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/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rcases t2c'.exists_isMinOn t2ne' pc.continuousOn with β¨β¨x, yβ©, m2, minβ© | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 ... | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [isMinOn_iff] at min | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | generalize xp : potential d x = p | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [xp] at min | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have m1 := t12' m2 | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have pb : p β€ b := by rw [β xp]; exact m1.2.1 | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have xm : x β multibrotExt d := ue m1.2.1 | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have ym : y β multibrotExt d := ue m1.2.2 | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have yp : potential d y = p := by rw [β abs_bottcher, β m1.1, abs_bottcher, xp] | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have p0i : p = 0 β x = β β§ y = β := by intro p0; rw [p0, potential_eq_zero] at xp yp; use xp, yp | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | by_cases xy : x β y | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrot... | case pos
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [not_not] at xy | case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [β xy] at m1 m2 p0i | case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | clear xy ym yp y | case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have db : mfderiv I I (bottcher d) x = 0 := by
contrapose m2; simp only [mem_closure_iff_frequently, Filter.not_frequently]
refine ((bottcherHolomorphic d _ xm).local_inj m2).mp (eventually_of_forall ?_)
intro β¨x, yβ© inj β¨xy, e, _β©; simp only at xy e inj; exact xy (inj e) | case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | by_cases p0 : p β 0 | case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | case pos
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rwa [β hb, potential_lt_one] | c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
β’ b < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
β’ b < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | refine β¨β¨x, yβ©, xy, bxy, ?_, ?_β© | c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 :... | case refine_1
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 ... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [mem_setOf, β hb, le_refl, u] | case refine_1
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [mem_setOf, β hb, β abs_bottcher, bxy, le_refl, u] | case refine_2
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | intro c m | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
β’ u β multibrotExt d | cβ : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
c : π
m : c β u
β’ c β multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
β’ u β m... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [β potential_lt_one] | cβ : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
c : π
m : c β u
β’ c β multibrotExt d | cβ : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
c : π
m : c β u
β’ potential d c < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
c : π... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | exact lt_of_le_of_lt m b1 | cβ : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
c : π
m : c β u
β’ potential d c < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
c : π... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [isClosed_iff_frequently] | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | intro β¨x, yβ© f | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have m0 : (x, y) β t0 :=
Filter.Frequently.mem_of_closed (f.mp (eventually_of_forall fun _ m β¦ t01 m)) t0c | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | refine β¨tendsto_nhds_unique_of_frequently_eq ?_ ?_
(f.mp (eventually_of_forall fun _ m β¦ m.1)), m0β© | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | case refine_1
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | exact (bottcherHolomorphic d _ (ue m0.1)).continuousAt.comp continuousAt_fst | case refine_1
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.No... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | exact (bottcherHolomorphic d _ (ue m0.2)).continuousAt.comp continuousAt_snd | case refine_2
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.No... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [β t1c.closure_eq] | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | exact closure_mono t12 | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [β xp] | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | exact m1.2.1 | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [β abs_bottcher, β m1.1, abs_bottcher, xp] | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | intro p0 | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [p0, potential_eq_zero] at xp yp | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | use xp, yp | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have p0 : p β 0 := by
contrapose xy; simp only [not_not] at xy β’; rcases p0i xy with β¨xi, yiβ©; rw [xi, yi] | case pos
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | case pos
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have f : βαΆ q : β Γ β in Filter.map
(fun q : π Γ π β¦ (bottcher d q.1, bottcher d q.2)) (π (x, y)),
q.1 = q.2 β§ abs q.1 < p := by
rw [nhds_prod_eq, β Filter.prod_map_map_eq, β (bottcherNontrivial xm).nhds_eq_map_nhds, β
(bottcherNontrivial ym).nhds_eq_map_nhds, m1.1, β nhds_prod_eq]
apply (continuous_... | case pos
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | case pos
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [Filter.frequently_map] at f | case pos
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | case pos
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rcases(f.and_eventually (Ne.eventually_ne xy)).exists with β¨β¨v, wβ©, β¨bvw, pvβ©, vwβ© | case pos
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [not_lt, abs_bottcher] at vw bvw pv β’ | case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
... | case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have pw : potential d w < p := by rwa [β abs_bottcher, β bvw, abs_bottcher] | case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
... | case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have m : (v, w) β t2 := β¨vw, bvw, le_trans pv.le pb, le_trans pw.le pbβ© | case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
... | case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | contrapose pv | case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
... | case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | clear pv | case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
... | case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [not_lt] | case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
... | case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | exact min β¨v, wβ© (subset_closure m) | case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.mk.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | contrapose xy | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [not_not] at xy β’ | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rcases p0i xy with β¨xi, yiβ© | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | case intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : ... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [xi, yi] | case intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonem... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [nhds_prod_eq, β Filter.prod_map_map_eq, β (bottcherNontrivial xm).nhds_eq_map_nhds, β
(bottcherNontrivial ym).nhds_eq_map_nhds, m1.1, β nhds_prod_eq] | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | apply (continuous_id.prod_mk continuous_id).continuousAt.frequently | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [eq_self_iff_true, true_and_iff, β yp, β abs_bottcher] | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | apply frequently_smaller | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | case z0
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsC... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [β Complex.abs.ne_zero_iff, abs_bottcher, yp] | case z0
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsC... | case z0
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsC... | Please generate a tactic in lean4 to solve the state.
STATE:
case z0
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | exact p0 | case z0
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsC... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case z0
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rwa [β abs_bottcher, β bvw, abs_bottcher] | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | contrapose m2 | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [mem_closure_iff_frequently, Filter.not_frequently] | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | refine ((bottcherHolomorphic d _ xm).local_inj m2).mp (eventually_of_forall ?_) | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | intro β¨x, yβ© inj β¨xy, e, _β© | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only at xy e inj | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | exact xy (inj e) | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rcases not_local_inj_of_mfderiv_zero (bottcherHolomorphic d _ xm) db with β¨r, ra, rx, eβ© | case pos
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | case pos.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [eventually_nhdsWithin_iff, mem_compl_singleton_iff] at e | case pos.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12... | case pos.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [β xp, β abs_bottcher, Complex.abs.ne_zero_iff] at p0 | case pos.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12... | case pos.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have h := frequently_smaller p0 | case pos.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12... | case pos.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [(bottcherNontrivial xm).nhds_eq_map_nhds, Filter.frequently_map] at h | case pos.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12... | case pos.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have m : βαΆ z in π x, potential d z < p β§ (z, r z) β t2 := by
refine h.mp (e.mp (eventually_of_forall fun z e lt β¦ ?_))
have zx : z β x := by
contrapose lt; simp only [not_not, not_lt] at lt β’; simp only [lt, le_refl]
rw [abs_bottcher, abs_bottcher, xp] at lt
rcases e zx with β¨rz, eβ©
refine β¨lt, rz.symm,... | case pos.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12... | case pos.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rcases m.exists with β¨y, yp, mβ© | case pos.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12... | case pos.intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 :... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | linarith [min _ (subset_closure m)] | case pos.intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | refine h.mp (e.mp (eventually_of_forall fun z e lt β¦ ?_)) | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have zx : z β x := by
contrapose lt; simp only [not_not, not_lt] at lt β’; simp only [lt, le_refl] | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [abs_bottcher, abs_bottcher, xp] at lt | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rcases e zx with β¨rz, eβ© | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | case intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : ... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | refine β¨lt, rz.symm, e.symm, le_trans lt.le pb, ?_β© | case intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : ... | case intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonem... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [β abs_bottcher, β e, abs_bottcher] at lt | case intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : ... | case intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonem... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | exact le_trans lt.le pb | case intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonem... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | contrapose lt | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [not_not, not_lt] at lt β’ | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [lt, le_refl] | c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [not_not] at p0 | case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [(p0i p0).1] at db | case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | exact bottcher_mfderiv_inf_ne_zero db | case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : Is... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempt... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | rayHolomorphic | [158, 1] | [159, 73] | rw [β bottcher_surj d] | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ HolomorphicOn I I (ray d) (ball 0 1) | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ HolomorphicOn I I (ray d) (bottcher d '' multibrotExt d) | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ HolomorphicOn I I (ray d) (ball 0 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | rayHolomorphic | [158, 1] | [159, 73] | exact (Classical.choose_spec (ray_exists d)).1 | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ HolomorphicOn I I (ray d) (bottcher d '' multibrotExt d) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ HolomorphicOn I I (ray d) (bottcher d '' multibrotExt d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_ray | [166, 1] | [168, 49] | rw [β bottcher_surj d] at m | c : β
d : β
instβ : Fact (2 β€ d)
z : β
m : z β ball 0 1
β’ bottcher d (ray d z) = z | c : β
d : β
instβ : Fact (2 β€ d)
z : β
m : z β bottcher d '' multibrotExt d
β’ bottcher d (ray d z) = z | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
z : β
m : z β ball 0 1
β’ bottcher d (ray d z) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_ray | [166, 1] | [168, 49] | rcases m with β¨c, m, czβ© | c : β
d : β
instβ : Fact (2 β€ d)
z : β
m : z β bottcher d '' multibrotExt d
β’ bottcher d (ray d z) = z | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
z : β
c : π
m : c β multibrotExt d
cz : bottcher d c = z
β’ bottcher d (ray d z) = z | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
z : β
m : z β bottcher d '' multibrotExt d
β’ bottcher d (ray d z) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_ray | [166, 1] | [168, 49] | nth_rw 1 [β cz] | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
z : β
c : π
m : c β multibrotExt d
cz : bottcher d c = z
β’ bottcher d (ray d z) = z | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
z : β
c : π
m : c β multibrotExt d
cz : bottcher d c = z
β’ bottcher d (ray d (bottcher d c)) = z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
z : β
c : π
m : c β multibrotExt d
cz : bottcher d c = z
β’ bottcher d (ray d z) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_ray | [166, 1] | [168, 49] | rw [ray_bottcher m] | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
z : β
c : π
m : c β multibrotExt d
cz : bottcher d c = z
β’ bottcher d (ray d (bottcher d c)) = z | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
z : β
c : π
m : c β multibrotExt d
cz : bottcher d c = z
β’ bottcher d c = z | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
z : β
c : π
m : c β multibrotExt d
cz : bottcher d c = z
β’ bottcher d (ray d (bottcher d c)) = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_ray | [166, 1] | [168, 49] | exact cz | case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
z : β
c : π
m : c β multibrotExt d
cz : bottcher d c = z
β’ bottcher d c = z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
cβ : β
d : β
instβ : Fact (2 β€ d)
z : β
c : π
m : c β multibrotExt d
cz : bottcher d c = z
β’ bottcher d c = z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | ray_surj | [171, 1] | [174, 38] | rw [β bottcher_surj d] | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ ray d '' ball 0 1 = multibrotExt d | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ ray d '' (bottcher d '' multibrotExt d) = multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ ray d '' ball 0 1 = multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | ray_surj | [171, 1] | [174, 38] | apply Set.ext | c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ ray d '' (bottcher d '' multibrotExt d) = multibrotExt d | case h
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ β (x : π), x β ray d '' (bottcher d '' multibrotExt d) β x β multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ ray d '' (bottcher d '' multibrotExt d) = multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | ray_surj | [171, 1] | [174, 38] | intro c | case h
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ β (x : π), x β ray d '' (bottcher d '' multibrotExt d) β x β multibrotExt d | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : π
β’ c β ray d '' (bottcher d '' multibrotExt d) β c β multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
β’ β (x : π), x β ray d '' (bottcher d '' multibrotExt d) β x β multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | ray_surj | [171, 1] | [174, 38] | simp only [β image_comp, mem_image] | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : π
β’ c β ray d '' (bottcher d '' multibrotExt d) β c β multibrotExt d | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : π
β’ (β x β multibrotExt d, (ray d β bottcher d) x = c) β c β multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : π
β’ c β ray d '' (bottcher d '' multibrotExt d) β c β multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | ray_surj | [171, 1] | [174, 38] | constructor | case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : π
β’ (β x β multibrotExt d, (ray d β bottcher d) x = c) β c β multibrotExt d | case h.mp
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : π
β’ (β x β multibrotExt d, (ray d β bottcher d) x = c) β c β multibrotExt d
case h.mpr
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : π
β’ c β multibrotExt d β β x β multibrotExt d, (ray d β bottcher d) x = c | Please generate a tactic in lean4 to solve the state.
STATE:
case h
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : π
β’ (β x β multibrotExt d, (ray d β bottcher d) x = c) β c β multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | ray_surj | [171, 1] | [174, 38] | intro β¨e, m, ecβ© | case h.mp
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : π
β’ (β x β multibrotExt d, (ray d β bottcher d) x = c) β c β multibrotExt d | case h.mp
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c e : π
m : e β multibrotExt d
ec : (ray d β bottcher d) e = c
β’ c β multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : π
β’ (β x β multibrotExt d, (ray d β bottcher d) x = c) β c β multibrotExt d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | ray_surj | [171, 1] | [174, 38] | simp only [Function.comp, ray_bottcher m] at ec | case h.mp
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c e : π
m : e β multibrotExt d
ec : (ray d β bottcher d) e = c
β’ c β multibrotExt d | case h.mp
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c e : π
m : e β multibrotExt d
ec : e = c
β’ c β multibrotExt d | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c e : π
m : e β multibrotExt d
ec : (ray d β bottcher d) e = c
β’ c β multibrotExt d
TACTIC:
|
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