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stringlengths 7
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stringlengths 1
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stringlengths 6
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https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get!_parallel_colors' | [162, 1] | [183, 14] | rw [Nat.cast_sub (by omega), Nat.cast_mul, four, mul_zero, sub_zero, ke] | case neg
f : ℕ → Color UInt8
chunk : ℕ
four : ↑4 = 0
n : ℕ
i : ∀ m < n, ∀ (o k : ℕ), k < m * 4 → (parallel_colors' f m o chunk).get! k = (f (o + k / 4))[↑k]
o k : ℕ
lt : k < n * 4
h : 1 < n ∧ chunk < n
c : ¬k < n / 2 * 4
ke : o + n / 2 + (k - n / 2 * 4) / 4 = o + k / 4
⊢ (f (o + n / 2 + (k - n / 2 * 4) / 4))[↑(k - n / 2 * 4)] = (f (o + k / 4))[↑k] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
f : ℕ → Color UInt8
chunk : ℕ
four : ↑4 = 0
n : ℕ
i : ∀ m < n, ∀ (o k : ℕ), k < m * 4 → (parallel_colors' f m o chunk).get! k = (f (o + k / 4))[↑k]
o k : ℕ
lt : k < n * 4
h : 1 < n ∧ chunk < n
c : ¬k < n / 2 * 4
ke : o + n / 2 + (k - n / 2 * 4) / 4 = o + k / 4
⊢ (f (o + n / 2 + (k - n / 2 * 4) / 4))[↑(k - n / 2 * 4)] = (f (o + k / 4))[↑k]
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get!_parallel_colors' | [162, 1] | [183, 14] | omega | f : ℕ → Color UInt8
chunk : ℕ
four : ↑4 = 0
n : ℕ
i : ∀ m < n, ∀ (o k : ℕ), k < m * 4 → (parallel_colors' f m o chunk).get! k = (f (o + k / 4))[↑k]
o k : ℕ
lt : k < n * 4
h : 1 < n ∧ chunk < n
c : ¬k < n / 2 * 4
⊢ o + n / 2 + (k - n / 2 * 4) / 4 = o + k / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → Color UInt8
chunk : ℕ
four : ↑4 = 0
n : ℕ
i : ∀ m < n, ∀ (o k : ℕ), k < m * 4 → (parallel_colors' f m o chunk).get! k = (f (o + k / 4))[↑k]
o k : ℕ
lt : k < n * 4
h : 1 < n ∧ chunk < n
c : ¬k < n / 2 * 4
⊢ o + n / 2 + (k - n / 2 * 4) / 4 = o + k / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get!_parallel_colors' | [162, 1] | [183, 14] | omega | f : ℕ → Color UInt8
chunk : ℕ
four : ↑4 = 0
n : ℕ
i : ∀ m < n, ∀ (o k : ℕ), k < m * 4 → (parallel_colors' f m o chunk).get! k = (f (o + k / 4))[↑k]
o k : ℕ
lt : k < n * 4
h : 1 < n ∧ chunk < n
c : ¬k < n / 2 * 4
ke : o + n / 2 + (k - n / 2 * 4) / 4 = o + k / 4
⊢ n / 2 * 4 ≤ k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → Color UInt8
chunk : ℕ
four : ↑4 = 0
n : ℕ
i : ∀ m < n, ∀ (o k : ℕ), k < m * 4 → (parallel_colors' f m o chunk).get! k = (f (o + k / 4))[↑k]
o k : ℕ
lt : k < n * 4
h : 1 < n ∧ chunk < n
c : ¬k < n / 2 * 4
ke : o + n / 2 + (k - n / 2 * 4) / 4 = o + k / 4
⊢ n / 2 * 4 ≤ k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get!_parallel_colors' | [162, 1] | [183, 14] | omega | case neg.a
f : ℕ → Color UInt8
chunk : ℕ
four : ↑4 = 0
n : ℕ
i : ∀ m < n, ∀ (o k : ℕ), k < m * 4 → (parallel_colors' f m o chunk).get! k = (f (o + k / 4))[↑k]
o k : ℕ
lt : k < n * 4
h : 1 < n ∧ chunk < n
c : ¬k < n / 2 * 4
⊢ n - n / 2 < n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
f : ℕ → Color UInt8
chunk : ℕ
four : ↑4 = 0
n : ℕ
i : ∀ m < n, ∀ (o k : ℕ), k < m * 4 → (parallel_colors' f m o chunk).get! k = (f (o + k / 4))[↑k]
o k : ℕ
lt : k < n * 4
h : 1 < n ∧ chunk < n
c : ¬k < n / 2 * 4
⊢ n - n / 2 < n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get!_parallel_colors' | [162, 1] | [183, 14] | omega | case neg.lt
f : ℕ → Color UInt8
chunk : ℕ
four : ↑4 = 0
n : ℕ
i : ∀ m < n, ∀ (o k : ℕ), k < m * 4 → (parallel_colors' f m o chunk).get! k = (f (o + k / 4))[↑k]
o k : ℕ
lt : k < n * 4
h : 1 < n ∧ chunk < n
c : ¬k < n / 2 * 4
⊢ k - n / 2 * 4 < (n - n / 2) * 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.lt
f : ℕ → Color UInt8
chunk : ℕ
four : ↑4 = 0
n : ℕ
i : ∀ m < n, ∀ (o k : ℕ), k < m * 4 → (parallel_colors' f m o chunk).get! k = (f (o + k / 4))[↑k]
o k : ℕ
lt : k < n * 4
h : 1 < n ∧ chunk < n
c : ¬k < n / 2 * 4
⊢ k - n / 2 * 4 < (n - n / 2) * 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.size_parallel_colors | [185, 1] | [188, 53] | simp only [parallel_colors, size_parallel_colors'] | f : ℕ → Color UInt8
n chunk : ℕ
⊢ (parallel_colors f n chunk).size = n * 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → Color UInt8
n chunk : ℕ
⊢ (parallel_colors f n chunk).size = n * 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get!_parallel_colors | [190, 1] | [193, 80] | simp only [parallel_colors, zero_add, get!_parallel_colors' f n 0 chunk k lt] | f : ℕ → Color UInt8
n chunk k : ℕ
lt : k < n * 4
⊢ (parallel_colors f n chunk).get! k = (f (k / 4))[↑k] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → Color UInt8
n chunk k : ℕ
lt : k < n * 4
⊢ (parallel_colors f n chunk).get! k = (f (k / 4))[↑k]
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.width_ofFn | [204, 1] | [205, 49] | rw [ofFn] | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
⊢ (ofFn w h chunk f).width = w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
⊢ (ofFn w h chunk f).width = w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.height_ofFn | [206, 1] | [207, 50] | rw [ofFn] | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
⊢ (ofFn w h chunk f).height = h | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
⊢ (ofFn w h chunk f).height = h
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | rw [get] | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
⊢ (ofFn w h chunk f).get x y = f ↑x ↑y | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
⊢ (let b := base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y;
let_fun lt := ⋯;
{ r := (ofFn w h chunk f).data[b], g := (ofFn w h chunk f).data[b + 1], b := (ofFn w h chunk f).data[b + 2],
a := (ofFn w h chunk f).data[b + 3] }) =
f ↑x ↑y | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
⊢ (ofFn w h chunk f).get x y = f ↑x ↑y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | simp only [ByteArray.getElemNat_eq_get!] | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
⊢ (let b := base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y;
let_fun lt := ⋯;
{ r := (ofFn w h chunk f).data[b], g := (ofFn w h chunk f).data[b + 1], b := (ofFn w h chunk f).data[b + 2],
a := (ofFn w h chunk f).data[b + 3] }) =
f ↑x ↑y | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
⊢ { r := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y),
g := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 1),
b := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 2),
a := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 3) } =
f ↑x ↑y | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
⊢ (let b := base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y;
let_fun lt := ⋯;
{ r := (ofFn w h chunk f).data[b], g := (ofFn w h chunk f).data[b + 1], b := (ofFn w h chunk f).data[b + 2],
a := (ofFn w h chunk f).data[b + 3] }) =
f ↑x ↑y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | have xw := x.prop | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
⊢ { r := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y),
g := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 1),
b := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 2),
a := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 3) } =
f ↑x ↑y | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < (ofFn w h chunk f).width
⊢ { r := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y),
g := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 1),
b := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 2),
a := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 3) } =
f ↑x ↑y | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
⊢ { r := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y),
g := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 1),
b := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 2),
a := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 3) } =
f ↑x ↑y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | have yh := y.prop | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < (ofFn w h chunk f).width
⊢ { r := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y),
g := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 1),
b := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 2),
a := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 3) } =
f ↑x ↑y | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < (ofFn w h chunk f).width
yh : ↑y < (ofFn w h chunk f).height
⊢ { r := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y),
g := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 1),
b := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 2),
a := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 3) } =
f ↑x ↑y | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < (ofFn w h chunk f).width
⊢ { r := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y),
g := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 1),
b := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 2),
a := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 3) } =
f ↑x ↑y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | simp only [width_ofFn, height_ofFn, Color.ext_iff] at xw yh ⊢ | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < (ofFn w h chunk f).width
yh : ↑y < (ofFn w h chunk f).height
⊢ { r := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y),
g := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 1),
b := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 2),
a := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 3) } =
f ↑x ↑y | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y) = (f ↑x ↑y).r ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 1) = (f ↑x ↑y).g ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 2) = (f ↑x ↑y).b ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < (ofFn w h chunk f).width
yh : ↑y < (ofFn w h chunk f).height
⊢ { r := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y),
g := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 1),
b := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 2),
a := (ofFn w h chunk f).data.get! (base (ofFn w h chunk f).width (ofFn w h chunk f).height ↑x ↑y + 3) } =
f ↑x ↑y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | have w0 : 0 < w := by omega | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y) = (f ↑x ↑y).r ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 1) = (f ↑x ↑y).g ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 2) = (f ↑x ↑y).b ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y) = (f ↑x ↑y).r ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 1) = (f ↑x ↑y).g ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 2) = (f ↑x ↑y).b ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y) = (f ↑x ↑y).r ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 1) = (f ↑x ↑y).g ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 2) = (f ↑x ↑y).b ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | have yh' : y ≤ h - 1 := by omega | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y) = (f ↑x ↑y).r ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 1) = (f ↑x ↑y).g ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 2) = (f ↑x ↑y).b ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y) = (f ↑x ↑y).r ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 1) = (f ↑x ↑y).g ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 2) = (f ↑x ↑y).b ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y) = (f ↑x ↑y).r ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 1) = (f ↑x ↑y).g ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 2) = (f ↑x ↑y).b ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | have m0 : ∀ x : Fin 4, (x * 4 : Fin 4) = 0 := by
intro x
have e : (4 : Fin 4) = 0 := rfl
simp only [e, mul_zero] | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y) = (f ↑x ↑y).r ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 1) = (f ↑x ↑y).g ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 2) = (f ↑x ↑y).b ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y) = (f ↑x ↑y).r ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 1) = (f ↑x ↑y).g ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 2) = (f ↑x ↑y).b ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y) = (f ↑x ↑y).r ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 1) = (f ↑x ↑y).g ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 2) = (f ↑x ↑y).b ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | have f0 : 0 < 4 := by norm_num | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y) = (f ↑x ↑y).r ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 1) = (f ↑x ↑y).g ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 2) = (f ↑x ↑y).b ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y) = (f ↑x ↑y).r ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 1) = (f ↑x ↑y).g ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 2) = (f ↑x ↑y).b ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y) = (f ↑x ↑y).r ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 1) = (f ↑x ↑y).g ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 2) = (f ↑x ↑y).b ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | refine ⟨?_, ?_, ?_, ?_⟩ | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y) = (f ↑x ↑y).r ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 1) = (f ↑x ↑y).g ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 2) = (f ↑x ↑y).b ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a | case refine_1
