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stringclasses 147
values | file_path
stringlengths 7
101
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stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
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| tactic
stringlengths 1
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stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
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stringlengths 73
2.09M
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|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | have h : ↑n < t * (2 * π)⁻¹ - 1 / 2 + 1 := by rw [← hn]; exact Int.ceil_lt_add_one _ | case left
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
⊢ 2 * π * ↑n < π + t | case left
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
h : ↑n < t * (2 * π)⁻¹ - 1 / 2 + 1
⊢ 2 * π * ↑n < π + t | Please generate a tactic in lean4 to solve the state.
STATE:
case left
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
⊢ 2 * π * ↑n < π + t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | calc 2 * π * ↑n
_ < 2 * π * (t * (2 * π)⁻¹ - 1 / 2 + 1) := by bound
_ = π + 2 * π * (2 * π)⁻¹ * t := by ring
_ = π + t := by field_simp [Real.two_pi_pos.ne'] | case left
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
h : ↑n < t * (2 * π)⁻¹ - 1 / 2 + 1
⊢ 2 * π * ↑n < π + t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
h : ↑n < t * (2 * π)⁻¹ - 1 / 2 + 1
⊢ 2 * π * ↑n < π + t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | rw [← hn] | t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
⊢ ↑n < t * (2 * π)⁻¹ - 1 / 2 + 1 | t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
⊢ ↑⌈t / (2 * π) - 1 / 2⌉ < t * (2 * π)⁻¹ - 1 / 2 + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
⊢ ↑n < t * (2 * π)⁻¹ - 1 / 2 + 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | exact Int.ceil_lt_add_one _ | t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
⊢ ↑⌈t / (2 * π) - 1 / 2⌉ < t * (2 * π)⁻¹ - 1 / 2 + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
⊢ ↑⌈t / (2 * π) - 1 / 2⌉ < t * (2 * π)⁻¹ - 1 / 2 + 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | bound | t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
h : ↑n < t * (2 * π)⁻¹ - 1 / 2 + 1
⊢ 2 * π * ↑n < 2 * π * (t * (2 * π)⁻¹ - 1 / 2 + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
h : ↑n < t * (2 * π)⁻¹ - 1 / 2 + 1
⊢ 2 * π * ↑n < 2 * π * (t * (2 * π)⁻¹ - 1 / 2 + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | ring | t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
h : ↑n < t * (2 * π)⁻¹ - 1 / 2 + 1
⊢ 2 * π * (t * (2 * π)⁻¹ - 1 / 2 + 1) = π + 2 * π * (2 * π)⁻¹ * t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
h : ↑n < t * (2 * π)⁻¹ - 1 / 2 + 1
⊢ 2 * π * (t * (2 * π)⁻¹ - 1 / 2 + 1) = π + 2 * π * (2 * π)⁻¹ * t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | field_simp [Real.two_pi_pos.ne'] | t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
h : ↑n < t * (2 * π)⁻¹ - 1 / 2 + 1
⊢ π + 2 * π * (2 * π)⁻¹ * t = π + t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
h : ↑n < t * (2 * π)⁻¹ - 1 / 2 + 1
⊢ π + 2 * π * (2 * π)⁻¹ * t = π + t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | have h : ↑n ≥ t * (2 * π)⁻¹ - 1 / 2 := by rw [← hn]; exact Int.le_ceil _ | case right
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
⊢ t ≤ π + 2 * π * ↑n | case right
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
h : ↑n ≥ t * (2 * π)⁻¹ - 1 / 2
⊢ t ≤ π + 2 * π * ↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case right
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
⊢ t ≤ π + 2 * π * ↑n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | calc π + 2 * π * ↑n
_ ≥ π + 2 * π * (t * (2 * π)⁻¹ - 1 / 2) := by bound
_ = 2 * π * (2 * π)⁻¹ * t := by ring
_ = t := by field_simp [Real.two_pi_pos.ne'] | case right
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
h : ↑n ≥ t * (2 * π)⁻¹ - 1 / 2
⊢ t ≤ π + 2 * π * ↑n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
h : ↑n ≥ t * (2 * π)⁻¹ - 1 / 2
⊢ t ≤ π + 2 * π * ↑n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | rw [← hn] | t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
⊢ ↑n ≥ t * (2 * π)⁻¹ - 1 / 2 | t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
⊢ ↑⌈t / (2 * π) - 1 / 2⌉ ≥ t * (2 * π)⁻¹ - 1 / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
⊢ ↑n ≥ t * (2 * π)⁻¹ - 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | exact Int.le_ceil _ | t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
⊢ ↑⌈t / (2 * π) - 1 / 2⌉ ≥ t * (2 * π)⁻¹ - 1 / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
⊢ ↑⌈t / (2 * π) - 1 / 2⌉ ≥ t * (2 * π)⁻¹ - 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | bound | t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
h : ↑n ≥ t * (2 * π)⁻¹ - 1 / 2
⊢ π + 2 * π * ↑n ≥ π + 2 * π * (t * (2 * π)⁻¹ - 1 / 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
h : ↑n ≥ t * (2 * π)⁻¹ - 1 / 2
⊢ π + 2 * π * ↑n ≥ π + 2 * π * (t * (2 * π)⁻¹ - 1 / 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | ring | t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
h : ↑n ≥ t * (2 * π)⁻¹ - 1 / 2
⊢ π + 2 * π * (t * (2 * π)⁻¹ - 1 / 2) = 2 * π * (2 * π)⁻¹ * t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
h : ↑n ≥ t * (2 * π)⁻¹ - 1 / 2
⊢ π + 2 * π * (t * (2 * π)⁻¹ - 1 / 2) = 2 * π * (2 * π)⁻¹ * t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | arg_exp_of_im | [186, 1] | [206, 98] | field_simp [Real.two_pi_pos.ne'] | t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
h : ↑n ≥ t * (2 * π)⁻¹ - 1 / 2
⊢ 2 * π * (2 * π)⁻¹ * t = t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
t : ℝ
n : ℤ
hn : ⌈t / (2 * π) - 1 / 2⌉ = n
en : (2 * ↑π * ↑n * I).exp = 1
e : (↑t * I).exp = (↑(t - 2 * π * ↑n) * I).exp
h : ↑n ≥ t * (2 * π)⁻¹ - 1 / 2
⊢ 2 * π * (2 * π)⁻¹ * t = t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | intro x xs y ys e | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
⊢ InjOn (realCircleMap c) (square r0 r1) | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : x ∈ square r0 r1
y : ℝ × ℝ
ys : y ∈ square r0 r1
e : realCircleMap c x = realCircleMap c y
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
⊢ InjOn (realCircleMap c) (square r0 r1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | simp [square] at xs ys | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : x ∈ square r0 r1
y : ℝ × ℝ
ys : y ∈ square r0 r1
e : realCircleMap c x = realCircleMap c y
⊢ x = y | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
e : realCircleMap c x = realCircleMap c y
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : x ∈ square r0 r1
y : ℝ × ℝ
ys : y ∈ square r0 r1
e : realCircleMap c x = realCircleMap c y
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | simp_rw [realCircleMap_eq_circleMap, Equiv.apply_eq_iff_eq] at e | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
e : realCircleMap c x = realCircleMap c y
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
⊢ x = y | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : circleMap c x.1 x.2 = circleMap c y.1 y.2
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
e : realCircleMap c x = realCircleMap c y
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | simp_rw [circleMap] at e | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : circleMap c x.1 x.2 = circleMap c y.1 y.2
⊢ x = y | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : c + ↑x.1 * (↑x.2 * I).exp = c + ↑y.1 * (↑y.2 * I).exp
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : circleMap c x.1 x.2 = circleMap c y.1 y.2
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | simp at e | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : c + ↑x.1 * (↑x.2 * I).exp = c + ↑y.1 * (↑y.2 * I).exp
⊢ x = y | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : c + ↑x.1 * (↑x.2 * I).exp = c + ↑y.1 * (↑y.2 * I).exp
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have re : abs (↑x.1 * exp (x.2 * I)) = abs (↑y.1 * exp (y.2 * I)) := by rw [e] | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
⊢ x = y | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
re : Complex.abs (↑x.1 * (↑x.2 * I).exp) = Complex.abs (↑y.1 * (↑y.2 * I).exp)
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have x0 : 0 < x.1 := by linarith | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
re : Complex.abs (↑x.1 * (↑x.2 * I).exp) = Complex.abs (↑y.1 * (↑y.2 * I).exp)
⊢ x = y | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
re : Complex.abs (↑x.1 * (↑x.2 * I).exp) = Complex.abs (↑y.1 * (↑y.2 * I).exp)
x0 : 0 < x.1
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
re : Complex.abs (↑x.1 * (↑x.2 * I).exp) = Complex.abs (↑y.1 * (↑y.2 * I).exp)
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have y0 : 0 < y.1 := by linarith | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
re : Complex.abs (↑x.1 * (↑x.2 * I).exp) = Complex.abs (↑y.1 * (↑y.2 * I).exp)
x0 : 0 < x.1
⊢ x = y | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
re : Complex.abs (↑x.1 * (↑x.2 * I).exp) = Complex.abs (↑y.1 * (↑y.2 * I).exp)
x0 : 0 < x.1
y0 : 0 < y.1
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
re : Complex.abs (↑x.1 * (↑x.2 * I).exp) = Complex.abs (↑y.1 * (↑y.2 * I).exp)
x0 : 0 < x.1
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | simp only [map_mul, Complex.abs_ofReal, abs_of_pos x0, Complex.abs_exp_ofReal_mul_I, mul_one,
abs_of_pos y0] at re | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
re : Complex.abs (↑x.1 * (↑x.2 * I).exp) = Complex.abs (↑y.1 * (↑y.2 * I).exp)
x0 : 0 < x.1
y0 : 0 < y.1
⊢ x = y | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
re : Complex.abs (↑x.1 * (↑x.2 * I).exp) = Complex.abs (↑y.1 * (↑y.2 * I).exp)
x0 : 0 < x.1
y0 : 0 < y.1
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have ae : arg (↑x.1 * exp (x.2 * I)) = arg (↑y.1 * exp (y.2 * I)) := by rw [e] | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
⊢ x = y | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (↑x.1 * (↑x.2 * I).exp).arg = (↑y.1 * (↑y.2 * I).exp).arg
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | simp [Complex.arg_real_mul _ x0, Complex.arg_real_mul _ y0] at ae | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (↑x.1 * (↑x.2 * I).exp).arg = (↑y.1 * (↑y.2 * I).exp).arg
⊢ x = y | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (↑x.2 * I).exp.arg = (↑y.2 * I).exp.arg
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (↑x.1 * (↑x.2 * I).exp).arg = (↑y.1 * (↑y.2 * I).exp).arg
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rcases arg_exp_of_im x.2 with ⟨nx, hx⟩ | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (↑x.2 * I).exp.arg = (↑y.2 * I).exp.arg
⊢ x = y | case intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (↑x.2 * I).exp.arg = (↑y.2 * I).exp.arg
nx : ℤ
hx : (↑x.2 * I).exp.arg = x.2 - 2 * π * ↑nx
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (↑x.2 * I).exp.arg = (↑y.2 * I).exp.arg
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rcases arg_exp_of_im y.2 with ⟨ny, h⟩ | case intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (↑x.2 * I).exp.arg = (↑y.2 * I).exp.arg
nx : ℤ
hx : (↑x.2 * I).exp.arg = x.2 - 2 * π * ↑nx
⊢ x = y | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (↑x.2 * I).exp.arg = (↑y.2 * I).exp.arg
nx : ℤ
hx : (↑x.2 * I).exp.arg = x.2 - 2 * π * ↑nx
ny : ℤ
h : (↑y.2 * I).exp.arg = y.2 - 2 * π * ↑ny
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (↑x.2 * I).exp.arg = (↑y.2 * I).exp.arg
nx : ℤ
hx : (↑x.2 * I).exp.arg = x.2 - 2 * π * ↑nx
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rw [← ae, hx] at h | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (↑x.2 * I).exp.arg = (↑y.2 * I).exp.arg
nx : ℤ
hx : (↑x.2 * I).exp.arg = x.2 - 2 * π * ↑nx
ny : ℤ
h : (↑y.2 * I).exp.arg = y.2 - 2 * π * ↑ny
⊢ x = y | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (↑x.2 * I).exp.arg = (↑y.2 * I).exp.arg
nx : ℤ
hx : (↑x.2 * I).exp.arg = x.2 - 2 * π * ↑nx
ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (↑x.2 * I).exp.arg = (↑y.2 * I).exp.arg
nx : ℤ
hx : (↑x.2 * I).exp.arg = x.2 - 2 * π * ↑nx
ny : ℤ
h : (↑y.2 * I).exp.arg = y.2 - 2 * π * ↑ny
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | clear e ae hx | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (↑x.2 * I).exp.arg = (↑y.2 * I).exp.arg
nx : ℤ
hx : (↑x.2 * I).exp.arg = x.2 - 2 * π * ↑nx
ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
⊢ x = y | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
ae : (↑x.2 * I).exp.arg = (↑y.2 * I).exp.arg
nx : ℤ
hx : (↑x.2 * I).exp.arg = x.2 - 2 * π * ↑nx
ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have n0 : 2 * π * (nx - ny) < 2 * π * 1 := by linarith | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
⊢ x = y | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have n1 : 2 * π * -1 < 2 * π * (nx - ny) := by linarith | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
⊢ x = y | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
n1 : 2 * π * -1 < 2 * π * (↑nx - ↑ny)
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have hn : (nx : ℝ) - ny = ↑(nx - ny) := by simp only [Int.cast_sub] | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
n1 : 2 * π * -1 < 2 * π * (↑nx - ↑ny)
⊢ x = y | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
n1 : 2 * π * -1 < 2 * π * (↑nx - ↑ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
n1 : 2 * π * -1 < 2 * π * (↑nx - ↑ny)
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have hn1 : (-1 : ℝ) = ↑(-1 : ℤ) := by norm_num | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
n1 : 2 * π * -1 < 2 * π * (↑nx - ↑ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
⊢ x = y | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
n1 : 2 * π * -1 < 2 * π * (↑nx - ↑ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
n1 : 2 * π * -1 < 2 * π * (↑nx - ↑ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have h1 : (1 : ℝ) = ↑(1 : ℤ) := by norm_num | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
n1 : 2 * π * -1 < 2 * π * (↑nx - ↑ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
⊢ x = y | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
n1 : 2 * π * -1 < 2 * π * (↑nx - ↑ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
n1 : 2 * π * -1 < 2 * π * (↑nx - ↑ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rw [mul_lt_mul_left Real.two_pi_pos, hn] at n0 n1 | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
n1 : 2 * π * -1 < 2 * π * (↑nx - ↑ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
⊢ x = y | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : ↑(nx - ny) < 1
n1 : -1 < ↑(nx - ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
n1 : 2 * π * -1 < 2 * π * (↑nx - ↑ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rw [hn1] at n1 | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : ↑(nx - ny) < 1
n1 : -1 < ↑(nx - ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
⊢ x = y | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : ↑(nx - ny) < 1
n1 : ↑(-1) < ↑(nx - ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : ↑(nx - ny) < 1
n1 : -1 < ↑(nx - ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rw [h1] at n0 | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : ↑(nx - ny) < 1
n1 : ↑(-1) < ↑(nx - ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
⊢ x = y | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : ↑(nx - ny) < ↑1
n1 : ↑(-1) < ↑(nx - ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : ↑(nx - ny) < 1
n1 : ↑(-1) < ↑(nx - ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rw [Int.cast_lt] at n0 n1 | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : ↑(nx - ny) < ↑1
n1 : ↑(-1) < ↑(nx - ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
⊢ x = y | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : ↑(nx - ny) < ↑1
n1 : ↑(-1) < ↑(nx - ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have n : nx = ny := by linarith | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
⊢ x = y | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
n : nx = ny
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rw [n] at h | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
n : nx = ny
⊢ x = y | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑ny = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
n : nx = ny
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
n : nx = ny
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have i : x.2 = y.2 := by linarith | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑ny = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
n : nx = ny
⊢ x = y | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑ny = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
n : nx = ny
i : x.2 = y.2
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑ny = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
n : nx = ny
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | have g : (x.1, x.2) = (y.1, y.2) := by rw [re, i] | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑ny = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
n : nx = ny
i : x.2 = y.2
⊢ x = y | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑ny = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
n : nx = ny
i : x.2 = y.2
g : (x.1, x.2) = (y.1, y.2)
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑ny = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
n : nx = ny
i : x.2 = y.2
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | simp only [Prod.mk.eta] at g | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑ny = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
n : nx = ny
i : x.2 = y.2
g : (x.1, x.2) = (y.1, y.2)
⊢ x = y | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑ny = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
n : nx = ny
i : x.2 = y.2
g : x = y
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑ny = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
n : nx = ny
i : x.2 = y.2
g : (x.1, x.2) = (y.1, y.2)
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | exact g | case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑ny = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
n : nx = ny
i : x.2 = y.2
g : x = y
⊢ x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑ny = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
n : nx = ny
i : x.2 = y.2
g : x = y
⊢ x = y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rw [e] | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
⊢ Complex.abs (↑x.1 * (↑x.2 * I).exp) = Complex.abs (↑y.1 * (↑y.2 * I).exp) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
⊢ Complex.abs (↑x.1 * (↑x.2 * I).exp) = Complex.abs (↑y.1 * (↑y.2 * I).exp)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | linarith | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
re : Complex.abs (↑x.1 * (↑x.2 * I).exp) = Complex.abs (↑y.1 * (↑y.2 * I).exp)
⊢ 0 < x.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
re : Complex.abs (↑x.1 * (↑x.2 * I).exp) = Complex.abs (↑y.1 * (↑y.2 * I).exp)
⊢ 0 < x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | linarith | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
re : Complex.abs (↑x.1 * (↑x.2 * I).exp) = Complex.abs (↑y.1 * (↑y.2 * I).exp)
x0 : 0 < x.1
⊢ 0 < y.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
re : Complex.abs (↑x.1 * (↑x.2 * I).exp) = Complex.abs (↑y.1 * (↑y.2 * I).exp)
x0 : 0 < x.1
⊢ 0 < y.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rw [e] | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
⊢ (↑x.1 * (↑x.2 * I).exp).arg = (↑y.1 * (↑y.2 * I).exp).arg | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
e : ↑x.1 * (↑x.2 * I).exp = ↑y.1 * (↑y.2 * I).exp
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
⊢ (↑x.1 * (↑x.2 * I).exp).arg = (↑y.1 * (↑y.2 * I).exp).arg
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | linarith | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
⊢ 2 * π * (↑nx - ↑ny) < 2 * π * 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
⊢ 2 * π * (↑nx - ↑ny) < 2 * π * 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | linarith | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
⊢ 2 * π * -1 < 2 * π * (↑nx - ↑ny) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
⊢ 2 * π * -1 < 2 * π * (↑nx - ↑ny)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | simp only [Int.cast_sub] | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
n1 : 2 * π * -1 < 2 * π * (↑nx - ↑ny)
⊢ ↑nx - ↑ny = ↑(nx - ny) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
n1 : 2 * π * -1 < 2 * π * (↑nx - ↑ny)
⊢ ↑nx - ↑ny = ↑(nx - ny)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | norm_num | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
n1 : 2 * π * -1 < 2 * π * (↑nx - ↑ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
