Datasets:
pkuAI4M
/
lean_github

Dataset card Data Studio Files
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https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
rw [← dist_pos] at xy
case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : x ≠ y ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
Please generate a tactic in lean4 to solve the state. STATE: case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : x ≠ y ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
have h : ¬Ioo 0 (dist x y) ⊆ R := by by_contra h; exact not_countable_Ioo xy (rc.mono h)
case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y h : ¬Ioo 0 (dist x y) ⊆ R ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
Please generate a tactic in lean4 to solve the state. STATE: case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
simp only [not_subset, mem_Ioo] at h
case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y h : ¬Ioo 0 (dist x y) ⊆ R ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y h : ∃ a, (0 < a ∧ a < dist x y) ∧ a ∉ R ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
Please generate a tactic in lean4 to solve the state. STATE: case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y h : ¬Ioo 0 (dist x y) ⊆ R ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
rcases h with ⟨r, ⟨rp, rxy⟩, rr⟩
case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y h : ∃ a, (0 < a ∧ a < dist x y) ∧ a ∉ R ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
case hX.intro.intro.intro X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
Please generate a tactic in lean4 to solve the state. STATE: case hX X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y h : ∃ a, (0 < a ∧ a < dist x y) ∧ a ∉ R ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
have e : ball x r = closedBall x r := by apply Set.ext; intro z; simp only [mem_ball, mem_closedBall] simp only [mem_setOf, not_exists, ← hR] at rr; simp only [Ne.le_iff_lt (rr z x)]
case hX.intro.intro.intro X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
case hX.intro.intro.intro X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
Please generate a tactic in lean4 to solve the state. STATE: case hX.intro.intro.intro X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
refine ⟨ball x r, ⟨?_, isOpen_ball⟩, ?_⟩
case hX.intro.intro.intro X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U
case hX.intro.intro.intro.refine_1 X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ IsClosed (ball x r) case hX.intro.intro.intro.refine_2 X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ x ∈ ball x r ∧ y ∉ ball x r
Please generate a tactic in lean4 to solve the state. STATE: case hX.intro.intro.intro X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
rw [e]
case hX.intro.intro.intro.refine_1 X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ IsClosed (ball x r) case hX.intro.intro.intro.refine_2 X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ x ∈ ball x r ∧ y ∉ ball x r
case hX.intro.intro.intro.refine_1 X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ IsClosed (closedBall x r) case hX.intro.intro.intro.refine_2 X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ x ∈ ball x r ∧ y ∉ ball x r
Please generate a tactic in lean4 to solve the state. STATE: case hX.intro.intro.intro.refine_1 X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ IsClosed (ball x r) case hX.intro.intro.intro.refine_2 X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ x ∈ ball x r ∧ y ∉ ball x r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
exact isClosed_ball
case hX.intro.intro.intro.refine_1 X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ IsClosed (closedBall x r) case hX.intro.intro.intro.refine_2 X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ x ∈ ball x r ∧ y ∉ ball x r
case hX.intro.intro.intro.refine_2 X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ x ∈ ball x r ∧ y ∉ ball x r
Please generate a tactic in lean4 to solve the state. STATE: case hX.intro.intro.intro.refine_1 X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ IsClosed (closedBall x r) case hX.intro.intro.intro.refine_2 X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ x ∈ ball x r ∧ y ∉ ball x r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
use mem_ball_self rp
case hX.intro.intro.intro.refine_2 X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ x ∈ ball x r ∧ y ∉ ball x r
case right X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ y ∉ ball x r
Please generate a tactic in lean4 to solve the state. STATE: case hX.intro.intro.intro.refine_2 X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ x ∈ ball x r ∧ y ∉ ball x r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
simp only [mem_ball, not_lt]
case right X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ y ∉ ball x r
case right X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ r ≤ dist y x
Please generate a tactic in lean4 to solve the state. STATE: case right X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ y ∉ ball x r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
rw [dist_comm]
case right X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ r ≤ dist y x
case right X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ r ≤ dist x y
Please generate a tactic in lean4 to solve the state. STATE: case right X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ r ≤ dist y x TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
exact rxy.le
case right X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ r ≤ dist x y
no goals
Please generate a tactic in lean4 to solve the state. STATE: case right X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y e : ball x r = closedBall x r ⊢ r ≤ dist x y TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
have e : R = range (uncurry dist) := by apply Set.ext; intro r; simp only [mem_setOf, mem_range, Prod.exists, uncurry, ← hR]; rfl
X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R ⊢ R.Countable
X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R e : R = range (uncurry dist) ⊢ R.Countable
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R ⊢ R.Countable TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
rw [e]
X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R e : R = range (uncurry dist) ⊢ R.Countable
X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R e : R = range (uncurry dist) ⊢ (range (uncurry dist)).Countable
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R e : R = range (uncurry dist) ⊢ R.Countable TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
exact countable_range _
X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R e : R = range (uncurry dist) ⊢ (range (uncurry dist)).Countable
no goals
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R e : R = range (uncurry dist) ⊢ (range (uncurry dist)).Countable TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
apply Set.ext
X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R ⊢ R = range (uncurry dist)
case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R ⊢ ∀ (x : ℝ), x ∈ R ↔ x ∈ range (uncurry dist)
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R ⊢ R = range (uncurry dist) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
intro r
case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R ⊢ ∀ (x : ℝ), x ∈ R ↔ x ∈ range (uncurry dist)
case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R r : ℝ ⊢ r ∈ R ↔ r ∈ range (uncurry dist)
Please generate a tactic in lean4 to solve the state. STATE: case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R ⊢ ∀ (x : ℝ), x ∈ R ↔ x ∈ range (uncurry dist) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
simp only [mem_setOf, mem_range, Prod.exists, uncurry, ← hR]
case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R r : ℝ ⊢ r ∈ R ↔ r ∈ range (uncurry dist)
case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R r : ℝ ⊢ (∃ x y, dist x y = r) ↔ ∃ a b, dist a b = r
Please generate a tactic in lean4 to solve the state. STATE: case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R r : ℝ ⊢ r ∈ R ↔ r ∈ range (uncurry dist) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
rfl
case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R r : ℝ ⊢ (∃ x y, dist x y = r) ↔ ∃ a b, dist a b = r
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R r : ℝ ⊢ (∃ x y, dist x y = r) ↔ ∃ a b, dist a b = r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
by_contra h
X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y ⊢ ¬Ioo 0 (dist x y) ⊆ R
X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y h : Ioo 0 (dist x y) ⊆ R ⊢ False
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y ⊢ ¬Ioo 0 (dist x y) ⊆ R TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
exact not_countable_Ioo xy (rc.mono h)
X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y h : Ioo 0 (dist x y) ⊆ R ⊢ False
no goals
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y h : Ioo 0 (dist x y) ⊆ R ⊢ False TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
apply Set.ext
X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y ⊢ ball x r = closedBall x r
case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y ⊢ ∀ (x_1 : X), x_1 ∈ ball x r ↔ x_1 ∈ closedBall x r
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y ⊢ ball x r = closedBall x r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
intro z
case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y ⊢ ∀ (x_1 : X), x_1 ∈ ball x r ↔ x_1 ∈ closedBall x r
case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y z : X ⊢ z ∈ ball x r ↔ z ∈ closedBall x r
Please generate a tactic in lean4 to solve the state. STATE: case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y ⊢ ∀ (x_1 : X), x_1 ∈ ball x r ↔ x_1 ∈ closedBall x r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
simp only [mem_ball, mem_closedBall]
case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y z : X ⊢ z ∈ ball x r ↔ z ∈ closedBall x r
case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y z : X ⊢ dist z x < r ↔ dist z x ≤ r
Please generate a tactic in lean4 to solve the state. STATE: case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y z : X ⊢ z ∈ ball x r ↔ z ∈ closedBall x r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
simp only [mem_setOf, not_exists, ← hR] at rr
case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y z : X ⊢ dist z x < r ↔ dist z x ≤ r
case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rp : 0 < r rxy : r < dist x y z : X rr : ∀ (x x_1 : X), ¬dist x x_1 = r ⊢ dist z x < r ↔ dist z x ≤ r
Please generate a tactic in lean4 to solve the state. STATE: case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rr : r ∉ R rp : 0 < r rxy : r < dist x y z : X ⊢ dist z x < r ↔ dist z x ≤ r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
Countable.totallyDisconnectedSpace
[62, 1]
[78, 61]
simp only [Ne.le_iff_lt (rr z x)]
case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rp : 0 < r rxy : r < dist x y z : X rr : ∀ (x x_1 : X), ¬dist x x_1 = r ⊢ dist z x < r ↔ dist z x ≤ r
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h X : Type inst✝¹ : MetricSpace X inst✝ : Countable X R : Set ℝ hR : {r | ∃ x y, dist x y = r} = R rc : R.Countable x y : X xy : 0 < dist x y r : ℝ rp : 0 < r rxy : r < dist x y z : X rr : ∀ (x x_1 : X), ¬dist x x_1 = r ⊢ dist z x < r ↔ dist z x ≤ r TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
IsCountable.isTotallyDisconnected
[81, 1]
[84, 74]
rw [← isTotallyDisconnected_iff_totally_disconnected_subtype]
X : Type inst✝ : MetricSpace X s : Set X h : s.Countable ⊢ IsTotallyDisconnected s
X : Type inst✝ : MetricSpace X s : Set X h : s.Countable ⊢ TotallyDisconnectedSpace ↑s
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝ : MetricSpace X s : Set X h : s.Countable ⊢ IsTotallyDisconnected s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Misc/TotallyDisconnected.lean
IsCountable.isTotallyDisconnected
[81, 1]
[84, 74]
exact @Countable.totallyDisconnectedSpace _ _ (countable_coe_iff.mpr h)
X : Type inst✝ : MetricSpace X s : Set X h : s.Countable ⊢ TotallyDisconnectedSpace ↑s
no goals
Please generate a tactic in lean4 to solve the state. STATE: X : Type inst✝ : MetricSpace X s : Set X h : s.Countable ⊢ TotallyDisconnectedSpace ↑s TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
d_pos
[41, 1]
[41, 71]
linarith [two_le_d d]
c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ 0 < d
no goals
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) ⊢ 0 < d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
d_gt_one
[43, 1]
[43, 74]
linarith [two_le_d d]
c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ 1 < d
no goals
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) ⊢ 1 < d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
d_minus_one_pos
[45, 1]
[45, 91]
have h := two_le_d d
c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ 0 < d - 1
c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) h : 2 ≤ d ⊢ 0 < d - 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) ⊢ 0 < d - 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
d_minus_one_pos
[45, 1]
[45, 91]
omega
c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) h : 2 ≤ d ⊢ 0 < d - 1
no goals
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) h : 2 ≤ d ⊢ 0 < d - 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
one_le_d_minus_one
[46, 1]
[46, 94]
have h := two_le_d d
c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ 1 ≤ d - 1
c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) h : 2 ≤ d ⊢ 1 ≤ d - 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) ⊢ 1 ≤ d - 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
one_le_d_minus_one
[46, 1]
[46, 94]
omega
c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) h : 2 ≤ d ⊢ 1 ≤ d - 1
no goals
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) h : 2 ≤ d ⊢ 1 ≤ d - 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
two_le_cast_d
[47, 1]
[48, 56]
norm_num
c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ 2 ≤ ↑2
no goals
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) ⊢ 2 ≤ ↑2 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
multibrotExt_coe
[77, 1]
[80, 72]
simp only [multibrotExt, mem_union, mem_singleton_iff, coe_eq_inf_iff, or_false_iff, mem_image, mem_compl_iff, coe_eq_coe, not_iff_not]
c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ ↑c ∈ multibrotExt d ↔ c ∉ multibrot d
c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ (∃ x ∈ multibrot d, x = c) ↔ c ∈ multibrot d
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ ↑c ∈ multibrotExt d ↔ c ∉ multibrot d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
multibrotExt_coe
[77, 1]
[80, 72]
constructor
c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ (∃ x ∈ multibrot d, x = c) ↔ c ∈ multibrot d
case mp c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ (∃ x ∈ multibrot d, x = c) → c ∈ multibrot d case mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ (∃ x ∈ multibrot d, x = c) ↔ c ∈ multibrot d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
multibrotExt_coe
[77, 1]
[80, 72]
intro ⟨x, m, e⟩
case mp c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ (∃ x ∈ multibrot d, x = c) → c ∈ multibrot d case mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c
case mp c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c x : ℂ m : x ∈ multibrot d e : x = c ⊢ c ∈ multibrot d case mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c
Please generate a tactic in lean4 to solve the state. STATE: case mp c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ (∃ x ∈ multibrot d, x = c) → c ∈ multibrot d case mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
multibrotExt_coe
[77, 1]
[80, 72]
rw [e] at m
case mp c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c x : ℂ m : x ∈ multibrot d e : x = c ⊢ c ∈ multibrot d case mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c
case mp c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c x : ℂ m : c ∈ multibrot d e : x = c ⊢ c ∈ multibrot d case mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c
Please generate a tactic in lean4 to solve the state. STATE: case mp c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c x : ℂ m : x ∈ multibrot d e : x = c ⊢ c ∈ multibrot d case mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
multibrotExt_coe
[77, 1]
[80, 72]
exact m
case mp c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c x : ℂ m : c ∈ multibrot d e : x = c ⊢ c ∈ multibrot d case mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c
case mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c
Please generate a tactic in lean4 to solve the state. STATE: case mp c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c x : ℂ m : c ∈ multibrot d e : x = c ⊢ c ∈ multibrot d case mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
multibrotExt_coe
[77, 1]
[80, 72]
intro m
case mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c
case mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ m : c ∈ multibrot d ⊢ ∃ x ∈ multibrot d, x = c
Please generate a tactic in lean4 to solve the state. STATE: case mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ c ∈ multibrot d → ∃ x ∈ multibrot d, x = c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
multibrotExt_coe
[77, 1]
[80, 72]
use c, m
case mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ m : c ∈ multibrot d ⊢ ∃ x ∈ multibrot d, x = c
no goals
Please generate a tactic in lean4 to solve the state. STATE: case mpr c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ m : c ∈ multibrot d ⊢ ∃ x ∈ multibrot d, x = c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
coe_preimage_multibrotExt
[81, 1]
[82, 84]
apply Set.ext
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ (fun z => ↑z) ⁻¹' multibrotExt d = (multibrot d)ᶜ
case h c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ∀ (x : ℂ), x ∈ (fun z => ↑z) ⁻¹' multibrotExt d ↔ x ∈ (multibrot d)ᶜ
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ (fun z => ↑z) ⁻¹' multibrotExt d = (multibrot d)ᶜ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
coe_preimage_multibrotExt
[81, 1]
[82, 84]
intro z
case h c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ∀ (x : ℂ), x ∈ (fun z => ↑z) ⁻¹' multibrotExt d ↔ x ∈ (multibrot d)ᶜ
case h c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ ⊢ z ∈ (fun z => ↑z) ⁻¹' multibrotExt d ↔ z ∈ (multibrot d)ᶜ
Please generate a tactic in lean4 to solve the state. STATE: case h c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ∀ (x : ℂ), x ∈ (fun z => ↑z) ⁻¹' multibrotExt d ↔ x ∈ (multibrot d)ᶜ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
coe_preimage_multibrotExt
[81, 1]
[82, 84]
simp only [mem_compl_iff, mem_preimage, multibrotExt_coe]
case h c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ ⊢ z ∈ (fun z => ↑z) ⁻¹' multibrotExt d ↔ z ∈ (multibrot d)ᶜ
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ ⊢ z ∈ (fun z => ↑z) ⁻¹' multibrotExt d ↔ z ∈ (multibrot d)ᶜ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
f_0'
[91, 1]
[92, 62]
simp only [lift_coe', f', zero_pow (d_ne_zero _), zero_add]
c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ f' d c 0 = c
no goals
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) ⊢ f' d c 0 = c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
f_0
[94, 1]
[95, 77]
simp only [f, ← coe_zero, lift_coe', f', zero_pow (d_ne_zero _), zero_add]
c : ℂ d✝ : ℕ inst✝¹ : Fact (2 ≤ d✝) d : ℕ inst✝ : Fact (2 ≤ d) ⊢ f d c 0 = ↑c
no goals
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) ⊢ f d c 0 = ↑c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
deriv_f'
[100, 1]
[103, 43]
have h : HasDerivAt (f' d c) (d * z ^ (d - 1) + 0) z := (hasDerivAt_pow _ _).add (hasDerivAt_const _ _)
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ ⊢ deriv (f' d c) z = ↑d * z ^ (d - 1)
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ h : HasDerivAt (f' d c) (↑d * z ^ (d - 1) + 0) z ⊢ deriv (f' d c) z = ↑d * z ^ (d - 1)
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ ⊢ deriv (f' d c) z = ↑d * z ^ (d - 1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
deriv_f'
[100, 1]
[103, 43]
simp only [add_zero] at h
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ h : HasDerivAt (f' d c) (↑d * z ^ (d - 1) + 0) z ⊢ deriv (f' d c) z = ↑d * z ^ (d - 1)
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ h : HasDerivAt (f' d c) (↑d * z ^ (d - 1)) z ⊢ deriv (f' d c) z = ↑d * z ^ (d - 1)
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ h : HasDerivAt (f' d c) (↑d * z ^ (d - 1) + 0) z ⊢ deriv (f' d c) z = ↑d * z ^ (d - 1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
deriv_f'
[100, 1]
[103, 43]
exact h.deriv
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ h : HasDerivAt (f' d c) (↑d * z ^ (d - 1)) z ⊢ deriv (f' d c) z = ↑d * z ^ (d - 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ h : HasDerivAt (f' d c) (↑d * z ^ (d - 1)) z ⊢ deriv (f' d c) z = ↑d * z ^ (d - 1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
simp only [atInf_basis.tendsto_right_iff, Complex.norm_eq_abs, Set.mem_setOf_eq, forall_true_left, uncurry, Metric.eventually_nhds_prod_iff]
c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ Tendsto (uncurry (f' d)) (𝓝 c ×ˢ atInf) atInf
c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ ∀ (i : ℝ), ∃ ε > 0, ∃ pa, (∀ᶠ (i : ℂ) in atInf, pa i) ∧ ∀ {x : ℂ}, dist x c < ε → ∀ {i_1 : ℂ}, pa i_1 → i < Complex.abs (f' d x i_1)
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ Tendsto (uncurry (f' d)) (𝓝 c ×ˢ atInf) atInf TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
intro r
c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ ∀ (i : ℝ), ∃ ε > 0, ∃ pa, (∀ᶠ (i : ℂ) in atInf, pa i) ∧ ∀ {x : ℂ}, dist x c < ε → ∀ {i_1 : ℂ}, pa i_1 → i < Complex.abs (f' d x i_1)
c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ ⊢ ∃ ε > 0, ∃ pa, (∀ᶠ (i : ℂ) in atInf, pa i) ∧ ∀ {x : ℂ}, dist x c < ε → ∀ {i : ℂ}, pa i → r < Complex.abs (f' d x i)
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ ⊢ ∀ (i : ℝ), ∃ ε > 0, ∃ pa, (∀ᶠ (i : ℂ) in atInf, pa i) ∧ ∀ {x : ℂ}, dist x c < ε → ∀ {i_1 : ℂ}, pa i_1 → i < Complex.abs (f' d x i_1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
use 1, zero_lt_one, fun z ↦ max r 0 + abs c + 1 < abs z
c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ ⊢ ∃ ε > 0, ∃ pa, (∀ᶠ (i : ℂ) in atInf, pa i) ∧ ∀ {x : ℂ}, dist x c < ε → ∀ {i : ℂ}, pa i → r < Complex.abs (f' d x i)
case h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ ⊢ (∀ᶠ (i : ℂ) in atInf, max r 0 + Complex.abs c + 1 < Complex.abs i) ∧ ∀ {x : ℂ}, dist x c < 1 → ∀ {i : ℂ}, max r 0 + Complex.abs c + 1 < Complex.abs i → r < Complex.abs (f' d x i)
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ ⊢ ∃ ε > 0, ∃ pa, (∀ᶠ (i : ℂ) in atInf, pa i) ∧ ∀ {x : ℂ}, dist x c < ε → ∀ {i : ℂ}, pa i → r < Complex.abs (f' d x i) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
constructor
case h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ ⊢ (∀ᶠ (i : ℂ) in atInf, max r 0 + Complex.abs c + 1 < Complex.abs i) ∧ ∀ {x : ℂ}, dist x c < 1 → ∀ {i : ℂ}, max r 0 + Complex.abs c + 1 < Complex.abs i → r < Complex.abs (f' d x i)
case h.left c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ ⊢ ∀ᶠ (i : ℂ) in atInf, max r 0 + Complex.abs c + 1 < Complex.abs i case h.right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ ⊢ ∀ {x : ℂ}, dist x c < 1 → ∀ {i : ℂ}, max r 0 + Complex.abs c + 1 < Complex.abs i → r < Complex.abs (f' d x i)
Please generate a tactic in lean4 to solve the state. STATE: case h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ ⊢ (∀ᶠ (i : ℂ) in atInf, max r 0 + Complex.abs c + 1 < Complex.abs i) ∧ ∀ {x : ℂ}, dist x c < 1 → ∀ {i : ℂ}, max r 0 + Complex.abs c + 1 < Complex.abs i → r < Complex.abs (f' d x i) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
refine (eventually_atInf (max r 0 + abs c + 1)).mp (eventually_of_forall fun w h ↦ ?_)
case h.left c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ ⊢ ∀ᶠ (i : ℂ) in atInf, max r 0 + Complex.abs c + 1 < Complex.abs i
case h.left c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ w : ℂ h : ‖w‖ > max r 0 + Complex.abs c + 1 ⊢ max r 0 + Complex.abs c + 1 < Complex.abs w
Please generate a tactic in lean4 to solve the state. STATE: case h.left c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ ⊢ ∀ᶠ (i : ℂ) in atInf, max r 0 + Complex.abs c + 1 < Complex.abs i TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
simp only [Complex.norm_eq_abs] at h
case h.left c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ w : ℂ h : ‖w‖ > max r 0 + Complex.abs c + 1 ⊢ max r 0 + Complex.abs c + 1 < Complex.abs w
case h.left c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ w : ℂ h : Complex.abs w > max r 0 + Complex.abs c + 1 ⊢ max r 0 + Complex.abs c + 1 < Complex.abs w
Please generate a tactic in lean4 to solve the state. STATE: case h.left c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ w : ℂ h : ‖w‖ > max r 0 + Complex.abs c + 1 ⊢ max r 0 + Complex.abs c + 1 < Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
exact h
case h.left c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ w : ℂ h : Complex.abs w > max r 0 + Complex.abs c + 1 ⊢ max r 0 + Complex.abs c + 1 < Complex.abs w
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.left c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ w : ℂ h : Complex.abs w > max r 0 + Complex.abs c + 1 ⊢ max r 0 + Complex.abs c + 1 < Complex.abs w TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
intro e ec z h
case h.right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ ⊢ ∀ {x : ℂ}, dist x c < 1 → ∀ {i : ℂ}, max r 0 + Complex.abs c + 1 < Complex.abs i → r < Complex.abs (f' d x i)
case h.right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : dist e c < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z ⊢ r < Complex.abs (f' d e z)
Please generate a tactic in lean4 to solve the state. STATE: case h.right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ ⊢ ∀ {x : ℂ}, dist x c < 1 → ∀ {i : ℂ}, max r 0 + Complex.abs c + 1 < Complex.abs i → r < Complex.abs (f' d x i) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
simp only [Complex.dist_eq] at ec
case h.right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : dist e c < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z ⊢ r < Complex.abs (f' d e z)
case h.right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z ⊢ r < Complex.abs (f' d e z)
Please generate a tactic in lean4 to solve the state. STATE: case h.right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : dist e c < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z ⊢ r < Complex.abs (f' d e z) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
have zz : abs z ≤ abs (z ^ d) := by rw [Complex.abs.map_pow] refine le_self_pow ?_ (d_ne_zero _) exact le_trans (le_add_of_nonneg_left (add_nonneg (le_max_right _ _) (Complex.abs.nonneg _))) h.le
case h.right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z ⊢ r < Complex.abs (f' d e z)
case h.right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z zz : Complex.abs z ≤ Complex.abs (z ^ d) ⊢ r < Complex.abs (f' d e z)
Please generate a tactic in lean4 to solve the state. STATE: case h.right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z ⊢ r < Complex.abs (f' d e z) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
calc abs (f' d e z) _ = abs (z ^ d + e) := rfl _ = abs (z ^ d + (c + (e - c))) := by ring_nf _ ≥ abs (z ^ d) - abs (c + (e - c)) := by bound _ ≥ abs (z ^ d) - (abs c + abs (e - c)) := by bound _ ≥ abs z - (abs c + 1) := by bound _ > max r 0 + abs c + 1 - (abs c + 1) := by bound _ = max r 0 := by ring_nf _ ≥ r := le_max_left _ _
case h.right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z zz : Complex.abs z ≤ Complex.abs (z ^ d) ⊢ r < Complex.abs (f' d e z)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.right c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z zz : Complex.abs z ≤ Complex.abs (z ^ d) ⊢ r < Complex.abs (f' d e z) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
rw [Complex.abs.map_pow]
c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z ⊢ Complex.abs z ≤ Complex.abs (z ^ d)
c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z ⊢ Complex.abs z ≤ Complex.abs z ^ d
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z ⊢ Complex.abs z ≤ Complex.abs (z ^ d) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
refine le_self_pow ?_ (d_ne_zero _)
c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z ⊢ Complex.abs z ≤ Complex.abs z ^ d
c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z ⊢ 1 ≤ Complex.abs z
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z ⊢ Complex.abs z ≤ Complex.abs z ^ d TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
exact le_trans (le_add_of_nonneg_left (add_nonneg (le_max_right _ _) (Complex.abs.nonneg _))) h.le
c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z ⊢ 1 ≤ Complex.abs z
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z ⊢ 1 ≤ Complex.abs z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
ring_nf
c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z zz : Complex.abs z ≤ Complex.abs (z ^ d) ⊢ Complex.abs (z ^ d + e) = Complex.abs (z ^ d + (c + (e - c)))
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z zz : Complex.abs z ≤ Complex.abs (z ^ d) ⊢ Complex.abs (z ^ d + e) = Complex.abs (z ^ d + (c + (e - c))) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
bound
c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z zz : Complex.abs z ≤ Complex.abs (z ^ d) ⊢ Complex.abs (z ^ d + (c + (e - c))) ≥ Complex.abs (z ^ d) - Complex.abs (c + (e - c))
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z zz : Complex.abs z ≤ Complex.abs (z ^ d) ⊢ Complex.abs (z ^ d + (c + (e - c))) ≥ Complex.abs (z ^ d) - Complex.abs (c + (e - c)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
bound
c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z zz : Complex.abs z ≤ Complex.abs (z ^ d) ⊢ Complex.abs (z ^ d) - Complex.abs (c + (e - c)) ≥ Complex.abs (z ^ d) - (Complex.abs c + Complex.abs (e - c))
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z zz : Complex.abs z ≤ Complex.abs (z ^ d) ⊢ Complex.abs (z ^ d) - Complex.abs (c + (e - c)) ≥ Complex.abs (z ^ d) - (Complex.abs c + Complex.abs (e - c)) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
bound
c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z zz : Complex.abs z ≤ Complex.abs (z ^ d) ⊢ Complex.abs (z ^ d) - (Complex.abs c + Complex.abs (e - c)) ≥ Complex.abs z - (Complex.abs c + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z zz : Complex.abs z ≤ Complex.abs (z ^ d) ⊢ Complex.abs (z ^ d) - (Complex.abs c + Complex.abs (e - c)) ≥ Complex.abs z - (Complex.abs c + 1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
bound
c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z zz : Complex.abs z ≤ Complex.abs (z ^ d) ⊢ Complex.abs z - (Complex.abs c + 1) > max r 0 + Complex.abs c + 1 - (Complex.abs c + 1)
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z zz : Complex.abs z ≤ Complex.abs (z ^ d) ⊢ Complex.abs z - (Complex.abs c + 1) > max r 0 + Complex.abs c + 1 - (Complex.abs c + 1) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
tendsto_f'_atInf
[105, 1]
[125, 31]
ring_nf
c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z zz : Complex.abs z ≤ Complex.abs (z ^ d) ⊢ max r 0 + Complex.abs c + 1 - (Complex.abs c + 1) = max r 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c : ℂ r : ℝ e : ℂ ec : Complex.abs (e - c) < 1 z : ℂ h : max r 0 + Complex.abs c + 1 < Complex.abs z zz : Complex.abs z ≤ Complex.abs (z ^ d) ⊢ max r 0 + Complex.abs c + 1 - (Complex.abs c + 1) = max r 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
writtenInExtChartAt_coe_f
[130, 1]
[132, 94]
simp only [writtenInExtChartAt, f, Function.comp, lift_coe', RiemannSphere.extChartAt_coe, PartialEquiv.symm_symm, coePartialEquiv_apply, coePartialEquiv_symm_apply, toComplex_coe]
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ ⊢ writtenInExtChartAt I I (↑z) (f d c) = f' d c
no goals
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ ⊢ writtenInExtChartAt I I (↑z) (f d c) = f' d c TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
fl_f
[134, 1]
[155, 10]
funext c z
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ fl (f d) ∞ = fun c z => z ^ d / (1 + c * z ^ d)
case h.h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ ⊢ fl (f d) ∞ c z = z ^ d / (1 + c * z ^ d)
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ fl (f d) ∞ = fun c z => z ^ d / (1 + c * z ^ d) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
fl_f
[134, 1]
[155, 10]
simp only [fl, RiemannSphere.extChartAt_inf, Function.comp, invEquiv_apply, PartialEquiv.trans_apply, Equiv.toPartialEquiv_apply, PartialEquiv.coe_trans_symm, coePartialEquiv_symm_apply, PartialEquiv.symm_symm, coePartialEquiv_apply, Equiv.toPartialEquiv_symm_apply, invEquiv_symm, RiemannSphere.inv_inf, toComplex_zero, add_zero, sub_zero]
case h.h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ ⊢ fl (f d) ∞ c z = z ^ d / (1 + c * z ^ d)
case h.h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ ⊢ (f d c (↑z)⁻¹)⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
Please generate a tactic in lean4 to solve the state. STATE: case h.h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ ⊢ fl (f d) ∞ c z = z ^ d / (1 + c * z ^ d) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
fl_f
[134, 1]
[155, 10]
by_cases z0 : z = 0
case h.h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ ⊢ (f d c (↑z)⁻¹)⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
case pos c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : z = 0 ⊢ (f d c (↑z)⁻¹)⁻¹.toComplex = z ^ d / (1 + c * z ^ d) case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 ⊢ (f d c (↑z)⁻¹)⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
Please generate a tactic in lean4 to solve the state. STATE: case h.h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ ⊢ (f d c (↑z)⁻¹)⁻¹.toComplex = z ^ d / (1 + c * z ^ d) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
fl_f
[134, 1]
[155, 10]
simp only [f, f', inv_coe z0, lift_coe', inv_pow]
case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 ⊢ (f d c (↑z)⁻¹)⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 ⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
Please generate a tactic in lean4 to solve the state. STATE: case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 ⊢ (f d c (↑z)⁻¹)⁻¹.toComplex = z ^ d / (1 + c * z ^ d) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
fl_f
[134, 1]
[155, 10]
have zd := pow_ne_zero d z0
case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 ⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 ⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
Please generate a tactic in lean4 to solve the state. STATE: case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 ⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
fl_f
[134, 1]
[155, 10]
by_cases h : (z ^ d)⁻¹ + c = 0
case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 ⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
case pos c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : (z ^ d)⁻¹ + c = 0 ⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d) case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : ¬(z ^ d)⁻¹ + c = 0 ⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
Please generate a tactic in lean4 to solve the state. STATE: case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 ⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
fl_f
[134, 1]
[155, 10]
rw [inv_coe h, toComplex_coe, eq_div_iff, inv_mul_eq_iff_eq_mul₀ h, right_distrib, inv_mul_cancel zd]
case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : ¬(z ^ d)⁻¹ + c = 0 ⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : ¬(z ^ d)⁻¹ + c = 0 ⊢ 1 + c * z ^ d ≠ 0
Please generate a tactic in lean4 to solve the state. STATE: case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : ¬(z ^ d)⁻¹ + c = 0 ⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
fl_f
[134, 1]
[155, 10]
contrapose h
case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : ¬(z ^ d)⁻¹ + c = 0 ⊢ 1 + c * z ^ d ≠ 0
case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : ¬1 + c * z ^ d ≠ 0 ⊢ ¬¬(z ^ d)⁻¹ + c = 0
Please generate a tactic in lean4 to solve the state. STATE: case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : ¬(z ^ d)⁻¹ + c = 0 ⊢ 1 + c * z ^ d ≠ 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
fl_f
[134, 1]
[155, 10]
rw [not_not]
case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : ¬1 + c * z ^ d ≠ 0 ⊢ ¬¬(z ^ d)⁻¹ + c = 0
case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : ¬1 + c * z ^ d ≠ 0 ⊢ (z ^ d)⁻¹ + c = 0
Please generate a tactic in lean4 to solve the state. STATE: case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : ¬1 + c * z ^ d ≠ 0 ⊢ ¬¬(z ^ d)⁻¹ + c = 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
fl_f
[134, 1]
[155, 10]
rw [not_not, add_comm, add_eq_zero_iff_eq_neg, ← eq_div_iff zd, neg_div, ← inv_eq_one_div, ← add_eq_zero_iff_eq_neg, add_comm] at h
case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : ¬1 + c * z ^ d ≠ 0 ⊢ (z ^ d)⁻¹ + c = 0
case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : (z ^ d)⁻¹ + c = 0 ⊢ (z ^ d)⁻¹ + c = 0
Please generate a tactic in lean4 to solve the state. STATE: case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : ¬1 + c * z ^ d ≠ 0 ⊢ (z ^ d)⁻¹ + c = 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
fl_f
[134, 1]
[155, 10]
exact h
case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : (z ^ d)⁻¹ + c = 0 ⊢ (z ^ d)⁻¹ + c = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : (z ^ d)⁻¹ + c = 0 ⊢ (z ^ d)⁻¹ + c = 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
fl_f
[134, 1]
[155, 10]
simp only [z0, coe_zero, inv_zero', f, lift_inf', RiemannSphere.inv_inf, toComplex_zero, zero_pow (d_ne_zero _), zero_div]
case pos c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : z = 0 ⊢ (f d c (↑z)⁻¹)⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : z = 0 ⊢ (f d c (↑z)⁻¹)⁻¹.toComplex = z ^ d / (1 + c * z ^ d) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
fl_f
[134, 1]
[155, 10]
simp only [h, coe_zero, inv_zero', toComplex_inf]
case pos c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : (z ^ d)⁻¹ + c = 0 ⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d)
case pos c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : (z ^ d)⁻¹ + c = 0 ⊢ 0 = z ^ d / (1 + c * z ^ d)
Please generate a tactic in lean4 to solve the state. STATE: case pos c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : (z ^ d)⁻¹ + c = 0 ⊢ (↑((z ^ d)⁻¹ + c))⁻¹.toComplex = z ^ d / (1 + c * z ^ d) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
fl_f
[134, 1]
[155, 10]
simp only [← add_eq_zero_iff_neg_eq.mp h, neg_mul, inv_mul_cancel zd, ← sub_eq_add_neg, sub_self, div_zero]
case pos c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : (z ^ d)⁻¹ + c = 0 ⊢ 0 = z ^ d / (1 + c * z ^ d)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ z0 : ¬z = 0 zd : z ^ d ≠ 0 h : (z ^ d)⁻¹ + c = 0 ⊢ 0 = z ^ d / (1 + c * z ^ d) TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
gl_f
[161, 1]
[166, 13]
simp only [fl_f, gl, g]
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ ⊢ g (fl (f d) ∞ c) d z = gl d c z
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ ⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ ⊢ g (fl (f d) ∞ c) d z = gl d c z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
gl_f
[161, 1]
[166, 13]
by_cases z0 : z = 0
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ ⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹
case pos c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ z0 : z = 0 ⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹ case neg c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ z0 : ¬z = 0 ⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ ⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
gl_f
[161, 1]
[166, 13]
simp only [if_pos, z0, zero_pow (d_ne_zero _), MulZeroClass.mul_zero, add_zero, inv_one]
case pos c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ z0 : z = 0 ⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹ case neg c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ z0 : ¬z = 0 ⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹
case neg c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ z0 : ¬z = 0 ⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹
Please generate a tactic in lean4 to solve the state. STATE: case pos c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ z0 : z = 0 ⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹ case neg c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ z0 : ¬z = 0 ⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
gl_f
[161, 1]
[166, 13]
rw [if_neg z0, div_eq_mul_inv _ (_ + _), mul_comm, mul_div_assoc, div_self (pow_ne_zero _ z0), mul_one]
case neg c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ z0 : ¬z = 0 ⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) z : ℂ z0 : ¬z = 0 ⊢ (if z = 0 then 1 else z ^ d / (1 + c * z ^ d) / z ^ d) = (1 + c * z ^ d)⁻¹ TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
analyticAt_gl
[168, 1]
[171, 36]
apply (analyticAt_const.add (analyticAt_const.mul ((analyticAt_id _ _).pow _))).inv
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ AnalyticAt ℂ (gl d c) 0
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ((fun x => 1) + fun x => c * id x ^ d) 0 ≠ 0
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ AnalyticAt ℂ (gl d c) 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
analyticAt_gl
[168, 1]
[171, 36]
simp only [Pi.pow_apply, id_eq, Pi.add_apply, ne_eq, zero_pow (d_ne_zero _), mul_zero, add_zero, one_ne_zero, not_false_eq_true]
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ((fun x => 1) + fun x => c * id x ^ d) 0 ≠ 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ((fun x => 1) + fun x => c * id x ^ d) 0 ≠ 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
fl_f'
[173, 1]
[174, 85]
funext c z
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ fl (f d) ∞ = fun c z => (z - 0) ^ d • gl d c z
case h.h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ ⊢ fl (f d) ∞ c z = (z - 0) ^ d • gl d c z
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ fl (f d) ∞ = fun c z => (z - 0) ^ d • gl d c z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
fl_f'
[173, 1]
[174, 85]
simp only [fl_f, gl, sub_zero, Algebra.id.smul_eq_mul, div_eq_mul_inv]
case h.h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ ⊢ fl (f d) ∞ c z = (z - 0) ^ d • gl d c z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.h c✝ : ℂ d : ℕ inst✝ : Fact (2 ≤ d) c z : ℂ ⊢ fl (f d) ∞ c z = (z - 0) ^ d • gl d c z TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
gl_zero
[176, 1]
[177, 74]
simp only [gl, zero_pow (d_ne_zero _), MulZeroClass.mul_zero]
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ gl d c 0 = 1
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ (1 + 0)⁻¹ = 1
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ gl d c 0 = 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
gl_zero
[176, 1]
[177, 74]
norm_num
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ (1 + 0)⁻¹ = 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ (1 + 0)⁻¹ = 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
gl_frequently_ne_zero
[179, 1]
[181, 20]
refine (analyticAt_gl.continuousAt.eventually_ne ?_).frequently
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ∃ᶠ (z : ℂ) in 𝓝 0, gl d c z ≠ 0
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ gl d c 0 ≠ 0
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ ∃ᶠ (z : ℂ) in 𝓝 0, gl d c z ≠ 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
gl_frequently_ne_zero
[179, 1]
[181, 20]
simp only [gl_zero]
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ gl d c 0 ≠ 0
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ 1 ≠ 0
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ gl d c 0 ≠ 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
gl_frequently_ne_zero
[179, 1]
[181, 20]
exact one_ne_zero
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ 1 ≠ 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ 1 ≠ 0 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
fc_f
[183, 1]
[185, 49]
rw [fl_f', analyticAt_gl.monomial_mul_leadingCoeff gl_frequently_ne_zero, leadingCoeff_of_ne_zero]
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ leadingCoeff (fl (f d) ∞ c) 0 = 1
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ gl d c 0 = 1 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ gl d c 0 ≠ 0
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ leadingCoeff (fl (f d) ∞ c) 0 = 1 TACTIC:
https://github.com/girving/ray.git
0be790285dd0fce78913b0cb9bddaffa94bd25f9
Ray/Dynamics/Multibrot/Basic.lean
fc_f
[183, 1]
[185, 49]
exact gl_zero
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ gl d c 0 = 1 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ gl d c 0 ≠ 0
c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ gl d c 0 ≠ 0
Please generate a tactic in lean4 to solve the state. STATE: c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ gl d c 0 = 1 c : ℂ d : ℕ inst✝ : Fact (2 ≤ d) ⊢ gl d c 0 ≠ 0 TACTIC: