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/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | rcases choose_spec (h _ i.mem) with β¨_, _, _, sβ© | f : β β β β β
d : β
u : Set β
tβ : Set (β Γ β)
s : SuperAtC f d u
w : Set (β Γ β)
wo : IsOpen w
wc : β c β u, (c, 0) β w
h : β c β u, β r > 0, ball c r β u β§ ball (c, 0) r β w β§ SuperNearC f d (ball c r) (ball (c, 0) r)
r : βu β β := fun c => choose β―
v : βu β Set β := fun c => ball (βc) (r c)
t : βu β Set (β Γ β) := fun c => ball (βc, 0) (r c)
e : u = β c, v c
tw : β c, t c β w
i : βu
β’ SuperNearC f d (v i) (t i) | case intro.intro.intro
f : β β β β β
d : β
u : Set β
tβ : Set (β Γ β)
sβ : SuperAtC f d u
w : Set (β Γ β)
wo : IsOpen w
wc : β c β u, (c, 0) β w
h : β c β u, β r > 0, ball c r β u β§ ball (c, 0) r β w β§ SuperNearC f d (ball c r) (ball (c, 0) r)
r : βu β β := fun c => choose β―
v : βu β Set β := fun c => ball (βc) (r c)
t : βu β Set (β Γ β) := fun c => ball (βc, 0) (r c)
e : u = β c, v c
tw : β c, t c β w
i : βu
leftβΒ² : choose β― > 0
leftβΒΉ : ball (βi) (choose β―) β u
leftβ : ball (βi, 0) (choose β―) β w
s : SuperNearC f d (ball (βi) (choose β―)) (ball (βi, 0) (choose β―))
β’ SuperNearC f d (v i) (t i) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
tβ : Set (β Γ β)
s : SuperAtC f d u
w : Set (β Γ β)
wo : IsOpen w
wc : β c β u, (c, 0) β w
h : β c β u, β r > 0, ball c r β u β§ ball (c, 0) r β w β§ SuperNearC f d (ball c r) (ball (c, 0) r)
r : βu β β := fun c => choose β―
v : βu β Set β := fun c => ball (βc) (r c)
t : βu β Set (β Γ β) := fun c => ball (βc, 0) (r c)
e : u = β c, v c
tw : β c, t c β w
i : βu
β’ SuperNearC f d (v i) (t i)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC' | [576, 1] | [630, 16] | exact s | case intro.intro.intro
f : β β β β β
d : β
u : Set β
tβ : Set (β Γ β)
sβ : SuperAtC f d u
w : Set (β Γ β)
wo : IsOpen w
wc : β c β u, (c, 0) β w
h : β c β u, β r > 0, ball c r β u β§ ball (c, 0) r β w β§ SuperNearC f d (ball c r) (ball (c, 0) r)
r : βu β β := fun c => choose β―
v : βu β Set β := fun c => ball (βc) (r c)
t : βu β Set (β Γ β) := fun c => ball (βc, 0) (r c)
e : u = β c, v c
tw : β c, t c β w
i : βu
leftβΒ² : choose β― > 0
leftβΒΉ : ball (βi) (choose β―) β u
leftβ : ball (βi, 0) (choose β―) β w
s : SuperNearC f d (ball (βi) (choose β―)) (ball (βi, 0) (choose β―))
β’ SuperNearC f d (v i) (t i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f : β β β β β
d : β
u : Set β
tβ : Set (β Γ β)
sβ : SuperAtC f d u
w : Set (β Γ β)
wo : IsOpen w
wc : β c β u, (c, 0) β w
h : β c β u, β r > 0, ball c r β u β§ ball (c, 0) r β w β§ SuperNearC f d (ball c r) (ball (c, 0) r)
r : βu β β := fun c => choose β―
v : βu β Set β := fun c => ball (βc) (r c)
t : βu β Set (β Γ β) := fun c => ball (βc, 0) (r c)
e : u = β c, v c
tw : β c, t c β w
i : βu
leftβΒ² : choose β― > 0
leftβΒΉ : ball (βi) (choose β―) β u
leftβ : ball (βi, 0) (choose β―) β w
s : SuperNearC f d (ball (βi) (choose β―)) (ball (βi, 0) (choose β―))
β’ SuperNearC f d (v i) (t i)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC | [633, 1] | [634, 89] | rcases s.superNearC' isOpen_univ fun _ _ β¦ Set.mem_univ _ with β¨t, _, sβ© | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperAtC f d u
β’ β t, SuperNearC f d u t | case intro.intro
f : β β β β β
d : β
u : Set β
tβ : Set (β Γ β)
sβ : SuperAtC f d u
t : Set (β Γ β)
leftβ : t β univ
s : SuperNearC f d u t
β’ β t, SuperNearC f d u t | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperAtC f d u
β’ β t, SuperNearC f d u t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | SuperAtC.superNearC | [633, 1] | [634, 89] | exact β¨t, sβ© | case intro.intro
f : β β β β β
d : β
u : Set β
tβ : Set (β Γ β)
sβ : SuperAtC f d u
t : Set (β Γ β)
leftβ : t β univ
s : SuperNearC f d u t
β’ β t, SuperNearC f d u t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β β β
d : β
u : Set β
tβ : Set (β Γ β)
sβ : SuperAtC f d u
t : Set (β Γ β)
leftβ : t β univ
s : SuperNearC f d u t
β’ β t, SuperNearC f d u t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | induction' n with n nh | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
β’ AnalyticAt β (fun c => (f c)^[n] z) c | case zero
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun c => (f c)^[0] z) c
case succ
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
n : β
nh : AnalyticAt β (fun c => (f c)^[n] z) c
β’ AnalyticAt β (fun c => (f c)^[n + 1] z) c | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
β’ AnalyticAt β (fun c => (f c)^[n] z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | simp only [Function.iterate_zero, id] | case zero
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun c => (f c)^[0] z) c | case zero
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun c => z) c | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun c => (f c)^[0] z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | exact analyticAt_const | case zero
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun c => z) c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun c => z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | simp_rw [Function.iterate_succ'] | case succ
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
n : β
nh : AnalyticAt β (fun c => (f c)^[n] z) c
β’ AnalyticAt β (fun c => (f c)^[n + 1] z) c | case succ
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
n : β
nh : AnalyticAt β (fun c => (f c)^[n] z) c
β’ AnalyticAt β (fun c => (f c β (f c)^[n]) z) c | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
n : β
nh : AnalyticAt β (fun c => (f c)^[n] z) c
β’ AnalyticAt β (fun c => (f c)^[n + 1] z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | simp only [Function.comp_apply] | case succ
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
n : β
nh : AnalyticAt β (fun c => (f c)^[n] z) c
β’ AnalyticAt β (fun c => (f c β (f c)^[n]) z) c | case succ
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
n : β
nh : AnalyticAt β (fun c => (f c)^[n] z) c
β’ AnalyticAt β (fun c => f c ((f c)^[n] z)) c | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
n : β
nh : AnalyticAt β (fun c => (f c)^[n] z) c
β’ AnalyticAt β (fun c => (f c β (f c)^[n]) z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | refine (s.fa _ ?_).comp ((analyticAt_id _ _).prod nh) | case succ
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
n : β
nh : AnalyticAt β (fun c => (f c)^[n] z) c
β’ AnalyticAt β (fun c => f c ((f c)^[n] z)) c | case succ
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
n : β
nh : AnalyticAt β (fun c => (f c)^[n] z) c
β’ (id c, (f c)^[n] z) β t | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
n : β
nh : AnalyticAt β (fun c => (f c)^[n] z) c
β’ AnalyticAt β (fun c => f c ((f c)^[n] z)) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | iterates_analytic_c | [636, 1] | [641, 30] | exact (s.ts m).mapsTo n m | case succ
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
n : β
nh : AnalyticAt β (fun c => (f c)^[n] z) c
β’ (id c, (f c)^[n] z) β t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
n : β
nh : AnalyticAt β (fun c => (f c)^[n] z) c
β’ (id c, (f c)^[n] z) β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | refine AnalyticAt.cpow ?_ analyticAt_const ?_ | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
β’ AnalyticAt β (fun c => term (f c) d n z) c | case refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
β’ AnalyticAt β (fun c => g (f c) d ((f c)^[n] z)) c
case refine_2
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
β’ g (f c) d ((f c)^[n] z) β Complex.slitPlane | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
β’ AnalyticAt β (fun c => term (f c) d n z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | have e : (fun c β¦ g (f c) d ((f c)^[n] z)) = fun c β¦ g2 f d (c, (f c)^[n] z) := rfl | case refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
β’ AnalyticAt β (fun c => g (f c) d ((f c)^[n] z)) c | case refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
β’ AnalyticAt β (fun c => g (f c) d ((f c)^[n] z)) c | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
β’ AnalyticAt β (fun c => g (f c) d ((f c)^[n] z)) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | rw [e] | case refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
β’ AnalyticAt β (fun c => g (f c) d ((f c)^[n] z)) c | case refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
β’ AnalyticAt β (fun c => g2 f d (c, (f c)^[n] z)) c | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
β’ AnalyticAt β (fun c => g (f c) d ((f c)^[n] z)) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | refine (s.ga _ ?_).comp ?_ | case refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
β’ AnalyticAt β (fun c => g2 f d (c, (f c)^[n] z)) c | case refine_1.refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
β’ (c, (f c)^[n] z) β t
case refine_1.refine_2
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
β’ AnalyticAt β (fun c => (c, (f c)^[n] z)) c | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
β’ AnalyticAt β (fun c => g2 f d (c, (f c)^[n] z)) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | exact (s.ts m).mapsTo n m | case refine_1.refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
β’ (c, (f c)^[n] z) β t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
β’ (c, (f c)^[n] z) β t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | apply (analyticAt_id _ _).prod (iterates_analytic_c s n m) | case refine_1.refine_2
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
β’ AnalyticAt β (fun c => (c, (f c)^[n] z)) c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.refine_2
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
e : (fun c => g (f c) d ((f c)^[n] z)) = fun c => g2 f d (c, (f c)^[n] z)
β’ AnalyticAt β (fun c => (c, (f c)^[n] z)) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | refine mem_slitPlane_of_near_one ?_ | case refine_2
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
β’ g (f c) d ((f c)^[n] z) β Complex.slitPlane | case refine_2
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
β’ Complex.abs (g (f c) d ((f c)^[n] z) - 1) < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
β’ g (f c) d ((f c)^[n] z) β Complex.slitPlane
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | exact lt_of_le_of_lt ((s.ts m).gs ((s.ts m).mapsTo n m)) (by norm_num) | case refine_2
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
β’ Complex.abs (g (f c) d ((f c)^[n] z) - 1) < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
β’ Complex.abs (g (f c) d ((f c)^[n] z) - 1) < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_analytic_c | [643, 1] | [652, 75] | norm_num | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
β’ 1 / 4 < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
n : β
m : (c, z) β t
β’ 1 / 4 < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | have c12 : (1 / 2 : β) β€ 1 / 2 := by norm_num | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun c => β' (n : β), term (f c) d n z) c | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
β’ AnalyticAt β (fun c => β' (n : β), term (f c) d n z) c | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun c => β' (n : β), term (f c) d n z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | have a0 : 0 β€ (1 / 2 : β) := by norm_num | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
β’ AnalyticAt β (fun c => β' (n : β), term (f c) d n z) c | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
a0 : 0 β€ 1 / 2
β’ AnalyticAt β (fun c => β' (n : β), term (f c) d n z) c | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
β’ AnalyticAt β (fun c => β' (n : β), term (f c) d n z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | set t' := {c | (c, z) β t} | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
a0 : 0 β€ 1 / 2
β’ AnalyticAt β (fun c => β' (n : β), term (f c) d n z) c | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
a0 : 0 β€ 1 / 2
t' : Set β := {c | (c, z) β t}
β’ AnalyticAt β (fun c => β' (n : β), term (f c) d n z) c | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
a0 : 0 β€ 1 / 2
β’ AnalyticAt β (fun c => β' (n : β), term (f c) d n z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | have o' : IsOpen t' := s.o.preimage (by continuity) | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
a0 : 0 β€ 1 / 2
t' : Set β := {c | (c, z) β t}
β’ AnalyticAt β (fun c => β' (n : β), term (f c) d n z) c | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
a0 : 0 β€ 1 / 2
t' : Set β := {c | (c, z) β t}
o' : IsOpen t'
β’ AnalyticAt β (fun c => β' (n : β), term (f c) d n z) c | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
a0 : 0 β€ 1 / 2
t' : Set β := {c | (c, z) β t}
β’ AnalyticAt β (fun c => β' (n : β), term (f c) d n z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | refine (fast_products_converge' o' c12 a0 (by linarith) ?_
fun n c m β¦ term_converges (s.ts m) n m).2.1 _ m | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
a0 : 0 β€ 1 / 2
t' : Set β := {c | (c, z) β t}
o' : IsOpen t'
β’ AnalyticAt β (fun c => β' (n : β), term (f c) d n z) c | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
a0 : 0 β€ 1 / 2
t' : Set β := {c | (c, z) β t}
o' : IsOpen t'
β’ β (n : β), AnalyticOn β (fun c => term (f (c, z).1) d n z) t' | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
a0 : 0 β€ 1 / 2
t' : Set β := {c | (c, z) β t}
o' : IsOpen t'
β’ AnalyticAt β (fun c => β' (n : β), term (f c) d n z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | exact fun n c m β¦ term_analytic_c s n m | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
a0 : 0 β€ 1 / 2
t' : Set β := {c | (c, z) β t}
o' : IsOpen t'
β’ β (n : β), AnalyticOn β (fun c => term (f (c, z).1) d n z) t' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
a0 : 0 β€ 1 / 2
t' : Set β := {c | (c, z) β t}
o' : IsOpen t'
β’ β (n : β), AnalyticOn β (fun c => term (f (c, z).1) d n z) t'
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | norm_num | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ 1 / 2 β€ 1 / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ 1 / 2 β€ 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | norm_num | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
β’ 0 β€ 1 / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
β’ 0 β€ 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | continuity | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
a0 : 0 β€ 1 / 2
t' : Set β := {c | (c, z) β t}
β’ Continuous fun c => (c, z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
a0 : 0 β€ 1 / 2
t' : Set β := {c | (c, z) β t}
β’ Continuous fun c => (c, z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic_c | [655, 1] | [663, 42] | linarith | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
a0 : 0 β€ 1 / 2
t' : Set β := {c | (c, z) β t}
o' : IsOpen t'
β’ 1 / 2 < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
c12 : 1 / 2 β€ 1 / 2
a0 : 0 β€ 1 / 2
t' : Set β := {c | (c, z) β t}
o' : IsOpen t'
β’ 1 / 2 < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic | [666, 1] | [670, 68] | refine Pair.hartogs s.o ?_ ?_ | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
β’ AnalyticOn β (fun p => β' (n : β), term (f p.1) d n p.2) t | case refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
β’ β (c0 c1 : β), (c0, c1) β t β AnalyticAt β (fun z0 => β' (n : β), term (f (z0, c1).1) d n (z0, c1).2) c0
case refine_2
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
β’ β (c0 c1 : β), (c0, c1) β t β AnalyticAt β (fun z1 => β' (n : β), term (f (c0, z1).1) d n (c0, z1).2) c1 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
β’ AnalyticOn β (fun p => β' (n : β), term (f p.1) d n p.2) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic | [666, 1] | [670, 68] | intro c z m | case refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
β’ β (c0 c1 : β), (c0, c1) β t β AnalyticAt β (fun z0 => β' (n : β), term (f (z0, c1).1) d n (z0, c1).2) c0 | case refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun z0 => β' (n : β), term (f (z0, z).1) d n (z0, z).2) c | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
β’ β (c0 c1 : β), (c0, c1) β t β AnalyticAt β (fun z0 => β' (n : β), term (f (z0, c1).1) d n (z0, c1).2) c0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic | [666, 1] | [670, 68] | simp only | case refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun z0 => β' (n : β), term (f (z0, z).1) d n (z0, z).2) c | case refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun z0 => β' (n : β), term (f z0) d n z) c | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun z0 => β' (n : β), term (f (z0, z).1) d n (z0, z).2) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic | [666, 1] | [670, 68] | exact term_prod_analytic_c s m | case refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun z0 => β' (n : β), term (f z0) d n z) c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun z0 => β' (n : β), term (f z0) d n z) c
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic | [666, 1] | [670, 68] | intro c z m | case refine_2
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
β’ β (c0 c1 : β), (c0, c1) β t β AnalyticAt β (fun z1 => β' (n : β), term (f (c0, z1).1) d n (c0, z1).2) c1 | case refine_2
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun z1 => β' (n : β), term (f (c, z1).1) d n (c, z1).2) z | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
β’ β (c0 c1 : β), (c0, c1) β t β AnalyticAt β (fun z1 => β' (n : β), term (f (c0, z1).1) d n (c0, z1).2) c1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic | [666, 1] | [670, 68] | simp only | case refine_2
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun z1 => β' (n : β), term (f (c, z1).1) d n (c, z1).2) z | case refine_2
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun z1 => β' (n : β), term (f c) d n z1) z | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun z1 => β' (n : β), term (f (c, z1).1) d n (c, z1).2) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | term_prod_analytic | [666, 1] | [670, 68] | exact term_prod_analytic_z (s.ts m) _ m | case refine_2
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun z1 => β' (n : β), term (f c) d n z1) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c z : β
m : (c, z) β t
β’ AnalyticAt β (fun z1 => β' (n : β), term (f c) d n z1) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | have df : β e z, (e, z) β t β
deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z := by
intro e z m; apply HasDerivAt.deriv
have fg : f e = fun z β¦ z ^ d * g (f e) d z := by funext; rw [(s.ts m).fg]
nth_rw 1 [fg]
apply HasDerivAt.mul; apply hasDerivAt_pow
rw [hasDerivAt_deriv_iff]; exact ((s.ts m).ga _ m).differentiableAt | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
β’ βαΆ (p : β Γ β) in π (c, 0), deriv (f p.1) p.2 = 0 β p.2 = 0 | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
β’ βαΆ (p : β Γ β) in π (c, 0), deriv (f p.1) p.2 = 0 β p.2 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
β’ βαΆ (p : β Γ β) in π (c, 0), deriv (f p.1) p.2 = 0 β p.2 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | apply small.mp | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
small : βαΆ (p : β Γ β) in π (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (βd * g (f p.1) d p.2)
β’ βαΆ (p : β Γ β) in π (c, 0), deriv (f p.1) p.2 = 0 β p.2 = 0 | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
small : βαΆ (p : β Γ β) in π (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (βd * g (f p.1) d p.2)
β’ βαΆ (x : β Γ β) in π (c, 0),
Complex.abs (x.2 * deriv (g (f x.1) d) x.2) < Complex.abs (βd * g (f x.1) d x.2) β (deriv (f x.1) x.2 = 0 β x.2 = 0) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
small : βαΆ (p : β Γ β) in π (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (βd * g (f p.1) d p.2)
β’ βαΆ (p : β Γ β) in π (c, 0), deriv (f p.1) p.2 = 0 β p.2 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | apply (s.o.eventually_mem (s.s m).t0).mp | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
small : βαΆ (p : β Γ β) in π (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (βd * g (f p.1) d p.2)
β’ βαΆ (x : β Γ β) in π (c, 0),
Complex.abs (x.2 * deriv (g (f x.1) d) x.2) < Complex.abs (βd * g (f x.1) d x.2) β (deriv (f x.1) x.2 = 0 β x.2 = 0) | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
small : βαΆ (p : β Γ β) in π (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (βd * g (f p.1) d p.2)
β’ βαΆ (x : β Γ β) in π (c, 0),
x β t β
Complex.abs (x.2 * deriv (g (f x.1) d) x.2) < Complex.abs (βd * g (f x.1) d x.2) β
(deriv (f x.1) x.2 = 0 β x.2 = 0) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
small : βαΆ (p : β Γ β) in π (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (βd * g (f p.1) d p.2)
β’ βαΆ (x : β Γ β) in π (c, 0),
Complex.abs (x.2 * deriv (g (f x.1) d) x.2) < Complex.abs (βd * g (f x.1) d x.2) β (deriv (f x.1) x.2 = 0 β x.2 = 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | apply Filter.eventually_of_forall | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
small : βαΆ (p : β Γ β) in π (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (βd * g (f p.1) d p.2)
β’ βαΆ (x : β Γ β) in π (c, 0),
x β t β
Complex.abs (x.2 * deriv (g (f x.1) d) x.2) < Complex.abs (βd * g (f x.1) d x.2) β
(deriv (f x.1) x.2 = 0 β x.2 = 0) | case hp
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
small : βαΆ (p : β Γ β) in π (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (βd * g (f p.1) d p.2)
β’ β x β t,
Complex.abs (x.2 * deriv (g (f x.1) d) x.2) < Complex.abs (βd * g (f x.1) d x.2) β (deriv (f x.1) x.2 = 0 β x.2 = 0) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
small : βαΆ (p : β Γ β) in π (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (βd * g (f p.1) d p.2)
β’ βαΆ (x : β Γ β) in π (c, 0),
x β t β
Complex.abs (x.2 * deriv (g (f x.1) d) x.2) < Complex.abs (βd * g (f x.1) d x.2) β
(deriv (f x.1) x.2 = 0 β x.2 = 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | clear small | case hp
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
small : βαΆ (p : β Γ β) in π (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (βd * g (f p.1) d p.2)
β’ β x β t,
Complex.abs (x.2 * deriv (g (f x.1) d) x.2) < Complex.abs (βd * g (f x.1) d x.2) β (deriv (f x.1) x.2 = 0 β x.2 = 0) | case hp
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
β’ β x β t,
Complex.abs (x.2 * deriv (g (f x.1) d) x.2) < Complex.abs (βd * g (f x.1) d x.2) β (deriv (f x.1) x.2 = 0 β x.2 = 0) | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
small : βαΆ (p : β Γ β) in π (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (βd * g (f p.1) d p.2)
β’ β x β t,
Complex.abs (x.2 * deriv (g (f x.1) d) x.2) < Complex.abs (βd * g (f x.1) d x.2) β (deriv (f x.1) x.2 = 0 β x.2 = 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | intro β¨e, wβ© m' small | case hp
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
β’ β x β t,
Complex.abs (x.2 * deriv (g (f x.1) d) x.2) < Complex.abs (βd * g (f x.1) d x.2) β (deriv (f x.1) x.2 = 0 β x.2 = 0) | case hp
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : β
m' : (e, w) β t
small : Complex.abs ((e, w).2 * deriv (g (f (e, w).1) d) (e, w).2) < Complex.abs (βd * g (f (e, w).1) d (e, w).2)
β’ deriv (f (e, w).1) (e, w).2 = 0 β (e, w).2 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
β’ β x β t,
Complex.abs (x.2 * deriv (g (f x.1) d) x.2) < Complex.abs (βd * g (f x.1) d x.2) β (deriv (f x.1) x.2 = 0 β x.2 = 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | simp only [df _ _ m'] at small β’ | case hp
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : β
m' : (e, w) β t
small : Complex.abs ((e, w).2 * deriv (g (f (e, w).1) d) (e, w).2) < Complex.abs (βd * g (f (e, w).1) d (e, w).2)
β’ deriv (f (e, w).1) (e, w).2 = 0 β (e, w).2 = 0 | case hp
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : β
m' : (e, w) β t
small : Complex.abs (w * deriv (g (f e) d) w) < Complex.abs (βd * g (f e) d w)
β’ βd * w ^ (d - 1) * g (f e) d w + w ^ d * deriv (g (f e) d) w = 0 β w = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : β
m' : (e, w) β t
small : Complex.abs ((e, w).2 * deriv (g (f (e, w).1) d) (e, w).2) < Complex.abs (βd * g (f (e, w).1) d (e, w).2)
β’ deriv (f (e, w).1) (e, w).2 = 0 β (e, w).2 = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | nth_rw 4 [β Nat.sub_add_cancel (Nat.succ_le_of_lt (s.s m).dp)] | case hp
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : β
m' : (e, w) β t
small : Complex.abs (w * deriv (g (f e) d) w) < Complex.abs (βd * g (f e) d w)
β’ βd * w ^ (d - 1) * g (f e) d w + w ^ d * deriv (g (f e) d) w = 0 β w = 0 | case hp
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : β
m' : (e, w) β t
small : Complex.abs (w * deriv (g (f e) d) w) < Complex.abs (βd * g (f e) d w)
β’ βd * w ^ (d - 1) * g (f e) d w + w ^ (d - Nat.succ 0 + Nat.succ 0) * deriv (g (f e) d) w = 0 β w = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : β
m' : (e, w) β t
small : Complex.abs (w * deriv (g (f e) d) w) < Complex.abs (βd * g (f e) d w)
β’ βd * w ^ (d - 1) * g (f e) d w + w ^ d * deriv (g (f e) d) w = 0 β w = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | simp only [pow_add, pow_one, mul_comm _ (w ^ (d - 1)), mul_assoc (w ^ (d - 1)) _ _, β
left_distrib, mul_eq_zero, pow_eq_zero_iff (Nat.sub_pos_of_lt (s.s m).d2).ne'] | case hp
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : β
m' : (e, w) β t
small : Complex.abs (w * deriv (g (f e) d) w) < Complex.abs (βd * g (f e) d w)
β’ βd * w ^ (d - 1) * g (f e) d w + w ^ (d - Nat.succ 0 + Nat.succ 0) * deriv (g (f e) d) w = 0 β w = 0 | case hp
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : β
m' : (e, w) β t
small : Complex.abs (w * deriv (g (f e) d) w) < Complex.abs (βd * g (f e) d w)
β’ w = 0 β¨ βd * g (f e) d w + w * deriv (g (f e) d) w = 0 β w = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : β
m' : (e, w) β t
small : Complex.abs (w * deriv (g (f e) d) w) < Complex.abs (βd * g (f e) d w)
β’ βd * w ^ (d - 1) * g (f e) d w + w ^ (d - Nat.succ 0 + Nat.succ 0) * deriv (g (f e) d) w = 0 β w = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | exact or_iff_left (add_ne_zero_of_abs_lt small) | case hp
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : β
m' : (e, w) β t
small : Complex.abs (w * deriv (g (f e) d) w) < Complex.abs (βd * g (f e) d w)
β’ w = 0 β¨ βd * g (f e) d w + w * deriv (g (f e) d) w = 0 β w = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hp
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
e w : β
m' : (e, w) β t
small : Complex.abs (w * deriv (g (f e) d) w) < Complex.abs (βd * g (f e) d w)
β’ w = 0 β¨ βd * g (f e) d w + w * deriv (g (f e) d) w = 0 β w = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | intro e z m | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
β’ β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
β’ deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
β’ β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | apply HasDerivAt.deriv | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
β’ deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z | case h
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
β’ HasDerivAt (f e) (βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z) z | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
β’ deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | have fg : f e = fun z β¦ z ^ d * g (f e) d z := by funext; rw [(s.ts m).fg] | case h
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
β’ HasDerivAt (f e) (βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z) z | case h
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
fg : f e = fun z => z ^ d * g (f e) d z
β’ HasDerivAt (f e) (βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z) z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
β’ HasDerivAt (f e) (βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | nth_rw 1 [fg] | case h
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
fg : f e = fun z => z ^ d * g (f e) d z
β’ HasDerivAt (f e) (βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z) z | case h
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
fg : f e = fun z => z ^ d * g (f e) d z
β’ HasDerivAt (fun z => z ^ d * g (f e) d z) (βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z) z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
fg : f e = fun z => z ^ d * g (f e) d z
β’ HasDerivAt (f e) (βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | apply HasDerivAt.mul | case h
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
fg : f e = fun z => z ^ d * g (f e) d z
β’ HasDerivAt (fun z => z ^ d * g (f e) d z) (βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z) z | case h.hc
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
fg : f e = fun z => z ^ d * g (f e) d z
β’ HasDerivAt (fun y => y ^ d) (βd * z ^ (d - 1)) z
case h.hd
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
fg : f e = fun z => z ^ d * g (f e) d z
β’ HasDerivAt (fun y => g (f e) d y) (deriv (g (f e) d) z) z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
fg : f e = fun z => z ^ d * g (f e) d z
β’ HasDerivAt (fun z => z ^ d * g (f e) d z) (βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | apply hasDerivAt_pow | case h.hc
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
fg : f e = fun z => z ^ d * g (f e) d z
β’ HasDerivAt (fun y => y ^ d) (βd * z ^ (d - 1)) z
case h.hd
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
fg : f e = fun z => z ^ d * g (f e) d z
β’ HasDerivAt (fun y => g (f e) d y) (deriv (g (f e) d) z) z | case h.hd
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
fg : f e = fun z => z ^ d * g (f e) d z
β’ HasDerivAt (fun y => g (f e) d y) (deriv (g (f e) d) z) z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hc
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
fg : f e = fun z => z ^ d * g (f e) d z
β’ HasDerivAt (fun y => y ^ d) (βd * z ^ (d - 1)) z
case h.hd
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
fg : f e = fun z => z ^ d * g (f e) d z
β’ HasDerivAt (fun y => g (f e) d y) (deriv (g (f e) d) z) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | rw [hasDerivAt_deriv_iff] | case h.hd
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
fg : f e = fun z => z ^ d * g (f e) d z
β’ HasDerivAt (fun y => g (f e) d y) (deriv (g (f e) d) z) z | case h.hd
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
fg : f e = fun z => z ^ d * g (f e) d z
β’ DifferentiableAt β (fun y => g (f e) d y) z | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hd
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
fg : f e = fun z => z ^ d * g (f e) d z
β’ HasDerivAt (fun y => g (f e) d y) (deriv (g (f e) d) z) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | exact ((s.ts m).ga _ m).differentiableAt | case h.hd
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
fg : f e = fun z => z ^ d * g (f e) d z
β’ DifferentiableAt β (fun y => g (f e) d y) z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hd
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
fg : f e = fun z => z ^ d * g (f e) d z
β’ DifferentiableAt β (fun y => g (f e) d y) z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | funext | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
β’ f e = fun z => z ^ d * g (f e) d z | case h
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
xβ : β
β’ f e xβ = xβ ^ d * g (f e) d xβ | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
β’ f e = fun z => z ^ d * g (f e) d z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | rw [(s.ts m).fg] | case h
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
xβ : β
β’ f e xβ = xβ ^ d * g (f e) d xβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
mβ : c β u
e z : β
m : (e, z) β t
xβ : β
β’ f e xβ = xβ ^ d * g (f e) d xβ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | have ga : AnalyticAt β (uncurry fun c z β¦ g (f c) d z) (c, 0) := s.ga _ (s.s m).t0 | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
β’ βαΆ (p : β Γ β) in π (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (βd * g (f p.1) d p.2) | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt β (uncurry fun c z => g (f c) d z) (c, 0)
β’ βαΆ (p : β Γ β) in π (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (βd * g (f p.1) d p.2) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
β’ βαΆ (p : β Γ β) in π (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (βd * g (f p.1) d p.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | apply ContinuousAt.eventually_lt | f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt β (uncurry fun c z => g (f c) d z) (c, 0)
β’ βαΆ (p : β Γ β) in π (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (βd * g (f p.1) d p.2) | case hf
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt β (uncurry fun c z => g (f c) d z) (c, 0)
β’ ContinuousAt (fun x => Complex.abs (x.2 * deriv (g (f x.1) d) x.2)) (c, 0)
case hg
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt β (uncurry fun c z => g (f c) d z) (c, 0)
β’ ContinuousAt (fun x => Complex.abs (βd * g (f x.1) d x.2)) (c, 0)
case hfg
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt β (uncurry fun c z => g (f c) d z) (c, 0)
β’ Complex.abs ((c, 0).2 * deriv (g (f (c, 0).1) d) (c, 0).2) < Complex.abs (βd * g (f (c, 0).1) d (c, 0).2) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt β (uncurry fun c z => g (f c) d z) (c, 0)
β’ βαΆ (p : β Γ β) in π (c, 0), Complex.abs (p.2 * deriv (g (f p.1) d) p.2) < Complex.abs (βd * g (f p.1) d p.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | exact Complex.continuous_abs.continuousAt.comp (continuousAt_snd.mul ga.deriv2.continuousAt) | case hf
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt β (uncurry fun c z => g (f c) d z) (c, 0)
β’ ContinuousAt (fun x => Complex.abs (x.2 * deriv (g (f x.1) d) x.2)) (c, 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt β (uncurry fun c z => g (f c) d z) (c, 0)
β’ ContinuousAt (fun x => Complex.abs (x.2 * deriv (g (f x.1) d) x.2)) (c, 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | exact Complex.continuous_abs.continuousAt.comp (continuousAt_const.mul ga.continuousAt) | case hg
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt β (uncurry fun c z => g (f c) d z) (c, 0)
β’ ContinuousAt (fun x => Complex.abs (βd * g (f x.1) d x.2)) (c, 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt β (uncurry fun c z => g (f c) d z) (c, 0)
β’ ContinuousAt (fun x => Complex.abs (βd * g (f x.1) d x.2)) (c, 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | simp only [g0, MulZeroClass.zero_mul, Complex.abs.map_zero, Complex.abs.map_mul,
Complex.abs_natCast, Complex.abs.map_one, mul_one, Nat.cast_pos] | case hfg
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt β (uncurry fun c z => g (f c) d z) (c, 0)
β’ Complex.abs ((c, 0).2 * deriv (g (f (c, 0).1) d) (c, 0).2) < Complex.abs (βd * g (f (c, 0).1) d (c, 0).2) | case hfg
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt β (uncurry fun c z => g (f c) d z) (c, 0)
β’ 0 < d | Please generate a tactic in lean4 to solve the state.
STATE:
case hfg
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt β (uncurry fun c z => g (f c) d z) (c, 0)
β’ Complex.abs ((c, 0).2 * deriv (g (f (c, 0).1) d) (c, 0).2) < Complex.abs (βd * g (f (c, 0).1) d (c, 0).2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/BottcherNear.lean | df_ne_zero | [683, 1] | [708, 50] | exact (s.s m).dp | case hfg
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt β (uncurry fun c z => g (f c) d z) (c, 0)
β’ 0 < d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hfg
f : β β β β β
d : β
u : Set β
t : Set (β Γ β)
s : SuperNearC f d u t
c : β
m : c β u
df : β (e z : β), (e, z) β t β deriv (f e) z = βd * z ^ (d - 1) * g (f e) d z + z ^ d * deriv (g (f e) d) z
ga : AnalyticAt β (uncurry fun c z => g (f c) d z) (c, 0)
β’ 0 < d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | by_contra bad | c : β
d : β
instβ : Fact (2 β€ d)
β’ InjOn (bottcher d) (multibrotExt d) | c : β
d : β
instβ : Fact (2 β€ d)
bad : Β¬InjOn (bottcher d) (multibrotExt d)
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
β’ InjOn (bottcher d) (multibrotExt d)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [InjOn, not_forall, β ne_eq] at bad | c : β
d : β
instβ : Fact (2 β€ d)
bad : Β¬InjOn (bottcher d) (multibrotExt d)
β’ False | c : β
d : β
instβ : Fact (2 β€ d)
bad : β x, β (_ : x β multibrotExt d), β x_1, β (_ : x_1 β multibrotExt d) (_ : bottcher d x = bottcher d x_1), x β x_1
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
bad : Β¬InjOn (bottcher d) (multibrotExt d)
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rcases bad with β¨x, xm, y, ym, bxy, xyβ© | c : β
d : β
instβ : Fact (2 β€ d)
bad : β x, β (_ : x β multibrotExt d), β x_1, β (_ : x_1 β multibrotExt d) (_ : bottcher d x = bottcher d x_1), x β x_1
β’ False | case intro.intro.intro.intro.intro
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
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
bad : β x, β (_ : x β multibrotExt d), β x_1, β (_ : x_1 β multibrotExt d) (_ : bottcher d x = bottcher d x_1), x β x_1
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | generalize hb : potential d x = b | case intro.intro.intro.intro.intro
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
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
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
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have b1 : b < 1 := by rwa [β hb, potential_lt_one] | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
β’ False | case intro.intro.intro.intro.intro
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
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | set u := {c | potential d c β€ b} | case intro.intro.intro.intro.intro
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
β’ False | case intro.intro.intro.intro.intro
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}
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
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
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | set t0 := u ΓΛ’ u | case intro.intro.intro.intro.intro
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}
β’ False | case intro.intro.intro.intro.intro
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
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
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}
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | set t1 := {q : π Γ π | bottcher d q.1 = bottcher d q.2 β§ q β t0} | case intro.intro.intro.intro.intro
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
β’ False | case intro.intro.intro.intro.intro
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}
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
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
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | set t2 := {q : π Γ π | q.1 β q.2 β§ q β t1} | case intro.intro.intro.intro.intro
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}
β’ False | case intro.intro.intro.intro.intro
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’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
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}
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | clear x xm y ym bxy xy hb | case intro.intro.intro.intro.intro
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}
t2ne : t2.Nonempty
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
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}
t2ne : t2.Nonempty
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have ue : u β multibrotExt d := by intro c m; rw [β potential_lt_one]; exact lt_of_le_of_lt m b1 | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have t01 : t1 β t0 := inter_subset_right _ _ | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have t12 : t2 β t1 := inter_subset_right _ _ | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have uc : IsClosed u := isClosed_le potential_continuous continuous_const | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have t0c : IsClosed t0 := uc.prod uc | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have t12' : closure t2 β t1 := by rw [β t1c.closure_eq]; exact closure_mono t12 | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ 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’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have t2c' : IsCompact (closure t2) := isClosed_closure.isCompact | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ 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
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have t2ne' : (closure t2).Nonempty := t2ne.closure | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : IsClosed u
t0c : IsClosed t0
t1c : IsClosed t1
t12' : closure t2 β t1
t2c' : IsCompact (closure t2)
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 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
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ 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)
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have pc : Continuous fun q : π Γ π β¦ potential d q.1 := potential_continuous.comp continuous_fst | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 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
β’ False | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 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
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ 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
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rcases t2c'.exists_isMinOn t2ne' pc.continuousOn with β¨β¨x, yβ©, m2, minβ© | case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 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
β’ False | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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
min : IsMinOn (fun q => potential d q.1) (closure t2) (x, y)
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ 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
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | simp only [isMinOn_iff] at min | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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
min : IsMinOn (fun q => potential d q.1) (closure t2) (x, y)
β’ False | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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
min : β x_1 β closure t2, potential d x β€ potential d x_1.1
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 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
min : IsMinOn (fun q => potential d q.1) (closure t2) (x, y)
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | generalize xp : potential d x = p | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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
min : β x_1 β closure t2, potential d x β€ potential d x_1.1
β’ False | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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
min : β x_1 β closure t2, potential d x β€ potential d x_1.1
p : β
xp : potential d x = p
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 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
min : β x_1 β closure t2, potential d x β€ potential d x_1.1
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [xp] at min | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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
min : β x_1 β closure t2, potential d x β€ potential d x_1.1
p : β
xp : potential d x = p
β’ False | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 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
min : β x_1 β closure t2, potential d x β€ potential d x_1.1
p : β
xp : potential d x = p
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have m1 := t12' m2 | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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
β’ False | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 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
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have pb : p β€ b := by rw [β xp]; exact m1.2.1 | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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’ False | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 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’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have xm : x β multibrotExt d := ue m1.2.1 | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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
β’ False | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 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
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have ym : y β multibrotExt d := ue m1.2.2 | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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
β’ False | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 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
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have yp : potential d y = p := by rw [β abs_bottcher, β m1.1, abs_bottcher, xp] | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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’ False | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 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’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have p0i : p = 0 β x = β β§ y = β := by intro p0; rw [p0, potential_eq_zero] at xp yp; use xp, yp | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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’ False | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 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’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | by_cases xy : x β y | case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β 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
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
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 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¬x β y
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.mk.intro
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 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
β’ False
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 | 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 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¬x β y
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 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 = y
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 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¬x β y
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rw [β xy] at m1 m2 p0i | case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : 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 = y
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 y : 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
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ x = β
xy : x = y
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 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 = y
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | clear xy ym yp y | case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : 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, 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
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ x = β
xy : x = y
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
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 y : 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
ym : y β multibrotExt d
yp : potential d y = p
p0i : p = 0 β x = β β§ x = β
xy : x = y
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | have db : mfderiv I I (bottcher d) x = 0 := by
contrapose m2; simp only [mem_closure_iff_frequently, Filter.not_frequently]
refine ((bottcherHolomorphic d _ xm).local_inj m2).mp (eventually_of_forall ?_)
intro β¨x, yβ© inj β¨xy, e, _β©; simp only at xy e inj; exact xy (inj e) | case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : 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’ 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
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
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | by_cases p0 : p β 0 | case neg
c : β
d : β
instβ : Fact (2 β€ d)
b : β
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
t2ne : t2.Nonempty
ue : u β multibrotExt d
t01 : t1 β t0
t12 : t2 β t1
uc : 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
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 : 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 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 | 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
β’ False
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | rwa [β hb, potential_lt_one] | c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
β’ b < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
β’ b < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Isomorphism.lean | bottcher_inj | [51, 1] | [142, 42] | refine β¨β¨x, yβ©, xy, bxy, ?_, ?_β© | c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.1 = bottcher d q.2 β§ q β t0}
t2 : Set (π Γ π) := {q | q.1 β q.2 β§ q β t1}
β’ t2.Nonempty | 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
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 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
d : β
instβ : Fact (2 β€ d)
x : π
xm : x β multibrotExt d
y : π
ym : y β multibrotExt d
bxy : bottcher d x = bottcher d y
xy : x β y
b : β
hb : potential d x = b
b1 : b < 1
u : Set π := {c | potential d c β€ b}
t0 : Set (π Γ π) := u ΓΛ’ u
t1 : Set (π Γ π) := {q | bottcher d q.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’ t2.Nonempty
TACTIC:
|
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