Datasets:
pkuAI4M
/
lean_github

Dataset card Data Studio Files
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1
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commit
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147 values
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7
101
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stringlengths
1
94
start
stringlengths
6
10
end
stringlengths
6
11
tactic
stringlengths
1
11.2k
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3
2.09M
state_after
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2.09M
input
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73
2.09M
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log1p_small'
[298, 1]
[312, 44]
bound
z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z ⊢ 1 / (1 - r) > 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z ⊢ 1 / (1 - r) > 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log1p_small'
[298, 1]
[312, 44]
bound
z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z w : ℂ wr : Complex.abs w < r ⊢ r < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ r : ℝ r1 : r < 1 zr : Complex.abs z ≤ r r0 : 0 < r fc : ContinuousAt (fun z => (1 + z).log) z w : ℂ wr : Complex.abs w < r ⊢ r < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log1p_small
[314, 1]
[315, 68]
norm_num
z : ℂ zs : Complex.abs z ≤ 1 / 2 ⊢ 1 / 2 < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 / 2 ⊢ 1 / 2 < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log1p_small
[314, 1]
[315, 68]
norm_num
z : ℂ zs : Complex.abs z ≤ 1 / 2 ⊢ 1 / (1 - 1 / 2) * Complex.abs z = 2 * Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 / 2 ⊢ 1 / (1 - 1 / 2) * Complex.abs z = 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Real.log1p_small'
[318, 1]
[324, 62]
set z := (x : ℂ)
x r : ℝ r1 : r < 1 xr : |x| ≤ r ⊢ |(1 + x).log| ≤ 1 / (1 - r) * |x|
x r : ℝ r1 : r < 1 xr : |x| ≤ r z : ℂ := ↑x ⊢ |(1 + x).log| ≤ 1 / (1 - r) * |x|
Please generate a tactic in lean4 to solve the state. STATE: x r : ℝ r1 : r < 1 xr : |x| ≤ r ⊢ |(1 + x).log| ≤ 1 / (1 - r) * |x| TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Real.log1p_small'
[318, 1]
[324, 62]
have zx : abs z = |x| := Complex.abs_ofReal _
x r : ℝ r1 : r < 1 xr : |x| ≤ r z : ℂ := ↑x ⊢ |(1 + x).log| ≤ 1 / (1 - r) * |x|
x r : ℝ r1 : r < 1 xr : |x| ≤ r z : ℂ := ↑x zx : Complex.abs z = |x| ⊢ |(1 + x).log| ≤ 1 / (1 - r) * |x|
Please generate a tactic in lean4 to solve the state. STATE: x r : ℝ r1 : r < 1 xr : |x| ≤ r z : ℂ := ↑x ⊢ |(1 + x).log| ≤ 1 / (1 - r) * |x| TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Real.log1p_small'
[318, 1]
[324, 62]
simp only [← Complex.log_ofReal_re, ← zx] at xr ⊢
x r : ℝ r1 : r < 1 xr : |x| ≤ r z : ℂ := ↑x zx : Complex.abs z = |x| ⊢ |(1 + x).log| ≤ 1 / (1 - r) * |x|
x r : ℝ r1 : r < 1 z : ℂ := ↑x zx : Complex.abs z = |x| xr : Complex.abs z ≤ r ⊢ |(↑(1 + x)).log.re| ≤ 1 / (1 - r) * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: x r : ℝ r1 : r < 1 xr : |x| ≤ r z : ℂ := ↑x zx : Complex.abs z = |x| ⊢ |(1 + x).log| ≤ 1 / (1 - r) * |x| TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Real.log1p_small'
[318, 1]
[324, 62]
refine _root_.trans (_root_.trans (Complex.abs_re_le_abs _) ?_) (_root_.log1p_small' r1 xr)
x r : ℝ r1 : r < 1 z : ℂ := ↑x zx : Complex.abs z = |x| xr : Complex.abs z ≤ r ⊢ |(↑(1 + x)).log.re| ≤ 1 / (1 - r) * Complex.abs z
x r : ℝ r1 : r < 1 z : ℂ := ↑x zx : Complex.abs z = |x| xr : Complex.abs z ≤ r ⊢ Complex.abs (↑(1 + x)).log ≤ Complex.abs (1 + z).log
Please generate a tactic in lean4 to solve the state. STATE: x r : ℝ r1 : r < 1 z : ℂ := ↑x zx : Complex.abs z = |x| xr : Complex.abs z ≤ r ⊢ |(↑(1 + x)).log.re| ≤ 1 / (1 - r) * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Real.log1p_small'
[318, 1]
[324, 62]
simp only [Complex.ofReal_add, Complex.ofReal_one, le_refl]
x r : ℝ r1 : r < 1 z : ℂ := ↑x zx : Complex.abs z = |x| xr : Complex.abs z ≤ r ⊢ Complex.abs (↑(1 + x)).log ≤ Complex.abs (1 + z).log
no goals
Please generate a tactic in lean4 to solve the state. STATE: x r : ℝ r1 : r < 1 z : ℂ := ↑x zx : Complex.abs z = |x| xr : Complex.abs z ≤ r ⊢ Complex.abs (↑(1 + x)).log ≤ Complex.abs (1 + z).log TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Real.log1p_small
[327, 1]
[328, 73]
norm_num
x : ℝ xr : |x| ≤ 1 / 2 ⊢ 1 / 2 < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ xr : |x| ≤ 1 / 2 ⊢ 1 / 2 < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Real.log1p_small
[327, 1]
[328, 73]
norm_num
x : ℝ xr : |x| ≤ 1 / 2 ⊢ 1 / (1 - 1 / 2) * |x| = 2 * |x|
no goals
Please generate a tactic in lean4 to solve the state. STATE: x : ℝ xr : |x| ≤ 1 / 2 ⊢ 1 / (1 - 1 / 2) * |x| = 2 * |x| TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log_small
[331, 1]
[333, 56]
generalize zw : z - 1 = z1
