url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/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:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | exact _root_.trans (exp_small (by linarith)) (by 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
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 | no goals | 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.sum g).exp - 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] | intro 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
⊢ ∀ n ∈ s, Complex.abs (f n - 1) ≤ 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
n : ℕ
m : n ∈ s
⊢ Complex.abs (f n - 1) ≤ 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
⊢ ∀ n ∈ s, Complex.abs (f n - 1) ≤ c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | refine _root_.trans ?_ le | 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
n : ℕ
m : n ∈ s
⊢ Complex.abs (f n - 1) ≤ 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
n : ℕ
m : n ∈ s
⊢ Complex.abs (f n - 1) ≤ s.sum fun n => Complex.abs (f n - 1) | 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
n : ℕ
m : n ∈ s
⊢ Complex.abs (f n - 1) ≤ c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | 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
n : ℕ
m : n ∈ s
⊢ Complex.abs (f n - 1) ≤ s.sum fun n => Complex.abs (f n - 1) | no goals | 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
n : ℕ
m : n ∈ s
⊢ Complex.abs (f n - 1) ≤ s.sum fun n => Complex.abs (f n - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | intro 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
⊢ ∀ n ∈ s, f n ≠ 0 | 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
n : ℕ
m : n ∈ s
⊢ f n ≠ 0 | 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
⊢ ∀ n ∈ s, f n ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | specialize b 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
n : ℕ
m : n ∈ s
⊢ f n ≠ 0 | 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
n : ℕ
m : n ∈ s
b : Complex.abs (f n - 1) ≤ c
⊢ f n ≠ 0 | 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
n : ℕ
m : n ∈ s
⊢ f n ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | contrapose b | 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
n : ℕ
m : n ∈ s
b : Complex.abs (f n - 1) ≤ c
⊢ f n ≠ 0 | 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
n : ℕ
m : n ∈ s
b : ¬f n ≠ 0
⊢ ¬Complex.abs (f n - 1) ≤ 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
n : ℕ
m : n ∈ s
b : Complex.abs (f n - 1) ≤ c
⊢ f n ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | simp only [not_not] at b | 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
n : ℕ
m : n ∈ s
b : ¬f n ≠ 0
⊢ ¬Complex.abs (f n - 1) ≤ 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
n : ℕ
m : n ∈ s
b : f n = 0
⊢ ¬Complex.abs (f n - 1) ≤ 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
n : ℕ
m : n ∈ s
b : ¬f n ≠ 0
⊢ ¬Complex.abs (f n - 1) ≤ c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | simp only [b, not_le] | 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
n : ℕ
m : n ∈ s
b : f n = 0
⊢ ¬Complex.abs (f n - 1) ≤ 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
n : ℕ
m : n ∈ s
b : f n = 0
⊢ c < Complex.abs (0 - 1) | 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
n : ℕ
m : n ∈ s
b : f n = 0
⊢ ¬Complex.abs (f n - 1) ≤ c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | norm_num | 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
n : ℕ
m : n ∈ s
b : f n = 0
⊢ c < Complex.abs (0 - 1) | 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
n : ℕ
m : n ∈ s
b : f n = 0
⊢ c < 1 | 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
n : ℕ
m : n ∈ s
b : f n = 0
⊢ c < Complex.abs (0 - 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | 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
n : ℕ
m : n ∈ s
b : f n = 0
⊢ c < 1 | no goals | 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
n : ℕ
m : n ∈ s
b : f n = 0
⊢ c < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | refine _root_.trans (Complex.abs.sum_le _ _) ?_ | 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.sum g) ≤ 2 * 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
⊢ (s.sum fun i => Complex.abs (f i).log) ≤ 2 * 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.sum g) ≤ 2 * c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | refine _root_.trans (Finset.sum_le_sum (fun n m ↦ log_small (_root_.trans (b n m) c1))) ?_ | 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
⊢ (s.sum fun i => Complex.abs (f i).log) ≤ 2 * 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
⊢ (s.sum fun i => 2 * Complex.abs (f i - 1)) ≤ 2 * 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
⊢ (s.sum fun i => Complex.abs (f i).log) ≤ 2 * c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | rw [← Finset.mul_sum] | 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
⊢ (s.sum fun i => 2 * Complex.abs (f i - 1)) ≤ 2 * 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
⊢ (2 * s.sum fun i => Complex.abs (f i - 1)) ≤ 2 * 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
⊢ (s.sum fun i => 2 * Complex.abs (f i - 1)) ≤ 2 * c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | 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
⊢ (2 * s.sum fun i => Complex.abs (f i - 1)) ≤ 2 * c | no goals | 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
⊢ (2 * s.sum fun i => Complex.abs (f i - 1)) ≤ 2 * c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | rw [Complex.exp_sum] | 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
⊢ s.prod f = (s.sum g).exp | 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
⊢ s.prod f = s.prod fun x => (f x).log.exp | 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
⊢ s.prod f = (s.sum g).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | apply Finset.prod_congr rfl | 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
⊢ s.prod f = s.prod fun x => (f x).log.exp | 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
⊢ ∀ x ∈ s, f x = (f x).log.exp | 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
⊢ s.prod f = s.prod fun x => (f x).log.exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | intro 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
⊢ ∀ x ∈ s, f x = (f x).log.exp | 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
n : ℕ
m : n ∈ s
⊢ f n = (f n).log.exp | 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
⊢ ∀ x ∈ s, f x = (f x).log.exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | 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
n : ℕ
m : n ∈ s
⊢ f n = (f n).log.exp | no goals | 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
n : ℕ
m : n ∈ s
⊢ f n = (f n).log.exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | 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
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) ≤ 1 | no goals | 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.sum g) ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | dist_prod_one_le_abs_sum | [410, 1] | [426, 69] | 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
f0 : ∀ n ∈ s, f n ≠ 0
sg : Complex.abs (s.sum g) ≤ 2 * c
e : s.prod f = (s.sum g).exp
⊢ 2 * Complex.abs (s.sum g) ≤ 4 * c | no goals | 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
⊢ 2 * Complex.abs (s.sum g) ≤ 4 * c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | intro x | A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
⊢ ∀ (x : M), IsClosed {x} | A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ IsClosed {x} | Please generate a tactic in lean4 to solve the state.
STATE:
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
⊢ ∀ (x : M), IsClosed {x}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | rw [←compl_compl ({x} : Set M)] | A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ IsClosed {x} | A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ IsClosed {x}ᶜᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ IsClosed {x}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | apply IsOpen.isClosed_compl | A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ IsClosed {x}ᶜᶜ | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ IsOpen {x}ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ IsClosed {x}ᶜᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | rw [isOpen_iff_mem_nhds] | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ IsOpen {x}ᶜ | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ ∀ x_1 ∈ {x}ᶜ, {x}ᶜ ∈ 𝓝 x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ IsOpen {x}ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | intro y m | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ ∀ x_1 ∈ {x}ᶜ, {x}ᶜ ∈ 𝓝 x_1 | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ∈ {x}ᶜ
⊢ {x}ᶜ ∈ 𝓝 y | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x : M
⊢ ∀ x_1 ∈ {x}ᶜ, {x}ᶜ ∈ 𝓝 x_1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | simp only [mem_compl_singleton_iff] at m | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ∈ {x}ᶜ
⊢ {x}ᶜ ∈ 𝓝 y | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
⊢ {x}ᶜ ∈ 𝓝 y | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ∈ {x}ᶜ
⊢ {x}ᶜ ∈ 𝓝 y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | simp only [mem_nhds_iff, subset_compl_singleton_iff] | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
⊢ {x}ᶜ ∈ 𝓝 y | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
⊢ {x}ᶜ ∈ 𝓝 y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | by_cases xm : x ∈ (chartAt A y).source | case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t | case pos
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t
case neg
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∉ (_root_.chartAt A y).source
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case a
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | set t := (chartAt A y).source \ {x} | case pos
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t | case pos
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | have e : t = (chartAt A y).source ∩ chartAt A y ⁻¹'
((chartAt A y).target \ {chartAt A y x}) := by
apply Set.ext; intro z
simp only [mem_diff, mem_singleton_iff, mem_inter_iff, mem_preimage]; constructor
intro ⟨zm, zx⟩; use zm, PartialEquiv.map_source _ zm, (PartialEquiv.injOn _).ne zm xm zx
intro ⟨zm, _, zx⟩; use zm, ((PartialEquiv.injOn _).ne_iff zm xm).mp zx | case pos
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t | case pos
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
e :
t = (_root_.chartAt A y).source ∩ ↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x})
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | use t | case pos
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
e :
t = (_root_.chartAt A y).source ∩ ↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x})
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t | case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
e :
t = (_root_.chartAt A y).source ∩ ↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x})
⊢ x ∉ t ∧ IsOpen t ∧ y ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
e :
t = (_root_.chartAt A y).source ∩ ↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x})
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | apply Set.ext | A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
⊢ t = (_root_.chartAt A y).source ∩ ↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x}) | case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
⊢ ∀ (x_1 : M),
x_1 ∈ t ↔
x_1 ∈
(_root_.chartAt A y).source ∩
↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x}) | Please generate a tactic in lean4 to solve the state.
STATE:
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
⊢ t = (_root_.chartAt A y).source ∩ ↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | intro z | case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
⊢ ∀ (x_1 : M),
x_1 ∈ t ↔
x_1 ∈
(_root_.chartAt A y).source ∩
↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x}) | case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ t ↔
z ∈
(_root_.chartAt A y).source ∩ ↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x}) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
⊢ ∀ (x_1 : M),
x_1 ∈ t ↔
x_1 ∈
(_root_.chartAt A y).source ∩
↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | simp only [mem_diff, mem_singleton_iff, mem_inter_iff, mem_preimage] | case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ t ↔
z ∈
(_root_.chartAt A y).source ∩ ↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x}) | case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ t ↔
z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ t ↔
z ∈
(_root_.chartAt A y).source ∩ ↑(_root_.chartAt A y) ⁻¹' ((_root_.chartAt A y).target \ {↑(_root_.chartAt A y) x})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | constructor | case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ t ↔
z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x | case h.mp
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ t →
z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x →
z ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ t ↔
z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | intro ⟨zm, zx⟩ | case h.mp
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ t →
z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x →
z ∈ t | case h.mp
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
zm : z ∈ (_root_.chartAt A y).source
zx : z ∉ {x}
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x →
z ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ t →
z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x →
z ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | use zm, PartialEquiv.map_source _ zm, (PartialEquiv.injOn _).ne zm xm zx | case h.mp
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
zm : z ∈ (_root_.chartAt A y).source
zx : z ∉ {x}
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x →
z ∈ t | case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x →
z ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
zm : z ∈ (_root_.chartAt A y).source
zx : z ∉ {x}
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x →
z ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | intro ⟨zm, _, zx⟩ | case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x →
z ∈ t | case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
zm : z ∈ (_root_.chartAt A y).source
left✝ : ↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target
zx : ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
⊢ z ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
⊢ z ∈ (_root_.chartAt A y).source ∧
↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target ∧ ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x →
z ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | use zm, ((PartialEquiv.injOn _).ne_iff zm xm).mp zx | case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
zm : z ∈ (_root_.chartAt A y).source
left✝ : ↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target
zx : ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
⊢ z ∈ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∈ (_root_.chartAt A y).source
t : Set M := (_root_.chartAt A y).source \ {x}
z : M
zm : z ∈ (_root_.chartAt A y).source
left✝ : ↑(_root_.chartAt A y) z ∈ (_root_.chartAt A y).target
zx : ¬↑(_root_.chartAt A y) z = ↑(_root_.chartAt A y) x
⊢ z ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.t1Space | [16, 1] | [39, 83] | use(chartAt A y).source, xm, (chartAt A y).open_source, mem_chart_source A y | case neg
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∉ (_root_.chartAt A y).source
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
A M : Type
inst✝³ : TopologicalSpace A
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace A M
inst✝ : T1Space A
x y : M
m : y ≠ x
xm : x ∉ (_root_.chartAt A y).source
⊢ ∃ t, x ∉ t ∧ IsOpen t ∧ y ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | apply RegularSpace.ofExistsMemNhdsIsClosedSubset | A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
⊢ RegularSpace M | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
⊢ ∀ (x : M), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
⊢ RegularSpace M
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | intro x s n | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
⊢ ∀ (x : M), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
⊢ ∀ (x : M), ∀ s ∈ 𝓝 x, ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | set t := (chartAt A x).target ∩ (chartAt A x).symm ⁻¹' ((chartAt A x).source ∩ s) | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | have cn : (chartAt A x).source ∈ 𝓝 x :=
(chartAt A x).open_source.mem_nhds (mem_chart_source A x) | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | have tn : t ∈ 𝓝 (chartAt A x x) := by
apply Filter.inter_mem ((chartAt A x).open_target.mem_nhds (mem_chart_target A x))
apply ((chartAt A x).continuousAt_symm (mem_chart_target _ _)).preimage_mem_nhds
rw [(chartAt A x).left_inv (mem_chart_source _ _)]; exact Filter.inter_mem cn n | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
tn : t ∈ 𝓝 (↑(_root_.chartAt A x) x)
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | rcases local_compact_nhds tn with ⟨u, un, ut, uc⟩ | case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
tn : t ∈ 𝓝 (↑(_root_.chartAt A x) x)
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | case h.intro.intro.intro
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
tn : t ∈ 𝓝 (↑(_root_.chartAt A x) x)
u : Set A
un : u ∈ 𝓝 (↑(_root_.chartAt A x) x)
ut : u ⊆ t
uc : IsCompact u
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
tn : t ∈ 𝓝 (↑(_root_.chartAt A x) x)
⊢ ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | apply Filter.inter_mem ((chartAt A x).open_target.mem_nhds (mem_chart_target A x)) | A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ t ∈ 𝓝 (↑(_root_.chartAt A x) x) | A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s) ∈ 𝓝 (↑(_root_.chartAt A x) x) | Please generate a tactic in lean4 to solve the state.
STATE:
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ t ∈ 𝓝 (↑(_root_.chartAt A x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | apply ((chartAt A x).continuousAt_symm (mem_chart_target _ _)).preimage_mem_nhds | A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s) ∈ 𝓝 (↑(_root_.chartAt A x) x) | A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ (_root_.chartAt A x).source ∩ s ∈ 𝓝 (↑(_root_.chartAt A x).symm (↑(_root_.chartAt A x) x)) | Please generate a tactic in lean4 to solve the state.
STATE:
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s) ∈ 𝓝 (↑(_root_.chartAt A x) x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | rw [(chartAt A x).left_inv (mem_chart_source _ _)] | A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ (_root_.chartAt A x).source ∩ s ∈ 𝓝 (↑(_root_.chartAt A x).symm (↑(_root_.chartAt A x) x)) | A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ (_root_.chartAt A x).source ∩ s ∈ 𝓝 x | Please generate a tactic in lean4 to solve the state.
STATE:
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ (_root_.chartAt A x).source ∩ s ∈ 𝓝 (↑(_root_.chartAt A x).symm (↑(_root_.chartAt A x) x))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/ChartedSpace.lean | ChartedSpace.regularSpace | [43, 1] | [62, 74] | exact Filter.inter_mem cn n | A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ (_root_.chartAt A x).source ∩ s ∈ 𝓝 x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A M : Type
inst✝⁴ : TopologicalSpace A
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace A M
inst✝¹ : T2Space M
inst✝ : LocallyCompactSpace A
x : M
s : Set M
n : s ∈ 𝓝 x
t : Set A := (_root_.chartAt A x).target ∩ ↑(_root_.chartAt A x).symm ⁻¹' ((_root_.chartAt A x).source ∩ s)
cn : (_root_.chartAt A x).source ∈ 𝓝 x
⊢ (_root_.chartAt A x).source ∩ s ∈ 𝓝 x
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.