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stringclasses 147
values | file_path
stringlengths 7
101
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stringlengths 1
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stringlengths 6
10
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stringlengths 6
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stringlengths 1
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|---|---|---|---|---|---|---|---|---|---|
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:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | UpperSemicontinuous.potential | [202, 1] | [208, 87] | simp only [uncurry, UpperSemicontinuousAt, s.potential_eq_one (not_exists.mp 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
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
⊢ ∀ (y : ℝ), 1 < y → ∀ᶠ (x' : ℂ × S) in 𝓝 (c, z), s.potential x'.1 x'.2 < y | 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
r : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ 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 fun y y1 ↦ eventually_of_forall fun p ↦ lt_of_le_of_lt s.potential_le_one y1 | 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
⊢ ∀ (y : ℝ), 1 < y → ∀ᶠ (x' : ℂ × S) in 𝓝 (c, z), s.potential x'.1 x'.2 < y | 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
r : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ ∀ (y : ℝ), 1 < y → ∀ᶠ (x' : ℂ × S) in 𝓝 (c, z), s.potential x'.1 x'.2 < y
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.preimage_eq' | [216, 1] | [217, 69] | have e := o.eq_a 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
o : OnePreimage s
⊢ f c z = a ↔ z = 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
o : OnePreimage s
e : f c z = a → z = a
⊢ f c z = a ↔ z = a | 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
o : OnePreimage s
⊢ f c z = a ↔ z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.preimage_eq' | [216, 1] | [217, 69] | refine ⟨e, ?_⟩ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
o : OnePreimage s
e : f c z = a → z = a
⊢ f c z = a ↔ z = 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
o : OnePreimage s
e : f c z = a → z = a
⊢ z = a → f c z = a | 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
o : OnePreimage s
e : f c z = a → z = a
⊢ f c z = a ↔ z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.preimage_eq' | [216, 1] | [217, 69] | intro e | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
o : OnePreimage s
e : f c z = a → z = a
⊢ z = a → f c z = 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
o : OnePreimage s
e✝ : f c z = a → z = a
e : z = a
⊢ f c z = a | 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
o : OnePreimage s
e : f c z = a → z = a
⊢ z = a → f c z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.preimage_eq' | [216, 1] | [217, 69] | simp only [e, s.f0] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
o : OnePreimage s
e✝ : f c z = a → z = a
e : z = a
⊢ f c z = a | 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
o : OnePreimage s
e✝ : f c z = a → z = a
e : z = a
⊢ f c z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.preimage_eq | [219, 1] | [222, 62] | 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
o : OnePreimage s
n : ℕ
⊢ (f c)^[n] z = a ↔ z = a | 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
o : OnePreimage s
⊢ (f c)^[0] z = a ↔ z = a
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
o : OnePreimage s
n : ℕ
h : (f c)^[n] z = a ↔ z = a
⊢ (f c)^[n + 1] z = a ↔ z = a | 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
o : OnePreimage s
n : ℕ
⊢ (f c)^[n] z = a ↔ z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.preimage_eq | [219, 1] | [222, 62] | simp only [Function.iterate_zero_apply] | 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
o : OnePreimage s
⊢ (f c)^[0] z = a ↔ z = a
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
o : OnePreimage s
n : ℕ
h : (f c)^[n] z = a ↔ z = a
⊢ (f c)^[n + 1] z = a ↔ z = a | 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
o : OnePreimage s
n : ℕ
h : (f c)^[n] z = a ↔ z = a
⊢ (f c)^[n + 1] z = a ↔ z = a | 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
o : OnePreimage s
⊢ (f c)^[0] z = a ↔ z = a
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
o : OnePreimage s
n : ℕ
h : (f c)^[n] z = a ↔ z = a
⊢ (f c)^[n + 1] z = a ↔ z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.preimage_eq | [219, 1] | [222, 62] | simp only [Function.iterate_succ_apply', s.preimage_eq', 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
o : OnePreimage s
n : ℕ
h : (f c)^[n] z = a ↔ z = a
⊢ (f c)^[n + 1] z = a ↔ z = a | 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
o : OnePreimage s
n : ℕ
h : (f c)^[n] z = a ↔ z = a
⊢ (f c)^[n + 1] z = a ↔ z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero_of_onePreimage | [224, 1] | [228, 42] | constructor | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ s.potential c z = 0 ↔ z = a | case mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ s.potential c z = 0 → z = a
case mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ z = a → s.potential c z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ s.potential c z = 0 ↔ z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero_of_onePreimage | [224, 1] | [228, 42] | intro h | case mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ s.potential c z = 0 → z = a | case mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : s.potential c z = 0
⊢ z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ s.potential c z = 0 → z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero_of_onePreimage | [224, 1] | [228, 42] | rw [s.potential_eq_zero] at h | case mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : s.potential c z = 0
⊢ z = a | case mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : ∃ n, (f c)^[n] z = a
⊢ z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : s.potential c z = 0
⊢ z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero_of_onePreimage | [224, 1] | [228, 42] | rcases h with ⟨n, h⟩ | case mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : ∃ n, (f c)^[n] z = a
⊢ z = a | case mp.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : ℕ
h : (f c)^[n] z = a
⊢ z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : ∃ n, (f c)^[n] z = a
⊢ z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero_of_onePreimage | [224, 1] | [228, 42] | rw [s.preimage_eq] at h | case mp.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : ℕ
h : (f c)^[n] z = a
⊢ z = a | case mp.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : ℕ
h : z = a
⊢ z = a | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : ℕ
h : (f c)^[n] z = a
⊢ z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero_of_onePreimage | [224, 1] | [228, 42] | exact h | case mp.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : ℕ
h : z = a
⊢ z = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : ℕ
h : z = a
⊢ z = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero_of_onePreimage | [224, 1] | [228, 42] | intro h | case mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ z = a → s.potential c z = 0 | case mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : z = a
⊢ s.potential c z = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ z = a → s.potential c z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_eq_zero_of_onePreimage | [224, 1] | [228, 42] | simp only [h, s.potential_a] | case mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : z = a
⊢ s.potential c z = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
h : z = a
⊢ s.potential c z = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_ne_zero | [230, 1] | [231, 89] | simp only [Ne, s.potential_eq_zero_of_onePreimage] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ s.potential c z ≠ 0 ↔ z ≠ a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ s.potential c z ≠ 0 ↔ z ≠ a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_pos | [233, 1] | [236, 61] | rw [← s.potential_ne_zero c] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ 0 < s.potential c z ↔ z ≠ a | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ 0 < s.potential c z ↔ s.potential c z ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ 0 < s.potential c z ↔ z ≠ a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.potential_pos | [233, 1] | [236, 61] | use ne_of_gt, fun ne ↦ ne.symm.lt_of_le s.potential_nonneg | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ 0 < s.potential c z ↔ s.potential c z ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
⊢ 0 < s.potential c z ↔ s.potential c z ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | have h : ∀ q : ℂ × S, f q.1 q.2 = a → q.2 = a := fun _ ↦ by simp only [s.preimage_eq', imp_self] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), ∀ (q : ℂ × S), p = s.fp q → q ∈ n | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), ∀ (q : ℂ × S), p = s.fp q → q ∈ n | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), ∀ (q : ℂ × S), p = s.fp q → q ∈ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | contrapose h | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), ∀ (q : ℂ × S), p = s.fp q → q ∈ n | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ¬∀ᶠ (p : ℂ × S) in 𝓝 (c, a), ∀ (q : ℂ × S), p = s.fp q → q ∈ n
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a
⊢ ∀ᶠ (p : ℂ × S) in 𝓝 (c, a), ∀ (q : ℂ × S), p = s.fp q → q ∈ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | simp only [Filter.not_eventually, not_forall, exists_prop] at h | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ¬∀ᶠ (p : ℂ × S) in 𝓝 (c, a), ∀ (q : ℂ × S), p = s.fp q → q ∈ n
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ¬∀ᶠ (p : ℂ × S) in 𝓝 (c, a), ∀ (q : ℂ × S), p = s.fp q → q ∈ n
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | set t := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | have tc : IsClosed t := by
refine (IsCompact.image ?_ s.fpa.continuous).isClosed
exact ((isCompact_closedBall _ _).prod isCompact_univ).inter_right no.isClosed_compl | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | have th : ∃ᶠ p in 𝓝 (c, a), p ∈ t := by
have mb : ∀ᶠ p : ℂ × S in 𝓝 (c, a), p.1 ∈ closedBall c 1 :=
continuousAt_fst.eventually_mem_nhd (Metric.closedBall_mem_nhds _ zero_lt_one)
refine (h.and_eventually mb).mp (eventually_of_forall fun p i ↦ ?_)
rcases i with ⟨⟨q, qp, m⟩, b⟩
simp only [Prod.ext_iff] at qp; simp only [qp.1] at b
simp only [Set.mem_image, Set.mem_compl_iff, Set.mem_inter_iff, Set.mem_prod_eq, Set.mem_univ,
and_true_iff, Prod.ext_iff, t]
use q, ⟨b, m⟩, qp.1.symm, qp.2.symm | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | have m := th.mem_of_closed tc | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m : (c, a) ∈ t
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | rcases(Set.mem_image _ _ _).mp m with ⟨p, m, pa⟩ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m : (c, a) ∈ t
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a | case intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m✝ : (c, a) ∈ t
p : ℂ × S
m : p ∈ closedBall c 1 ×ˢ univ ∩ nᶜ
pa : s.fp p = (c, a)
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m : (c, a) ∈ t
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | simp only [Super.fp, Prod.mk.inj_iff] at pa | case intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m✝ : (c, a) ∈ t
p : ℂ × S
m : p ∈ closedBall c 1 ×ˢ univ ∩ nᶜ
pa : s.fp p = (c, a)
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a | case intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m✝ : (c, a) ∈ t
p : ℂ × S
m : p ∈ closedBall c 1 ×ˢ univ ∩ nᶜ
pa : p.1 = c ∧ f p.1 p.2 = a
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m✝ : (c, a) ∈ t
p : ℂ × S
m : p ∈ closedBall c 1 ×ˢ univ ∩ nᶜ
pa : s.fp p = (c, a)
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | simp only [not_forall] | case intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m✝ : (c, a) ∈ t
p : ℂ × S
m : p ∈ closedBall c 1 ×ˢ univ ∩ nᶜ
pa : p.1 = c ∧ f p.1 p.2 = a
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a | case intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m✝ : (c, a) ∈ t
p : ℂ × S
m : p ∈ closedBall c 1 ×ˢ univ ∩ nᶜ
pa : p.1 = c ∧ f p.1 p.2 = a
⊢ ∃ x, ∃ (_ : f x.1 x.2 = a), ¬x.2 = a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m✝ : (c, a) ∈ t
p : ℂ × S
m : p ∈ closedBall c 1 ×ˢ univ ∩ nᶜ
pa : p.1 = c ∧ f p.1 p.2 = a
⊢ ¬∀ (q : ℂ × S), f q.1 q.2 = a → q.2 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | use p, pa.2 | case intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m✝ : (c, a) ∈ t
p : ℂ × S
m : p ∈ closedBall c 1 ×ˢ univ ∩ nᶜ
pa : p.1 = c ∧ f p.1 p.2 = a
⊢ ∃ x, ∃ (_ : f x.1 x.2 = a), ¬x.2 = a | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m✝ : (c, a) ∈ t
p : ℂ × S
m : p ∈ closedBall c 1 ×ˢ univ ∩ nᶜ
pa : p.1 = c ∧ f p.1 p.2 = a
⊢ ¬p.2 = a | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m✝ : (c, a) ∈ t
p : ℂ × S
m : p ∈ closedBall c 1 ×ˢ univ ∩ nᶜ
pa : p.1 = c ∧ f p.1 p.2 = a
⊢ ∃ x, ∃ (_ : f x.1 x.2 = a), ¬x.2 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | contrapose m | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m✝ : (c, a) ∈ t
p : ℂ × S
m : p ∈ closedBall c 1 ×ˢ univ ∩ nᶜ
pa : p.1 = c ∧ f p.1 p.2 = a
⊢ ¬p.2 = a | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m✝ : (c, a) ∈ t
p : ℂ × S
pa : p.1 = c ∧ f p.1 p.2 = a
m : ¬¬p.2 = a
⊢ p ∉ closedBall c 1 ×ˢ univ ∩ nᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m✝ : (c, a) ∈ t
p : ℂ × S
m : p ∈ closedBall c 1 ×ˢ univ ∩ nᶜ
pa : p.1 = c ∧ f p.1 p.2 = a
⊢ ¬p.2 = a
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | simp only [not_not, Set.mem_compl_iff] at m ⊢ | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m✝ : (c, a) ∈ t
p : ℂ × S
pa : p.1 = c ∧ f p.1 p.2 = a
m : ¬¬p.2 = a
⊢ p ∉ closedBall c 1 ×ˢ univ ∩ nᶜ | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m✝ : (c, a) ∈ t
p : ℂ × S
pa : p.1 = c ∧ f p.1 p.2 = a
m : p.2 = a
⊢ p ∉ closedBall c 1 ×ˢ univ ∩ nᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m✝ : (c, a) ∈ t
p : ℂ × S
pa : p.1 = c ∧ f p.1 p.2 = a
m : ¬¬p.2 = a
⊢ p ∉ closedBall c 1 ×ˢ univ ∩ nᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | rw [← @Prod.mk.eta _ _ p, pa.1, m] | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m✝ : (c, a) ∈ t
p : ℂ × S
pa : p.1 = c ∧ f p.1 p.2 = a
m : p.2 = a
⊢ p ∉ closedBall c 1 ×ˢ univ ∩ nᶜ | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m✝ : (c, a) ∈ t
p : ℂ × S
pa : p.1 = c ∧ f p.1 p.2 = a
m : p.2 = a
⊢ (c, a) ∉ closedBall c 1 ×ˢ univ ∩ nᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
h : ∃ᶠ (x : ℂ × S) in 𝓝 (c, a), ∃ x_1, x = s.fp x_1 ∧ x_1 ∉ n
t : Set (ℂ × S) := s.fp '' (closedBall c 1 ×ˢ univ ∩ nᶜ)
tc : IsClosed t
th : ∃ᶠ (p : ℂ × S) in 𝓝 (c, a), p ∈ t
m✝ : (c, a) ∈ t
p : ℂ × S
pa : p.1 = c ∧ f p.1 p.2 = a
m : p.2 = a
⊢ p ∉ closedBall c 1 ×ˢ univ ∩ nᶜ
TACTIC:
|
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