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stringclasses 147
values | file_path
stringlengths 7
101
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stringlengths 1
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| start
stringlengths 6
10
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stringlengths 6
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stringlengths 1
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2.09M
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|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | simp | X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : NormedAddCommGroup Y
f : ℕ → X → Y
s : Set X
fc : ∀ (n : ℕ), ContinuousOn (f n) s
sc : IsCompact s
c : ℕ → ℝ
hc : (fun n => Classical.choose ⋯) = c
cs : ∀ (n : ℕ), 0 ≤ c n ∧ ∀ x ∈ s, ‖f n x‖ ≤ c n
N : ℕ
bs : Finset ℝ
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 ∈ bs
b : ℝ
hb : 1 + bs.max' ⋯ = b
n : ℕ
x : X
xs : x ∈ s
nN : N < n
H : dist (f N x) (f n x) < 1
⊢ N < N + 1 ∧ c N = c N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : NormedAddCommGroup Y
f : ℕ → X → Y
s : Set X
fc : ∀ (n : ℕ), ContinuousOn (f n) s
sc : IsCompact s
c : ℕ → ℝ
hc : (fun n => Classical.choose ⋯) = c
cs : ∀ (n : ℕ), 0 ≤ c n ∧ ∀ x ∈ s, ‖f n x‖ ≤ c n
N : ℕ
bs : Finset ℝ
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 ∈ bs
b : ℝ
hb : 1 + bs.max' ⋯ = b
n : ℕ
x : X
xs : x ∈ s
nN : N < n
H : dist (f N x) (f n x) < 1
⊢ N < N + 1 ∧ c N = c N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | rw [sub_sub_cancel] | X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : NormedAddCommGroup Y
f : ℕ → X → Y
s : Set X
fc : ∀ (n : ℕ), ContinuousOn (f n) s
sc : IsCompact s
c : ℕ → ℝ
hc : (fun n => Classical.choose ⋯) = c
cs : ∀ (n : ℕ), 0 ≤ c n ∧ ∀ x ∈ s, ‖f n x‖ ≤ c n
N : ℕ
bs : Finset ℝ
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 ∈ bs
b : ℝ
hb : 1 + bs.max' ⋯ = b
n : ℕ
x : X
xs : x ∈ s
nN : N < n
H : ‖f N x - f n x‖ < 1
cN : c N ∈ bs
bN : ‖f N x‖ ≤ bs.max' ⋯
⊢ ‖f n x‖ = ‖f N x - (f N x - f n x)‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : NormedAddCommGroup Y
f : ℕ → X → Y
s : Set X
fc : ∀ (n : ℕ), ContinuousOn (f n) s
sc : IsCompact s
c : ℕ → ℝ
hc : (fun n => Classical.choose ⋯) = c
cs : ∀ (n : ℕ), 0 ≤ c n ∧ ∀ x ∈ s, ‖f n x‖ ≤ c n
N : ℕ
bs : Finset ℝ
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 ∈ bs
b : ℝ
hb : 1 + bs.max' ⋯ = b
n : ℕ
x : X
xs : x ∈ s
nN : N < n
H : ‖f N x - f n x‖ < 1
cN : c N ∈ bs
bN : ‖f N x‖ ≤ bs.max' ⋯
⊢ ‖f n x‖ = ‖f N x - (f N x - f n x)‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | ring | X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : NormedAddCommGroup Y
f : ℕ → X → Y
s : Set X
fc : ∀ (n : ℕ), ContinuousOn (f n) s
sc : IsCompact s
c : ℕ → ℝ
hc : (fun n => Classical.choose ⋯) = c
cs : ∀ (n : ℕ), 0 ≤ c n ∧ ∀ x ∈ s, ‖f n x‖ ≤ c n
N : ℕ
bs : Finset ℝ
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 ∈ bs
b : ℝ
hb : 1 + bs.max' ⋯ = b
n : ℕ
x : X
xs : x ∈ s
nN : N < n
H : ‖f N x - f n x‖ < 1
cN : c N ∈ bs
bN : ‖f N x‖ ≤ bs.max' ⋯
⊢ bs.max' ⋯ + 1 = 1 + bs.max' ?m.16989 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : NormedAddCommGroup Y
f : ℕ → X → Y
s : Set X
fc : ∀ (n : ℕ), ContinuousOn (f n) s
sc : IsCompact s
c : ℕ → ℝ
hc : (fun n => Classical.choose ⋯) = c
cs : ∀ (n : ℕ), 0 ≤ c n ∧ ∀ x ∈ s, ‖f n x‖ ≤ c n
N : ℕ
bs : Finset ℝ
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 ∈ bs
b : ℝ
hb : 1 + bs.max' ⋯ = b
n : ℕ
x : X
xs : x ∈ s
nN : N < n
H : ‖f N x - f n x‖ < 1
cN : c N ∈ bs
bN : ‖f N x‖ ≤ bs.max' ⋯
⊢ bs.max' ⋯ + 1 = 1 + bs.max' ?m.16989
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | UniformCauchySeqOn.bounded | [21, 1] | [49, 35] | simp only [hb] | X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : NormedAddCommGroup Y
f : ℕ → X → Y
s : Set X
fc : ∀ (n : ℕ), ContinuousOn (f n) s
sc : IsCompact s
c : ℕ → ℝ
hc : (fun n => Classical.choose ⋯) = c
cs : ∀ (n : ℕ), 0 ≤ c n ∧ ∀ x ∈ s, ‖f n x‖ ≤ c n
N : ℕ
bs : Finset ℝ
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 ∈ bs
b : ℝ
hb : 1 + bs.max' ⋯ = b
n : ℕ
x : X
xs : x ∈ s
nN : N < n
H : ‖f N x - f n x‖ < 1
cN : c N ∈ bs
bN : ‖f N x‖ ≤ bs.max' ⋯
⊢ 1 + bs.max' ⋯ = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
inst✝¹ : TopologicalSpace X
inst✝ : NormedAddCommGroup Y
f : ℕ → X → Y
s : Set X
fc : ∀ (n : ℕ), ContinuousOn (f n) s
sc : IsCompact s
c : ℕ → ℝ
hc : (fun n => Classical.choose ⋯) = c
cs : ∀ (n : ℕ), 0 ≤ c n ∧ ∀ x ∈ s, ‖f n x‖ ≤ c n
N : ℕ
bs : Finset ℝ
hbs : Finset.image c (Finset.range (N + 1)) = bs
c0 : c 0 ∈ bs
b : ℝ
hb : 1 + bs.max' ⋯ = b
n : ℕ
x : X
xs : x ∈ s
nN : N < n
H : ‖f N x - f n x‖ < 1
cN : c N ∈ bs
bN : ‖f N x‖ ≤ bs.max' ⋯
⊢ 1 + bs.max' ⋯ = b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | tendsto_inv_iff_tendsto | [62, 1] | [67, 32] | refine ⟨fun h ↦ ?_, fun h ↦ h.inv₀ a0⟩ | A B : Type
inst✝ : NontriviallyNormedField B
l : Filter A
f : A → B
a : B
a0 : a ≠ 0
⊢ Tendsto (fun x => (f x)⁻¹) l (𝓝 a⁻¹) ↔ Tendsto f l (𝓝 a) | A B : Type
inst✝ : NontriviallyNormedField B
l : Filter A
f : A → B
a : B
a0 : a ≠ 0
h : Tendsto (fun x => (f x)⁻¹) l (𝓝 a⁻¹)
⊢ Tendsto f l (𝓝 a) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
inst✝ : NontriviallyNormedField B
l : Filter A
f : A → B
a : B
a0 : a ≠ 0
⊢ Tendsto (fun x => (f x)⁻¹) l (𝓝 a⁻¹) ↔ Tendsto f l (𝓝 a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | tendsto_inv_iff_tendsto | [62, 1] | [67, 32] | have h := h.inv₀ (inv_ne_zero a0) | A B : Type
inst✝ : NontriviallyNormedField B
l : Filter A
f : A → B
a : B
a0 : a ≠ 0
h : Tendsto (fun x => (f x)⁻¹) l (𝓝 a⁻¹)
⊢ Tendsto f l (𝓝 a) | A B : Type
inst✝ : NontriviallyNormedField B
l : Filter A
f : A → B
a : B
a0 : a ≠ 0
h✝ : Tendsto (fun x => (f x)⁻¹) l (𝓝 a⁻¹)
h : Tendsto (fun x => (f x)⁻¹⁻¹) l (𝓝 a⁻¹⁻¹)
⊢ Tendsto f l (𝓝 a) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
inst✝ : NontriviallyNormedField B
l : Filter A
f : A → B
a : B
a0 : a ≠ 0
h : Tendsto (fun x => (f x)⁻¹) l (𝓝 a⁻¹)
⊢ Tendsto f l (𝓝 a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | tendsto_inv_iff_tendsto | [62, 1] | [67, 32] | field_simp [a0] at h | A B : Type
inst✝ : NontriviallyNormedField B
l : Filter A
f : A → B
a : B
a0 : a ≠ 0
h✝ : Tendsto (fun x => (f x)⁻¹) l (𝓝 a⁻¹)
h : Tendsto (fun x => (f x)⁻¹⁻¹) l (𝓝 a⁻¹⁻¹)
⊢ Tendsto f l (𝓝 a) | A B : Type
inst✝ : NontriviallyNormedField B
l : Filter A
f : A → B
a : B
a0 : a ≠ 0
h✝ : Tendsto (fun x => (f x)⁻¹) l (𝓝 a⁻¹)
h : Tendsto (fun x => f x) l (𝓝 a)
⊢ Tendsto f l (𝓝 a) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
inst✝ : NontriviallyNormedField B
l : Filter A
f : A → B
a : B
a0 : a ≠ 0
h✝ : Tendsto (fun x => (f x)⁻¹) l (𝓝 a⁻¹)
h : Tendsto (fun x => (f x)⁻¹⁻¹) l (𝓝 a⁻¹⁻¹)
⊢ Tendsto f l (𝓝 a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | tendsto_inv_iff_tendsto | [62, 1] | [67, 32] | exact h | A B : Type
inst✝ : NontriviallyNormedField B
l : Filter A
f : A → B
a : B
a0 : a ≠ 0
h✝ : Tendsto (fun x => (f x)⁻¹) l (𝓝 a⁻¹)
h : Tendsto (fun x => f x) l (𝓝 a)
⊢ Tendsto f l (𝓝 a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
inst✝ : NontriviallyNormedField B
l : Filter A
f : A → B
a : B
a0 : a ≠ 0
h✝ : Tendsto (fun x => (f x)⁻¹) l (𝓝 a⁻¹)
h : Tendsto (fun x => f x) l (𝓝 a)
⊢ Tendsto f l (𝓝 a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | generalize hs' : ofDual ⁻¹' s = s' | X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
⊢ Icc a b ⊆ s | X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
⊢ Icc a b ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
⊢ Icc a b ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | intro x m | X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
rev : Icc (toDual b) (toDual a) ⊆ s'
⊢ Icc a b ⊆ s | X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
rev : Icc (toDual b) (toDual a) ⊆ s'
x : X
m : x ∈ Icc a b
⊢ x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
rev : Icc (toDual b) (toDual a) ⊆ s'
⊢ Icc a b ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | simp only [Set.mem_Icc] at m | X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
rev : Icc (toDual b) (toDual a) ⊆ s'
x : X
m : x ∈ Icc a b
⊢ x ∈ s | X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
rev : Icc (toDual b) (toDual a) ⊆ s'
x : X
m : a ≤ x ∧ x ≤ b
⊢ x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
rev : Icc (toDual b) (toDual a) ⊆ s'
x : X
m : x ∈ Icc a b
⊢ x ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | specialize @rev (toDual x) | X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
rev : Icc (toDual b) (toDual a) ⊆ s'
x : X
m : a ≤ x ∧ x ≤ b
⊢ x ∈ s | X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : X
m : a ≤ x ∧ x ≤ b
rev : toDual x ∈ Icc (toDual b) (toDual a) → toDual x ∈ s'
⊢ x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
rev : Icc (toDual b) (toDual a) ⊆ s'
x : X
m : a ≤ x ∧ x ≤ b
⊢ x ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | simp only [Set.dual_Icc, Set.mem_preimage, Set.mem_Icc, and_imp, OrderDual.ofDual_toDual,
← hs'] at rev | X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : X
m : a ≤ x ∧ x ≤ b
rev : toDual x ∈ Icc (toDual b) (toDual a) → toDual x ∈ s'
⊢ x ∈ s | X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : X
m : a ≤ x ∧ x ≤ b
rev : a ≤ x → x ≤ b → x ∈ s
⊢ x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : X
m : a ≤ x ∧ x ≤ b
rev : toDual x ∈ Icc (toDual b) (toDual a) → toDual x ∈ s'
⊢ x ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | exact rev m.1 m.2 | X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : X
m : a ≤ x ∧ x ≤ b
rev : a ≤ x → x ≤ b → x ∈ s
⊢ x ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : X
m : a ≤ x ∧ x ≤ b
rev : a ≤ x → x ≤ b → x ∈ s
⊢ x ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | apply IsClosed.Icc_subset_of_forall_mem_nhdsWithin | X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
⊢ Icc (toDual b) (toDual a) ⊆ s' | case hs
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
⊢ IsClosed (s' ∩ Icc (toDual b) (toDual a))
case ha
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
⊢ toDual b ∈ s'
case hgt
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
⊢ ∀ x ∈ s' ∩ Ico (toDual b) (toDual a), s' ∈ 𝓝[>] x | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
⊢ Icc (toDual b) (toDual a) ⊆ s'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | have e : s' ∩ Icc (toDual b) (toDual a) = ofDual ⁻¹' (s ∩ Icc a b) := by
apply Set.ext; intro x; simp only [Set.dual_Icc, Set.preimage_inter, ← hs'] | case hs
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
⊢ IsClosed (s' ∩ Icc (toDual b) (toDual a)) | case hs
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
e : s' ∩ Icc (toDual b) (toDual a) = ⇑ofDual ⁻¹' (s ∩ Icc a b)
⊢ IsClosed (s' ∩ Icc (toDual b) (toDual a)) | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
⊢ IsClosed (s' ∩ Icc (toDual b) (toDual a))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | rw [e] | case hs
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
e : s' ∩ Icc (toDual b) (toDual a) = ⇑ofDual ⁻¹' (s ∩ Icc a b)
⊢ IsClosed (s' ∩ Icc (toDual b) (toDual a)) | case hs
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
e : s' ∩ Icc (toDual b) (toDual a) = ⇑ofDual ⁻¹' (s ∩ Icc a b)
⊢ IsClosed (⇑ofDual ⁻¹' (s ∩ Icc a b)) | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
e : s' ∩ Icc (toDual b) (toDual a) = ⇑ofDual ⁻¹' (s ∩ Icc a b)
⊢ IsClosed (s' ∩ Icc (toDual b) (toDual a))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | exact IsClosed.preimage continuous_ofDual sc | case hs
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
e : s' ∩ Icc (toDual b) (toDual a) = ⇑ofDual ⁻¹' (s ∩ Icc a b)
⊢ IsClosed (⇑ofDual ⁻¹' (s ∩ Icc a b)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
e : s' ∩ Icc (toDual b) (toDual a) = ⇑ofDual ⁻¹' (s ∩ Icc a b)
⊢ IsClosed (⇑ofDual ⁻¹' (s ∩ Icc a b))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | apply Set.ext | X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
⊢ s' ∩ Icc (toDual b) (toDual a) = ⇑ofDual ⁻¹' (s ∩ Icc a b) | case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
⊢ ∀ (x : Xᵒᵈ), x ∈ s' ∩ Icc (toDual b) (toDual a) ↔ x ∈ ⇑ofDual ⁻¹' (s ∩ Icc a b) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
⊢ s' ∩ Icc (toDual b) (toDual a) = ⇑ofDual ⁻¹' (s ∩ Icc a b)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | intro x | case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
⊢ ∀ (x : Xᵒᵈ), x ∈ s' ∩ Icc (toDual b) (toDual a) ↔ x ∈ ⇑ofDual ⁻¹' (s ∩ Icc a b) | case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
⊢ x ∈ s' ∩ Icc (toDual b) (toDual a) ↔ x ∈ ⇑ofDual ⁻¹' (s ∩ Icc a b) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
⊢ ∀ (x : Xᵒᵈ), x ∈ s' ∩ Icc (toDual b) (toDual a) ↔ x ∈ ⇑ofDual ⁻¹' (s ∩ Icc a b)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | simp only [Set.dual_Icc, Set.preimage_inter, ← hs'] | case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
⊢ x ∈ s' ∩ Icc (toDual b) (toDual a) ↔ x ∈ ⇑ofDual ⁻¹' (s ∩ Icc a b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
⊢ x ∈ s' ∩ Icc (toDual b) (toDual a) ↔ x ∈ ⇑ofDual ⁻¹' (s ∩ Icc a b)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | simp only [Set.mem_preimage, OrderDual.ofDual_toDual, sb, ← hs'] | case ha
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
⊢ toDual b ∈ s' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
⊢ toDual b ∈ s'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | intro x m | case hgt
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
⊢ ∀ x ∈ s' ∩ Ico (toDual b) (toDual a), s' ∈ 𝓝[>] x | case hgt
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
m : x ∈ s' ∩ Ico (toDual b) (toDual a)
⊢ s' ∈ 𝓝[>] x | Please generate a tactic in lean4 to solve the state.
STATE:
case hgt
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
⊢ ∀ x ∈ s' ∩ Ico (toDual b) (toDual a), s' ∈ 𝓝[>] x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | simp only [Set.mem_preimage, Set.mem_inter_iff, Set.mem_Ico, OrderDual.toDual_le,
OrderDual.lt_toDual] at m | case hgt
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
m : x ∈ s' ∩ Ico (toDual b) (toDual a)
⊢ s' ∈ 𝓝[>] x | case hgt
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
m : x ∈ s' ∧ ofDual x ≤ b ∧ a < ofDual x
⊢ s' ∈ 𝓝[>] x | Please generate a tactic in lean4 to solve the state.
STATE:
case hgt
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
m : x ∈ s' ∩ Ico (toDual b) (toDual a)
⊢ s' ∈ 𝓝[>] x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | simp only [mem_nhdsWithin_iff_eventually, eventually_nhds_iff, Set.mem_inter_iff,
Set.mem_Ioc, ← hs'] at so m ⊢ | case hgt
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
m : x ∈ s' ∧ ofDual x ≤ b ∧ a < ofDual x
⊢ s' ∈ 𝓝[>] x | case hgt
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
⊢ ∃ t, (∀ x_1 ∈ t, x_1 ∈ Ioi x → x_1 ∈ ⇑ofDual ⁻¹' s) ∧ IsOpen t ∧ x ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case hgt
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
so : ∀ x ∈ s ∩ Ioc a b, s ∈ 𝓝[<] x
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
m : x ∈ s' ∧ ofDual x ≤ b ∧ a < ofDual x
⊢ s' ∈ 𝓝[>] x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | rcases so (ofDual x) ⟨m.1, m.2.2, m.2.1⟩ with ⟨n, h, o, nx⟩ | case hgt
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
⊢ ∃ t, (∀ x_1 ∈ t, x_1 ∈ Ioi x → x_1 ∈ ⇑ofDual ⁻¹' s) ∧ IsOpen t ∧ x ∈ t | case hgt.intro.intro.intro
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
h : ∀ x_1 ∈ n, x_1 ∈ Iio (ofDual x) → x_1 ∈ s
o : IsOpen n
nx : ofDual x ∈ n
⊢ ∃ t, (∀ x_1 ∈ t, x_1 ∈ Ioi x → x_1 ∈ ⇑ofDual ⁻¹' s) ∧ IsOpen t ∧ x ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case hgt
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
⊢ ∃ t, (∀ x_1 ∈ t, x_1 ∈ Ioi x → x_1 ∈ ⇑ofDual ⁻¹' s) ∧ IsOpen t ∧ x ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | use ofDual ⁻¹' n | case hgt.intro.intro.intro
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
h : ∀ x_1 ∈ n, x_1 ∈ Iio (ofDual x) → x_1 ∈ s
o : IsOpen n
nx : ofDual x ∈ n
⊢ ∃ t, (∀ x_1 ∈ t, x_1 ∈ Ioi x → x_1 ∈ ⇑ofDual ⁻¹' s) ∧ IsOpen t ∧ x ∈ t | case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
h : ∀ x_1 ∈ n, x_1 ∈ Iio (ofDual x) → x_1 ∈ s
o : IsOpen n
nx : ofDual x ∈ n
⊢ (∀ x_1 ∈ ⇑ofDual ⁻¹' n, x_1 ∈ Ioi x → x_1 ∈ ⇑ofDual ⁻¹' s) ∧ IsOpen (⇑ofDual ⁻¹' n) ∧ x ∈ ⇑ofDual ⁻¹' n | Please generate a tactic in lean4 to solve the state.
STATE:
case hgt.intro.intro.intro
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
h : ∀ x_1 ∈ n, x_1 ∈ Iio (ofDual x) → x_1 ∈ s
o : IsOpen n
nx : ofDual x ∈ n
⊢ ∃ t, (∀ x_1 ∈ t, x_1 ∈ Ioi x → x_1 ∈ ⇑ofDual ⁻¹' s) ∧ IsOpen t ∧ x ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | refine ⟨?_, o.preimage continuous_ofDual, mem_preimage.mpr nx⟩ | case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
h : ∀ x_1 ∈ n, x_1 ∈ Iio (ofDual x) → x_1 ∈ s
o : IsOpen n
nx : ofDual x ∈ n
⊢ (∀ x_1 ∈ ⇑ofDual ⁻¹' n, x_1 ∈ Ioi x → x_1 ∈ ⇑ofDual ⁻¹' s) ∧ IsOpen (⇑ofDual ⁻¹' n) ∧ x ∈ ⇑ofDual ⁻¹' n | case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
h : ∀ x_1 ∈ n, x_1 ∈ Iio (ofDual x) → x_1 ∈ s
o : IsOpen n
nx : ofDual x ∈ n
⊢ ∀ x_1 ∈ ⇑ofDual ⁻¹' n, x_1 ∈ Ioi x → x_1 ∈ ⇑ofDual ⁻¹' s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
h : ∀ x_1 ∈ n, x_1 ∈ Iio (ofDual x) → x_1 ∈ s
o : IsOpen n
nx : ofDual x ∈ n
⊢ (∀ x_1 ∈ ⇑ofDual ⁻¹' n, x_1 ∈ Ioi x → x_1 ∈ ⇑ofDual ⁻¹' s) ∧ IsOpen (⇑ofDual ⁻¹' n) ∧ x ∈ ⇑ofDual ⁻¹' n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | intro y m xy | case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
h : ∀ x_1 ∈ n, x_1 ∈ Iio (ofDual x) → x_1 ∈ s
o : IsOpen n
nx : ofDual x ∈ n
⊢ ∀ x_1 ∈ ⇑ofDual ⁻¹' n, x_1 ∈ Ioi x → x_1 ∈ ⇑ofDual ⁻¹' s | case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m✝ : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
h : ∀ x_1 ∈ n, x_1 ∈ Iio (ofDual x) → x_1 ∈ s
o : IsOpen n
nx : ofDual x ∈ n
y : Xᵒᵈ
m : y ∈ ⇑ofDual ⁻¹' n
xy : y ∈ Ioi x
⊢ y ∈ ⇑ofDual ⁻¹' s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
h : ∀ x_1 ∈ n, x_1 ∈ Iio (ofDual x) → x_1 ∈ s
o : IsOpen n
nx : ofDual x ∈ n
⊢ ∀ x_1 ∈ ⇑ofDual ⁻¹' n, x_1 ∈ Ioi x → x_1 ∈ ⇑ofDual ⁻¹' s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | simp only [Set.mem_Ioi] at xy | case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m✝ : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
h : ∀ x_1 ∈ n, x_1 ∈ Iio (ofDual x) → x_1 ∈ s
o : IsOpen n
nx : ofDual x ∈ n
y : Xᵒᵈ
m : y ∈ ⇑ofDual ⁻¹' n
xy : y ∈ Ioi x
⊢ y ∈ ⇑ofDual ⁻¹' s | case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m✝ : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
h : ∀ x_1 ∈ n, x_1 ∈ Iio (ofDual x) → x_1 ∈ s
o : IsOpen n
nx : ofDual x ∈ n
y : Xᵒᵈ
m : y ∈ ⇑ofDual ⁻¹' n
xy : x < y
⊢ y ∈ ⇑ofDual ⁻¹' s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m✝ : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
h : ∀ x_1 ∈ n, x_1 ∈ Iio (ofDual x) → x_1 ∈ s
o : IsOpen n
nx : ofDual x ∈ n
y : Xᵒᵈ
m : y ∈ ⇑ofDual ⁻¹' n
xy : y ∈ Ioi x
⊢ y ∈ ⇑ofDual ⁻¹' s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | simp only [Set.mem_preimage] | case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m✝ : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
h : ∀ x_1 ∈ n, x_1 ∈ Iio (ofDual x) → x_1 ∈ s
o : IsOpen n
nx : ofDual x ∈ n
y : Xᵒᵈ
m : y ∈ ⇑ofDual ⁻¹' n
xy : x < y
⊢ y ∈ ⇑ofDual ⁻¹' s | case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m✝ : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
h : ∀ x_1 ∈ n, x_1 ∈ Iio (ofDual x) → x_1 ∈ s
o : IsOpen n
nx : ofDual x ∈ n
y : Xᵒᵈ
m : y ∈ ⇑ofDual ⁻¹' n
xy : x < y
⊢ ofDual y ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m✝ : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
h : ∀ x_1 ∈ n, x_1 ∈ Iio (ofDual x) → x_1 ∈ s
o : IsOpen n
nx : ofDual x ∈ n
y : Xᵒᵈ
m : y ∈ ⇑ofDual ⁻¹' n
xy : x < y
⊢ y ∈ ⇑ofDual ⁻¹' s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | simp only [Set.mem_Iio, Set.mem_preimage, OrderDual.ofDual_lt_ofDual] at h | case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m✝ : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
h : ∀ x_1 ∈ n, x_1 ∈ Iio (ofDual x) → x_1 ∈ s
o : IsOpen n
nx : ofDual x ∈ n
y : Xᵒᵈ
m : y ∈ ⇑ofDual ⁻¹' n
xy : x < y
⊢ ofDual y ∈ s | case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m✝ : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
o : IsOpen n
nx : ofDual x ∈ n
y : Xᵒᵈ
m : y ∈ ⇑ofDual ⁻¹' n
xy : x < y
h : ∀ x_1 ∈ n, x_1 < ofDual x → x_1 ∈ s
⊢ ofDual y ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m✝ : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
h : ∀ x_1 ∈ n, x_1 ∈ Iio (ofDual x) → x_1 ∈ s
o : IsOpen n
nx : ofDual x ∈ n
y : Xᵒᵈ
m : y ∈ ⇑ofDual ⁻¹' n
xy : x < y
⊢ ofDual y ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsClosed.Icc_subset_of_forall_mem_nhds_within' | [76, 1] | [101, 20] | exact h _ m xy | case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m✝ : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
o : IsOpen n
nx : ofDual x ∈ n
y : Xᵒᵈ
m : y ∈ ⇑ofDual ⁻¹' n
xy : x < y
h : ∀ x_1 ∈ n, x_1 < ofDual x → x_1 ∈ s
⊢ ofDual y ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝³ : ConditionallyCompleteLinearOrder X
inst✝² : TopologicalSpace X
inst✝¹ : OrderTopology X
inst✝ : DenselyOrdered X
a b : X
s : Set X
sc : IsClosed (s ∩ Icc a b)
sb : b ∈ s
s' : Set Xᵒᵈ
hs' : ⇑ofDual ⁻¹' s = s'
x : Xᵒᵈ
so : ∀ (x : X), x ∈ s ∧ a < x ∧ x ≤ b → ∃ t, (∀ x_1 ∈ t, x_1 ∈ Iio x → x_1 ∈ s) ∧ IsOpen t ∧ x ∈ t
m✝ : x ∈ ⇑ofDual ⁻¹' s ∧ ofDual x ≤ b ∧ a < ofDual x
n : Set X
o : IsOpen n
nx : ofDual x ∈ n
y : Xᵒᵈ
m : y ∈ ⇑ofDual ⁻¹' n
xy : x < y
h : ∀ x_1 ∈ n, x_1 < ofDual x → x_1 ∈ s
⊢ ofDual y ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsPreconnected.sUnion_of_pairwise_exists_isPreconnected | [103, 1] | [119, 63] | refine isPreconnected_of_forall_pair fun x hx y hy ↦ ?_ | X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
⊢ IsPreconnected S.sUnion | X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
⊢ ∃ t ⊆ S.sUnion, x ∈ t ∧ y ∈ t ∧ IsPreconnected t | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
⊢ IsPreconnected S.sUnion
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsPreconnected.sUnion_of_pairwise_exists_isPreconnected | [103, 1] | [119, 63] | rcases mem_sUnion.1 hx with ⟨s, hs, hxs⟩ | X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
⊢ ∃ t ⊆ S.sUnion, x ∈ t ∧ y ∈ t ∧ IsPreconnected t | case intro.intro
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
⊢ ∃ t ⊆ S.sUnion, x ∈ t ∧ y ∈ t ∧ IsPreconnected t | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
⊢ ∃ t ⊆ S.sUnion, x ∈ t ∧ y ∈ t ∧ IsPreconnected t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsPreconnected.sUnion_of_pairwise_exists_isPreconnected | [103, 1] | [119, 63] | rcases mem_sUnion.1 hy with ⟨t, ht, hyt⟩ | case intro.intro
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
⊢ ∃ t ⊆ S.sUnion, x ∈ t ∧ y ∈ t ∧ IsPreconnected t | case intro.intro.intro.intro
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
⊢ ∃ t ⊆ S.sUnion, x ∈ t ∧ y ∈ t ∧ IsPreconnected t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
⊢ ∃ t ⊆ S.sUnion, x ∈ t ∧ y ∈ t ∧ IsPreconnected t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsPreconnected.sUnion_of_pairwise_exists_isPreconnected | [103, 1] | [119, 63] | rcases eq_or_ne s t with rfl | hst | case intro.intro.intro.intro
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
⊢ ∃ t ⊆ S.sUnion, x ∈ t ∧ y ∈ t ∧ IsPreconnected t | case intro.intro.intro.intro.inl
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
ht : s ∈ S
hyt : y ∈ s
⊢ ∃ t ⊆ S.sUnion, x ∈ t ∧ y ∈ t ∧ IsPreconnected t
case intro.intro.intro.intro.inr
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
⊢ ∃ t ⊆ S.sUnion, x ∈ t ∧ y ∈ t ∧ IsPreconnected t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
⊢ ∃ t ⊆ S.sUnion, x ∈ t ∧ y ∈ t ∧ IsPreconnected t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsPreconnected.sUnion_of_pairwise_exists_isPreconnected | [103, 1] | [119, 63] | exact ⟨s, subset_sUnion_of_mem hs, hxs, hyt, hSc s hs⟩ | case intro.intro.intro.intro.inl
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
ht : s ∈ S
hyt : y ∈ s
⊢ ∃ t ⊆ S.sUnion, x ∈ t ∧ y ∈ t ∧ IsPreconnected t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.inl
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
ht : s ∈ S
hyt : y ∈ s
⊢ ∃ t ⊆ S.sUnion, x ∈ t ∧ y ∈ t ∧ IsPreconnected t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsPreconnected.sUnion_of_pairwise_exists_isPreconnected | [103, 1] | [119, 63] | rcases h hs ht hst ⟨x, hxs⟩ ⟨y, hyt⟩ with ⟨u, huS, hsu, hut, hu⟩ | case intro.intro.intro.intro.inr
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
⊢ ∃ t ⊆ S.sUnion, x ∈ t ∧ y ∈ t ∧ IsPreconnected t | case intro.intro.intro.intro.inr.intro.intro.intro.intro
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
u : Set X
huS : u ⊆ S.sUnion
hsu : (s ∩ u).Nonempty
hut : (u ∩ t).Nonempty
hu : IsPreconnected u
⊢ ∃ t ⊆ S.sUnion, x ∈ t ∧ y ∈ t ∧ IsPreconnected t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.inr
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
⊢ ∃ t ⊆ S.sUnion, x ∈ t ∧ y ∈ t ∧ IsPreconnected t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsPreconnected.sUnion_of_pairwise_exists_isPreconnected | [103, 1] | [119, 63] | refine ⟨s ∪ u ∪ t, ?_, ?_, ?_, ?_⟩ | case intro.intro.intro.intro.inr.intro.intro.intro.intro
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
u : Set X
huS : u ⊆ S.sUnion
hsu : (s ∩ u).Nonempty
hut : (u ∩ t).Nonempty
hu : IsPreconnected u
⊢ ∃ t ⊆ S.sUnion, x ∈ t ∧ y ∈ t ∧ IsPreconnected t | case intro.intro.intro.intro.inr.intro.intro.intro.intro.refine_1
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
u : Set X
huS : u ⊆ S.sUnion
hsu : (s ∩ u).Nonempty
hut : (u ∩ t).Nonempty
hu : IsPreconnected u
⊢ s ∪ u ∪ t ⊆ S.sUnion
case intro.intro.intro.intro.inr.intro.intro.intro.intro.refine_2
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
u : Set X
huS : u ⊆ S.sUnion
hsu : (s ∩ u).Nonempty
hut : (u ∩ t).Nonempty
hu : IsPreconnected u
⊢ x ∈ s ∪ u ∪ t
case intro.intro.intro.intro.inr.intro.intro.intro.intro.refine_3
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
u : Set X
huS : u ⊆ S.sUnion
hsu : (s ∩ u).Nonempty
hut : (u ∩ t).Nonempty
hu : IsPreconnected u
⊢ y ∈ s ∪ u ∪ t
case intro.intro.intro.intro.inr.intro.intro.intro.intro.refine_4
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
u : Set X
huS : u ⊆ S.sUnion
hsu : (s ∩ u).Nonempty
hut : (u ∩ t).Nonempty
hu : IsPreconnected u
⊢ IsPreconnected (s ∪ u ∪ t) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.inr.intro.intro.intro.intro
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
u : Set X
huS : u ⊆ S.sUnion
hsu : (s ∩ u).Nonempty
hut : (u ∩ t).Nonempty
hu : IsPreconnected u
⊢ ∃ t ⊆ S.sUnion, x ∈ t ∧ y ∈ t ∧ IsPreconnected t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsPreconnected.sUnion_of_pairwise_exists_isPreconnected | [103, 1] | [119, 63] | simp [*, subset_sUnion_of_mem] | case intro.intro.intro.intro.inr.intro.intro.intro.intro.refine_1
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
u : Set X
huS : u ⊆ S.sUnion
hsu : (s ∩ u).Nonempty
hut : (u ∩ t).Nonempty
hu : IsPreconnected u
⊢ s ∪ u ∪ t ⊆ S.sUnion | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.inr.intro.intro.intro.intro.refine_1
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
u : Set X
huS : u ⊆ S.sUnion
hsu : (s ∩ u).Nonempty
hut : (u ∩ t).Nonempty
hu : IsPreconnected u
⊢ s ∪ u ∪ t ⊆ S.sUnion
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsPreconnected.sUnion_of_pairwise_exists_isPreconnected | [103, 1] | [119, 63] | simp [*] | case intro.intro.intro.intro.inr.intro.intro.intro.intro.refine_2
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
u : Set X
huS : u ⊆ S.sUnion
hsu : (s ∩ u).Nonempty
hut : (u ∩ t).Nonempty
hu : IsPreconnected u
⊢ x ∈ s ∪ u ∪ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.inr.intro.intro.intro.intro.refine_2
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
u : Set X
huS : u ⊆ S.sUnion
hsu : (s ∩ u).Nonempty
hut : (u ∩ t).Nonempty
hu : IsPreconnected u
⊢ x ∈ s ∪ u ∪ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsPreconnected.sUnion_of_pairwise_exists_isPreconnected | [103, 1] | [119, 63] | simp [*] | case intro.intro.intro.intro.inr.intro.intro.intro.intro.refine_3
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
u : Set X
huS : u ⊆ S.sUnion
hsu : (s ∩ u).Nonempty
hut : (u ∩ t).Nonempty
hu : IsPreconnected u
⊢ y ∈ s ∪ u ∪ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.inr.intro.intro.intro.intro.refine_3
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
u : Set X
huS : u ⊆ S.sUnion
hsu : (s ∩ u).Nonempty
hut : (u ∩ t).Nonempty
hu : IsPreconnected u
⊢ y ∈ s ∪ u ∪ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsPreconnected.sUnion_of_pairwise_exists_isPreconnected | [103, 1] | [119, 63] | refine ((hSc s hs).union' hsu hu).union' (hut.mono ?_) (hSc t ht) | case intro.intro.intro.intro.inr.intro.intro.intro.intro.refine_4
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
u : Set X
huS : u ⊆ S.sUnion
hsu : (s ∩ u).Nonempty
hut : (u ∩ t).Nonempty
hu : IsPreconnected u
⊢ IsPreconnected (s ∪ u ∪ t) | case intro.intro.intro.intro.inr.intro.intro.intro.intro.refine_4
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
u : Set X
huS : u ⊆ S.sUnion
hsu : (s ∩ u).Nonempty
hut : (u ∩ t).Nonempty
hu : IsPreconnected u
⊢ u ∩ t ⊆ (s ∪ u) ∩ t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.inr.intro.intro.intro.intro.refine_4
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
u : Set X
huS : u ⊆ S.sUnion
hsu : (s ∩ u).Nonempty
hut : (u ∩ t).Nonempty
hu : IsPreconnected u
⊢ IsPreconnected (s ∪ u ∪ t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsPreconnected.sUnion_of_pairwise_exists_isPreconnected | [103, 1] | [119, 63] | exact inter_subset_inter_left _ (subset_union_right _ _) | case intro.intro.intro.intro.inr.intro.intro.intro.intro.refine_4
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
u : Set X
huS : u ⊆ S.sUnion
hsu : (s ∩ u).Nonempty
hut : (u ∩ t).Nonempty
hu : IsPreconnected u
⊢ u ∩ t ⊆ (s ∪ u) ∩ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.inr.intro.intro.intro.intro.refine_4
X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
hSc : ∀ s ∈ S, IsPreconnected s
h :
S.Pairwise fun s t => s.Nonempty → t.Nonempty → ∃ u ⊆ S.sUnion, (s ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
x : X
hx : x ∈ S.sUnion
y : X
hy : y ∈ S.sUnion
s : Set X
hs : s ∈ S
hxs : x ∈ s
t : Set X
ht : t ∈ S
hyt : y ∈ t
hst : s ≠ t
u : Set X
huS : u ⊆ S.sUnion
hsu : (s ∩ u).Nonempty
hut : (u ∩ t).Nonempty
hu : IsPreconnected u
⊢ u ∩ t ⊆ (s ∪ u) ∩ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsPreconnected.iUnion_of_pairwise_exists_isPreconnected | [121, 1] | [128, 33] | apply IsPreconnected.sUnion_of_pairwise_exists_isPreconnected (forall_mem_range.2 hsc) | ι : Type u_1
X : Type u_2
inst✝ : TopologicalSpace X
s : ι → Set X
hsc : ∀ (i : ι), IsPreconnected (s i)
h :
Pairwise fun i j =>
(s i).Nonempty → (s j).Nonempty → ∃ u ⊆ ⋃ i, s i, (s i ∩ u).Nonempty ∧ (u ∩ s j).Nonempty ∧ IsPreconnected u
⊢ IsPreconnected (⋃ i, s i) | ι : Type u_1
X : Type u_2
inst✝ : TopologicalSpace X
s : ι → Set X
hsc : ∀ (i : ι), IsPreconnected (s i)
h :
Pairwise fun i j =>
(s i).Nonempty → (s j).Nonempty → ∃ u ⊆ ⋃ i, s i, (s i ∩ u).Nonempty ∧ (u ∩ s j).Nonempty ∧ IsPreconnected u
⊢ (range fun i => s i).Pairwise fun s_1 t =>
s_1.Nonempty →
t.Nonempty → ∃ u ⊆ (range fun i => s i).sUnion, (s_1 ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
X : Type u_2
inst✝ : TopologicalSpace X
s : ι → Set X
hsc : ∀ (i : ι), IsPreconnected (s i)
h :
Pairwise fun i j =>
(s i).Nonempty → (s j).Nonempty → ∃ u ⊆ ⋃ i, s i, (s i ∩ u).Nonempty ∧ (u ∩ s j).Nonempty ∧ IsPreconnected u
⊢ IsPreconnected (⋃ i, s i)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsPreconnected.iUnion_of_pairwise_exists_isPreconnected | [121, 1] | [128, 33] | rintro _ ⟨i, rfl⟩ _ ⟨j, rfl⟩ hij | ι : Type u_1
X : Type u_2
inst✝ : TopologicalSpace X
s : ι → Set X
hsc : ∀ (i : ι), IsPreconnected (s i)
h :
Pairwise fun i j =>
(s i).Nonempty → (s j).Nonempty → ∃ u ⊆ ⋃ i, s i, (s i ∩ u).Nonempty ∧ (u ∩ s j).Nonempty ∧ IsPreconnected u
⊢ (range fun i => s i).Pairwise fun s_1 t =>
s_1.Nonempty →
t.Nonempty → ∃ u ⊆ (range fun i => s i).sUnion, (s_1 ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u | case intro.intro
ι : Type u_1
X : Type u_2
inst✝ : TopologicalSpace X
s : ι → Set X
hsc : ∀ (i : ι), IsPreconnected (s i)
h :
Pairwise fun i j =>
(s i).Nonempty → (s j).Nonempty → ∃ u ⊆ ⋃ i, s i, (s i ∩ u).Nonempty ∧ (u ∩ s j).Nonempty ∧ IsPreconnected u
i j : ι
hij : (fun i => s i) i ≠ (fun i => s i) j
⊢ ((fun i => s i) i).Nonempty →
((fun i => s i) j).Nonempty →
∃ u ⊆ (range fun i => s i).sUnion,
((fun i => s i) i ∩ u).Nonempty ∧ (u ∩ (fun i => s i) j).Nonempty ∧ IsPreconnected u | Please generate a tactic in lean4 to solve the state.
STATE:
ι : Type u_1
X : Type u_2
inst✝ : TopologicalSpace X
s : ι → Set X
hsc : ∀ (i : ι), IsPreconnected (s i)
h :
Pairwise fun i j =>
(s i).Nonempty → (s j).Nonempty → ∃ u ⊆ ⋃ i, s i, (s i ∩ u).Nonempty ∧ (u ∩ s j).Nonempty ∧ IsPreconnected u
⊢ (range fun i => s i).Pairwise fun s_1 t =>
s_1.Nonempty →
t.Nonempty → ∃ u ⊆ (range fun i => s i).sUnion, (s_1 ∩ u).Nonempty ∧ (u ∩ t).Nonempty ∧ IsPreconnected u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | IsPreconnected.iUnion_of_pairwise_exists_isPreconnected | [121, 1] | [128, 33] | exact h (ne_of_apply_ne s hij) | case intro.intro
ι : Type u_1
X : Type u_2
inst✝ : TopologicalSpace X
s : ι → Set X
hsc : ∀ (i : ι), IsPreconnected (s i)
h :
Pairwise fun i j =>
(s i).Nonempty → (s j).Nonempty → ∃ u ⊆ ⋃ i, s i, (s i ∩ u).Nonempty ∧ (u ∩ s j).Nonempty ∧ IsPreconnected u
i j : ι
hij : (fun i => s i) i ≠ (fun i => s i) j
⊢ ((fun i => s i) i).Nonempty →
((fun i => s i) j).Nonempty →
∃ u ⊆ (range fun i => s i).sUnion,
((fun i => s i) i ∩ u).Nonempty ∧ (u ∩ (fun i => s i) j).Nonempty ∧ IsPreconnected u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
ι : Type u_1
X : Type u_2
inst✝ : TopologicalSpace X
s : ι → Set X
hsc : ∀ (i : ι), IsPreconnected (s i)
h :
Pairwise fun i j =>
(s i).Nonempty → (s j).Nonempty → ∃ u ⊆ ⋃ i, s i, (s i ∩ u).Nonempty ∧ (u ∩ s j).Nonempty ∧ IsPreconnected u
i j : ι
hij : (fun i => s i) i ≠ (fun i => s i) j
⊢ ((fun i => s i) i).Nonempty →
((fun i => s i) j).Nonempty →
∃ u ⊆ (range fun i => s i).sUnion,
((fun i => s i) i ∩ u).Nonempty ∧ (u ∩ (fun i => s i) j).Nonempty ∧ IsPreconnected u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | local_preconnected_nhdsSet | [132, 1] | [144, 57] | rw [← subset_interior_iff_mem_nhdsSet] at st | X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : s ∈ 𝓝ˢ t
⊢ ∃ c, IsOpen c ∧ t ⊆ c ∧ c ⊆ s ∧ IsPreconnected c | X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : t ⊆ interior s
⊢ ∃ c, IsOpen c ∧ t ⊆ c ∧ c ⊆ s ∧ IsPreconnected c | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : s ∈ 𝓝ˢ t
⊢ ∃ c, IsOpen c ∧ t ⊆ c ∧ c ⊆ s ∧ IsPreconnected c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | local_preconnected_nhdsSet | [132, 1] | [144, 57] | have hsub : t ⊆ ⋃ x : t, connectedComponentIn (interior s) x := fun x hx ↦
mem_iUnion.2 ⟨⟨x, hx⟩, mem_connectedComponentIn (st hx)⟩ | X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : t ⊆ interior s
⊢ ∃ c, IsOpen c ∧ t ⊆ c ∧ c ⊆ s ∧ IsPreconnected c | X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : t ⊆ interior s
hsub : t ⊆ ⋃ x, connectedComponentIn (interior s) ↑x
⊢ ∃ c, IsOpen c ∧ t ⊆ c ∧ c ⊆ s ∧ IsPreconnected c | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : t ⊆ interior s
⊢ ∃ c, IsOpen c ∧ t ⊆ c ∧ c ⊆ s ∧ IsPreconnected c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | local_preconnected_nhdsSet | [132, 1] | [144, 57] | refine ⟨_, isOpen_iUnion fun _ ↦ isOpen_interior.connectedComponentIn, hsub,
iUnion_subset fun x ↦ ?_, ?_⟩ | X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : t ⊆ interior s
hsub : t ⊆ ⋃ x, connectedComponentIn (interior s) ↑x
⊢ ∃ c, IsOpen c ∧ t ⊆ c ∧ c ⊆ s ∧ IsPreconnected c | case refine_1
X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : t ⊆ interior s
hsub : t ⊆ ⋃ x, connectedComponentIn (interior s) ↑x
x : ↑t
⊢ connectedComponentIn (interior s) ↑x ⊆ s
case refine_2
X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : t ⊆ interior s
hsub : t ⊆ ⋃ x, connectedComponentIn (interior s) ↑x
⊢ IsPreconnected (⋃ i, connectedComponentIn (interior s) ↑i) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : t ⊆ interior s
hsub : t ⊆ ⋃ x, connectedComponentIn (interior s) ↑x
⊢ ∃ c, IsOpen c ∧ t ⊆ c ∧ c ⊆ s ∧ IsPreconnected c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | local_preconnected_nhdsSet | [132, 1] | [144, 57] | exact (connectedComponentIn_subset _ _).trans interior_subset | case refine_1
X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : t ⊆ interior s
hsub : t ⊆ ⋃ x, connectedComponentIn (interior s) ↑x
x : ↑t
⊢ connectedComponentIn (interior s) ↑x ⊆ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : t ⊆ interior s
hsub : t ⊆ ⋃ x, connectedComponentIn (interior s) ↑x
x : ↑t
⊢ connectedComponentIn (interior s) ↑x ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | local_preconnected_nhdsSet | [132, 1] | [144, 57] | apply IsPreconnected.iUnion_of_pairwise_exists_isPreconnected | case refine_2
X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : t ⊆ interior s
hsub : t ⊆ ⋃ x, connectedComponentIn (interior s) ↑x
⊢ IsPreconnected (⋃ i, connectedComponentIn (interior s) ↑i) | case refine_2.hsc
X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : t ⊆ interior s
hsub : t ⊆ ⋃ x, connectedComponentIn (interior s) ↑x
⊢ ∀ (i : ↑t), IsPreconnected (connectedComponentIn (interior s) ↑i)
case refine_2.h
X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : t ⊆ interior s
hsub : t ⊆ ⋃ x, connectedComponentIn (interior s) ↑x
⊢ Pairwise fun i j =>
(connectedComponentIn (interior s) ↑i).Nonempty →
(connectedComponentIn (interior s) ↑j).Nonempty →
∃ u ⊆ ⋃ i, connectedComponentIn (interior s) ↑i,
(connectedComponentIn (interior s) ↑i ∩ u).Nonempty ∧
(u ∩ connectedComponentIn (interior s) ↑j).Nonempty ∧ IsPreconnected u | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : t ⊆ interior s
hsub : t ⊆ ⋃ x, connectedComponentIn (interior s) ↑x
⊢ IsPreconnected (⋃ i, connectedComponentIn (interior s) ↑i)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | local_preconnected_nhdsSet | [132, 1] | [144, 57] | exact fun _ ↦ isPreconnected_connectedComponentIn | case refine_2.hsc
X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : t ⊆ interior s
hsub : t ⊆ ⋃ x, connectedComponentIn (interior s) ↑x
⊢ ∀ (i : ↑t), IsPreconnected (connectedComponentIn (interior s) ↑i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.hsc
X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : t ⊆ interior s
hsub : t ⊆ ⋃ x, connectedComponentIn (interior s) ↑x
⊢ ∀ (i : ↑t), IsPreconnected (connectedComponentIn (interior s) ↑i)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | local_preconnected_nhdsSet | [132, 1] | [144, 57] | exact fun x y _ _ _ ↦ ⟨t, hsub, ⟨x, mem_connectedComponentIn (st x.2), x.2⟩,
⟨y, y.2, mem_connectedComponentIn (st y.2)⟩, tc⟩ | case refine_2.h
X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : t ⊆ interior s
hsub : t ⊆ ⋃ x, connectedComponentIn (interior s) ↑x
⊢ Pairwise fun i j =>
(connectedComponentIn (interior s) ↑i).Nonempty →
(connectedComponentIn (interior s) ↑j).Nonempty →
∃ u ⊆ ⋃ i, connectedComponentIn (interior s) ↑i,
(connectedComponentIn (interior s) ↑i ∩ u).Nonempty ∧
(u ∩ connectedComponentIn (interior s) ↑j).Nonempty ∧ IsPreconnected u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.h
X : Type
inst✝ : TopologicalSpace X
lc : LocallyConnectedSpace X
s t : Set X
tc : IsPreconnected t
st : t ⊆ interior s
hsub : t ⊆ ⋃ x, connectedComponentIn (interior s) ↑x
⊢ Pairwise fun i j =>
(connectedComponentIn (interior s) ↑i).Nonempty →
(connectedComponentIn (interior s) ↑j).Nonempty →
∃ u ⊆ ⋃ i, connectedComponentIn (interior s) ↑i,
(connectedComponentIn (interior s) ↑i ∩ u).Nonempty ∧
(u ∩ connectedComponentIn (interior s) ↑j).Nonempty ∧ IsPreconnected u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | Prod.frequently | [170, 1] | [173, 6] | simp only [frequently_iff_neBot, ← prod_neBot, ← prod_inf_prod, prod_principal_principal] | A B : Type
f : Filter A
g : Filter B
p : A → Prop
q : B → Prop
⊢ (∃ᶠ (x : A × B) in f ×ˢ g, p x.1 ∧ q x.2) ↔ (∃ᶠ (a : A) in f, p a) ∧ ∃ᶠ (b : B) in g, q b | A B : Type
f : Filter A
g : Filter B
p : A → Prop
q : B → Prop
⊢ (f ×ˢ g ⊓ 𝓟 {x | p x.1 ∧ q x.2}).NeBot ↔ (f ×ˢ g ⊓ 𝓟 ({a | p a} ×ˢ {b | q b})).NeBot | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : Filter A
g : Filter B
p : A → Prop
q : B → Prop
⊢ (∃ᶠ (x : A × B) in f ×ˢ g, p x.1 ∧ q x.2) ↔ (∃ᶠ (a : A) in f, p a) ∧ ∃ᶠ (b : B) in g, q b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | Prod.frequently | [170, 1] | [173, 6] | rfl | A B : Type
f : Filter A
g : Filter B
p : A → Prop
q : B → Prop
⊢ (f ×ˢ g ⊓ 𝓟 {x | p x.1 ∧ q x.2}).NeBot ↔ (f ×ˢ g ⊓ 𝓟 ({a | p a} ×ˢ {b | q b})).NeBot | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : Filter A
g : Filter B
p : A → Prop
q : B → Prop
⊢ (f ×ˢ g ⊓ 𝓟 {x | p x.1 ∧ q x.2}).NeBot ↔ (f ×ˢ g ⊓ 𝓟 ({a | p a} ×ˢ {b | q b})).NeBot
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | MapClusterPt.prod | [176, 1] | [184, 67] | rw [mapClusterPt_iff] at fa ⊢ | A B C : Type
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace C
f : A → B
g : A → C
a : Filter A
b : B
c : C
fa : MapClusterPt b a f
ga : Tendsto g a (𝓝 c)
⊢ MapClusterPt (b, c) a fun x => (f x, g x) | A B C : Type
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace C
f : A → B
g : A → C
a : Filter A
b : B
c : C
fa : ∀ s ∈ 𝓝 b, ∃ᶠ (a : A) in a, f a ∈ s
ga : Tendsto g a (𝓝 c)
⊢ ∀ s ∈ 𝓝 (b, c), ∃ᶠ (a : A) in a, (f a, g a) ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace C
f : A → B
g : A → C
a : Filter A
b : B
c : C
fa : MapClusterPt b a f
ga : Tendsto g a (𝓝 c)
⊢ MapClusterPt (b, c) a fun x => (f x, g x)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | MapClusterPt.prod | [176, 1] | [184, 67] | intro s n | A B C : Type
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace C
f : A → B
g : A → C
a : Filter A
b : B
c : C
fa : ∀ s ∈ 𝓝 b, ∃ᶠ (a : A) in a, f a ∈ s
ga : Tendsto g a (𝓝 c)
⊢ ∀ s ∈ 𝓝 (b, c), ∃ᶠ (a : A) in a, (f a, g a) ∈ s | A B C : Type
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace C
f : A → B
g : A → C
a : Filter A
b : B
c : C
fa : ∀ s ∈ 𝓝 b, ∃ᶠ (a : A) in a, f a ∈ s
ga : Tendsto g a (𝓝 c)
s : Set (B × C)
n : s ∈ 𝓝 (b, c)
⊢ ∃ᶠ (a : A) in a, (f a, g a) ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace C
f : A → B
g : A → C
a : Filter A
b : B
c : C
fa : ∀ s ∈ 𝓝 b, ∃ᶠ (a : A) in a, f a ∈ s
ga : Tendsto g a (𝓝 c)
⊢ ∀ s ∈ 𝓝 (b, c), ∃ᶠ (a : A) in a, (f a, g a) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | MapClusterPt.prod | [176, 1] | [184, 67] | rcases mem_nhds_prod_iff.mp n with ⟨u, un, v, vn, sub⟩ | A B C : Type
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace C
f : A → B
g : A → C
a : Filter A
b : B
c : C
fa : ∀ s ∈ 𝓝 b, ∃ᶠ (a : A) in a, f a ∈ s
ga : Tendsto g a (𝓝 c)
s : Set (B × C)
n : s ∈ 𝓝 (b, c)
⊢ ∃ᶠ (a : A) in a, (f a, g a) ∈ s | case intro.intro.intro.intro
A B C : Type
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace C
f : A → B
g : A → C
a : Filter A
b : B
c : C
fa : ∀ s ∈ 𝓝 b, ∃ᶠ (a : A) in a, f a ∈ s
ga : Tendsto g a (𝓝 c)
s : Set (B × C)
n : s ∈ 𝓝 (b, c)
u : Set B
un : u ∈ 𝓝 b
v : Set C
vn : v ∈ 𝓝 c
sub : u ×ˢ v ⊆ s
⊢ ∃ᶠ (a : A) in a, (f a, g a) ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace C
f : A → B
g : A → C
a : Filter A
b : B
c : C
fa : ∀ s ∈ 𝓝 b, ∃ᶠ (a : A) in a, f a ∈ s
ga : Tendsto g a (𝓝 c)
s : Set (B × C)
n : s ∈ 𝓝 (b, c)
⊢ ∃ᶠ (a : A) in a, (f a, g a) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | MapClusterPt.prod | [176, 1] | [184, 67] | apply (fa _ un).mp | case intro.intro.intro.intro
A B C : Type
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace C
f : A → B
g : A → C
a : Filter A
b : B
c : C
fa : ∀ s ∈ 𝓝 b, ∃ᶠ (a : A) in a, f a ∈ s
ga : Tendsto g a (𝓝 c)
s : Set (B × C)
n : s ∈ 𝓝 (b, c)
u : Set B
un : u ∈ 𝓝 b
v : Set C
vn : v ∈ 𝓝 c
sub : u ×ˢ v ⊆ s
⊢ ∃ᶠ (a : A) in a, (f a, g a) ∈ s | case intro.intro.intro.intro
A B C : Type
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace C
f : A → B
g : A → C
a : Filter A
b : B
c : C
fa : ∀ s ∈ 𝓝 b, ∃ᶠ (a : A) in a, f a ∈ s
ga : Tendsto g a (𝓝 c)
s : Set (B × C)
n : s ∈ 𝓝 (b, c)
u : Set B
un : u ∈ 𝓝 b
v : Set C
vn : v ∈ 𝓝 c
sub : u ×ˢ v ⊆ s
⊢ ∀ᶠ (x : A) in a, f x ∈ u → (f x, g x) ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A B C : Type
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace C
f : A → B
g : A → C
a : Filter A
b : B
c : C
fa : ∀ s ∈ 𝓝 b, ∃ᶠ (a : A) in a, f a ∈ s
ga : Tendsto g a (𝓝 c)
s : Set (B × C)
n : s ∈ 𝓝 (b, c)
u : Set B
un : u ∈ 𝓝 b
v : Set C
vn : v ∈ 𝓝 c
sub : u ×ˢ v ⊆ s
⊢ ∃ᶠ (a : A) in a, (f a, g a) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | MapClusterPt.prod | [176, 1] | [184, 67] | apply (Filter.tendsto_iff_forall_eventually_mem.mp ga v vn).mp | case intro.intro.intro.intro
A B C : Type
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace C
f : A → B
g : A → C
a : Filter A
b : B
c : C
fa : ∀ s ∈ 𝓝 b, ∃ᶠ (a : A) in a, f a ∈ s
ga : Tendsto g a (𝓝 c)
s : Set (B × C)
n : s ∈ 𝓝 (b, c)
u : Set B
un : u ∈ 𝓝 b
v : Set C
vn : v ∈ 𝓝 c
sub : u ×ˢ v ⊆ s
⊢ ∀ᶠ (x : A) in a, f x ∈ u → (f x, g x) ∈ s | case intro.intro.intro.intro
A B C : Type
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace C
f : A → B
g : A → C
a : Filter A
b : B
c : C
fa : ∀ s ∈ 𝓝 b, ∃ᶠ (a : A) in a, f a ∈ s
ga : Tendsto g a (𝓝 c)
s : Set (B × C)
n : s ∈ 𝓝 (b, c)
u : Set B
un : u ∈ 𝓝 b
v : Set C
vn : v ∈ 𝓝 c
sub : u ×ˢ v ⊆ s
⊢ ∀ᶠ (x : A) in a, g x ∈ v → f x ∈ u → (f x, g x) ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A B C : Type
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace C
f : A → B
g : A → C
a : Filter A
b : B
c : C
fa : ∀ s ∈ 𝓝 b, ∃ᶠ (a : A) in a, f a ∈ s
ga : Tendsto g a (𝓝 c)
s : Set (B × C)
n : s ∈ 𝓝 (b, c)
u : Set B
un : u ∈ 𝓝 b
v : Set C
vn : v ∈ 𝓝 c
sub : u ×ˢ v ⊆ s
⊢ ∀ᶠ (x : A) in a, f x ∈ u → (f x, g x) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Topology.lean | MapClusterPt.prod | [176, 1] | [184, 67] | exact eventually_of_forall fun x gv fu ↦ sub (mk_mem_prod fu gv) | case intro.intro.intro.intro
A B C : Type
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace C
f : A → B
g : A → C
a : Filter A
b : B
c : C
fa : ∀ s ∈ 𝓝 b, ∃ᶠ (a : A) in a, f a ∈ s
ga : Tendsto g a (𝓝 c)
s : Set (B × C)
n : s ∈ 𝓝 (b, c)
u : Set B
un : u ∈ 𝓝 b
v : Set C
vn : v ∈ 𝓝 c
sub : u ×ˢ v ⊆ s
⊢ ∀ᶠ (x : A) in a, g x ∈ v → f x ∈ u → (f x, g x) ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
A B C : Type
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace C
f : A → B
g : A → C
a : Filter A
b : B
c : C
fa : ∀ s ∈ 𝓝 b, ∃ᶠ (a : A) in a, f a ∈ s
ga : Tendsto g a (𝓝 c)
s : Set (B × C)
n : s ∈ 𝓝 (b, c)
u : Set B
un : u ∈ 𝓝 b
v : Set C
vn : v ∈ 𝓝 c
sub : u ×ˢ v ⊆ s
⊢ ∀ᶠ (x : A) in a, g x ∈ v → f x ∈ u → (f x, g x) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | rw [push, pop] | N : Finset ℕ
⊢ push (pop N) = insert 0 N | N : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) (Finset.image (fun n => n - 1) (N.erase 0))) = insert 0 N | Please generate a tactic in lean4 to solve the state.
STATE:
N : Finset ℕ
⊢ push (pop N) = insert 0 N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | apply Finset.ext | N : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) (Finset.image (fun n => n - 1) (N.erase 0))) = insert 0 N | case a
N : Finset ℕ
⊢ ∀ (a : ℕ), a ∈ insert 0 (Finset.image (fun n => n + 1) (Finset.image (fun n => n - 1) (N.erase 0))) ↔ a ∈ insert 0 N | Please generate a tactic in lean4 to solve the state.
STATE:
N : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) (Finset.image (fun n => n - 1) (N.erase 0))) = insert 0 N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | simp | case a
N : Finset ℕ
⊢ ∀ (a : ℕ), a ∈ insert 0 (Finset.image (fun n => n + 1) (Finset.image (fun n => n - 1) (N.erase 0))) ↔ a ∈ insert 0 N | case a
N : Finset ℕ
⊢ ∀ (a : ℕ), (a = 0 ∨ ∃ a_1, (¬a_1 = 0 ∧ a_1 ∈ N) ∧ a_1 - 1 + 1 = a) ↔ a = 0 ∨ a ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case a
N : Finset ℕ
⊢ ∀ (a : ℕ), a ∈ insert 0 (Finset.image (fun n => n + 1) (Finset.image (fun n => n - 1) (N.erase 0))) ↔ a ∈ insert 0 N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | intro n | case a
N : Finset ℕ
⊢ ∀ (a : ℕ), (a = 0 ∨ ∃ a_1, (¬a_1 = 0 ∧ a_1 ∈ N) ∧ a_1 - 1 + 1 = a) ↔ a = 0 ∨ a ∈ N | case a
N : Finset ℕ
n : ℕ
⊢ (n = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n = 0 ∨ n ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case a
N : Finset ℕ
⊢ ∀ (a : ℕ), (a = 0 ∨ ∃ a_1, (¬a_1 = 0 ∧ a_1 ∈ N) ∧ a_1 - 1 + 1 = a) ↔ a = 0 ∨ a ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | by_cases n0 : n = 0 | case a
N : Finset ℕ
n : ℕ
⊢ (n = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n = 0 ∨ n ∈ N | case pos
N : Finset ℕ
n : ℕ
n0 : n = 0
⊢ (n = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n = 0 ∨ n ∈ N
case neg
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ (n = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n = 0 ∨ n ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case a
N : Finset ℕ
n : ℕ
⊢ (n = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n = 0 ∨ n ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | simp_rw [or_iff_right n0] | case neg
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ (n = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n = 0 ∨ n ∈ N | case neg
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ (∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ (n = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n = 0 ∨ n ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | constructor | case neg
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ (∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n ∈ N | case neg.mp
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ (∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) → n ∈ N
case neg.mpr
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ n ∈ N → ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ (∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | rw [n0] | case pos
N : Finset ℕ
n : ℕ
n0 : n = 0
⊢ (n = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n = 0 ∨ n ∈ N | case pos
N : Finset ℕ
n : ℕ
n0 : n = 0
⊢ (0 = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = 0) ↔ 0 = 0 ∨ 0 ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
N : Finset ℕ
n : ℕ
n0 : n = 0
⊢ (n = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) ↔ n = 0 ∨ n ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | simp | case pos
N : Finset ℕ
n : ℕ
n0 : n = 0
⊢ (0 = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = 0) ↔ 0 = 0 ∨ 0 ∈ N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
N : Finset ℕ
n : ℕ
n0 : n = 0
⊢ (0 = 0 ∨ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = 0) ↔ 0 = 0 ∨ 0 ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | intro h | case neg.mp
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ (∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) → n ∈ N | case neg.mp
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n
⊢ n ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.mp
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ (∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n) → n ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | rcases h with ⟨x, ⟨x0, xN⟩, xn⟩ | case neg.mp
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n
⊢ n ∈ N | case neg.mp.intro.intro.intro
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
x : ℕ
xn : x - 1 + 1 = n
x0 : ¬x = 0
xN : x ∈ N
⊢ n ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.mp
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n
⊢ n ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | rw [Nat.sub_add_cancel (Nat.one_le_iff_ne_zero.mpr x0)] at xn | case neg.mp.intro.intro.intro
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
x : ℕ
xn : x - 1 + 1 = n
x0 : ¬x = 0
xN : x ∈ N
⊢ n ∈ N | case neg.mp.intro.intro.intro
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
x : ℕ
xn : x = n
x0 : ¬x = 0
xN : x ∈ N
⊢ n ∈ N | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.mp.intro.intro.intro
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
x : ℕ
xn : x - 1 + 1 = n
x0 : ¬x = 0
xN : x ∈ N
⊢ n ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | rwa [←xn] | case neg.mp.intro.intro.intro
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
x : ℕ
xn : x = n
x0 : ¬x = 0
xN : x ∈ N
⊢ n ∈ N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.mp.intro.intro.intro
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
x : ℕ
xn : x = n
x0 : ¬x = 0
xN : x ∈ N
⊢ n ∈ N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | intro h | case neg.mpr
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ n ∈ N → ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n | case neg.mpr
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : n ∈ N
⊢ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.mpr
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
⊢ n ∈ N → ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | exists n | case neg.mpr
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : n ∈ N
⊢ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n | case neg.mpr
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : n ∈ N
⊢ (¬n = 0 ∧ n ∈ N) ∧ n - 1 + 1 = n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.mpr
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : n ∈ N
⊢ ∃ a, (¬a = 0 ∧ a ∈ N) ∧ a - 1 + 1 = n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | use ⟨n0,h⟩ | case neg.mpr
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : n ∈ N
⊢ (¬n = 0 ∧ n ∈ N) ∧ n - 1 + 1 = n | case right
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : n ∈ N
⊢ n - 1 + 1 = n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.mpr
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : n ∈ N
⊢ (¬n = 0 ∧ n ∈ N) ∧ n - 1 + 1 = n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_pop | [26, 1] | [35, 61] | exact Nat.sub_add_cancel (Nat.one_le_iff_ne_zero.mpr n0) | case right
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : n ∈ N
⊢ n - 1 + 1 = n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
N : Finset ℕ
n : ℕ
n0 : ¬n = 0
h : n ∈ N
⊢ n - 1 + 1 = n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | simp | A B : Finset ℕ
⊢ push A ≤ push B ↔ A ≤ B | A B : Finset ℕ
⊢ push A ⊆ push B ↔ A ⊆ B | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
⊢ push A ≤ push B ↔ A ≤ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | rw [push] | A B : Finset ℕ
⊢ push A ⊆ push B ↔ A ⊆ B | A B : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ push B ↔ A ⊆ B | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
⊢ push A ⊆ push B ↔ A ⊆ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | rw [push] | A B : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ push B ↔ A ⊆ B | A B : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) ↔ A ⊆ B | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ push B ↔ A ⊆ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | constructor | A B : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) ↔ A ⊆ B | case mp
A B : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) → A ⊆ B
case mpr
A B : Finset ℕ
⊢ A ⊆ B → insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) ↔ A ⊆ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | intro AB | case mp
A B : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) → A ⊆ B | case mp
A B : Finset ℕ
AB : insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B)
⊢ A ⊆ B | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A B : Finset ℕ
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) → A ⊆ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | rw [Finset.subset_iff] at AB ⊢ | case mp
A B : Finset ℕ
AB : insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B)
⊢ A ⊆ B | case mp
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
⊢ ∀ ⦃x : ℕ⦄, x ∈ A → x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A B : Finset ℕ
AB : insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B)
⊢ A ⊆ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | intro x xA | case mp
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
⊢ ∀ ⦃x : ℕ⦄, x ∈ A → x ∈ B | case mp
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
x : ℕ
xA : x ∈ A
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
⊢ ∀ ⦃x : ℕ⦄, x ∈ A → x ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | have h : x + 1 ∈ insert 0 (Finset.image (fun n : ℕ ↦ n + 1) A) := by simpa | case mp
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
x : ℕ
xA : x ∈ A
⊢ x ∈ B | case mp
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
x : ℕ
xA : x ∈ A
h : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
x : ℕ
xA : x ∈ A
⊢ x ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | specialize AB h | case mp
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
x : ℕ
xA : x ∈ A
h : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
⊢ x ∈ B | case mp
A B : Finset ℕ
x : ℕ
xA : x ∈ A
h : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
AB : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) B)
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
x : ℕ
xA : x ∈ A
h : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
⊢ x ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | simp at AB | case mp
A B : Finset ℕ
x : ℕ
xA : x ∈ A
h : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
AB : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) B)
⊢ x ∈ B | case mp
A B : Finset ℕ
x : ℕ
xA : x ∈ A
h : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
AB : x ∈ B
⊢ x ∈ B | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A B : Finset ℕ
x : ℕ
xA : x ∈ A
h : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
AB : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) B)
⊢ x ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | assumption | case mp
A B : Finset ℕ
x : ℕ
xA : x ∈ A
h : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
AB : x ∈ B
⊢ x ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A B : Finset ℕ
x : ℕ
xA : x ∈ A
h : x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
AB : x ∈ B
⊢ x ∈ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | simpa | A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
x : ℕ
xA : x ∈ A
⊢ x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Finset ℕ
AB : ∀ ⦃x : ℕ⦄, x ∈ insert 0 (Finset.image (fun n => n + 1) A) → x ∈ insert 0 (Finset.image (fun n => n + 1) B)
x : ℕ
xA : x ∈ A
⊢ x + 1 ∈ insert 0 (Finset.image (fun n => n + 1) A)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | intro AB | case mpr
A B : Finset ℕ
⊢ A ⊆ B → insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) | case mpr
A B : Finset ℕ
AB : A ⊆ B
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A B : Finset ℕ
⊢ A ⊆ B → insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | apply Finset.insert_subset_insert | case mpr
A B : Finset ℕ
AB : A ⊆ B
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B) | case mpr.h
A B : Finset ℕ
AB : A ⊆ B
⊢ Finset.image (fun n => n + 1) A ⊆ Finset.image (fun n => n + 1) B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A B : Finset ℕ
AB : A ⊆ B
⊢ insert 0 (Finset.image (fun n => n + 1) A) ⊆ insert 0 (Finset.image (fun n => n + 1) B)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | apply Finset.image_mono | case mpr.h
A B : Finset ℕ
AB : A ⊆ B
⊢ Finset.image (fun n => n + 1) A ⊆ Finset.image (fun n => n + 1) B | case mpr.h.a
A B : Finset ℕ
AB : A ⊆ B
⊢ A ≤ B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.h
A B : Finset ℕ
AB : A ⊆ B
⊢ Finset.image (fun n => n + 1) A ⊆ Finset.image (fun n => n + 1) B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_le_push | [38, 1] | [44, 85] | assumption | case mpr.h.a
A B : Finset ℕ
AB : A ⊆ B
⊢ A ≤ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.h.a
A B : Finset ℕ
AB : A ⊆ B
⊢ A ≤ B
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_sum | [47, 1] | [49, 23] | rw [push] | X : Type
inst✝ : AddCommGroup X
a : X
f : ℕ → X
N : Finset ℕ
⊢ a + N.sum f = (push N).sum (cons a f) | X : Type
inst✝ : AddCommGroup X
a : X
f : ℕ → X
N : Finset ℕ
⊢ a + N.sum f = (insert 0 (Finset.image (fun n => n + 1) N)).sum (cons a f) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : AddCommGroup X
a : X
f : ℕ → X
N : Finset ℕ
⊢ a + N.sum f = (push N).sum (cons a f)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_sum | [47, 1] | [49, 23] | simp | X : Type
inst✝ : AddCommGroup X
a : X
f : ℕ → X
N : Finset ℕ
⊢ a + N.sum f = (insert 0 (Finset.image (fun n => n + 1) N)).sum (cons a f) | X : Type
inst✝ : AddCommGroup X
a : X
f : ℕ → X
N : Finset ℕ
⊢ a + N.sum f = cons a f 0 + N.sum fun x => cons a f (x + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : AddCommGroup X
a : X
f : ℕ → X
N : Finset ℕ
⊢ a + N.sum f = (insert 0 (Finset.image (fun n => n + 1) N)).sum (cons a f)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_sum | [47, 1] | [49, 23] | rfl | X : Type
inst✝ : AddCommGroup X
a : X
f : ℕ → X
N : Finset ℕ
⊢ a + N.sum f = cons a f 0 + N.sum fun x => cons a f (x + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : AddCommGroup X
a : X
f : ℕ → X
N : Finset ℕ
⊢ a + N.sum f = cons a f 0 + N.sum fun x => cons a f (x + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Finset.lean | push_prod | [52, 1] | [53, 23] | rw [push] | a : ℂ
f : ℕ → ℂ
N : Finset ℕ
⊢ a * N.prod f = (push N).prod (cons a f) | a : ℂ
f : ℕ → ℂ
N : Finset ℕ
⊢ a * N.prod f = (insert 0 (Finset.image (fun n => n + 1) N)).prod (cons a f) | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℂ
f : ℕ → ℂ
N : Finset ℕ
⊢ a * N.prod f = (push N).prod (cons a f)
TACTIC:
|
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