Datasets:
pkuAI4M
/
lean_github

Dataset card Data Studio Files
xet
Community
1
url
stringclasses
147 values
commit
stringclasses
147 values
file_path
stringlengths
7
101
full_name
stringlengths
1
94
start
stringlengths
6
10
end
stringlengths
6
11
tactic
stringlengths
1
11.2k
state_before
stringlengths
3
2.09M
state_after
stringlengths
6
2.09M
input
stringlengths
73
2.09M
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
rw [← hy, duals]
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k hy : TopologicalSpace.denseSeq E n = y ⊢ duals n = dualVector y
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k hy : TopologicalSpace.denseSeq E n = y ⊢ duals n = dualVector y TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
have k11 : (k : ℝ≥0) = k * 1 * 1 := by norm_num
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s ⊢ LipschitzWith k fun x => g (Complex.abs ((dualVector y) x))
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s k11 : k = k * 1 * 1 ⊢ LipschitzWith k fun x => g (Complex.abs ((dualVector y) x))
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s ⊢ LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
rw [k11]
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s k11 : k = k * 1 * 1 ⊢ LipschitzWith k fun x => g (Complex.abs ((dualVector y) x))
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s k11 : k = k * 1 * 1 ⊢ LipschitzWith (k * 1 * 1) fun x => g (Complex.abs ((dualVector y) x))
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s k11 : k = k * 1 * 1 ⊢ LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
simp_rw [←Complex.norm_eq_abs]
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s k11 : k = k * 1 * 1 ⊢ LipschitzWith (k * 1 * 1) fun x => g (Complex.abs ((dualVector y) x))
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s k11 : k = k * 1 * 1 ⊢ LipschitzWith (k * 1 * 1) fun x => g ‖(dualVector y) x‖
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s k11 : k = k * 1 * 1 ⊢ LipschitzWith (k * 1 * 1) fun x => g (Complex.abs ((dualVector y) x)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
apply (gk.comp lipschitzWith_one_norm).comp
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s k11 : k = k * 1 * 1 ⊢ LipschitzWith (k * 1 * 1) fun x => g ‖(dualVector y) x‖
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s k11 : k = k * 1 * 1 ⊢ LipschitzWith 1 fun x => (dualVector y) x
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s k11 : k = k * 1 * 1 ⊢ LipschitzWith (k * 1 * 1) fun x => g ‖(dualVector y) x‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
exact (dualVector y).lipschitz.weaken (dualVector_nnnorm y)
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s k11 : k = k * 1 * 1 ⊢ LipschitzWith 1 fun x => (dualVector y) x
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s k11 : k = k * 1 * 1 ⊢ LipschitzWith 1 fun x => (dualVector y) x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
norm_num
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s ⊢ k = k * 1 * 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s ⊢ k = k * 1 * 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
bound
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ g ‖y‖ + ↑k * 1 * dist x y ≤ g ‖y‖ + ↑k * 1 * (e / 2 / ↑k)
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ g ‖y‖ + ↑k * 1 * dist x y ≤ g ‖y‖ + ↑k * 1 * (e / 2 / ↑k) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
ring
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ g ‖y‖ + ↑k * 1 * (e / 2 / ↑k) = g ‖y‖ + ↑k / ↑k * e / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ g ‖y‖ + ↑k * 1 * (e / 2 / ↑k) = g ‖y‖ + ↑k / ↑k * e / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
bound
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ g ‖y‖ + ↑k / ↑k * e / 2 ≤ g ‖y‖ + 1 * e / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ g ‖y‖ + ↑k / ↑k * e / 2 ≤ g ‖y‖ + 1 * e / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
simp only [one_mul]
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ g ‖y‖ + 1 * e / 2 = g ‖y‖ + e / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ g ‖y‖ + 1 * e / 2 = g ‖y‖ + e / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
simp only [dualVector_apply, Complex.abs_ofReal, abs_norm]
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ g ‖y‖ + e / 2 = g (Complex.abs ((dualVector y) y)) + e / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ g ‖y‖ + e / 2 = g (Complex.abs ((dualVector y) y)) + e / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
bound [gk'.le]
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ g (Complex.abs ((dualVector y) y)) + e / 2 ≤ g (Complex.abs ((dualVector y) x)) + ↑k * dist y x + e / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ g (Complex.abs ((dualVector y) y)) + e / 2 ≤ g (Complex.abs ((dualVector y) x)) + ↑k * dist y x + e / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
linarith
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ g (Complex.abs ((dualVector y) x)) + ↑k * dist y x + e / 2 ≤ s + ↑k * dist y x + e / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ g (Complex.abs ((dualVector y) x)) + ↑k * dist y x + e / 2 ≤ s + ↑k * dist y x + e / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
rw [dist_comm]
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ s + ↑k * dist y x + e / 2 = s + ↑k * dist x y + e / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ s + ↑k * dist y x + e / 2 = s + ↑k * dist x y + e / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
bound
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ s + ↑k * dist x y + e / 2 ≤ s + ↑k * (e / 2 / ↑k) + e / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ s + ↑k * dist x y + e / 2 ≤ s + ↑k * (e / 2 / ↑k) + e / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
ring_nf
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ s + ↑k * (e / 2 / ↑k) + e / 2 = s + ↑k / ↑k * e / 2 + e / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ s + ↑k * (e / 2 / ↑k) + e / 2 = s + ↑k / ↑k * e / 2 + e / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
bound
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ s + ↑k / ↑k * e / 2 + e / 2 ≤ s + 1 * e / 2 + e / 2
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ s + ↑k / ↑k * e / 2 + e / 2 ≤ s + 1 * e / 2 + e / 2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
ring_nf
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ s + 1 * e / 2 + e / 2 = s + e
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k e : ℝ ep : 0 < e n : ℕ y : E nx : dist x y < e / 2 / ↑k s : ℝ h : g (Complex.abs ((dualVector y) x)) ≤ s gk' : LipschitzWith k fun x => g (Complex.abs ((dualVector y) x)) ⊢ s + 1 * e / 2 + e / 2 = s + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
apply ciSup_le
case neg.a G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k ⊢ ⨆ n, g ‖(duals n) x‖ ≤ g ‖x‖
case neg.a.H G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k ⊢ ∀ (x_1 : ℕ), g ‖(duals x_1) x‖ ≤ g ‖x‖
Please generate a tactic in lean4 to solve the state. STATE: case neg.a G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k ⊢ ⨆ n, g ‖(duals n) x‖ ≤ g ‖x‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
intro n
case neg.a.H G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k ⊢ ∀ (x_1 : ℕ), g ‖(duals x_1) x‖ ≤ g ‖x‖
case neg.a.H G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k n : ℕ ⊢ g ‖(duals n) x‖ ≤ g ‖x‖
Please generate a tactic in lean4 to solve the state. STATE: case neg.a.H G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k ⊢ ∀ (x_1 : ℕ), g ‖(duals x_1) x‖ ≤ g ‖x‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
apply gm
case neg.a.H G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k n : ℕ ⊢ g ‖(duals n) x‖ ≤ g ‖x‖
case neg.a.H.a G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k n : ℕ ⊢ ‖(duals n) x‖ ≤ ‖x‖
Please generate a tactic in lean4 to solve the state. STATE: case neg.a.H G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k n : ℕ ⊢ g ‖(duals n) x‖ ≤ g ‖x‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
simp only [Complex.norm_eq_abs]
case neg.a.H.a G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k n : ℕ ⊢ ‖(duals n) x‖ ≤ ‖x‖
case neg.a.H.a G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k n : ℕ ⊢ Complex.abs ((duals n) x) ≤ ‖x‖
Please generate a tactic in lean4 to solve the state. STATE: case neg.a.H.a G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k n : ℕ ⊢ ‖(duals n) x‖ ≤ ‖x‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
apply dualVector_le
case neg.a.H.a G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k n : ℕ ⊢ Complex.abs ((duals n) x) ≤ ‖x‖
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg.a.H.a G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E k0 : ¬k = 0 kp : 0 < ↑k n : ℕ ⊢ Complex.abs ((duals n) x) ≤ ‖x‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_iSup
[115, 1]
[117, 43]
have h := norm_eq_duals_supr' (@monotone_id ℝ _) LipschitzWith.id x
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E x : E ⊢ ‖x‖ = ⨆ n, ‖(duals n) x‖
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E x : E h : id ‖x‖ = ⨆ n, id ‖(duals n) x‖ ⊢ ‖x‖ = ⨆ n, ‖(duals n) x‖
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E x : E ⊢ ‖x‖ = ⨆ n, ‖(duals n) x‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_iSup
[115, 1]
[117, 43]
simpa only [Complex.norm_eq_abs] using h
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E x : E h : id ‖x‖ = ⨆ n, id ‖(duals n) x‖ ⊢ ‖x‖ = ⨆ n, ‖(duals n) x‖
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E x : E h : id ‖x‖ = ⨆ n, id ‖(duals n) x‖ ⊢ ‖x‖ = ⨆ n, ‖(duals n) x‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
rw [Metric.tendsto_atTop]
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) ⊢ Filter.Tendsto (fun n => (partialSups s) n) atTop (𝓝 (⨆ n, s n))
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) ⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist ((partialSups s) n) (⨆ n, s n) < ε
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) ⊢ Filter.Tendsto (fun n => (partialSups s) n) atTop (𝓝 (⨆ n, s n)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
intro e ep
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) ⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist ((partialSups s) n) (⨆ n, s n) < ε
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 ⊢ ∃ N, ∀ n ≥ N, dist ((partialSups s) n) (⨆ n, s n) < e
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) ⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist ((partialSups s) n) (⨆ n, s n) < ε TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
generalize hb : (⨆ n, s n) - e = b
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 ⊢ ∃ N, ∀ n ≥ N, dist ((partialSups s) n) (⨆ n, s n) < e
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b ⊢ ∃ N, ∀ n ≥ N, dist ((partialSups s) n) (⨆ n, s n) < e
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 ⊢ ∃ N, ∀ n ≥ N, dist ((partialSups s) n) (⨆ n, s n) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
have bs : b < ⨆ n, s n := by rw [← hb]; exact sub_lt_self _ (by linarith)
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b ⊢ ∃ N, ∀ n ≥ N, dist ((partialSups s) n) (⨆ n, s n) < e
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n ⊢ ∃ N, ∀ n ≥ N, dist ((partialSups s) n) (⨆ n, s n) < e
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b ⊢ ∃ N, ∀ n ≥ N, dist ((partialSups s) n) (⨆ n, s n) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
rcases exists_lt_of_lt_ciSup bs with ⟨N, sN⟩
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n ⊢ ∃ N, ∀ n ≥ N, dist ((partialSups s) n) (⨆ n, s n) < e
case intro G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N ⊢ ∃ N, ∀ n ≥ N, dist ((partialSups s) n) (⨆ n, s n) < e
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n ⊢ ∃ N, ∀ n ≥ N, dist ((partialSups s) n) (⨆ n, s n) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
use N
case intro G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N ⊢ ∃ N, ∀ n ≥ N, dist ((partialSups s) n) (⨆ n, s n) < e
case h G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N ⊢ ∀ n ≥ N, dist ((partialSups s) n) (⨆ n, s n) < e
Please generate a tactic in lean4 to solve the state. STATE: case intro G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N ⊢ ∃ N, ∀ n ≥ N, dist ((partialSups s) n) (⨆ n, s n) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
intro n nN
case h G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N ⊢ ∀ n ≥ N, dist ((partialSups s) n) (⨆ n, s n) < e
case h G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N ⊢ dist ((partialSups s) n) (⨆ n, s n) < e
Please generate a tactic in lean4 to solve the state. STATE: case h G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N ⊢ ∀ n ≥ N, dist ((partialSups s) n) (⨆ n, s n) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
rw [Real.dist_eq]
case h G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N ⊢ dist ((partialSups s) n) (⨆ n, s n) < e
case h G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N ⊢ |(partialSups s) n - ⨆ n, s n| < e
Please generate a tactic in lean4 to solve the state. STATE: case h G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N ⊢ dist ((partialSups s) n) (⨆ n, s n) < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
rw [abs_lt]
case h G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N ⊢ |(partialSups s) n - ⨆ n, s n| < e
case h G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N ⊢ -e < (partialSups s) n - ⨆ n, s n ∧ (partialSups s) n - ⨆ n, s n < e
Please generate a tactic in lean4 to solve the state. STATE: case h G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N ⊢ |(partialSups s) n - ⨆ n, s n| < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
constructor
case h G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N ⊢ -e < (partialSups s) n - ⨆ n, s n ∧ (partialSups s) n - ⨆ n, s n < e
case h.left G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N ⊢ -e < (partialSups s) n - ⨆ n, s n case h.right G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N ⊢ (partialSups s) n - ⨆ n, s n < e
Please generate a tactic in lean4 to solve the state. STATE: case h G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N ⊢ -e < (partialSups s) n - ⨆ n, s n ∧ (partialSups s) n - ⨆ n, s n < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
rw [← hb]
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b ⊢ b < ⨆ n, s n
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b ⊢ (⨆ n, s n) - e < ⨆ n, s n
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b ⊢ b < ⨆ n, s n TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
exact sub_lt_self _ (by linarith)
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b ⊢ (⨆ n, s n) - e < ⨆ n, s n
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b ⊢ (⨆ n, s n) - e < ⨆ n, s n TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
linarith
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b ⊢ 0 < e
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b ⊢ 0 < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
simp only [neg_lt_sub_iff_lt_add]
case h.left G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N ⊢ -e < (partialSups s) n - ⨆ n, s n
case h.left G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N ⊢ ⨆ n, s n < e + (partialSups s) n
Please generate a tactic in lean4 to solve the state. STATE: case h.left G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N ⊢ -e < (partialSups s) n - ⨆ n, s n TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
simp only [←hb] at sN
case h.left G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N ⊢ ⨆ n, s n < e + (partialSups s) n
case h.left G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N n : ℕ nN : n ≥ N sN : (⨆ n, s n) - e < s N ⊢ ⨆ n, s n < e + (partialSups s) n
Please generate a tactic in lean4 to solve the state. STATE: case h.left G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N ⊢ ⨆ n, s n < e + (partialSups s) n TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
calc iSup s _ = iSup s - e + e := by ring _ < s N + e := by linarith _ ≤ partialSups s n + e := by linarith [le_partialSups_of_le s nN] _ = e + partialSups s n := by ring
case h.left G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N n : ℕ nN : n ≥ N sN : (⨆ n, s n) - e < s N ⊢ ⨆ n, s n < e + (partialSups s) n
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.left G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N n : ℕ nN : n ≥ N sN : (⨆ n, s n) - e < s N ⊢ ⨆ n, s n < e + (partialSups s) n TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
ring
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N n : ℕ nN : n ≥ N sN : (⨆ n, s n) - e < s N ⊢ iSup s = iSup s - e + e
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N n : ℕ nN : n ≥ N sN : (⨆ n, s n) - e < s N ⊢ iSup s = iSup s - e + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
linarith
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N n : ℕ nN : n ≥ N sN : (⨆ n, s n) - e < s N ⊢ iSup s - e + e < s N + e
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N n : ℕ nN : n ≥ N sN : (⨆ n, s n) - e < s N ⊢ iSup s - e + e < s N + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
linarith [le_partialSups_of_le s nN]
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N n : ℕ nN : n ≥ N sN : (⨆ n, s n) - e < s N ⊢ s N + e ≤ (partialSups s) n + e
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N n : ℕ nN : n ≥ N sN : (⨆ n, s n) - e < s N ⊢ s N + e ≤ (partialSups s) n + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
ring
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N n : ℕ nN : n ≥ N sN : (⨆ n, s n) - e < s N ⊢ (partialSups s) n + e = e + (partialSups s) n
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N n : ℕ nN : n ≥ N sN : (⨆ n, s n) - e < s N ⊢ (partialSups s) n + e = e + (partialSups s) n TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
have rs : partialSups s n ≤ iSup s := partialSups_le _ _ _ fun a _ ↦ le_ciSup ba a
case h.right G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N ⊢ (partialSups s) n - ⨆ n, s n < e
case h.right G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N rs : (partialSups s) n ≤ iSup s ⊢ (partialSups s) n - ⨆ n, s n < e
Please generate a tactic in lean4 to solve the state. STATE: case h.right G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N ⊢ (partialSups s) n - ⨆ n, s n < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
calc partialSups s n - iSup s _ ≤ iSup s - iSup s := by linarith _ = 0 := by ring _ < e := ep
case h.right G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N rs : (partialSups s) n ≤ iSup s ⊢ (partialSups s) n - ⨆ n, s n < e
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.right G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N rs : (partialSups s) n ≤ iSup s ⊢ (partialSups s) n - ⨆ n, s n < e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
linarith
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N rs : (partialSups s) n ≤ iSup s ⊢ (partialSups s) n - iSup s ≤ iSup s - iSup s
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N rs : (partialSups s) n ≤ iSup s ⊢ (partialSups s) n - iSup s ≤ iSup s - iSup s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
Csupr.has_lim
[124, 1]
[141, 18]
ring
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N rs : (partialSups s) n ≤ iSup s ⊢ iSup s - iSup s = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E s : ℕ → ℝ ba : BddAbove (range s) e : ℝ ep : e > 0 b : ℝ hb : (⨆ n, s n) - e = b bs : b < ⨆ n, s n N : ℕ sN : b < s N n : ℕ nN : n ≥ N rs : (partialSups s) n ≤ iSup s ⊢ iSup s - iSup s = 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
duals_lim_tendsto_maxLog_norm
[144, 1]
[146, 95]
rw [maxLog_norm_eq_duals_iSup]
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E b : ℝ x : E ⊢ Filter.Tendsto (⇑(partialSups fun k => maxLog b ‖(duals k) x‖)) atTop (𝓝 (maxLog b ‖x‖))
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E b : ℝ x : E ⊢ Filter.Tendsto (⇑(partialSups fun k => maxLog b ‖(duals k) x‖)) atTop (𝓝 (⨆ n, maxLog b ‖(duals n) x‖))
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E b : ℝ x : E ⊢ Filter.Tendsto (⇑(partialSups fun k => maxLog b ‖(duals k) x‖)) atTop (𝓝 (maxLog b ‖x‖)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
duals_lim_tendsto_maxLog_norm
[144, 1]
[146, 95]
exact Csupr.has_lim _ (duals_bddAbove (monotone_maxLog _) _)
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E b : ℝ x : E ⊢ Filter.Tendsto (⇑(partialSups fun k => maxLog b ‖(duals k) x‖)) atTop (𝓝 (⨆ n, maxLog b ‖(duals n) x‖))
no goals
Please generate a tactic in lean4 to solve the state. STATE: G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E b : ℝ x : E ⊢ Filter.Tendsto (⇑(partialSups fun k => maxLog b ‖(duals k) x‖)) atTop (𝓝 (⨆ n, maxLog b ‖(duals n) x‖)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Set.Nonempty.left_invCoe
[21, 1]
[23, 76]
intro ⟨x, m⟩
X : Type s : Set X ne : s.Nonempty ⊢ ∀ (x : ↑s), ne.invCoe ↑x = x
X : Type s : Set X ne : s.Nonempty x : X m : x ∈ s ⊢ ne.invCoe ↑⟨x, m⟩ = ⟨x, m⟩
Please generate a tactic in lean4 to solve the state. STATE: X : Type s : Set X ne : s.Nonempty ⊢ ∀ (x : ↑s), ne.invCoe ↑x = x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Set.Nonempty.left_invCoe
[21, 1]
[23, 76]
simp only [Set.Nonempty.invCoe, Subtype.coe_mk, m, dif_pos]
X : Type s : Set X ne : s.Nonempty x : X m : x ∈ s ⊢ ne.invCoe ↑⟨x, m⟩ = ⟨x, m⟩
no goals
Please generate a tactic in lean4 to solve the state. STATE: X : Type s : Set X ne : s.Nonempty x : X m : x ∈ s ⊢ ne.invCoe ↑⟨x, m⟩ = ⟨x, m⟩ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Set.Nonempty.right_invCoe
[25, 1]
[27, 73]
intro x m
X : Type s : Set X ne : s.Nonempty ⊢ ∀ x ∈ s, ↑(ne.invCoe x) = x
X : Type s : Set X ne : s.Nonempty x : X m : x ∈ s ⊢ ↑(ne.invCoe x) = x
Please generate a tactic in lean4 to solve the state. STATE: X : Type s : Set X ne : s.Nonempty ⊢ ∀ x ∈ s, ↑(ne.invCoe x) = x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Set.Nonempty.right_invCoe
[25, 1]
[27, 73]
simp only [Set.Nonempty.invCoe, m, dif_pos, Subtype.coe_mk]
X : Type s : Set X ne : s.Nonempty x : X m : x ∈ s ⊢ ↑(ne.invCoe x) = x
no goals
Please generate a tactic in lean4 to solve the state. STATE: X : Type s : Set X ne : s.Nonempty x : X m : x ∈ s ⊢ ↑(ne.invCoe x) = x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Set.Nonempty.continuousOn_invCoe
[29, 1]
[32, 53]
rw [embedding_subtype_val.continuousOn_iff]
X : Type s : Set X ne : s.Nonempty inst✝ : TopologicalSpace X ⊢ ContinuousOn ne.invCoe s
X : Type s : Set X ne : s.Nonempty inst✝ : TopologicalSpace X ⊢ ContinuousOn (Subtype.val ∘ ne.invCoe) s
Please generate a tactic in lean4 to solve the state. STATE: X : Type s : Set X ne : s.Nonempty inst✝ : TopologicalSpace X ⊢ ContinuousOn ne.invCoe s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Set.Nonempty.continuousOn_invCoe
[29, 1]
[32, 53]
apply continuousOn_id.congr
X : Type s : Set X ne : s.Nonempty inst✝ : TopologicalSpace X ⊢ ContinuousOn (Subtype.val ∘ ne.invCoe) s
X : Type s : Set X ne : s.Nonempty inst✝ : TopologicalSpace X ⊢ EqOn (Subtype.val ∘ ne.invCoe) id s
Please generate a tactic in lean4 to solve the state. STATE: X : Type s : Set X ne : s.Nonempty inst✝ : TopologicalSpace X ⊢ ContinuousOn (Subtype.val ∘ ne.invCoe) s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Set.Nonempty.continuousOn_invCoe
[29, 1]
[32, 53]
intro x m
X : Type s : Set X ne : s.Nonempty inst✝ : TopologicalSpace X ⊢ EqOn (Subtype.val ∘ ne.invCoe) id s
X : Type s : Set X ne : s.Nonempty inst✝ : TopologicalSpace X x : X m : x ∈ s ⊢ (Subtype.val ∘ ne.invCoe) x = id x
Please generate a tactic in lean4 to solve the state. STATE: X : Type s : Set X ne : s.Nonempty inst✝ : TopologicalSpace X ⊢ EqOn (Subtype.val ∘ ne.invCoe) id s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Set.Nonempty.continuousOn_invCoe
[29, 1]
[32, 53]
simp only [Function.comp, ne.right_invCoe _ m, id]
X : Type s : Set X ne : s.Nonempty inst✝ : TopologicalSpace X x : X m : x ∈ s ⊢ (Subtype.val ∘ ne.invCoe) x = id x
no goals
Please generate a tactic in lean4 to solve the state. STATE: X : Type s : Set X ne : s.Nonempty inst✝ : TopologicalSpace X x : X m : x ∈ s ⊢ (Subtype.val ∘ ne.invCoe) x = id x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
constructor
X : Type inst✝ : TopologicalSpace X s : Set X ⊢ TotallyDisconnectedSpace ↑s ↔ IsTotallyDisconnected s
case mp X : Type inst✝ : TopologicalSpace X s : Set X ⊢ TotallyDisconnectedSpace ↑s → IsTotallyDisconnected s case mpr X : Type inst✝ : TopologicalSpace X s : Set X ⊢ IsTotallyDisconnected s → TotallyDisconnectedSpace ↑s
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝ : TopologicalSpace X s : Set X ⊢ TotallyDisconnectedSpace ↑s ↔ IsTotallyDisconnected s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
intro h
case mp X : Type inst✝ : TopologicalSpace X s : Set X ⊢ TotallyDisconnectedSpace ↑s → IsTotallyDisconnected s
case mp X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ⊢ IsTotallyDisconnected s
Please generate a tactic in lean4 to solve the state. STATE: case mp X : Type inst✝ : TopologicalSpace X s : Set X ⊢ TotallyDisconnectedSpace ↑s → IsTotallyDisconnected s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
by_cases ne : s.Nonempty
case mp X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ⊢ IsTotallyDisconnected s
case pos X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty ⊢ IsTotallyDisconnected s case neg X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : ¬s.Nonempty ⊢ IsTotallyDisconnected s
Please generate a tactic in lean4 to solve the state. STATE: case mp X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ⊢ IsTotallyDisconnected s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
intro t ts tc
case pos X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty ⊢ IsTotallyDisconnected s
case pos X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t ⊢ t.Subsingleton
Please generate a tactic in lean4 to solve the state. STATE: case pos X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty ⊢ IsTotallyDisconnected s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
set t' := ne.invCoe '' t
case pos X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t ⊢ t.Subsingleton
case pos X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t ⊢ t.Subsingleton
Please generate a tactic in lean4 to solve the state. STATE: case pos X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t ⊢ t.Subsingleton TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
have tc' : IsPreconnected t' := tc.image _ (ne.continuousOn_invCoe.mono ts)
case pos X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t ⊢ t.Subsingleton
case pos X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' ⊢ t.Subsingleton
Please generate a tactic in lean4 to solve the state. STATE: case pos X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t ⊢ t.Subsingleton TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
have q := h.isTotallyDisconnected_univ _ (subset_univ _) tc'
case pos X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' ⊢ t.Subsingleton
case pos X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton ⊢ t.Subsingleton
Please generate a tactic in lean4 to solve the state. STATE: case pos X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' ⊢ t.Subsingleton TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
rw [e]
case pos X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton e : t = (fun x => ↑x) '' t' ⊢ t.Subsingleton
case pos X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton e : t = (fun x => ↑x) '' t' ⊢ ((fun x => ↑x) '' t').Subsingleton
Please generate a tactic in lean4 to solve the state. STATE: case pos X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton e : t = (fun x => ↑x) '' t' ⊢ t.Subsingleton TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
exact q.image _
case pos X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton e : t = (fun x => ↑x) '' t' ⊢ ((fun x => ↑x) '' t').Subsingleton
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton e : t = (fun x => ↑x) '' t' ⊢ ((fun x => ↑x) '' t').Subsingleton TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
apply Set.ext
X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton ⊢ t = (fun x => ↑x) '' t'
case h X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton ⊢ ∀ (x : X), x ∈ t ↔ x ∈ (fun x => ↑x) '' t'
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton ⊢ t = (fun x => ↑x) '' t' TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
intro x
case h X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton ⊢ ∀ (x : X), x ∈ t ↔ x ∈ (fun x => ↑x) '' t'
case h X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X ⊢ x ∈ t ↔ x ∈ (fun x => ↑x) '' t'
Please generate a tactic in lean4 to solve the state. STATE: case h X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton ⊢ ∀ (x : X), x ∈ t ↔ x ∈ (fun x => ↑x) '' t' TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
simp only [mem_image]
case h X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X ⊢ x ∈ t ↔ x ∈ (fun x => ↑x) '' t'
case h X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X ⊢ x ∈ t ↔ ∃ x_1 ∈ t', ↑x_1 = x
Please generate a tactic in lean4 to solve the state. STATE: case h X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X ⊢ x ∈ t ↔ x ∈ (fun x => ↑x) '' t' TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
constructor
case h X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X ⊢ x ∈ t ↔ ∃ x_1 ∈ t', ↑x_1 = x
case h.mp X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X ⊢ x ∈ t → ∃ x_1 ∈ t', ↑x_1 = x case h.mpr X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X ⊢ (∃ x_1 ∈ t', ↑x_1 = x) → x ∈ t
Please generate a tactic in lean4 to solve the state. STATE: case h X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X ⊢ x ∈ t ↔ ∃ x_1 ∈ t', ↑x_1 = x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
intro xt
case h.mp X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X ⊢ x ∈ t → ∃ x_1 ∈ t', ↑x_1 = x
case h.mp X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X xt : x ∈ t ⊢ ∃ x_1 ∈ t', ↑x_1 = x
Please generate a tactic in lean4 to solve the state. STATE: case h.mp X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X ⊢ x ∈ t → ∃ x_1 ∈ t', ↑x_1 = x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
use ⟨x, ts xt⟩
case h.mp X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X xt : x ∈ t ⊢ ∃ x_1 ∈ t', ↑x_1 = x
case h X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X xt : x ∈ t ⊢ ⟨x, ⋯⟩ ∈ t' ∧ ↑⟨x, ⋯⟩ = x
Please generate a tactic in lean4 to solve the state. STATE: case h.mp X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X xt : x ∈ t ⊢ ∃ x_1 ∈ t', ↑x_1 = x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
refine ⟨⟨x,xt,?_⟩,?_⟩
case h X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X xt : x ∈ t ⊢ ⟨x, ⋯⟩ ∈ t' ∧ ↑⟨x, ⋯⟩ = x
case h.refine_1 X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X xt : x ∈ t ⊢ ne.invCoe x = ⟨x, ⋯⟩ case h.refine_2 X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X xt : x ∈ t ⊢ ↑⟨x, ⋯⟩ = x
Please generate a tactic in lean4 to solve the state. STATE: case h X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X xt : x ∈ t ⊢ ⟨x, ⋯⟩ ∈ t' ∧ ↑⟨x, ⋯⟩ = x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
simp only [Subtype.ext_iff, Subtype.coe_mk, ne.right_invCoe _ (ts xt)]
case h.refine_1 X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X xt : x ∈ t ⊢ ne.invCoe x = ⟨x, ⋯⟩ case h.refine_2 X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X xt : x ∈ t ⊢ ↑⟨x, ⋯⟩ = x
case h.refine_2 X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X xt : x ∈ t ⊢ ↑⟨x, ⋯⟩ = x
Please generate a tactic in lean4 to solve the state. STATE: case h.refine_1 X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X xt : x ∈ t ⊢ ne.invCoe x = ⟨x, ⋯⟩ case h.refine_2 X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X xt : x ∈ t ⊢ ↑⟨x, ⋯⟩ = x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
rw [Subtype.coe_mk]
case h.refine_2 X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X xt : x ∈ t ⊢ ↑⟨x, ⋯⟩ = x
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.refine_2 X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X xt : x ∈ t ⊢ ↑⟨x, ⋯⟩ = x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
intro ⟨⟨y, ys⟩, ⟨z, zt, zy⟩, yx⟩
case h.mpr X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X ⊢ (∃ x_1 ∈ t', ↑x_1 = x) → x ∈ t
case h.mpr X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x y : X ys : y ∈ s z : X zt : z ∈ t zy : ne.invCoe z = ⟨y, ys⟩ yx : ↑⟨y, ys⟩ = x ⊢ x ∈ t
Please generate a tactic in lean4 to solve the state. STATE: case h.mpr X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x : X ⊢ (∃ x_1 ∈ t', ↑x_1 = x) → x ∈ t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
simp only [Subtype.coe_mk, Subtype.ext_iff, ne.right_invCoe _ (ts zt)] at yx zy
case h.mpr X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x y : X ys : y ∈ s z : X zt : z ∈ t zy : ne.invCoe z = ⟨y, ys⟩ yx : ↑⟨y, ys⟩ = x ⊢ x ∈ t
case h.mpr X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x y : X ys : y ∈ s z : X zt : z ∈ t yx : y = x zy : z = y ⊢ x ∈ t
Please generate a tactic in lean4 to solve the state. STATE: case h.mpr X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x y : X ys : y ∈ s z : X zt : z ∈ t zy : ne.invCoe z = ⟨y, ys⟩ yx : ↑⟨y, ys⟩ = x ⊢ x ∈ t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
rw [← yx, ← zy]
case h.mpr X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x y : X ys : y ∈ s z : X zt : z ∈ t yx : y = x zy : z = y ⊢ x ∈ t
case h.mpr X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x y : X ys : y ∈ s z : X zt : z ∈ t yx : y = x zy : z = y ⊢ z ∈ t
Please generate a tactic in lean4 to solve the state. STATE: case h.mpr X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x y : X ys : y ∈ s z : X zt : z ∈ t yx : y = x zy : z = y ⊢ x ∈ t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
exact zt
case h.mpr X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x y : X ys : y ∈ s z : X zt : z ∈ t yx : y = x zy : z = y ⊢ z ∈ t
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.mpr X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s.Nonempty t : Set X ts : t ⊆ s tc : IsPreconnected t t' : Set ↑s := ne.invCoe '' t tc' : IsPreconnected t' q : t'.Subsingleton x y : X ys : y ∈ s z : X zt : z ∈ t yx : y = x zy : z = y ⊢ z ∈ t TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
simp only [not_nonempty_iff_eq_empty] at ne
case neg X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : ¬s.Nonempty ⊢ IsTotallyDisconnected s
case neg X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s = ∅ ⊢ IsTotallyDisconnected s
Please generate a tactic in lean4 to solve the state. STATE: case neg X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : ¬s.Nonempty ⊢ IsTotallyDisconnected s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
rw [ne]
case neg X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s = ∅ ⊢ IsTotallyDisconnected s
case neg X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s = ∅ ⊢ IsTotallyDisconnected ∅
Please generate a tactic in lean4 to solve the state. STATE: case neg X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s = ∅ ⊢ IsTotallyDisconnected s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
exact isTotallyDisconnected_empty
case neg X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s = ∅ ⊢ IsTotallyDisconnected ∅
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg X : Type inst✝ : TopologicalSpace X s : Set X h : TotallyDisconnectedSpace ↑s ne : s = ∅ ⊢ IsTotallyDisconnected ∅ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
intro h
case mpr X : Type inst✝ : TopologicalSpace X s : Set X ⊢ IsTotallyDisconnected s → TotallyDisconnectedSpace ↑s
case mpr X : Type inst✝ : TopologicalSpace X s : Set X h : IsTotallyDisconnected s ⊢ TotallyDisconnectedSpace ↑s
Please generate a tactic in lean4 to solve the state. STATE: case mpr X : Type inst✝ : TopologicalSpace X s : Set X ⊢ IsTotallyDisconnected s → TotallyDisconnectedSpace ↑s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
refine ⟨?_⟩
case mpr X : Type inst✝ : TopologicalSpace X s : Set X h : IsTotallyDisconnected s ⊢ TotallyDisconnectedSpace ↑s
case mpr X : Type inst✝ : TopologicalSpace X s : Set X h : IsTotallyDisconnected s ⊢ IsTotallyDisconnected univ
Please generate a tactic in lean4 to solve the state. STATE: case mpr X : Type inst✝ : TopologicalSpace X s : Set X h : IsTotallyDisconnected s ⊢ TotallyDisconnectedSpace ↑s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
apply embedding_subtype_val.isTotallyDisconnected
case mpr X : Type inst✝ : TopologicalSpace X s : Set X h : IsTotallyDisconnected s ⊢ IsTotallyDisconnected univ
case mpr X : Type inst✝ : TopologicalSpace X s : Set X h : IsTotallyDisconnected s ⊢ IsTotallyDisconnected (Subtype.val '' univ)
Please generate a tactic in lean4 to solve the state. STATE: case mpr X : Type inst✝ : TopologicalSpace X s : Set X h : IsTotallyDisconnected s ⊢ IsTotallyDisconnected univ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
rw [Subtype.coe_image_univ]
case mpr X : Type inst✝ : TopologicalSpace X s : Set X h : IsTotallyDisconnected s ⊢ IsTotallyDisconnected (Subtype.val '' univ)
case mpr X : Type inst✝ : TopologicalSpace X s : Set X h : IsTotallyDisconnected s ⊢ IsTotallyDisconnected s
Please generate a tactic in lean4 to solve the state. STATE: case mpr X : Type inst✝ : TopologicalSpace X s : Set X h : IsTotallyDisconnected s ⊢ IsTotallyDisconnected (Subtype.val '' univ) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
isTotallyDisconnected_iff_totally_disconnected_subtype
[35, 1]
[55, 41]
exact h
case mpr X : Type inst✝ : TopologicalSpace X s : Set X h : IsTotallyDisconnected s ⊢ IsTotallyDisconnected s
no goals
Please generate a tactic in lean4 to solve the state. STATE: case mpr X : Type inst✝ : TopologicalSpace X s : Set X h : IsTotallyDisconnected s ⊢ IsTotallyDisconnected s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
not_countable_Ioo
[58, 1]
[59, 101]
rw [← Cardinal.le_aleph0_iff_set_countable, not_le, Cardinal.mk_Ioo_real h]
a b : ℝ h : a < b ⊢ ¬(Ioo a b).Countable
a b : ℝ h : a < b ⊢ Cardinal.aleph0 < Cardinal.continuum
Please generate a tactic in lean4 to solve the state. STATE: a b : ℝ h : a < b ⊢ ¬(Ioo a b).Countable TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
not_countable_Ioo
[58, 1]
[59, 101]
apply Cardinal.cantor
a b : ℝ h : a < b ⊢ Cardinal.aleph0 < Cardinal.continuum
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b : ℝ h : a < b ⊢ Cardinal.aleph0 < Cardinal.continuum TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
generalize hR : {r | ∃ x y : X, dist x y = r} = R
X : Type inst✝¹ : MetricSpace X inst✝ : Countable X ⊢ TotallyDisconnectedSpace X
X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R ⊢ TotallyDisconnectedSpace X
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝¹ : MetricSpace X inst✝ : Countable X ⊢ TotallyDisconnectedSpace X TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
have rc : R.Countable := by have e : R = range (uncurry dist) := by apply Set.ext; intro r; simp only [mem_setOf, mem_range, Prod.exists, uncurry, ← hR]; rfl rw [e]; exact countable_range _
X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R ⊢ TotallyDisconnectedSpace X
X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable ⊢ TotallyDisconnectedSpace X
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R ⊢ TotallyDisconnectedSpace X TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
refine ⟨?_⟩
X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable ⊢ TotallyDisconnectedSpace X
X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable ⊢ IsTotallyDisconnected univ
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable ⊢ TotallyDisconnectedSpace X TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
apply isTotallyDisconnected_of_isClopen_set
X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable ⊢ IsTotallyDisconnected univ
case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable ⊢ Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable ⊢ IsTotallyDisconnected univ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
intro x y xy
case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable ⊢ Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : x ≠ y ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
Please generate a tactic in lean4 to solve the state. STATE: case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable ⊢ Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
rw [← dist_pos] at xy
case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : x ≠ y ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
Please generate a tactic in lean4 to solve the state. STATE: case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : x ≠ y ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
have h : ¬Ioo 0 (dist x y) ⊆ R := by by_contra h; exact not_countable_Ioo xy (rc.mono h)
case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y h : ¬Ioo 0 (dist x y) ⊆ R ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
Please generate a tactic in lean4 to solve the state. STATE: case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
simp only [not_subset, mem_Ioo] at h
case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y h : ¬Ioo 0 (dist x y) ⊆ R ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y h : ∃ a, (0 < a ∧ a < dist x y) ∧ a ∉ R ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
Please generate a tactic in lean4 to solve the state. STATE: case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y h : ¬Ioo 0 (dist x y) ⊆ R ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U TACTIC: