url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
| input
stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | intro p m | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
⊢ AnalyticOn ℂ (uncurry f) tu | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
p : ℂ × ℂ
m : p ∈ tu
⊢ AnalyticAt ℂ (uncurry f) p | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
⊢ AnalyticOn ℂ (uncurry f) tu
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | rcases Set.mem_iUnion.mp m with ⟨i, m⟩ | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
p : ℂ × ℂ
m : p ∈ tu
⊢ AnalyticAt ℂ (uncurry f) p | case intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
p : ℂ × ℂ
m✝ : p ∈ tu
i : I
m : p ∈ t i
⊢ AnalyticAt ℂ (uncurry f) p | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
p : ℂ × ℂ
m : p ∈ tu
⊢ AnalyticAt ℂ (uncurry f) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | exact (s i).fa _ m | case intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
p : ℂ × ℂ
m✝ : p ∈ tu
i : I
m : p ∈ t i
⊢ AnalyticAt ℂ (uncurry f) p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
p : ℂ × ℂ
m✝ : p ∈ tu
i : I
m : p ∈ t i
⊢ AnalyticAt ℂ (uncurry f) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | intro c m | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
⊢ ∀ {c : ℂ}, c ∈ ⋃ i, u i → SuperNear (f c) d {z | (c, z) ∈ tu} | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m : c ∈ ⋃ i, u i
⊢ SuperNear (f c) d {z | (c, z) ∈ tu} | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
⊢ ∀ {c : ℂ}, c ∈ ⋃ i, u i → SuperNear (f c) d {z | (c, z) ∈ tu}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | rcases Set.mem_iUnion.mp m with ⟨i, m⟩ | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m : c ∈ ⋃ i, u i
⊢ SuperNear (f c) d {z | (c, z) ∈ tu} | case intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝ : c ∈ ⋃ i, u i
i : I
m : c ∈ u i
⊢ SuperNear (f c) d {z | (c, z) ∈ tu} | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m : c ∈ ⋃ i, u i
⊢ SuperNear (f c) d {z | (c, z) ∈ tu}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | have s := (s i).s m | case intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝ : c ∈ ⋃ i, u i
i : I
m : c ∈ u i
⊢ SuperNear (f c) d {z | (c, z) ∈ tu} | case intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝ : c ∈ ⋃ i, u i
i : I
m : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
⊢ SuperNear (f c) d {z | (c, z) ∈ tu} | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝ : c ∈ ⋃ i, u i
i : I
m : c ∈ u i
⊢ SuperNear (f c) d {z | (c, z) ∈ tu}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | exact
{ d2 := s.d2
fa0 := s.fa0
fd := s.fd
fc := s.fc
o := o.snd_preimage c
t0 := Set.subset_iUnion _ i s.t0
t2 := by intro z m; rcases sm m with ⟨u, m, _, s⟩; exact s.t2 m
fa := by intro z m; rcases sm m with ⟨u, m, _, s⟩; exact s.fa _ m
ft := by intro z m; rcases sm m with ⟨u, m, us, s⟩; exact us (s.ft m)
gs' := by intro z z0 m; rcases sm m with ⟨u, m, _, s⟩; exact s.gs' z0 m } | case intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝ : c ∈ ⋃ i, u i
i : I
m : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
⊢ SuperNear (f c) d {z | (c, z) ∈ tu} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝ : c ∈ ⋃ i, u i
i : I
m : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
⊢ SuperNear (f c) d {z | (c, z) ∈ tu}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | intro z m | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝ : c ∈ ⋃ i, u i
i : I
m : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
⊢ ∀ {z : ℂ}, z ∈ {z | (c, z) ∈ tu} → Complex.abs z ≤ 1 / 2 | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝¹ : c ∈ ⋃ i, u i
i : I
m✝ : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
m : z ∈ {z | (c, z) ∈ tu}
⊢ Complex.abs z ≤ 1 / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝ : c ∈ ⋃ i, u i
i : I
m : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
⊢ ∀ {z : ℂ}, z ∈ {z | (c, z) ∈ tu} → Complex.abs z ≤ 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | rcases sm m with ⟨u, m, _, s⟩ | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝¹ : c ∈ ⋃ i, u i
i : I
m✝ : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
m : z ∈ {z | (c, z) ∈ tu}
⊢ Complex.abs z ≤ 1 / 2 | case intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝¹ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u✝ : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝¹ : ∀ (i : I), SuperNearC f d (u✝ i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝² : c ∈ ⋃ i, u✝ i
i : I
m✝¹ : c ∈ u✝ i
s✝ : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
m✝ : z ∈ {z | (c, z) ∈ tu}
u : Set ℂ
m : z ∈ u
left✝ : u ⊆ {z | (c, z) ∈ tu}
s : SuperNear (f c) d u
⊢ Complex.abs z ≤ 1 / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝¹ : c ∈ ⋃ i, u i
i : I
m✝ : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
m : z ∈ {z | (c, z) ∈ tu}
⊢ Complex.abs z ≤ 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | exact s.t2 m | case intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝¹ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u✝ : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝¹ : ∀ (i : I), SuperNearC f d (u✝ i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝² : c ∈ ⋃ i, u✝ i
i : I
m✝¹ : c ∈ u✝ i
s✝ : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
m✝ : z ∈ {z | (c, z) ∈ tu}
u : Set ℂ
m : z ∈ u
left✝ : u ⊆ {z | (c, z) ∈ tu}
s : SuperNear (f c) d u
⊢ Complex.abs z ≤ 1 / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝¹ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u✝ : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝¹ : ∀ (i : I), SuperNearC f d (u✝ i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝² : c ∈ ⋃ i, u✝ i
i : I
m✝¹ : c ∈ u✝ i
s✝ : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
m✝ : z ∈ {z | (c, z) ∈ tu}
u : Set ℂ
m : z ∈ u
left✝ : u ⊆ {z | (c, z) ∈ tu}
s : SuperNear (f c) d u
⊢ Complex.abs z ≤ 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | intro z m | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝ : c ∈ ⋃ i, u i
i : I
m : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
⊢ AnalyticOn ℂ (f c) {z | (c, z) ∈ tu} | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝¹ : c ∈ ⋃ i, u i
i : I
m✝ : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
m : z ∈ {z | (c, z) ∈ tu}
⊢ AnalyticAt ℂ (f c) z | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝ : c ∈ ⋃ i, u i
i : I
m : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
⊢ AnalyticOn ℂ (f c) {z | (c, z) ∈ tu}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | rcases sm m with ⟨u, m, _, s⟩ | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝¹ : c ∈ ⋃ i, u i
i : I
m✝ : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
m : z ∈ {z | (c, z) ∈ tu}
⊢ AnalyticAt ℂ (f c) z | case intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝¹ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u✝ : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝¹ : ∀ (i : I), SuperNearC f d (u✝ i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝² : c ∈ ⋃ i, u✝ i
i : I
m✝¹ : c ∈ u✝ i
s✝ : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
m✝ : z ∈ {z | (c, z) ∈ tu}
u : Set ℂ
m : z ∈ u
left✝ : u ⊆ {z | (c, z) ∈ tu}
s : SuperNear (f c) d u
⊢ AnalyticAt ℂ (f c) z | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝¹ : c ∈ ⋃ i, u i
i : I
m✝ : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
m : z ∈ {z | (c, z) ∈ tu}
⊢ AnalyticAt ℂ (f c) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | exact s.fa _ m | case intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝¹ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u✝ : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝¹ : ∀ (i : I), SuperNearC f d (u✝ i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝² : c ∈ ⋃ i, u✝ i
i : I
m✝¹ : c ∈ u✝ i
s✝ : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
m✝ : z ∈ {z | (c, z) ∈ tu}
u : Set ℂ
m : z ∈ u
left✝ : u ⊆ {z | (c, z) ∈ tu}
s : SuperNear (f c) d u
⊢ AnalyticAt ℂ (f c) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝¹ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u✝ : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝¹ : ∀ (i : I), SuperNearC f d (u✝ i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝² : c ∈ ⋃ i, u✝ i
i : I
m✝¹ : c ∈ u✝ i
s✝ : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
m✝ : z ∈ {z | (c, z) ∈ tu}
u : Set ℂ
m : z ∈ u
left✝ : u ⊆ {z | (c, z) ∈ tu}
s : SuperNear (f c) d u
⊢ AnalyticAt ℂ (f c) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | intro z m | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝ : c ∈ ⋃ i, u i
i : I
m : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
⊢ MapsTo (f c) {z | (c, z) ∈ tu} {z | (c, z) ∈ tu} | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝¹ : c ∈ ⋃ i, u i
i : I
m✝ : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
m : z ∈ {z | (c, z) ∈ tu}
⊢ f c z ∈ {z | (c, z) ∈ tu} | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝ : c ∈ ⋃ i, u i
i : I
m : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
⊢ MapsTo (f c) {z | (c, z) ∈ tu} {z | (c, z) ∈ tu}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | rcases sm m with ⟨u, m, us, s⟩ | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝¹ : c ∈ ⋃ i, u i
i : I
m✝ : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
m : z ∈ {z | (c, z) ∈ tu}
⊢ f c z ∈ {z | (c, z) ∈ tu} | case intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝¹ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u✝ : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝¹ : ∀ (i : I), SuperNearC f d (u✝ i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝² : c ∈ ⋃ i, u✝ i
i : I
m✝¹ : c ∈ u✝ i
s✝ : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
m✝ : z ∈ {z | (c, z) ∈ tu}
u : Set ℂ
m : z ∈ u
us : u ⊆ {z | (c, z) ∈ tu}
s : SuperNear (f c) d u
⊢ f c z ∈ {z | (c, z) ∈ tu} | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝¹ : c ∈ ⋃ i, u i
i : I
m✝ : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
m : z ∈ {z | (c, z) ∈ tu}
⊢ f c z ∈ {z | (c, z) ∈ tu}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | exact us (s.ft m) | case intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝¹ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u✝ : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝¹ : ∀ (i : I), SuperNearC f d (u✝ i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝² : c ∈ ⋃ i, u✝ i
i : I
m✝¹ : c ∈ u✝ i
s✝ : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
m✝ : z ∈ {z | (c, z) ∈ tu}
u : Set ℂ
m : z ∈ u
us : u ⊆ {z | (c, z) ∈ tu}
s : SuperNear (f c) d u
⊢ f c z ∈ {z | (c, z) ∈ tu} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝¹ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u✝ : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝¹ : ∀ (i : I), SuperNearC f d (u✝ i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝² : c ∈ ⋃ i, u✝ i
i : I
m✝¹ : c ∈ u✝ i
s✝ : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
m✝ : z ∈ {z | (c, z) ∈ tu}
u : Set ℂ
m : z ∈ u
us : u ⊆ {z | (c, z) ∈ tu}
s : SuperNear (f c) d u
⊢ f c z ∈ {z | (c, z) ∈ tu}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | intro z z0 m | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝ : c ∈ ⋃ i, u i
i : I
m : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
⊢ ∀ {z : ℂ}, z ≠ 0 → z ∈ {z | (c, z) ∈ tu} → Complex.abs (f c z / z ^ d - 1) ≤ 1 / 4 | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝¹ : c ∈ ⋃ i, u i
i : I
m✝ : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
z0 : z ≠ 0
m : z ∈ {z | (c, z) ∈ tu}
⊢ Complex.abs (f c z / z ^ d - 1) ≤ 1 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝ : c ∈ ⋃ i, u i
i : I
m : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
⊢ ∀ {z : ℂ}, z ≠ 0 → z ∈ {z | (c, z) ∈ tu} → Complex.abs (f c z / z ^ d - 1) ≤ 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | rcases sm m with ⟨u, m, _, s⟩ | f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝¹ : c ∈ ⋃ i, u i
i : I
m✝ : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
z0 : z ≠ 0
m : z ∈ {z | (c, z) ∈ tu}
⊢ Complex.abs (f c z / z ^ d - 1) ≤ 1 / 4 | case intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝¹ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u✝ : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝¹ : ∀ (i : I), SuperNearC f d (u✝ i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝² : c ∈ ⋃ i, u✝ i
i : I
m✝¹ : c ∈ u✝ i
s✝ : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
z0 : z ≠ 0
m✝ : z ∈ {z | (c, z) ∈ tu}
u : Set ℂ
m : z ∈ u
left✝ : u ⊆ {z | (c, z) ∈ tu}
s : SuperNear (f c) d u
⊢ Complex.abs (f c z / z ^ d - 1) ≤ 1 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u✝ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝ : ∀ (i : I), SuperNearC f d (u i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝¹ : c ∈ ⋃ i, u i
i : I
m✝ : c ∈ u i
s : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
z0 : z ≠ 0
m : z ∈ {z | (c, z) ∈ tu}
⊢ Complex.abs (f c z / z ^ d - 1) ≤ 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperNearC.union | [544, 1] | [573, 88] | exact s.gs' z0 m | case intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝¹ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u✝ : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝¹ : ∀ (i : I), SuperNearC f d (u✝ i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝² : c ∈ ⋃ i, u✝ i
i : I
m✝¹ : c ∈ u✝ i
s✝ : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
z0 : z ≠ 0
m✝ : z ∈ {z | (c, z) ∈ tu}
u : Set ℂ
m : z ∈ u
left✝ : u ⊆ {z | (c, z) ∈ tu}
s : SuperNear (f c) d u
⊢ Complex.abs (f c z / z ^ d - 1) ≤ 1 / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u✝¹ : Set ℂ
t✝ : Set (ℂ × ℂ)
I : Type
u✝ : I → Set ℂ
t : I → Set (ℂ × ℂ)
s✝¹ : ∀ (i : I), SuperNearC f d (u✝ i) (t i)
tu : Set (ℂ × ℂ) := ⋃ i, t i
o : IsOpen tu
sm : ∀ {c z : ℂ}, (c, z) ∈ tu → ∃ u, z ∈ u ∧ u ⊆ {z | (c, z) ∈ tu} ∧ SuperNear (f c) d u
c : ℂ
m✝² : c ∈ ⋃ i, u✝ i
i : I
m✝¹ : c ∈ u✝ i
s✝ : SuperNear (f c) d {z | (c, z) ∈ t i}
z : ℂ
z0 : z ≠ 0
m✝ : z ∈ {z | (c, z) ∈ tu}
u : Set ℂ
m : z ∈ u
left✝ : u ⊆ {z | (c, z) ∈ tu}
s : SuperNear (f c) d u
⊢ Complex.abs (f c z / z ^ d - 1) ≤ 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | set r := fun c : u ↦ choose (h _ c.mem) | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
⊢ ∃ t ⊆ w, SuperNearC f d u t | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
⊢ ∃ t ⊆ w, SuperNearC f d u t | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
⊢ ∃ t ⊆ w, SuperNearC f d u t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | set v := fun c : u ↦ ball (c : ℂ) (r c) | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
⊢ ∃ t ⊆ w, SuperNearC f d u t | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
⊢ ∃ t ⊆ w, SuperNearC f d u t | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
⊢ ∃ t ⊆ w, SuperNearC f d u t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | set t := fun c : u ↦ ball ((c : ℂ), (0 : ℂ)) (r c) | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
⊢ ∃ t ⊆ w, SuperNearC f d u t | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
⊢ ∃ t ⊆ w, SuperNearC f d u t | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
⊢ ∃ t ⊆ w, SuperNearC f d u t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | use⋃ c : u, t c | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
⊢ ∃ t ⊆ w, SuperNearC f d u t | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
⊢ ⋃ c, t c ⊆ w ∧ SuperNearC f d u (⋃ c, t c) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
⊢ ∃ t ⊆ w, SuperNearC f d u t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | have tw : (⋃ c : u, t c) ⊆ w := by
apply Set.iUnion_subset; intro i; rcases choose_spec (h _ i.mem) with ⟨_, _, rw, _⟩; exact rw | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
⊢ ⋃ c, t c ⊆ w ∧ SuperNearC f d u (⋃ c, t c) | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
⊢ ⋃ c, t c ⊆ w ∧ SuperNearC f d u (⋃ c, t c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
⊢ ⋃ c, t c ⊆ w ∧ SuperNearC f d u (⋃ c, t c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | have si : ∀ c : u, SuperNearC f d (v c) (t c) := by
intro i; rcases choose_spec (h _ i.mem) with ⟨_, _, _, s⟩; exact s | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
⊢ ⋃ c, t c ⊆ w ∧ SuperNearC f d u (⋃ c, t c) | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
si : ∀ (c : ↑u), SuperNearC f d (v c) (t c)
⊢ ⋃ c, t c ⊆ w ∧ SuperNearC f d u (⋃ c, t c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
⊢ ⋃ c, t c ⊆ w ∧ SuperNearC f d u (⋃ c, t c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | have s := SuperNearC.union si | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
si : ∀ (c : ↑u), SuperNearC f d (v c) (t c)
⊢ ⋃ c, t c ⊆ w ∧ SuperNearC f d u (⋃ c, t c) | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s✝ : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
si : ∀ (c : ↑u), SuperNearC f d (v c) (t c)
s : SuperNearC f d (⋃ i, v i) (⋃ i, t i)
⊢ ⋃ c, t c ⊆ w ∧ SuperNearC f d u (⋃ c, t c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
si : ∀ (c : ↑u), SuperNearC f d (v c) (t c)
⊢ ⋃ c, t c ⊆ w ∧ SuperNearC f d u (⋃ c, t c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | rw [← e] at s | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s✝ : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
si : ∀ (c : ↑u), SuperNearC f d (v c) (t c)
s : SuperNearC f d (⋃ i, v i) (⋃ i, t i)
⊢ ⋃ c, t c ⊆ w ∧ SuperNearC f d u (⋃ c, t c) | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s✝ : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
si : ∀ (c : ↑u), SuperNearC f d (v c) (t c)
s : SuperNearC f d u (⋃ i, t i)
⊢ ⋃ c, t c ⊆ w ∧ SuperNearC f d u (⋃ c, t c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s✝ : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
si : ∀ (c : ↑u), SuperNearC f d (v c) (t c)
s : SuperNearC f d (⋃ i, v i) (⋃ i, t i)
⊢ ⋃ c, t c ⊆ w ∧ SuperNearC f d u (⋃ c, t c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | exact ⟨tw, s⟩ | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s✝ : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
si : ∀ (c : ↑u), SuperNearC f d (v c) (t c)
s : SuperNearC f d u (⋃ i, t i)
⊢ ⋃ c, t c ⊆ w ∧ SuperNearC f d u (⋃ c, t c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s✝ : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
si : ∀ (c : ↑u), SuperNearC f d (v c) (t c)
s : SuperNearC f d u (⋃ i, t i)
⊢ ⋃ c, t c ⊆ w ∧ SuperNearC f d u (⋃ c, t c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | intro c m | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
⊢ ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
⊢ ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | rcases(s.fa m).exists_ball_analyticOn with ⟨r0, r0p, fa⟩ | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | case intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | rcases Metric.isOpen_iff.mp s.o c m with ⟨r1, r1p, rc⟩ | case intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | case intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc : ball c r1 ⊆ u
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | set r2 := min r0 r1 | case intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc : ball c r1 ⊆ u
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | case intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc : ball c r1 ⊆ u
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | have fa := fa.mono (Metric.ball_subset_ball (min_le_left r0 r1)) | case intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | case intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | have rc : ball c r2 ⊆ u := le_trans (Metric.ball_subset_ball (by bound)) rc | case intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | case intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | have ga := s.ga_of_fa isOpen_ball fa
(by intro p m; simp only [← ball_prod_same, Set.mem_prod] at m; exact rc m.1) | case intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | case intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | rcases Metric.isOpen_iff.mp wo (c, 0) (wc c m) with ⟨r3, r3p, rw⟩ | case intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | case intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | rcases Metric.continuousAt_iff.mp (ga (c, 0) (mem_ball_self (by bound))).continuousAt
(1 / 4) (by norm_num) with ⟨r4, r4p, gs⟩ | case intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | set r := min (min r2 r3) (min r4 (1 / 2)) | case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | have rp : 0 < r := by bound | case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | have rh : r ≤ 1 / 2 := le_trans (min_le_right _ _) (min_le_right _ _) | case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | have rr4 : r ≤ r4 := le_trans (min_le_right _ _) (min_le_left r4 _) | case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | have rc : ball c r ⊆ u := le_trans (Metric.ball_subset_ball (by bound)) rc | case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | have rw : ball (c, 0) r ⊆ w :=
_root_.trans (Metric.ball_subset_ball (le_trans (min_le_left _ _) (min_le_right _ _))) rw | case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | use r, rp, rc, rw | case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r) | case right
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
⊢ SuperNearC f d (ball c r) (ball (c, 0) r) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
⊢ ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | bound | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
⊢ r2 ≤ r1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
⊢ r2 ≤ r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | intro p m | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
⊢ ∀ {p : ℂ × ℂ}, p ∈ ball (c, 0) (min r0 r1) → p.1 ∈ u | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
p : ℂ × ℂ
m : p ∈ ball (c, 0) (min r0 r1)
⊢ p.1 ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
⊢ ∀ {p : ℂ × ℂ}, p ∈ ball (c, 0) (min r0 r1) → p.1 ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | simp only [← ball_prod_same, Set.mem_prod] at m | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
p : ℂ × ℂ
m : p ∈ ball (c, 0) (min r0 r1)
⊢ p.1 ∈ u | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
p : ℂ × ℂ
m : p.1 ∈ ball c (min r0 r1) ∧ p.2 ∈ ball 0 (min r0 r1)
⊢ p.1 ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
p : ℂ × ℂ
m : p ∈ ball (c, 0) (min r0 r1)
⊢ p.1 ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | exact rc m.1 | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
p : ℂ × ℂ
m : p.1 ∈ ball c (min r0 r1) ∧ p.2 ∈ ball 0 (min r0 r1)
⊢ p.1 ∈ u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
p : ℂ × ℂ
m : p.1 ∈ ball c (min r0 r1) ∧ p.2 ∈ ball 0 (min r0 r1)
⊢ p.1 ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | bound | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
⊢ 0 < min r0 r1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
⊢ 0 < min r0 r1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | norm_num | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
⊢ 1 / 4 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
⊢ 1 / 4 > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | bound | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
⊢ 0 < r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
⊢ 0 < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | bound | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
⊢ r ≤ r2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
⊢ r ≤ r2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | intro p m | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
⊢ ∀ {p : ℂ × ℂ}, p ∈ ball (c, 0) r → p.1 ∈ ball c r | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
p : ℂ × ℂ
m : p ∈ ball (c, 0) r
⊢ p.1 ∈ ball c r | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
⊢ ∀ {p : ℂ × ℂ}, p ∈ ball (c, 0) r → p.1 ∈ ball c r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | simp only [← ball_prod_same, Set.mem_prod] at m | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
p : ℂ × ℂ
m : p ∈ ball (c, 0) r
⊢ p.1 ∈ ball c r | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
p : ℂ × ℂ
m : p.1 ∈ ball c r ∧ p.2 ∈ ball 0 r
⊢ p.1 ∈ ball c r | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
p : ℂ × ℂ
m : p ∈ ball (c, 0) r
⊢ p.1 ∈ ball c r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | exact Metric.ball_subset_ball (by linarith) m.1 | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
p : ℂ × ℂ
m : p.1 ∈ ball c r ∧ p.2 ∈ ball 0 r
⊢ p.1 ∈ ball c r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
p : ℂ × ℂ
m : p.1 ∈ ball c r ∧ p.2 ∈ ball 0 r
⊢ p.1 ∈ ball c r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | linarith | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
p : ℂ × ℂ
m : p.1 ∈ ball c r ∧ p.2 ∈ ball 0 r
⊢ r ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
p : ℂ × ℂ
m : p.1 ∈ ball c r ∧ p.2 ∈ ball 0 r
⊢ r ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | intro c' m | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
⊢ ∀ {c_1 : ℂ}, c_1 ∈ ball c r → SuperNear (f c_1) d {z | (c_1, z) ∈ ball (c, 0) r} | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
⊢ SuperNear (f c') d {z | (c', z) ∈ ball (c, 0) r} | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
⊢ ∀ {c_1 : ℂ}, c_1 ∈ ball c r → SuperNear (f c_1) d {z | (c_1, z) ∈ ball (c, 0) r}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | simp only [← ball_prod_same, Set.mem_prod, m, true_and_iff] | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
⊢ SuperNear (f c') d {z | (c', z) ∈ ball (c, 0) r} | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
⊢ SuperNear (f c') d {z | z ∈ ball 0 r} | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
⊢ SuperNear (f c') d {z | (c', z) ∈ ball (c, 0) r}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | apply (s.s (rc m)).super_on_ball rp rh | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
⊢ SuperNear (f c') d {z | z ∈ ball 0 r} | case fa
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
⊢ AnalyticOn ℂ (f c') (ball 0 r)
case gs
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
⊢ ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g (f c') d z - 1) < 1 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
⊢ SuperNear (f c') d {z | z ∈ ball 0 r}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | apply fa.comp₂ analyticOn_const (analyticOn_id _) | case fa
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
⊢ AnalyticOn ℂ (f c') (ball 0 r) | case fa
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
⊢ ∀ x ∈ ball 0 r, (c', x) ∈ ball (c, 0) (min r0 r1) | Please generate a tactic in lean4 to solve the state.
STATE:
case fa
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
⊢ AnalyticOn ℂ (f c') (ball 0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | intro z zm | case fa
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
⊢ ∀ x ∈ ball 0 r, (c', x) ∈ ball (c, 0) (min r0 r1) | case fa
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
z : ℂ
zm : z ∈ ball 0 r
⊢ (c', z) ∈ ball (c, 0) (min r0 r1) | Please generate a tactic in lean4 to solve the state.
STATE:
case fa
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
⊢ ∀ x ∈ ball 0 r, (c', x) ∈ ball (c, 0) (min r0 r1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | apply Metric.ball_subset_ball (by bound : r ≤ r2) | case fa
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
z : ℂ
zm : z ∈ ball 0 r
⊢ (c', z) ∈ ball (c, 0) (min r0 r1) | case fa.a
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
z : ℂ
zm : z ∈ ball 0 r
⊢ (c', z) ∈ ball (c, 0) r | Please generate a tactic in lean4 to solve the state.
STATE:
case fa
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
z : ℂ
zm : z ∈ ball 0 r
⊢ (c', z) ∈ ball (c, 0) (min r0 r1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | simp only [← ball_prod_same, Set.mem_prod, m, true_and_iff] | case fa.a
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
z : ℂ
zm : z ∈ ball 0 r
⊢ (c', z) ∈ ball (c, 0) r | case fa.a
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
z : ℂ
zm : z ∈ ball 0 r
⊢ z ∈ ball 0 r | Please generate a tactic in lean4 to solve the state.
STATE:
case fa.a
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
z : ℂ
zm : z ∈ ball 0 r
⊢ (c', z) ∈ ball (c, 0) r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | exact zm | case fa.a
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
z : ℂ
zm : z ∈ ball 0 r
⊢ z ∈ ball 0 r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case fa.a
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
z : ℂ
zm : z ∈ ball 0 r
⊢ z ∈ ball 0 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | bound | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
z : ℂ
zm : z ∈ ball 0 r
⊢ r ≤ r2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
z : ℂ
zm : z ∈ ball 0 r
⊢ r ≤ r2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | simp only [Complex.dist_eq, Prod.dist_eq, sub_zero, max_lt_iff, and_imp, g2, g0] at gs | case gs
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
⊢ ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g (f c') d z - 1) < 1 / 4 | case gs
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
gs : ∀ {x : ℂ × ℂ}, Complex.abs (x.1 - c) < r4 → Complex.abs x.2 < r4 → Complex.abs (g (f x.1) d x.2 - 1) < 1 / 4
⊢ ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g (f c') d z - 1) < 1 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case gs
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
gs : ∀ {x : ℂ × ℂ}, dist x (c, 0) < r4 → dist (g2 f d x) (g2 f d (c, 0)) < 1 / 4
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
⊢ ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g (f c') d z - 1) < 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | simp only [Metric.mem_ball, Complex.dist_eq] at m | case gs
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
gs : ∀ {x : ℂ × ℂ}, Complex.abs (x.1 - c) < r4 → Complex.abs x.2 < r4 → Complex.abs (g (f x.1) d x.2 - 1) < 1 / 4
⊢ ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g (f c') d z - 1) < 1 / 4 | case gs
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
gs : ∀ {x : ℂ × ℂ}, Complex.abs (x.1 - c) < r4 → Complex.abs x.2 < r4 → Complex.abs (g (f x.1) d x.2 - 1) < 1 / 4
m : Complex.abs (c' - c) < r
⊢ ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g (f c') d z - 1) < 1 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case gs
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
m : c' ∈ ball c r
gs : ∀ {x : ℂ × ℂ}, Complex.abs (x.1 - c) < r4 → Complex.abs x.2 < r4 → Complex.abs (g (f x.1) d x.2 - 1) < 1 / 4
⊢ ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g (f c') d z - 1) < 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | intro z zr | case gs
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
gs : ∀ {x : ℂ × ℂ}, Complex.abs (x.1 - c) < r4 → Complex.abs x.2 < r4 → Complex.abs (g (f x.1) d x.2 - 1) < 1 / 4
m : Complex.abs (c' - c) < r
⊢ ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g (f c') d z - 1) < 1 / 4 | case gs
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
gs : ∀ {x : ℂ × ℂ}, Complex.abs (x.1 - c) < r4 → Complex.abs x.2 < r4 → Complex.abs (g (f x.1) d x.2 - 1) < 1 / 4
m : Complex.abs (c' - c) < r
z : ℂ
zr : Complex.abs z < r
⊢ Complex.abs (g (f c') d z - 1) < 1 / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
case gs
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
gs : ∀ {x : ℂ × ℂ}, Complex.abs (x.1 - c) < r4 → Complex.abs x.2 < r4 → Complex.abs (g (f x.1) d x.2 - 1) < 1 / 4
m : Complex.abs (c' - c) < r
⊢ ∀ {z : ℂ}, Complex.abs z < r → Complex.abs (g (f c') d z - 1) < 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | exact @gs ⟨c', z⟩ (lt_of_lt_of_le m rr4) (lt_of_lt_of_le zr rr4) | case gs
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
gs : ∀ {x : ℂ × ℂ}, Complex.abs (x.1 - c) < r4 → Complex.abs x.2 < r4 → Complex.abs (g (f x.1) d x.2 - 1) < 1 / 4
m : Complex.abs (c' - c) < r
z : ℂ
zr : Complex.abs z < r
⊢ Complex.abs (g (f c') d z - 1) < 1 / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case gs
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
c : ℂ
m✝ : c ∈ u
r0 : ℝ
r0p : 0 < r0
fa✝ : AnalyticOn ℂ (uncurry f) (ball (c, 0) r0)
r1 : ℝ
r1p : r1 > 0
rc✝¹ : ball c r1 ⊆ u
r2 : ℝ := min r0 r1
fa : AnalyticOn ℂ (uncurry f) (ball (c, 0) (min r0 r1))
rc✝ : ball c r2 ⊆ u
ga : AnalyticOn ℂ (g2 f d) (ball (c, 0) (min r0 r1))
r3 : ℝ
r3p : r3 > 0
rw✝ : ball (c, 0) r3 ⊆ w
r4 : ℝ
r4p : r4 > 0
r : ℝ := min (min r2 r3) (min r4 (1 / 2))
rp : 0 < r
rh : r ≤ 1 / 2
rr4 : r ≤ r4
rc : ball c r ⊆ u
rw : ball (c, 0) r ⊆ w
c' : ℂ
gs : ∀ {x : ℂ × ℂ}, Complex.abs (x.1 - c) < r4 → Complex.abs x.2 < r4 → Complex.abs (g (f x.1) d x.2 - 1) < 1 / 4
m : Complex.abs (c' - c) < r
z : ℂ
zr : Complex.abs z < r
⊢ Complex.abs (g (f c') d z - 1) < 1 / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | apply Set.ext | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
⊢ u = ⋃ c, v c | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
⊢ ∀ (x : ℂ), x ∈ u ↔ x ∈ ⋃ c, v c | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
⊢ u = ⋃ c, v c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | intro c | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
⊢ ∀ (x : ℂ), x ∈ u ↔ x ∈ ⋃ c, v c | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
⊢ c ∈ u ↔ c ∈ ⋃ c, v c | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
⊢ ∀ (x : ℂ), x ∈ u ↔ x ∈ ⋃ c, v c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | rw [Set.mem_iUnion] | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
⊢ c ∈ u ↔ c ∈ ⋃ c, v c | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
⊢ c ∈ u ↔ ∃ i, c ∈ v i | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
⊢ c ∈ u ↔ c ∈ ⋃ c, v c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | constructor | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
⊢ c ∈ u ↔ ∃ i, c ∈ v i | case h.mp
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
⊢ c ∈ u → ∃ i, c ∈ v i
case h.mpr
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
⊢ (∃ i, c ∈ v i) → c ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
⊢ c ∈ u ↔ ∃ i, c ∈ v i
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | intro m | case h.mp
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
⊢ c ∈ u → ∃ i, c ∈ v i | case h.mp
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
m : c ∈ u
⊢ ∃ i, c ∈ v i | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
⊢ c ∈ u → ∃ i, c ∈ v i
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | use⟨c, m⟩ | case h.mp
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
m : c ∈ u
⊢ ∃ i, c ∈ v i | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
m : c ∈ u
⊢ c ∈ v ⟨c, m⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
m : c ∈ u
⊢ ∃ i, c ∈ v i
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | rcases choose_spec (h c m) with ⟨rp, _, _⟩ | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
m : c ∈ u
⊢ c ∈ v ⟨c, m⟩ | case h.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
m : c ∈ u
rp : choose ⋯ > 0
left✝ : ball c (choose ⋯) ⊆ u
right✝ : ball (c, 0) (choose ⋯) ⊆ w ∧ SuperNearC f d (ball c (choose ⋯)) (ball (c, 0) (choose ⋯))
⊢ c ∈ v ⟨c, m⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
m : c ∈ u
⊢ c ∈ v ⟨c, m⟩
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | exact mem_ball_self rp | case h.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
m : c ∈ u
rp : choose ⋯ > 0
left✝ : ball c (choose ⋯) ⊆ u
right✝ : ball (c, 0) (choose ⋯) ⊆ w ∧ SuperNearC f d (ball c (choose ⋯)) (ball (c, 0) (choose ⋯))
⊢ c ∈ v ⟨c, m⟩ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
m : c ∈ u
rp : choose ⋯ > 0
left✝ : ball c (choose ⋯) ⊆ u
right✝ : ball (c, 0) (choose ⋯) ⊆ w ∧ SuperNearC f d (ball c (choose ⋯)) (ball (c, 0) (choose ⋯))
⊢ c ∈ v ⟨c, m⟩
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | intro m | case h.mpr
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
⊢ (∃ i, c ∈ v i) → c ∈ u | case h.mpr
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
m : ∃ i, c ∈ v i
⊢ c ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
⊢ (∃ i, c ∈ v i) → c ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | rcases m with ⟨i, m⟩ | case h.mpr
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
m : ∃ i, c ∈ v i
⊢ c ∈ u | case h.mpr.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
i : ↑u
m : c ∈ v i
⊢ c ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
m : ∃ i, c ∈ v i
⊢ c ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | rcases choose_spec (h _ i.mem) with ⟨_, us, _⟩ | case h.mpr.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
i : ↑u
m : c ∈ v i
⊢ c ∈ u | case h.mpr.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
i : ↑u
m : c ∈ v i
left✝ : choose ⋯ > 0
us : ball (↑i) (choose ⋯) ⊆ u
right✝ : ball (↑i, 0) (choose ⋯) ⊆ w ∧ SuperNearC f d (ball (↑i) (choose ⋯)) (ball (↑i, 0) (choose ⋯))
⊢ c ∈ u | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
i : ↑u
m : c ∈ v i
⊢ c ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | exact us m | case h.mpr.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
i : ↑u
m : c ∈ v i
left✝ : choose ⋯ > 0
us : ball (↑i) (choose ⋯) ⊆ u
right✝ : ball (↑i, 0) (choose ⋯) ⊆ w ∧ SuperNearC f d (ball (↑i) (choose ⋯)) (ball (↑i, 0) (choose ⋯))
⊢ c ∈ u | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
c : ℂ
i : ↑u
m : c ∈ v i
left✝ : choose ⋯ > 0
us : ball (↑i) (choose ⋯) ⊆ u
right✝ : ball (↑i, 0) (choose ⋯) ⊆ w ∧ SuperNearC f d (ball (↑i) (choose ⋯)) (ball (↑i, 0) (choose ⋯))
⊢ c ∈ u
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | apply Set.iUnion_subset | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
⊢ ⋃ c, t c ⊆ w | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
⊢ ∀ (i : ↑u), t i ⊆ w | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
⊢ ⋃ c, t c ⊆ w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | intro i | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
⊢ ∀ (i : ↑u), t i ⊆ w | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
i : ↑u
⊢ t i ⊆ w | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
⊢ ∀ (i : ↑u), t i ⊆ w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | rcases choose_spec (h _ i.mem) with ⟨_, _, rw, _⟩ | case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
i : ↑u
⊢ t i ⊆ w | case h.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
i : ↑u
left✝¹ : choose ⋯ > 0
left✝ : ball (↑i) (choose ⋯) ⊆ u
rw : ball (↑i, 0) (choose ⋯) ⊆ w
right✝ : SuperNearC f d (ball (↑i) (choose ⋯)) (ball (↑i, 0) (choose ⋯))
⊢ t i ⊆ w | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
i : ↑u
⊢ t i ⊆ w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | exact rw | case h.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
i : ↑u
left✝¹ : choose ⋯ > 0
left✝ : ball (↑i) (choose ⋯) ⊆ u
rw : ball (↑i, 0) (choose ⋯) ⊆ w
right✝ : SuperNearC f d (ball (↑i) (choose ⋯)) (ball (↑i, 0) (choose ⋯))
⊢ t i ⊆ w | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
i : ↑u
left✝¹ : choose ⋯ > 0
left✝ : ball (↑i) (choose ⋯) ⊆ u
rw : ball (↑i, 0) (choose ⋯) ⊆ w
right✝ : SuperNearC f d (ball (↑i) (choose ⋯)) (ball (↑i, 0) (choose ⋯))
⊢ t i ⊆ w
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | intro i | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
⊢ ∀ (c : ↑u), SuperNearC f d (v c) (t c) | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
i : ↑u
⊢ SuperNearC f d (v i) (t i) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
⊢ ∀ (c : ↑u), SuperNearC f d (v c) (t c)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | rcases choose_spec (h _ i.mem) with ⟨_, _, _, s⟩ | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
i : ↑u
⊢ SuperNearC f d (v i) (t i) | case intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s✝ : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
i : ↑u
left✝² : choose ⋯ > 0
left✝¹ : ball (↑i) (choose ⋯) ⊆ u
left✝ : ball (↑i, 0) (choose ⋯) ⊆ w
s : SuperNearC f d (ball (↑i) (choose ⋯)) (ball (↑i, 0) (choose ⋯))
⊢ SuperNearC f d (v i) (t i) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
i : ↑u
⊢ SuperNearC f d (v i) (t i)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | exact s | case intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s✝ : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
i : ↑u
left✝² : choose ⋯ > 0
left✝¹ : ball (↑i) (choose ⋯) ⊆ u
left✝ : ball (↑i, 0) (choose ⋯) ⊆ w
s : SuperNearC f d (ball (↑i) (choose ⋯)) (ball (↑i, 0) (choose ⋯))
⊢ SuperNearC f d (v i) (t i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s✝ : SuperAtC f d u
w : Set (ℂ × ℂ)
wo : IsOpen w
wc : ∀ c ∈ u, (c, 0) ∈ w
h : ∀ c ∈ u, ∃ r > 0, ball c r ⊆ u ∧ ball (c, 0) r ⊆ w ∧ SuperNearC f d (ball c r) (ball (c, 0) r)
r : ↑u → ℝ := fun c => choose ⋯
v : ↑u → Set ℂ := fun c => ball (↑c) (r c)
t : ↑u → Set (ℂ × ℂ) := fun c => ball (↑c, 0) (r c)
e : u = ⋃ c, v c
tw : ⋃ c, t c ⊆ w
i : ↑u
left✝² : choose ⋯ > 0
left✝¹ : ball (↑i) (choose ⋯) ⊆ u
left✝ : ball (↑i, 0) (choose ⋯) ⊆ w
s : SuperNearC f d (ball (↑i) (choose ⋯)) (ball (↑i, 0) (choose ⋯))
⊢ SuperNearC f d (v i) (t i)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC | [633, 1] | [634, 89] | rcases s.superNearC' isOpen_univ fun _ _ ↦ Set.mem_univ _ with ⟨t, _, s⟩ | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
⊢ ∃ t, SuperNearC f d u t | case intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s✝ : SuperAtC f d u
t : Set (ℂ × ℂ)
left✝ : t ⊆ univ
s : SuperNearC f d u t
⊢ ∃ t, SuperNearC f d u t | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperAtC f d u
⊢ ∃ t, SuperNearC f d u t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC | [633, 1] | [634, 89] | exact ⟨t, s⟩ | case intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s✝ : SuperAtC f d u
t : Set (ℂ × ℂ)
left✝ : t ⊆ univ
s : SuperNearC f d u t
⊢ ∃ t, SuperNearC f d u t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t✝ : Set (ℂ × ℂ)
s✝ : SuperAtC f d u
t : Set (ℂ × ℂ)
left✝ : t ⊆ univ
s : SuperNearC f d u t
⊢ ∃ t, SuperNearC f d u t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | induction' n with n nh | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => (f c)^[n] z) c | case zero
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => (f c)^[0] z) c
case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ AnalyticAt ℂ (fun c => (f c)^[n + 1] z) c | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => (f c)^[n] z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | simp only [Function.iterate_zero, id] | case zero
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => (f c)^[0] z) c | case zero
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => z) c | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => (f c)^[0] z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | exact analyticAt_const | case zero
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => z) c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | simp_rw [Function.iterate_succ'] | case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ AnalyticAt ℂ (fun c => (f c)^[n + 1] z) c | case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ AnalyticAt ℂ (fun c => (f c ∘ (f c)^[n]) z) c | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ AnalyticAt ℂ (fun c => (f c)^[n + 1] z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | simp only [Function.comp_apply] | case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ AnalyticAt ℂ (fun c => (f c ∘ (f c)^[n]) z) c | case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ AnalyticAt ℂ (fun c => f c ((f c)^[n] z)) c | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ AnalyticAt ℂ (fun c => (f c ∘ (f c)^[n]) z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | refine (s.fa _ ?_).comp ((analyticAt_id _ _).prod nh) | case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ AnalyticAt ℂ (fun c => f c ((f c)^[n] z)) c | case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ (id c, (f c)^[n] z) ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ AnalyticAt ℂ (fun c => f c ((f c)^[n] z)) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | exact (s.ts m).mapsTo n m | case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ (id c, (f c)^[n] z) ∈ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
m : (c, z) ∈ t
n : ℕ
nh : AnalyticAt ℂ (fun c => (f c)^[n] z) c
⊢ (id c, (f c)^[n] z) ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | refine AnalyticAt.cpow ?_ analyticAt_const ?_ | f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => term (f c) d n z) c | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => g (f c) d ((f c)^[n] z)) c
case refine_2
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
⊢ g (f c) d ((f c)^[n] z) ∈ Complex.slitPlane | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => term (f c) d n z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | have e : (fun c ↦ g (f c) d ((f c)^[n] z)) = fun c ↦ g2 f d (c, (f c)^[n] z) := rfl | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => g (f c) d ((f c)^[n] z)) c | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
⊢ AnalyticAt ℂ (fun c => g (f c) d ((f c)^[n] z)) c | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
⊢ AnalyticAt ℂ (fun c => g (f c) d ((f c)^[n] z)) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | rw [e] | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
⊢ AnalyticAt ℂ (fun c => g (f c) d ((f c)^[n] z)) c | case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
⊢ AnalyticAt ℂ (fun c => g2 f d (c, (f c)^[n] z)) c | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : ℂ → ℂ → ℂ
d : ℕ
u : Set ℂ
t : Set (ℂ × ℂ)
s : SuperNearC f d u t
c z : ℂ
n : ℕ
m : (c, z) ∈ t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
⊢ AnalyticAt ℂ (fun c => g (f c) d ((f c)^[n] z)) c
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.