⊢ -1 = ↑(-1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
n1 : 2 * π * -1 < 2 * π * (↑nx - ↑ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
⊢ -1 = ↑(-1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | norm_num | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
n1 : 2 * π * -1 < 2 * π * (↑nx - ↑ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
⊢ 1 = ↑1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : 2 * π * (↑nx - ↑ny) < 2 * π * 1
n1 : 2 * π * -1 < 2 * π * (↑nx - ↑ny)
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
⊢ 1 = ↑1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | linarith | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
⊢ nx = ny | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑nx = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
⊢ nx = ny
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | linarith | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑ny = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
n : nx = ny
⊢ x.2 = y.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑ny = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
n : nx = ny
⊢ x.2 = y.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm_inj | [209, 1] | [234, 40] | rw [re, i] | c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑ny = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
n : nx = ny
i : x.2 = y.2
⊢ (x.1, x.2) = (y.1, y.2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
r0 r1 : ℝ
r0p : 0 ≤ r0
x y : ℝ × ℝ
xs : (r0 < x.1 ∧ x.1 ≤ r1) ∧ 0 < x.2 ∧ x.2 ≤ 2 * π
ys : (r0 < y.1 ∧ y.1 ≤ r1) ∧ 0 < y.2 ∧ y.2 ≤ 2 * π
x0 : 0 < x.1
y0 : 0 < y.1
re : x.1 = y.1
nx ny : ℤ
h : x.2 - 2 * π * ↑ny = y.2 - 2 * π * ↑ny
n0 : nx - ny < 1
n1 : -1 < nx - ny
hn : ↑nx - ↑ny = ↑(nx - ny)
hn1 : -1 = ↑(-1)
h1 : 1 = ↑1
n : nx = ny
i : x.2 = y.2
⊢ (x.1, x.2) = (y.1, y.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | measurable_symm_equiv_inverse | [239, 1] | [245, 55] | simp only [Complex.equivRealProd_apply] | z : ℂ
⊢ Complex.measurableEquivRealProd.symm (Complex.equivRealProd z) = z | z : ℂ
⊢ Complex.measurableEquivRealProd.symm (z.re, z.im) = z | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ Complex.measurableEquivRealProd.symm (Complex.equivRealProd z) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | measurable_symm_equiv_inverse | [239, 1] | [245, 55] | rw [Complex.measurableEquivRealProd, Homeomorph.toMeasurableEquiv_symm_coe] | z : ℂ
⊢ Complex.measurableEquivRealProd.symm (z.re, z.im) = z | z : ℂ
⊢ Complex.equivRealProdCLM.toHomeomorph.symm (z.re, z.im) = z | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ Complex.measurableEquivRealProd.symm (z.re, z.im) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | measurable_symm_equiv_inverse | [239, 1] | [245, 55] | simp only [ContinuousLinearEquiv.symm_toHomeomorph, ContinuousLinearEquiv.coe_toHomeomorph] | z : ℂ
⊢ Complex.equivRealProdCLM.toHomeomorph.symm (z.re, z.im) = z | z : ℂ
⊢ Complex.equivRealProdCLM.symm (z.re, z.im) = z | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ Complex.equivRealProdCLM.toHomeomorph.symm (z.re, z.im) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | measurable_symm_equiv_inverse | [239, 1] | [245, 55] | apply Complex.ext | z : ℂ
⊢ Complex.equivRealProdCLM.symm (z.re, z.im) = z | case a
z : ℂ
⊢ (Complex.equivRealProdCLM.symm (z.re, z.im)).re = z.re
case a
z : ℂ
⊢ (Complex.equivRealProdCLM.symm (z.re, z.im)).im = z.im | Please generate a tactic in lean4 to solve the state.
STATE:
z : ℂ
⊢ Complex.equivRealProdCLM.symm (z.re, z.im) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | measurable_symm_equiv_inverse | [239, 1] | [245, 55] | simp only [Complex.equivRealProdCLM_symm_apply_re] | case a
z : ℂ
⊢ (Complex.equivRealProdCLM.symm (z.re, z.im)).re = z.re | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
z : ℂ
⊢ (Complex.equivRealProdCLM.symm (z.re, z.im)).re = z.re
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | measurable_symm_equiv_inverse | [239, 1] | [245, 55] | simp only [Complex.equivRealProdCLM_symm_apply_im] | case a
z : ℂ
⊢ (Complex.equivRealProdCLM.symm (z.re, z.im)).im = z.im | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
z : ℂ
⊢ (Complex.equivRealProdCLM.symm (z.re, z.im)).im = z.im
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | continuous_circleMap_full | [248, 1] | [249, 13] | continuity | c : ℂ
⊢ Continuous fun x => circleMap c x.1 x.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
⊢ Continuous fun x => circleMap c x.1 x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | invert_toReal | [252, 1] | [255, 56] | intro h | x : ENNReal
y : ℝ
yp : y > 0
⊢ x.toReal = y → x = ENNReal.ofReal y | x : ENNReal
y : ℝ
yp : y > 0
h : x.toReal = y
⊢ x = ENNReal.ofReal y | Please generate a tactic in lean4 to solve the state.
STATE:
x : ENNReal
y : ℝ
yp : y > 0
⊢ x.toReal = y → x = ENNReal.ofReal y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | invert_toReal | [252, 1] | [255, 56] | rw [← h] | x : ENNReal
y : ℝ
yp : y > 0
h : x.toReal = y
⊢ x = ENNReal.ofReal y | x : ENNReal
y : ℝ
yp : y > 0
h : x.toReal = y
⊢ x = ENNReal.ofReal x.toReal | Please generate a tactic in lean4 to solve the state.
STATE:
x : ENNReal
y : ℝ
yp : y > 0
h : x.toReal = y
⊢ x = ENNReal.ofReal y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | invert_toReal | [252, 1] | [255, 56] | refine (ENNReal.ofReal_toReal ?_).symm | x : ENNReal
y : ℝ
yp : y > 0
h : x.toReal = y
⊢ x = ENNReal.ofReal x.toReal | x : ENNReal
y : ℝ
yp : y > 0
h : x.toReal = y
⊢ x ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
x : ENNReal
y : ℝ
yp : y > 0
h : x.toReal = y
⊢ x = ENNReal.ofReal x.toReal
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | invert_toReal | [252, 1] | [255, 56] | contrapose yp | x : ENNReal
y : ℝ
yp : y > 0
h : x.toReal = y
⊢ x ≠ ⊤ | x : ENNReal
y : ℝ
h : x.toReal = y
yp : ¬x ≠ ⊤
⊢ ¬y > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ENNReal
y : ℝ
yp : y > 0
h : x.toReal = y
⊢ x ≠ ⊤
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | invert_toReal | [252, 1] | [255, 56] | simp only [ne_eq, not_not] at yp | x : ENNReal
y : ℝ
h : x.toReal = y
yp : ¬x ≠ ⊤
⊢ ¬y > 0 | x : ENNReal
y : ℝ
h : x.toReal = y
yp : x = ⊤
⊢ ¬y > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ENNReal
y : ℝ
h : x.toReal = y
yp : ¬x ≠ ⊤
⊢ ¬y > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | invert_toReal | [252, 1] | [255, 56] | simp only [yp, ENNReal.top_toReal] at h | x : ENNReal
y : ℝ
h : x.toReal = y
yp : x = ⊤
⊢ ¬y > 0 | x : ENNReal
y : ℝ
yp : x = ⊤
h : 0 = y
⊢ ¬y > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
x : ENNReal
y : ℝ
h : x.toReal = y
yp : x = ⊤
⊢ ¬y > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | invert_toReal | [252, 1] | [255, 56] | simp only [← h, lt_self_iff_false, not_false_eq_true] | x : ENNReal
y : ℝ
yp : x = ⊤
h : 0 = y
⊢ ¬y > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ENNReal
y : ℝ
yp : x = ⊤
h : 0 = y
⊢ ¬y > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | have im := MeasurePreserving.symm _ Complex.volume_preserving_equiv_real_prod | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (z : ℂ) in annulus_oc c r0 r1, f z = ∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
im : MeasurePreserving (⇑Complex.measurableEquivRealProd.symm) volume volume
⊢ ∫ (z : ℂ) in annulus_oc c r0 r1, f z = ∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (z : ℂ) in annulus_oc c r0 r1, f z = ∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | rw [←MeasurePreserving.setIntegral_preimage_emb im
Complex.measurableEquivRealProd.symm.measurableEmbedding f _] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
im : MeasurePreserving (⇑Complex.measurableEquivRealProd.symm) volume volume
⊢ ∫ (z : ℂ) in annulus_oc c r0 r1, f z = ∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
im : MeasurePreserving (⇑Complex.measurableEquivRealProd.symm) volume volume
⊢ ∫ (x : ℝ × ℝ) in ⇑Complex.measurableEquivRealProd.symm ⁻¹' annulus_oc c r0 r1,
f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
im : MeasurePreserving (⇑Complex.measurableEquivRealProd.symm) volume volume
⊢ ∫ (z : ℂ) in annulus_oc c r0 r1, f z = ∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | clear im | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
im : MeasurePreserving (⇑Complex.measurableEquivRealProd.symm) volume volume
⊢ ∫ (x : ℝ × ℝ) in ⇑Complex.measurableEquivRealProd.symm ⁻¹' annulus_oc c r0 r1,
f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in ⇑Complex.measurableEquivRealProd.symm ⁻¹' annulus_oc c r0 r1,
f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
im : MeasurePreserving (⇑Complex.measurableEquivRealProd.symm) volume volume
⊢ ∫ (x : ℝ × ℝ) in ⇑Complex.measurableEquivRealProd.symm ⁻¹' annulus_oc c r0 r1,
f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | rw [square_eq r0p] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in ⇑Complex.measurableEquivRealProd.symm ⁻¹' annulus_oc c r0 r1,
f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in realCircleMap c '' square r0 r1, f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in ⇑Complex.measurableEquivRealProd.symm ⁻¹' annulus_oc c r0 r1,
f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | have dc : ∀ x, x ∈ square r0 r1 →
HasFDerivWithinAt (realCircleMap c) (rcmDeriv x) (square r0 r1) x :=
fun _ _ ↦ realCircleMap.fderiv.hasFDerivWithinAt | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in realCircleMap c '' square r0 r1, f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
dc : ∀ x ∈ square r0 r1, HasFDerivWithinAt (realCircleMap c) (rcmDeriv x) (square r0 r1) x
⊢ ∫ (x : ℝ × ℝ) in realCircleMap c '' square r0 r1, f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in realCircleMap c '' square r0 r1, f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | rw [integral_image_eq_integral_abs_det_fderiv_smul volume Measurable.square dc (rcm_inj r0p)] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
dc : ∀ x ∈ square r0 r1, HasFDerivWithinAt (realCircleMap c) (rcmDeriv x) (square r0 r1) x
⊢ ∫ (x : ℝ × ℝ) in realCircleMap c '' square r0 r1, f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
dc : ∀ x ∈ square r0 r1, HasFDerivWithinAt (realCircleMap c) (rcmDeriv x) (square r0 r1) x
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |(rcmDeriv x).det| • f (Complex.measurableEquivRealProd.symm (realCircleMap c x)) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
dc : ∀ x ∈ square r0 r1, HasFDerivWithinAt (realCircleMap c) (rcmDeriv x) (square r0 r1) x
⊢ ∫ (x : ℝ × ℝ) in realCircleMap c '' square r0 r1, f (Complex.measurableEquivRealProd.symm x) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | clear dc | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
dc : ∀ x ∈ square r0 r1, HasFDerivWithinAt (realCircleMap c) (rcmDeriv x) (square r0 r1) x
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |(rcmDeriv x).det| • f (Complex.measurableEquivRealProd.symm (realCircleMap c x)) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |(rcmDeriv x).det| • f (Complex.measurableEquivRealProd.symm (realCircleMap c x)) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
dc : ∀ x ∈ square r0 r1, HasFDerivWithinAt (realCircleMap c) (rcmDeriv x) (square r0 r1) x