z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ⊢ Complex.abs z.log ≤ 2 * Complex.abs (z - 1)
z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 ⊢ Complex.abs z.log ≤ 2 * Complex.abs z1
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ⊢ Complex.abs z.log ≤ 2 * Complex.abs (z - 1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log_small
[331, 1]
[333, 56]
have wz : z = 1 + z1 := by rw [← zw]; ring
z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 ⊢ Complex.abs z.log ≤ 2 * Complex.abs z1
z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs z.log ≤ 2 * Complex.abs z1
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 ⊢ Complex.abs z.log ≤ 2 * Complex.abs z1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log_small
[331, 1]
[333, 56]
rw [wz]
z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs z.log ≤ 2 * Complex.abs z1
z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs (1 + z1).log ≤ 2 * Complex.abs z1
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs z.log ≤ 2 * Complex.abs z1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log_small
[331, 1]
[333, 56]
refine log1p_small ?_
z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs (1 + z1).log ≤ 2 * Complex.abs z1
z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs z1 ≤ 1 / 2
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs (1 + z1).log ≤ 2 * Complex.abs z1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log_small
[331, 1]
[333, 56]
rw [← zw]
z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs z1 ≤ 1 / 2
z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs (z - 1) ≤ 1 / 2
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs z1 ≤ 1 / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log_small
[331, 1]
[333, 56]
assumption
z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs (z - 1) ≤ 1 / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs (z - 1) ≤ 1 / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log_small
[331, 1]
[333, 56]
rw [← zw]
z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 ⊢ z = 1 + z1
z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 ⊢ z = 1 + (z - 1)
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 ⊢ z = 1 + z1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log_small
[331, 1]
[333, 56]
ring
z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 ⊢ z = 1 + (z - 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 z1 : ℂ zw : z - 1 = z1 ⊢ z = 1 + (z - 1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_exp_small
[335, 1]
[341, 26]
have hr : 0 ≤ (1 : ℝ) := by norm_num
z : ℂ h : Complex.abs z < 1 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_exp_small
[335, 1]
[341, 26]
have L := Complex.locally_lipschitz_exp hr (by bound) 0 z (by simpa only [sub_zero, Complex.norm_eq_abs])
z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : ‖z.exp - exp 0‖ ≤ (1 + 1) * ‖exp 0‖ * ‖z - 0‖ ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_exp_small
[335, 1]
[341, 26]
simp only [Complex.exp_zero, Complex.norm_eq_abs, norm_one, mul_one, sub_zero] at L
z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : ‖z.exp - exp 0‖ ≤ (1 + 1) * ‖exp 0‖ * ‖z - 0‖ ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ (1 + 1) * Complex.abs z ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : ‖z.exp - exp 0‖ ≤ (1 + 1) * ‖exp 0‖ * ‖z - 0‖ ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_exp_small
[335, 1]
[341, 26]
have t : 1 + 1 = (2 : ℝ) := by norm_num
z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ (1 + 1) * Complex.abs z ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ (1 + 1) * Complex.abs z t : 1 + 1 = 2 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ (1 + 1) * Complex.abs z ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_exp_small
[335, 1]
[341, 26]
rw [t] at L
z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ (1 + 1) * Complex.abs z t : 1 + 1 = 2 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z t : 1 + 1 = 2 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ (1 + 1) * Complex.abs z t : 1 + 1 = 2 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_exp_small
[335, 1]
[341, 26]
assumption
z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z t : 1 + 1 = 2 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z t : 1 + 1 = 2 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_exp_small
[335, 1]
[341, 26]
norm_num
z : ℂ h : Complex.abs z < 1 ⊢ 0 ≤ 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 ⊢ 0 ≤ 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_exp_small
[335, 1]
[341, 26]
bound
z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 ⊢ 1 ≤ 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 ⊢ 1 ≤ 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_exp_small
[335, 1]
[341, 26]
