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 | 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/Hartogs/FubiniBall.lean | realCircleMap_eq_circleMap | [35, 1] | [39, 66] | simp only [realCircleMap, circleMap, Complex.equivRealProd_apply, Complex.add_re, Complex.mul_re,
Complex.ofReal_re, Complex.exp_ofReal_mul_I_re, Complex.ofReal_im, Complex.exp_ofReal_mul_I_im,
zero_mul, sub_zero, Complex.add_im, Complex.mul_im, add_zero] | c : β
x : β Γ β
β’ realCircleMap c x = Complex.equivRealProd (circleMap c x.1 x.2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
x : β Γ β
β’ realCircleMap c x = Complex.equivRealProd (circleMap c x.1 x.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | realCircleMap.fderiv | [50, 1] | [54, 73] | simp_rw [realCircleMap] | c : β
x : β Γ β
β’ HasFDerivAt (fun x => realCircleMap c x) (rcmDeriv x) x | c : β
x : β Γ β
β’ HasFDerivAt (fun x => (c.re + x.1 * x.2.cos, c.im + x.1 * x.2.sin)) (rcmDeriv x) x | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
x : β Γ β
β’ HasFDerivAt (fun x => realCircleMap c x) (rcmDeriv x) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | realCircleMap.fderiv | [50, 1] | [54, 73] | apply_rules [hasFDerivAt_const, hasFDerivAt_fst, hasFDerivAt_snd, HasFDerivAt.cos,
HasFDerivAt.sin, HasFDerivAt.add, HasFDerivAt.mul, HasFDerivAt.prod] | c : β
x : β Γ β
β’ HasFDerivAt (fun x => (c.re + x.1 * x.2.cos, c.im + x.1 * x.2.sin)) (rcmDeriv x) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
x : β Γ β
β’ HasFDerivAt (fun x => (c.re + x.1 * x.2.cos, c.im + x.1 * x.2.sin)) (rcmDeriv x) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm00 | [59, 1] | [60, 53] | simp [rcmMatrix, rcmDeriv, toMatrix_apply, d1, d2] | x : β Γ β
β’ rcmMatrix x 0 0 = x.2.cos | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β Γ β
β’ rcmMatrix x 0 0 = x.2.cos
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm01 | [61, 1] | [62, 53] | simp [rcmMatrix, rcmDeriv, toMatrix_apply, d1, d2] | x : β Γ β
β’ rcmMatrix x 0 1 = -x.1 * x.2.sin | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β Γ β
β’ rcmMatrix x 0 1 = -x.1 * x.2.sin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm10 | [63, 1] | [64, 53] | simp [rcmMatrix, rcmDeriv, toMatrix_apply, d1, d2] | x : β Γ β
β’ rcmMatrix x 1 0 = x.2.sin | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β Γ β
β’ rcmMatrix x 1 0 = x.2.sin
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcm11 | [65, 1] | [66, 53] | simp [rcmMatrix, rcmDeriv, toMatrix_apply, d1, d2] | x : β Γ β
β’ rcmMatrix x 1 1 = x.1 * x.2.cos | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β Γ β
β’ rcmMatrix x 1 1 = x.1 * x.2.cos
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcmDeriv.det | [68, 1] | [74, 62] | rw [ContinuousLinearMap.det, β LinearMap.det_toMatrix (Basis.finTwoProd β), βrcmMatrix] | x : β Γ β
β’ (rcmDeriv x).det = x.1 | x : β Γ β
β’ (rcmMatrix x).det = x.1 | Please generate a tactic in lean4 to solve the state.
STATE:
x : β Γ β
β’ (rcmDeriv x).det = x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcmDeriv.det | [68, 1] | [74, 62] | rw [Matrix.det_fin_two, rcm00, rcm01, rcm10, rcm11] | x : β Γ β
β’ (rcmMatrix x).det = x.1 | x : β Γ β
β’ x.2.cos * (x.1 * x.2.cos) - -x.1 * x.2.sin * x.2.sin = x.1 | Please generate a tactic in lean4 to solve the state.
