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/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | simp only at o | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → ... | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | rw [nhds_prod_eq, o] | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → ... | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | simp only [Filter.prod_map_map_eq, Filter.eventually_map] | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → ... | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | refine (fg.prod_mk fg).mp (eventually_of_forall ?_) | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → ... | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | intro ⟨x, y⟩ ⟨ex, ey⟩ h1 h2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → ... | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | simp only at h1 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → ... | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | simp only [h1] at ex ey h2 ⊢ | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → ... | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | simp only [ex, ey] at h2 | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → ... | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | simp only [h2] | case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | have e : (c, z) = (c, g c (f c z)) := by rw [gf.self_of_nhds] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | rw [e] at fa | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : Holomo... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | refine (NontrivialHolomorphicAt.anti ?_ fa.along_snd ga.along_snd).2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : Holomo... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : Holomo... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : Holomo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | refine (nontrivialHolomorphicAt_id _).congr ?_ | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : Holomo... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : Holomo... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : Holomo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | refine ((continuousAt_const.prod continuousAt_id).eventually fg).mp (eventually_of_forall ?_) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : Holomo... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : Holomo... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : Holomo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | exact fun _ e ↦ e.symm | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, g c (f c z))
ga : Holomo... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
fa : Holomo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj'' | [39, 1] | [54, 90] | rw [gf.self_of_nhds] | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ → T → S
ga : HolomorphicAt (I... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | set g : ℂ × S × S → (ℂ × S) × ℂ × S := fun p ↦ ((p.1, p.2.1), (p.1, p.2.2)) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
⊢ ∀ᶠ (p : ℂ × S × S) in 𝓝 (c, z, z... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := ... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | refine (t.eventually (fa.local_inj'' nc)).mp (eventually_of_forall ?_) | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := ... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := ... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | intro ⟨e, x, y⟩ inj fe | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := ... | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := ... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | exact (Prod.ext_iff.mp (inj rfl fe)).2 | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | apply Continuous.continuousAt | S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ × S := ... | case h
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ ... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | apply Continuous.prod_mk | case h
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) × ℂ ... | case h.hf
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) ×... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c,... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | exact continuous_fst.prod_mk (continuous_fst.comp continuous_snd) | case h.hf
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) ×... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hf
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/AnalyticManifold/LocalInj.lean | HolomorphicAt.local_inj' | [58, 1] | [67, 65] | exact continuous_fst.prod_mk (continuous_snd.comp continuous_snd) | case h.hg
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) (c, z)
nc : mfderiv I I (f c) z ≠ 0
g : ℂ × S × S → (ℂ × S) ×... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hg
S : Type
inst✝³ : TopologicalSpace S
inst✝² : ChartedSpace ℂ S
cms : AnalyticManifold I S
T : Type
inst✝¹ : TopologicalSpace T
inst✝ : ChartedSpace ℂ T
cmt : AnalyticManifold I T
f : ℂ → S → T
c : ℂ
z : S
fa : HolomorphicAt (I.prod I) I (uncurry f) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_slice | [51, 1] | [53, 96] | apply Set.ext | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
c : ℂ
⊢ {x | (c, x) ∈ s.ext} = ball 0 (s.p c) | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
c : ℂ
⊢ ∀ (x : ℂ), x ∈ {x | (c, x) ∈ s.ext} ↔ x ∈ ball 0 (s.p c) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
c : ℂ
⊢ {x | (c, x) ∈ s.ext} = ball 0 (s.p... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_slice | [51, 1] | [53, 96] | intro x | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
c : ℂ
⊢ ∀ (x : ℂ), x ∈ {x | (c, x) ∈ s.ext} ↔ x ∈ ball 0 (s.p c) | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
c x : ℂ
⊢ x ∈ {x | (c, x) ∈ s.ext} ↔ x ∈ ball 0 (s.p c) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
c : ℂ
⊢ ∀ (x : ℂ), x ∈ {x | (c, x) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_slice | [51, 1] | [53, 96] | simp only [Super.ext, mem_ball, mem_setOf, Complex.dist_eq, sub_zero] | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
c x : ℂ
⊢ x ∈ {x | (c, x) ∈ s.ext} ↔ x ∈ ball 0 (s.p c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
c x : ℂ
⊢ x ∈ {x | (c, x) ∈ s.ext}... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.isOpen_ext | [56, 1] | [63, 37] | set f := fun y : ℂ × ℂ ↦ s.p y.1 - abs y.2 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ IsOpen s.ext | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f✝ : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f✝ d a
y : ℂ × ℂ
s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × ℂ → ℝ := fun y => s.p y.1 - Complex.abs y.2
⊢ IsOpen s.ext | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ IsOpen s.ext
TACTI... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.isOpen_ext | [56, 1] | [63, 37] | have fc : LowerSemicontinuous f :=
(s.lowerSemicontinuous_p.comp continuous_fst).add
(Complex.continuous_abs.comp continuous_snd).neg.lowerSemicontinuous | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f✝ : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f✝ d a
y : ℂ × ℂ
s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × ℂ → ℝ := fun y => s.p y.1 - Complex.abs y.2
⊢ IsOpen s.ext | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f✝ : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f✝ d a
y : ℂ × ℂ
s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × ℂ → ℝ := fun y => s.p y.1 - Complex.abs y.2
fc : LowerSemicontinuous f... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f✝ : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f✝ d a
y : ℂ × ℂ
s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × ℂ → ℝ := ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.isOpen_ext | [56, 1] | [63, 37] | have e : s.ext = f ⁻¹' Ioi 0 :=
Set.ext fun _ ↦ by simp only [Super.ext, mem_setOf, mem_preimage, mem_Ioi, sub_pos, f] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f✝ : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f✝ d a
y : ℂ × ℂ
s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × ℂ → ℝ := fun y => s.p y.1 - Complex.abs y.2
fc : LowerSemicontinuous f... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f✝ : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f✝ d a
y : ℂ × ℂ
s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × ℂ → ℝ := fun y => s.p y.1 - Complex.abs y.2
fc : LowerSemicontinuous f... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f✝ : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f✝ d a
y : ℂ × ℂ
s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × ℂ → ℝ := ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.isOpen_ext | [56, 1] | [63, 37] | rw [e] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f✝ : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f✝ d a
y : ℂ × ℂ
s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × ℂ → ℝ := fun y => s.p y.1 - Complex.abs y.2
fc : LowerSemicontinuous f... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f✝ : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f✝ d a
y : ℂ × ℂ
s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × ℂ → ℝ := fun y => s.p y.1 - Complex.abs y.2
fc : LowerSemicontinuous f... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f✝ : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f✝ d a
y : ℂ × ℂ
s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × ℂ → ℝ := ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.isOpen_ext | [56, 1] | [63, 37] | exact fc.isOpen_preimage _ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f✝ : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f✝ d a
y : ℂ × ℂ
s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × ℂ → ℝ := fun y => s.p y.1 - Complex.abs y.2
fc : LowerSemicontinuous f... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f✝ : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f✝ d a
y : ℂ × ℂ
s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × ℂ → ℝ := ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.isOpen_ext | [56, 1] | [63, 37] | simp only [Super.ext, mem_setOf, mem_preimage, mem_Ioi, sub_pos, f] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f✝ : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f✝ d a
y : ℂ × ℂ
s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × ℂ → ℝ := fun y => s.p y.1 - Complex.abs y.2
fc : LowerSemicontinuous f... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f✝ : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f✝ d a
y : ℂ × ℂ
s : Super f✝ d a
inst✝ : OnePreimage s
f : ℂ × ℂ → ℝ := ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.mem_ext | [66, 1] | [67, 68] | simp only [Super.ext, mem_setOf, Complex.abs.map_zero, s.p_pos c] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ (c, 0) ∈ s.ext | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ (c, 0) ∈ s.... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_slice_connected | [70, 1] | [73, 93] | rw [s.ext_slice c] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ IsConnected {x | (c, x) ∈ s.ext} | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ IsConnected (ball 0 (s.p c)) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ IsConnected... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_slice_connected | [70, 1] | [73, 93] | exact ⟨⟨(0 : ℂ), mem_ball_self (s.p_pos c)⟩, (convex_ball (0 : ℂ) (s.p c)).isPreconnected⟩ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ IsConnected (ball 0 (s.p c)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ IsConnected... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | refine ⟨⟨(0, 0), s.mem_ext 0⟩, isPreconnected_of_forall (0, 0) ?_⟩ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ IsConnected s.ext | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ ∀ y ∈ s.ext, ∃ t ⊆ s.ext, (0, 0) ∈ t ∧ y ∈ t ∧ IsPreconnected t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ IsConnected s.ext
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | intro ⟨c, x⟩ m | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ ∀ y ∈ s.ext, ∃ t ⊆ s.ext, (0, 0) ∈ t ∧ y ∈ t ∧ IsPreconnected t | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ ∃ t ⊆ s.ext, (0, 0) ∈ t ∧ (c, x) ∈ t ∧ IsPreconnec... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ ∀ y ∈ s.ext, ∃ t ⊆... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | use(fun x ↦ (c, x)) '' {x | (c, x) ∈ s.ext} ∪ univ ×ˢ {0} | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ ∃ t ⊆ s.ext, (0, 0) ∈ t ∧ (c, x) ∈ t ∧ IsPreconnec... | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ (fun x => (c, x)) '' {x | (c, x) ∈ s.ext} ∪... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | simp only [mem_image, mem_union, union_subset_iff, mem_setOf, mem_prod_eq, mem_univ,
true_and_iff, mem_singleton_iff, eq_self_iff_true, or_true_iff] | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ (fun x => (c, x)) '' {x | (c, x) ∈ s.ext} ∪... | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ ((fun x => (c, x)) '' {x | (c, x) ∈ s.ext} ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | refine ⟨⟨?_, ?_⟩, ?_, ?_⟩ | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ ((fun x => (c, x)) '' {x | (c, x) ∈ s.ext} ... | case h.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ (fun x => (c, x)) '' {x | (c, x) ∈... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | intro y n | case h.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ (fun x => (c, x)) '' {x | (c, x) ∈... | case h.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y✝ : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
y : ℂ × ℂ
n : y ∈ (fun x => (c, x)... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | simp only [mem_image, mem_setOf] at n | case h.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y✝ : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
y : ℂ × ℂ
n : y ∈ (fun x => (c, x)... | case h.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y✝ : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
y : ℂ × ℂ
n : ∃ x, (c, x) ∈ s.ext ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y✝ : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | rcases n with ⟨x, m, e⟩ | case h.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y✝ : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
y : ℂ × ℂ
n : ∃ x, (c, x) ∈ s.ext ... | case h.refine_1.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝¹ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y✝ : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x✝ : ℂ
m✝ : (c, x✝) ∈ s.ext
y : ℂ × ℂ
x : ℂ
m :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y✝ : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | rw [e] at m | case h.refine_1.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝¹ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y✝ : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x✝ : ℂ
m✝ : (c, x✝) ∈ s.ext
y : ℂ × ℂ
x : ℂ
m :... | case h.refine_1.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝¹ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y✝ : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x✝ : ℂ
m✝ : (c, x✝) ∈ s.ext
y : ℂ × ℂ
x : ℂ
m :... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝¹ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y✝ : ℂ × ℂ
s : Super f d a
inst✝ : On... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | exact m | case h.refine_1.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝¹ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y✝ : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x✝ : ℂ
m✝ : (c, x✝) ∈ s.ext
y : ℂ × ℂ
x : ℂ
m :... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝¹ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y✝ : ℂ × ℂ
s : Super f d a
inst✝ : On... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | intro ⟨c, x⟩ m | case h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ univ ×ˢ {0} ⊆ s.ext | case h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝¹ x✝¹ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c✝ x✝ : ℂ
m✝ : (c✝, x✝) ∈ s.ext
c x : ℂ
m : (c, x) ∈ univ ×ˢ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | simp only [mem_prod_eq, mem_singleton_iff] at m | case h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝¹ x✝¹ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c✝ x✝ : ℂ
m✝ : (c✝, x✝) ∈ s.ext
c x : ℂ
m : (c, x) ∈ univ ×ˢ ... | case h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝¹ x✝¹ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c✝ x✝ : ℂ
m✝ : (c✝, x✝) ∈ s.ext
c x : ℂ
m : c ∈ univ ∧ x = 0
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝¹ x✝¹ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | rw [m.2] | case h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝¹ x✝¹ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c✝ x✝ : ℂ
m✝ : (c✝, x✝) ∈ s.ext
c x : ℂ
m : c ∈ univ ∧ x = 0
... | case h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝¹ x✝¹ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c✝ x✝ : ℂ
m✝ : (c✝, x✝) ∈ s.ext
c x : ℂ
m : c ∈ univ ∧ x = 0
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝¹ x✝¹ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | exact s.mem_ext c | case h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝¹ x✝¹ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c✝ x✝ : ℂ
m✝ : (c✝, x✝) ∈ s.ext
c x : ℂ
m : c ∈ univ ∧ x = 0
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝¹ x✝¹ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | left | case h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ (∃ x_1, (c, x_1) ∈ s.ext ∧ (c, x_1... | case h.refine_3.h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ ∃ x_1, (c, x_1) ∈ s.ext ∧ (c, x_... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | exact ⟨x, m, rfl⟩ | case h.refine_3.h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ ∃ x_1, (c, x_1) ∈ s.ext ∧ (c, x_... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_3.h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | refine IsPreconnected.union (c, 0) ?_ ?_ ?_ ?_ | case h.refine_4
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ IsPreconnected ((fun x => (c, x)) ... | case h.refine_4.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ (c, 0) ∈ (fun x => (c, x)... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_4
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | use 0, s.mem_ext c | case h.refine_4.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ (c, 0) ∈ (fun x => (c, x)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_4.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePrei... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | exact mk_mem_prod (mem_univ _) rfl | case h.refine_4.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ (c, 0) ∈ univ ×ˢ {0} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_4.refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePrei... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | exact IsPreconnected.image (s.ext_slice_connected c).isPreconnected _
(Continuous.Prod.mk _).continuousOn | case h.refine_4.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ IsPreconnected ((fun x =>... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_4.refine_3
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePrei... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ext_connected | [76, 1] | [90, 62] | exact isPreconnected_univ.prod isPreconnected_singleton | case h.refine_4.refine_4
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ IsPreconnected (univ ×ˢ {... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_4.refine_4
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePrei... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_holomorphicOn | [112, 1] | [113, 93] | intro ⟨c, x⟩ m | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ HolomorphicOn (I.prod I) I (uncurry s.ray) s.ext | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ HolomorphicAt (I.prod I) I (uncurry s.ray) (c, x) | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ HolomorphicOn (I.p... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_holomorphicOn | [112, 1] | [113, 93] | exact s.ray_holomorphic m | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x) ∈ s.ext
⊢ HolomorphicAt (I.prod I) I (uncurry s.ray) (c, x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c x : ℂ
m : (c, x)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_post | [147, 1] | [149, 84] | simp only [Super.post, Postcritical, mem_setOf, s.ray_potential post] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
⊢ (c, s.ray c x) ∈ s.post | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
⊢ Complex.abs x < s.p c | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ex... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_post | [147, 1] | [149, 84] | exact post | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
⊢ Complex.abs x < s.p c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ex... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical_zero | [152, 1] | [163, 62] | have h : mfderiv I I (s.bottcherNear c ∘ s.ray c) 0 ≠ 0 := by
have e : s.bottcherNear c ∘ s.ray c =ᶠ[𝓝 0] id :=
(continuousAt_const.prod continuousAt_id).eventually (s.ray_eqn_zero c)
rw [e.mfderiv_eq]; exact id_mderiv_ne_zero | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ mfderiv I I (s.ray c) 0 ≠ 0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : mfderiv I I (s.bottcherNear c ∘ s.ray c) 0 ≠ 0
⊢ mfderiv I I (s.ray c)... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ mfderiv I I... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical_zero | [152, 1] | [163, 62] | contrapose h | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : mfderiv I I (s.bottcherNear c ∘ s.ray c) 0 ≠ 0
⊢ mfderiv I I (s.ray c)... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : ¬mfderiv I I (s.ray c) 0 ≠ 0
⊢ ¬mfderiv I I (s.bottcherNear c ∘ s.ray ... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : mfderiv I... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical_zero | [152, 1] | [163, 62] | simp only [not_not] at h ⊢ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : ¬mfderiv I I (s.ray c) 0 ≠ 0
⊢ ¬mfderiv I I (s.bottcherNear c ∘ s.ray ... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : mfderiv I I (s.ray c) 0 = 0
⊢ mfderiv I I (s.bottcherNear c ∘ s.ray c)... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : ¬mfderiv ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical_zero | [152, 1] | [163, 62] | have hb : MDifferentiableAt I I (s.bottcherNear c) (s.ray c 0) := by
rw [s.ray_zero]; exact (s.bottcherNear_holomorphic _ (s.mem_near c)).along_snd.mdifferentiableAt | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : mfderiv I I (s.ray c) 0 = 0
⊢ mfderiv I I (s.bottcherNear c ∘ s.ray c)... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : mfderiv I I (s.ray c) 0 = 0
hb : MDifferentiableAt I I (s.bottcherNear... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : mfderiv I... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical_zero | [152, 1] | [163, 62] | have hr : MDifferentiableAt I I (s.ray c) 0 :=
(s.ray_holomorphic (s.mem_ext c)).along_snd.mdifferentiableAt | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : mfderiv I I (s.ray c) 0 = 0
hb : MDifferentiableAt I I (s.bottcherNear... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : mfderiv I I (s.ray c) 0 = 0
hb : MDifferentiableAt I I (s.bottcherNear... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : mfderiv I... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical_zero | [152, 1] | [163, 62] | rw [mfderiv_comp 0 hb hr, h, ContinuousLinearMap.comp_zero] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : mfderiv I I (s.ray c) 0 = 0
hb : MDifferentiableAt I I (s.bottcherNear... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : mfderiv I... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical_zero | [152, 1] | [163, 62] | have e : s.bottcherNear c ∘ s.ray c =ᶠ[𝓝 0] id :=
(continuousAt_const.prod continuousAt_id).eventually (s.ray_eqn_zero c) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ mfderiv I I (s.bottcherNear c ∘ s.ray c) 0 ≠ 0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
e : (𝓝 0).EventuallyEq (s.bottcherNear c ∘ s.ray c) id
⊢ mfderiv I I (s.b... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ mfderiv I I... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical_zero | [152, 1] | [163, 62] | rw [e.mfderiv_eq] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
e : (𝓝 0).EventuallyEq (s.bottcherNear c ∘ s.ray c) id
⊢ mfderiv I I (s.b... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
e : (𝓝 0).EventuallyEq (s.bottcherNear c ∘ s.ray c) id
⊢ mfderiv I I id 0... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
e : (𝓝 0).Ev... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical_zero | [152, 1] | [163, 62] | exact id_mderiv_ne_zero | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
e : (𝓝 0).EventuallyEq (s.bottcherNear c ∘ s.ray c) id
⊢ mfderiv I I id 0... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
e : (𝓝 0).Ev... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical_zero | [152, 1] | [163, 62] | rw [s.ray_zero] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : mfderiv I I (s.ray c) 0 = 0
⊢ MDifferentiableAt I I (s.bottcherNear c)... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : mfderiv I I (s.ray c) 0 = 0
⊢ MDifferentiableAt I I (s.bottcherNear c)... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : mfderiv I... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical_zero | [152, 1] | [163, 62] | exact (s.bottcherNear_holomorphic _ (s.mem_near c)).along_snd.mdifferentiableAt | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : mfderiv I I (s.ray c) 0 = 0
⊢ MDifferentiableAt I I (s.bottcherNear c)... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : mfderiv I... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical | [166, 1] | [185, 12] | by_cases x0 : x = 0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
⊢ mfderiv I I (s.ray c) x ≠ 0 | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
x0 : x = 0
⊢ mfderiv I I (s.ray c) x ≠ 0
case neg... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ex... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical | [166, 1] | [185, 12] | rw [x0] | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
x0 : x = 0
⊢ mfderiv I I (s.ray c) x ≠ 0
case neg... | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
x0 : x = 0
⊢ mfderiv I I (s.ray c) 0 ≠ 0
case neg... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical | [166, 1] | [185, 12] | exact s.ray_noncritical_zero c | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
x0 : x = 0
⊢ mfderiv I I (s.ray c) 0 ≠ 0
case neg... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
x0 : ¬x = 0
⊢ mfderiv I I (s.ray c) x ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical | [166, 1] | [185, 12] | set n := s.np c (abs x) | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
x0 : ¬x = 0
⊢ mfderiv I I (s.ray c) x ≠ 0 | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
x0 : ¬x = 0
n : ℕ := s.np c (Complex.abs x)
⊢ mfd... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical | [166, 1] | [185, 12] | have h : mfderiv I I (s.bottcherNearIter n c ∘ s.ray c) x ≠ 0 := by
have e : s.bottcherNearIter n c ∘ s.ray c =ᶠ[𝓝 x] fun x ↦ x ^ d ^ n :=
(continuousAt_const.prod continuousAt_id).eventually (s.ray_eqn_iter post)
rw [e.mfderiv_eq]; contrapose x0; simp only [not_not] at x0 ⊢
rw [mfderiv_eq_fderiv] at x0
ha... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
x0 : ¬x = 0
n : ℕ := s.np c (Complex.abs x)
⊢ mfd... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
x0 : ¬x = 0
n : ℕ := s.np c (Complex.abs x)
h : m... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c,... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical | [166, 1] | [185, 12] | simp only [mfderiv_comp x
(s.bottcherNearIter_holomorphic (s.ray_near post)).along_snd.mdifferentiableAt
(s.ray_holomorphic post).along_snd.mdifferentiableAt,
Ne, mderiv_comp_eq_zero_iff, not_or] at h | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
x0 : ¬x = 0
n : ℕ := s.np c (Complex.abs x)
h : m... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
x0 : ¬x = 0
n : ℕ := s.np c (Complex.abs x)
h :
... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c,... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical | [166, 1] | [185, 12] | exact h.2 | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
x0 : ¬x = 0
n : ℕ := s.np c (Complex.abs x)
h :
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c,... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical | [166, 1] | [185, 12] | have e : s.bottcherNearIter n c ∘ s.ray c =ᶠ[𝓝 x] fun x ↦ x ^ d ^ n :=
(continuousAt_const.prod continuousAt_id).eventually (s.ray_eqn_iter post) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
x0 : ¬x = 0
n : ℕ := s.np c (Complex.abs x)
⊢ mfderiv I I ... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
x0 : ¬x = 0
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).Eve... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.e... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical | [166, 1] | [185, 12] | rw [e.mfderiv_eq] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
x0 : ¬x = 0
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).Eve... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
x0 : ¬x = 0
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).Eve... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.e... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical | [166, 1] | [185, 12] | contrapose x0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
x0 : ¬x = 0
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).Eve... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).EventuallyEq (s... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.e... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical | [166, 1] | [185, 12] | simp only [not_not] at x0 ⊢ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).EventuallyEq (s... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).EventuallyEq (s... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.e... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical | [166, 1] | [185, 12] | rw [mfderiv_eq_fderiv] at x0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).EventuallyEq (s... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).EventuallyEq (s... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.e... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical | [166, 1] | [185, 12] | have d := (differentiableAt_pow (x := x) (d ^ n)).hasFDerivAt.hasDerivAt.deriv | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).EventuallyEq (s... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d✝ n✝ : ℕ
s✝ : Super f d✝ a
y : ℂ × ℂ
s : Super f d✝ a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).EventuallyEq... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n✝ : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
post : (c, x) ∈ s.e... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical | [166, 1] | [185, 12] | apply_fun (fun x ↦ x 1) at x0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d✝ n✝ : ℕ
s✝ : Super f d✝ a
y : ℂ × ℂ
s : Super f d✝ a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).EventuallyEq... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d✝ n✝ : ℕ
s✝ : Super f d✝ a
y : ℂ × ℂ
s : Super f d✝ a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).EventuallyEq... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d✝ n✝ : ℕ
s✝ : Super f d✝ a
y : ℂ × ℂ
s : Super f d✝ a
inst✝ : OnePreimage s
post : (c, x) ∈ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical | [166, 1] | [185, 12] | rw [x0] at d | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d✝ n✝ : ℕ
s✝ : Super f d✝ a
y : ℂ × ℂ
s : Super f d✝ a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).EventuallyEq... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d✝ n✝ : ℕ
s✝ : Super f d✝ a
y : ℂ × ℂ
s : Super f d✝ a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).EventuallyEq... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d✝ n✝ : ℕ
s✝ : Super f d✝ a
y : ℂ × ℂ
s : Super f d✝ a
inst✝ : OnePreimage s
post : (c, x) ∈ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical | [166, 1] | [185, 12] | replace d := Eq.trans d (ContinuousLinearMap.zero_apply _) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d✝ n✝ : ℕ
s✝ : Super f d✝ a
y : ℂ × ℂ
s : Super f d✝ a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).EventuallyEq... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d✝ n✝ : ℕ
s✝ : Super f d✝ a
y : ℂ × ℂ
s : Super f d✝ a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).EventuallyEq... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d✝ n✝ : ℕ
s✝ : Super f d✝ a
y : ℂ × ℂ
s : Super f d✝ a
inst✝ : OnePreimage s
post : (c, x) ∈ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical | [166, 1] | [185, 12] | rw [deriv_pow, mul_eq_zero, Nat.cast_eq_zero, pow_eq_zero_iff', pow_eq_zero_iff'] at d | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d✝ n✝ : ℕ
s✝ : Super f d✝ a
y : ℂ × ℂ
s : Super f d✝ a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).EventuallyEq... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d✝ n✝ : ℕ
s✝ : Super f d✝ a
y : ℂ × ℂ
s : Super f d✝ a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).EventuallyEq... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d✝ n✝ : ℕ
s✝ : Super f d✝ a
y : ℂ × ℂ
s : Super f d✝ a
inst✝ : OnePreimage s
post : (c, x) ∈ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical | [166, 1] | [185, 12] | simp only [s.d0, false_and_iff, false_or_iff] at d | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d✝ n✝ : ℕ
s✝ : Super f d✝ a
y : ℂ × ℂ
s : Super f d✝ a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).EventuallyEq... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d✝ n✝ : ℕ
s✝ : Super f d✝ a
y : ℂ × ℂ
s : Super f d✝ a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).EventuallyEq... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d✝ n✝ : ℕ
s✝ : Super f d✝ a
y : ℂ × ℂ
s : Super f d✝ a
inst✝ : OnePreimage s
post : (c, x) ∈ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_noncritical | [166, 1] | [185, 12] | exact d.1 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d✝ n✝ : ℕ
s✝ : Super f d✝ a
y : ℂ × ℂ
s : Super f d✝ a
inst✝ : OnePreimage s
post : (c, x) ∈ s.ext
n : ℕ := s.np c (Complex.abs x)
e : (𝓝 x).EventuallyEq... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d✝ n✝ : ℕ
s✝ : Super f d✝ a
y : ℂ × ℂ
s : Super f d✝ a
inst✝ : OnePreimage s
post : (c, x) ∈ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_inj | [203, 1] | [282, 65] | intro p0 p1 e | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
⊢ (c, x0) ∈ s.ext → (c, x1) ∈ s.ext → s.ray c x0 = s.ray c x1 → x0 = x1 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p0 : (c, x0) ∈ s.ext
p1 : (c, x1) ∈ s.ext
e : s.ray c x0 = s.ray c x1
⊢... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
⊢ (c, x0) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_inj | [203, 1] | [282, 65] | have ax : abs x0 = abs x1 := by simp only [← s.ray_potential p0, ← s.ray_potential p1, e] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p0 : (c, x0) ∈ s.ext
p1 : (c, x1) ∈ s.ext
e : s.ray c x0 = s.ray c x1
⊢... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p0 : (c, x0) ∈ s.ext
p1 : (c, x1) ∈ s.ext
e : s.ray c x0 = s.ray c x1
a... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p0 : (c, x... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_inj | [203, 1] | [282, 65] | by_cases x00 : x0 = 0 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p0 : (c, x0) ∈ s.ext
p1 : (c, x1) ∈ s.ext
e : s.ray c x0 = s.ray c x1
a... | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p0 : (c, x0) ∈ s.ext
p1 : (c, x1) ∈ s.ext
e : s.ray c x0 = s.r... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p0 : (c, x... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_inj | [203, 1] | [282, 65] | have tc : ∀ (x : ℂ) (t), ContinuousAt (fun t : ℝ ↦ ↑t * x) t := fun x t ↦
Complex.continuous_ofReal.continuousAt.mul continuousAt_const | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p0 : (c, x0) ∈ s.ext
p1 : (c, x1) ∈ s.ext
e : s.ray c x0 = s.r... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p0 : (c, x0) ∈ s.ext
p1 : (c, x1) ∈ s.ext
e : s.ray c x0 = s.r... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_inj | [203, 1] | [282, 65] | have pt : ∀ {x : ℂ} {t : ℝ}, (c, x) ∈ s.ext → t ∈ Ioc (0 : ℝ) 1 → (c, ↑t * x) ∈ s.ext := by
intro x t p m
simp only [Super.ext, mem_setOf, Complex.abs.map_mul, Complex.abs_ofReal, abs_of_pos m.1] at p ⊢
exact lt_of_le_of_lt (mul_le_of_le_one_left (Complex.abs.nonneg _) m.2) p | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p0 : (c, x0) ∈ s.ext
p1 : (c, x1) ∈ s.ext
e : s.ray c x0 = s.r... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p0 : (c, x0) ∈ s.ext
p1 : (c, x1) ∈ s.ext
e : s.ray c x0 = s.r... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_inj | [203, 1] | [282, 65] | set u : Set ℝ := {t : ℝ | s.ray c (t * x0) = s.ray c (t * x1)} | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p0 : (c, x0) ∈ s.ext
p1 : (c, x1) ∈ s.ext
e : s.ray c x0 = s.r... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p0 : (c, x0) ∈ s.ext
p1 : (c, x1) ∈ s.ext
e : s.ray c x0 = s.r... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_inj | [203, 1] | [282, 65] | suffices h : Ioc (0 : ℝ) 1 ⊆ interior u by
replace h := _root_.trans h interior_subset
replace tc := (tc x0 0).prod_mk (tc x1 0); simp only [← nhds_prod_eq] at tc
simp only [ContinuousAt, Complex.ofReal_zero, MulZeroClass.zero_mul] at tc
have inj := tc.eventually ((s.ray_holomorphic (s.mem_ext c)).along_snd.loc... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p0 : (c, x0) ∈ s.ext
p1 : (c, x1) ∈ s.ext
e : s.ray c x0 = s.r... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p0 : (c, x0) ∈ s.ext
p1 : (c, x1) ∈ s.ext
e : s.ray c x0 = s.r... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_inj | [203, 1] | [282, 65] | refine isPreconnected_Ioc.relative_clopen ?_ ?_ ?_ | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p0 : (c, x0) ∈ s.ext
p1 : (c, x1) ∈ s.ext
e : s.ray c x0 = s.r... | case neg.refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p0 : (c, x0) ∈ s.ext
p1 : (c, x1) ∈ s.ext
e : s.ray c... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_inj | [203, 1] | [282, 65] | simp only [← s.ray_potential p0, ← s.ray_potential p1, e] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p0 : (c, x0) ∈ s.ext
p1 : (c, x1) ∈ s.ext
e : s.ray c x0 = s.ray c x1
⊢... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
x0 x1 : ℂ
p0 : (c, x... |
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