simpa only [sub_zero, Complex.norm_eq_abs]
z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 ⊢ ‖z - 0‖ < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 ⊢ ‖z - 0‖ < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
weak_exp_small
[335, 1]
[341, 26]
norm_num
z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ (1 + 1) * Complex.abs z ⊢ 1 + 1 = 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ h : Complex.abs z < 1 hr : 0 ≤ 1 L : Complex.abs (z.exp - 1) ≤ (1 + 1) * Complex.abs z ⊢ 1 + 1 = 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
exp_small
[344, 1]
[351, 38]
have rp : (1 : ℝ) > 0 := by norm_num
z : ℂ zs : Complex.abs z ≤ 1 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
exp_small
[344, 1]
[351, 38]
have cp : (2 : ℝ) > 0 := by norm_num
z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
exp_small
[344, 1]
[351, 38]
have fc : ContinuousAt (fun z ↦ exp z - 1) z := by apply ContinuousAt.sub; apply ContinuousAt.cexp exact continuousAt_id; exact continuousAt_const
z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 fc : ContinuousAt (fun z => z.exp - 1) z ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
exp_small
[344, 1]
[351, 38]
apply weak_to_strong_small rp cp zs fc
z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 fc : ContinuousAt (fun z => z.exp - 1) z ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 fc : ContinuousAt (fun z => z.exp - 1) z ⊢ ∀ (z : ℂ), Complex.abs z < 1 → Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 fc : ContinuousAt (fun z => z.exp - 1) z ⊢ Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
exp_small
[344, 1]
[351, 38]
intro w wr
z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 fc : ContinuousAt (fun z => z.exp - 1) z ⊢ ∀ (z : ℂ), Complex.abs z < 1 → Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z
z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 fc : ContinuousAt (fun z => z.exp - 1) z w : ℂ wr : Complex.abs w < 1 ⊢ Complex.abs (w.exp - 1) ≤ 2 * Complex.abs w
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 fc : ContinuousAt (fun z => z.exp - 1) z ⊢ ∀ (z : ℂ), Complex.abs z < 1 → Complex.abs (z.exp - 1) ≤ 2 * Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
exp_small
[344, 1]
[351, 38]
exact weak_exp_small wr
z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 fc : ContinuousAt (fun z => z.exp - 1) z w : ℂ wr : Complex.abs w < 1 ⊢ Complex.abs (w.exp - 1) ≤ 2 * Complex.abs w
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 fc : ContinuousAt (fun z => z.exp - 1) z w : ℂ wr : Complex.abs w < 1 ⊢ Complex.abs (w.exp - 1) ≤ 2 * Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
exp_small
[344, 1]
[351, 38]
norm_num
z : ℂ zs : Complex.abs z ≤ 1 ⊢ 1 > 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 ⊢ 1 > 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
exp_small
[344, 1]
[351, 38]
norm_num
z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 ⊢ 2 > 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 ⊢ 2 > 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
exp_small
[344, 1]
[351, 38]
apply ContinuousAt.sub
z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun z => z.exp - 1) z
case hf z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => x.exp) z case hg z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => 1) z
Please generate a tactic in lean4 to solve the state. STATE: z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun z => z.exp - 1) z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
exp_small
[344, 1]
[351, 38]
apply ContinuousAt.cexp
case hf z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => x.exp) z case hg z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => 1) z
case hf.h z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun y => y) z case hg z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => 1) z
Please generate a tactic in lean4 to solve the state. STATE: case hf z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => x.exp) z case hg z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => 1) z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
exp_small
[344, 1]
[351, 38]
exact continuousAt_id
case hf.h z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun y => y) z case hg z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => 1) z
case hg z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => 1) z
Please generate a tactic in lean4 to solve the state. STATE: case hf.h z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun y => y) z case hg z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => 1) z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
exp_small
[344, 1]
[351, 38]
exact continuousAt_const