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y) = (f ↑x ↑y).r
case refine_2
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y + 1) = (f ↑x ↑y).g
case refine_3
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y + 2) = (f ↑x ↑y).b
case refine_4
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y) = (f ↑x ↑y).r ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 1) = (f ↑x ↑y).g ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 2) = (f ↑x ↑y).b ∧
(ofFn w h chunk f).data.get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | omega | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
⊢ 0 < w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
⊢ 0 < w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | omega | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
⊢ ↑y ≤ h - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
⊢ ↑y ≤ h - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | intro x | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
⊢ ∀ (x : Fin 4), x * 4 = 0 | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x✝ : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x✝ < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
x : Fin 4
⊢ x * 4 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
⊢ ∀ (x : Fin 4), x * 4 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | have e : (4 : Fin 4) = 0 := rfl | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x✝ : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x✝ < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
x : Fin 4
⊢ x * 4 = 0 | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x✝ : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x✝ < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
x : Fin 4
e : 4 = 0
⊢ x * 4 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x✝ : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x✝ < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
x : Fin 4
⊢ x * 4 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | simp only [e, mul_zero] | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x✝ : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x✝ < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
x : Fin 4
e : 4 = 0
⊢ x * 4 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x✝ : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x✝ < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
x : Fin 4
e : 4 = 0
⊢ x * 4 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | norm_num | f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
⊢ 0 < 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
⊢ 0 < 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | nth_rw 1 [ofFn] | case refine_4
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a | case refine_4
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ { data := parallel_colors (fun i => f (i % w) (h - 1 - i / w)) (h * w) chunk, width := w, height := h,
size_eq := ⋯ }.data.get!
(base w h ↑x ↑y + 3) =
(f ↑x ↑y).a | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_4
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ (ofFn w h chunk f).data.get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | simp only | case refine_4
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ { data := parallel_colors (fun i => f (i % w) (h - 1 - i / w)) (h * w) chunk, width := w, height := h,
size_eq := ⋯ }.data.get!
(base w h ↑x ↑y + 3) =
(f ↑x ↑y).a | case refine_4
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ (parallel_colors (fun i => f (i % w) (h - 1 - i / w)) (h * w) chunk).get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_4
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ { data := parallel_colors (fun i => f (i % w) (h - 1 - i / w)) (h * w) chunk, width := w, height := h,
size_eq := ⋯ }.data.get!
(base w h ↑x ↑y + 3) =
(f ↑x ↑y).a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | rw [get!_parallel_colors] | case refine_4
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ (parallel_colors (fun i => f (i % w) (h - 1 - i / w)) (h * w) chunk).get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a | case refine_4
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ (f ((base w h ↑x ↑y + 3) / 4 % w) (h - 1 - (base w h ↑x ↑y + 3) / 4 / w))[↑(base w h ↑x ↑y + 3)] = (f ↑x ↑y).a
case refine_4.lt
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ base w h ↑x ↑y + 3 < h * w * 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_4
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ (parallel_colors (fun i => f (i % w) (h - 1 - i / w)) (h * w) chunk).get! (base w h ↑x ↑y + 3) = (f ↑x ↑y).a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | simp [base, add_comm _ (x : ℕ), Nat.add_mul_div_right _ _ w0, Nat.div_eq_of_lt xw,
Nat.sub_sub_self yh', Nat.mod_eq_of_lt xw, m0, add_comm (_ * 4),
Nat.add_mul_div_right _ _ f0] | case refine_4
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ (f ((base w h ↑x ↑y + 3) / 4 % w) (h - 1 - (base w h ↑x ↑y + 3) / 4 / w))[↑(base w h ↑x ↑y + 3)] = (f ↑x ↑y).a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_4
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ (f ((base w h ↑x ↑y + 3) / 4 % w) (h - 1 - (base w h ↑x ↑y + 3) / 4 / w))[↑(base w h ↑x ↑y + 3)] = (f ↑x ↑y).a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | have le := base_le xw yh | case refine_4.lt
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ base w h ↑x ↑y + 3 < h * w * 4 | case refine_4.lt
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
le : base w h ↑x ↑y + 4 ≤ h * w * 4
⊢ base w h ↑x ↑y + 3 < h * w * 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_4.lt
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
⊢ base w h ↑x ↑y + 3 < h * w * 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Image.lean | Image.get_ofFn | [209, 1] | [233, 14] | omega | case refine_4.lt
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
le : base w h ↑x ↑y + 4 ≤ h * w * 4
⊢ base w h ↑x ↑y + 3 < h * w * 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_4.lt
f : ℕ → ℕ → Color UInt8
w h chunk : ℕ
x : Fin (ofFn w h chunk f).width
y : Fin (ofFn w h chunk f).height
xw : ↑x < w
yh : ↑y < h
w0 : 0 < w
yh' : ↑y ≤ h - 1
m0 : ∀ (x : Fin 4), x * 4 = 0
f0 : 0 < 4
le : base w h ↑x ↑y + 4 ≤ h * w * 4
⊢ base w h ↑x ↑y + 3 < h * w * 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | closure_inter_subset_compl | [25, 1] | [28, 82] | rw [← vo.isClosed_compl.closure_eq] | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
vo : IsOpen v
d : Disjoint u v
⊢ closure (s ∩ u) ⊆ vᶜ | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
vo : IsOpen v
d : Disjoint u v
⊢ closure (s ∩ u) ⊆ closure vᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
vo : IsOpen v
d : Disjoint u v
⊢ closure (s ∩ u) ⊆ vᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | closure_inter_subset_compl | [25, 1] | [28, 82] | apply closure_mono | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
vo : IsOpen v
d : Disjoint u v
⊢ closure (s ∩ u) ⊆ closure vᶜ | case h
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
vo : IsOpen v
d : Disjoint u v
⊢ s ∩ u ⊆ vᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
vo : IsOpen v
d : Disjoint u v
⊢ closure (s ∩ u) ⊆ closure vᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | closure_inter_subset_compl | [25, 1] | [28, 82] | exact _root_.trans (inter_subset_right _ _) (Disjoint.subset_compl_left d.symm) | case h
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
vo : IsOpen v
d : Disjoint u v
⊢ s ∩ u ⊆ vᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
vo : IsOpen v
d : Disjoint u v
⊢ s ∩ u ⊆ vᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | rw [←closure_subset_iff_isClosed, ←diff_eq_empty] | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
⊢ IsClosed (s ∩ u) | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
⊢ closure (s ∩ u) \ (s ∩ u) = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
⊢ IsClosed (s ∩ u)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | by_contra h | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
⊢ closure (s ∩ u) \ (s ∩ u) = ∅ | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
h : ¬closure (s ∩ u) \ (s ∩ u) = ∅
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
⊢ closure (s ∩ u) \ (s ∩ u) = ∅
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | simp only [← ne_eq, ← nonempty_iff_ne_empty] at h | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
h : ¬closure (s ∩ u) \ (s ∩ u) = ∅
⊢ False | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
h : (closure (s ∩ u) \ (s ∩ u)).Nonempty
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
h : ¬closure (s ∩ u) \ (s ∩ u) = ∅
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | rcases h with ⟨x, h⟩ | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
h : (closure (s ∩ u) \ (s ∩ u)).Nonempty
⊢ False | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) \ (s ∩ u)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
h : (closure (s ∩ u) \ (s ∩ u)).Nonempty
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | simp only [mem_diff, mem_inter_iff, not_and] at h | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) \ (s ∩ u)
⊢ False | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) \ (s ∩ u)
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | have sus : closure (s ∩ u) ⊆ s := by
nth_rw 2 [← sc.closure_eq]; apply closure_mono; apply inter_subset_left | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ False | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | have xs := sus h.1 | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
⊢ False | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
xs : x ∈ s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | have m := not_or.mpr ⟨h.2 xs, not_mem_of_mem_compl (closure_inter_subset_compl vo d h.1)⟩ | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
xs : x ∈ s
⊢ False | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
xs : x ∈ s
m : ¬(x ∈ u ∨ x ∈ v)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
xs : x ∈ s
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | rw [← mem_union _ _ _] at m | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
xs : x ∈ s
m : ¬(x ∈ u ∨ x ∈ v)
⊢ False | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
xs : x ∈ s
m : x ∉ u ∪ v
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
xs : x ∈ s
m : ¬(x ∈ u ∨ x ∈ v)
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | exact not_mem_subset suv m xs | case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
xs : x ∈ s
m : x ∉ u ∪ v
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
sus : closure (s ∩ u) ⊆ s
xs : x ∈ s
m : x ∉ u ∪ v
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | nth_rw 2 [← sc.closure_eq] | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ closure (s ∩ u) ⊆ s | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ closure (s ∩ u) ⊆ closure s | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ closure (s ∩ u) ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | apply closure_mono | X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ closure (s ∩ u) ⊆ closure s | case h
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ s ∩ u ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ closure (s ∩ u) ⊆ closure s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isClosed_closed_inter | [30, 1] | [40, 61] | apply inter_subset_left | case h
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ s ∩ u ⊆ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁴ : TopologicalSpace X
I : Type
inst✝³ : TopologicalSpace I
inst✝² : ConditionallyCompleteLinearOrder I
inst✝¹ : DenselyOrdered I
inst✝ : OrderTopology I
s u v : Set X
sc : IsClosed s
vo : IsOpen v
d : Disjoint u v
suv : s ⊆ u ∪ v
x : X
h : x ∈ closure (s ∩ u) ∧ (x ∈ s → x ∉ u)
⊢ s ∩ u ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | rw [isPreconnected_iff_subset_of_fully_disjoint_closed sc] | X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ IsPreconnected s ↔ ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v | X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ (∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v) ↔
∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ IsPreconnected s ↔ ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | constructor | X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ (∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v) ↔
∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ (∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v) →
∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
case mpr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ (∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v) →
∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ (∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v) ↔
∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | intro h u v uo vo suv uv | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ (∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v) →
∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
⊢ s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ (∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v) →
∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | have suc : IsClosed (s ∩ u) := isClosed_closed_inter sc vo uv suv | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
⊢ s ⊆ u ∨ s ⊆ v | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
⊢ s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | have svc : IsClosed (s ∩ v) := isClosed_closed_inter sc uo uv.symm ((union_comm u v).subst suv) | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
⊢ s ⊆ u ∨ s ⊆ v | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
⊢ s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | have h0 : s ⊆ s ∩ u ∪ s ∩ v := by
simp only [←inter_union_distrib_left]; exact subset_inter (subset_refl _) suv | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
⊢ s ⊆ u ∨ s ⊆ v | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
⊢ s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | have h1 : Disjoint (s ∩ u) (s ∩ v) := Disjoint.inter_left' _ (Disjoint.inter_right' _ uv) | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
⊢ s ⊆ u ∨ s ⊆ v | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
⊢ s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | cases' h (s ∩ u) (s ∩ v) suc svc h0 h1 with su sv | case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
⊢ s ⊆ u ∨ s ⊆ v | case mp.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
su : s ⊆ s ∩ u
⊢ s ⊆ u ∨ s ⊆ v
case mp.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
sv : s ⊆ s ∩ v
⊢ s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | simp only [←inter_union_distrib_left] | X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
⊢ s ⊆ s ∩ u ∪ s ∩ v | X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
⊢ s ⊆ s ∩ (u ∪ v) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
⊢ s ⊆ s ∩ u ∪ s ∩ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | exact subset_inter (subset_refl _) suv | X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
⊢ s ⊆ s ∩ (u ∪ v) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
⊢ s ⊆ s ∩ (u ∪ v)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | left | case mp.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
su : s ⊆ s ∩ u
⊢ s ⊆ u ∨ s ⊆ v | case mp.inl.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
su : s ⊆ s ∩ u
⊢ s ⊆ u | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
su : s ⊆ s ∩ u
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | exact (subset_inter_iff.mp su).2 | case mp.inl.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
su : s ⊆ s ∩ u
⊢ s ⊆ u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inl.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
su : s ⊆ s ∩ u
⊢ s ⊆ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | right | case mp.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
sv : s ⊆ s ∩ v
⊢ s ⊆ u ∨ s ⊆ v | case mp.inr.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
sv : s ⊆ s ∩ v
⊢ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
sv : s ⊆ s ∩ v
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | exact (subset_inter_iff.mp sv).2 | case mp.inr.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
sv : s ⊆ s ∩ v
⊢ s ⊆ v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.inr.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s ⊆ u ∪ v
uv : Disjoint u v
suc : IsClosed (s ∩ u)
svc : IsClosed (s ∩ v)
h0 : s ⊆ s ∩ u ∪ s ∩ v
h1 : Disjoint (s ∩ u) (s ∩ v)
sv : s ⊆ s ∩ v
⊢ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | intro h u v uc vc suv uv | case mpr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ (∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v) →
∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v | case mpr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
⊢ s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
⊢ (∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v) →
∀ (u v : Set X), IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | rcases NormalSpace.normal u v uc vc uv with ⟨u', v', uo, vo, uu, vv, uv'⟩ | case mpr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
⊢ s ⊆ u ∨ s ⊆ v | case mpr.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
⊢ s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | cases' h u' v' uo vo (_root_.trans suv (union_subset_union uu vv)) uv' with h h | case mpr.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
⊢ s ⊆ u ∨ s ⊆ v | case mpr.intro.intro.intro.intro.intro.intro.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
⊢ s ⊆ u ∨ s ⊆ v
case mpr.intro.intro.intro.intro.intro.intro.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
⊢ s ⊆ u ∨ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | left | case mpr.intro.intro.intro.intro.intro.intro.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
⊢ s ⊆ u ∨ s ⊆ v | case mpr.intro.intro.intro.intro.intro.intro.inl.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
⊢ s ⊆ u | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | intro x m | case mpr.intro.intro.intro.intro.intro.intro.inl.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
⊢ s ⊆ u | case mpr.intro.intro.intro.intro.intro.intro.inl.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
⊢ x ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inl.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
⊢ s ⊆ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | cases' (mem_union _ _ _).mp (suv m) with mu mv | case mpr.intro.intro.intro.intro.intro.intro.inl.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
⊢ x ∈ u | case mpr.intro.intro.intro.intro.intro.intro.inl.h.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mu : x ∈ u
⊢ x ∈ u
case mpr.intro.intro.intro.intro.intro.intro.inl.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inl.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
⊢ x ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | exact mu | case mpr.intro.intro.intro.intro.intro.intro.inl.h.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mu : x ∈ u
⊢ x ∈ u
case mpr.intro.intro.intro.intro.intro.intro.inl.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ u | case mpr.intro.intro.intro.intro.intro.intro.inl.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inl.h.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mu : x ∈ u
⊢ x ∈ u
case mpr.intro.intro.intro.intro.intro.intro.inl.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | exfalso | case mpr.intro.intro.intro.intro.intro.intro.inl.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ u | case mpr.intro.intro.intro.intro.intro.intro.inl.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mv : x ∈ v
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inl.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | exact disjoint_left.mp uv' (h m) (vv mv) | case mpr.intro.intro.intro.intro.intro.intro.inl.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mv : x ∈ v
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inl.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ u'
x : X
m : x ∈ s
mv : x ∈ v
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | right | case mpr.intro.intro.intro.intro.intro.intro.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
⊢ s ⊆ u ∨ s ⊆ v | case mpr.intro.intro.intro.intro.intro.intro.inr.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
⊢ s ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
⊢ s ⊆ u ∨ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | intro x m | case mpr.intro.intro.intro.intro.intro.intro.inr.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
⊢ s ⊆ v | case mpr.intro.intro.intro.intro.intro.intro.inr.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
⊢ x ∈ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inr.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
⊢ s ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | cases' (mem_union _ _ _).mp (suv m) with mu mv | case mpr.intro.intro.intro.intro.intro.intro.inr.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
⊢ x ∈ v | case mpr.intro.intro.intro.intro.intro.intro.inr.h.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mu : x ∈ u
⊢ x ∈ v
case mpr.intro.intro.intro.intro.intro.intro.inr.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inr.h
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
⊢ x ∈ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | exfalso | case mpr.intro.intro.intro.intro.intro.intro.inr.h.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mu : x ∈ u
⊢ x ∈ v
case mpr.intro.intro.intro.intro.intro.intro.inr.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ v | case mpr.intro.intro.intro.intro.intro.intro.inr.h.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mu : x ∈ u
⊢ False
case mpr.intro.intro.intro.intro.intro.intro.inr.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inr.h.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mu : x ∈ u
⊢ x ∈ v
case mpr.intro.intro.intro.intro.intro.intro.inr.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | exact disjoint_right.mp uv' (h m) (uu mu) | case mpr.intro.intro.intro.intro.intro.intro.inr.h.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mu : x ∈ u
⊢ False
case mpr.intro.intro.intro.intro.intro.intro.inr.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ v | case mpr.intro.intro.intro.intro.intro.intro.inr.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ v | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inr.h.inl
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mu : x ∈ u
⊢ False
case mpr.intro.intro.intro.intro.intro.intro.inr.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | isPreconnected_iff_subset_of_fully_disjoint_open | [45, 1] | [64, 67] | exact mv | case mpr.intro.intro.intro.intro.intro.intro.inr.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.intro.intro.intro.intro.inr.h.inr
X : Type
inst✝⁵ : TopologicalSpace X
I : Type
inst✝⁴ : TopologicalSpace I
inst✝³ : ConditionallyCompleteLinearOrder I
inst✝² : DenselyOrdered I
inst✝¹ : OrderTopology I
inst✝ : NormalSpace X
s : Set X
sc : IsClosed s
h✝ : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v
u v : Set X
uc : IsClosed u
vc : IsClosed v
suv : s ⊆ u ∪ v
uv : Disjoint u v
u' v' : Set X
uo : IsOpen u'
vo : IsOpen v'
uu : u ⊆ u'
vv : v ⊆ v'
uv' : Disjoint u' v'
h : s ⊆ v'
x : X
m : x ∈ s
mv : x ∈ v
⊢ x ∈ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | contrapose p | X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
p : ∀ (a : I), IsPreconnected (s a)
c : ∀ (a : I), IsCompact (s a)
⊢ IsPreconnected (⋂ a, s a) | X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
p : ¬IsPreconnected (⋂ a, s a)
⊢ ¬∀ (a : I), IsPreconnected (s a) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
p : ∀ (a : I), IsPreconnected (s a)
c : ∀ (a : I), IsCompact (s a)
⊢ IsPreconnected (⋂ a, s a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | have ci : IsClosed (⋂ a, s a) := isClosed_iInter fun i ↦ (c i).isClosed | X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
p : ¬IsPreconnected (⋂ a, s a)
⊢ ¬∀ (a : I), IsPreconnected (s a) | X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
p : ¬IsPreconnected (⋂ a, s a)
ci : IsClosed (⋂ a, s a)
⊢ ¬∀ (a : I), IsPreconnected (s a) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
p : ¬IsPreconnected (⋂ a, s a)
⊢ ¬∀ (a : I), IsPreconnected (s a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | simp only [isPreconnected_iff_subset_of_fully_disjoint_open ci, not_forall] at p | X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
p : ¬IsPreconnected (⋂ a, s a)
ci : IsClosed (⋂ a, s a)
⊢ ¬∀ (a : I), IsPreconnected (s a) | X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
p :
∃ x x_1,
∃ (_ : IsOpen x) (_ : IsOpen x_1) (_ : ⋂ a, s a ⊆ x ∪ x_1) (_ : Disjoint x x_1), ¬(⋂ a, s a ⊆ x ∨ ⋂ a, s a ⊆ x_1)
⊢ ¬∀ (a : I), IsPreconnected (s a) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
p : ¬IsPreconnected (⋂ a, s a)
ci : IsClosed (⋂ a, s a)
⊢ ¬∀ (a : I), IsPreconnected (s a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | simp only [isPreconnected_iff_subset_of_fully_disjoint_open (c _).isClosed, not_forall] | X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
p :
∃ x x_1,
∃ (_ : IsOpen x) (_ : IsOpen x_1) (_ : ⋂ a, s a ⊆ x ∪ x_1) (_ : Disjoint x x_1), ¬(⋂ a, s a ⊆ x ∨ ⋂ a, s a ⊆ x_1)
⊢ ¬∀ (a : I), IsPreconnected (s a) | X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
p :
∃ x x_1,
∃ (_ : IsOpen x) (_ : IsOpen x_1) (_ : ⋂ a, s a ⊆ x ∪ x_1) (_ : Disjoint x x_1), ¬(⋂ a, s a ⊆ x ∨ ⋂ a, s a ⊆ x_1)
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
p :
∃ x x_1,
∃ (_ : IsOpen x) (_ : IsOpen x_1) (_ : ⋂ a, s a ⊆ x ∪ x_1) (_ : Disjoint x x_1), ¬(⋂ a, s a ⊆ x ∨ ⋂ a, s a ⊆ x_1)
⊢ ¬∀ (a : I), IsPreconnected (s a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | rcases p with ⟨u, v, uo, vo, suv, uv, no⟩ | X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
p :
∃ x x_1,
∃ (_ : IsOpen x) (_ : IsOpen x_1) (_ : ⋂ a, s a ⊆ x ∪ x_1) (_ : Disjoint x x_1), ¬(⋂ a, s a ⊆ x ∨ ⋂ a, s a ⊆ x_1)
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2) | case intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
p :
∃ x x_1,
∃ (_ : IsOpen x) (_ : IsOpen x_1) (_ : ⋂ a, s a ⊆ x ∪ x_1) (_ : Disjoint x x_1), ¬(⋂ a, s a ⊆ x ∨ ⋂ a, s a ⊆ x_1)
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | have e : ∃ a, s a ⊆ u ∪ v := by
by_contra h; simp only [not_exists, Set.not_subset] at h
suffices n : (⋂ a, s a \ (u ∪ v)).Nonempty by
rcases n with ⟨x, n⟩; simp only [mem_iInter, mem_diff, forall_and, forall_const] at n
rw [← mem_iInter] at n; simp only [suv n.1, not_true, imp_false] at n; exact n.2
apply IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed
intro a b; rcases d a b with ⟨c, ac, bc⟩
use c, diff_subset_diff_left ac, diff_subset_diff_left bc
intro a; rcases h a with ⟨x, xa, xuv⟩; exact ⟨x, mem_diff_of_mem xa xuv⟩
intro a; exact (c a).diff (uo.union vo)
intro a; exact ((c a).diff (uo.union vo)).isClosed | case intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2) | case intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
e : ∃ a, s a ⊆ u ∪ v
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | rcases e with ⟨a, auv⟩ | case intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
e : ∃ a, s a ⊆ u ∪ v
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2) | case intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
a : I
auv : s a ⊆ u ∪ v
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
e : ∃ a, s a ⊆ u ∪ v
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | use a, u, v, uo, vo, auv, uv | case intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
a : I
auv : s a ⊆ u ∪ v
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2) | case h
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
a : I
auv : s a ⊆ u ∪ v
⊢ ¬(s a ⊆ u ∨ s a ⊆ v) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
a : I
auv : s a ⊆ u ∪ v
⊢ ∃ x x_1 x_2,
∃ (_ : IsOpen x_1) (_ : IsOpen x_2) (_ : s x ⊆ x_1 ∪ x_2) (_ : Disjoint x_1 x_2), ¬(s x ⊆ x_1 ∨ s x ⊆ x_2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | contrapose no | case h
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
a : I
auv : s a ⊆ u ∪ v
⊢ ¬(s a ⊆ u ∨ s a ⊆ v) | case h
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
no : ¬¬(s a ⊆ u ∨ s a ⊆ v)
⊢ ¬¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
a : I
auv : s a ⊆ u ∪ v
⊢ ¬(s a ⊆ u ∨ s a ⊆ v)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | simp only [not_not] at no ⊢ | case h
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
no : ¬¬(s a ⊆ u ∨ s a ⊆ v)
⊢ ¬¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v) | case h
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
no : s a ⊆ u ∨ s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
no : ¬¬(s a ⊆ u ∨ s a ⊆ v)
⊢ ¬¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | cases' no with su sv | case h
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
no : s a ⊆ u ∨ s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v | case h.inl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
su : s a ⊆ u
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v
case h.inr
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
no : s a ⊆ u ∨ s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | left | case h.inl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
su : s a ⊆ u
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v
case h.inr
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v | case h.inl.h
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
su : s a ⊆ u
⊢ ⋂ a, s a ⊆ u
case h.inr
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
su : s a ⊆ u
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v
case h.inr
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | exact _root_.trans (iInter_subset _ _) su | case h.inl.h
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
su : s a ⊆ u
⊢ ⋂ a, s a ⊆ u
case h.inr
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v | case h.inr
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inl.h
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
su : s a ⊆ u
⊢ ⋂ a, s a ⊆ u
case h.inr
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | right | case h.inr
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v | case h.inr.h
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ v | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | exact _root_.trans (iInter_subset _ _) sv | case h.inr.h
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.inr.h
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
a : I
auv : s a ⊆ u ∪ v
sv : s a ⊆ v
⊢ ⋂ a, s a ⊆ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | by_contra h | X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
⊢ ∃ a, s a ⊆ u ∪ v | X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ¬∃ a, s a ⊆ u ∪ v
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
⊢ ∃ a, s a ⊆ u ∪ v
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | simp only [not_exists, Set.not_subset] at h | X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ¬∃ a, s a ⊆ u ∪ v
⊢ False | X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ¬∃ a, s a ⊆ u ∪ v
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | suffices n : (⋂ a, s a \ (u ∪ v)).Nonempty by
rcases n with ⟨x, n⟩; simp only [mem_iInter, mem_diff, forall_and, forall_const] at n
rw [← mem_iInter] at n; simp only [suv n.1, not_true, imp_false] at n; exact n.2 | X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ False | X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ (⋂ a, s a \ (u ∪ v)).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | apply IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed | X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ (⋂ a, s a \ (u ∪ v)).Nonempty | case htd
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ Directed (fun x x_1 => x ⊇ x_1) fun i => s i \ (u ∪ v)
case htn
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), (s i \ (u ∪ v)).Nonempty
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ (⋂ a, s a \ (u ∪ v)).Nonempty
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | intro a b | case htd
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ Directed (fun x x_1 => x ⊇ x_1) fun i => s i \ (u ∪ v)
case htn
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), (s i \ (u ∪ v)).Nonempty
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v)) | case htd
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
a b : I
⊢ ∃ z,
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) a) ((fun i => s i \ (u ∪ v)) z) ∧
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) b) ((fun i => s i \ (u ∪ v)) z)
case htn
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), (s i \ (u ∪ v)).Nonempty
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v)) | Please generate a tactic in lean4 to solve the state.
STATE:
case htd
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ Directed (fun x x_1 => x ⊇ x_1) fun i => s i \ (u ∪ v)
case htn
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), (s i \ (u ∪ v)).Nonempty
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | rcases d a b with ⟨c, ac, bc⟩ | case htd
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
a b : I
⊢ ∃ z,
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) a) ((fun i => s i \ (u ∪ v)) z) ∧
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) b) ((fun i => s i \ (u ∪ v)) z)
case htn
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), (s i \ (u ∪ v)).Nonempty
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v)) | case htd.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c✝ : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
a b c : I
ac : s a ⊇ s c
bc : s b ⊇ s c
⊢ ∃ z,
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) a) ((fun i => s i \ (u ∪ v)) z) ∧
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) b) ((fun i => s i \ (u ∪ v)) z)
case htn
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), (s i \ (u ∪ v)).Nonempty
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v)) | Please generate a tactic in lean4 to solve the state.
STATE:
case htd
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
a b : I
⊢ ∃ z,
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) a) ((fun i => s i \ (u ∪ v)) z) ∧
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) b) ((fun i => s i \ (u ∪ v)) z)
case htn
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), (s i \ (u ∪ v)).Nonempty
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | use c, diff_subset_diff_left ac, diff_subset_diff_left bc | case htd.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c✝ : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
a b c : I
ac : s a ⊇ s c
bc : s b ⊇ s c
⊢ ∃ z,
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) a) ((fun i => s i \ (u ∪ v)) z) ∧
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) b) ((fun i => s i \ (u ∪ v)) z)
case htn
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), (s i \ (u ∪ v)).Nonempty
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v)) | case htn
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), (s i \ (u ∪ v)).Nonempty
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v)) | Please generate a tactic in lean4 to solve the state.
STATE:
case htd.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c✝ : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
a b c : I
ac : s a ⊇ s c
bc : s b ⊇ s c
⊢ ∃ z,
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) a) ((fun i => s i \ (u ∪ v)) z) ∧
(fun x x_1 => x ⊇ x_1) ((fun i => s i \ (u ∪ v)) b) ((fun i => s i \ (u ∪ v)) z)
case htn
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), (s i \ (u ∪ v)).Nonempty
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | intro a | case htn
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), (s i \ (u ∪ v)).Nonempty
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v)) | case htn
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
a : I
⊢ (s a \ (u ∪ v)).Nonempty
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v)) | Please generate a tactic in lean4 to solve the state.
STATE:
case htn
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), (s i \ (u ∪ v)).Nonempty
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Connected.lean | IsPreconnected.directed_iInter | [67, 1] | [91, 51] | rcases h a with ⟨x, xa, xuv⟩ | case htn
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
a : I
⊢ (s a \ (u ∪ v)).Nonempty
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v)) | case htn.intro.intro
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
a : I
x : X
xa : x ∈ s a
xuv : x ∉ u ∪ v
⊢ (s a \ (u ∪ v)).Nonempty
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v)) | Please generate a tactic in lean4 to solve the state.
STATE:
case htn
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
a : I
⊢ (s a \ (u ∪ v)).Nonempty
case htc
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsCompact (s i \ (u ∪ v))
case htcl
X : Type
inst✝⁶ : TopologicalSpace X
I✝ : Type
inst✝⁵ : TopologicalSpace I✝
inst✝⁴ : ConditionallyCompleteLinearOrder I✝
inst✝³ : DenselyOrdered I✝
inst✝² : OrderTopology I✝
I : Type
s : I → Set X
inst✝¹ : Nonempty I
inst✝ : T4Space X
d : Directed Superset s
c : ∀ (a : I), IsCompact (s a)
ci : IsClosed (⋂ a, s a)
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : ⋂ a, s a ⊆ u ∪ v
uv : Disjoint u v
no : ¬(⋂ a, s a ⊆ u ∨ ⋂ a, s a ⊆ v)
h : ∀ (x : I), ∃ a ∈ s x, a ∉ u ∪ v
⊢ ∀ (i : I), IsClosed (s i \ (u ∪ v))
TACTIC:
|
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