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/Multibrot/Connected.lean | isPathConnected_potential_levelset | [36, 1] | [48, 76] | simp only [mem_ball, Complex.dist_eq, sub_zero, ep, p1] | case h.mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
e : ℂ
ep : Complex.abs e = p
ec : ray d e = c
⊢ e ∈ ball 0 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c✝ : ℂ
d : ℕ
inst✝ : Fact (2 ≤ d)
p : ℝ
p0 : 0 ≤ p
p1 : p < 1
c : 𝕊
e : ℂ
ep : Complex.abs e = p
ec : ray d e = c
⊢ e ∈ ball 0 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | refine ⟨⟨((0 : ℂ) : 𝕊),?_⟩,?_⟩ | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsConnected (multibrotExt d)ᶜ | case refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ↑0 ∈ (multibrotExt d)ᶜ
case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsPreconnected (multibrotExt d)ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsConnected (multibrotExt d)ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | have e : (multibrotExt d)ᶜ = ⋂ p : Ico 0 (1 : ℝ), potential d ⁻¹' Ici p := by
apply Set.ext; intro z
simp only [mem_compl_iff, ← potential_lt_one, mem_iInter, mem_preimage, not_lt, mem_Ici]
constructor; intro p1 ⟨q, m⟩; simp only [Subtype.coe_mk, mem_Ico] at m ⊢; linarith
intro h; contrapose h; simp only [not_le, not_forall] at h ⊢
rcases exists_between h with ⟨y, py, y1⟩
exact ⟨⟨y, ⟨le_trans potential_nonneg py.le, y1⟩⟩, py⟩ | case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsPreconnected (multibrotExt d)ᶜ | case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
⊢ IsPreconnected (multibrotExt d)ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsPreconnected (multibrotExt d)ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | rw [e] | case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
⊢ IsPreconnected (multibrotExt d)ᶜ | case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
⊢ IsPreconnected (⋂ p, potential d ⁻¹' Ici ↑p) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
⊢ IsPreconnected (multibrotExt d)ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | refine @IsPreconnected.directed_iInter _ _ _ _ ?_ _ ?_ ?_ ?_ | case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
⊢ IsPreconnected (⋂ p, potential d ⁻¹' Ici ↑p) | case refine_2.refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
⊢ Nonempty ↑(Ico 0 1)
case refine_2.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
⊢ Directed Superset fun p => potential d ⁻¹' Ici ↑p
case refine_2.refine_3
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
⊢ ∀ (a : ↑(Ico 0 1)), IsPreconnected (potential d ⁻¹' Ici ↑a)
case refine_2.refine_4
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
⊢ ∀ (a : ↑(Ico 0 1)), IsCompact (potential d ⁻¹' Ici ↑a) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
⊢ IsPreconnected (⋂ p, potential d ⁻¹' Ici ↑p)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | simp only [mem_compl_iff, multibrotExt_coe, not_not, multibrot_zero] | case refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ↑0 ∈ (multibrotExt d)ᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ↑0 ∈ (multibrotExt d)ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | apply Set.ext | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ (x : 𝕊), x ∈ (multibrotExt d)ᶜ ↔ x ∈ ⋂ p, potential d ⁻¹' Ici ↑p | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | intro z | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ (x : 𝕊), x ∈ (multibrotExt d)ᶜ ↔ x ∈ ⋂ p, potential d ⁻¹' Ici ↑p | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ z ∈ (multibrotExt d)ᶜ ↔ z ∈ ⋂ p, potential d ⁻¹' Ici ↑p | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ (x : 𝕊), x ∈ (multibrotExt d)ᶜ ↔ x ∈ ⋂ p, potential d ⁻¹' Ici ↑p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | simp only [mem_compl_iff, ← potential_lt_one, mem_iInter, mem_preimage, not_lt, mem_Ici] | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ z ∈ (multibrotExt d)ᶜ ↔ z ∈ ⋂ p, potential d ⁻¹' Ici ↑p | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ 1 ≤ potential d z ↔ ∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ z ∈ (multibrotExt d)ᶜ ↔ z ∈ ⋂ p, potential d ⁻¹' Ici ↑p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | constructor | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ 1 ≤ potential d z ↔ ∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z | case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ 1 ≤ potential d z → ∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ (∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z) → 1 ≤ potential d z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ 1 ≤ potential d z ↔ ∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | intro p1 ⟨q, m⟩ | case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ 1 ≤ potential d z → ∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ (∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z) → 1 ≤ potential d z | case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
p1 : 1 ≤ potential d z
q : ℝ
m : q ∈ Ico 0 1
⊢ ↑⟨q, m⟩ ≤ potential d z
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ (∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z) → 1 ≤ potential d z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ 1 ≤ potential d z → ∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ (∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z) → 1 ≤ potential d z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | simp only [Subtype.coe_mk, mem_Ico] at m ⊢ | case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
p1 : 1 ≤ potential d z
q : ℝ
m : q ∈ Ico 0 1
⊢ ↑⟨q, m⟩ ≤ potential d z
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ (∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z) → 1 ≤ potential d z | case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
p1 : 1 ≤ potential d z
q : ℝ
m : 0 ≤ q ∧ q < 1
⊢ q ≤ potential d z
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ (∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z) → 1 ≤ potential d z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
p1 : 1 ≤ potential d z
q : ℝ
m : q ∈ Ico 0 1
⊢ ↑⟨q, m⟩ ≤ potential d z
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ (∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z) → 1 ≤ potential d z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | linarith | case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
p1 : 1 ≤ potential d z
q : ℝ
m : 0 ≤ q ∧ q < 1
⊢ q ≤ potential d z
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ (∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z) → 1 ≤ potential d z | case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ (∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z) → 1 ≤ potential d z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
p1 : 1 ≤ potential d z
q : ℝ
m : 0 ≤ q ∧ q < 1
⊢ q ≤ potential d z
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ (∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z) → 1 ≤ potential d z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | intro h | case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ (∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z) → 1 ≤ potential d z | case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
h : ∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z
⊢ 1 ≤ potential d z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
⊢ (∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z) → 1 ≤ potential d z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | contrapose h | case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
h : ∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z
⊢ 1 ≤ potential d z | case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
h : ¬1 ≤ potential d z
⊢ ¬∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
h : ∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z
⊢ 1 ≤ potential d z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | simp only [not_le, not_forall] at h ⊢ | case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
h : ¬1 ≤ potential d z
⊢ ¬∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z | case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
h : potential d z < 1
⊢ ∃ x, potential d z < ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
h : ¬1 ≤ potential d z
⊢ ¬∀ (i : ↑(Ico 0 1)), ↑i ≤ potential d z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | rcases exists_between h with ⟨y, py, y1⟩ | case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
h : potential d z < 1
⊢ ∃ x, potential d z < ↑x | case h.mpr.intro.intro
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
h : potential d z < 1
y : ℝ
py : potential d z < y
y1 : y < 1
⊢ ∃ x, potential d z < ↑x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
h : potential d z < 1
⊢ ∃ x, potential d z < ↑x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | exact ⟨⟨y, ⟨le_trans potential_nonneg py.le, y1⟩⟩, py⟩ | case h.mpr.intro.intro
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
h : potential d z < 1
y : ℝ
py : potential d z < y
y1 : y < 1
⊢ ∃ x, potential d z < ↑x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.intro
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : 𝕊
h : potential d z < 1
y : ℝ
py : potential d z < y
y1 : y < 1
⊢ ∃ x, potential d z < ↑x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | exact Zero.instNonempty | case refine_2.refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
⊢ Nonempty ↑(Ico 0 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
⊢ Nonempty ↑(Ico 0 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | intro ⟨a, a0, a1⟩ ⟨b, b0, b1⟩ | case refine_2.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
⊢ Directed Superset fun p => potential d ⁻¹' Ici ↑p | case refine_2.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
⊢ ∃ z,
(fun p => potential d ⁻¹' Ici ↑p) ⟨a, ⋯⟩ ⊇ (fun p => potential d ⁻¹' Ici ↑p) z ∧
(fun p => potential d ⁻¹' Ici ↑p) ⟨b, ⋯⟩ ⊇ (fun p => potential d ⁻¹' Ici ↑p) z | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
⊢ Directed Superset fun p => potential d ⁻¹' Ici ↑p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | refine ⟨⟨max a b, mem_Ico.mpr ⟨le_max_of_le_left a0, max_lt a1 b1⟩⟩, ?_, ?_⟩ | case refine_2.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
⊢ ∃ z,
(fun p => potential d ⁻¹' Ici ↑p) ⟨a, ⋯⟩ ⊇ (fun p => potential d ⁻¹' Ici ↑p) z ∧
(fun p => potential d ⁻¹' Ici ↑p) ⟨b, ⋯⟩ ⊇ (fun p => potential d ⁻¹' Ici ↑p) z | case refine_2.refine_2.refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
⊢ (fun p => potential d ⁻¹' Ici ↑p) ⟨a, ⋯⟩ ⊇ (fun p => potential d ⁻¹' Ici ↑p) ⟨max a b, ⋯⟩
case refine_2.refine_2.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
⊢ (fun p => potential d ⁻¹' Ici ↑p) ⟨b, ⋯⟩ ⊇ (fun p => potential d ⁻¹' Ici ↑p) ⟨max a b, ⋯⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
⊢ ∃ z,
(fun p => potential d ⁻¹' Ici ↑p) ⟨a, ⋯⟩ ⊇ (fun p => potential d ⁻¹' Ici ↑p) z ∧
(fun p => potential d ⁻¹' Ici ↑p) ⟨b, ⋯⟩ ⊇ (fun p => potential d ⁻¹' Ici ↑p) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | intro z h | case refine_2.refine_2.refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
⊢ (fun p => potential d ⁻¹' Ici ↑p) ⟨a, ⋯⟩ ⊇ (fun p => potential d ⁻¹' Ici ↑p) ⟨max a b, ⋯⟩ | case refine_2.refine_2.refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
z : 𝕊
h : z ∈ (fun p => potential d ⁻¹' Ici ↑p) ⟨max a b, ⋯⟩
⊢ z ∈ (fun p => potential d ⁻¹' Ici ↑p) ⟨a, ⋯⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2.refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
⊢ (fun p => potential d ⁻¹' Ici ↑p) ⟨a, ⋯⟩ ⊇ (fun p => potential d ⁻¹' Ici ↑p) ⟨max a b, ⋯⟩
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | simp only [mem_preimage, mem_Ici, Subtype.coe_mk, max_le_iff] at h ⊢ | case refine_2.refine_2.refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
z : 𝕊
h : z ∈ (fun p => potential d ⁻¹' Ici ↑p) ⟨max a b, ⋯⟩
⊢ z ∈ (fun p => potential d ⁻¹' Ici ↑p) ⟨a, ⋯⟩ | case refine_2.refine_2.refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
z : 𝕊
h : a ≤ potential d z ∧ b ≤ potential d z
⊢ a ≤ potential d z | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2.refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
z : 𝕊
h : z ∈ (fun p => potential d ⁻¹' Ici ↑p) ⟨max a b, ⋯⟩
⊢ z ∈ (fun p => potential d ⁻¹' Ici ↑p) ⟨a, ⋯⟩
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | exact h.1 | case refine_2.refine_2.refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
z : 𝕊
h : a ≤ potential d z ∧ b ≤ potential d z
⊢ a ≤ potential d z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2.refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
z : 𝕊
h : a ≤ potential d z ∧ b ≤ potential d z
⊢ a ≤ potential d z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | intro z h | case refine_2.refine_2.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
⊢ (fun p => potential d ⁻¹' Ici ↑p) ⟨b, ⋯⟩ ⊇ (fun p => potential d ⁻¹' Ici ↑p) ⟨max a b, ⋯⟩ | case refine_2.refine_2.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
z : 𝕊
h : z ∈ (fun p => potential d ⁻¹' Ici ↑p) ⟨max a b, ⋯⟩
⊢ z ∈ (fun p => potential d ⁻¹' Ici ↑p) ⟨b, ⋯⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
⊢ (fun p => potential d ⁻¹' Ici ↑p) ⟨b, ⋯⟩ ⊇ (fun p => potential d ⁻¹' Ici ↑p) ⟨max a b, ⋯⟩
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | simp only [mem_preimage, mem_Ici, Subtype.coe_mk, max_le_iff] at h ⊢ | case refine_2.refine_2.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
z : 𝕊
h : z ∈ (fun p => potential d ⁻¹' Ici ↑p) ⟨max a b, ⋯⟩
⊢ z ∈ (fun p => potential d ⁻¹' Ici ↑p) ⟨b, ⋯⟩ | case refine_2.refine_2.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
z : 𝕊
h : a ≤ potential d z ∧ b ≤ potential d z
⊢ b ≤ potential d z | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
z : 𝕊
h : z ∈ (fun p => potential d ⁻¹' Ici ↑p) ⟨max a b, ⋯⟩
⊢ z ∈ (fun p => potential d ⁻¹' Ici ↑p) ⟨b, ⋯⟩
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | exact h.2 | case refine_2.refine_2.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
z : 𝕊
h : a ≤ potential d z ∧ b ≤ potential d z
⊢ b ≤ potential d z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2.refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
a : ℝ
a0 : 0 ≤ a
a1 : a < 1
b : ℝ
b0 : 0 ≤ b
b1 : b < 1
z : 𝕊
h : a ≤ potential d z ∧ b ≤ potential d z
⊢ b ≤ potential d z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | intro ⟨p, m⟩ | case refine_2.refine_3
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
⊢ ∀ (a : ↑(Ico 0 1)), IsPreconnected (potential d ⁻¹' Ici ↑a) | case refine_2.refine_3
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsPreconnected (potential d ⁻¹' Ici ↑⟨p, m⟩) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_3
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
⊢ ∀ (a : ↑(Ico 0 1)), IsPreconnected (potential d ⁻¹' Ici ↑a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | simp only [Subtype.coe_mk] | case refine_2.refine_3
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsPreconnected (potential d ⁻¹' Ici ↑⟨p, m⟩) | case refine_2.refine_3
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsPreconnected (potential d ⁻¹' Ici p) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_3
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsPreconnected (potential d ⁻¹' Ici ↑⟨p, m⟩)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | refine IsConnected.isPreconnected (IsPathConnected.isConnected ?_) | case refine_2.refine_3
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsPreconnected (potential d ⁻¹' Ici p) | case refine_2.refine_3
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsPathConnected (potential d ⁻¹' Ici p) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_3
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsPreconnected (potential d ⁻¹' Ici p)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | apply IsPathConnected.of_frontier | case refine_2.refine_3
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsPathConnected (potential d ⁻¹' Ici p) | case refine_2.refine_3.pc
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsPathConnected (potential d ⁻¹' frontier (Ici p))
case refine_2.refine_3.fc
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ Continuous (potential d)
case refine_2.refine_3.sc
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsClosed (Ici p) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_3
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsPathConnected (potential d ⁻¹' Ici p)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | rw [frontier_Ici] | case refine_2.refine_3.pc
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsPathConnected (potential d ⁻¹' frontier (Ici p)) | case refine_2.refine_3.pc
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsPathConnected (potential d ⁻¹' {p}) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_3.pc
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsPathConnected (potential d ⁻¹' frontier (Ici p))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | exact isPathConnected_potential_levelset _ m.1 m.2 | case refine_2.refine_3.pc
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsPathConnected (potential d ⁻¹' {p}) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_3.pc
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsPathConnected (potential d ⁻¹' {p})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | exact potential_continuous | case refine_2.refine_3.fc
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ Continuous (potential d) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_3.fc
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ Continuous (potential d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | exact isClosed_Ici | case refine_2.refine_3.sc
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsClosed (Ici p) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_3.sc
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsClosed (Ici p)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | intro ⟨p, m⟩ | case refine_2.refine_4
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
⊢ ∀ (a : ↑(Ico 0 1)), IsCompact (potential d ⁻¹' Ici ↑a) | case refine_2.refine_4
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsCompact (potential d ⁻¹' Ici ↑⟨p, m⟩) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_4
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
⊢ ∀ (a : ↑(Ico 0 1)), IsCompact (potential d ⁻¹' Ici ↑a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrotExt | [52, 1] | [74, 79] | exact (isClosed_Ici.preimage potential_continuous).isCompact | case refine_2.refine_4
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsCompact (potential d ⁻¹' Ici ↑⟨p, m⟩) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_4
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : (multibrotExt d)ᶜ = ⋂ p, potential d ⁻¹' Ici ↑p
p : ℝ
m : p ∈ Ico 0 1
⊢ IsCompact (potential d ⁻¹' Ici ↑⟨p, m⟩)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | rw [e] | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
⊢ IsConnected (multibrot d) | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
⊢ IsConnected ((fun z => z.toComplex) '' (multibrotExt d)ᶜ) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
⊢ IsConnected (multibrot d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | apply (isConnected_compl_multibrotExt d).image | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
⊢ IsConnected ((fun z => z.toComplex) '' (multibrotExt d)ᶜ) | case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
⊢ ContinuousOn (fun z => z.toComplex) (multibrotExt d)ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
⊢ IsConnected ((fun z => z.toComplex) '' (multibrotExt d)ᶜ)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | refine continuousOn_toComplex.mono ?_ | case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
⊢ ContinuousOn (fun z => z.toComplex) (multibrotExt d)ᶜ | case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
⊢ (multibrotExt d)ᶜ ⊆ {∞}ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
⊢ ContinuousOn (fun z => z.toComplex) (multibrotExt d)ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | intro z m | case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
⊢ (multibrotExt d)ᶜ ⊆ {∞}ᶜ | case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
z : 𝕊
m : z ∈ (multibrotExt d)ᶜ
⊢ z ∈ {∞}ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
⊢ (multibrotExt d)ᶜ ⊆ {∞}ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | contrapose m | case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
z : 𝕊
m : z ∈ (multibrotExt d)ᶜ
⊢ z ∈ {∞}ᶜ | case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
z : 𝕊
m : z ∉ {∞}ᶜ
⊢ z ∉ (multibrotExt d)ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
z : 𝕊
m : z ∈ (multibrotExt d)ᶜ
⊢ z ∈ {∞}ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | simp only [mem_compl_iff, mem_singleton_iff, not_not] at m | case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
z : 𝕊
m : z ∉ {∞}ᶜ
⊢ z ∉ (multibrotExt d)ᶜ | case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
z : 𝕊
m : z = ∞
⊢ z ∉ (multibrotExt d)ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
z : 𝕊
m : z ∉ {∞}ᶜ
⊢ z ∉ (multibrotExt d)ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | simp only [m, not_mem_compl_iff, multibrotExt_inf] | case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
z : 𝕊
m : z = ∞
⊢ z ∉ (multibrotExt d)ᶜ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
e : multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
z : 𝕊
m : z = ∞
⊢ z ∉ (multibrotExt d)ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | apply Set.ext | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ (x : ℂ), x ∈ multibrot d ↔ x ∈ (fun z => z.toComplex) '' (multibrotExt d)ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ multibrot d = (fun z => z.toComplex) '' (multibrotExt d)ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | intro z | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ (x : ℂ), x ∈ multibrot d ↔ x ∈ (fun z => z.toComplex) '' (multibrotExt d)ᶜ | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ z ∈ multibrot d ↔ z ∈ (fun z => z.toComplex) '' (multibrotExt d)ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ∀ (x : ℂ), x ∈ multibrot d ↔ x ∈ (fun z => z.toComplex) '' (multibrotExt d)ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | simp only [mem_image, mem_compl_iff] | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ z ∈ multibrot d ↔ z ∈ (fun z => z.toComplex) '' (multibrotExt d)ᶜ | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ z ∈ multibrot d ↔ ∃ x ∉ multibrotExt d, x.toComplex = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ z ∈ multibrot d ↔ z ∈ (fun z => z.toComplex) '' (multibrotExt d)ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | constructor | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ z ∈ multibrot d ↔ ∃ x ∉ multibrotExt d, x.toComplex = z | case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ z ∈ multibrot d → ∃ x ∉ multibrotExt d, x.toComplex = z
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ (∃ x ∉ multibrotExt d, x.toComplex = z) → z ∈ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ z ∈ multibrot d ↔ ∃ x ∉ multibrotExt d, x.toComplex = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | intro m | case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ z ∈ multibrot d → ∃ x ∉ multibrotExt d, x.toComplex = z
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ (∃ x ∉ multibrotExt d, x.toComplex = z) → z ∈ multibrot d | case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
m : z ∈ multibrot d
⊢ ∃ x ∉ multibrotExt d, x.toComplex = z
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ (∃ x ∉ multibrotExt d, x.toComplex = z) → z ∈ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ z ∈ multibrot d → ∃ x ∉ multibrotExt d, x.toComplex = z
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ (∃ x ∉ multibrotExt d, x.toComplex = z) → z ∈ multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | use z | case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
m : z ∈ multibrot d
⊢ ∃ x ∉ multibrotExt d, x.toComplex = z
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ (∃ x ∉ multibrotExt d, x.toComplex = z) → z ∈ multibrot d | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
m : z ∈ multibrot d
⊢ ↑z ∉ multibrotExt d ∧ (↑z).toComplex = z
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ (∃ x ∉ multibrotExt d, x.toComplex = z) → z ∈ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
m : z ∈ multibrot d
⊢ ∃ x ∉ multibrotExt d, x.toComplex = z
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ (∃ x ∉ multibrotExt d, x.toComplex = z) → z ∈ multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | simp only [multibrotExt_coe, not_not, m, toComplex_coe, true_and_iff,
eq_self_iff_true] | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
m : z ∈ multibrot d
⊢ ↑z ∉ multibrotExt d ∧ (↑z).toComplex = z
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ (∃ x ∉ multibrotExt d, x.toComplex = z) → z ∈ multibrot d | case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ (∃ x ∉ multibrotExt d, x.toComplex = z) → z ∈ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
m : z ∈ multibrot d
⊢ ↑z ∉ multibrotExt d ∧ (↑z).toComplex = z
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ (∃ x ∉ multibrotExt d, x.toComplex = z) → z ∈ multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | intro ⟨w, m, wz⟩ | case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ (∃ x ∉ multibrotExt d, x.toComplex = z) → z ∈ multibrot d | case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
w : 𝕊
m : w ∉ multibrotExt d
wz : w.toComplex = z
⊢ z ∈ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
⊢ (∃ x ∉ multibrotExt d, x.toComplex = z) → z ∈ multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | induction w using OnePoint.rec | case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
w : 𝕊
m : w ∉ multibrotExt d
wz : w.toComplex = z
⊢ z ∈ multibrot d | case h.mpr.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
m : ∞ ∉ multibrotExt d
wz : ∞.toComplex = z
⊢ z ∈ multibrot d
case h.mpr.h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z x✝ : ℂ
m : ↑x✝ ∉ multibrotExt d
wz : (↑x✝).toComplex = z
⊢ z ∈ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
w : 𝕊
m : w ∉ multibrotExt d
wz : w.toComplex = z
⊢ z ∈ multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | contrapose m | case h.mpr.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
m : ∞ ∉ multibrotExt d
wz : ∞.toComplex = z
⊢ z ∈ multibrot d | case h.mpr.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
wz : ∞.toComplex = z
m : ¬z ∈ multibrot d
⊢ ¬∞ ∉ multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
m : ∞ ∉ multibrotExt d
wz : ∞.toComplex = z
⊢ z ∈ multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | clear m | case h.mpr.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
wz : ∞.toComplex = z
m : ¬z ∈ multibrot d
⊢ ¬∞ ∉ multibrotExt d | case h.mpr.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
wz : ∞.toComplex = z
⊢ ¬∞ ∉ multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
wz : ∞.toComplex = z
m : ¬z ∈ multibrot d
⊢ ¬∞ ∉ multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | simp only [not_not, multibrotExt_inf] | case h.mpr.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
wz : ∞.toComplex = z
⊢ ¬∞ ∉ multibrotExt d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z : ℂ
wz : ∞.toComplex = z
⊢ ¬∞ ∉ multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | simp only [multibrotExt_coe, not_not, toComplex_coe] at m wz | case h.mpr.h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z x✝ : ℂ
m : ↑x✝ ∉ multibrotExt d
wz : (↑x✝).toComplex = z
⊢ z ∈ multibrot d | case h.mpr.h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z x✝ : ℂ
wz : x✝ = z
m : x✝ ∈ multibrot d
⊢ z ∈ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z x✝ : ℂ
m : ↑x✝ ∉ multibrotExt d
wz : (↑x✝).toComplex = z
⊢ z ∈ multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_multibrot | [77, 1] | [89, 53] | rwa [← wz] | case h.mpr.h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z x✝ : ℂ
wz : x✝ = z
m : x✝ ∈ multibrot d
⊢ z ∈ multibrot d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
z x✝ : ℂ
wz : x✝ = z
m : x✝ ∈ multibrot d
⊢ z ∈ multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | rw [e] | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
e : (multibrot d)ᶜ = (fun z => z.toComplex) '' (multibrotExt d \ {∞})
⊢ IsConnected (multibrot d)ᶜ | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
e : (multibrot d)ᶜ = (fun z => z.toComplex) '' (multibrotExt d \ {∞})
⊢ IsConnected ((fun z => z.toComplex) '' (multibrotExt d \ {∞})) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
e : (multibrot d)ᶜ = (fun z => z.toComplex) '' (multibrotExt d \ {∞})
⊢ IsConnected (multibrot d)ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | apply dc.image | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
e : (multibrot d)ᶜ = (fun z => z.toComplex) '' (multibrotExt d \ {∞})
⊢ IsConnected ((fun z => z.toComplex) '' (multibrotExt d \ {∞})) | case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
e : (multibrot d)ᶜ = (fun z => z.toComplex) '' (multibrotExt d \ {∞})
⊢ ContinuousOn (fun z => z.toComplex) (multibrotExt d \ {∞}) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
e : (multibrot d)ᶜ = (fun z => z.toComplex) '' (multibrotExt d \ {∞})
⊢ IsConnected ((fun z => z.toComplex) '' (multibrotExt d \ {∞}))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | refine continuousOn_toComplex.mono ?_ | case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
e : (multibrot d)ᶜ = (fun z => z.toComplex) '' (multibrotExt d \ {∞})
⊢ ContinuousOn (fun z => z.toComplex) (multibrotExt d \ {∞}) | case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
e : (multibrot d)ᶜ = (fun z => z.toComplex) '' (multibrotExt d \ {∞})
⊢ multibrotExt d \ {∞} ⊆ {∞}ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
e : (multibrot d)ᶜ = (fun z => z.toComplex) '' (multibrotExt d \ {∞})
⊢ ContinuousOn (fun z => z.toComplex) (multibrotExt d \ {∞})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | intro z ⟨_, i⟩ | case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
e : (multibrot d)ᶜ = (fun z => z.toComplex) '' (multibrotExt d \ {∞})
⊢ multibrotExt d \ {∞} ⊆ {∞}ᶜ | case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
e : (multibrot d)ᶜ = (fun z => z.toComplex) '' (multibrotExt d \ {∞})
z : 𝕊
left✝ : z ∈ multibrotExt d
i : z ∉ {∞}
⊢ z ∈ {∞}ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
e : (multibrot d)ᶜ = (fun z => z.toComplex) '' (multibrotExt d \ {∞})
⊢ multibrotExt d \ {∞} ⊆ {∞}ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | simp only [mem_singleton_iff, mem_compl_iff] at i ⊢ | case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
e : (multibrot d)ᶜ = (fun z => z.toComplex) '' (multibrotExt d \ {∞})
z : 𝕊
left✝ : z ∈ multibrotExt d
i : z ∉ {∞}
⊢ z ∈ {∞}ᶜ | case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
e : (multibrot d)ᶜ = (fun z => z.toComplex) '' (multibrotExt d \ {∞})
z : 𝕊
left✝ : z ∈ multibrotExt d
i : ¬z = ∞
⊢ ¬z = ∞ | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
e : (multibrot d)ᶜ = (fun z => z.toComplex) '' (multibrotExt d \ {∞})
z : 𝕊
left✝ : z ∈ multibrotExt d
i : z ∉ {∞}
⊢ z ∈ {∞}ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | exact i | case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
e : (multibrot d)ᶜ = (fun z => z.toComplex) '' (multibrotExt d \ {∞})
z : 𝕊
left✝ : z ∈ multibrotExt d
i : ¬z = ∞
⊢ ¬z = ∞ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
e : (multibrot d)ᶜ = (fun z => z.toComplex) '' (multibrotExt d \ {∞})
z : 𝕊
left✝ : z ∈ multibrotExt d
i : ¬z = ∞
⊢ ¬z = ∞
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | refine ⟨⟨(((3 : ℝ) : ℂ) : 𝕊),?_⟩,?_⟩ | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsConnected (multibrotExt d \ {∞}) | case refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ↑↑3 ∈ multibrotExt d \ {∞}
case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsPreconnected (multibrotExt d \ {∞}) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsConnected (multibrotExt d \ {∞})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | constructor | case refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ↑↑3 ∈ multibrotExt d \ {∞}
case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsPreconnected (multibrotExt d \ {∞}) | case refine_1.left
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ↑↑3 ∈ multibrotExt d
case refine_1.right
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ↑↑3 ∉ {∞}
case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsPreconnected (multibrotExt d \ {∞}) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ↑↑3 ∈ multibrotExt d \ {∞}
case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsPreconnected (multibrotExt d \ {∞})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | simp only [multibrotExt_coe, mem_compl_iff] | case refine_1.left
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ↑↑3 ∈ multibrotExt d | case refine_1.left
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ↑3 ∉ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.left
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ↑↑3 ∈ multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | apply multibrot_two_lt | case refine_1.left
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ↑3 ∉ multibrot d | case refine_1.left.a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 2 < Complex.abs ↑3 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.left
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ↑3 ∉ multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | rw [Complex.abs_ofReal, abs_of_pos] | case refine_1.left.a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 2 < Complex.abs ↑3 | case refine_1.left.a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 2 < 3
case refine_1.left.a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 0 < 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.left.a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 2 < Complex.abs ↑3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | norm_num | case refine_1.left.a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 2 < 3
case refine_1.left.a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 0 < 3 | case refine_1.left.a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 0 < 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.left.a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 2 < 3
case refine_1.left.a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 0 < 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | norm_num | case refine_1.left.a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 0 < 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.left.a
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ 0 < 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | simp only [mem_singleton_iff, coe_ne_inf, not_false_iff] | case refine_1.right
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ↑↑3 ∉ {∞} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.right
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ ↑↑3 ∉ {∞}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | exact (isPathConnected_multibrotExt d).isConnected.isPreconnected.open_diff_singleton
isOpen_multibrotExt _ | case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsPreconnected (multibrotExt d \ {∞}) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
⊢ IsPreconnected (multibrotExt d \ {∞})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | apply Set.ext | c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
⊢ (multibrot d)ᶜ = (fun z => z.toComplex) '' (multibrotExt d \ {∞}) | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
⊢ ∀ (x : ℂ), x ∈ (multibrot d)ᶜ ↔ x ∈ (fun z => z.toComplex) '' (multibrotExt d \ {∞}) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
⊢ (multibrot d)ᶜ = (fun z => z.toComplex) '' (multibrotExt d \ {∞})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | intro z | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
⊢ ∀ (x : ℂ), x ∈ (multibrot d)ᶜ ↔ x ∈ (fun z => z.toComplex) '' (multibrotExt d \ {∞}) | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
⊢ z ∈ (multibrot d)ᶜ ↔ z ∈ (fun z => z.toComplex) '' (multibrotExt d \ {∞}) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
⊢ ∀ (x : ℂ), x ∈ (multibrot d)ᶜ ↔ x ∈ (fun z => z.toComplex) '' (multibrotExt d \ {∞})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | simp only [mem_compl_iff, mem_image] | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
⊢ z ∈ (multibrot d)ᶜ ↔ z ∈ (fun z => z.toComplex) '' (multibrotExt d \ {∞}) | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
⊢ z ∉ multibrot d ↔ ∃ x ∈ multibrotExt d \ {∞}, x.toComplex = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
⊢ z ∈ (multibrot d)ᶜ ↔ z ∈ (fun z => z.toComplex) '' (multibrotExt d \ {∞})
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | constructor | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
⊢ z ∉ multibrot d ↔ ∃ x ∈ multibrotExt d \ {∞}, x.toComplex = z | case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
⊢ z ∉ multibrot d → ∃ x ∈ multibrotExt d \ {∞}, x.toComplex = z
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
⊢ (∃ x ∈ multibrotExt d \ {∞}, x.toComplex = z) → z ∉ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
⊢ z ∉ multibrot d ↔ ∃ x ∈ multibrotExt d \ {∞}, x.toComplex = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | intro m | case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
⊢ z ∉ multibrot d → ∃ x ∈ multibrotExt d \ {∞}, x.toComplex = z | case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
m : z ∉ multibrot d
⊢ ∃ x ∈ multibrotExt d \ {∞}, x.toComplex = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
⊢ z ∉ multibrot d → ∃ x ∈ multibrotExt d \ {∞}, x.toComplex = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | use z | case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
m : z ∉ multibrot d
⊢ ∃ x ∈ multibrotExt d \ {∞}, x.toComplex = z | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
m : z ∉ multibrot d
⊢ ↑z ∈ multibrotExt d \ {∞} ∧ (↑z).toComplex = z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
m : z ∉ multibrot d
⊢ ∃ x ∈ multibrotExt d \ {∞}, x.toComplex = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | simp only [multibrotExt_coe, m, true_and_iff, toComplex_coe, not_false_iff, true_and_iff,
mem_diff, eq_self_iff_true, and_true_iff, mem_singleton_iff, coe_ne_inf] | case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
m : z ∉ multibrot d
⊢ ↑z ∈ multibrotExt d \ {∞} ∧ (↑z).toComplex = z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
m : z ∉ multibrot d
⊢ ↑z ∈ multibrotExt d \ {∞} ∧ (↑z).toComplex = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | intro ⟨w, ⟨m, wi⟩, wz⟩ | case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
⊢ (∃ x ∈ multibrotExt d \ {∞}, x.toComplex = z) → z ∉ multibrot d | case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
w : 𝕊
m : w ∈ multibrotExt d
wi : w ∉ {∞}
wz : w.toComplex = z
⊢ z ∉ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
⊢ (∃ x ∈ multibrotExt d \ {∞}, x.toComplex = z) → z ∉ multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | induction w using OnePoint.rec | case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
w : 𝕊
m : w ∈ multibrotExt d
wi : w ∉ {∞}
wz : w.toComplex = z
⊢ z ∉ multibrot d | case h.mpr.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
m : ∞ ∈ multibrotExt d
wi : ∞ ∉ {∞}
wz : ∞.toComplex = z
⊢ z ∉ multibrot d
case h.mpr.h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z x✝ : ℂ
m : ↑x✝ ∈ multibrotExt d
wi : ↑x✝ ∉ {∞}
wz : (↑x✝).toComplex = z
⊢ z ∉ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
w : 𝕊
m : w ∈ multibrotExt d
wi : w ∉ {∞}
wz : w.toComplex = z
⊢ z ∉ multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | contrapose wi | case h.mpr.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
m : ∞ ∈ multibrotExt d
wi : ∞ ∉ {∞}
wz : ∞.toComplex = z
⊢ z ∉ multibrot d | case h.mpr.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
m : ∞ ∈ multibrotExt d
wz : ∞.toComplex = z
wi : ¬z ∉ multibrot d
⊢ ¬∞ ∉ {∞} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
m : ∞ ∈ multibrotExt d
wi : ∞ ∉ {∞}
wz : ∞.toComplex = z
⊢ z ∉ multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | clear wi | case h.mpr.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
m : ∞ ∈ multibrotExt d
wz : ∞.toComplex = z
wi : ¬z ∉ multibrot d
⊢ ¬∞ ∉ {∞} | case h.mpr.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
m : ∞ ∈ multibrotExt d
wz : ∞.toComplex = z
⊢ ¬∞ ∉ {∞} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
m : ∞ ∈ multibrotExt d
wz : ∞.toComplex = z
wi : ¬z ∉ multibrot d
⊢ ¬∞ ∉ {∞}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | simp only [mem_singleton_iff, not_not] | case h.mpr.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
m : ∞ ∈ multibrotExt d
wz : ∞.toComplex = z
⊢ ¬∞ ∉ {∞} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.h₁
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z : ℂ
m : ∞ ∈ multibrotExt d
wz : ∞.toComplex = z
⊢ ¬∞ ∉ {∞}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | simp only [multibrotExt_coe, toComplex_coe, mem_diff] at m wz | case h.mpr.h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z x✝ : ℂ
m : ↑x✝ ∈ multibrotExt d
wi : ↑x✝ ∉ {∞}
wz : (↑x✝).toComplex = z
⊢ z ∉ multibrot d | case h.mpr.h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z x✝ : ℂ
wi : ↑x✝ ∉ {∞}
wz : x✝ = z
m : x✝ ∉ multibrot d
⊢ z ∉ multibrot d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z x✝ : ℂ
m : ↑x✝ ∈ multibrotExt d
wi : ↑x✝ ∉ {∞}
wz : (↑x✝).toComplex = z
⊢ z ∉ multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Connected.lean | isConnected_compl_multibrot | [92, 1] | [111, 63] | rwa [← wz] | case h.mpr.h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z x✝ : ℂ
wi : ↑x✝ ∉ {∞}
wz : x✝ = z
m : x✝ ∉ multibrot d
⊢ z ∉ multibrot d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.h₂
c : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
dc : IsConnected (multibrotExt d \ {∞})
z x✝ : ℂ
wi : ↑x✝ ∉ {∞}
wz : x✝ = z
m : x✝ ∉ multibrot d
⊢ z ∉ multibrot d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_iff_disjoint_range | [27, 1] | [30, 83] | simp only [Late, ge_iff_le, Finset.disjoint_iff_ne, Finset.mem_range, ne_eq] | m : ℕ
A : Finset ℕ
⊢ Late A m ↔ Disjoint A (Finset.range m) | m : ℕ
A : Finset ℕ
⊢ (∀ n ∈ A, m ≤ n) ↔ ∀ a ∈ A, ∀ b < m, ¬a = b | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
A : Finset ℕ
⊢ Late A m ↔ Disjoint A (Finset.range m)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_iff_disjoint_range | [27, 1] | [30, 83] | constructor | m : ℕ
A : Finset ℕ
⊢ (∀ n ∈ A, m ≤ n) ↔ ∀ a ∈ A, ∀ b < m, ¬a = b | case mp
m : ℕ
A : Finset ℕ
⊢ (∀ n ∈ A, m ≤ n) → ∀ a ∈ A, ∀ b < m, ¬a = b
case mpr
m : ℕ
A : Finset ℕ
⊢ (∀ a ∈ A, ∀ b < m, ¬a = b) → ∀ n ∈ A, m ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
A : Finset ℕ
⊢ (∀ n ∈ A, m ≤ n) ↔ ∀ a ∈ A, ∀ b < m, ¬a = b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_iff_disjoint_range | [27, 1] | [30, 83] | intro h n na b bm | case mp
m : ℕ
A : Finset ℕ
⊢ (∀ n ∈ A, m ≤ n) → ∀ a ∈ A, ∀ b < m, ¬a = b | case mp
m : ℕ
A : Finset ℕ
h : ∀ n ∈ A, m ≤ n
n : ℕ
na : n ∈ A
b : ℕ
bm : b < m
⊢ ¬n = b | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
m : ℕ
A : Finset ℕ
⊢ (∀ n ∈ A, m ≤ n) → ∀ a ∈ A, ∀ b < m, ¬a = b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_iff_disjoint_range | [27, 1] | [30, 83] | linarith [h _ na] | case mp
m : ℕ
A : Finset ℕ
h : ∀ n ∈ A, m ≤ n
n : ℕ
na : n ∈ A
b : ℕ
bm : b < m
⊢ ¬n = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
m : ℕ
A : Finset ℕ
h : ∀ n ∈ A, m ≤ n
n : ℕ
na : n ∈ A
b : ℕ
bm : b < m
⊢ ¬n = b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_iff_disjoint_range | [27, 1] | [30, 83] | intro h n na | case mpr
m : ℕ
A : Finset ℕ
⊢ (∀ a ∈ A, ∀ b < m, ¬a = b) → ∀ n ∈ A, m ≤ n | case mpr
m : ℕ
A : Finset ℕ
h : ∀ a ∈ A, ∀ b < m, ¬a = b
n : ℕ
na : n ∈ A
⊢ m ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
m : ℕ
A : Finset ℕ
⊢ (∀ a ∈ A, ∀ b < m, ¬a = b) → ∀ n ∈ A, m ≤ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_iff_disjoint_range | [27, 1] | [30, 83] | specialize h n na n | case mpr
m : ℕ
A : Finset ℕ
h : ∀ a ∈ A, ∀ b < m, ¬a = b
n : ℕ
na : n ∈ A
⊢ m ≤ n | case mpr
m : ℕ
A : Finset ℕ
n : ℕ
na : n ∈ A
h : n < m → ¬n = n
⊢ m ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
m : ℕ
A : Finset ℕ
h : ∀ a ∈ A, ∀ b < m, ¬a = b
n : ℕ
na : n ∈ A
⊢ m ≤ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | late_iff_disjoint_range | [27, 1] | [30, 83] | simpa [not_true, imp_false, not_lt] using h | case mpr
m : ℕ
A : Finset ℕ
n : ℕ
na : n ∈ A
h : n < m → ¬n = n
⊢ m ≤ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
m : ℕ
A : Finset ℕ
n : ℕ
na : n ∈ A
h : n < m → ¬n = n
⊢ m ≤ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sdiff_late | [32, 1] | [38, 17] | intro Bm n nAB | m : ℕ
B A : Finset ℕ
⊢ B ≥ Finset.range m → Late (A \ B) m | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A \ B
⊢ n ≥ m | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
B A : Finset ℕ
⊢ B ≥ Finset.range m → Late (A \ B) m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sdiff_late | [32, 1] | [38, 17] | rw [Finset.mem_sdiff] at nAB | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A \ B
⊢ n ≥ m | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
⊢ n ≥ m | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A \ B
⊢ n ≥ m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sdiff_late | [32, 1] | [38, 17] | by_contra h | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
⊢ n ≥ m | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : ¬n ≥ m
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
⊢ n ≥ m
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sdiff_late | [32, 1] | [38, 17] | simp only [not_le] at h | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : ¬n ≥ m
⊢ False | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : n < m
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : ¬n ≥ m
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sdiff_late | [32, 1] | [38, 17] | have nr := Finset.mem_range.mpr h | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : n < m
⊢ False | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : n < m
nr : n ∈ Finset.range m
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : n < m
⊢ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Bounds.lean | sdiff_late | [32, 1] | [38, 17] | have nB := Finset.mem_of_subset Bm nr | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : n < m
nr : n ∈ Finset.range m
⊢ False | m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : n < m
nr : n ∈ Finset.range m
nB : n ∈ B
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
m : ℕ
B A : Finset ℕ
Bm : B ≥ Finset.range m
n : ℕ
nAB : n ∈ A ∧ n ∉ B
h : n < m
nr : n ∈ Finset.range m
⊢ False
TACTIC:
|
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