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/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | refine _root_.trans (csInf_le s.bddBelow_ps ?_) zp | case intro.mk.intro.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✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 :... | case intro.mk.intro.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✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro.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✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | right | case intro.mk.intro.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✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1 :... | case intro.mk.intro.intro.intro.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✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro.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✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.lowerSemicontinuous_p | [89, 1] | [115, 66] | use za, z, rfl, zc | case intro.mk.intro.intro.intro.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✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
p : ℝ
h : ∃ᶠ (x : ℂ) in 𝓝 c, s.p x ≤ p
q : ℝ
pq : p < q
q1... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk.intro.intro.intro.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✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical | [131, 1] | [138, 86] | contrapose p | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p : Postcritical s c z
p0 : s.potential c z ≠ 0
⊢ ¬Precritical (f c) z | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
p : ¬¬Precritical (f c) z
⊢ ¬Postcritical s c z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p : Postcritical s c z
p0 : s.potential c z ≠ 0
⊢ ¬Precritical (f ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical | [131, 1] | [138, 86] | simp only [Postcritical, not_not, not_forall, not_lt] at p ⊢ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
p : ¬¬Precritical (f c) z
⊢ ¬Postcritical s c z | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
p : Precritical (f c) z
⊢ s.p c ≤ s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
p : ¬¬Precritical (f c) z
⊢ ¬Postcritical... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical | [131, 1] | [138, 86] | rcases p with ⟨n, p⟩ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
p : Precritical (f c) z
⊢ s.p c ≤ s.potential c z | case intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
n : ℕ
p : Critical (f c) ((f c)^[n] z)
⊢ s.p c ≤ s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
p : Precritical (f c) z
⊢ s.p c ≤ s.poten... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical | [131, 1] | [138, 86] | trans s.potential c ((f c)^[n] z) | case intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
n : ℕ
p : Critical (f c) ((f c)^[n] z)
⊢ s.p c ≤ s.potential c z | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
n : ℕ
p : Critical (f c) ((f c)^[n] z)
⊢ s.p c ≤ s.potential c ((f c)^[n] z)
S : Type
inst✝⁴ : Topol... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
n : ℕ
p : Critical (f c) ((f ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical | [131, 1] | [138, 86] | refine csInf_le s.bddBelow_ps (Or.inr ⟨?_, (f c)^[n] z, rfl, p⟩) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
n : ℕ
p : Critical (f c) ((f c)^[n] z)
⊢ s.p c ≤ s.potential c ((f c)^[n] z) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
n : ℕ
p : Critical (f c) ((f c)^[n] z)
⊢ s.potential c ((f c)^[n] z) ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
n : ℕ
p : Critical (f c) ((f c)^[n] z)
⊢... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical | [131, 1] | [138, 86] | simp only [s.potential_eqn_iter] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
n : ℕ
p : Critical (f c) ((f c)^[n] z)
⊢ s.potential c ((f c)^[n] z) ≠ 0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
n : ℕ
p : Critical (f c) ((f c)^[n] z)
⊢ s.potential c z ^ d ^ n ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
n : ℕ
p : Critical (f c) ((f c)^[n] z)
⊢... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical | [131, 1] | [138, 86] | exact pow_ne_zero _ p0 | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
n : ℕ
p : Critical (f c) ((f c)^[n] z)
⊢ s.potential c z ^ d ^ n ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
n : ℕ
p : Critical (f c) ((f c)^[n] z)
⊢... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical | [131, 1] | [138, 86] | simp only [s.potential_eqn_iter] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
n : ℕ
p : Critical (f c) ((f c)^[n] z)
⊢ s.potential c ((f c)^[n] z) ≤ s.potential c z | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
n : ℕ
p : Critical (f c) ((f c)^[n] z)
⊢ s.potential c z ^ d ^ n ≤ s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
n : ℕ
p : Critical (f c) ((f c)^[n] z)
⊢... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical | [131, 1] | [138, 86] | exact pow_le_of_le_one s.potential_nonneg s.potential_le_one (pow_ne_zero _ s.d0) | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
n : ℕ
p : Critical (f c) ((f c)^[n] z)
⊢ s.potential c z ^ d ^ n ≤ s.potential c z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s : Super f d a
p0 : s.potential c z ≠ 0
n : ℕ
p : Critical (f c) ((f c)^[n] z)
⊢... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical' | [141, 1] | [143, 88] | apply p.not_precritical | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p : Postcritical s c z
za : z ≠ a
inst✝ : OnePreimage s
⊢ ¬Precritical (f 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 : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p : Postcritical s c z
za : z ≠ a
inst✝ : OnePreimage s
⊢ s.potential c z ≠ 0 | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p : Postcritical s c z
za : z ≠ a
inst✝ : OnePreimage s
⊢ ¬Precri... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical' | [141, 1] | [143, 88] | simp only [Ne, s.potential_eq_zero_of_onePreimage] | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p : Postcritical s c z
za : z ≠ a
inst✝ : OnePreimage s
⊢ s.potential c z ≠ 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 : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p : Postcritical s c z
za : z ≠ a
inst✝ : OnePreimage s
⊢ ¬z = a | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p : Postcritical s c z
za : z ≠ a
inst✝ : OnePreimage s
⊢ s.poten... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.not_precritical' | [141, 1] | [143, 88] | exact za | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p : Postcritical s c z
za : z ≠ a
inst✝ : OnePreimage s
⊢ ¬z = a | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p : Postcritical s c z
za : z ≠ a
inst✝ : OnePreimage s
⊢ ¬z = a
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.isOpen_post | [150, 1] | [158, 37] | set f := fun x : ℂ × S ↦ s.p x.1 - s.potential x.1 x.2 | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
⊢ IsOpen 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × S → ℝ := fun x => s.p x.1 - s.potential x.1 x.2
⊢ IsOpen 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
⊢ IsOpen s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.isOpen_post | [150, 1] | [158, 37] | have fc : LowerSemicontinuous f :=
(s.lowerSemicontinuous_p.comp continuous_fst).add
(Continuous.potential s).neg.lowerSemicontinuous | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × S → ℝ := fun x => s.p x.1 - s.potential x.1 x.2
⊢ IsOpen 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × S → ℝ := fun x => s.p x.1 - s.potential x.1 x.2
fc : LowerSemicontinuous f
⊢ IsOpen 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × S → ℝ := fun x => s.p x.1 - s.... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.isOpen_post | [150, 1] | [158, 37] | have e : s.post = f ⁻¹' Ioi 0 :=
Set.ext fun _ ↦ by
simp only [Super.post, mem_setOf, Postcritical, mem_preimage, mem_Ioi, sub_pos, f] | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × S → ℝ := fun x => s.p x.1 - s.potential x.1 x.2
fc : LowerSemicontinuous f
⊢ IsOpen 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × S → ℝ := fun x => s.p x.1 - s.potential x.1 x.2
fc : LowerSemicontinuous f
e : s.post = f ⁻... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × S → ℝ := fun x => s.p x.1 - s.... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.isOpen_post | [150, 1] | [158, 37] | rw [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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × S → ℝ := fun x => s.p x.1 - s.potential x.1 x.2
fc : LowerSemicontinuous f
e : s.post = f ⁻... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × S → ℝ := fun x => s.p x.1 - s.potential x.1 x.2
fc : LowerSemicontinuous f
e : s.post = f ⁻... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × S → ℝ := fun x => s.p x.1 - s.... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.isOpen_post | [150, 1] | [158, 37] | exact fc.isOpen_preimage _ | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × S → ℝ := fun x => s.p x.1 - s.potential x.1 x.2
fc : LowerSemicontinuous f
e : s.post = f ⁻... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × S → ℝ := fun x => s.p x.1 - s.... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.isOpen_post | [150, 1] | [158, 37] | simp only [Super.post, mem_setOf, Postcritical, mem_preimage, mem_Ioi, sub_pos, f] | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × S → ℝ := fun x => s.p x.1 - s.potential x.1 x.2
fc : LowerSemicontinuous f
x✝ : ℂ × S
⊢ x✝ ... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × S → ℝ := fun x => s.p x.1 - s.... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.eventually | [161, 1] | [163, 90] | refine (s.isOpen_post.eventually_mem ?_).mp (eventually_of_forall fun _ m ↦ 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 : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p : Postcritical s c z
inst✝ : OnePreimage s
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), Postcritical s p.1 p.2 | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p : Postcritical s c z
inst✝ : OnePreimage s
⊢ (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 : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p : Postcritical s c z
inst✝ : OnePreimage s
⊢ ∀ᶠ (p : ℂ × S) in ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Postcritical.eventually | [161, 1] | [163, 90] | exact 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 : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p : Postcritical s c z
inst✝ : OnePreimage s
⊢ (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 : ℂ
a z z0 z1 : S
d n : ℕ
s : Super f d a
p : Postcritical s c z
inst✝ : OnePreimage s
⊢ (c, z) ∈ s.post
TA... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.post_a | [174, 1] | [175, 82] | simp only [Super.post, Postcritical, s.potential_a, mem_setOf] | 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✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ (c, a) ∈ 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✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ 0 < s.p c | 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✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ (c, a) ∈ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.post_a | [174, 1] | [175, 82] | exact s.p_pos c | 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✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ 0 < s.p c | 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✝ : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ 0 < s.p c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.stays_post | [178, 1] | [181, 87] | rcases p with ⟨c, z⟩ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
p : ℂ × S
m : p ∈ s.post
⊢ (p.1, f p.1 p.2) ∈ s.post | case mk
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ ((c, z).1, f (c, z).1 (c, z).2) ∈ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
p : ℂ × S
m : p ∈ s.post
⊢ (p.1, f p.1 p.2) ∈ s.post
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.stays_post | [178, 1] | [181, 87] | simp only [Super.post, mem_setOf, Postcritical, s.potential_eqn] | case mk
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ ((c, z).1, f (c, z).1 (c, z).2) ∈ s.post | case mk
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ s.potential c z ^ d < s.p c | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ ((c, z).1, f (c, z)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.stays_post | [178, 1] | [181, 87] | exact lt_of_le_of_lt (pow_le_of_le_one s.potential_nonneg s.potential_le_one s.d0) m | case mk
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ s.potential c z ^ d < s.p c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ z0 z1 : S
d n : ℕ
s✝ s : Super f d a
c : ℂ
z : S
m : (c, z) ∈ s.post
⊢ s.potential c z ^ d... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.iter_stays_post | [184, 1] | [187, 65] | induction' n with n h | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s✝ s : Super f d a
p : ℂ × S
m : p ∈ s.post
n : ℕ
⊢ (p.1, (f p.1)^[n] p.2) ∈ s.post | case zero
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
p : ℂ × S
m : p ∈ s.post
⊢ (p.1, (f p.1)^[0] p.2) ∈ s.post
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ ... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s✝ s : Super f d a
p : ℂ × S
m : p ∈ s.post
n : ℕ
⊢ (p.1, (f p.1)^[n] p.2) ∈ s.po... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.iter_stays_post | [184, 1] | [187, 65] | simp only [Function.iterate_zero_apply] | case zero
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
p : ℂ × S
m : p ∈ s.post
⊢ (p.1, (f p.1)^[0] p.2) ∈ s.post
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ ... | case zero
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
p : ℂ × S
m : p ∈ s.post
⊢ (p.1, p.2) ∈ s.post
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpa... | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
p : ℂ × S
m : p ∈ s.post
⊢ (p.1, (f p.1)^[0] p.2) ∈ s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.iter_stays_post | [184, 1] | [187, 65] | exact m | case zero
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
p : ℂ × S
m : p ∈ s.post
⊢ (p.1, p.2) ∈ s.post
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpa... | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s✝ s : Super f d a
p : ℂ × S
m : p ∈ s.post
n : ℕ
h : (p.1, (f p.1)^[n] p.2) ∈ s.post
⊢ (p.1, (f p.1)^[n + 1] p.2) ∈ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
p : ℂ × S
m : p ∈ s.post
⊢ (p.1, p.2) ∈ s.post
case ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.iter_stays_post | [184, 1] | [187, 65] | simp only [Function.iterate_succ_apply'] | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s✝ s : Super f d a
p : ℂ × S
m : p ∈ s.post
n : ℕ
h : (p.1, (f p.1)^[n] p.2) ∈ s.post
⊢ (p.1, (f p.1)^[n + 1] p.2) ∈ s.post | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s✝ s : Super f d a
p : ℂ × S
m : p ∈ s.post
n : ℕ
h : (p.1, (f p.1)^[n] p.2) ∈ s.post
⊢ (p.1, f p.1 ((f p.1)^[n] p.2)) ∈ s.post | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s✝ s : Super f d a
p : ℂ × S
m : p ∈ s.post
n : ℕ
h : (p.1, (f p.1)^[n]... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.iter_stays_post | [184, 1] | [187, 65] | exact s.stays_post h | case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s✝ s : Super f d a
p : ℂ × S
m : p ∈ s.post
n : ℕ
h : (p.1, (f p.1)^[n] p.2) ∈ s.post
⊢ (p.1, f p.1 ((f p.1)^[n] p.2)) ∈ s.post | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z z0 z1 : S
d n✝ : ℕ
s✝ s : Super f d a
p : ℂ × S
m : p ∈ s.post
n : ℕ
h : (p.1, (f p.1)^[n]... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.basin_post | [190, 1] | [194, 67] | rcases tendsto_atTop_nhds.mp (s.basin_attracts m) {z | (c, z) ∈ s.post} (s.post_a c)
(s.isOpen_post.snd_preimage c) 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.basin
⊢ ∃ n, (c, (f c)^[n] 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 : ℂ
a z z0 z1 : S
d n✝ : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.basin
n : ℕ
h : ∀ (n_1 : ℕ), n ≤ n_1 → (f c)^[n_1] z ∈ {z | (c, z) ∈ s.pos... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.basin
⊢ ∃ n, (c, (f c)^[n... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.basin_post | [190, 1] | [194, 67] | specialize h n (le_refl 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 : ℂ
a z z0 z1 : S
d n✝ : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.basin
n : ℕ
h : ∀ (n_1 : ℕ), n ≤ n_1 → (f c)^[n_1] z ∈ {z | (c, z) ∈ s.pos... | 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 : ℂ
a z z0 z1 : S
d n✝ : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.basin
n : ℕ
h : (f c)^[n] z ∈ {z | (c, z) ∈ s.post}
⊢ ∃ n, (c, (f c)^[n] z... | 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 : ℂ
a z z0 z1 : S
d n✝ : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.basin
n : ℕ
h... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.basin_post | [190, 1] | [194, 67] | simp only [mem_setOf] at h | 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 : ℂ
a z z0 z1 : S
d n✝ : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.basin
n : ℕ
h : (f c)^[n] z ∈ {z | (c, z) ∈ s.post}
⊢ ∃ n, (c, (f c)^[n] z... | 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 : ℂ
a z z0 z1 : S
d n✝ : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.basin
n : ℕ
h : (c, (f c)^[n] z) ∈ s.post
⊢ ∃ n, (c, (f c)^[n] 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 : ℂ
a z z0 z1 : S
d n✝ : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.basin
n : ℕ
h... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.basin_post | [190, 1] | [194, 67] | use n, h | 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 : ℂ
a z z0 z1 : S
d n✝ : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.basin
n : ℕ
h : (c, (f c)^[n] z) ∈ s.post
⊢ ∃ n, (c, (f c)^[n] z) ∈ s.post | no goals | 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 : ℂ
a z z0 z1 : S
d n✝ : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
m : (c, z) ∈ s.basin
n : ℕ
h... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | rcases((Filter.eventually_ge_atTop n).and (s.eventually_noncritical ⟨_, r⟩)).exists with
⟨m, nm, mc⟩ | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
⊢ NontrivialHolomorphicAt (s.bottcherNearIter n ... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : O... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | have r' := s.iter_stays_near' r nm | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritica... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | replace h := h.nonconst | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritica... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | refine ⟨(s.bottcherNearIter_holomorphic r).along_snd, ?_⟩ | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritica... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | contrapose h | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritica... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | simp only [Filter.not_frequently, not_not] at h ⊢ | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritica... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | rw [← Nat.sub_add_cancel nm] | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritica... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | generalize hk : m - n = k | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritica... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | clear hk nm mc r' p 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
inst✝ : OnePreimage s
h : ∀ᶠ (x : S) in 𝓝 z, s.bottcherNearIter n c x = s.b... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritica... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | have er : ∀ᶠ w in 𝓝 z, (c, (f c)^[n] w) ∈ s.near :=
(continuousAt_const.prod (s.continuousAt_iter continuousAt_const
continuousAt_id)).eventually_mem (s.isOpen_near.mem_nhds r) | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
inst✝ : OnePreimage s
h : ∀ᶠ (x : S) in 𝓝 z, s.bottcherNearIter n c x = s.b... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
inst✝ : OnePreimage s
h : ∀ᶠ (x : S) in 𝓝 z, s.bottcherNearIter n c x = s.b... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
inst✝ : OnePrei... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | refine (h.and er).mp (eventually_of_forall ?_) | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
inst✝ : OnePreimage s
h : ∀ᶠ (x : S) in 𝓝 z, s.bottcherNearIter n c x = s.b... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
inst✝ : OnePreimage s
h : ∀ᶠ (x : S) in 𝓝 z, s.bottcherNearIter n c x = s.b... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
inst✝ : OnePrei... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | intro x ⟨e, 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
inst✝ : OnePreimage s
h : ∀ᶠ (x : S) in 𝓝 z, s.bottcherNearIter n c x = s.b... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
inst✝ : OnePreimage s
h : ∀ᶠ (x : S) in 𝓝 z, s.bottcherNearIter n c x = s.b... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
inst✝ : OnePrei... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | simp only [Super.bottcherNearIter] at e | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
inst✝ : OnePreimage s
h : ∀ᶠ (x : S) in 𝓝 z, s.bottcherNearIter n c x = s.b... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
inst✝ : OnePreimage s
h : ∀ᶠ (x : S) in 𝓝 z, s.bottcherNearIter n c x = s.b... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
inst✝ : OnePrei... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | simp only [Super.bottcherNearIter, Function.iterate_add_apply, s.bottcherNear_eqn_iter m,
s.bottcherNear_eqn_iter r, e] | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
inst✝ : OnePreimage s
h : ∀ᶠ (x : S) in 𝓝 z, s.bottcherNearIter n c x = s.b... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
inst✝ : OnePrei... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | by_cases p0 : s.potential c z = 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I I (s.bottcherNea... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I I (s.bo... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : O... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | rw [s.potential_eq_zero_of_onePreimage] at p0 | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I I (s.bo... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I I (s.bo... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | rw [p0] | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I I (s.bo... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I I (s.bo... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | exact s.bottcherNearIter_nontrivial_a | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I I (s.bo... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.bottcherNearIterNontrivial | [197, 1] | [219, 34] | exact nontrivialHolomorphicAt_of_mfderiv_ne_zero (s.bottcherNearIter_holomorphic r').along_snd
(s.bottcherNearIter_mfderiv_ne_zero mc (p.not_precritical p0)) | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
inst✝ : OnePreimage s
m : ℕ
nm : n ≤ m
mc : mfderiv I I (s.bo... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
r : (c, (f c)^[n] z) ∈ s.near
p : Postcritical s c z
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ∀ᶠ (w : S) in 𝓝 z, s.potential c z ≤ s.potential c w
⊢ z = 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
⊢ ¬∀ᶠ (w : S) in 𝓝 z, s.potential c z ≤ s.potential c w | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ∀ᶠ (w : S) in... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | simp only [Filter.not_eventually, not_le] | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
⊢ ¬∀ᶠ (w : S) in 𝓝 z, s.potential c z ≤ s.potential c w | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
⊢ ∃ᶠ (x : S) in 𝓝 z, s.potential c x < 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
⊢ ¬∀ᶠ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | rcases s.nice_nz p.basin z (le_refl _) with ⟨near, nc⟩ | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
⊢ ∃ᶠ (x : S) in 𝓝 z, s.potential c x < s.potential c z | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
near : (c, (f c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ℕ), ... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
⊢ ∃ᶠ (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | set f : S → ℂ := s.bottcherNearIter (s.nz c z) c | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
near : (c, (f c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ℕ... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | have o : 𝓝 (f z) = Filter.map f (𝓝 z) :=
(nontrivialHolomorphicAt_of_mfderiv_ne_zero (s.bottcherNearIter_holomorphic near).along_snd
(s.bottcherNearIter_mfderiv_ne_zero (nc _ (le_refl _))
(p.not_precritical ((s.potential_ne_zero _).mpr m)))).nhds_eq_map_nhds | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ℕ... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | have e : ∃ᶠ x : ℂ in 𝓝 (f z), abs x < abs (f z) := by
apply frequently_smaller; contrapose m; simp only [not_not] at m ⊢
replace m := (s.bottcherNear_eq_zero near).mp m
rw [s.preimage_eq] at m; exact m | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ℕ... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | rw [o, Filter.frequently_map] at e | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ℕ... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | apply e.mp | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ℕ... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | apply (((s.isOpen_preimage _).snd_preimage c).eventually_mem near).mp | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ℕ... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | refine eventually_of_forall fun w m lt ↦ ?_ | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m✝ : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | rw [mem_setOf, mem_setOf] at m | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m✝ : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m✝ : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m✝ :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | simp only at m | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m✝ : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m✝ : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m✝ :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | simp only [s.potential_eq m, s.potential_eq near, Super.potential'] | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m✝ : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m✝ : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m✝ :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | exact Real.rpow_lt_rpow (Complex.abs.nonneg _) lt
(inv_pos.mpr (pow_pos (Nat.cast_pos.mpr s.dp) _)) | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m✝ : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ... | no goals | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m✝ :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | apply frequently_smaller | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ℕ), s.nz c z... | case z0
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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ℕ), ... | 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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
near... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | contrapose m | case z0
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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z = a
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ℕ), ... | case z0
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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ℕ), s.nz c z ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case z0
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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
m : ¬z ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | simp only [not_not] at m ⊢ | case z0
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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ℕ), s.nz c z ≤ ... | case z0
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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ℕ), s.nz c z ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case z0
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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
near : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | replace m := (s.bottcherNear_eq_zero near).mp m | case z0
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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ℕ), s.nz c z ≤ ... | case z0
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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ℕ), s.nz c z ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case z0
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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
near : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | rw [s.preimage_eq] at m | case z0
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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ℕ), s.nz c z ≤ ... | case z0
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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ℕ), s.nz c z ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case z0
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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
near : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Postcritical.lean | Super.potential_minima_only_a | [222, 1] | [242, 54] | exact m | case z0
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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
near : (c, (f✝ c)^[s.nz c z] z) ∈ s.near
nc : ∀ (k : ℕ), s.nz c z ≤ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case z0
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 : ℂ
a z z0 z1 : S
d n : ℕ
s✝ s : Super f✝ d a
inst✝ : OnePreimage s
p : Postcritical s c z
near : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | set s := superF 2 | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
m : (c, ↑z) ∈ ⋯.basin
⊢ ∃ n, Complex.abs ((f' 2 c)^[n + 1] z) ∈ Ioc 4 20 | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
⊢ ∃ n, Complex.abs ((f' 2 c)^[n + 1] z) ∈ Ioc 4 20 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
m : (c, ↑z) ∈ ⋯.basin
⊢ ∃ n, Complex.abs ((f' 2 c)^[n + 1] z) ∈ Ioc 4 20
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | simp only [s.basin_iff_attracts, Attracts, RiemannSphere.tendsto_inf_iff_tendsto_atInf, f_f'_iter,
tendsto_atInf_iff_norm_tendsto_atTop, Complex.norm_eq_abs] at m | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
⊢ ∃ n, Complex.abs ((f' 2 c)^[n + 1] z) ∈ Ioc 4 20 | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
⊢ ∃ n, Complex.abs ((f' 2 c)^[n + 1] z) ∈ Ioc 4 20 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : (c, ↑z) ∈ s.basin
⊢ ∃ n, Complex.abs ((f' 2 c)^[n + 1] z) ∈ Ioc 4 20
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | rcases Filter.tendsto_atTop_atTop.mp m 5 with ⟨n,h⟩ | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
⊢ ∃ n, Complex.abs ((f' 2 c)^[n + 1] z) ∈ Ioc 4 20 | case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
h : ∀ (a : ℕ), n ≤ a → 5 ≤ Complex.abs ((f' 2 c)^[a] z)
⊢ ∃ n, Complex.abs ((f' 2 c)^[n + 1] z) ∈ Ioc 4 20 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
⊢ ∃ n, Complex.abs ((f' 2 c)^[n + 1] z) ∈ Ioc 4 20
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | generalize hp : (fun n ↦ 4 < abs ((f' 2 c)^[n] z)) = p | case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
h : ∀ (a : ℕ), n ≤ a → 5 ≤ Complex.abs ((f' 2 c)^[a] z)
⊢ ∃ n, Complex.abs ((f' 2 c)^[n + 1] z) ∈ Ioc 4 20 | case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
h : ∀ (a : ℕ), n ≤ a → 5 ≤ Complex.abs ((f' 2 c)^[a] z)
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) =... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
h : ∀ (a : ℕ), n ≤ a → 5 ≤ Complex.abs ((f' 2 c)^[a] z)
⊢ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | replace h : p n := by rw [←hp]; linarith [h n (by linarith)] | case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
h : ∀ (a : ℕ), n ≤ a → 5 ≤ Complex.abs ((f' 2 c)^[a] z)
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) =... | case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
⊢ ∃ n, Complex.abs ((f' 2 c)^[n + 1] z) ∈ Ioc... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
h : ∀ (a : ℕ), n ≤ a → 5 ≤ Complex.abs ((f' 2 c)^[a] z)
p ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | generalize hk : Nat.find (p := p) ⟨_,h⟩ = k | case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
⊢ ∃ n, Complex.abs ((f' 2 c)^[n + 1] z) ∈ Ioc... | case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
⊢ ∃ n, Complex.abs ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | have k4 : p k := by rw [←hk]; exact Nat.find_spec (p := p) ⟨_,h⟩ | case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
⊢ ∃ n, Complex.abs ... | case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
⊢ ∃ n, Com... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | have k0 : k ≠ 0 := by
contrapose k4
simp only [not_not] at k4
simp only [k4, ←hp, not_lt, Function.iterate_zero_apply, z4] | case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
⊢ ∃ n, Com... | case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
k0 : k ≠ 0... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | have k1 : 1 ≤ k := Nat.pos_iff_ne_zero.mpr k0 | case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
k0 : k ≠ 0... | case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
k0 : k ≠ 0... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | use k-1 | case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
k0 : k ≠ 0... | case h
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
k0 : k ≠ 0
k1 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | have lt : ¬p (k-1) := by apply Nat.find_min; rw [hk]; omega | case h
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
k0 : k ≠ 0
k1 ... | case h
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
k0 : k ≠ 0
k1 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | simp only [Nat.sub_add_cancel k1, ←hp, not_lt] at k4 k1 lt ⊢ | case h
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
k0 : k ≠ 0
k1 ... | case h
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k0 : k ≠ 0
k1 : 1 ≤ k
k... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | use k4 | case h
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k0 : k ≠ 0
k1 : 1 ≤ k
k... | case right
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k0 : k ≠ 0
k1 : 1 ≤... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | have fs := iter_small 2 c ((f' 2 c)^[k-1] z) | case right
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k0 : k ≠ 0
k1 : 1 ≤... | case right
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k0 : k ≠ 0
k1 : 1 ≤... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | simp only [←Function.iterate_succ_apply', Nat.succ_eq_add_one, Nat.sub_add_cancel k1] at fs | case right
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k0 : k ≠ 0
k1 : 1 ≤... | case right
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k0 : k ≠ 0
k1 : 1 ≤... | Please generate a tactic in lean4 to solve the state.
STATE:
case right
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | exact le_trans fs (le_trans (add_le_add (pow_le_pow_left (Complex.abs.nonneg _) lt 2) c4)
(by norm_num)) | case right
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k0 : k ≠ 0
k1 : 1 ≤... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | rw [←hp] | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
h : ∀ (a : ℕ), n ≤ a → 5 ≤ Complex.abs ((f' 2 c)^[a] z)
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
⊢ p n | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
h : ∀ (a : ℕ), n ≤ a → 5 ≤ Complex.abs ((f' 2 c)^[a] z)
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
⊢ (fun n... | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
h : ∀ (a : ℕ), n ≤ a → 5 ≤ Complex.abs ((f' 2 c)^[a] z)
p : ℕ → Prop
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | linarith [h n (by linarith)] | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
h : ∀ (a : ℕ), n ≤ a → 5 ≤ Complex.abs ((f' 2 c)^[a] z)
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
⊢ (fun n... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
h : ∀ (a : ℕ), n ≤ a → 5 ≤ Complex.abs ((f' 2 c)^[a] z)
p : ℕ → Prop
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | linarith | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
h : ∀ (a : ℕ), n ≤ a → 5 ≤ Complex.abs ((f' 2 c)^[a] z)
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
⊢ n ≤ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
h : ∀ (a : ℕ), n ≤ a → 5 ≤ Complex.abs ((f' 2 c)^[a] z)
p : ℕ → Prop
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | rw [←hk] | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
⊢ p k | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
⊢ p (Nat.find ⋯) | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | exact Nat.find_spec (p := p) ⟨_,h⟩ | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
⊢ p (Nat.find ⋯) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | contrapose k4 | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : p k
⊢ k ≠ 0 | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : ¬k ≠ 0
⊢ ¬p k | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | simp only [not_not] at k4 | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : ¬k ≠ 0
⊢ ¬p k | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : k = 0
⊢ ¬p k | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/PotentialLower.lean | pass_through | [28, 1] | [54, 19] | simp only [k4, ←hp, not_lt, Function.iterate_zero_apply, z4] | c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h : p n
k : ℕ
hk : Nat.find ⋯ = k
k4 : k = 0
⊢ ¬p k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : ℂ
c4 : Complex.abs c ≤ 4
z4 : Complex.abs z ≤ 4
s : Super (f 2) 2 OnePoint.infty := superF 2
m : Filter.Tendsto (fun x => Complex.abs ((f' 2 c)^[x] z)) Filter.atTop Filter.atTop
n : ℕ
p : ℕ → Prop
hp : (fun n => 4 < Complex.abs ((f' 2 c)^[n] z)) = p
h :... |
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