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/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | refine (HolomorphicOn.eq_of_locally_eq ?_ (fun z m ↦
(s.bottcher_holomorphicOn (c, z) m).along_snd.pow) (s.post_slice_connected c).isPreconnected
⟨a, s.post_a c, e⟩).self_of_nhdsSet m | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d
⊢ HolomorphicOn I I (fun x => s.bottcher c (f c x)) fun z => s.post (c, z) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | intro z m | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d
⊢ HolomorphicOn I I (fun x => s.bottcher c (f c x)) fun z => s.post (c, z) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z✝¹ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z✝ : S
m✝ : (c, z✝) ∈ s.post
e : ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d
z : S
m : z ∈ fun z => s.post (c, z)
⊢ HolomorphicAt I I (fun x => s.bottcher c (f c x)) z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
e : ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d
⊢ HolomorphicOn I I (fun x => s.bottcher c (f c x)) fun z => s.post (c, z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | exact (s.bottcher_holomorphicOn _ (s.stays_post m)).along_snd.comp (s.fa _).along_snd | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z✝¹ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z✝ : S
m✝ : (c, z✝) ∈ s.post
e : ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d
z : S
m : z ∈ fun z => s.post (c, z)
⊢ HolomorphicAt I I (fun x => s.bottcher c (f c x)) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z✝¹ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z✝ : S
m✝ : (c, z✝) ∈ s.post
e : ∀ᶠ (w : S) in 𝓝 a, s.bottcher c (f c w) = s.bottcher c w ^ d
z : S
m : z ∈ fun z => s.post (c, z)
⊢ HolomorphicAt I I (fun x => s.bottcher c (f c x)) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | have e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post := s.basin_post m | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | have e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post := by
rcases e0 with ⟨n, e0⟩; use n
simp only [← Function.iterate_succ_apply, Function.iterate_succ_apply']
exact s.stays_post e0 | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | simp only [Super.bottcher, e0, e1, dif_pos] | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
⊢ (fun w => w ^ (↑d)⁻¹)^[Nat.find ⋯] (s.bottcherPost c ((f c)^[Nat.find ⋯] (f c z))) =
(fun w => w ^ (↑d)⁻¹)^[Nat.find ⋯] (s.bottcherPost c ((f c)^[Nat.find ⋯] z)) ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
⊢ s.bottcher c (f c z) = s.bottcher c z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | generalize hk0 : Nat.find e0 = k0 | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
⊢ (fun w => w ^ (↑d)⁻¹)^[Nat.find ⋯] (s.bottcherPost c ((f c)^[Nat.find ⋯] (f c z))) =
(fun w => w ^ (↑d)⁻¹)^[Nat.find ⋯] (s.bottcherPost c ((f c)^[Nat.find ⋯] z)) ^ d | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
⊢ (fun w => w ^ (↑d)⁻¹)^[Nat.find ⋯] (s.bottcherPost c ((f c)^[Nat.find ⋯] (f c z))) =
(fun w => w ^ (↑d)⁻¹)^[k0] (s.bottcherPost c ((f c)^[k0] z)) ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
⊢ (fun w => w ^ (↑d)⁻¹)^[Nat.find ⋯] (s.bottcherPost c ((f c)^[Nat.find ⋯] (f c z))) =
(fun w => w ^ (↑d)⁻¹)^[Nat.find ⋯] (s.bottcherPost c ((f c)^[Nat.find ⋯] z)) ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | generalize hk1 : Nat.find e1 = k1 | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
⊢ (fun w => w ^ (↑d)⁻¹)^[Nat.find ⋯] (s.bottcherPost c ((f c)^[Nat.find ⋯] (f c z))) =
(fun w => w ^ (↑d)⁻¹)^[k0] (s.bottcherPost c ((f c)^[k0] z)) ^ d | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ (fun w => w ^ (↑d)⁻¹)^[k1] (s.bottcherPost c ((f c)^[k1] (f c z))) =
(fun w => w ^ (↑d)⁻¹)^[k0] (s.bottcherPost c ((f c)^[k0] z)) ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
⊢ (fun w => w ^ (↑d)⁻¹)^[Nat.find ⋯] (s.bottcherPost c ((f c)^[Nat.find ⋯] (f c z))) =
(fun w => w ^ (↑d)⁻¹)^[k0] (s.bottcherPost c ((f c)^[k0] z)) ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | simp only [kk, ← Function.iterate_succ_apply, Function.iterate_succ_apply'] | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
kk : k0 = k1 + 1
⊢ (fun w => w ^ (↑d)⁻¹)^[k1] (s.bottcherPost c ((f c)^[k1] (f c z))) =
(fun w => w ^ (↑d)⁻¹)^[k0] (s.bottcherPost c ((f c)^[k0] z)) ^ d | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
kk : k0 = k1 + 1
⊢ (fun w => w ^ (↑d)⁻¹)^[k1] (s.bottcherPost c (f c ((f c)^[k1] z))) =
((fun w => w ^ (↑d)⁻¹)^[k1] (s.bottcherPost c (f c ((f c)^[k1] z))) ^ (↑d)⁻¹) ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
kk : k0 = k1 + 1
⊢ (fun w => w ^ (↑d)⁻¹)^[k1] (s.bottcherPost c ((f c)^[k1] (f c z))) =
(fun w => w ^ (↑d)⁻¹)^[k0] (s.bottcherPost c ((f c)^[k0] z)) ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | rw [Complex.cpow_nat_inv_pow _ s.d0] | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
kk : k0 = k1 + 1
⊢ (fun w => w ^ (↑d)⁻¹)^[k1] (s.bottcherPost c (f c ((f c)^[k1] z))) =
((fun w => w ^ (↑d)⁻¹)^[k1] (s.bottcherPost c (f c ((f c)^[k1] z))) ^ (↑d)⁻¹) ^ d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
kk : k0 = k1 + 1
⊢ (fun w => w ^ (↑d)⁻¹)^[k1] (s.bottcherPost c (f c ((f c)^[k1] z))) =
((fun w => w ^ (↑d)⁻¹)^[k1] (s.bottcherPost c (f c ((f c)^[k1] z))) ^ (↑d)⁻¹) ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | rcases e0 with ⟨n, e0⟩ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post | case intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
n : ℕ
e0 : (c, (f c)^[n] z) ∈ s.post
⊢ ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
⊢ ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | use n | case intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
n : ℕ
e0 : (c, (f c)^[n] z) ∈ s.post
⊢ ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
n : ℕ
e0 : (c, (f c)^[n] z) ∈ s.post
⊢ (c, (f c)^[n] (f c z)) ∈ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
n : ℕ
e0 : (c, (f c)^[n] z) ∈ s.post
⊢ ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | simp only [← Function.iterate_succ_apply, Function.iterate_succ_apply'] | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
n : ℕ
e0 : (c, (f c)^[n] z) ∈ s.post
⊢ (c, (f c)^[n] (f c z)) ∈ s.post | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
n : ℕ
e0 : (c, (f c)^[n] z) ∈ s.post
⊢ (c, f c ((f c)^[n] z)) ∈ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
n : ℕ
e0 : (c, (f c)^[n] z) ∈ s.post
⊢ (c, (f c)^[n] (f c z)) ∈ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | exact s.stays_post e0 | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
n : ℕ
e0 : (c, (f c)^[n] z) ∈ s.post
⊢ (c, f c ((f c)^[n] z)) ∈ s.post | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
n : ℕ
e0 : (c, (f c)^[n] z) ∈ s.post
⊢ (c, f c ((f c)^[n] z)) ∈ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | rw [← hk0, ← hk1] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ k0 = k1 + 1 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ Nat.find e0 = Nat.find e1 + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ k0 = k1 + 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | apply le_antisymm | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ Nat.find e0 = Nat.find e1 + 1 | case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ Nat.find e0 ≤ Nat.find e1 + 1
case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ Nat.find e1 + 1 ≤ Nat.find e0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ Nat.find e0 = Nat.find e1 + 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | apply Nat.find_le | case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ Nat.find e0 ≤ Nat.find e1 + 1 | case a.hn
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ (c, (f c)^[Nat.find e1 + 1] z) ∈ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ Nat.find e0 ≤ Nat.find e1 + 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | simp only [Function.iterate_succ_apply] | case a.hn
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ (c, (f c)^[Nat.find e1 + 1] z) ∈ s.post | case a.hn
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ (c, (f c)^[Nat.find e1] (f c z)) ∈ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hn
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ (c, (f c)^[Nat.find e1 + 1] z) ∈ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | exact Nat.find_spec e1 | case a.hn
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ (c, (f c)^[Nat.find e1] (f c z)) ∈ s.post | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hn
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ (c, (f c)^[Nat.find e1] (f c z)) ∈ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | rw [Nat.succ_le_iff, Nat.lt_find_iff] | case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ Nat.find e1 + 1 ≤ Nat.find e0 | case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ ∀ m ≤ Nat.find e1, (c, (f c)^[m] z) ∉ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ Nat.find e1 + 1 ≤ Nat.find e0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | intro n n1 | case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ ∀ m ≤ Nat.find e1, (c, (f c)^[m] z) ∉ s.post | case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : n ≤ Nat.find e1
⊢ (c, (f c)^[n] z) ∉ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
⊢ ∀ m ≤ Nat.find e1, (c, (f c)^[m] z) ∉ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | contrapose n1 | case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : n ≤ Nat.find e1
⊢ (c, (f c)^[n] z) ∉ s.post | case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : ¬(c, (f c)^[n] z) ∉ s.post
⊢ ¬n ≤ Nat.find e1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : n ≤ Nat.find e1
⊢ (c, (f c)^[n] z) ∉ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | simp only [not_not, not_le] at n1 ⊢ | case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : ¬(c, (f c)^[n] z) ∉ s.post
⊢ ¬n ≤ Nat.find e1 | case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
⊢ Nat.find e1 < n | Please generate a tactic in lean4 to solve the state.
STATE:
case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : ¬(c, (f c)^[n] z) ∉ s.post
⊢ ¬n ≤ Nat.find e1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | have n0 : n ≠ 0 := by
contrapose p; simp only [not_not] at p ⊢
simp only [p, Function.iterate_zero_apply] at n1; exact n1 | case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
⊢ Nat.find e1 < n | case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
n0 : n ≠ 0
⊢ Nat.find e1 < n | Please generate a tactic in lean4 to solve the state.
STATE:
case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
⊢ Nat.find e1 < n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | rw [← Nat.succ_le_iff, Nat.succ_eq_add_one, ← Nat.sub_add_cancel (Nat.pos_of_ne_zero n0)] | case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
n0 : n ≠ 0
⊢ Nat.find e1 < n | case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
n0 : n ≠ 0
⊢ Nat.find e1 + 1 ≤ n - Nat.succ 0 + Nat.succ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
n0 : n ≠ 0
⊢ Nat.find e1 < n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | apply Nat.succ_le_succ | case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
n0 : n ≠ 0
⊢ Nat.find e1 + 1 ≤ n - Nat.succ 0 + Nat.succ 0 | case a.a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
n0 : n ≠ 0
⊢ Nat.find e1 ≤ n - Nat.succ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
n0 : n ≠ 0
⊢ Nat.find e1 + 1 ≤ n - Nat.succ 0 + Nat.succ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | apply Nat.find_le | case a.a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
n0 : n ≠ 0
⊢ Nat.find e1 ≤ n - Nat.succ 0 | case a.a.hn
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
n0 : n ≠ 0
⊢ (c, (f c)^[n - Nat.succ 0] (f c z)) ∈ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
n0 : n ≠ 0
⊢ Nat.find e1 ≤ n - Nat.succ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | simp only [← Function.iterate_succ_apply, Nat.succ_eq_add_one,
Nat.sub_add_cancel (Nat.pos_of_ne_zero n0), n1, zero_add] | case a.a.hn
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
n0 : n ≠ 0
⊢ (c, (f c)^[n - Nat.succ 0] (f c z)) ∈ s.post | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a.hn
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
n0 : n ≠ 0
⊢ (c, (f c)^[n - Nat.succ 0] (f c z)) ∈ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | contrapose p | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
⊢ n ≠ 0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
p : ¬n ≠ 0
⊢ ¬(c, z) ∉ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
⊢ n ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | simp only [not_not] at p ⊢ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
p : ¬n ≠ 0
⊢ ¬(c, z) ∉ s.post | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
p : n = 0
⊢ (c, z) ∈ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
p : ¬n ≠ 0
⊢ ¬(c, z) ∉ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | simp only [p, Function.iterate_zero_apply] at n1 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
p : n = 0
⊢ (c, z) ∈ s.post | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
p : n = 0
n1 : (c, z) ∈ s.post
⊢ (c, z) ∈ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
n1 : (c, (f c)^[n] z) ∈ s.post
p : n = 0
⊢ (c, z) ∈ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | exact n1 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
p : n = 0
n1 : (c, z) ∈ s.post
⊢ (c, z) ∈ s.post | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
m : (c, z) ∈ s.basin
e0 : ∃ n, (c, (f c)^[n] z) ∈ s.post
e1 : ∃ n, (c, (f c)^[n] (f c z)) ∈ s.post
k0 : ℕ
hk0 : Nat.find e0 = k0
k1 : ℕ
hk1 : Nat.find e1 = k1
n : ℕ
p : n = 0
n1 : (c, z) ∈ s.post
⊢ (c, z) ∈ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | contrapose m | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∉ s.basin
⊢ (c, f c z) ∉ s.basin | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : ¬(c, f c z) ∉ s.basin
⊢ ¬(c, z) ∉ s.basin | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, z) ∉ s.basin
⊢ (c, f c z) ∉ s.basin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | simp only [not_not] at m ⊢ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : ¬(c, f c z) ∉ s.basin
⊢ ¬(c, z) ∉ s.basin | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, f c z) ∈ s.basin
⊢ (c, z) ∈ s.basin | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : ¬(c, f c z) ∉ s.basin
⊢ ¬(c, z) ∉ s.basin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | rcases m with ⟨n, m⟩ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, f c z) ∈ s.basin
⊢ (c, z) ∈ s.basin | case intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
n : ℕ
m : ((c, f c z).1, (f (c, f c z).1)^[n] (c, f c z).2) ∈ s.near
⊢ (c, z) ∈ s.basin | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
m : (c, f c z) ∈ s.basin
⊢ (c, z) ∈ s.basin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | use n + 1 | case intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
n : ℕ
m : ((c, f c z).1, (f (c, f c z).1)^[n] (c, f c z).2) ∈ s.near
⊢ (c, z) ∈ s.basin | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
n : ℕ
m : ((c, f c z).1, (f (c, f c z).1)^[n] (c, f c z).2) ∈ s.near
⊢ ((c, z).1, (f (c, z).1)^[n + 1] (c, z).2) ∈ s.near | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
n : ℕ
m : ((c, f c z).1, (f (c, f c z).1)^[n] (c, f c z).2) ∈ s.near
⊢ (c, z) ∈ s.basin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | simp only at m ⊢ | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
n : ℕ
m : ((c, f c z).1, (f (c, f c z).1)^[n] (c, f c z).2) ∈ s.near
⊢ ((c, z).1, (f (c, z).1)^[n + 1] (c, z).2) ∈ s.near | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
n : ℕ
m : (c, (f c)^[n] (f c z)) ∈ s.near
⊢ (c, (f c)^[n + 1] z) ∈ s.near | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
n : ℕ
m : ((c, f c z).1, (f (c, f c z).1)^[n] (c, f c z).2) ∈ s.near
⊢ ((c, z).1, (f (c, z).1)^[n + 1] (c, z).2) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn | [174, 1] | [218, 71] | rwa [Function.iterate_succ_apply] | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
n : ℕ
m : (c, (f c)^[n] (f c z)) ∈ s.near
⊢ (c, (f c)^[n + 1] z) ∈ s.near | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
h0 : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → s.bottcher c (f c z) = s.bottcher c z ^ d
p : (c, z) ∉ s.post
n : ℕ
m : (c, (f c)^[n] (f c z)) ∈ s.near
⊢ (c, (f c)^[n + 1] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn_iter | [221, 1] | [224, 83] | induction' n with n h | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
n : ℕ
⊢ s.bottcher c ((f c)^[n] z) = s.bottcher c z ^ d ^ n | case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ s.bottcher c ((f c)^[0] z) = s.bottcher c z ^ d ^ 0
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
n : ℕ
h : s.bottcher c ((f c)^[n] z) = s.bottcher c z ^ d ^ n
⊢ s.bottcher c ((f c)^[n + 1] z) = s.bottcher c z ^ d ^ (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
n : ℕ
⊢ s.bottcher c ((f c)^[n] z) = s.bottcher c z ^ d ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn_iter | [221, 1] | [224, 83] | simp only [Function.iterate_zero_apply, pow_zero, pow_one] | case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ s.bottcher c ((f c)^[0] z) = s.bottcher c z ^ d ^ 0
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
n : ℕ
h : s.bottcher c ((f c)^[n] z) = s.bottcher c z ^ d ^ n
⊢ s.bottcher c ((f c)^[n + 1] z) = s.bottcher c z ^ d ^ (n + 1) | case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
n : ℕ
h : s.bottcher c ((f c)^[n] z) = s.bottcher c z ^ d ^ n
⊢ s.bottcher c ((f c)^[n + 1] z) = s.bottcher c z ^ d ^ (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ s.bottcher c ((f c)^[0] z) = s.bottcher c z ^ d ^ 0
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
n : ℕ
h : s.bottcher c ((f c)^[n] z) = s.bottcher c z ^ d ^ n
⊢ s.bottcher c ((f c)^[n + 1] z) = s.bottcher c z ^ d ^ (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_eqn_iter | [221, 1] | [224, 83] | simp only [Function.iterate_succ_apply', s.bottcher_eqn, h, ← pow_mul, pow_succ] | case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
n : ℕ
h : s.bottcher c ((f c)^[n] z) = s.bottcher c z ^ d ^ n
⊢ s.bottcher c ((f c)^[n + 1] z) = s.bottcher c z ^ d ^ (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
n : ℕ
h : s.bottcher c ((f c)^[n] z) = s.bottcher c z ^ d ^ n
⊢ s.bottcher c ((f c)^[n + 1] z) = s.bottcher c z ^ d ^ (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.abs_bottcher | [227, 1] | [237, 83] | have base : ∀ {c z}, (c, z) ∈ s.post → abs (s.bottcher c z) = s.potential c z := by
intro c z m; rcases s.ray_surj m with ⟨x, m, e⟩; rw [← e, s.bottcher_ray m, s.ray_potential m] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ Complex.abs (s.bottcher c z) = s.potential c z | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
base : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z
⊢ Complex.abs (s.bottcher c z) = s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ Complex.abs (s.bottcher c z) = s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.abs_bottcher | [227, 1] | [237, 83] | by_cases m : (c, z) ∈ s.basin | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
base : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z
⊢ Complex.abs (s.bottcher c z) = s.potential c z | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
base : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z
m : (c, z) ∈ s.basin
⊢ Complex.abs (s.bottcher c z) = s.potential c z
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
base : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z
m : (c, z) ∉ s.basin
⊢ Complex.abs (s.bottcher c z) = s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
base : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z
⊢ Complex.abs (s.bottcher c z) = s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.abs_bottcher | [227, 1] | [237, 83] | intro c z m | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ Complex.abs (s.bottcher c z) = s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.abs_bottcher | [227, 1] | [237, 83] | rcases s.ray_surj m with ⟨x, m, e⟩ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ Complex.abs (s.bottcher c z) = s.potential c z | case intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m✝ : (c, z) ∈ s.post
x : ℂ
m : (c, x) ∈ s.ext
e : s.ray c x = z
⊢ Complex.abs (s.bottcher c z) = s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ Complex.abs (s.bottcher c z) = s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.abs_bottcher | [227, 1] | [237, 83] | rw [← e, s.bottcher_ray m, s.ray_potential m] | case intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m✝ : (c, z) ∈ s.post
x : ℂ
m : (c, x) ∈ s.ext
e : s.ray c x = z
⊢ Complex.abs (s.bottcher c z) = s.potential c z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
m✝ : (c, z) ∈ s.post
x : ℂ
m : (c, x) ∈ s.ext
e : s.ray c x = z
⊢ Complex.abs (s.bottcher c z) = s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.abs_bottcher | [227, 1] | [237, 83] | rcases s.basin_post m with ⟨n, p⟩ | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
base : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z
m : (c, z) ∈ s.basin
⊢ Complex.abs (s.bottcher c z) = s.potential c z | case pos.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
base : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z
m : (c, z) ∈ s.basin
n : ℕ
p : (c, (f c)^[n] z) ∈ s.post
⊢ Complex.abs (s.bottcher c z) = s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
base : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z
m : (c, z) ∈ s.basin
⊢ Complex.abs (s.bottcher c z) = s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.abs_bottcher | [227, 1] | [237, 83] | rw [← Real.pow_rpow_inv_natCast (Complex.abs.nonneg _) (pow_ne_zero n s.d0), ←
Complex.abs.map_pow, ← s.bottcher_eqn_iter n, base p, s.potential_eqn_iter,
Real.pow_rpow_inv_natCast s.potential_nonneg (pow_ne_zero n s.d0)] | case pos.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
base : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z
m : (c, z) ∈ s.basin
n : ℕ
p : (c, (f c)^[n] z) ∈ s.post
⊢ Complex.abs (s.bottcher c z) = s.potential c z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
base : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z
m : (c, z) ∈ s.basin
n : ℕ
p : (c, (f c)^[n] z) ∈ s.post
⊢ Complex.abs (s.bottcher c z) = s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.abs_bottcher | [227, 1] | [237, 83] | have m' := m | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
base : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z
m : (c, z) ∉ s.basin
⊢ Complex.abs (s.bottcher c z) = s.potential c z | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
base : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z
m m' : (c, z) ∉ s.basin
⊢ Complex.abs (s.bottcher c z) = s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
base : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z
m : (c, z) ∉ s.basin
⊢ Complex.abs (s.bottcher c z) = s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.abs_bottcher | [227, 1] | [237, 83] | simp only [Super.basin, not_exists, mem_setOf] at m' | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
base : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z
m m' : (c, z) ∉ s.basin
⊢ Complex.abs (s.bottcher c z) = s.potential c z | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
base : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z
m : (c, z) ∉ s.basin
m' : ∀ (x : ℕ), (c, (f c)^[x] z) ∉ s.near
⊢ Complex.abs (s.bottcher c z) = s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
base : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z
m m' : (c, z) ∉ s.basin
⊢ Complex.abs (s.bottcher c z) = s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.abs_bottcher | [227, 1] | [237, 83] | simp only [s.bottcher_not_basin m, Complex.abs.map_one, s.potential_eq_one m'] | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
base : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z
m : (c, z) ∉ s.basin
m' : ∀ (x : ℕ), (c, (f c)^[x] z) ∉ s.near
⊢ Complex.abs (s.bottcher c z) = s.potential c z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
base : ∀ {c : ℂ} {z : S}, (c, z) ∈ s.post → Complex.abs (s.bottcher c z) = s.potential c z
m : (c, z) ∉ s.basin
m' : ∀ (x : ℕ), (c, (f c)^[x] z) ∉ s.near
⊢ Complex.abs (s.bottcher c z) = s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_lt_one | [240, 1] | [244, 36] | replace m := s.bottcher_ext m | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
⊢ Complex.abs (s.bottcher c z) < 1 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, s.bottcher c z) ∈ s.ext
⊢ Complex.abs (s.bottcher c z) < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.post
⊢ Complex.abs (s.bottcher c z) < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_lt_one | [240, 1] | [244, 36] | simp only [Super.ext, mem_setOf] at m | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, s.bottcher c z) ∈ s.ext
⊢ Complex.abs (s.bottcher c z) < 1 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : Complex.abs (s.bottcher c z) < s.p c
⊢ Complex.abs (s.bottcher c z) < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : (c, s.bottcher c z) ∈ s.ext
⊢ Complex.abs (s.bottcher c z) < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Bottcher.lean | Super.bottcher_lt_one | [240, 1] | [244, 36] | exact lt_of_lt_of_le m s.p_le_one | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : Complex.abs (s.bottcher c z) < s.p c
⊢ Complex.abs (s.bottcher c z) < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
m : Complex.abs (s.bottcher c z) < s.p c
⊢ Complex.abs (s.bottcher c z) < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | swap_swap | [25, 1] | [28, 49] | ext x | A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
⊢ swap '' (swap '' s) = s | case h
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
⊢ x ∈ swap '' (swap '' s) ↔ x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
⊢ swap '' (swap '' s) = s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | swap_swap | [25, 1] | [28, 49] | simp only [Set.mem_image, Prod.exists] | case h
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
⊢ x ∈ swap '' (swap '' s) ↔ x ∈ s | case h
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
⊢ (∃ a b, (∃ a_1 b_1, (a_1, b_1) ∈ s ∧ (a_1, b_1).swap = (a, b)) ∧ (a, b).swap = x) ↔ x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
⊢ x ∈ swap '' (swap '' s) ↔ x ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | swap_swap | [25, 1] | [28, 49] | constructor | case h
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
⊢ (∃ a b, (∃ a_1 b_1, (a_1, b_1) ∈ s ∧ (a_1, b_1).swap = (a, b)) ∧ (a, b).swap = x) ↔ x ∈ s | case h.mp
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
⊢ (∃ a b, (∃ a_1 b_1, (a_1, b_1) ∈ s ∧ (a_1, b_1).swap = (a, b)) ∧ (a, b).swap = x) → x ∈ s
case h.mpr
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
⊢ x ∈ s → ∃ a b, (∃ a_1 b_1, (a_1, b_1) ∈ s ∧ (a_1, b_1).swap = (a, b)) ∧ (a, b).swap = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
⊢ (∃ a b, (∃ a_1 b_1, (a_1, b_1) ∈ s ∧ (a_1, b_1).swap = (a, b)) ∧ (a, b).swap = x) ↔ x ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | swap_swap | [25, 1] | [28, 49] | intro ⟨a,b,⟨⟨c,d,e,f⟩,g⟩⟩ | case h.mp
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
⊢ (∃ a b, (∃ a_1 b_1, (a_1, b_1) ∈ s ∧ (a_1, b_1).swap = (a, b)) ∧ (a, b).swap = x) → x ∈ s | case h.mp
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
a : B
b c : A
d : B
e : (c, d) ∈ s
f : (c, d).swap = (a, b)
g : (a, b).swap = x
⊢ x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
⊢ (∃ a b, (∃ a_1 b_1, (a_1, b_1) ∈ s ∧ (a_1, b_1).swap = (a, b)) ∧ (a, b).swap = x) → x ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | swap_swap | [25, 1] | [28, 49] | rw [←g, ←f] | case h.mp
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
a : B
b c : A
d : B
e : (c, d) ∈ s
f : (c, d).swap = (a, b)
g : (a, b).swap = x
⊢ x ∈ s | case h.mp
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
a : B
b c : A
d : B
e : (c, d) ∈ s
f : (c, d).swap = (a, b)
g : (a, b).swap = x
⊢ (c, d).swap.swap ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
a : B
b c : A
d : B
e : (c, d) ∈ s
f : (c, d).swap = (a, b)
g : (a, b).swap = x
⊢ x ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | swap_swap | [25, 1] | [28, 49] | simpa only [swap] | case h.mp
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
a : B
b c : A
d : B
e : (c, d) ∈ s
f : (c, d).swap = (a, b)
g : (a, b).swap = x
⊢ (c, d).swap.swap ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
a : B
b c : A
d : B
e : (c, d) ∈ s
f : (c, d).swap = (a, b)
g : (a, b).swap = x
⊢ (c, d).swap.swap ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | swap_swap | [25, 1] | [28, 49] | intro m | case h.mpr
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
⊢ x ∈ s → ∃ a b, (∃ a_1 b_1, (a_1, b_1) ∈ s ∧ (a_1, b_1).swap = (a, b)) ∧ (a, b).swap = x | case h.mpr
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
m : x ∈ s
⊢ ∃ a b, (∃ a_1 b_1, (a_1, b_1) ∈ s ∧ (a_1, b_1).swap = (a, b)) ∧ (a, b).swap = x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
⊢ x ∈ s → ∃ a b, (∃ a_1 b_1, (a_1, b_1) ∈ s ∧ (a_1, b_1).swap = (a, b)) ∧ (a, b).swap = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | swap_swap | [25, 1] | [28, 49] | exact ⟨x.2,x.1,⟨x.1,x.2,m,rfl⟩,rfl⟩ | case h.mpr
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
m : x ∈ s
⊢ ∃ a b, (∃ a_1 b_1, (a_1, b_1) ∈ s ∧ (a_1, b_1).swap = (a, b)) ∧ (a, b).swap = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
s : Set (A × B)
x : A × B
m : x ∈ s
⊢ ∃ a b, (∃ a_1 b_1, (a_1, b_1) ∈ s ∧ (a_1, b_1).swap = (a, b)) ∧ (a, b).swap = x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | isOpen_swap | [41, 1] | [43, 78] | rw [Set.image_swap_eq_preimage_swap] | A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
s : Set (A × B)
inst✝¹ : TopologicalSpace A
inst✝ : TopologicalSpace B
⊢ IsOpen s → IsOpen (swap '' s) | A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
s : Set (A × B)
inst✝¹ : TopologicalSpace A
inst✝ : TopologicalSpace B
⊢ IsOpen s → IsOpen (swap ⁻¹' s) | Please generate a tactic in lean4 to solve the state.
STATE:
A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
s : Set (A × B)
inst✝¹ : TopologicalSpace A
inst✝ : TopologicalSpace B
⊢ IsOpen s → IsOpen (swap '' s)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | isOpen_swap | [41, 1] | [43, 78] | exact IsOpen.preimage continuous_swap | A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
s : Set (A × B)
inst✝¹ : TopologicalSpace A
inst✝ : TopologicalSpace B
⊢ IsOpen s → IsOpen (swap ⁻¹' s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
s : Set (A × B)
inst✝¹ : TopologicalSpace A
inst✝ : TopologicalSpace B
⊢ IsOpen s → IsOpen (swap ⁻¹' s)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | swap_mem | [45, 1] | [47, 47] | constructor | A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
⊢ (b, a) ∈ swap '' s ↔ (a, b) ∈ s | case mp
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
⊢ (b, a) ∈ swap '' s → (a, b) ∈ s
case mpr
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
⊢ (a, b) ∈ s → (b, a) ∈ swap '' s | Please generate a tactic in lean4 to solve the state.
STATE:
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
⊢ (b, a) ∈ swap '' s ↔ (a, b) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | swap_mem | [45, 1] | [47, 47] | intro m | case mp
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
⊢ (b, a) ∈ swap '' s → (a, b) ∈ s | case mp
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
m : (b, a) ∈ swap '' s
⊢ (a, b) ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
⊢ (b, a) ∈ swap '' s → (a, b) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | swap_mem | [45, 1] | [47, 47] | simp at m | case mp
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
m : (b, a) ∈ swap '' s
⊢ (a, b) ∈ s | case mp
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
m : ∃ a_1 b_1, (a_1, b_1) ∈ s ∧ b_1 = b ∧ a_1 = a
⊢ (a, b) ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
m : (b, a) ∈ swap '' s
⊢ (a, b) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | swap_mem | [45, 1] | [47, 47] | rcases m with ⟨a', b', m, hb, ha⟩ | case mp
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
m : ∃ a_1 b_1, (a_1, b_1) ∈ s ∧ b_1 = b ∧ a_1 = a
⊢ (a, b) ∈ s | case mp.intro.intro.intro.intro
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
a' : A
b' : B
m : (a', b') ∈ s
hb : b' = b
ha : a' = a
⊢ (a, b) ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
m : ∃ a_1 b_1, (a_1, b_1) ∈ s ∧ b_1 = b ∧ a_1 = a
⊢ (a, b) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | swap_mem | [45, 1] | [47, 47] | rwa [← ha, ← hb] | case mp.intro.intro.intro.intro
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
a' : A
b' : B
m : (a', b') ∈ s
hb : b' = b
ha : a' = a
⊢ (a, b) ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro.intro.intro
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
a' : A
b' : B
m : (a', b') ∈ s
hb : b' = b
ha : a' = a
⊢ (a, b) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | swap_mem | [45, 1] | [47, 47] | intro m | case mpr
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
⊢ (a, b) ∈ s → (b, a) ∈ swap '' s | case mpr
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
m : (a, b) ∈ s
⊢ (b, a) ∈ swap '' s | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
⊢ (a, b) ∈ s → (b, a) ∈ swap '' s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | swap_mem | [45, 1] | [47, 47] | exact Set.mem_image_of_mem swap m | case mpr
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
m : (a, b) ∈ s
⊢ (b, a) ∈ swap '' s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
a : A
b : B
s : Set (A × B)
m : (a, b) ∈ s
⊢ (b, a) ∈ swap '' s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | swap_mem' | [49, 1] | [50, 62] | have h := @swap_mem _ _ x.snd x.fst s | A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
x : A × B
s : Set (B × A)
⊢ x ∈ swap '' s ↔ x.swap ∈ s | A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
x : A × B
s : Set (B × A)
h : (x.1, x.2) ∈ swap '' s ↔ (x.2, x.1) ∈ s
⊢ x ∈ swap '' s ↔ x.swap ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
x : A × B
s : Set (B × A)
⊢ x ∈ swap '' s ↔ x.swap ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | swap_mem' | [49, 1] | [50, 62] | simp at h ⊢ | A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
x : A × B
s : Set (B × A)
h : (x.1, x.2) ∈ swap '' s ↔ (x.2, x.1) ∈ s
⊢ x ∈ swap '' s ↔ x.swap ∈ s | A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
x : A × B
s : Set (B × A)
h : (∃ a b, (a, b) ∈ s ∧ (b, a) = x) ↔ (x.2, x.1) ∈ s
⊢ (∃ a b, (a, b) ∈ s ∧ (b, a) = x) ↔ x.swap ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
x : A × B
s : Set (B × A)
h : (x.1, x.2) ∈ swap '' s ↔ (x.2, x.1) ∈ s
⊢ x ∈ swap '' s ↔ x.swap ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | swap_mem' | [49, 1] | [50, 62] | exact h | A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
x : A × B
s : Set (B × A)
h : (∃ a b, (a, b) ∈ s ∧ (b, a) = x) ↔ (x.2, x.1) ∈ s
⊢ (∃ a b, (a, b) ∈ s ∧ (b, a) = x) ↔ x.swap ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B C 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
x : A × B
s : Set (B × A)
h : (∃ a b, (a, b) ∈ s ∧ (b, a) = x) ↔ (x.2, x.1) ∈ s
⊢ (∃ a b, (a, b) ∈ s ∧ (b, a) = x) ↔ x.swap ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | ball_prod_same' | [52, 1] | [55, 45] | have s := ball_prod_same x.fst x.snd r | A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x : A × B
r : ℝ
⊢ ball x r = ball x.1 r ×ˢ ball x.2 r | A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x : A × B
r : ℝ
s : ball x.1 r ×ˢ ball x.2 r = ball (x.1, x.2) r
⊢ ball x r = ball x.1 r ×ˢ ball x.2 r | Please generate a tactic in lean4 to solve the state.
STATE:
A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x : A × B
r : ℝ
⊢ ball x r = ball x.1 r ×ˢ ball x.2 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | ball_prod_same' | [52, 1] | [55, 45] | simp only [Prod.mk.eta] at s | A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x : A × B
r : ℝ
s : ball x.1 r ×ˢ ball x.2 r = ball (x.1, x.2) r
⊢ ball x r = ball x.1 r ×ˢ ball x.2 r | A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x : A × B
r : ℝ
s : ball x.1 r ×ˢ ball x.2 r = ball x r
⊢ ball x r = ball x.1 r ×ˢ ball x.2 r | Please generate a tactic in lean4 to solve the state.
STATE:
A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x : A × B
r : ℝ
s : ball x.1 r ×ˢ ball x.2 r = ball (x.1, x.2) r
⊢ ball x r = ball x.1 r ×ˢ ball x.2 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | ball_prod_same' | [52, 1] | [55, 45] | exact s.symm | A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x : A × B
r : ℝ
s : ball x.1 r ×ˢ ball x.2 r = ball x r
⊢ ball x r = ball x.1 r ×ˢ ball x.2 r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x : A × B
r : ℝ
s : ball x.1 r ×ˢ ball x.2 r = ball x r
⊢ ball x r = ball x.1 r ×ˢ ball x.2 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | ball_swap | [57, 1] | [61, 76] | apply Set.ext | A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x : A × B
r : ℝ
⊢ ball x.swap r = swap '' ball x r | case h
A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x : A × B
r : ℝ
⊢ ∀ (x_1 : B × A), x_1 ∈ ball x.swap r ↔ x_1 ∈ swap '' ball x r | Please generate a tactic in lean4 to solve the state.
STATE:
A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x : A × B
r : ℝ
⊢ ball x.swap r = swap '' ball x r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | ball_swap | [57, 1] | [61, 76] | intro y | case h
A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x : A × B
r : ℝ
⊢ ∀ (x_1 : B × A), x_1 ∈ ball x.swap r ↔ x_1 ∈ swap '' ball x r | case h
A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x : A × B
r : ℝ
y : B × A
⊢ y ∈ ball x.swap r ↔ y ∈ swap '' ball x r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x : A × B
r : ℝ
⊢ ∀ (x_1 : B × A), x_1 ∈ ball x.swap r ↔ x_1 ∈ swap '' ball x r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | ball_swap | [57, 1] | [61, 76] | rw [swap_mem', Metric.mem_ball, Metric.mem_ball, Prod.dist_eq, Prod.dist_eq] | case h
A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x : A × B
r : ℝ
y : B × A
⊢ y ∈ ball x.swap r ↔ y ∈ swap '' ball x r | case h
A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x : A × B
r : ℝ
y : B × A
⊢ max (dist y.1 x.swap.1) (dist y.2 x.swap.2) < r ↔ max (dist y.swap.1 x.1) (dist y.swap.2 x.2) < r | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x : A × B
r : ℝ
y : B × A
⊢ y ∈ ball x.swap r ↔ y ∈ swap '' ball x r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | ball_swap | [57, 1] | [61, 76] | simp only [ge_iff_le, max_lt_iff, Prod.fst_swap, Prod.snd_swap, and_comm] | case h
A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x : A × B
r : ℝ
y : B × A
⊢ max (dist y.1 x.swap.1) (dist y.2 x.swap.2) < r ↔ max (dist y.swap.1 x.1) (dist y.swap.2 x.2) < r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x : A × B
r : ℝ
y : B × A
⊢ max (dist y.1 x.swap.1) (dist y.2 x.swap.2) < r ↔ max (dist y.swap.1 x.1) (dist y.swap.2 x.2) < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | dist_swap | [63, 1] | [65, 97] | rw [Prod.dist_eq, Prod.dist_eq] | A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x y : A × B
⊢ dist x.swap y.swap = dist x y | A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x y : A × B
⊢ max (dist x.swap.1 y.swap.1) (dist x.swap.2 y.swap.2) = max (dist x.1 y.1) (dist x.2 y.2) | Please generate a tactic in lean4 to solve the state.
STATE:
A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x y : A × B
⊢ dist x.swap y.swap = dist x y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Prod.lean | dist_swap | [63, 1] | [65, 97] | simp only [Prod.fst_swap, Prod.snd_swap, ge_iff_le, max_comm] | A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x y : A × B
⊢ max (dist x.swap.1 y.swap.1) (dist x.swap.2 y.swap.2) = max (dist x.1 y.1) (dist x.2 y.2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B C 𝕜 : Type
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : PseudoMetricSpace A
inst✝ : PseudoMetricSpace B
x y : A × B
⊢ max (dist x.swap.1 y.swap.1) (dist x.swap.2 y.swap.2) = max (dist x.1 y.1) (dist x.2 y.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | rcases complex_inverse_fun' fa nc with ⟨g, ga, gf, fg⟩ | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | have n : NontrivialHolomorphicAt g (f z) := by
rw [← gf.self_of_nhds] at fa
refine (NontrivialHolomorphicAt.anti ?_ fa ga).2
exact (nontrivialHolomorphicAt_id _).congr (Filter.EventuallyEq.symm fg) | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | have o := n.nhds_eq_map_nhds | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 (g (f z)) = Filter.map g (𝓝 (f z))
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | rw [gf.self_of_nhds] at o | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 (g (f z)) = Filter.map g (𝓝 (f z))
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 (g (f z)) = Filter.map g (𝓝 (f z))
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | simp only [nhds_prod_eq, o, Filter.prod_map_map_eq, Filter.eventually_map] | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
⊢ ∀ᶠ (a : T × T) in 𝓝 (f z) ×ˢ 𝓝 (f z), f (g a.1) = f (g a.2) → g a.1 = g a.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
⊢ ∀ᶠ (p : S × S) in 𝓝 (z, z), f p.1 = f p.2 → p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | refine (fg.prod_mk fg).mp (eventually_of_forall ?_) | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
⊢ ∀ᶠ (a : T × T) in 𝓝 (f z) ×ˢ 𝓝 (f z), f (g a.1) = f (g a.2) → g a.1 = g a.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
⊢ ∀ (x : T × T), f (g x.1) = x.1 ∧ f (g x.2) = x.2 → f (g x.1) = f (g x.2) → g x.1 = g x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
⊢ ∀ᶠ (a : T × T) in 𝓝 (f z) ×ˢ 𝓝 (f z), f (g a.1) = f (g a.2) → g a.1 = g a.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | intro ⟨x, y⟩ ⟨ex, ey⟩ h | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
⊢ ∀ (x : T × T), f (g x.1) = x.1 ∧ f (g x.2) = x.2 → f (g x.1) = f (g x.2) → g x.1 = g x.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
x y : T
ex : f (g (x, y).1) = (x, y).1
ey : f (g (x, y).2) = (x, y).2
h : f (g (x, y).1) = f (g (x, y).2)
⊢ g (x, y).1 = g (x, y).2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
⊢ ∀ (x : T × T), f (g x.1) = x.1 ∧ f (g x.2) = x.2 → f (g x.1) = f (g x.2) → g x.1 = g x.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | simp only at ex ey | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
x y : T
ex : f (g (x, y).1) = (x, y).1
ey : f (g (x, y).2) = (x, y).2
h : f (g (x, y).1) = f (g (x, y).2)
⊢ g (x, y).1 = g (x, y).2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
x y : T
ex : f (g x) = x
ey : f (g y) = y
h : f (g (x, y).1) = f (g (x, y).2)
⊢ g (x, y).1 = g (x, y).2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
x y : T
ex : f (g (x, y).1) = (x, y).1
ey : f (g (x, y).2) = (x, y).2
h : f (g (x, y).1) = f (g (x, y).2)
⊢ g (x, y).1 = g (x, y).2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | simp only [ex, ey] at h | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
x y : T
ex : f (g x) = x
ey : f (g y) = y
h : f (g (x, y).1) = f (g (x, y).2)
⊢ g (x, y).1 = g (x, y).2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
x y : T
ex : f (g x) = x
ey : f (g y) = y
h : x = y
⊢ g (x, y).1 = g (x, y).2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
x y : T
ex : f (g x) = x
ey : f (g y) = y
h : f (g (x, y).1) = f (g (x, y).2)
⊢ g (x, y).1 = g (x, y).2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | simp only [h] | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
x y : T
ex : f (g x) = x
ey : f (g y) = y
h : x = y
⊢ g (x, y).1 = g (x, y).2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
n : NontrivialHolomorphicAt g (f z)
o : 𝓝 z = Filter.map g (𝓝 (f z))
x y : T
ex : f (g x) = x
ey : f (g y) = y
h : x = y
⊢ g (x, y).1 = g (x, y).2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | rw [← gf.self_of_nhds] at fa | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
⊢ NontrivialHolomorphicAt g (f z) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
nc : mfderiv I I f z ≠ 0
g : T → S
fa : HolomorphicAt I I f (g (f z))
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
⊢ NontrivialHolomorphicAt g (f z) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
fa : HolomorphicAt I I f z
nc : mfderiv I I f z ≠ 0
g : T → S
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
⊢ NontrivialHolomorphicAt g (f z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | refine (NontrivialHolomorphicAt.anti ?_ fa ga).2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
nc : mfderiv I I f z ≠ 0
g : T → S
fa : HolomorphicAt I I f (g (f z))
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
⊢ NontrivialHolomorphicAt g (f z) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
nc : mfderiv I I f z ≠ 0
g : T → S
fa : HolomorphicAt I I f (g (f z))
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
⊢ NontrivialHolomorphicAt (fun z => f (g z)) (f z) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
nc : mfderiv I I f z ≠ 0
g : T → S
fa : HolomorphicAt I I f (g (f z))
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
⊢ NontrivialHolomorphicAt g (f z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj | [24, 1] | [35, 61] | exact (nontrivialHolomorphicAt_id _).congr (Filter.EventuallyEq.symm fg) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
nc : mfderiv I I f z ≠ 0
g : T → S
fa : HolomorphicAt I I f (g (f z))
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
⊢ NontrivialHolomorphicAt (fun z => f (g z)) (f z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : S → T
z : S
nc : mfderiv I I f z ≠ 0
g : T → S
fa : HolomorphicAt I I f (g (f z))
ga : HolomorphicAt I I g (f z)
gf : ∀ᶠ (x : S) in 𝓝 z, g (f x) = x
fg : ∀ᶠ (x : T) in 𝓝 (f z), f (g x) = x
⊢ NontrivialHolomorphicAt (fun z => f (g z)) (f z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | rcases complex_inverse_fun fa nc with ⟨g, ga, gf, fg⟩ | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | have n : NontrivialHolomorphicAt (g c) (f c z) := by
have e : (c, z) = (c, g c (f c z)) := by rw [gf.self_of_nhds]
rw [e] at fa
refine (NontrivialHolomorphicAt.anti ?_ fa.along_snd ga.along_snd).2
refine (nontrivialHolomorphicAt_id _).congr ?_
refine ((continuousAt_const.prod continuousAt_id).eventually fg).mp (eventually_of_forall ?_)
exact fun _ e ↦ e.symm | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | have o := n.nhds_eq_map_nhds_param ga | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, g c (f c z)) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | rw [gf.self_of_nhds] at o | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, g c (f c z)) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, (c, z).2) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I.prod I) I (uncurry g) (c, f c z)
gf : ∀ᶠ (x : ℂ × S) in 𝓝 (c, z), g x.1 (f x.1 x.2) = x.2
fg : ∀ᶠ (x : ℂ × T) in 𝓝 (c, f c z), f x.1 (g x.1 x.2) = x.2
n : NontrivialHolomorphicAt (g c) (f c z)
o : 𝓝 (c, g c (f c z)) = Filter.map (fun p => (p.1, g p.1 p.2)) (𝓝 (c, f c z))
⊢ ∀ᶠ (p : (ℂ × S) × ℂ × S) in 𝓝 ((c, z), c, z), p.1.1 = p.2.1 → f p.1.1 p.1.2 = f p.2.1 p.2.2 → p.1 = p.2
TACTIC:
|
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