case hg z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => 1) z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case hg z : ℂ zs : Complex.abs z ≤ 1 rp : 1 > 0 cp : 2 > 0 ⊢ ContinuousAt (fun x => 1) z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
have z1 : 1 + z ≠ 0 := by rw [←Complex.abs.ne_zero_iff]; apply ne_of_gt calc abs (1 + z) ≥ abs (1 : ℂ) - abs z := by bound _ ≥ abs (1 : ℂ) - 1/2 := by bound _ > 0 := by norm_num
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ Complex.abs ((1 + z) ^ w - 1) ≤ 4 * Complex.abs z * Complex.abs w
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ⊢ Complex.abs ((1 + z) ^ w - 1) ≤ 4 * Complex.abs z * Complex.abs w
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ Complex.abs ((1 + z) ^ w - 1) ≤ 4 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
rw [Complex.cpow_def_of_ne_zero z1]
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ⊢ Complex.abs ((1 + z) ^ w - 1) ≤ 4 * Complex.abs z * Complex.abs w
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ⊢ Complex.abs ((1 + z) ^ w - 1) ≤ 4 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
have ls := log1p_small zs
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
have eas : abs (log (1 + z) * w) ≤ 1 := by rw [Complex.abs.map_mul] calc abs (log (1 + z)) * abs w ≤ 2 * abs z * abs w := by bound _ ≤ 2 * (1/2) * 1 := by bound _ = 1 := by norm_num
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
have es := exp_small eas
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs ((1 + z).log * w) ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
rw [Complex.abs.map_mul, ←mul_assoc] at es
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs ((1 + z).log * w) ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs ((1 + z).log * w) ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
trans 2 * abs (log (1 + z)) * abs w
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ 2 * Complex.abs (1 + z).log * Complex.abs w ≤ 4 * Complex.abs z * Complex.abs w
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 4 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
exact es
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ 2 * Complex.abs (1 + z).log * Complex.abs w ≤ 4 * Complex.abs z * Complex.abs w
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ 2 * Complex.abs (1 + z).log * Complex.abs w ≤ 4 * Complex.abs z * Complex.abs w
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ 2 * Complex.abs (1 + z).log * Complex.abs w ≤ 4 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
calc 2 * abs (log (1 + z)) * abs w ≤ 2 * (2 * abs z) * abs w := by bound _ = 4 * abs z * abs w := by ring
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ 2 * Complex.abs (1 + z).log * Complex.abs w ≤ 4 * Complex.abs z * Complex.abs w
no goals
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ 2 * Complex.abs (1 + z).log * Complex.abs w ≤ 4 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
rw [←Complex.abs.ne_zero_iff]
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ 1 + z ≠ 0
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ Complex.abs (1 + z) ≠ 0
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ 1 + z ≠ 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
apply ne_of_gt
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ Complex.abs (1 + z) ≠ 0
case h z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ 0 < Complex.abs (1 + z)
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ Complex.abs (1 + z) ≠ 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
calc abs (1 + z) ≥ abs (1 : ℂ) - abs z := by bound _ ≥ abs (1 : ℂ) - 1/2 := by bound _ > 0 := by norm_num
case h z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ 0 < Complex.abs (1 + z)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ 0 < Complex.abs (1 + z) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
bound
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ Complex.abs (1 + z) ≥ Complex.abs 1 - Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ Complex.abs (1 + z) ≥ Complex.abs 1 - Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
bound
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ Complex.abs 1 - Complex.abs z ≥ Complex.abs 1 - 1 / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ Complex.abs 1 - Complex.abs z ≥ Complex.abs 1 - 1 / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
norm_num
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ Complex.abs 1 - 1 / 2 > 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ Complex.abs 1 - 1 / 2 > 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
rw [Complex.abs.map_mul]
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z ⊢ Complex.abs ((1 + z).log * w) ≤ 1
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z ⊢ Complex.abs (1 + z).log * Complex.abs w ≤ 1
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z ⊢ Complex.abs ((1 + z).log * w) ≤ 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
calc abs (log (1 + z)) * abs w ≤ 2 * abs z * abs w := by bound _ ≤ 2 * (1/2) * 1 := by bound _ = 1 := by norm_num
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z ⊢ Complex.abs (1 + z).log * Complex.abs w ≤ 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z ⊢ Complex.abs (1 + z).log * Complex.abs w ≤ 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
bound
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z ⊢ Complex.abs (1 + z).log * Complex.abs w ≤ 2 * Complex.abs z * Complex.abs w
no goals
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z ⊢ Complex.abs (1 + z).log * Complex.abs w ≤ 2 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
bound
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z ⊢ 2 * Complex.abs z * Complex.abs w ≤ 2 * (1 / 2) * 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z ⊢ 2 * Complex.abs z * Complex.abs w ≤ 2 * (1 / 2) * 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
norm_num
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z ⊢ 2 * (1 / 2) * 1 = 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z ⊢ 2 * (1 / 2) * 1 = 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
bound
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ 2 * Complex.abs (1 + z).log * Complex.abs w ≤ 2 * (2 * Complex.abs z) * Complex.abs w
no goals
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ 2 * Complex.abs (1 + z).log * Complex.abs w ≤ 2 * (2 * Complex.abs z) * Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow1p_small
[353, 1]
[372, 37]
ring
z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ 2 * (2 * Complex.abs z) * Complex.abs w = 4 * Complex.abs z * Complex.abs w
no goals
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs z ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : 1 + z ≠ 0 ls : Complex.abs (1 + z).log ≤ 2 * Complex.abs z eas : Complex.abs ((1 + z).log * w) ≤ 1 es : Complex.abs (((1 + z).log * w).exp - 1) ≤ 2 * Complex.abs (1 + z).log * Complex.abs w ⊢ 2 * (2 * Complex.abs z) * Complex.abs w = 4 * Complex.abs z * Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow_small
[375, 1]
[378, 59]
generalize zw : z - 1 = z1
z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ Complex.abs (z ^ w - 1) ≤ 4 * Complex.abs (z - 1) * Complex.abs w
z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 ⊢ Complex.abs (z ^ w - 1) ≤ 4 * Complex.abs z1 * Complex.abs w
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 ⊢ Complex.abs (z ^ w - 1) ≤ 4 * Complex.abs (z - 1) * Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow_small
[375, 1]
[378, 59]
have wz : z = 1 + z1 := by rw [← zw]; ring
z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 ⊢ Complex.abs (z ^ w - 1) ≤ 4 * Complex.abs z1 * Complex.abs w
z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs (z ^ w - 1) ≤ 4 * Complex.abs z1 * Complex.abs w
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 ⊢ Complex.abs (z ^ w - 1) ≤ 4 * Complex.abs z1 * Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow_small
[375, 1]
[378, 59]
rw [wz]
z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs (z ^ w - 1) ≤ 4 * Complex.abs z1 * Complex.abs w
z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs ((1 + z1) ^ w - 1) ≤ 4 * Complex.abs z1 * Complex.abs w
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs (z ^ w - 1) ≤ 4 * Complex.abs z1 * Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow_small
[375, 1]
[378, 59]
refine pow1p_small ?_ ws
z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs ((1 + z1) ^ w - 1) ≤ 4 * Complex.abs z1 * Complex.abs w
z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs z1 ≤ 1 / 2
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs ((1 + z1) ^ w - 1) ≤ 4 * Complex.abs z1 * Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow_small
[375, 1]
[378, 59]
rw [← zw]
z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs z1 ≤ 1 / 2
z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs (z - 1) ≤ 1 / 2
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs z1 ≤ 1 / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow_small
[375, 1]
[378, 59]
assumption
z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs (z - 1) ≤ 1 / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 wz : z = 1 + z1 ⊢ Complex.abs (z - 1) ≤ 1 / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow_small
[375, 1]
[378, 59]
rw [← zw]
z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 ⊢ z = 1 + z1
z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 ⊢ z = 1 + (z - 1)
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 ⊢ z = 1 + z1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
pow_small
[375, 1]
[378, 59]
ring
z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 ⊢ z = 1 + (z - 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: z w : ℂ zs : Complex.abs (z - 1) ≤ 1 / 2 ws : Complex.abs w ≤ 1 z1 : ℂ zw : z - 1 = z1 ⊢ z = 1 + (z - 1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
add_ne_zero_of_abs_lt
[381, 1]
[384, 56]
have e : a + b = a - -b := by abel
a b : ℂ h : Complex.abs b < Complex.abs a ⊢ a + b ≠ 0
a b : ℂ h : Complex.abs b < Complex.abs a e : a + b = a - -b ⊢ a + b ≠ 0
Please generate a tactic in lean4 to solve the state. STATE: a b : ℂ h : Complex.abs b < Complex.abs a ⊢ a + b ≠ 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
add_ne_zero_of_abs_lt
[381, 1]
[384, 56]
rw [e, sub_ne_zero]
a b : ℂ h : Complex.abs b < Complex.abs a e : a + b = a - -b ⊢ a + b ≠ 0
a b : ℂ h : Complex.abs b < Complex.abs a e : a + b = a - -b ⊢ a ≠ -b
Please generate a tactic in lean4 to solve the state. STATE: a b : ℂ h : Complex.abs b < Complex.abs a e : a + b = a - -b ⊢ a + b ≠ 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
add_ne_zero_of_abs_lt
[381, 1]
[384, 56]
contrapose h
a b : ℂ h : Complex.abs b < Complex.abs a e : a + b = a - -b ⊢ a ≠ -b
a b : ℂ e : a + b = a - -b h : ¬a ≠ -b ⊢ ¬Complex.abs b < Complex.abs a
Please generate a tactic in lean4 to solve the state. STATE: a b : ℂ h : Complex.abs b < Complex.abs a e : a + b = a - -b ⊢ a ≠ -b TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
add_ne_zero_of_abs_lt
[381, 1]
[384, 56]
simp only [not_not] at h
a b : ℂ e : a + b = a - -b h : ¬a ≠ -b ⊢ ¬Complex.abs b < Complex.abs a
a b : ℂ e : a + b = a - -b h : a = -b ⊢ ¬Complex.abs b < Complex.abs a
Please generate a tactic in lean4 to solve the state. STATE: a b : ℂ e : a + b = a - -b h : ¬a ≠ -b ⊢ ¬Complex.abs b < Complex.abs a TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
add_ne_zero_of_abs_lt
[381, 1]
[384, 56]
simp only [h, not_lt, AbsoluteValue.map_neg, le_refl]
a b : ℂ e : a + b = a - -b h : a = -b ⊢ ¬Complex.abs b < Complex.abs a
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b : ℂ e : a + b = a - -b h : a = -b ⊢ ¬Complex.abs b < Complex.abs a TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
add_ne_zero_of_abs_lt
[381, 1]
[384, 56]
abel
a b : ℂ h : Complex.abs b < Complex.abs a ⊢ a + b = a - -b
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b : ℂ h : Complex.abs b < Complex.abs a ⊢ a + b = a - -b TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Real.exp_one_lt_3
[387, 1]
[388, 48]
norm_num
⊢ 2.7182818286 < 3
no goals
Please generate a tactic in lean4 to solve the state. STATE: ⊢ 2.7182818286 < 3 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log_add
[390, 1]
[393, 84]
have d0 : 0 < 1 + b/a := by field_simp [a0.ne']; bound
a b : ℝ a0 : 0 < a ab0 : 0 < a + b ⊢ (a + b).log = a.log + (1 + b / a).log
a b : ℝ a0 : 0 < a ab0 : 0 < a + b d0 : 0 < 1 + b / a ⊢ (a + b).log = a.log + (1 + b / a).log
Please generate a tactic in lean4 to solve the state. STATE: a b : ℝ a0 : 0 < a ab0 : 0 < a + b ⊢ (a + b).log = a.log + (1 + b / a).log TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log_add
[390, 1]
[393, 84]
rw [←Real.log_mul a0.ne' d0.ne', left_distrib, mul_one, mul_div_cancel₀ _ a0.ne']
a b : ℝ a0 : 0 < a ab0 : 0 < a + b d0 : 0 < 1 + b / a ⊢ (a + b).log = a.log + (1 + b / a).log
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b : ℝ a0 : 0 < a ab0 : 0 < a + b d0 : 0 < 1 + b / a ⊢ (a + b).log = a.log + (1 + b / a).log TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log_add
[390, 1]
[393, 84]
field_simp [a0.ne']
a b : ℝ a0 : 0 < a ab0 : 0 < a + b ⊢ 0 < 1 + b / a
a b : ℝ a0 : 0 < a ab0 : 0 < a + b ⊢ 0 < a
Please generate a tactic in lean4 to solve the state. STATE: a b : ℝ a0 : 0 < a ab0 : 0 < a + b ⊢ 0 < 1 + b / a TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log_add
[390, 1]
[393, 84]
bound
a b : ℝ a0 : 0 < a ab0 : 0 < a + b ⊢ 0 < a
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b : ℝ a0 : 0 < a ab0 : 0 < a + b ⊢ 0 < a TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log_abs_add
[396, 1]
[401, 96]
have d0 : 1 + b/a ≠ 0 := by field_simp [a0, ab0]
a b : ℂ a0 : a ≠ 0 ab0 : a + b ≠ 0 ⊢ (Complex.abs (a + b)).log = (Complex.abs a).log + (Complex.abs (1 + b / a)).log
a b : ℂ a0 : a ≠ 0 ab0 : a + b ≠ 0 d0 : 1 + b / a ≠ 0 ⊢ (Complex.abs (a + b)).log = (Complex.abs a).log + (Complex.abs (1 + b / a)).log
Please generate a tactic in lean4 to solve the state. STATE: a b : ℂ a0 : a ≠ 0 ab0 : a + b ≠ 0 ⊢ (Complex.abs (a + b)).log = (Complex.abs a).log + (Complex.abs (1 + b / a)).log TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log_abs_add
[396, 1]
[401, 96]
have a0' : abs a ≠ 0 := Complex.abs.ne_zero a0
a b : ℂ a0 : a ≠ 0 ab0 : a + b ≠ 0 d0 : 1 + b / a ≠ 0 ⊢ (Complex.abs (a + b)).log = (Complex.abs a).log + (Complex.abs (1 + b / a)).log
a b : ℂ a0 : a ≠ 0 ab0 : a + b ≠ 0 d0 : 1 + b / a ≠ 0 a0' : Complex.abs a ≠ 0 ⊢ (Complex.abs (a + b)).log = (Complex.abs a).log + (Complex.abs (1 + b / a)).log
Please generate a tactic in lean4 to solve the state. STATE: a b : ℂ a0 : a ≠ 0 ab0 : a + b ≠ 0 d0 : 1 + b / a ≠ 0 ⊢ (Complex.abs (a + b)).log = (Complex.abs a).log + (Complex.abs (1 + b / a)).log TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log_abs_add
[396, 1]
[401, 96]
have d0' : abs (1 + b / a) ≠ 0 := Complex.abs.ne_zero d0
a b : ℂ a0 : a ≠ 0 ab0 : a + b ≠ 0 d0 : 1 + b / a ≠ 0 a0' : Complex.abs a ≠ 0 ⊢ (Complex.abs (a + b)).log = (Complex.abs a).log + (Complex.abs (1 + b / a)).log
a b : ℂ a0 : a ≠ 0 ab0 : a + b ≠ 0 d0 : 1 + b / a ≠ 0 a0' : Complex.abs a ≠ 0 d0' : Complex.abs (1 + b / a) ≠ 0 ⊢ (Complex.abs (a + b)).log = (Complex.abs a).log + (Complex.abs (1 + b / a)).log
Please generate a tactic in lean4 to solve the state. STATE: a b : ℂ a0 : a ≠ 0 ab0 : a + b ≠ 0 d0 : 1 + b / a ≠ 0 a0' : Complex.abs a ≠ 0 ⊢ (Complex.abs (a + b)).log = (Complex.abs a).log + (Complex.abs (1 + b / a)).log TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log_abs_add
[396, 1]
[401, 96]
rw [←Real.log_mul a0' d0', ←Complex.abs.map_mul, left_distrib, mul_one, mul_div_cancel₀ _ a0]
a b : ℂ a0 : a ≠ 0 ab0 : a + b ≠ 0 d0 : 1 + b / a ≠ 0 a0' : Complex.abs a ≠ 0 d0' : Complex.abs (1 + b / a) ≠ 0 ⊢ (Complex.abs (a + b)).log = (Complex.abs a).log + (Complex.abs (1 + b / a)).log
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b : ℂ a0 : a ≠ 0 ab0 : a + b ≠ 0 d0 : 1 + b / a ≠ 0 a0' : Complex.abs a ≠ 0 d0' : Complex.abs (1 + b / a) ≠ 0 ⊢ (Complex.abs (a + b)).log = (Complex.abs a).log + (Complex.abs (1 + b / a)).log TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
log_abs_add
[396, 1]
[401, 96]
field_simp [a0, ab0]
a b : ℂ a0 : a ≠ 0 ab0 : a + b ≠ 0 ⊢ 1 + b / a ≠ 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b : ℂ a0 : a ≠ 0 ab0 : a + b ≠ 0 ⊢ 1 + b / a ≠ 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Real.exp_forth_lt_four_thirds
[404, 1]
[407, 54]
rw [←Real.exp_one_rpow, one_div, ←@Real.pow_rpow_inv_natCast (4/3) 4 (by norm_num) (by norm_num)]
⊢ (1 / 4).exp < 4 / 3
⊢ rexp 1 ^ 4⁻¹ < ((4 / 3) ^ 4) ^ (↑4)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: ⊢ (1 / 4).exp < 4 / 3 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Real.exp_forth_lt_four_thirds
[404, 1]
[407, 54]
refine Real.rpow_lt_rpow (Real.exp_pos _).le ?_ (by norm_num)
⊢ rexp 1 ^ 4⁻¹ < ((4 / 3) ^ 4) ^ (↑4)⁻¹
⊢ rexp 1 < (4 / 3) ^ 4
Please generate a tactic in lean4 to solve the state. STATE: ⊢ rexp 1 ^ 4⁻¹ < ((4 / 3) ^ 4) ^ (↑4)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Real.exp_forth_lt_four_thirds
[404, 1]
[407, 54]
exact _root_.trans Real.exp_one_lt_d9 (by norm_num)
⊢ rexp 1 < (4 / 3) ^ 4
no goals
Please generate a tactic in lean4 to solve the state. STATE: ⊢ rexp 1 < (4 / 3) ^ 4 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Real.exp_forth_lt_four_thirds
[404, 1]
[407, 54]
norm_num
⊢ 0 ≤ 4 / 3
no goals
Please generate a tactic in lean4 to solve the state. STATE: ⊢ 0 ≤ 4 / 3 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Real.exp_forth_lt_four_thirds
[404, 1]
[407, 54]
norm_num
⊢ 4 ≠ 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: ⊢ 4 ≠ 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Real.exp_forth_lt_four_thirds
[404, 1]
[407, 54]
norm_num
⊢ 0 < 4⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: ⊢ 0 < 4⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
Real.exp_forth_lt_four_thirds
[404, 1]
[407, 54]
norm_num
⊢ 2.7182818286 < (4 / 3) ^ 4
no goals
Please generate a tactic in lean4 to solve the state. STATE: ⊢ 2.7182818286 < (4 / 3) ^ 4 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
dist_prod_one_le_abs_sum
[410, 1]
[426, 69]
set g := fun n ↦ Complex.log (f n)
f : ℕ → ℂ s : Finset ℕ c : ℝ le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c c1 : c ≤ 1 / 2 ⊢ Complex.abs (s.prod f - 1) ≤ 4 * c
f : ℕ → ℂ s : Finset ℕ c : ℝ le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c c1 : c ≤ 1 / 2 g : ℕ → ℂ := fun n => (f n).log ⊢ Complex.abs (s.prod f - 1) ≤ 4 * c
Please generate a tactic in lean4 to solve the state. STATE: f : ℕ → ℂ s : Finset ℕ c : ℝ le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c c1 : c ≤ 1 / 2 ⊢ Complex.abs (s.prod f - 1) ≤ 4 * c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
dist_prod_one_le_abs_sum
[410, 1]
[426, 69]
have b : ∀ n, n ∈ s → abs (f n - 1) ≤ c := by intro n m; refine _root_.trans ?_ le exact Finset.single_le_sum (f := fun n ↦ abs (f n - 1)) (fun _ _ ↦ Complex.abs.nonneg _) m
f : ℕ → ℂ s : Finset ℕ c : ℝ le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c c1 : c ≤ 1 / 2 g : ℕ → ℂ := fun n => (f n).log ⊢ Complex.abs (s.prod f - 1) ≤ 4 * c
f : ℕ → ℂ s : Finset ℕ c : ℝ le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c c1 : c ≤ 1 / 2 g : ℕ → ℂ := fun n => (f n).log b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c ⊢ Complex.abs (s.prod f - 1) ≤ 4 * c
Please generate a tactic in lean4 to solve the state. STATE: f : ℕ → ℂ s : Finset ℕ c : ℝ le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c c1 : c ≤ 1 / 2 g : ℕ → ℂ := fun n => (f n).log ⊢ Complex.abs (s.prod f - 1) ≤ 4 * c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
dist_prod_one_le_abs_sum
[410, 1]
[426, 69]
have f0 : ∀ n, n ∈ s → f n ≠ 0 := by intro n m; specialize b n m; contrapose b; simp only [not_not] at b simp only [b, not_le]; norm_num; linarith
f : ℕ → ℂ s : Finset ℕ c : ℝ le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c c1 : c ≤ 1 / 2 g : ℕ → ℂ := fun n => (f n).log b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c ⊢ Complex.abs (s.prod f - 1) ≤ 4 * c
f : ℕ → ℂ s : Finset ℕ c : ℝ le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c c1 : c ≤ 1 / 2 g : ℕ → ℂ := fun n => (f n).log b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c f0 : ∀ n ∈ s, f n ≠ 0 ⊢ Complex.abs (s.prod f - 1) ≤ 4 * c
Please generate a tactic in lean4 to solve the state. STATE: f : ℕ → ℂ s : Finset ℕ c : ℝ le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c c1 : c ≤ 1 / 2 g : ℕ → ℂ := fun n => (f n).log b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c ⊢ Complex.abs (s.prod f - 1) ≤ 4 * c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
dist_prod_one_le_abs_sum
[410, 1]
[426, 69]
have sg : abs (s.sum g) ≤ 2 * c := by refine _root_.trans (Complex.abs.sum_le _ _) ?_ refine _root_.trans (Finset.sum_le_sum (fun n m ↦ log_small (_root_.trans (b n m) c1))) ?_ rw [← Finset.mul_sum]; bound
f : ℕ → ℂ s : Finset ℕ c : ℝ le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c c1 : c ≤ 1 / 2 g : ℕ → ℂ := fun n => (f n).log b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c f0 : ∀ n ∈ s, f n ≠ 0 ⊢ Complex.abs (s.prod f - 1) ≤ 4 * c
f : ℕ → ℂ s : Finset ℕ c : ℝ le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c c1 : c ≤ 1 / 2 g : ℕ → ℂ := fun n => (f n).log b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c f0 : ∀ n ∈ s, f n ≠ 0 sg : Complex.abs (s.sum g) ≤ 2 * c ⊢ Complex.abs (s.prod f - 1) ≤ 4 * c
Please generate a tactic in lean4 to solve the state. STATE: f : ℕ → ℂ s : Finset ℕ c : ℝ le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c c1 : c ≤ 1 / 2 g : ℕ → ℂ := fun n => (f n).log b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c f0 : ∀ n ∈ s, f n ≠ 0 ⊢ Complex.abs (s.prod f - 1) ≤ 4 * c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
dist_prod_one_le_abs_sum
[410, 1]
[426, 69]
have e : s.prod f = Complex.exp (s.sum g) := by rw [Complex.exp_sum]; apply Finset.prod_congr rfl intro n m; rw [Complex.exp_log (f0 n m)]
f : ℕ → ℂ s : Finset ℕ c : ℝ le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c c1 : c ≤ 1 / 2 g : ℕ → ℂ := fun n => (f n).log b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c f0 : ∀ n ∈ s, f n ≠ 0 sg : Complex.abs (s.sum g) ≤ 2 * c ⊢ Complex.abs (s.prod f - 1) ≤ 4 * c
f : ℕ → ℂ s : Finset ℕ c : ℝ le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c c1 : c ≤ 1 / 2 g : ℕ → ℂ := fun n => (f n).log b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c f0 : ∀ n ∈ s, f n ≠ 0 sg : Complex.abs (s.sum g) ≤ 2 * c e : s.prod f = (s.sum g).exp ⊢ Complex.abs (s.prod f - 1) ≤ 4 * c
Please generate a tactic in lean4 to solve the state. STATE: f : ℕ → ℂ s : Finset ℕ c : ℝ le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c c1 : c ≤ 1 / 2 g : ℕ → ℂ := fun n => (f n).log b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c f0 : ∀ n ∈ s, f n ≠ 0 sg : Complex.abs (s.sum g) ≤ 2 * c ⊢ Complex.abs (s.prod f - 1) ≤ 4 * c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/Bounds.lean
dist_prod_one_le_abs_sum
[410, 1]
[426, 69]
rw [e]
f : ℕ → ℂ s : Finset ℕ c : ℝ le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c c1 : c ≤ 1 / 2 g : ℕ → ℂ := fun n => (f n).log b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c f0 : ∀ n ∈ s, f n ≠ 0 sg : Complex.abs (s.sum g) ≤ 2 * c e : s.prod f = (s.sum g).exp ⊢ Complex.abs (s.prod f - 1) ≤ 4 * c
f : ℕ → ℂ s : Finset ℕ c : ℝ le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c c1 : c ≤ 1 / 2 g : ℕ → ℂ := fun n => (f n).log b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c f0 : ∀ n ∈ s, f n ≠ 0 sg : Complex.abs (s.sum g) ≤ 2 * c e : s.prod f = (s.sum g).exp ⊢ Complex.abs ((s.sum g).exp - 1) ≤ 4 * c
Please generate a tactic in lean4 to solve the state. STATE: f : ℕ → ℂ s : Finset ℕ c : ℝ le : (s.sum fun n => Complex.abs (f n - 1)) ≤ c c1 : c ≤ 1 / 2 g : ℕ → ℂ := fun n => (f n).log b : ∀ n ∈ s, Complex.abs (f n - 1) ≤ c f0 : ∀ n ∈ s, f n ≠ 0 sg : Complex.abs (s.sum g) ≤ 2 * c e : s.prod f = (s.sum g).exp ⊢ Complex.abs (s.prod f - 1) ≤ 4 * c TACTIC: