Datasets:
pkuAI4M
/
lean_github

Dataset card Data Studio Files
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https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
duals_bddAbove
[62, 1]
[67, 54]
apply dualVector_le
case 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 gm : Monotone g x : E y✝ : ℝ x✝ : ℕ h : g (Complex.abs ((duals x✝) x)) = y✝ ⊢ Complex.abs ((duals x✝) x) ≤ ‖x‖
no goals
Please generate a tactic in lean4 to solve the state. STATE: case 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 gm : Monotone g x : E y✝ : ℝ x✝ : ℕ h : g (Complex.abs ((duals x✝) x)) = y✝ ⊢ Complex.abs ((duals x✝) x) ≤ ‖x‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
LipschitzWith.le
[70, 1]
[76, 63]
calc f x _ = f y + (f x - f y) := by ring_nf _ ≤ f y + |f x - f y| := by bound _ = f y + dist (f x) (f y) := by rw [Real.dist_eq] _ ≤ f y + k * dist x y := by linarith [fk.dist_le_mul x y]
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E f : G → ℝ k : ℝ≥0 fk : LipschitzWith k f x y : G ⊢ f x ≤ f y + ↑k * dist x 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 f : G → ℝ k : ℝ≥0 fk : LipschitzWith k f x y : G ⊢ f x ≤ f y + ↑k * dist x y TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
LipschitzWith.le
[70, 1]
[76, 63]
ring_nf
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E f : G → ℝ k : ℝ≥0 fk : LipschitzWith k f x y : G ⊢ f x = f y + (f x - f 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 f : G → ℝ k : ℝ≥0 fk : LipschitzWith k f x y : G ⊢ f x = f y + (f x - f y) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
LipschitzWith.le
[70, 1]
[76, 63]
bound
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E f : G → ℝ k : ℝ≥0 fk : LipschitzWith k f x y : G ⊢ f y + (f x - f y) ≤ f y + |f x - f 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 f : G → ℝ k : ℝ≥0 fk : LipschitzWith k f x y : G ⊢ f y + (f x - f y) ≤ f y + |f x - f y| TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
LipschitzWith.le
[70, 1]
[76, 63]
rw [Real.dist_eq]
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E f : G → ℝ k : ℝ≥0 fk : LipschitzWith k f x y : G ⊢ f y + |f x - f y| = f y + dist (f x) (f 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 f : G → ℝ k : ℝ≥0 fk : LipschitzWith k f x y : G ⊢ f y + |f x - f y| = f y + dist (f x) (f y) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
LipschitzWith.le
[70, 1]
[76, 63]
linarith [fk.dist_le_mul x y]
G : Type inst✝³ : NormedAddCommGroup G E : Type inst✝² : NormedAddCommGroup E inst✝¹ : NormedSpace ℂ E inst✝ : SecondCountableTopology E f : G → ℝ k : ℝ≥0 fk : LipschitzWith k f x y : G ⊢ f y + dist (f x) (f y) ≤ f y + ↑k * dist x 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 f : G → ℝ k : ℝ≥0 fk : LipschitzWith k f x y : G ⊢ f y + dist (f x) (f y) ≤ f y + ↑k * dist x y TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
by_cases k0 : k = 0
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 ⊢ g ‖x‖ = ⨆ n, g ‖(duals n) x‖
case pos 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 ⊢ g ‖x‖ = ⨆ n, g ‖(duals n) x‖ case neg 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 ⊢ g ‖x‖ = ⨆ n, g ‖(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 g : ℝ → ℝ k : ℝ≥0 gm : Monotone g gk : LipschitzWith k g x : E ⊢ g ‖x‖ = ⨆ n, g ‖(duals n) x‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
have kp : 0 < (k : ℝ) := by simp only [NNReal.coe_pos]; exact Ne.bot_lt k0
case neg 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 ⊢ g ‖x‖ = ⨆ n, g ‖(duals n) x‖
case neg 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 ⊢ g ‖x‖ = ⨆ n, g ‖(duals n) x‖
Please generate a tactic in lean4 to solve the state. STATE: case neg 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 ⊢ g ‖x‖ = ⨆ n, g ‖(duals n) x‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
apply le_antisymm
case neg 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 ⊢ g ‖x‖ = ⨆ n, g ‖(duals n) x‖
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 ⊢ g ‖x‖ ≤ ⨆ n, g ‖(duals n) x‖ 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‖
Please generate a tactic in lean4 to solve the state. STATE: case neg 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 ⊢ g ‖x‖ = ⨆ n, g ‖(duals n) x‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
rw [k0] at gk
case pos 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 ⊢ g ‖x‖ = ⨆ n, g ‖(duals n) x‖
case pos 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 0 g x : E k0 : k = 0 ⊢ g ‖x‖ = ⨆ n, g ‖(duals n) x‖
Please generate a tactic in lean4 to solve the state. STATE: case pos 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 ⊢ g ‖x‖ = ⨆ n, g ‖(duals n) x‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
have g0 := gk.is_const 0
case pos 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 0 g x : E k0 : k = 0 ⊢ g ‖x‖ = ⨆ n, g ‖(duals n) x‖
case pos 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 0 g x : E k0 : k = 0 g0 : ∀ (y : ℝ), g 0 = g y ⊢ g ‖x‖ = ⨆ n, g ‖(duals n) x‖
Please generate a tactic in lean4 to solve the state. STATE: case pos 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 0 g x : E k0 : k = 0 ⊢ g ‖x‖ = ⨆ n, g ‖(duals n) x‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
simp only [← g0 _, ciSup_const]
case pos 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 0 g x : E k0 : k = 0 g0 : ∀ (y : ℝ), g 0 = g y ⊢ g ‖x‖ = ⨆ n, g ‖(duals n) x‖
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos 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 0 g x : E k0 : k = 0 g0 : ∀ (y : ℝ), g 0 = g y ⊢ g ‖x‖ = ⨆ n, g ‖(duals n) x‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
simp only [NNReal.coe_pos]
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 ⊢ 0 < ↑k
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 ⊢ 0 < k
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 ⊢ 0 < ↑k TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
exact Ne.bot_lt k0
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 ⊢ 0 < 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 ⊢ 0 < k TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
apply le_of_forall_pos_le_add
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 ⊢ g ‖x‖ ≤ ⨆ n, g ‖(duals n) 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 ⊢ ∀ (ε : ℝ), 0 < ε → g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + ε
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 ⊢ g ‖x‖ ≤ ⨆ n, g ‖(duals n) x‖ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
intro e ep
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 ⊢ ∀ (ε : ℝ), 0 < ε → g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + ε
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 e : ℝ ep : 0 < e ⊢ g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + e
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 ⊢ ∀ (ε : ℝ), 0 < ε → g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + ε TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
rcases Metric.denseRange_iff.mp (TopologicalSpace.denseRange_denseSeq E) x (e / 2 / k) (by bound) with ⟨n, nx⟩
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 e : ℝ ep : 0 < e ⊢ g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + e
case neg.a.h.intro 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 nx : dist x (TopologicalSpace.denseSeq E n) < e / 2 / ↑k ⊢ g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + e
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 e : ℝ ep : 0 < e ⊢ g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
generalize hy : TopologicalSpace.denseSeq E n = y
case neg.a.h.intro 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 nx : dist x (TopologicalSpace.denseSeq E n) < e / 2 / ↑k ⊢ g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + e
case neg.a.h.intro 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 nx : dist x (TopologicalSpace.denseSeq E n) < e / 2 / ↑k y : E hy : TopologicalSpace.denseSeq E n = y ⊢ g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + e
Please generate a tactic in lean4 to solve the state. STATE: case neg.a.h.intro 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 nx : dist x (TopologicalSpace.denseSeq E n) < e / 2 / ↑k ⊢ g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
rw [hy] at nx
case neg.a.h.intro 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 nx : dist x (TopologicalSpace.denseSeq E n) < e / 2 / ↑k y : E hy : TopologicalSpace.denseSeq E n = y ⊢ g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + e
case neg.a.h.intro 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 ⊢ g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + e
Please generate a tactic in lean4 to solve the state. STATE: case neg.a.h.intro 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 nx : dist x (TopologicalSpace.denseSeq E n) < e / 2 / ↑k y : E hy : TopologicalSpace.denseSeq E n = y ⊢ g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
have hn : duals n = dualVector y := by rw [← hy, duals]
case neg.a.h.intro 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 ⊢ g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + e
case neg.a.h.intro 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 hn : duals n = dualVector y ⊢ g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + e
Please generate a tactic in lean4 to solve the state. STATE: case neg.a.h.intro 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 ⊢ g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
have h := le_ciSup (duals_bddAbove gm x) n
case neg.a.h.intro 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 hn : duals n = dualVector y ⊢ g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + e
case neg.a.h.intro 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 hn : duals n = dualVector y h : g ‖(duals n) x‖ ≤ ⨆ n, g ‖(duals n) x‖ ⊢ g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + e
Please generate a tactic in lean4 to solve the state. STATE: case neg.a.h.intro 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 hn : duals n = dualVector y ⊢ g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
generalize hs : ⨆ n, g ‖duals n x‖ = s
case neg.a.h.intro 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 hn : duals n = dualVector y h : g ‖(duals n) x‖ ≤ ⨆ n, g ‖(duals n) x‖ ⊢ g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + e
case neg.a.h.intro 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 hn : duals n = dualVector y h : g ‖(duals n) x‖ ≤ ⨆ n, g ‖(duals n) x‖ s : ℝ hs : ⨆ n, g ‖(duals n) x‖ = s ⊢ g ‖x‖ ≤ s + e
Please generate a tactic in lean4 to solve the state. STATE: case neg.a.h.intro 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 hn : duals n = dualVector y h : g ‖(duals n) x‖ ≤ ⨆ n, g ‖(duals n) x‖ ⊢ g ‖x‖ ≤ (⨆ n, g ‖(duals n) x‖) + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
simp_rw [hs, hn] at h
case neg.a.h.intro 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 hn : duals n = dualVector y h : g ‖(duals n) x‖ ≤ ⨆ n, g ‖(duals n) x‖ s : ℝ hs : ⨆ n, g ‖(duals n) x‖ = s ⊢ g ‖x‖ ≤ s + e
case neg.a.h.intro 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 hn : duals n = dualVector y s : ℝ hs : ⨆ n, g ‖(duals n) x‖ = s h : g ‖(dualVector y) x‖ ≤ s ⊢ g ‖x‖ ≤ s + e
Please generate a tactic in lean4 to solve the state. STATE: case neg.a.h.intro 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 hn : duals n = dualVector y h : g ‖(duals n) x‖ ≤ ⨆ n, g ‖(duals n) x‖ s : ℝ hs : ⨆ n, g ‖(duals n) x‖ = s ⊢ g ‖x‖ ≤ s + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
clear hs hn hy
case neg.a.h.intro 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 hn : duals n = dualVector y s : ℝ hs : ⨆ n, g ‖(duals n) x‖ = s h : g ‖(dualVector y) x‖ ≤ s ⊢ g ‖x‖ ≤ s + e
case neg.a.h.intro 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 ‖(dualVector y) x‖ ≤ s ⊢ g ‖x‖ ≤ s + e
Please generate a tactic in lean4 to solve the state. STATE: case neg.a.h.intro 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 hn : duals n = dualVector y s : ℝ hs : ⨆ n, g ‖(duals n) x‖ = s h : g ‖(dualVector y) x‖ ≤ s ⊢ g ‖x‖ ≤ s + e 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] at h
case neg.a.h.intro 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 ‖(dualVector y) x‖ ≤ s ⊢ g ‖x‖ ≤ s + e
case neg.a.h.intro 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 ⊢ g ‖x‖ ≤ s + e
Please generate a tactic in lean4 to solve the state. STATE: case neg.a.h.intro 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 ‖(dualVector y) x‖ ≤ s ⊢ g ‖x‖ ≤ s + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
have gk' : LipschitzWith k fun x ↦ g (abs (dualVector y x)) := by have k11 : (k : ℝ≥0) = k * 1 * 1 := by norm_num rw [k11] simp_rw [←Complex.norm_eq_abs]; apply (gk.comp lipschitzWith_one_norm).comp exact (dualVector y).lipschitz.weaken (dualVector_nnnorm y)
case neg.a.h.intro 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 ⊢ g ‖x‖ ≤ s + e
case neg.a.h.intro 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 ‖x‖ ≤ s + e
Please generate a tactic in lean4 to solve the state. STATE: case neg.a.h.intro 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 ⊢ g ‖x‖ ≤ s + e TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Hartogs/Duals.lean
norm_eq_duals_supr'
[79, 1]
[112, 92]
calc g ‖x‖ _ ≤ g ‖y‖ + k * 1 * dist x y := (gk.comp lipschitzWith_one_norm).le x y _ ≤ g ‖y‖ + k * 1 * (e / 2 / k) := by bound _ = g ‖y‖ + k / k * e / 2 := by ring _ ≤ g ‖y‖ + 1 * e / 2 := by bound _ = g ‖y‖ + e / 2 := by simp only [one_mul] _ = g (abs (dualVector y y)) + e / 2 := by simp only [dualVector_apply, Complex.abs_ofReal, abs_norm] _ ≤ g (abs (dualVector y x)) + k * dist y x + e / 2 := by bound [gk'.le] _ ≤ s + k * dist y x + e / 2 := by linarith _ = s + k * dist x y + e / 2 := by rw [dist_comm] _ ≤ s + k * (e / 2 / k) + e / 2 := by bound _ = s + k / k * e / 2 + e / 2 := by ring_nf _ ≤ s + 1 * e / 2 + e / 2 := by bound _ = s + e := by ring_nf
case neg.a.h.intro 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 ‖x‖ ≤ s + e
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg.a.h.intro 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 ‖x‖ ≤ s + e 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 ⊢ e / 2 / ↑k > 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 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 ⊢ e / 2 / ↑k > 0 TACTIC:
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/Dynamics/Multibrot/Connected.lean
isPathConnected_multibrotExt
[30, 1]
[33, 40]
rw [← ray_surj d]
c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ IsPathConnected (multibrotExt d)
c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ IsPathConnected (ray d '' ball 0 1)
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ IsPathConnected (multibrotExt d) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Connected.lean
isPathConnected_multibrotExt
[30, 1]
[33, 40]
apply IsPathConnected.image_of_continuousOn
c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ IsPathConnected (ray d '' ball 0 1)
case sc c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ IsPathConnected (ball 0 1) case fc c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ContinuousOn (ray d) (ball 0 1)
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ IsPathConnected (ray d '' ball 0 1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Connected.lean
isPathConnected_multibrotExt
[30, 1]
[33, 40]
exact (convex_ball _ _).isPathConnected (Metric.nonempty_ball.mpr one_pos)
case sc c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ IsPathConnected (ball 0 1) case fc c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ContinuousOn (ray d) (ball 0 1)
case fc c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ContinuousOn (ray d) (ball 0 1)
Please generate a tactic in lean4 to solve the state. STATE: case sc c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ IsPathConnected (ball 0 1) case fc c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ContinuousOn (ray d) (ball 0 1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Connected.lean
isPathConnected_multibrotExt
[30, 1]
[33, 40]
exact (rayHolomorphic d).continuousOn
case fc c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ContinuousOn (ray d) (ball 0 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case fc c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ContinuousOn (ray d) (ball 0 1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Connected.lean
isPathConnected_potential_levelset
[36, 1]
[48, 76]
rw [e]
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 e : potential d ⁻¹' {p} = ray d '' sphere 0 p ⊢ IsPathConnected (potential d ⁻¹' {p})
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 e : potential d ⁻¹' {p} = ray d '' sphere 0 p ⊢ IsPathConnected (ray d '' sphere 0 p)
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 e : potential d ⁻¹' {p} = ray d '' sphere 0 p ⊢ IsPathConnected (potential d ⁻¹' {p}) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Connected.lean
isPathConnected_potential_levelset
[36, 1]
[48, 76]
apply (isPathConnected_sphere p0).image_of_continuousOn
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 e : potential d ⁻¹' {p} = ray d '' sphere 0 p ⊢ IsPathConnected (ray d '' sphere 0 p)
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 e : potential d ⁻¹' {p} = ray d '' sphere 0 p ⊢ ContinuousOn (ray d) (sphere 0 p)
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 e : potential d ⁻¹' {p} = ray d '' sphere 0 p ⊢ IsPathConnected (ray d '' sphere 0 p) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Connected.lean
isPathConnected_potential_levelset
[36, 1]
[48, 76]
exact (rayHolomorphic d).continuousOn.mono (Metric.sphere_subset_ball p1)
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 e : potential d ⁻¹' {p} = ray d '' sphere 0 p ⊢ ContinuousOn (ray d) (sphere 0 p)
no goals
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 e : potential d ⁻¹' {p} = ray d '' sphere 0 p ⊢ ContinuousOn (ray d) (sphere 0 p) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Connected.lean
isPathConnected_potential_levelset
[36, 1]
[48, 76]
apply Set.ext
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 ⊢ potential d ⁻¹' {p} = ray d '' sphere 0 p
case h c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 ⊢ ∀ (x : 𝕊), x ∈ potential d ⁻¹' {p} ↔ x ∈ ray d '' sphere 0 p
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 ⊢ potential d ⁻¹' {p} = ray d '' sphere 0 p TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Connected.lean
isPathConnected_potential_levelset
[36, 1]
[48, 76]
intro c
case h c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 ⊢ ∀ (x : 𝕊), x ∈ potential d ⁻¹' {p} ↔ x ∈ ray d '' sphere 0 p
case h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 ⊢ c ∈ potential d ⁻¹' {p} ↔ c ∈ ray d '' sphere 0 p
Please generate a tactic in lean4 to solve the state. STATE: case h c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 ⊢ ∀ (x : 𝕊), x ∈ potential d ⁻¹' {p} ↔ x ∈ ray d '' sphere 0 p TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Connected.lean
isPathConnected_potential_levelset
[36, 1]
[48, 76]
simp only [mem_preimage, mem_singleton_iff, ← abs_bottcher, mem_image, mem_sphere, Complex.dist_eq, sub_zero]
case h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 ⊢ c ∈ potential d ⁻¹' {p} ↔ c ∈ ray d '' sphere 0 p
case h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 ⊢ Complex.abs (bottcher d c) = p ↔ ∃ x, Complex.abs x = p ∧ ray d x = c
Please generate a tactic in lean4 to solve the state. STATE: case h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 ⊢ c ∈ potential d ⁻¹' {p} ↔ c ∈ ray d '' sphere 0 p TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Connected.lean
isPathConnected_potential_levelset
[36, 1]
[48, 76]
constructor
case h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 ⊢ Complex.abs (bottcher d c) = p ↔ ∃ x, Complex.abs x = p ∧ ray d x = c
case h.mp c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 ⊢ Complex.abs (bottcher d c) = p → ∃ x, Complex.abs x = p ∧ ray d x = c case h.mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 ⊢ (∃ x, Complex.abs x = p ∧ ray d x = c) → Complex.abs (bottcher d c) = p
Please generate a tactic in lean4 to solve the state. STATE: case h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 ⊢ Complex.abs (bottcher d c) = p ↔ ∃ x, Complex.abs x = p ∧ ray d x = c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Connected.lean
isPathConnected_potential_levelset
[36, 1]
[48, 76]
intro h
case h.mp c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 ⊢ Complex.abs (bottcher d c) = p → ∃ x, Complex.abs x = p ∧ ray d x = c
case h.mp c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 h : Complex.abs (bottcher d c) = p ⊢ ∃ x, Complex.abs x = p ∧ ray d x = c
Please generate a tactic in lean4 to solve the state. STATE: case h.mp c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 ⊢ Complex.abs (bottcher d c) = p → ∃ x, Complex.abs x = p ∧ ray d x = c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Connected.lean
isPathConnected_potential_levelset
[36, 1]
[48, 76]
use bottcher d c
case h.mp c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 h : Complex.abs (bottcher d c) = p ⊢ ∃ x, Complex.abs x = p ∧ ray d x = c
case h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 h : Complex.abs (bottcher d c) = p ⊢ Complex.abs (bottcher d c) = p ∧ ray d (bottcher d c) = c
Please generate a tactic in lean4 to solve the state. STATE: case h.mp c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 h : Complex.abs (bottcher d c) = p ⊢ ∃ x, Complex.abs x = p ∧ ray d x = c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Connected.lean
isPathConnected_potential_levelset
[36, 1]
[48, 76]
use h
case h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 h : Complex.abs (bottcher d c) = p ⊢ Complex.abs (bottcher d c) = p ∧ ray d (bottcher d c) = c
case right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 h : Complex.abs (bottcher d c) = p ⊢ ray d (bottcher d c) = c
Please generate a tactic in lean4 to solve the state. STATE: case h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 h : Complex.abs (bottcher d c) = p ⊢ Complex.abs (bottcher d c) = p ∧ ray d (bottcher d c) = c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Connected.lean
isPathConnected_potential_levelset
[36, 1]
[48, 76]
rw [ray_bottcher]
case right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 h : Complex.abs (bottcher d c) = p ⊢ ray d (bottcher d c) = c
case right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 h : Complex.abs (bottcher d c) = p ⊢ c ∈ multibrotExt d
Please generate a tactic in lean4 to solve the state. STATE: case right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 h : Complex.abs (bottcher d c) = p ⊢ ray d (bottcher d c) = c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Connected.lean
isPathConnected_potential_levelset
[36, 1]
[48, 76]
rw [← potential_lt_one, ← abs_bottcher, h]
case right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 h : Complex.abs (bottcher d c) = p ⊢ c ∈ multibrotExt d
case right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 h : Complex.abs (bottcher d c) = p ⊢ p < 1
Please generate a tactic in lean4 to solve the state. STATE: case right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 h : Complex.abs (bottcher d c) = p ⊢ c ∈ multibrotExt d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Connected.lean
isPathConnected_potential_levelset
[36, 1]
[48, 76]
exact p1
case right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 h : Complex.abs (bottcher d c) = p ⊢ p < 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 h : Complex.abs (bottcher d c) = p ⊢ p < 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Connected.lean
isPathConnected_potential_levelset
[36, 1]
[48, 76]
intro ⟨e, ep, ec⟩
case h.mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 ⊢ (∃ x, Complex.abs x = p ∧ ray d x = c) → Complex.abs (bottcher d c) = p
case h.mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 e : ℂ ep : Complex.abs e = p ec : ray d e = c ⊢ Complex.abs (bottcher d c) = p
Please generate a tactic in lean4 to solve the state. STATE: case h.mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 ⊢ (∃ x, Complex.abs x = p ∧ ray d x = c) → Complex.abs (bottcher d c) = p TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Connected.lean
isPathConnected_potential_levelset
[36, 1]
[48, 76]
rw [← ec, bottcher_ray]
case h.mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 e : ℂ ep : Complex.abs e = p ec : ray d e = c ⊢ Complex.abs (bottcher d c) = p
case h.mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 e : ℂ ep : Complex.abs e = p ec : ray d e = c ⊢ Complex.abs e = p case h.mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 e : ℂ ep : Complex.abs e = p ec : ray d e = c ⊢ e ∈ ball 0 1
Please generate a tactic in lean4 to solve the state. STATE: case h.mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 e : ℂ ep : Complex.abs e = p ec : ray d e = c ⊢ Complex.abs (bottcher d c) = p TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Connected.lean
isPathConnected_potential_levelset
[36, 1]
[48, 76]
exact ep
case h.mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 e : ℂ ep : Complex.abs e = p ec : ray d e = c ⊢ Complex.abs e = p case h.mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 e : ℂ ep : Complex.abs e = p ec : ray d e = c ⊢ e ∈ ball 0 1
case h.mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 e : ℂ ep : Complex.abs e = p ec : ray d e = c ⊢ e ∈ ball 0 1
Please generate a tactic in lean4 to solve the state. STATE: case h.mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 e : ℂ ep : Complex.abs e = p ec : ray d e = c ⊢ Complex.abs e = p case h.mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) p : ℝ p0 : 0 ≤ p p1 : p < 1 c : 𝕊 e : ℂ ep : Complex.abs e = p ec : ray d e = c ⊢ e ∈ ball 0 1 TACTIC: