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] | 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 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
β’ (x, y).1 β u | 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’ 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}
β’ (x, y).1 β u
TACTIC:
|
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 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
β’ (x, y).2 β u | 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’ 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}
β’ (x, y).2 β u
TACTIC:
|
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 multibrotExt d
TACTIC:
|
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
m : c β u
β’ c β multibrotExt d
TACTIC:
|
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
m : c β u
β’ potential d c < 1
TACTIC:
|
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
t0c : IsClosed t0
β’ IsClosed t1 | 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
t0c : IsClosed t0
β’ β (x : π Γ π), (βαΆ (y : π Γ π) in π x, y β t1) β x β t1 | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
β’ IsClosed t1
TACTIC:
|
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
t0c : IsClosed t0
β’ β (x : π Γ π), (βαΆ (y : π Γ π) in π x, y β t1) β x β t1 | 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
t0c : IsClosed t0
x y : π
f : βαΆ (y : π Γ π) in π (x, y), y β t1
β’ (x, y) β t1 | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
β’ β (x : π Γ π), (βαΆ (y : π Γ π) in π x, y β t1) β x β t1
TACTIC:
|
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
t0c : IsClosed t0
x y : π
f : βαΆ (y : π Γ π) in π (x, y), y β t1
β’ (x, y) β t1 | 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
t0c : IsClosed t0
x y : π
f : βαΆ (y : π Γ π) in π (x, y), y β t1
m0 : (x, y) β t0
β’ (x, y) β t1 | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
x y : π
f : βαΆ (y : π Γ π) in π (x, y), y β t1
β’ (x, y) β t1
TACTIC:
|
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
t0c : IsClosed t0
x y : π
f : βαΆ (y : π Γ π) in π (x, y), y β t1
m0 : (x, y) β t0
β’ (x, y) β t1 | 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 : IsClosed u
t0c : IsClosed t0
x y : π
f : βαΆ (y : π Γ π) in π (x, y), y β t1
m0 : (x, y) β t0
β’ Tendsto (fun x => bottcher d x.1) (π (x, y)) (π (bottcher d (x, y).1))
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 : IsClosed u
t0c : IsClosed t0
x y : π
f : βαΆ (y : π Γ π) in π (x, y), y β t1
m0 : (x, y) β t0
β’ Tendsto (fun x => bottcher d x.2) (π (x, y)) (π (bottcher d (x, y).2)) | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
x y : π
f : βαΆ (y : π Γ π) in π (x, y), y β t1
m0 : (x, y) β t0
β’ (x, y) β t1
TACTIC:
|
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 : IsClosed u
t0c : IsClosed t0
x y : π
f : βαΆ (y : π Γ π) in π (x, y), y β t1
m0 : (x, y) β t0
β’ Tendsto (fun x => bottcher d x.1) (π (x, y)) (π (bottcher d (x, y).1)) | 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.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
x y : π
f : βαΆ (y : π Γ π) in π (x, y), y β t1
m0 : (x, y) β t0
β’ Tendsto (fun x => bottcher d x.1) (π (x, y)) (π (bottcher d (x, y).1))
TACTIC:
|
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 : IsClosed u
t0c : IsClosed t0
x y : π
f : βαΆ (y : π Γ π) in π (x, y), y β t1
m0 : (x, y) β t0
β’ Tendsto (fun x => bottcher d x.2) (π (x, y)) (π (bottcher d (x, y).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)
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
t0c : IsClosed t0
x y : π
f : βαΆ (y : π Γ π) in π (x, y), y β t1
m0 : (x, y) β t0
β’ Tendsto (fun x => bottcher d x.2) (π (x, y)) (π (bottcher d (x, y).2))
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
β’ closure t2 β t1 | 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
t0c : IsClosed t0
t1c : IsClosed t1
β’ closure t2 β closure t1 | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
β’ closure t2 β t1
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
β’ closure t2 β closure t1 | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
β’ closure t2 β closure t1
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
β’ p β€ b | 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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
β’ potential d x β€ b | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
β’ p β€ b
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
β’ potential d x β€ b | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
β’ potential d x β€ b
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
β’ potential d y = p | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
β’ potential d y = p
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
β’ p = 0 β x = β β§ y = 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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0 : p = 0
β’ x = β β§ y = β | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
β’ p = 0 β x = β β§ y = β
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0 : p = 0
β’ x = β β§ y = 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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : x = β
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : y = β
p0 : p = 0
β’ x = β β§ y = β | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0 : p = 0
β’ x = β β§ y = β
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : x = β
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : y = β
p0 : p = 0
β’ x = β β§ y = β | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : x = β
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : y = β
p0 : p = 0
β’ x = β β§ y = β
TACTIC:
|
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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
β’ False | 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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ False | 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.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
β’ False
TACTIC:
|
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_id.prod_mk continuous_id).continuousAt.frequently
simp only [eq_self_iff_true, true_and_iff, β yp, β abs_bottcher]; apply frequently_smaller
rw [β Complex.abs.ne_zero_iff, abs_bottcher, yp]; exact p0 | 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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ False | 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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (q : β Γ β) in Filter.map (fun q => (bottcher d q.1, bottcher d q.2)) (π (x, y)), q.1 = q.2 β§ Complex.abs q.1 < p
β’ False | 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.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ False
TACTIC:
|
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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (q : β Γ β) in Filter.map (fun q => (bottcher d q.1, bottcher d q.2)) (π (x, y)), q.1 = q.2 β§ Complex.abs q.1 < p
β’ False | 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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
β’ False | 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.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (q : β Γ β) in Filter.map (fun q => (bottcher d q.1, bottcher d q.2)) (π (x, y)), q.1 = q.2 β§ Complex.abs q.1 < p
β’ False
TACTIC:
|
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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
β’ False | 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
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : (v, w).1 β (v, w).2
bvw : bottcher d (v, w).1 = bottcher d (v, w).2
pv : Complex.abs (bottcher d (v, w).1) < p
β’ False | 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.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
β’ False
TACTIC:
|
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
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : (v, w).1 β (v, w).2
bvw : bottcher d (v, w).1 = bottcher d (v, w).2
pv : Complex.abs (bottcher d (v, w).1) < p
β’ False | 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
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pv : potential d v < p
β’ False | 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 t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : (v, w).1 β (v, w).2
bvw : bottcher d (v, w).1 = bottcher d (v, w).2
pv : Complex.abs (bottcher d (v, w).1) < p
β’ False
TACTIC:
|
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
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pv : potential d v < p
β’ False | 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
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pv : potential d v < p
pw : potential d w < p
β’ False | 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 t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pv : potential d v < p
β’ False
TACTIC:
|
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
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pv : potential d v < p
pw : potential d w < p
β’ False | 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
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pv : potential d v < p
pw : potential d w < p
m : (v, w) β t2
β’ False | 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 t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pv : potential d v < p
pw : potential d w < p
β’ False
TACTIC:
|
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
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pv : potential d v < p
pw : potential d w < p
m : (v, w) β t2
β’ False | 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
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pw : potential d w < p
m : (v, w) β t2
pv : Β¬False
β’ Β¬potential d v < p | 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 t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pv : potential d v < p
pw : potential d w < p
m : (v, w) β t2
β’ False
TACTIC:
|
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
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pw : potential d w < p
m : (v, w) β t2
pv : Β¬False
β’ Β¬potential d v < p | 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
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pw : potential d w < p
m : (v, w) β t2
β’ Β¬potential d v < p | 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 t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pw : potential d w < p
m : (v, w) β t2
pv : Β¬False
β’ Β¬potential d v < p
TACTIC:
|
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
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pw : potential d w < p
m : (v, w) β t2
β’ Β¬potential d v < p | 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
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pw : potential d w < p
m : (v, w) β t2
β’ p β€ potential d v | 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 t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pw : potential d w < p
m : (v, w) β t2
β’ Β¬potential d v < p
TACTIC:
|
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
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pw : potential d w < p
m : (v, w) β t2
β’ p β€ potential d v | 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 t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pw : potential d w < p
m : (v, w) β t2
β’ p β€ potential d v
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
β’ p β 0 | 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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : Β¬p β 0
β’ Β¬x β y | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
β’ p β 0
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : Β¬p β 0
β’ Β¬x β y | 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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : p = 0
β’ x = y | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : Β¬p β 0
β’ Β¬x β y
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : p = 0
β’ x = y | 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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : p = 0
xi : x = β
yi : y = β
β’ x = y | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : p = 0
β’ x = y
TACTIC:
|
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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : p = 0
xi : x = β
yi : y = β
β’ x = y | 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.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : p = 0
xi : x = β
yi : y = β
β’ x = y
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ βαΆ (q : β Γ β) in Filter.map (fun q => (bottcher d q.1, bottcher d q.2)) (π (x, y)), q.1 = q.2 β§ Complex.abs q.1 < p | 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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ βαΆ (q : β Γ β) in π (bottcher d (x, y).2, bottcher d y), q.1 = q.2 β§ Complex.abs q.1 < p | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ βαΆ (q : β Γ β) in Filter.map (fun q => (bottcher d q.1, bottcher d q.2)) (π (x, y)), q.1 = q.2 β§ Complex.abs q.1 < p
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ βαΆ (q : β Γ β) in π (bottcher d (x, y).2, bottcher d y), q.1 = q.2 β§ Complex.abs q.1 < p | 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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ βαΆ (x : β) in π (fill (bottcher' d) 0 (x, y).2), (id x, id x).1 = (id x, id x).2 β§ Complex.abs (id x, id x).1 < p | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ βαΆ (q : β Γ β) in π (bottcher d (x, y).2, bottcher d y), q.1 = q.2 β§ Complex.abs q.1 < p
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ βαΆ (x : β) in π (fill (bottcher' d) 0 (x, y).2), (id x, id x).1 = (id x, id x).2 β§ Complex.abs (id x, id x).1 < p | 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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ βαΆ (x : β) in π (fill (bottcher' d) 0 y), Complex.abs (id x) < Complex.abs (bottcher d y) | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ βαΆ (x : β) in π (fill (bottcher' d) 0 (x, y).2), (id x, id x).1 = (id x, id x).2 β§ Complex.abs (id x, id x).1 < p
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ βαΆ (x : β) in π (fill (bottcher' d) 0 y), Complex.abs (id x) < Complex.abs (bottcher d y) | 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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ bottcher d y β 0 | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ βαΆ (x : β) in π (fill (bottcher' d) 0 y), Complex.abs (id x) < Complex.abs (bottcher d y)
TACTIC:
|
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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ bottcher d y β 0 | 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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ p β 0 | 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
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ bottcher d y β 0
TACTIC:
|
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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ p β 0 | 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
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
β’ p β 0
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pv : potential d v < p
β’ potential d w < p | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x y : π
m2 : (x, y) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, y) β t1
pb : p β€ b
xm : x β multibrotExt d
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ y = β
xy : x β y
p0 : p β 0
f : βαΆ (a : π Γ π) in π (x, y), bottcher d a.1 = bottcher d a.2 β§ Complex.abs (bottcher d a.1) < p
v w : π
vw : v β w
bvw : bottcher d v = bottcher d w
pv : potential d v < p
β’ potential d w < p
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
β’ mfderiv I I (bottcher d) x = 0 | 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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
m2 : Β¬mfderiv I I (bottcher d) x = 0
β’ (x, x) β closure t2 | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
β’ mfderiv I I (bottcher d) x = 0
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
m2 : Β¬mfderiv I I (bottcher d) x = 0
β’ (x, x) β closure t2 | 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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
m2 : Β¬mfderiv I I (bottcher d) x = 0
β’ βαΆ (x : π Γ π) in π (x, x), x β t2 | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
m2 : Β¬mfderiv I I (bottcher d) x = 0
β’ (x, x) β closure t2
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
m2 : Β¬mfderiv I I (bottcher d) x = 0
β’ βαΆ (x : π Γ π) in π (x, x), x β t2 | 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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
m2 : Β¬mfderiv I I (bottcher d) x = 0
β’ β (x : π Γ π), (bottcher d x.1 = bottcher d x.2 β x.1 = x.2) β x β t2 | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
m2 : Β¬mfderiv I I (bottcher d) x = 0
β’ βαΆ (x : π Γ π) in π (x, x), x β t2
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
m2 : Β¬mfderiv I I (bottcher d) x = 0
β’ β (x : π Γ π), (bottcher d x.1 = bottcher d x.2 β x.1 = x.2) β x β t2 | 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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
xβ : π
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d xβ = p
m1 : (xβ, xβ) β t1
pb : p β€ b
xm : xβ β multibrotExt d
p0i : p = 0 β xβ = β β§ xβ = β
m2 : Β¬mfderiv I I (bottcher d) xβ = 0
x y : π
inj : bottcher d (x, y).1 = bottcher d (x, y).2 β (x, y).1 = (x, y).2
xy : (x, y).1 β (x, y).2
e : bottcher d (x, y).1 = bottcher d (x, y).2
rightβ : (x, y) β t0
β’ False | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
m2 : Β¬mfderiv I I (bottcher d) x = 0
β’ β (x : π Γ π), (bottcher d x.1 = bottcher d x.2 β x.1 = x.2) β x β t2
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
xβ : π
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d xβ = p
m1 : (xβ, xβ) β t1
pb : p β€ b
xm : xβ β multibrotExt d
p0i : p = 0 β xβ = β β§ xβ = β
m2 : Β¬mfderiv I I (bottcher d) xβ = 0
x y : π
inj : bottcher d (x, y).1 = bottcher d (x, y).2 β (x, y).1 = (x, y).2
xy : (x, y).1 β (x, y).2
e : bottcher d (x, y).1 = bottcher d (x, y).2
rightβ : (x, y) β t0
β’ False | 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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
xβ : π
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d xβ = p
m1 : (xβ, xβ) β t1
pb : p β€ b
xm : xβ β multibrotExt d
p0i : p = 0 β xβ = β β§ xβ = β
m2 : Β¬mfderiv I I (bottcher d) xβ = 0
x y : π
inj : bottcher d x = bottcher d y β x = y
xy : x β y
e : bottcher d x = bottcher d y
rightβ : (x, y) β t0
β’ False | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
xβ : π
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d xβ = p
m1 : (xβ, xβ) β t1
pb : p β€ b
xm : xβ β multibrotExt d
p0i : p = 0 β xβ = β β§ xβ = β
m2 : Β¬mfderiv I I (bottcher d) xβ = 0
x y : π
inj : bottcher d (x, y).1 = bottcher d (x, y).2 β (x, y).1 = (x, y).2
xy : (x, y).1 β (x, y).2
e : bottcher d (x, y).1 = bottcher d (x, y).2
rightβ : (x, y) β t0
β’ False
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
xβ : π
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d xβ = p
m1 : (xβ, xβ) β t1
pb : p β€ b
xm : xβ β multibrotExt d
p0i : p = 0 β xβ = β β§ xβ = β
m2 : Β¬mfderiv I I (bottcher d) xβ = 0
x y : π
inj : bottcher d x = bottcher d y β x = y
xy : x β y
e : bottcher d x = bottcher d y
rightβ : (x, y) β t0
β’ False | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
xβ : π
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d xβ = p
m1 : (xβ, xβ) β t1
pb : p β€ b
xm : xβ β multibrotExt d
p0i : p = 0 β xβ = β β§ xβ = β
m2 : Β¬mfderiv I I (bottcher d) xβ = 0
x y : π
inj : bottcher d x = bottcher d y β x = y
xy : x β y
e : bottcher d x = bottcher d y
rightβ : (x, y) β t0
β’ False
TACTIC:
|
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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : p β 0
β’ False | 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 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : p β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (z : π) in π[β ] x, r z β z β§ bottcher d (r z) = bottcher d z
β’ False | 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.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : p β 0
β’ False
TACTIC:
|
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 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : p β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (z : π) in π[β ] x, r z β z β§ bottcher d (r z) = bottcher d z
β’ False | 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 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : p β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
β’ False | 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}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : p β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (z : π) in π[β ] x, r z β z β§ bottcher d (r z) = bottcher d z
β’ False
TACTIC:
|
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 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : p β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
β’ False | 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 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
β’ False | 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}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : p β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
β’ False
TACTIC:
|
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 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
β’ False | 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 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (w : β) in π (bottcher d x), Complex.abs w < Complex.abs (bottcher d x)
β’ False | 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}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
β’ False
TACTIC:
|
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 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (w : β) in π (bottcher d x), Complex.abs w < Complex.abs (bottcher d x)
β’ False | 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 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
β’ False | 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}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (w : β) in π (bottcher d x), Complex.abs w < Complex.abs (bottcher d x)
β’ False
TACTIC:
|
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, e.symm, le_trans lt.le pb, ?_β©
rw [β abs_bottcher, β e, abs_bottcher] at lt; exact le_trans lt.le pb | 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 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
β’ False | 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 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
m : βαΆ (z : π) in π x, potential d z < p β§ (z, r z) β t2
β’ False | 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}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
β’ False
TACTIC:
|
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 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
m : βαΆ (z : π) in π x, potential d z < p β§ (z, r z) β t2
β’ False | 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 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
mβ : βαΆ (z : π) in π x, potential d z < p β§ (z, r z) β t2
y : π
yp : potential d y < p
m : (y, r y) β t2
β’ False | 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}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
m : βαΆ (z : π) in π x, potential d z < p β§ (z, r z) β t2
β’ False
TACTIC:
|
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 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
mβ : βαΆ (z : π) in π x, potential d z < p β§ (z, r z) β t2
y : π
yp : potential d y < p
m : (y, r y) β t2
β’ False | 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.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
mβ : βαΆ (z : π) in π x, potential d z < p β§ (z, r z) β t2
y : π
yp : potential d y < p
m : (y, r y) β t2
β’ False
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
β’ βαΆ (z : π) in π x, potential d z < p β§ (z, r z) β t2 | 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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
e : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : Complex.abs (bottcher d z) < Complex.abs (bottcher d x)
β’ potential d z < p β§ (z, r z) β t2 | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
e : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
β’ βαΆ (z : π) in π x, potential d z < p β§ (z, r z) β t2
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
e : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : Complex.abs (bottcher d z) < Complex.abs (bottcher d x)
β’ potential d z < p β§ (z, r z) β t2 | 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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
e : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : Complex.abs (bottcher d z) < Complex.abs (bottcher d x)
zx : z β x
β’ potential d z < p β§ (z, r z) β t2 | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
e : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : Complex.abs (bottcher d z) < Complex.abs (bottcher d x)
β’ potential d z < p β§ (z, r z) β t2
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
e : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : Complex.abs (bottcher d z) < Complex.abs (bottcher d x)
zx : z β x
β’ potential d z < p β§ (z, r z) β t2 | 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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
e : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : potential d z < p
zx : z β x
β’ potential d z < p β§ (z, r z) β t2 | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
e : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : Complex.abs (bottcher d z) < Complex.abs (bottcher d x)
zx : z β x
β’ potential d z < p β§ (z, r z) β t2
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
e : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : potential d z < p
zx : z β x
β’ potential d z < p β§ (z, r z) β t2 | 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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβΒΉ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
eβ : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : potential d z < p
zx : z β x
rz : r z β z
e : bottcher d (r z) = bottcher d z
β’ potential d z < p β§ (z, r z) β t2 | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
e : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : potential d z < p
zx : z β x
β’ potential d z < p β§ (z, r z) β t2
TACTIC:
|
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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβΒΉ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
eβ : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : potential d z < p
zx : z β x
rz : r z β z
e : bottcher d (r z) = bottcher d z
β’ potential d z < p β§ (z, r z) β t2 | 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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβΒΉ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
eβ : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : potential d z < p
zx : z β x
rz : r z β z
e : bottcher d (r z) = bottcher d z
β’ (z, r z).2 β u | 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.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβΒΉ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
eβ : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : potential d z < p
zx : z β x
rz : r z β z
e : bottcher d (r z) = bottcher d z
β’ potential d z < p β§ (z, r z) β t2
TACTIC:
|
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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβΒΉ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
eβ : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : potential d z < p
zx : z β x
rz : r z β z
e : bottcher d (r z) = bottcher d z
β’ (z, r z).2 β 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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβΒΉ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
eβ : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : potential d (r z) < p
zx : z β x
rz : r z β z
e : bottcher d (r z) = bottcher d z
β’ (z, r z).2 β u | 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.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβΒΉ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
eβ : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : potential d z < p
zx : z β x
rz : r z β z
e : bottcher d (r z) = bottcher d z
β’ (z, r z).2 β u
TACTIC:
|
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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβΒΉ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
eβ : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : potential d (r z) < p
zx : z β x
rz : r z β z
e : bottcher d (r z) = bottcher d z
β’ (z, r z).2 β u | 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.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβΒΉ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
eβ : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : potential d (r z) < p
zx : z β x
rz : r z β z
e : bottcher d (r z) = bottcher d z
β’ (z, r z).2 β u
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
e : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : Complex.abs (bottcher d z) < Complex.abs (bottcher d x)
β’ z β x | 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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
e : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : Β¬z β x
β’ Β¬Complex.abs (bottcher d z) < Complex.abs (bottcher d x) | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
e : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : Complex.abs (bottcher d z) < Complex.abs (bottcher d x)
β’ z β x
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
e : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : Β¬z β x
β’ Β¬Complex.abs (bottcher d z) < Complex.abs (bottcher d x) | 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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
e : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : z = x
β’ Complex.abs (bottcher d x) β€ Complex.abs (bottcher d z) | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
e : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : Β¬z β x
β’ Β¬Complex.abs (bottcher d z) < Complex.abs (bottcher d x)
TACTIC:
|
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
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
e : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : z = x
β’ Complex.abs (bottcher d x) β€ Complex.abs (bottcher d z) | 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 β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : bottcher d x β 0
r : π β π
ra : HolomorphicAt I I r x
rx : r x = x
eβ : βαΆ (x_1 : π) in π x, x_1 β x β r x_1 β x_1 β§ bottcher d (r x_1) = bottcher d x_1
h : βαΆ (a : π) in π x, Complex.abs (bottcher d a) < Complex.abs (bottcher d x)
z : π
e : z β x β r z β z β§ bottcher d (r z) = bottcher d z
lt : z = x
β’ Complex.abs (bottcher d x) β€ Complex.abs (bottcher d z)
TACTIC:
|
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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : Β¬p β 0
β’ False | 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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : p = 0
β’ False | 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.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : Β¬p β 0
β’ False
TACTIC:
|
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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : p = 0
β’ False | 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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) β = 0
p0 : p = 0
β’ False | 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.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) x = 0
p0 : p = 0
β’ False
TACTIC:
|
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 : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) β = 0
p0 : p = 0
β’ False | 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.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
t2ne' : (closure t2).Nonempty
pc : Continuous fun q => potential d q.1
x : π
m2 : (x, x) β closure t2
p : β
min : β x β closure t2, p β€ potential d x.1
xp : potential d x = p
m1 : (x, x) β t1
pb : p β€ b
xm : x β multibrotExt d
p0i : p = 0 β x = β β§ x = β
db : mfderiv I I (bottcher d) β = 0
p0 : p = 0
β’ False
TACTIC:
|
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:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | ray_surj | [171, 1] | [174, 38] | rwa [β 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 : e = c
β’ c β multibrotExt d | no goals | 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 : e = 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 m | 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 | case h.mpr
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : π
m : 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.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
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | ray_surj | [171, 1] | [174, 38] | use c, m, ray_bottcher m | case h.mpr
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
β’ β x β multibrotExt d, (ray d β bottcher d) x = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
cβ : β
dβ : β
instβΒΉ : Fact (2 β€ dβ)
d : β
instβ : Fact (2 β€ d)
c : π
m : c β multibrotExt d
β’ β x β multibrotExt d, (ray d β bottcher d) x = c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | mem_analyticGroupoid | [57, 1] | [67, 6] | rfl | π : Type
instβΒ³ : NontriviallyNormedField π
E A : Type
instβΒ² : NormedAddCommGroup E
instβΒΉ : NormedSpace π E
instβ : TopologicalSpace A
I : ModelWithCorners π E A
f : PartialHomeomorph A A
β’ f β analyticGroupoid I β
f β contDiffGroupoid β€ I β§
(AnalyticOn π (βI β βf β βI.symm) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI β βf β βI.symm '' (βI.symm β»ΒΉ' f.source β© interior (range βI)) β interior (range βI)) β§
AnalyticOn π (βI β βf.symm β βI.symm) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI β βf.symm β βI.symm '' (βI.symm β»ΒΉ' f.target β© interior (range βI)) β interior (range βI) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type
instβΒ³ : NontriviallyNormedField π
E A : Type
instβΒ² : NormedAddCommGroup E
instβΒΉ : NormedSpace π E
instβ : TopologicalSpace A
I : ModelWithCorners π E A
f : PartialHomeomorph A A
β’ f β analyticGroupoid I β
f β contDiffGroupoid β€ I β§
(AnalyticOn π (βI β βf β βI.symm) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI β βf β βI.symm '' (βI.symm β»ΒΉ' f.source β© interior (range βI)) β interior (range βI)) β§
AnalyticOn π (βI β βf.symm β βI.symm) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI β βf.symm β βI.symm '' (βI.symm β»ΒΉ' f.target β© interior (range βI)) β interior (range βI)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | have er : (range fun x : A Γ B β¦ (I x.1, J x.2)) = range I ΓΛ’ range J := range_prod_map | π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f β analyticGroupoid I
ga : g β analyticGroupoid J
β’ f.prod g β analyticGroupoid (I.prod J) | π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f β analyticGroupoid I
ga : g β analyticGroupoid J
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
β’ f.prod g β analyticGroupoid (I.prod J) | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f β analyticGroupoid I
ga : g β analyticGroupoid J
β’ f.prod g β analyticGroupoid (I.prod J)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | have ei : interior (range fun x : A Γ B β¦ (I x.1, J x.2)) =
interior (range I) ΓΛ’ interior (range J) := by rw [er, interior_prod_eq] | π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f β analyticGroupoid I
ga : g β analyticGroupoid J
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
β’ f.prod g β analyticGroupoid (I.prod J) | π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f β analyticGroupoid I
ga : g β analyticGroupoid J
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
β’ f.prod g β analyticGroupoid (I.prod J) | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f β analyticGroupoid I
ga : g β analyticGroupoid J
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
β’ f.prod g β analyticGroupoid (I.prod J)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | simp only [mem_analyticGroupoid, Function.comp, image_subset_iff] at fa ga β’ | π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f β analyticGroupoid I
ga : g β analyticGroupoid J
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
β’ f.prod g β analyticGroupoid (I.prod J) | π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ f.prod g β contDiffGroupoid β€ (I.prod J) β§
(AnalyticOn π (fun x => β(I.prod J) (β(f.prod g) (β(I.prod J).symm x)))
(β(I.prod J).symm β»ΒΉ' (f.prod g).source β© interior (range β(I.prod J))) β§
β(I.prod J).symm β»ΒΉ' (f.prod g).source β© interior (range β(I.prod J)) β
(fun x => β(I.prod J) (β(f.prod g) (β(I.prod J).symm x))) β»ΒΉ' interior (range β(I.prod J))) β§
AnalyticOn π (fun x => β(I.prod J) (β(f.prod g).symm (β(I.prod J).symm x)))
(β(I.prod J).symm β»ΒΉ' (f.prod g).target β© interior (range β(I.prod J))) β§
β(I.prod J).symm β»ΒΉ' (f.prod g).target β© interior (range β(I.prod J)) β
(fun x => β(I.prod J) (β(f.prod g).symm (β(I.prod J).symm x))) β»ΒΉ' interior (range β(I.prod J)) | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f β analyticGroupoid I
ga : g β analyticGroupoid J
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
β’ f.prod g β analyticGroupoid (I.prod J)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | refine β¨contDiffGroupoid_prod fa.1 ga.1, β¨?_, ?_β©, β¨?_, ?_β©β© | π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ f.prod g β contDiffGroupoid β€ (I.prod J) β§
(AnalyticOn π (fun x => β(I.prod J) (β(f.prod g) (β(I.prod J).symm x)))
(β(I.prod J).symm β»ΒΉ' (f.prod g).source β© interior (range β(I.prod J))) β§
β(I.prod J).symm β»ΒΉ' (f.prod g).source β© interior (range β(I.prod J)) β
(fun x => β(I.prod J) (β(f.prod g) (β(I.prod J).symm x))) β»ΒΉ' interior (range β(I.prod J))) β§
AnalyticOn π (fun x => β(I.prod J) (β(f.prod g).symm (β(I.prod J).symm x)))
(β(I.prod J).symm β»ΒΉ' (f.prod g).target β© interior (range β(I.prod J))) β§
β(I.prod J).symm β»ΒΉ' (f.prod g).target β© interior (range β(I.prod J)) β
(fun x => β(I.prod J) (β(f.prod g).symm (β(I.prod J).symm x))) β»ΒΉ' interior (range β(I.prod J)) | case refine_1
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ AnalyticOn π (fun x => β(I.prod J) (β(f.prod g) (β(I.prod J).symm x)))
(β(I.prod J).symm β»ΒΉ' (f.prod g).source β© interior (range β(I.prod J)))
case refine_2
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ β(I.prod J).symm β»ΒΉ' (f.prod g).source β© interior (range β(I.prod J)) β
(fun x => β(I.prod J) (β(f.prod g) (β(I.prod J).symm x))) β»ΒΉ' interior (range β(I.prod J))
case refine_3
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ AnalyticOn π (fun x => β(I.prod J) (β(f.prod g).symm (β(I.prod J).symm x)))
(β(I.prod J).symm β»ΒΉ' (f.prod g).target β© interior (range β(I.prod J)))
case refine_4
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ β(I.prod J).symm β»ΒΉ' (f.prod g).target β© interior (range β(I.prod J)) β
(fun x => β(I.prod J) (β(f.prod g).symm (β(I.prod J).symm x))) β»ΒΉ' interior (range β(I.prod J)) | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ f.prod g β contDiffGroupoid β€ (I.prod J) β§
(AnalyticOn π (fun x => β(I.prod J) (β(f.prod g) (β(I.prod J).symm x)))
(β(I.prod J).symm β»ΒΉ' (f.prod g).source β© interior (range β(I.prod J))) β§
β(I.prod J).symm β»ΒΉ' (f.prod g).source β© interior (range β(I.prod J)) β
(fun x => β(I.prod J) (β(f.prod g) (β(I.prod J).symm x))) β»ΒΉ' interior (range β(I.prod J))) β§
AnalyticOn π (fun x => β(I.prod J) (β(f.prod g).symm (β(I.prod J).symm x)))
(β(I.prod J).symm β»ΒΉ' (f.prod g).target β© interior (range β(I.prod J))) β§
β(I.prod J).symm β»ΒΉ' (f.prod g).target β© interior (range β(I.prod J)) β
(fun x => β(I.prod J) (β(f.prod g).symm (β(I.prod J).symm x))) β»ΒΉ' interior (range β(I.prod J))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | rw [er, interior_prod_eq] | π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f β analyticGroupoid I
ga : g β analyticGroupoid J
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
β’ interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
fa : f β analyticGroupoid I
ga : g β analyticGroupoid J
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
β’ interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | apply AnalyticOn.prod | case refine_1
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ AnalyticOn π (fun x => β(I.prod J) (β(f.prod g) (β(I.prod J).symm x)))
(β(I.prod J).symm β»ΒΉ' (f.prod g).source β© interior (range β(I.prod J))) | case refine_1.hf
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ AnalyticOn π (fun x => βI (β(f.prod g) (β(I.prod J).symm x)).1)
(β(I.prod J).symm β»ΒΉ' (f.prod g).source β© interior (range β(I.prod J)))
case refine_1.hg
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ AnalyticOn π (fun x => βJ (β(f.prod g) (β(I.prod J).symm x)).2)
(β(I.prod J).symm β»ΒΉ' (f.prod g).source β© interior (range β(I.prod J))) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ AnalyticOn π (fun x => β(I.prod J) (β(f.prod g) (β(I.prod J).symm x)))
(β(I.prod J).symm β»ΒΉ' (f.prod g).source β© interior (range β(I.prod J)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | simp only [ModelWithCorners.symm, modelWithCorners_prod_toPartialEquiv,
PartialEquiv.prod_symm, PartialEquiv.prod_coe, ModelWithCorners.toPartialEquiv_coe_symm,
PartialHomeomorph.prod_apply, PartialHomeomorph.prod_toPartialEquiv,
PartialEquiv.prod_source] | case refine_1.hf
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ AnalyticOn π (fun x => βI (β(f.prod g) (β(I.prod J).symm x)).1)
(β(I.prod J).symm β»ΒΉ' (f.prod g).source β© interior (range β(I.prod J))) | case refine_1.hf
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ AnalyticOn π (fun x => βI (βf (βI.symm x.1)))
((fun p => (βI.symm p.1, βJ.symm p.2)) β»ΒΉ' f.source ΓΛ’ g.source β© interior (range β(I.prod J))) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hf
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ AnalyticOn π (fun x => βI (β(f.prod g) (β(I.prod J).symm x)).1)
(β(I.prod J).symm β»ΒΉ' (f.prod g).source β© interior (range β(I.prod J)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | refine fa.2.1.1.comp (analyticOn_fst _) ?_ | case refine_1.hf
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ AnalyticOn π (fun x => βI (βf (βI.symm x.1)))
((fun p => (βI.symm p.1, βJ.symm p.2)) β»ΒΉ' f.source ΓΛ’ g.source β© interior (range β(I.prod J))) | case refine_1.hf
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ MapsTo (fun x => x.1) ((fun p => (βI.symm p.1, βJ.symm p.2)) β»ΒΉ' f.source ΓΛ’ g.source β© interior (range β(I.prod J)))
(βI.symm β»ΒΉ' f.source β© interior (range βI)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hf
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ AnalyticOn π (fun x => βI (βf (βI.symm x.1)))
((fun p => (βI.symm p.1, βJ.symm p.2)) β»ΒΉ' f.source ΓΛ’ g.source β© interior (range β(I.prod J)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | intro x m | case refine_1.hf
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ MapsTo (fun x => x.1) ((fun p => (βI.symm p.1, βJ.symm p.2)) β»ΒΉ' f.source ΓΛ’ g.source β© interior (range β(I.prod J)))
(βI.symm β»ΒΉ' f.source β© interior (range βI)) | case refine_1.hf
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
x : E Γ F
m : x β (fun p => (βI.symm p.1, βJ.symm p.2)) β»ΒΉ' f.source ΓΛ’ g.source β© interior (range β(I.prod J))
β’ (fun x => x.1) x β βI.symm β»ΒΉ' f.source β© interior (range βI) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hf
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ MapsTo (fun x => x.1) ((fun p => (βI.symm p.1, βJ.symm p.2)) β»ΒΉ' f.source ΓΛ’ g.source β© interior (range β(I.prod J)))
(βI.symm β»ΒΉ' f.source β© interior (range βI))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | simp only [ModelWithCorners.prod, ModelWithCorners.source_eq, mem_univ, and_self, setOf_true,
PartialEquiv.prod_target, ModelWithCorners.target_eq, ModelWithCorners.mk_coe,
mem_inter_iff, mem_preimage, mem_prod] at m β’ | case refine_1.hf
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
x : E Γ F
m : x β (fun p => (βI.symm p.1, βJ.symm p.2)) β»ΒΉ' f.source ΓΛ’ g.source β© interior (range β(I.prod J))
β’ (fun x => x.1) x β βI.symm β»ΒΉ' f.source β© interior (range βI) | case refine_1.hf
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
x : E Γ F
m : (βI.symm x.1 β f.source β§ βJ.symm x.2 β g.source) β§ x β interior (range fun x => (βI x.1, βJ x.2))
β’ βI.symm x.1 β f.source β§ x.1 β interior (range βI) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hf
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
x : E Γ F
m : x β (fun p => (βI.symm p.1, βJ.symm p.2)) β»ΒΉ' f.source ΓΛ’ g.source β© interior (range β(I.prod J))
β’ (fun x => x.1) x β βI.symm β»ΒΉ' f.source β© interior (range βI)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | exact β¨m.1.1, (subset_of_eq ei m.2).1β© | case refine_1.hf
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
x : E Γ F
m : (βI.symm x.1 β f.source β§ βJ.symm x.2 β g.source) β§ x β interior (range fun x => (βI x.1, βJ x.2))
β’ βI.symm x.1 β f.source β§ x.1 β interior (range βI) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hf
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
x : E Γ F
m : (βI.symm x.1 β f.source β§ βJ.symm x.2 β g.source) β§ x β interior (range fun x => (βI x.1, βJ x.2))
β’ βI.symm x.1 β f.source β§ x.1 β interior (range βI)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | simp only [ModelWithCorners.symm, modelWithCorners_prod_toPartialEquiv,
PartialEquiv.prod_symm, PartialEquiv.prod_coe, ModelWithCorners.toPartialEquiv_coe_symm,
PartialHomeomorph.prod_apply, PartialHomeomorph.prod_toPartialEquiv,
PartialEquiv.prod_source] | case refine_1.hg
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ AnalyticOn π (fun x => βJ (β(f.prod g) (β(I.prod J).symm x)).2)
(β(I.prod J).symm β»ΒΉ' (f.prod g).source β© interior (range β(I.prod J))) | case refine_1.hg
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ AnalyticOn π (fun x => βJ (βg (βJ.symm x.2)))
((fun p => (βI.symm p.1, βJ.symm p.2)) β»ΒΉ' f.source ΓΛ’ g.source β© interior (range β(I.prod J))) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hg
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ AnalyticOn π (fun x => βJ (β(f.prod g) (β(I.prod J).symm x)).2)
(β(I.prod J).symm β»ΒΉ' (f.prod g).source β© interior (range β(I.prod J)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | refine ga.2.1.1.comp (analyticOn_snd _) ?_ | case refine_1.hg
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ AnalyticOn π (fun x => βJ (βg (βJ.symm x.2)))
((fun p => (βI.symm p.1, βJ.symm p.2)) β»ΒΉ' f.source ΓΛ’ g.source β© interior (range β(I.prod J))) | case refine_1.hg
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ MapsTo (fun x => x.2) ((fun p => (βI.symm p.1, βJ.symm p.2)) β»ΒΉ' f.source ΓΛ’ g.source β© interior (range β(I.prod J)))
(βJ.symm β»ΒΉ' g.source β© interior (range βJ)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hg
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ AnalyticOn π (fun x => βJ (βg (βJ.symm x.2)))
((fun p => (βI.symm p.1, βJ.symm p.2)) β»ΒΉ' f.source ΓΛ’ g.source β© interior (range β(I.prod J)))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/AnalyticManifold.lean | analyticGroupoid_prod | [79, 1] | [150, 72] | intro x m | case refine_1.hg
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ MapsTo (fun x => x.2) ((fun p => (βI.symm p.1, βJ.symm p.2)) β»ΒΉ' f.source ΓΛ’ g.source β© interior (range β(I.prod J)))
(βJ.symm β»ΒΉ' g.source β© interior (range βJ)) | case refine_1.hg
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
x : E Γ F
m : x β (fun p => (βI.symm p.1, βJ.symm p.2)) β»ΒΉ' f.source ΓΛ’ g.source β© interior (range β(I.prod J))
β’ (fun x => x.2) x β βJ.symm β»ΒΉ' g.source β© interior (range βJ) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.hg
π : Type
instββΆ : NontriviallyNormedField π
E A : Type
instββ΅ : NormedAddCommGroup E
instββ΄ : NormedSpace π E
instβΒ³ : TopologicalSpace A
F B : Type
instβΒ² : NormedAddCommGroup F
instβΒΉ : NormedSpace π F
instβ : TopologicalSpace B
I : ModelWithCorners π E A
J : ModelWithCorners π F B
f : PartialHomeomorph A A
g : PartialHomeomorph B B
er : (range fun x => (βI x.1, βJ x.2)) = range βI ΓΛ’ range βJ
ei : interior (range fun x => (βI x.1, βJ x.2)) = interior (range βI) ΓΛ’ interior (range βJ)
fa :
f β contDiffGroupoid β€ I β§
(AnalyticOn π (fun x => βI (βf (βI.symm x))) (βI.symm β»ΒΉ' f.source β© interior (range βI)) β§
βI.symm β»ΒΉ' f.source β© interior (range βI) β (fun x => βI (βf (βI.symm x))) β»ΒΉ' interior (range βI)) β§
AnalyticOn π (fun x => βI (βf.symm (βI.symm x))) (βI.symm β»ΒΉ' f.target β© interior (range βI)) β§
βI.symm β»ΒΉ' f.target β© interior (range βI) β (fun x => βI (βf.symm (βI.symm x))) β»ΒΉ' interior (range βI)
ga :
g β contDiffGroupoid β€ J β§
(AnalyticOn π (fun x => βJ (βg (βJ.symm x))) (βJ.symm β»ΒΉ' g.source β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.source β© interior (range βJ) β (fun x => βJ (βg (βJ.symm x))) β»ΒΉ' interior (range βJ)) β§
AnalyticOn π (fun x => βJ (βg.symm (βJ.symm x))) (βJ.symm β»ΒΉ' g.target β© interior (range βJ)) β§
βJ.symm β»ΒΉ' g.target β© interior (range βJ) β (fun x => βJ (βg.symm (βJ.symm x))) β»ΒΉ' interior (range βJ)
β’ MapsTo (fun x => x.2) ((fun p => (βI.symm p.1, βJ.symm p.2)) β»ΒΉ' f.source ΓΛ’ g.source β© interior (range β(I.prod J)))
(βJ.symm β»ΒΉ' g.source β© interior (range βJ))
TACTIC:
|
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