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/Render/Iterate.lean | ne_nan_of_iterate | [117, 1] | [121, 74] | contrapose e | c z : Box
rs : Floating
n : ℕ
e : (iterate c z rs n).exit ≠ Exit.nan
⊢ rs ≠ nan | c z : Box
rs : Floating
n : ℕ
e : ¬rs ≠ nan
⊢ ¬(iterate c z rs n).exit ≠ Exit.nan | Please generate a tactic in lean4 to solve the state.
STATE:
c z : Box
rs : Floating
n : ℕ
e : (iterate c z rs n).exit ≠ Exit.nan
⊢ rs ≠ nan
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | ne_nan_of_iterate | [117, 1] | [121, 74] | simp only [ne_eq, not_not] at e | c z : Box
rs : Floating
n : ℕ
e : ¬rs ≠ nan
⊢ ¬(iterate c z rs n).exit ≠ Exit.nan | c z : Box
rs : Floating
n : ℕ
e : rs = nan
⊢ ¬(iterate c z rs n).exit ≠ Exit.nan | Please generate a tactic in lean4 to solve the state.
STATE:
c z : Box
rs : Floating
n : ℕ
e : ¬rs ≠ nan
⊢ ¬(iterate c z rs n).exit ≠ Exit.nan
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Iterate.lean | ne_nan_of_iterate | [117, 1] | [121, 74] | simp only [e, iterate_nan, ne_eq, not_true_eq_false, not_false_eq_true] | c z : Box
rs : Floating
n : ℕ
e : rs = nan
⊢ ¬(iterate c z rs n).exit ≠ Exit.nan | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c z : Box
rs : Floating
n : ℕ
e : rs = nan
⊢ ¬(iterate c z rs n).exit ≠ Exit.nan
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | clear n0s n1s n0 n1 | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n0s : (c, (f c)^[n0] z) ∈ s.near
n1s : (c, (f c)^[n1] z) ∈ s.near
⊢ ∀ {n0 n1 : ℕ},
n0 ≤ n1 → (c, (f c)^[n0] z) ∈ s.near → (c, (f c)^[n1] z) ∈ s.near → s.potential' c z n0 = s.potential' c z n1 | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
⊢ ∀ {n0 n1 : ℕ},
n0 ≤ n1 → (c, (f c)^[n0] z) ∈ s.near → (c, (f c)^[n1] z) ∈ s.near → s.potential' c z n0 = s.potential' c z n1 | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n0s : (c, (f c)^[n0] z) ∈ s.near
n1s : (c, (f c)^[n1] z) ∈ s.near
⊢ ∀ {n0 n1 : ℕ},
n0 ≤ n1 → (c, (f c)^[n0] z) ∈ s.near → (c, (f c)^[n1] z) ∈ s.near → s.potential' c z n0 = s.potential' c z n1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | intro n0 n1 n01 n0s _ | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
⊢ ∀ {n0 n1 : ℕ},
n0 ≤ n1 → (c, (f c)^[n0] z) ∈ s.near → (c, (f c)^[n1] z) ∈ s.near → s.potential' c z n0 = s.potential' c z n1 | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
⊢ s.potential' c z n0 = s.potential' c z n1 | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
⊢ ∀ {n0 n1 : ℕ},
n0 ≤ n1 → (c, (f c)^[n0] z) ∈ s.near → (c, (f c)^[n1] z) ∈ s.near → s.potential' c z n0 = s.potential' c z n1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | rw [← Nat.sub_add_cancel n01] | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
⊢ s.potential' c z n0 = s.potential' c z n1 | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
⊢ s.potential' c z n0 = s.potential' c z (n1 - n0 + n0) | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
⊢ s.potential' c z n0 = s.potential' c z n1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | have m : ∀ k, (c, (f c)^[n0 + k] z) ∈ s.near := by
intro k; rw [Nat.add_comm]
simp only [Function.iterate_add, s.iter_stays_near n0s k, Function.comp] | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
⊢ s.potential' c z n0 = s.potential' c z (n1 - n0 + n0) | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
⊢ s.potential' c z n0 = s.potential' c z (n1 - n0 + n0) | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
⊢ s.potential' c z n0 = s.potential' c z (n1 - n0 + n0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | generalize hk : n1 - n0 = k | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
⊢ s.potential' c z n0 = s.potential' c z (n1 - n0 + n0) | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
hk : n1 - n0 = k
⊢ s.potential' c z n0 = s.potential' c z (k + n0) | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
⊢ s.potential' c z n0 = s.potential' c z (n1 - n0 + n0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | rw [Nat.add_comm] | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
hk : n1 - n0 = k
⊢ s.potential' c z n0 = s.potential' c z (k + n0) | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
hk : n1 - n0 = k
⊢ s.potential' c z n0 = s.potential' c z (n0 + k) | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
hk : n1 - n0 = k
⊢ s.potential' c z n0 = s.potential' c z (k + n0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | clear hk | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
hk : n1 - n0 = k
⊢ s.potential' c z n0 = s.potential' c z (n0 + k) | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
⊢ s.potential' c z n0 = s.potential' c z (n0 + k) | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
hk : n1 - n0 = k
⊢ s.potential' c z n0 = s.potential' c z (n0 + k)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | induction' k with k 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
⊢ s.potential' c z n0 = s.potential' c z (n0 + k) | case zero
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
⊢ s.potential' c z n0 = s.potential' c z (n0 + 0)
case succ
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ s.potential' c z n0 = s.potential' c z (n0 + (k + 1)) | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
⊢ s.potential' c z n0 = s.potential' c z (n0 + k)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | by_cases n01 : n0 ≤ n1 | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n0s : (c, (f c)^[n0] z) ∈ s.near
n1s : (c, (f c)^[n1] z) ∈ s.near
h :
∀ {n0 n1 : ℕ},
n0 ≤ n1 → (c, (f c)^[n0] z) ∈ s.near → (c, (f c)^[n1] z) ∈ s.near → s.potential' c z n0 = s.potential' c z n1
⊢ s.potential' c z n0 = s.potential' c z n1 | case pos
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n0s : (c, (f c)^[n0] z) ∈ s.near
n1s : (c, (f c)^[n1] z) ∈ s.near
h :
∀ {n0 n1 : ℕ},
n0 ≤ n1 → (c, (f c)^[n0] z) ∈ s.near → (c, (f c)^[n1] z) ∈ s.near → s.potential' c z n0 = s.potential' c z n1
n01 : n0 ≤ n1
⊢ s.potential' c z n0 = s.potential' c z n1
case neg
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n0s : (c, (f c)^[n0] z) ∈ s.near
n1s : (c, (f c)^[n1] z) ∈ s.near
h :
∀ {n0 n1 : ℕ},
n0 ≤ n1 → (c, (f c)^[n0] z) ∈ s.near → (c, (f c)^[n1] z) ∈ s.near → s.potential' c z n0 = s.potential' c z n1
n01 : ¬n0 ≤ n1
⊢ s.potential' c z n0 = s.potential' c z n1 | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n0s : (c, (f c)^[n0] z) ∈ s.near
n1s : (c, (f c)^[n1] z) ∈ s.near
h :
∀ {n0 n1 : ℕ},
n0 ≤ n1 → (c, (f c)^[n0] z) ∈ s.near → (c, (f c)^[n1] z) ∈ s.near → s.potential' c z n0 = s.potential' c z n1
⊢ s.potential' c z n0 = s.potential' c z n1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | exact h n01 n0s n1s | case pos
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n0s : (c, (f c)^[n0] z) ∈ s.near
n1s : (c, (f c)^[n1] z) ∈ s.near
h :
∀ {n0 n1 : ℕ},
n0 ≤ n1 → (c, (f c)^[n0] z) ∈ s.near → (c, (f c)^[n1] z) ∈ s.near → s.potential' c z n0 = s.potential' c z n1
n01 : n0 ≤ n1
⊢ s.potential' c z n0 = s.potential' c z n1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n0s : (c, (f c)^[n0] z) ∈ s.near
n1s : (c, (f c)^[n1] z) ∈ s.near
h :
∀ {n0 n1 : ℕ},
n0 ≤ n1 → (c, (f c)^[n0] z) ∈ s.near → (c, (f c)^[n1] z) ∈ s.near → s.potential' c z n0 = s.potential' c z n1
n01 : n0 ≤ n1
⊢ s.potential' c z n0 = s.potential' c z n1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | exact (h (not_le.mp n01).le n1s n0s).symm | case neg
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n0s : (c, (f c)^[n0] z) ∈ s.near
n1s : (c, (f c)^[n1] z) ∈ s.near
h :
∀ {n0 n1 : ℕ},
n0 ≤ n1 → (c, (f c)^[n0] z) ∈ s.near → (c, (f c)^[n1] z) ∈ s.near → s.potential' c z n0 = s.potential' c z n1
n01 : ¬n0 ≤ n1
⊢ s.potential' c z n0 = s.potential' c z n1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n0s : (c, (f c)^[n0] z) ∈ s.near
n1s : (c, (f c)^[n1] z) ∈ s.near
h :
∀ {n0 n1 : ℕ},
n0 ≤ n1 → (c, (f c)^[n0] z) ∈ s.near → (c, (f c)^[n1] z) ∈ s.near → s.potential' c z n0 = s.potential' c z n1
n01 : ¬n0 ≤ n1
⊢ s.potential' c z n0 = s.potential' c z n1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | intro k | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
⊢ ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
k : ℕ
⊢ (c, (f c)^[n0 + k] z) ∈ s.near | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
⊢ ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | rw [Nat.add_comm] | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
k : ℕ
⊢ (c, (f c)^[n0 + k] z) ∈ s.near | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
k : ℕ
⊢ (c, (f c)^[k + n0] z) ∈ s.near | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
k : ℕ
⊢ (c, (f c)^[n0 + k] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | simp only [Function.iterate_add, s.iter_stays_near n0s k, Function.comp] | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
k : ℕ
⊢ (c, (f c)^[k + n0] z) ∈ s.near | no goals | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
k : ℕ
⊢ (c, (f c)^[k + n0] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | simp only [Nat.zero_eq, add_zero] | case zero
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
⊢ s.potential' c z n0 = s.potential' c z (n0 + 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
⊢ s.potential' c z n0 = s.potential' c z (n0 + 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | simp only [Nat.add_succ, Function.iterate_succ', Super.potential', Function.comp] | case succ
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ s.potential' c z n0 = s.potential' c z (n0 + (k + 1)) | case succ
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ Complex.abs (s.bottcherNear c ((f c)^[n0] z)) ^ (↑d ^ n0)⁻¹ =
Complex.abs (s.bottcherNear c (f c ((f c)^[n0 + k] z))) ^ (↑d ^ (n0 + k).succ)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ s.potential' c z n0 = s.potential' c z (n0 + (k + 1))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | rw [s.bottcherNear_eqn (m k)] | case succ
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ Complex.abs (s.bottcherNear c ((f c)^[n0] z)) ^ (↑d ^ n0)⁻¹ =
Complex.abs (s.bottcherNear c (f c ((f c)^[n0 + k] z))) ^ (↑d ^ (n0 + k).succ)⁻¹ | case succ
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ Complex.abs (s.bottcherNear c ((f c)^[n0] z)) ^ (↑d ^ n0)⁻¹ =
Complex.abs (s.bottcherNear c ((f c)^[n0 + k] z) ^ d) ^ (↑d ^ (n0 + k).succ)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ Complex.abs (s.bottcherNear c ((f c)^[n0] z)) ^ (↑d ^ n0)⁻¹ =
Complex.abs (s.bottcherNear c (f c ((f c)^[n0 + k] z))) ^ (↑d ^ (n0 + k).succ)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | rw [pow_succ' _ (n0 + k), mul_inv, Complex.abs.map_pow, Real.rpow_mul, ← Real.rpow_natCast _ d] | case succ
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ Complex.abs (s.bottcherNear c ((f c)^[n0] z)) ^ (↑d ^ n0)⁻¹ =
Complex.abs (s.bottcherNear c ((f c)^[n0 + k] z) ^ d) ^ (↑d ^ (n0 + k).succ)⁻¹ | case succ
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ Complex.abs (s.bottcherNear c ((f c)^[n0] z)) ^ (↑d ^ n0)⁻¹ =
((Complex.abs (s.bottcherNear c ((f c)^[n0 + k] z)) ^ ↑d) ^ (↑d)⁻¹) ^ (↑d ^ (n0 + k))⁻¹
case succ.hx
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ 0 ≤ Complex.abs (s.bottcherNear c ((f c)^[n0 + k] z)) ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ Complex.abs (s.bottcherNear c ((f c)^[n0] z)) ^ (↑d ^ n0)⁻¹ =
Complex.abs (s.bottcherNear c ((f c)^[n0 + k] z) ^ d) ^ (↑d ^ (n0 + k).succ)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | rw [← Real.rpow_mul (Complex.abs.nonneg _) _ d⁻¹,
mul_inv_cancel (s.superAtC.s (Set.mem_univ c)).drz, Real.rpow_one] | case succ
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ Complex.abs (s.bottcherNear c ((f c)^[n0] z)) ^ (↑d ^ n0)⁻¹ =
((Complex.abs (s.bottcherNear c ((f c)^[n0 + k] z)) ^ ↑d) ^ (↑d)⁻¹) ^ (↑d ^ (n0 + k))⁻¹
case succ.hx
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ 0 ≤ Complex.abs (s.bottcherNear c ((f c)^[n0 + k] z)) ^ d | case succ
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ Complex.abs (s.bottcherNear c ((f c)^[n0] z)) ^ (↑d ^ n0)⁻¹ =
Complex.abs (s.bottcherNear c ((f c)^[n0 + k] z)) ^ (↑d ^ (n0 + k))⁻¹
case succ.hx
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ 0 ≤ Complex.abs (s.bottcherNear c ((f c)^[n0 + k] z)) ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ Complex.abs (s.bottcherNear c ((f c)^[n0] z)) ^ (↑d ^ n0)⁻¹ =
((Complex.abs (s.bottcherNear c ((f c)^[n0 + k] z)) ^ ↑d) ^ (↑d)⁻¹) ^ (↑d ^ (n0 + k))⁻¹
case succ.hx
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ 0 ≤ Complex.abs (s.bottcherNear c ((f c)^[n0 + k] z)) ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | exact h | case succ
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ Complex.abs (s.bottcherNear c ((f c)^[n0] z)) ^ (↑d ^ n0)⁻¹ =
Complex.abs (s.bottcherNear c ((f c)^[n0 + k] z)) ^ (↑d ^ (n0 + k))⁻¹
case succ.hx
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ 0 ≤ Complex.abs (s.bottcherNear c ((f c)^[n0 + k] z)) ^ d | case succ.hx
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ 0 ≤ Complex.abs (s.bottcherNear c ((f c)^[n0 + k] z)) ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ Complex.abs (s.bottcherNear c ((f c)^[n0] z)) ^ (↑d ^ n0)⁻¹ =
Complex.abs (s.bottcherNear c ((f c)^[n0 + k] z)) ^ (↑d ^ (n0 + k))⁻¹
case succ.hx
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ 0 ≤ Complex.abs (s.bottcherNear c ((f c)^[n0 + k] z)) ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq' | [58, 1] | [78, 19] | bound | case succ.hx
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ 0 ≤ Complex.abs (s.bottcherNear c ((f c)^[n0 + k] z)) ^ d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.hx
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
n0 n1 : ℕ
n01 : n0 ≤ n1
n0s : (c, (f c)^[n0] z) ∈ s.near
a✝ : (c, (f c)^[n1] z) ∈ s.near
m : ∀ (k : ℕ), (c, (f c)^[n0 + k] z) ∈ s.near
k : ℕ
h : s.potential' c z n0 = s.potential' c z (n0 + k)
⊢ 0 ≤ Complex.abs (s.bottcherNear c ((f c)^[n0 + k] z)) ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq | [85, 1] | [89, 45] | have h : ∃ k, (c, (f c)^[k] z) ∈ s.near := ⟨k,ks⟩ | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
k : ℕ
ks : (c, (f c)^[k] z) ∈ s.near
⊢ s.potential c z = s.potential' c z k | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
k : ℕ
ks : (c, (f c)^[k] z) ∈ s.near
h : ∃ k, (c, (f c)^[k] z) ∈ s.near
⊢ s.potential c z = s.potential' c z k | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
k : ℕ
ks : (c, (f c)^[k] z) ∈ s.near
⊢ s.potential c z = s.potential' c z k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq | [85, 1] | [89, 45] | simp only [Super.potential, h, dif_pos] | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
k : ℕ
ks : (c, (f c)^[k] z) ∈ s.near
h : ∃ k, (c, (f c)^[k] z) ∈ s.near
⊢ s.potential c z = s.potential' c z k | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
k : ℕ
ks : (c, (f c)^[k] z) ∈ s.near
h : ∃ k, (c, (f c)^[k] z) ∈ s.near
⊢ s.potential' c z (Nat.find ⋯) = s.potential' c z k | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
k : ℕ
ks : (c, (f c)^[k] z) ∈ s.near
h : ∃ k, (c, (f c)^[k] z) ∈ s.near
⊢ s.potential c z = s.potential' c z k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq | [85, 1] | [89, 45] | exact s.potential_eq' (Nat.find_spec h) ks | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
k : ℕ
ks : (c, (f c)^[k] z) ∈ s.near
h : ∃ k, (c, (f c)^[k] z) ∈ s.near
⊢ s.potential' c z (Nat.find ⋯) = s.potential' c z k | no goals | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
k : ℕ
ks : (c, (f c)^[k] z) ∈ s.near
h : ∃ k, (c, (f c)^[k] z) ∈ s.near
⊢ s.potential' c z (Nat.find ⋯) = s.potential' c z k
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.abs_bottcherNear | [92, 1] | [97, 50] | simp only [s.potential_eq r, Super.potential'] | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
⊢ Complex.abs (s.bottcherNear c ((f c)^[n] z)) = s.potential c z ^ d ^ n | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
⊢ Complex.abs (s.bottcherNear c ((f c)^[n] z)) = (Complex.abs (s.bottcherNear c ((f c)^[n] z)) ^ (↑d ^ n)⁻¹) ^ d ^ n | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
⊢ Complex.abs (s.bottcherNear c ((f c)^[n] z)) = s.potential c z ^ d ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.abs_bottcherNear | [92, 1] | [97, 50] | rw [← Real.rpow_natCast, ← Real.rpow_mul (Complex.abs.nonneg _), Nat.cast_pow, inv_mul_cancel,
Real.rpow_one] | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
⊢ Complex.abs (s.bottcherNear c ((f c)^[n] z)) = (Complex.abs (s.bottcherNear c ((f c)^[n] z)) ^ (↑d ^ n)⁻¹) ^ d ^ n | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
⊢ ↑d ^ n ≠ 0 | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
⊢ Complex.abs (s.bottcherNear c ((f c)^[n] z)) = (Complex.abs (s.bottcherNear c ((f c)^[n] z)) ^ (↑d ^ n)⁻¹) ^ d ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.abs_bottcherNear | [92, 1] | [97, 50] | exact pow_ne_zero _ (Nat.cast_ne_zero.mpr s.d0) | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
⊢ ↑d ^ n ≠ 0 | no goals | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
⊢ ↑d ^ n ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_a | [100, 1] | [103, 60] | have r : (c, (f c)^[0] a) ∈ s.near := by simp only [Function.iterate_zero, s.mem_near, id] | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ s.potential c a = 0 | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
r : (c, (f c)^[0] a) ∈ s.near
⊢ s.potential c a = 0 | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ s.potential c a = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_a | [100, 1] | [103, 60] | simp only [s.potential_eq r, Super.potential', Function.iterate_zero, id, s.bottcherNear_a,
Complex.abs.map_zero, pow_zero, inv_one, Real.rpow_one] | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
r : (c, (f c)^[0] a) ∈ s.near
⊢ s.potential c a = 0 | no goals | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
r : (c, (f c)^[0] a) ∈ s.near
⊢ s.potential c a = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_a | [100, 1] | [103, 60] | simp only [Function.iterate_zero, s.mem_near, id] | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ (c, (f c)^[0] a) ∈ s.near | no goals | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ (c, (f c)^[0] a) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_one | [106, 1] | [108, 87] | simp only [Super.potential, not_exists.mpr a, not_false_iff, dif_neg, and_false_iff] | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∀ (n : ℕ), (c, (f c)^[n] z) ∉ s.near
⊢ s.potential c z = 1 | no goals | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∀ (n : ℕ), (c, (f c)^[n] z) ∉ s.near
⊢ s.potential c z = 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_lt_one | [111, 1] | [115, 48] | simp only [Super.potential, a, dif_pos, Super.potential'] | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c z < 1 | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ Complex.abs (s.bottcherNear c ((f c)^[Nat.find ⋯] z)) ^ (↑d ^ Nat.find ⋯)⁻¹ < 1 | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c z < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_lt_one | [111, 1] | [115, 48] | refine Real.rpow_lt_one (Complex.abs.nonneg _) ?_ (by bound) | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ Complex.abs (s.bottcherNear c ((f c)^[Nat.find ⋯] z)) ^ (↑d ^ Nat.find ⋯)⁻¹ < 1 | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ Complex.abs (s.bottcherNear c ((f c)^[Nat.find ⋯] z)) < 1 | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ Complex.abs (s.bottcherNear c ((f c)^[Nat.find ⋯] z)) ^ (↑d ^ Nat.find ⋯)⁻¹ < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_lt_one | [111, 1] | [115, 48] | exact s.bottcherNear_lt_one (Nat.find_spec a) | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ Complex.abs (s.bottcherNear c ((f c)^[Nat.find ⋯] z)) < 1 | no goals | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ Complex.abs (s.bottcherNear c ((f c)^[Nat.find ⋯] z)) < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_lt_one | [111, 1] | [115, 48] | bound | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ 0 < (↑d ^ Nat.find ⋯)⁻¹ | no goals | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ 0 < (↑d ^ Nat.find ⋯)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_lt_one_iff | [118, 1] | [122, 69] | refine ⟨?_, s.potential_lt_one⟩ | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ s.potential c z < 1 ↔ ∃ n, (c, (f c)^[n] z) ∈ s.near | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ s.potential c z < 1 → ∃ n, (c, (f c)^[n] z) ∈ s.near | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ s.potential c z < 1 ↔ ∃ n, (c, (f c)^[n] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_lt_one_iff | [118, 1] | [122, 69] | intro 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ s.potential c z < 1 → ∃ n, (c, (f c)^[n] z) ∈ s.near | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : s.potential c z < 1
⊢ ∃ n, (c, (f c)^[n] z) ∈ s.near | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ s.potential c z < 1 → ∃ n, (c, (f c)^[n] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_lt_one_iff | [118, 1] | [122, 69] | contrapose 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : s.potential c z < 1
⊢ ∃ n, (c, (f c)^[n] z) ∈ s.near | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ ¬s.potential c z < 1 | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : s.potential c z < 1
⊢ ∃ n, (c, (f c)^[n] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_lt_one_iff | [118, 1] | [122, 69] | simp only [not_exists] at 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ ¬s.potential c z < 1 | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : ∀ (x : ℕ), (c, (f c)^[x] z) ∉ s.near
⊢ ¬s.potential c z < 1 | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ ¬s.potential c z < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_lt_one_iff | [118, 1] | [122, 69] | simp only [s.potential_eq_one h, lt_self_iff_false, not_false_iff] | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : ∀ (x : ℕ), (c, (f c)^[x] z) ∉ s.near
⊢ ¬s.potential c z < 1 | no goals | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : ∀ (x : ℕ), (c, (f c)^[x] z) ∉ s.near
⊢ ¬s.potential c z < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_le_one | [125, 1] | [128, 56] | by_cases a : ∃ n, (c, (f c)^[n] z) ∈ s.near | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ s.potential c z ≤ 1 | case pos
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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c z ≤ 1
case neg
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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c z ≤ 1 | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ s.potential c z ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_le_one | [125, 1] | [128, 56] | exact (s.potential_lt_one a).le | case pos
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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c z ≤ 1
case neg
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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c z ≤ 1 | case neg
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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c z ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c z ≤ 1
case neg
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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c z ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_le_one | [125, 1] | [128, 56] | exact le_of_eq (s.potential_eq_one (not_exists.mp a)) | case neg
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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c z ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c z ≤ 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_nonneg | [131, 10] | [134, 64] | by_cases r : ∃ n, (c, (f c)^[n] z) ∈ s.near | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ 0 ≤ s.potential c z | case pos
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
r : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ s.potential c z
case neg
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
r : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ s.potential c z | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ 0 ≤ s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_nonneg | [131, 10] | [134, 64] | rcases r with ⟨n, r⟩ | case pos
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
r : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ s.potential c z
case neg
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
r : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ s.potential c z | case pos.intro
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ s.potential c z
case neg
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
r : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
r : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ s.potential c z
case neg
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
r : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_nonneg | [131, 10] | [134, 64] | simp only [s.potential_eq r, Super.potential'] | case pos.intro
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ s.potential c z
case neg
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
r : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ s.potential c z | case pos.intro
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ Complex.abs (s.bottcherNear c ((f c)^[n] z)) ^ (↑d ^ n)⁻¹
case neg
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
r : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ s.potential c z
case neg
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
r : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_nonneg | [131, 10] | [134, 64] | bound | case pos.intro
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ Complex.abs (s.bottcherNear c ((f c)^[n] z)) ^ (↑d ^ n)⁻¹
case neg
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
r : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ s.potential c z | case neg
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
r : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ s.potential c z | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ Complex.abs (s.bottcherNear c ((f c)^[n] z)) ^ (↑d ^ n)⁻¹
case neg
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
r : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_nonneg | [131, 10] | [134, 64] | simp only [s.potential_eq_one (not_exists.mp r), zero_le_one] | case neg
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
r : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ s.potential c z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
r : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ 0 ≤ s.potential c z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eqn | [137, 1] | [149, 85] | by_cases a : ∃ n, (c, (f c)^[n] z) ∈ s.near | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ s.potential c (f c z) = s.potential c z ^ d | case pos
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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c (f c z) = s.potential c z ^ d
case neg
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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c (f c z) = s.potential c z ^ d | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ s.potential c (f c z) = s.potential c z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eqn | [137, 1] | [149, 85] | rcases a with ⟨n, a⟩ | case pos
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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c (f c z) = s.potential c z ^ d | case pos.intro
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c (f c z) = s.potential c z ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c (f c z) = s.potential c z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eqn | [137, 1] | [149, 85] | have a' : (c, (f c)^[n] (f c z)) ∈ s.near := by
simp only [← Function.iterate_succ_apply, Function.iterate_succ', s.stays_near a,
Function.comp] | case pos.intro
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c (f c z) = s.potential c z ^ d | case pos.intro
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
a' : (c, (f c)^[n] (f c z)) ∈ s.near
⊢ s.potential c (f c z) = s.potential c z ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c (f c z) = s.potential c z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eqn | [137, 1] | [149, 85] | simp only [s.potential_eq a, s.potential_eq a', Super.potential', ← Function.iterate_succ_apply,
Function.iterate_succ', s.bottcherNear_eqn a, Complex.abs.map_pow, ← Real.rpow_natCast, ←
Real.rpow_mul (Complex.abs.nonneg _), mul_comm, Function.comp] | case pos.intro
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
a' : (c, (f c)^[n] (f c z)) ∈ s.near
⊢ s.potential c (f c z) = s.potential c z ^ d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
a' : (c, (f c)^[n] (f c z)) ∈ s.near
⊢ s.potential c (f c z) = s.potential c z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eqn | [137, 1] | [149, 85] | simp only [← Function.iterate_succ_apply, Function.iterate_succ', s.stays_near a,
Function.comp] | 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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
⊢ (c, (f c)^[n] (f c z)) ∈ s.near | no goals | 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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
⊢ (c, (f c)^[n] (f c z)) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eqn | [137, 1] | [149, 85] | have a' : ∀ n, (c, (f c)^[n] (f c z)) ∉ s.near := by
contrapose a; simp only [not_forall, not_not, ← Function.iterate_succ_apply] at a ⊢
rcases a with ⟨n, a⟩; exact ⟨n + 1, a⟩ | case neg
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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c (f c z) = s.potential c z ^ d | case neg
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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
a' : ∀ (n : ℕ), (c, (f c)^[n] (f c z)) ∉ s.near
⊢ s.potential c (f c z) = s.potential c z ^ d | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ s.potential c (f c z) = s.potential c z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eqn | [137, 1] | [149, 85] | simp only [s.potential_eq_one (not_exists.mp a), s.potential_eq_one a', one_pow] | case neg
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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
a' : ∀ (n : ℕ), (c, (f c)^[n] (f c z)) ∉ s.near
⊢ s.potential c (f c z) = s.potential c z ^ d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
a' : ∀ (n : ℕ), (c, (f c)^[n] (f c z)) ∉ s.near
⊢ s.potential c (f c z) = s.potential c z ^ d
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eqn | [137, 1] | [149, 85] | contrapose a | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ ∀ (n : ℕ), (c, (f c)^[n] (f c z)) ∉ s.near | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ¬∀ (n : ℕ), (c, (f c)^[n] (f c z)) ∉ s.near
⊢ ¬¬∃ n, (c, (f c)^[n] z) ∈ s.near | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ ∀ (n : ℕ), (c, (f c)^[n] (f c z)) ∉ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eqn | [137, 1] | [149, 85] | simp only [not_forall, not_not, ← Function.iterate_succ_apply] at a ⊢ | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ¬∀ (n : ℕ), (c, (f c)^[n] (f c z)) ∉ s.near
⊢ ¬¬∃ n, (c, (f c)^[n] z) ∈ s.near | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ x, (c, (f c)^[x.succ] z) ∈ s.near
⊢ ∃ n, (c, (f c)^[n] z) ∈ s.near | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ¬∀ (n : ℕ), (c, (f c)^[n] (f c z)) ∉ s.near
⊢ ¬¬∃ n, (c, (f c)^[n] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eqn | [137, 1] | [149, 85] | rcases a with ⟨n, a⟩ | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ x, (c, (f c)^[x.succ] z) ∈ s.near
⊢ ∃ n, (c, (f c)^[n] z) ∈ s.near | case intro
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n.succ] z) ∈ s.near
⊢ ∃ n, (c, (f c)^[n] z) ∈ s.near | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ x, (c, (f c)^[x.succ] z) ∈ s.near
⊢ ∃ n, (c, (f c)^[n] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eqn | [137, 1] | [149, 85] | exact ⟨n + 1, a⟩ | case intro
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n.succ] z) ∈ s.near
⊢ ∃ n, (c, (f c)^[n] z) ∈ s.near | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n.succ] z) ∈ s.near
⊢ ∃ n, (c, (f c)^[n] z) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eqn_iter | [152, 1] | [157, 21] | induction' n with n 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
⊢ s.potential c ((f c)^[n] z) = s.potential c z ^ d ^ n | case zero
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ s.potential c ((f c)^[0] z) = s.potential c z ^ d ^ 0
case succ
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : s.potential c ((f c)^[n] z) = s.potential c z ^ d ^ n
⊢ s.potential c ((f c)^[n + 1] z) = s.potential c z ^ d ^ (n + 1) | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
⊢ s.potential c ((f c)^[n] z) = s.potential c z ^ d ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eqn_iter | [152, 1] | [157, 21] | simp only [Function.iterate_zero, id, pow_zero, pow_one] | case zero
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ s.potential c ((f c)^[0] z) = s.potential c z ^ d ^ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ s.potential c ((f c)^[0] z) = s.potential c z ^ d ^ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eqn_iter | [152, 1] | [157, 21] | simp only [Function.iterate_succ', Super.potential_eqn, h, ← pow_mul, ← pow_succ,
Function.comp] | case succ
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : s.potential c ((f c)^[n] z) = s.potential c z ^ d ^ n
⊢ s.potential c ((f c)^[n + 1] z) = s.potential c z ^ d ^ (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
h : s.potential c ((f c)^[n] z) = s.potential c z ^ d ^ n
⊢ s.potential c ((f c)^[n + 1] z) = s.potential c z ^ d ^ (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.iter_holomorphic' | [160, 1] | [164, 43] | intro p | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
⊢ Holomorphic (I.prod I) I fun p => (f p.1)^[n] p.2 | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : ℂ × S
⊢ HolomorphicAt (I.prod I) I (fun p => (f p.1)^[n] p.2) p | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
⊢ Holomorphic (I.prod I) I fun p => (f p.1)^[n] p.2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.iter_holomorphic' | [160, 1] | [164, 43] | induction' n with n 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : ℂ × S
⊢ HolomorphicAt (I.prod I) I (fun p => (f p.1)^[n] p.2) p | case zero
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
p : ℂ × S
⊢ HolomorphicAt (I.prod I) I (fun p => (f p.1)^[0] p.2) p
case succ
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
p : ℂ × S
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f p.1)^[n] p.2) p
⊢ HolomorphicAt (I.prod I) I (fun p => (f p.1)^[n + 1] p.2) p | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : ℂ × S
⊢ HolomorphicAt (I.prod I) I (fun p => (f p.1)^[n] p.2) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.iter_holomorphic' | [160, 1] | [164, 43] | simp [Function.iterate_zero, holomorphicAt_snd] | case zero
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
p : ℂ × S
⊢ HolomorphicAt (I.prod I) I (fun p => (f p.1)^[0] p.2) p
case succ
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
p : ℂ × S
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f p.1)^[n] p.2) p
⊢ HolomorphicAt (I.prod I) I (fun p => (f p.1)^[n + 1] p.2) p | case succ
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
p : ℂ × S
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f p.1)^[n] p.2) p
⊢ HolomorphicAt (I.prod I) I (fun p => (f p.1)^[n + 1] p.2) p | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
p : ℂ × S
⊢ HolomorphicAt (I.prod I) I (fun p => (f p.1)^[0] p.2) p
case succ
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
p : ℂ × S
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f p.1)^[n] p.2) p
⊢ HolomorphicAt (I.prod I) I (fun p => (f p.1)^[n + 1] p.2) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.iter_holomorphic' | [160, 1] | [164, 43] | simp only [Function.iterate_succ', Function.comp] | case succ
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
p : ℂ × S
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f p.1)^[n] p.2) p
⊢ HolomorphicAt (I.prod I) I (fun p => (f p.1)^[n + 1] p.2) p | case succ
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
p : ℂ × S
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f p.1)^[n] p.2) p
⊢ HolomorphicAt (I.prod I) I (fun p => f p.1 ((f p.1)^[n] p.2)) p | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
p : ℂ × S
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f p.1)^[n] p.2) p
⊢ HolomorphicAt (I.prod I) I (fun p => (f p.1)^[n + 1] p.2) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.iter_holomorphic' | [160, 1] | [164, 43] | exact (s.fa _).comp₂ holomorphicAt_fst h | case succ
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
p : ℂ × S
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f p.1)^[n] p.2) p
⊢ HolomorphicAt (I.prod I) I (fun p => f p.1 ((f p.1)^[n] p.2)) p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
p : ℂ × S
n : ℕ
h : HolomorphicAt (I.prod I) I (fun p => (f p.1)^[n] p.2) p
⊢ HolomorphicAt (I.prod I) I (fun p => f p.1 ((f p.1)^[n] p.2)) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.iter_holomorphic | [166, 1] | [168, 67] | intro p | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
⊢ Holomorphic (I.prod I) (I.prod I) fun p => (p.1, (f p.1)^[n] p.2) | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : ℂ × S
⊢ HolomorphicAt (I.prod I) (I.prod I) (fun p => (p.1, (f p.1)^[n] p.2)) p | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
⊢ Holomorphic (I.prod I) (I.prod I) fun p => (p.1, (f p.1)^[n] p.2)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.iter_holomorphic | [166, 1] | [168, 67] | apply holomorphicAt_fst.prod | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : ℂ × S
⊢ HolomorphicAt (I.prod I) (I.prod I) (fun p => (p.1, (f p.1)^[n] p.2)) p | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : ℂ × S
⊢ HolomorphicAt (I.prod I) I (fun x => (f x.1)^[n] x.2) p | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : ℂ × S
⊢ HolomorphicAt (I.prod I) (I.prod I) (fun p => (p.1, (f p.1)^[n] p.2)) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.iter_holomorphic | [166, 1] | [168, 67] | apply s.iter_holomorphic' | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : ℂ × S
⊢ HolomorphicAt (I.prod I) I (fun x => (f x.1)^[n] x.2) p | no goals | 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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : ℂ × S
⊢ HolomorphicAt (I.prod I) I (fun x => (f x.1)^[n] x.2) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | ContinuousAt.potential_of_reaches | [171, 1] | [184, 17] | rcases a with ⟨n, a⟩ | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ ContinuousAt (uncurry s.potential) (c, z) | case intro
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
⊢ ContinuousAt (uncurry s.potential) (c, z) | 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 : ℂ
a✝ z : S
d n : ℕ
s : Super f d a✝
a : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ ContinuousAt (uncurry s.potential) (c, z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | ContinuousAt.potential_of_reaches | [171, 1] | [184, 17] | have e : uncurry s.potential =ᶠ[𝓝 (c, z)] fun p : ℂ × S ↦ s.potential' p.1 p.2 n := by
have a' : ∀ᶠ p : ℂ × S in 𝓝 (c, z), (p.1, (f p.1)^[n] p.2) ∈ s.near :=
(s.iter_holomorphic n _).continuousAt.eventually_mem (s.isOpen_near.mem_nhds a)
refine a'.mp (eventually_of_forall fun p h ↦ ?_)
simp only [uncurry, s.potential_eq h] | case intro
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
⊢ ContinuousAt (uncurry s.potential) (c, z) | case intro
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ ContinuousAt (uncurry s.potential) (c, z) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
⊢ ContinuousAt (uncurry s.potential) (c, z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | ContinuousAt.potential_of_reaches | [171, 1] | [184, 17] | simp only [continuousAt_congr e, Super.potential'] | case intro
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ ContinuousAt (uncurry s.potential) (c, z) | case intro
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ ContinuousAt (fun p => Complex.abs (s.bottcherNear p.1 ((f p.1)^[n] p.2)) ^ (↑d ^ n)⁻¹) (c, z) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ ContinuousAt (uncurry s.potential) (c, z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | ContinuousAt.potential_of_reaches | [171, 1] | [184, 17] | refine ContinuousAt.rpow ?_ continuousAt_const ?_ | case intro
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ ContinuousAt (fun p => Complex.abs (s.bottcherNear p.1 ((f p.1)^[n] p.2)) ^ (↑d ^ n)⁻¹) (c, z) | case intro.refine_1
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ ContinuousAt (fun p => Complex.abs (s.bottcherNear p.1 ((f p.1)^[n] p.2))) (c, z)
case intro.refine_2
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ Complex.abs (s.bottcherNear (c, z).1 ((f (c, z).1)^[n] (c, z).2)) ≠ 0 ∨ 0 < (↑d ^ n)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ ContinuousAt (fun p => Complex.abs (s.bottcherNear p.1 ((f p.1)^[n] p.2)) ^ (↑d ^ n)⁻¹) (c, z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | ContinuousAt.potential_of_reaches | [171, 1] | [184, 17] | have a' : ∀ᶠ p : ℂ × S in 𝓝 (c, z), (p.1, (f p.1)^[n] p.2) ∈ s.near :=
(s.iter_holomorphic n _).continuousAt.eventually_mem (s.isOpen_near.mem_nhds a) | 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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
⊢ (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n | 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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
a' : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), (p.1, (f p.1)^[n] p.2) ∈ s.near
⊢ (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n | 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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
⊢ (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | ContinuousAt.potential_of_reaches | [171, 1] | [184, 17] | refine a'.mp (eventually_of_forall fun p 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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
a' : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), (p.1, (f p.1)^[n] p.2) ∈ s.near
⊢ (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n | 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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
a' : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), (p.1, (f p.1)^[n] p.2) ∈ s.near
p : ℂ × S
h : (p.1, (f p.1)^[n] p.2) ∈ s.near
⊢ uncurry s.potential p = (fun p => s.potential' p.1 p.2 n) p | 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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
a' : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), (p.1, (f p.1)^[n] p.2) ∈ s.near
⊢ (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | ContinuousAt.potential_of_reaches | [171, 1] | [184, 17] | simp only [uncurry, s.potential_eq 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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
a' : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), (p.1, (f p.1)^[n] p.2) ∈ s.near
p : ℂ × S
h : (p.1, (f p.1)^[n] p.2) ∈ s.near
⊢ uncurry s.potential p = (fun p => s.potential' p.1 p.2 n) p | no goals | 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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
a' : ∀ᶠ (p : ℂ × S) in 𝓝 (c, z), (p.1, (f p.1)^[n] p.2) ∈ s.near
p : ℂ × S
h : (p.1, (f p.1)^[n] p.2) ∈ s.near
⊢ uncurry s.potential p = (fun p => s.potential' p.1 p.2 n) p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | ContinuousAt.potential_of_reaches | [171, 1] | [184, 17] | apply Complex.continuous_abs.continuousAt.comp | case intro.refine_1
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ ContinuousAt (fun p => Complex.abs (s.bottcherNear p.1 ((f p.1)^[n] p.2))) (c, z) | case intro.refine_1
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ ContinuousAt (fun p => s.bottcherNear p.1 ((f p.1)^[n] p.2)) (c, z) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ ContinuousAt (fun p => Complex.abs (s.bottcherNear p.1 ((f p.1)^[n] p.2))) (c, z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | ContinuousAt.potential_of_reaches | [171, 1] | [184, 17] | refine ((s.bottcherNear_holomorphic _ ?_).comp (s.iter_holomorphic n (c, z))).continuousAt | case intro.refine_1
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ ContinuousAt (fun p => s.bottcherNear p.1 ((f p.1)^[n] p.2)) (c, z) | case intro.refine_1
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ ((c, z).1, (f (c, z).1)^[n] (c, z).2) ∈ s.near | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ ContinuousAt (fun p => s.bottcherNear p.1 ((f p.1)^[n] p.2)) (c, z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | ContinuousAt.potential_of_reaches | [171, 1] | [184, 17] | exact a | case intro.refine_1
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ ((c, z).1, (f (c, z).1)^[n] (c, z).2) ∈ s.near | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ ((c, z).1, (f (c, z).1)^[n] (c, z).2) ∈ s.near
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | ContinuousAt.potential_of_reaches | [171, 1] | [184, 17] | right | case intro.refine_2
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ Complex.abs (s.bottcherNear (c, z).1 ((f (c, z).1)^[n] (c, z).2)) ≠ 0 ∨ 0 < (↑d ^ n)⁻¹ | case intro.refine_2.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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ 0 < (↑d ^ n)⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2
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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ Complex.abs (s.bottcherNear (c, z).1 ((f (c, z).1)^[n] (c, z).2)) ≠ 0 ∨ 0 < (↑d ^ n)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | ContinuousAt.potential_of_reaches | [171, 1] | [184, 17] | bound | case intro.refine_2.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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ 0 < (↑d ^ n)⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.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 : ℂ
a✝ z : S
d n✝ : ℕ
s : Super f d a✝
n : ℕ
a : (c, (f c)^[n] z) ∈ s.near
e : (𝓝 (c, z)).EventuallyEq (uncurry s.potential) fun p => s.potential' p.1 p.2 n
⊢ 0 < (↑d ^ n)⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero | [187, 1] | [199, 78] | constructor | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ s.potential c z = 0 ↔ ∃ n, (f c)^[n] z = a | case mp
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ s.potential c z = 0 → ∃ n, (f c)^[n] z = a
case mpr
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ (∃ n, (f c)^[n] z = a) → s.potential c z = 0 | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ s.potential c z = 0 ↔ ∃ n, (f c)^[n] z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero | [187, 1] | [199, 78] | intro h | case mp
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ s.potential c z = 0 → ∃ n, (f c)^[n] z = a | case mp
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : s.potential c z = 0
⊢ ∃ n, (f c)^[n] z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ s.potential c z = 0 → ∃ n, (f c)^[n] z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero | [187, 1] | [199, 78] | by_cases r : ∃ n, (c, (f c)^[n] z) ∈ s.near | case mp
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : s.potential c z = 0
⊢ ∃ n, (f c)^[n] z = a | case pos
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : s.potential c z = 0
r : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ ∃ n, (f c)^[n] z = a
case neg
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : s.potential c z = 0
r : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ ∃ n, (f c)^[n] z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : s.potential c z = 0
⊢ ∃ n, (f c)^[n] z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero | [187, 1] | [199, 78] | rcases r with ⟨n, r⟩ | case pos
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : s.potential c z = 0
r : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ ∃ n, (f c)^[n] z = a | case pos.intro
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
h : s.potential c z = 0
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
⊢ ∃ n, (f c)^[n] z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : s.potential c z = 0
r : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ ∃ n, (f c)^[n] z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero | [187, 1] | [199, 78] | simp only [s.potential_eq r, Super.potential',
Real.rpow_eq_zero_iff_of_nonneg (Complex.abs.nonneg _), Complex.abs.eq_zero,
s.bottcherNear_eq_zero r] at h | case pos.intro
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
h : s.potential c z = 0
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
⊢ ∃ n, (f c)^[n] z = a | case pos.intro
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
h : (f c)^[n] z = a ∧ (↑d ^ n)⁻¹ ≠ 0
⊢ ∃ n, (f c)^[n] z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
h : s.potential c z = 0
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
⊢ ∃ n, (f c)^[n] z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero | [187, 1] | [199, 78] | use n, h.1 | case pos.intro
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
h : (f c)^[n] z = a ∧ (↑d ^ n)⁻¹ ≠ 0
⊢ ∃ n, (f c)^[n] z = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
r : (c, (f c)^[n] z) ∈ s.near
h : (f c)^[n] z = a ∧ (↑d ^ n)⁻¹ ≠ 0
⊢ ∃ n, (f c)^[n] z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero | [187, 1] | [199, 78] | rw [not_exists] at r | case neg
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : s.potential c z = 0
r : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ ∃ n, (f c)^[n] z = a | case neg
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : s.potential c z = 0
r : ∀ (x : ℕ), (c, (f c)^[x] z) ∉ s.near
⊢ ∃ n, (f c)^[n] z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : s.potential c z = 0
r : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ ∃ n, (f c)^[n] z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero | [187, 1] | [199, 78] | simp only [s.potential_eq_one r, one_ne_zero] at h | case neg
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : s.potential c z = 0
r : ∀ (x : ℕ), (c, (f c)^[x] z) ∉ s.near
⊢ ∃ n, (f c)^[n] z = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
h : s.potential c z = 0
r : ∀ (x : ℕ), (c, (f c)^[x] z) ∉ s.near
⊢ ∃ n, (f c)^[n] z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero | [187, 1] | [199, 78] | intro p | case mpr
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ (∃ n, (f c)^[n] z = a) → s.potential c z = 0 | case mpr
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
p : ∃ n, (f c)^[n] z = a
⊢ s.potential c z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ (∃ n, (f c)^[n] z = a) → s.potential c z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero | [187, 1] | [199, 78] | rcases p with ⟨n, p⟩ | case mpr
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
p : ∃ n, (f c)^[n] z = a
⊢ s.potential c z = 0 | case mpr.intro
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : (f c)^[n] z = a
⊢ s.potential c z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
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 : ℂ
a z : S
d n : ℕ
s : Super f d a
p : ∃ n, (f c)^[n] z = a
⊢ s.potential c z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero | [187, 1] | [199, 78] | have nz : d^n > 0 := pow_pos s.dp _ | case mpr.intro
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : (f c)^[n] z = a
⊢ s.potential c z = 0 | case mpr.intro
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : (f c)^[n] z = a
nz : d ^ n > 0
⊢ s.potential c z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : (f c)^[n] z = a
⊢ s.potential c z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero | [187, 1] | [199, 78] | rw [← pow_eq_zero_iff nz.ne', ← s.potential_eqn_iter n, p, s.potential_a] | case mpr.intro
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : (f c)^[n] z = a
nz : d ^ n > 0
⊢ s.potential c z = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
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 : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n : ℕ
p : (f c)^[n] z = a
nz : d ^ n > 0
⊢ s.potential c z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | UpperSemicontinuous.potential | [202, 1] | [208, 87] | intro ⟨c, z⟩ | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ UpperSemicontinuous (uncurry s.potential) | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
⊢ UpperSemicontinuousAt (uncurry s.potential) (c, z) | 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 : ℂ
a z : S
d n : ℕ
s : Super f d a
⊢ UpperSemicontinuous (uncurry s.potential)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | UpperSemicontinuous.potential | [202, 1] | [208, 87] | by_cases r : ∃ n, (c, (f c)^[n] z) ∈ s.near | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
⊢ UpperSemicontinuousAt (uncurry s.potential) (c, z) | case pos
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
r : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ UpperSemicontinuousAt (uncurry s.potential) (c, z)
case neg
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
r : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ UpperSemicontinuousAt (uncurry s.potential) (c, z) | 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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
⊢ UpperSemicontinuousAt (uncurry s.potential) (c, z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | UpperSemicontinuous.potential | [202, 1] | [208, 87] | exact (ContinuousAt.potential_of_reaches s r).upperSemicontinuousAt | case pos
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
r : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ UpperSemicontinuousAt (uncurry s.potential) (c, z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
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✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
c : ℂ
z : S
r : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ UpperSemicontinuousAt (uncurry s.potential) (c, z)
TACTIC:
|
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