STATE:
x : β Γ β
β’ (rcmMatrix x).det = x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcmDeriv.det | [68, 1] | [74, 62] | ring_nf | x : β Γ β
β’ x.2.cos * (x.1 * x.2.cos) - -x.1 * x.2.sin * x.2.sin = x.1 | x : β Γ β
β’ x.2.cos ^ 2 * x.1 + x.1 * x.2.sin ^ 2 = x.1 | Please generate a tactic in lean4 to solve the state.
STATE:
x : β Γ β
β’ x.2.cos * (x.1 * x.2.cos) - -x.1 * x.2.sin * x.2.sin = x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcmDeriv.det | [68, 1] | [74, 62] | calc cos x.2 ^ 2 * x.1 + x.1 * sin x.2 ^ 2
_ = x.1 * (cos x.2 ^ 2 + sin x.2 ^ 2) := by ring
_ = x.1 := by simp only [Real.cos_sq_add_sin_sq, mul_one] | x : β Γ β
β’ x.2.cos ^ 2 * x.1 + x.1 * x.2.sin ^ 2 = x.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β Γ β
β’ x.2.cos ^ 2 * x.1 + x.1 * x.2.sin ^ 2 = x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcmDeriv.det | [68, 1] | [74, 62] | ring | x : β Γ β
β’ x.2.cos ^ 2 * x.1 + x.1 * x.2.sin ^ 2 = x.1 * (x.2.cos ^ 2 + x.2.sin ^ 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β Γ β
β’ x.2.cos ^ 2 * x.1 + x.1 * x.2.sin ^ 2 = x.1 * (x.2.cos ^ 2 + x.2.sin ^ 2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | rcmDeriv.det | [68, 1] | [74, 62] | simp only [Real.cos_sq_add_sin_sq, mul_one] | x : β Γ β
β’ x.1 * (x.2.cos ^ 2 + x.2.sin ^ 2) = x.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β Γ β
β’ x.1 * (x.2.cos ^ 2 + x.2.sin ^ 2) = x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | constructor | S : Type
instβ : RCLike S
c : S
r : β
β’ sphere c r = β
β r < 0 | case mp
S : Type
instβ : RCLike S
c : S
r : β
β’ sphere c r = β
β r < 0
case mpr
S : Type
instβ : RCLike S
c : S
r : β
β’ r < 0 β sphere c r = β
| Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
instβ : RCLike S
c : S
r : β
β’ sphere c r = β
β r < 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | intro rp | case mp
S : Type
instβ : RCLike S
c : S
r : β
β’ sphere c r = β
β r < 0 | case mp
S : Type
instβ : RCLike S
c : S
r : β
rp : sphere c r = β
β’ r < 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
S : Type
instβ : RCLike S
c : S
r : β
β’ sphere c r = β
β r < 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | contrapose rp | case mp
S : Type
instβ : RCLike S
c : S
r : β
rp : sphere c r = β
β’ r < 0 | case mp
S : Type
instβ : RCLike S
c : S
r : β
rp : Β¬r < 0
β’ Β¬sphere c r = β
| Please generate a tactic in lean4 to solve the state.
STATE:
case mp
S : Type
instβ : RCLike S
c : S
r : β
rp : sphere c r = β
β’ r < 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | simp at rp | case mp
S : Type
instβ : RCLike S
c : S
r : β
rp : Β¬r < 0
β’ Β¬sphere c r = β
| case mp
S : Type
instβ : RCLike S
c : S
r : β
rp : 0 β€ r
β’ Β¬sphere c r = β
| Please generate a tactic in lean4 to solve the state.
STATE:
case mp
S : Type
instβ : RCLike S
c : S
r : β
rp : Β¬r < 0
β’ Β¬sphere c r = β
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | refine Nonempty.ne_empty β¨c + r, ?_β© | case mp
S : Type
instβ : RCLike S
c : S
r : β
rp : 0 β€ r
β’ Β¬sphere c r = β
| case mp
S : Type
instβ : RCLike S
c : S
r : β
rp : 0 β€ r
β’ c + βr β sphere c r | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
S : Type
instβ : RCLike S
c : S
r : β
rp : 0 β€ r
β’ Β¬sphere c r = β
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | simpa only [mem_sphere_iff_norm, add_sub_cancel_left, RCLike.norm_ofReal, abs_eq_self] | case mp
S : Type
instβ : RCLike S
c : S
r : β
rp : 0 β€ r
β’ c + βr β sphere c r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
S : Type
instβ : RCLike S
c : S
r : β
rp : 0 β€ r
β’ c + βr β sphere c r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | intro n | case mpr
S : Type
instβ : RCLike S
c : S
r : β
β’ r < 0 β sphere c r = β
| case mpr
S : Type
instβ : RCLike S
c : S
r : β
n : r < 0
β’ sphere c r = β
| Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
S : Type
instβ : RCLike S
c : S
r : β
β’ r < 0 β sphere c r = β
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | contrapose n | case mpr
S : Type
instβ : RCLike S
c : S
r : β
n : r < 0
β’ sphere c r = β
| case mpr
S : Type
instβ : RCLike S
c : S
r : β
n : Β¬sphere c r = β
β’ Β¬r < 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
S : Type
instβ : RCLike S
c : S
r : β
n : r < 0
β’ sphere c r = β
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | rw [β not_nonempty_iff_eq_empty] at n | case mpr
S : Type
instβ : RCLike S
c : S
r : β
n : Β¬sphere c r = β
β’ Β¬r < 0 | case mpr
S : Type
instβ : RCLike S
c : S
r : β
n : ¬¬(sphere c r).Nonempty
β’ Β¬r < 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
S : Type
instβ : RCLike S
c : S
r : β
n : Β¬sphere c r = β
β’ Β¬r < 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Metric.sphere_eq_empty | [80, 1] | [87, 69] | simpa only [not_lt, NormedSpace.sphere_nonempty, not_le] using n | case mpr
S : Type
instβ : RCLike S
c : S
r : β
n : ¬¬(sphere c r).Nonempty
β’ Β¬r < 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
S : Type
instβ : RCLike S
c : S
r : β
n : ¬¬(sphere c r).Nonempty
β’ Β¬r < 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | by_cases rp : r < 0 | c z : β
r : β
zs : z β sphere c r
β’ β t β Ioc 0 (2 * Ο), z = circleMap c r t | case pos
c z : β
r : β
zs : z β sphere c r
rp : r < 0
β’ β t β Ioc 0 (2 * Ο), z = circleMap c r t
case neg
c z : β
r : β
zs : z β sphere c r
rp : Β¬r < 0
β’ β t β Ioc 0 (2 * Ο), z = circleMap c r t | Please generate a tactic in lean4 to solve the state.
STATE:
c z : β
r : β
zs : z β sphere c r
β’ β t β Ioc 0 (2 * Ο), z = circleMap c r t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | simp only [not_lt] at rp | case neg
c z : β
r : β
zs : z β sphere c r
rp : Β¬r < 0
β’ β t β Ioc 0 (2 * Ο), z = circleMap c r t | case neg
c z : β
r : β
zs : z β sphere c r
rp : 0 β€ r
β’ β t β Ioc 0 (2 * Ο), z = circleMap c r t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c z : β
r : β
zs : z β sphere c r
rp : Β¬r < 0
β’ β t β Ioc 0 (2 * Ο), z = circleMap c r t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | rw [βabs_of_nonneg rp, β range_circleMap, mem_range] at zs | case neg
c z : β
r : β
zs : z β sphere c r
rp : 0 β€ r
β’ β t β Ioc 0 (2 * Ο), z = circleMap c r t | case neg
c z : β
r : β
zs : β y, circleMap c r y = z
rp : 0 β€ r
β’ β t β Ioc 0 (2 * Ο), z = circleMap c r t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c z : β
r : β
zs : z β sphere c r
rp : 0 β€ r
β’ β t β Ioc 0 (2 * Ο), z = circleMap c r t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | rcases zs with β¨t, htβ© | case neg
c z : β
r : β
zs : β y, circleMap c r y = z
rp : 0 β€ r
β’ β t β Ioc 0 (2 * Ο), z = circleMap c r t | case neg.intro
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
β’ β t β Ioc 0 (2 * Ο), z = circleMap c r t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c z : β
r : β
zs : β y, circleMap c r y = z
rp : 0 β€ r
β’ β t β Ioc 0 (2 * Ο), z = circleMap c r t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | generalize ha : 2 * Ο = a | case neg.intro
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
β’ β t β Ioc 0 (2 * Ο), z = circleMap c r t | case neg.intro
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
β’ β t β Ioc 0 a, z = circleMap c r t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
β’ β t β Ioc 0 (2 * Ο), z = circleMap c r t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | have ap : a > 0 := by rw [βha]; bound | case neg.intro
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
β’ β t β Ioc 0 a, z = circleMap c r t | case neg.intro
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
β’ β t β Ioc 0 a, z = circleMap c r t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
β’ β t β Ioc 0 a, z = circleMap c r t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | generalize hs : t + a - a * βt / aβ = s | case neg.intro
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
β’ β t β Ioc 0 a, z = circleMap c r t | case neg.intro
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ β t β Ioc 0 a, z = circleMap c r t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
β’ β t β Ioc 0 a, z = circleMap c r t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | use s | case neg.intro
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ β t β Ioc 0 a, z = circleMap c r t | case h
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ s β Ioc 0 a β§ z = circleMap c r s | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ β t β Ioc 0 a, z = circleMap c r t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | constructor | case h
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ s β Ioc 0 a β§ z = circleMap c r s | case h.left
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ s β Ioc 0 a
case h.right
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ z = circleMap c r s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ s β Ioc 0 a β§ z = circleMap c r s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | simp only [Metric.sphere_eq_empty.mpr rp, mem_empty_iff_false] at zs | case pos
c z : β
r : β
zs : z β sphere c r
rp : r < 0
β’ β t β Ioc 0 (2 * Ο), z = circleMap c r t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c z : β
r : β
zs : z β sphere c r
rp : r < 0
β’ β t β Ioc 0 (2 * Ο), z = circleMap c r t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | rw [βha] | c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
β’ a > 0 | c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
β’ 2 * Ο > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
β’ a > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | bound | c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
β’ 2 * Ο > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
β’ 2 * Ο > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | simp only [mem_Ioc, sub_pos, tsub_le_iff_right, β hs] | case h.left
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ s β Ioc 0 a | case h.left
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ a * ββt / aβ < t + a β§ t + a β€ a + a * ββt / aβ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ s β Ioc 0 a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | constructor | case h.left
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ a * ββt / aβ < t + a β§ t + a β€ a + a * ββt / aβ | case h.left.left
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ a * ββt / aβ < t + a
case h.left.right
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ t + a β€ a + a * ββt / aβ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ a * ββt / aβ < t + a β§ t + a β€ a + a * ββt / aβ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | calc a * βt / aβ
_ < a * (t / a + 1) := by bound
_ = a / a * t + a := by ring
_ = t + a := by field_simp [ap.ne'] | case h.left.left
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ a * ββt / aβ < t + a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.left
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ a * ββt / aβ < t + a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | bound | c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ a * ββt / aβ < a * (t / a + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ a * ββt / aβ < a * (t / a + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | ring | c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ a * (t / a + 1) = a / a * t + a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ a * (t / a + 1) = a / a * t + a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | field_simp [ap.ne'] | c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ a / a * t + a = t + a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ a / a * t + a = t + a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | calc a + a * βt / aβ
_ β₯ a + a * (t / a) := by bound
_ = a / a * t + a := by ring
_ = t + a := by field_simp [ap.ne'] | case h.left.right
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ t + a β€ a + a * ββt / aβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.right
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ t + a β€ a + a * ββt / aβ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | bound | c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ a + a * ββt / aβ β₯ a + a * (t / a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ a + a * ββt / aβ β₯ a + a * (t / a)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | ring | c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ a + a * (t / a) = a / a * t + a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ a + a * (t / a) = a / a * t + a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | simp only [βht, circleMap, Complex.ofReal_sub, Complex.ofReal_add, Complex.ofReal_mul,
Complex.ofReal_intCast, add_right_inj, mul_eq_mul_left_iff, Complex.ofReal_eq_zero, β hs] | case h.right
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ z = circleMap c r s | case h.right
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ (βt * I).exp = ((βt + βa - βa * ββt / aβ) * I).exp β¨ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ z = circleMap c r s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | rw [mul_sub_right_distrib, right_distrib, Complex.exp_sub, Complex.exp_add] | case h.right
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ (βt * I).exp = ((βt + βa - βa * ββt / aβ) * I).exp β¨ r = 0 | case h.right
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ (βt * I).exp = (βt * I).exp * (βa * I).exp / (βa * ββt / aβ * I).exp β¨ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ (βt * I).exp = ((βt + βa - βa * ββt / aβ) * I).exp β¨ r = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | rw [mul_comm _ (β_β:β), mul_assoc, Complex.exp_int_mul, β ha] | case h.right
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ (βt * I).exp = (βt * I).exp * (βa * I).exp / (βa * ββt / aβ * I).exp β¨ r = 0 | case h.right
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ (βt * I).exp = (βt * I).exp * (β(2 * Ο) * I).exp / (β(2 * Ο) * I).exp ^ βt / (2 * Ο)β β¨ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ (βt * I).exp = (βt * I).exp * (βa * I).exp / (βa * ββt / aβ * I).exp β¨ r = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | circleMap_Ioc | [90, 1] | [116, 34] | simp only [Complex.ofReal_mul, Complex.ofReal_ofNat, Complex.exp_two_pi_mul_I, mul_one,
one_zpow, div_one, true_or] | case h.right
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ (βt * I).exp = (βt * I).exp * (β(2 * Ο) * I).exp / (β(2 * Ο) * I).exp ^ βt / (2 * Ο)β β¨ r = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
c z : β
r : β
rp : 0 β€ r
t : β
ht : circleMap c r t = z
a : β
ha : 2 * Ο = a
ap : a > 0
s : β
hs : t + a - a * ββt / aβ = s
β’ (βt * I).exp = (βt * I).exp * (β(2 * Ο) * I).exp / (β(2 * Ο) * I).exp ^ βt / (2 * Ο)β β¨ r = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square.rp | [124, 1] | [127, 26] | simp only [square, gt_iff_lt, not_lt, ge_iff_le, zero_lt_two, mul_pos_iff_of_pos_left,
mem_prod, mem_Ioc, and_imp] | r0 r1 : β
x : β Γ β
r0p : 0 β€ r0
β’ x β square r0 r1 β 0 < x.1 | r0 r1 : β
x : β Γ β
r0p : 0 β€ r0
β’ r0 < x.1 β x.1 β€ r1 β 0 < x.2 β x.2 β€ 2 * Ο β 0 < x.1 | Please generate a tactic in lean4 to solve the state.
STATE:
r0 r1 : β
x : β Γ β
r0p : 0 β€ r0
β’ x β square r0 r1 β 0 < x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square.rp | [124, 1] | [127, 26] | intro h _ _ _ | r0 r1 : β
x : β Γ β
r0p : 0 β€ r0
β’ r0 < x.1 β x.1 β€ r1 β 0 < x.2 β x.2 β€ 2 * Ο β 0 < x.1 | r0 r1 : β
x : β Γ β
r0p : 0 β€ r0
h : r0 < x.1
aβΒ² : x.1 β€ r1
aβΒΉ : 0 < x.2
aβ : x.2 β€ 2 * Ο
β’ 0 < x.1 | Please generate a tactic in lean4 to solve the state.
STATE:
r0 r1 : β
x : β Γ β
r0p : 0 β€ r0
β’ r0 < x.1 β x.1 β€ r1 β 0 < x.2 β x.2 β€ 2 * Ο β 0 < x.1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | square.rp | [124, 1] | [127, 26] | linarith | r0 r1 : β
x : β Γ β
r0p : 0 β€ r0
h : r0 < x.1
aβΒ² : x.1 β€ r1
aβΒΉ : 0 < x.2
aβ : x.2 β€ 2 * Ο
β’ 0 < x.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
r0 r1 : β
x : β Γ β
r0p : 0 β€ r0
h : r0 < x.1
aβΒ² : x.1 β€ r1
aβΒΉ : 0 < x.2
aβ : x.2 β€ 2 * Ο
β’ 0 < x.1
TACTIC:
|
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