑n, RuesDiff n (↑k) z₀ * RuesDiff n (↑↑n - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesArgumentSumRule | [533, 1] | [540, 10] | ext k | case e_f
n : ℕ+
z₀ z₁ : ℂ
⊢ (fun k => RuesDiff n (↑k) z₀ * RuesDiff n (0 - ↑k) z₁) = fun k => RuesDiff n (↑k) z₀ * RuesDiff n (↑↑n - ↑k) z₁ | case e_f.h
n : ℕ+
z₀ z₁ : ℂ
k : ℕ
⊢ RuesDiff n (↑k) z₀ * RuesDiff n (0 - ↑k) z₁ = RuesDiff n (↑k) z₀ * RuesDiff n (↑↑n - ↑k) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f
n : ℕ+
z₀ z₁ : ℂ
⊢ (fun k => RuesDiff n (↑k) z₀ * RuesDiff n (0 - ↑k) z₁) = fun k => RuesDiff n (↑k) z₀ * RuesDiff n (↑↑n - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesArgumentSumRule | [533, 1] | [540, 10] | congr 1 | case e_f.h
n : ℕ+
z₀ z₁ : ℂ
k : ℕ
⊢ RuesDiff n (↑k) z₀ * RuesDiff n (0 - ↑k) z₁ = RuesDiff n (↑k) z₀ * RuesDiff n (↑↑n - ↑k) z₁ | case e_f.h.e_a
n : ℕ+
z₀ z₁ : ℂ
k : ℕ
⊢ RuesDiff n (0 - ↑k) z₁ = RuesDiff n (↑↑n - ↑k) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h
n : ℕ+
z₀ z₁ : ℂ
k : ℕ
⊢ RuesDiff n (↑k) z₀ * RuesDiff n (0 - ↑k) z₁ = RuesDiff n (↑k) z₀ * RuesDiff n (↑↑n - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesArgumentSumRule | [533, 1] | [540, 10] | rw [RuesDiffMPeriodic n (0 - ↑k) 1] | case e_f.h.e_a
n : ℕ+
z₀ z₁ : ℂ
k : ℕ
⊢ RuesDiff n (0 - ↑k) z₁ = RuesDiff n (↑↑n - ↑k) z₁ | case e_f.h.e_a
n : ℕ+
z₀ z₁ : ℂ
k : ℕ
⊢ RuesDiff n (0 - ↑k + 1 * ↑↑n) z₁ = RuesDiff n (↑↑n - ↑k) z₁ | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_a
n : ℕ+
z₀ z₁ : ℂ
k : ℕ
⊢ RuesDiff n (0 - ↑k) z₁ = RuesDiff n (↑↑n - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesArgumentSumRule | [533, 1] | [540, 10] | congr 1 | case e_f.h.e_a
n : ℕ+
z₀ z₁ : ℂ
k : ℕ
⊢ RuesDiff n (0 - ↑k + 1 * ↑↑n) z₁ = RuesDiff n (↑↑n - ↑k) z₁ | case e_f.h.e_a.e_m
n : ℕ+
z₀ z₁ : ℂ
k : ℕ
⊢ 0 - ↑k + 1 * ↑↑n = ↑↑n - ↑k | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_a
n : ℕ+
z₀ z₁ : ℂ
k : ℕ
⊢ RuesDiff n (0 - ↑k + 1 * ↑↑n) z₁ = RuesDiff n (↑↑n - ↑k) z₁
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesArgumentSumRule | [533, 1] | [540, 10] | ring_nf | case e_f.h.e_a.e_m
n : ℕ+
z₀ z₁ : ℂ
k : ℕ
⊢ 0 - ↑k + 1 * ↑↑n = ↑↑n - ↑k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_f.h.e_a.e_m
n : ℕ+
z₀ z₁ : ℂ
k : ℕ
⊢ 0 - ↑k + 1 * ↑↑n = ↑↑n - ↑k
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | RuesDiffZ0EqualsIte | [542, 1] | [543, 8] | sorry | n : ℕ+
m : ℤ
⊢ RuesDiff n m 0 = if ↑↑n ∣ m then 1 else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ+
m : ℤ
⊢ RuesDiff n m 0 = if ↑↑n ∣ m then 1 else 0
TACTIC:
|
https://github.com/Nazgand/NazgandLean4.git | a6c5455a06d14c59786b1c23c2d20dada7598be6 | NazgandLean4/RootOfUnityExponentialSum.lean | EqualsNthDerivRuesDiffSum | [545, 1] | [546, 8] | sorry | f : ℂ → ℂ
n : ℕ+
⊢ f = iteratedDeriv (↑n) f ↔ f = ∑ k ∈ range ↑n, (fun z => iteratedDeriv k f 0) * RuesDiff n (-↑k) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ
n : ℕ+
⊢ f = iteratedDeriv (↑n) f ↔ f = ∑ k ∈ range ↑n, (fun z => iteratedDeriv k f 0) * RuesDiff n (-↑k)
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | rcases hε with ⟨εpos, εle⟩ | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
hε : ε ∈ Set.Ioc 0 (2 * Real.pi)
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | case intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
hε : ε ∈ Set.Ioc 0 (2 * Real.pi)
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | set ε' := ε / 4 / C_control_approximation_effect ε with ε'def | case intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | case intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | have ε'pos : ε' > 0 := by
rw [ε'def]
apply div_pos _ (C_control_approximation_effect_pos εpos)