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |(rcmDeriv x).det| • f (Complex.measurableEquivRealProd.symm (realCircleMap c x)) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | simp_rw [rcmDeriv.det] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |(rcmDeriv x).det| • f (Complex.measurableEquivRealProd.symm (realCircleMap c x)) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |x.1| • f (Complex.measurableEquivRealProd.symm (realCircleMap c x)) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |(rcmDeriv x).det| • f (Complex.measurableEquivRealProd.symm (realCircleMap c x)) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | simp_rw [realCircleMap_eq_circleMap] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |x.1| • f (Complex.measurableEquivRealProd.symm (realCircleMap c x)) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1,
|x.1| • f (Complex.measurableEquivRealProd.symm (Complex.equivRealProd (circleMap c x.1 x.2))) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |x.1| • f (Complex.measurableEquivRealProd.symm (realCircleMap c x)) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | simp_rw [measurable_symm_equiv_inverse] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1,
|x.1| • f (Complex.measurableEquivRealProd.symm (Complex.equivRealProd (circleMap c x.1 x.2))) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |x.1| • f (circleMap c x.1 x.2) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1,
|x.1| • f (Complex.measurableEquivRealProd.symm (Complex.equivRealProd (circleMap c x.1 x.2))) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | have e : ∀ x : ℝ × ℝ, x ∈ square r0 r1 → |x.1| • f (circleMap c x.1 x.2) =
x.1 • f (circleMap c x.1 x.2) := by
intro x xs; rw [abs_of_pos (square.rp r0p xs)] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |x.1| • f (circleMap c x.1 x.2) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
e : ∀ x ∈ square r0 r1, |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2)
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |x.1| • f (circleMap c x.1 x.2) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |x.1| • f (circleMap c x.1 x.2) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | rw [MeasureTheory.setIntegral_congr Measurable.square e] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
e : ∀ x ∈ square r0 r1, |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2)
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |x.1| • f (circleMap c x.1 x.2) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
e : ∀ x ∈ square r0 r1, |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2)
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, x.1 • f (circleMap c x.1 x.2) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
e : ∀ x ∈ square r0 r1, |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2)
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, |x.1| • f (circleMap c x.1 x.2) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | clear e | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
e : ∀ x ∈ square r0 r1, |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2)
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, x.1 • f (circleMap c x.1 x.2) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, x.1 • f (circleMap c x.1 x.2) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
e : ∀ x ∈ square r0 r1, |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2)
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, x.1 • f (circleMap c x.1 x.2) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | rw [square, Measure.volume_eq_prod, MeasureTheory.setIntegral_prod] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, x.1 • f (circleMap c x.1 x.2) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ) in Ioc r0 r1, ∫ (y : ℝ) in Ioc 0 (2 * π), (x, y).1 • f (circleMap c (x, y).1 (x, y).2) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Ioc r0 r1 ×ˢ Ioc 0 (2 * π)) (volume.prod volume) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ × ℝ) in square r0 r1, x.1 • f (circleMap c x.1 x.2) ∂volume =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | simp [integral_smul] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ) in Ioc r0 r1, ∫ (y : ℝ) in Ioc 0 (2 * π), (x, y).1 • f (circleMap c (x, y).1 (x, y).2) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Ioc r0 r1 ×ˢ Ioc 0 (2 * π)) (volume.prod volume) | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Ioc r0 r1 ×ˢ Ioc 0 (2 * π)) (volume.prod volume) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∫ (x : ℝ) in Ioc r0 r1, ∫ (y : ℝ) in Ioc 0 (2 * π), (x, y).1 • f (circleMap c (x, y).1 (x, y).2) =
∫ (s : ℝ) in Ioc r0 r1, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Ioc r0 r1 ×ˢ Ioc 0 (2 * π)) (volume.prod volume)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | exact fi.mono_set (prod_mono Ioc_subset_Icc_self Ioc_subset_Icc_self) | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
fi : IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) volume
⊢ IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Ioc r0 r1 ×ˢ Ioc 0 (2 * π)) (volume.prod volume) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
fi : IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) volume
⊢ IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Ioc r0 r1 ×ˢ Ioc 0 (2 * π)) (volume.prod volume)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | intro x xs | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∀ x ∈ square r0 r1, |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : x ∈ square r0 r1
⊢ |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ∀ x ∈ square r0 r1, |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | rw [abs_of_pos (square.rp r0p xs)] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : x ∈ square r0 r1
⊢ |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : x ∈ square r0 r1
⊢ |x.1| • f (circleMap c x.1 x.2) = x.1 • f (circleMap c x.1 x.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | apply ContinuousOn.integrableOn_compact | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) volume | case hK
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ IsCompact (Icc r0 r1 ×ˢ Icc 0 (2 * π))
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ContinuousOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ IntegrableOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) volume
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | exact IsCompact.prod isCompact_Icc isCompact_Icc | case hK
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ IsCompact (Icc r0 r1 ×ˢ Icc 0 (2 * π)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hK
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ IsCompact (Icc r0 r1 ×ˢ Icc 0 (2 * π))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | apply ContinuousOn.smul continuous_fst.continuousOn | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ContinuousOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ContinuousOn (fun x => f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ContinuousOn (fun x => x.1 • f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | apply fc.comp continuous_circleMap_full.continuousOn | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ContinuousOn (fun x => f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ MapsTo (fun x => circleMap c x.1 x.2) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) (annulus_cc c r0 r1) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ ContinuousOn (fun x => f (circleMap c x.1 x.2)) (Icc r0 r1 ×ˢ Icc 0 (2 * π))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | intro x xs | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ MapsTo (fun x => circleMap c x.1 x.2) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) (annulus_cc c r0 r1) | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : x ∈ Icc r0 r1 ×ˢ Icc 0 (2 * π)
⊢ (fun x => circleMap c x.1 x.2) x ∈ annulus_cc c r0 r1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
⊢ MapsTo (fun x => circleMap c x.1 x.2) (Icc r0 r1 ×ˢ Icc 0 (2 * π)) (annulus_cc c r0 r1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | simp only [Icc_prod_Icc, mem_Icc, Prod.le_def] at xs | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : x ∈ Icc r0 r1 ×ˢ Icc 0 (2 * π)
⊢ (fun x => circleMap c x.1 x.2) x ∈ annulus_cc c r0 r1 | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : (r0 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ r1 ∧ x.2 ≤ 2 * π
⊢ (fun x => circleMap c x.1 x.2) x ∈ annulus_cc c r0 r1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : x ∈ Icc r0 r1 ×ˢ Icc 0 (2 * π)
⊢ (fun x => circleMap c x.1 x.2) x ∈ annulus_cc c r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | have x0 : 0 ≤ x.1 := by linarith | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : (r0 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ r1 ∧ x.2 ≤ 2 * π
⊢ (fun x => circleMap c x.1 x.2) x ∈ annulus_cc c r0 r1 | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : (r0 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ r1 ∧ x.2 ≤ 2 * π
x0 : 0 ≤ x.1
⊢ (fun x => circleMap c x.1 x.2) x ∈ annulus_cc c r0 r1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : (r0 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ r1 ∧ x.2 ≤ 2 * π
⊢ (fun x => circleMap c x.1 x.2) x ∈ annulus_cc c r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | simp only [circleMap, annulus_cc, mem_diff, Metric.mem_closedBall, dist_self_add_left,
norm_mul, Complex.norm_eq_abs, Complex.abs_ofReal, abs_of_nonneg x0,
Complex.abs_exp_ofReal_mul_I, mul_one, xs.2.1, Metric.mem_ball, not_lt, xs.1.1, and_self] | case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : (r0 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ r1 ∧ x.2 ≤ 2 * π
x0 : 0 ≤ x.1
⊢ (fun x => circleMap c x.1 x.2) x ∈ annulus_cc c r0 r1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : (r0 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ r1 ∧ x.2 ≤ 2 * π
x0 : 0 ≤ x.1
⊢ (fun x => circleMap c x.1 x.2) x ∈ annulus_cc c r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_annulus | [258, 1] | [293, 72] | linarith | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : (r0 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ r1 ∧ x.2 ≤ 2 * π
⊢ 0 ≤ x.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r0 r1 : ℝ
fc : ContinuousOn f (annulus_cc c r0 r1)
r0p : 0 ≤ r0
x : ℝ × ℝ
xs : (r0 ≤ x.1 ∧ 0 ≤ x.2) ∧ x.1 ≤ r1 ∧ x.2 ≤ 2 * π
⊢ 0 ≤ x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_ball | [296, 1] | [305, 8] | have center : closedBall c r =ᵐ[volume] (closedBall c r \ {c} : Set ℂ) := ae_minus_point | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ ∫ (z : ℂ) in closedBall c r, f z = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
center : volume.ae.EventuallyEq (closedBall c r) (closedBall c r \ {c})
⊢ ∫ (z : ℂ) in closedBall c r, f z = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ ∫ (z : ℂ) in closedBall c r, f z = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_ball | [296, 1] | [305, 8] | rw [MeasureTheory.setIntegral_congr_set_ae center] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
center : volume.ae.EventuallyEq (closedBall c r) (closedBall c r \ {c})
⊢ ∫ (z : ℂ) in closedBall c r, f z = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
center : volume.ae.EventuallyEq (closedBall c r) (closedBall c r \ {c})
⊢ ∫ (x : ℂ) in closedBall c r \ {c}, f x = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
center : volume.ae.EventuallyEq (closedBall c r) (closedBall c r \ {c})
⊢ ∫ (z : ℂ) in closedBall c r, f z = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_ball | [296, 1] | [305, 8] | clear center | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
center : volume.ae.EventuallyEq (closedBall c r) (closedBall c r \ {c})
⊢ ∫ (x : ℂ) in closedBall c r \ {c}, f x = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ ∫ (x : ℂ) in closedBall c r \ {c}, f x = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
center : volume.ae.EventuallyEq (closedBall c r) (closedBall c r \ {c})
⊢ ∫ (x : ℂ) in closedBall c r \ {c}, f x = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t)
TACTIC:
|
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