apply div_pos εpos (by norm_num) | case intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | case intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | obtain ⟨f₀, contDiff_f₀, periodic_f₀, hf₀⟩ := closeSmoothApproxPeriodic unicontf periodicf ε'pos | case intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | case intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | have ε4pos : ε / 4 > 0 := by linarith | case intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | case intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | obtain ⟨N₀, hN₀⟩ := fourierConv_ofTwiceDifferentiable periodic_f₀ ((contDiff_top.mp (contDiff_f₀)) 2) ε4pos | case intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | case intro.intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | set h := f₀ - f with hdef | case intro.intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | case intro.intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | have h_measurable : Measurable h := Continuous.measurable (Continuous.sub contDiff_f₀.continuous unicontf.continuous) | case intro.intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | case intro.intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | have h_periodic : Function.Periodic h (2 * Real.pi) := Function.Periodic.sub periodic_f₀ periodicf | case intro.intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | case intro.intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | have h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ ε' := by
intro x _
simp [hdef]
rw [←Complex.dist_eq, dist_comm, Complex.dist_eq]
exact hf₀ x | case intro.intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | case intro.intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ ε'
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | obtain ⟨E, Esubset, Emeasurable, Evolume, hE⟩ := control_approximation_effect' ⟨εpos, εle⟩ ε'pos h_measurable h_periodic h_bound | case intro.intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ ε'
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | case intro.intro.intro.intro.intro.intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ ε'
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | use E, Esubset, Emeasurable, Evolume, N₀ | case intro.intro.intro.intro.intro.intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | case h
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
⊢ ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
⊢ ∃ E ⊆ Set.Icc 0 (2 * Real.pi),
MeasurableSet E ∧
MeasureTheory.volume.real E ≤ ε ∧
∃ N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | intro x hx N NgtN₀ | case h
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
⊢ ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε | case h
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f x - partialFourierSum f N x) ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
⊢ ∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E, ∀ N > N₀, Complex.abs (f x - partialFourierSum f N x) ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | calc Complex.abs (f x - partialFourierSum f N x)
_ = Complex.abs ((f x - f₀ x) + (f₀ x - partialFourierSum f₀ N x) + (partialFourierSum f₀ N x - partialFourierSum f N x)) := by congr; ring
_ ≤ Complex.abs ((f x - f₀ x) + (f₀ x - partialFourierSum f₀ N x)) + Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) := by
apply AbsoluteValue.add_le
_ ≤ Complex.abs (f x - f₀ x) + Complex.abs (f₀ x - partialFourierSum f₀ N x) + Complex.abs (partialFourierSum f₀ N x - partialFourierSum f N x) := by
apply add_le_add_right
apply AbsoluteValue.add_le
_ ≤ ε' + (ε / 4) + (ε / 4) := by
gcongr
. exact hf₀ x
. exact hN₀ N NgtN₀ x hx.1
. have := hE x hx N
rw [hdef, partialFourierSum_sub (contDiff_f₀.continuous.intervalIntegrable 0 (2 * Real.pi)) (unicontf.continuous.intervalIntegrable 0 (2 * Real.pi))] at this
apply le_trans this
rw [ε'def, mul_div_cancel₀ _ (C_control_approximation_effect_pos εpos).ne.symm]
_ ≤ (ε / 2) + (ε / 4) + (ε / 4) := by
gcongr
rw [ε'def, div_div]
apply div_le_div_of_nonneg_left εpos.le (by norm_num)
rw [← div_le_iff' (by norm_num)]
apply le_trans' (lt_C_control_approximation_effect εpos).le (by linarith [Real.two_le_pi])
_ ≤ ε := by linarith | case h
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f x - partialFourierSum f N x) ≤ ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
h_bound : ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ ε'
E : Set ℝ
Esubset : E ⊆ Set.Icc 0 (2 * Real.pi)
Emeasurable : MeasurableSet E
Evolume : MeasureTheory.volume.real E ≤ ε
hE :
∀ x ∈ Set.Icc 0 (2 * Real.pi) \ E,
∀ (N : ℕ), Complex.abs (partialFourierSum h N x) ≤ C_control_approximation_effect ε * ε'
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi) \ E
N : ℕ
NgtN₀ : N > N₀
⊢ Complex.abs (f x - partialFourierSum f N x) ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | rw [ε'def] | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
⊢ ε' > 0 | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
⊢ ε / 4 / C_control_approximation_effect ε > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
⊢ ε' > 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | apply div_pos _ (C_control_approximation_effect_pos εpos) | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
⊢ ε / 4 / C_control_approximation_effect ε > 0 | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
⊢ 0 < ε / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
⊢ ε / 4 / C_control_approximation_effect ε > 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | apply div_pos εpos (by norm_num) | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
⊢ 0 < ε / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
⊢ 0 < ε / 4
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | norm_num | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
⊢ 0 < 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
⊢ 0 < 4
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | linarith | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
⊢ ε / 4 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
⊢ ε / 4 > 0
TACTIC:
|
https://github.com/fpvandoorn/carleson.git | 6d448ddfa1ff78506367ab09a8caac5351011ead | Carleson/Theorem1_1/ClassicalCarleson.lean | classical_carleson | [13, 1] | [67, 23] | intro x _ | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
⊢ ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ ε' | f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
x : ℝ
a✝ : x ∈ Set.Icc (-Real.pi) (3 * Real.pi)
⊢ Complex.abs (h x) ≤ ε' | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℂ
unicontf : UniformContinuous f
periodicf : Function.Periodic f (2 * Real.pi)
ε : ℝ
εpos : 0 < ε
εle : ε ≤ 2 * Real.pi
ε' : ℝ := ε / 4 / C_control_approximation_effect ε
ε'def : ε' = ε / 4 / C_control_approximation_effect ε
ε'pos : ε' > 0
f₀ : ℝ → ℂ
contDiff_f₀ : ContDiff ℝ ⊤ f₀
periodic_f₀ : Function.Periodic f₀ (2 * Real.pi)
hf₀ : ∀ (x : ℝ), Complex.abs (f x - f₀ x) ≤ ε'
ε4pos : ε / 4 > 0
N₀ : ℕ
hN₀ : ∀ N > N₀, ∀ x ∈ Set.Icc 0 (2 * Real.pi), Complex.abs (f₀ x - partialFourierSum f₀ N x) ≤ ε / 4
h : ℝ → ℂ := f₀ - f
hdef : h = f₀ - f
h_measurable : Measurable h
h_periodic : Function.Periodic h (2 * Real.pi)
⊢ ∀ x ∈ Set.Icc (-Real.pi) (3 * Real.pi), Complex.abs (h x) ≤ ε'
TACTIC:
|
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