url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | simp only [Set.mem_inter_iff, Set.prod_mk_mem_set_prod_eq, Metric.mem_closedBall, dist_self,
zero_le_one, Set.mem_univ, Set.mem_compl_iff, true_and_iff, Set.not_not_mem, not_not,
na] | 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 ... | no goals | 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 :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | 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
x✝ : ℂ × S
⊢ f x✝.1 x✝.2 = a → x✝.2 = 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | refine (IsCompact.image ?_ s.fpa.continuous).isClosed | S : Type
inst✝⁵ : TopologicalSpace 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... | S : Type
inst✝⁵ : TopologicalSpace 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... | 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)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | 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... | 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | 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) | S : Type
inst✝⁵ : TopologicalSpace 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... | S : Type
inst✝⁵ : TopologicalSpace 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... | 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)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | refine (h.and_eventually mb).mp (eventually_of_forall fun p i ↦ ?_) | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : 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... | S : Type
inst✝⁵ : TopologicalSpace 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... | 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)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | rcases i with ⟨⟨q, qp, m⟩, b⟩ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : 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... | case intro.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), ... | 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)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | simp only [Prod.ext_iff] at qp | case intro.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), ... | case intro.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), ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.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... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | simp only [qp.1] at b | case intro.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), ... | case intro.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), ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.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... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | 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] | case intro.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), ... | case intro.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), ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.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... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.no_jump | [239, 1] | [265, 8] | use q, ⟨b, m⟩, qp.1.symm, qp.2.symm | case intro.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), ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.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... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | set n' := n ∩ s.near | S : Type
inst✝⁵ : TopologicalSpace 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
⊢ ∃ t, Barrier s c n t | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
⊢ ∃ t, Barrier s c n t | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
⊢ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | have nn' : n' ∈ 𝓝 (c, a) :=
Filter.inter_mem (no.mem_nhds na) (s.isOpen_near.mem_nhds (s.mem_near 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
⊢ ∃ t, Barrier s c n t | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
⊢ ∃ t, Barr... | 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n'... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | rcases (Filter.hasBasis_iff.mp (compact_basis_nhds (c, a)) n').mp nn' with ⟨u, ⟨un, uc⟩, us⟩ | S : Type
inst✝⁵ : TopologicalSpace 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
⊢ ∃ t, Barr... | case intro.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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n'... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | simp only [Set.subset_inter_iff, n'] at us | case intro.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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ... | case intro.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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.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
n : Set (ℂ × S)
no : IsOp... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | rcases eventually_nhds_iff.mp
(s.no_jump c (interior u) isOpen_interior (mem_interior_iff_mem_nhds.mpr un)) with
⟨i, ih, io, ia⟩ | case intro.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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ... | case intro.intro.intro.intro.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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.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
n : Set (ℂ × S)
no : IsOp... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | rcases mem_nhds_prod_iff'.mp (Filter.inter_mem un (io.mem_nhds ia)) with
⟨i0, i1, i0o, i0m, i1o, i1m, ii⟩ | case intro.intro.intro.intro.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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ... | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.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
n : Set (ℂ × S)
no : IsOpen n
na... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.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
n : Set... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | simp only [Set.subset_inter_iff] at ii | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.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
n : Set (ℂ × S)
no : IsOpen n
na... | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.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
n : Set (ℂ × S)
no : IsOpen n
na... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.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... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | set t := u \ univ ×ˢ i1 | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.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
n : Set (ℂ × S)
no : IsOpen n
na... | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.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
n : Set (ℂ × S)
no : IsOpen n
na... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.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... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | have ta : ∀ e, (e, a) ∉ t := fun e ↦
Set.not_mem_diff_of_mem (Set.mk_mem_prod (Set.mem_univ _) i1m) | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.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
n : Set (ℂ × S)
no : IsOpen n
na... | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.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
n : Set (ℂ × S)
no : IsOpen n
na... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.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... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | use t | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.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
n : Set (ℂ × S)
no : IsOpen n
na... | 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
u : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.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... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | refine ⟨uc.diff (isOpen_univ.prod i1o), _root_.trans (Set.diff_subset _ _) us.1,
_root_.trans (Set.diff_subset _ _) us.2, ta, ?_⟩ | 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
u : ... | 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
u : ... | 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | rw [eventually_nhds_iff] | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
u : ... | 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
u : ... | 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | use i0 | 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
u : ... | 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
u : ... | 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | refine ⟨?_, i0o, i0m⟩ | 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
u : ... | 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
u : ... | 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | intro e em z zm za | 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
u : ... | 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
u :... | 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | rcases tendsto_atTop_nhds.mp za i1 i1m i1o with ⟨m, mh⟩ | 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
u :... | case h.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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ∈ 𝓝 (c, ... | 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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | have en : ∃ n, (f e)^[n] z ∈ i1 := ⟨m, mh m (le_refl _)⟩ | case h.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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ∈ 𝓝 (c, ... | case h.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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ∈ 𝓝 (c, ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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
n : Set (ℂ × S)
no : IsOpen n
na :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | set n := Nat.find en | case h.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
n : Set (ℂ × S)
no : IsOpen n
na : (c, a) ∈ n
n' : Set (ℂ × S) := n ∩ s.near
nn' : n' ∈ 𝓝 (c, ... | case h.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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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
n : Set (ℂ × S)
no : IsOpen n
na :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | use n - 1 | case h.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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝... | 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c, a... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
n... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | have ni1 : (f e)^[n] z ∈ i1 := Nat.find_spec en | 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c, 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c, a... | 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | have n0 : n ≠ 0 := by
contrapose zm; simp only [Set.not_not_mem]
simp only [Nat.sub, Ne, Nat.find_eq_zero en, Function.iterate_zero, id,
Set.not_not_mem] at zm
exact us.1 (ii.1 (Set.mk_mem_prod em zm)) | 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c, 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c, a... | 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | have nt : (f e)^[n-1] z ∉ i1 := Nat.find_min en (Nat.pred_lt n0) | 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c, 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c, a... | 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | apply Set.mem_diff_of_mem | 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c, a... | case h.h1
S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c... | 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | contrapose zm | S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
u : S... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
u : S... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | simp only [Set.not_not_mem] | S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
u : S... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
u : S... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | simp only [Nat.sub, Ne, Nat.find_eq_zero en, Function.iterate_zero, id,
Set.not_not_mem] at zm | S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
u : S... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
u : S... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z✝ : S
d n✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | exact us.1 (ii.1 (Set.mk_mem_prod em zm)) | S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c, a)
u : S... | 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | apply interior_subset | case h.h1
S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c... | case h.h1.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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1
S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | apply ih (e, (f e)^[n] z) (ii.2 (Set.mk_mem_prod em ni1)) | case h.h1.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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 ... | case h.h1.a.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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ �... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1.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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | simp only [Super.fp] | case h.h1.a.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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ �... | case h.h1.a.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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ �... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1.a.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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | rw [← Function.iterate_succ_apply' (f e) (n - 1)] | case h.h1.a.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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ �... | case h.h1.a.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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ �... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1.a.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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | simp only [Nat.succ_eq_add_one, Nat.sub_add_cancel (Nat.one_le_of_lt (Nat.pos_of_ne_zero n0))] | case h.h1.a.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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ �... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h1.a.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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | contrapose nt | case h.h2
S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c... | case h.h2
S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h2
S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | simp only [Set.prod_mk_mem_set_prod_eq, not_and, not_forall, Set.not_not_mem,
exists_prop] at nt ⊢ | case h.h2
S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c... | case h.h2
S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h2
S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Super.barrier | [277, 1] | [316, 15] | exact nt.2 | case h.h2
S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na : (c, a) ∈ n✝
n' : Set (ℂ × S) := n✝ ∩ s.near
nn' : n' ∈ 𝓝 (c... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h2
S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n✝ : Set (ℂ × S)
no : IsOpen n✝
na :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | by_cases t0 : t = ∅ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c n t
⊢ ∃ r > 0, ∀ (e : ℂ) (z : S), (e, z) ∈ t → r ≤ s.potential e z | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t = ∅
⊢ ∃ r > 0, ∀ (e : ℂ) (z : S), (e, z) ∈ t → r ≤ s.po... | 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
n t : Set (ℂ × S)
b : Barrier s c n t
⊢ ∃ r > 0,... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | simp only [← ne_eq, ← Set.nonempty_iff_ne_empty] at t0 | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : ¬t = ∅
⊢ ∃ r > 0, ∀ (e : ℂ) (z : S), (e, z) ∈ t → r ≤ s.p... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
⊢ ∃ r > 0, ∀ (e : ℂ) (z : S), (e, z) ∈ t → r ≤... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c n t
t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | have pc : ContinuousOn (uncurry s.potential) t := by
refine ContinuousOn.mono ?_ b.near
intro ⟨c, z⟩ m; apply ContinuousAt.continuousWithinAt
apply ContinuousAt.potential_of_reaches s; use 0; simpa only [Function.iterate_zero_apply] | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
⊢ ∃ r > 0, ∀ (e : ℂ) (z : S), (e, z) ∈ t → r ≤... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) t
⊢ ∃ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c n t
t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | rcases b.compact.exists_isMinOn t0 pc with ⟨⟨e, z⟩, ps, pm⟩ | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) t
⊢ ∃ ... | case neg.intro.mk.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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.p... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c n t
t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | use s.potential e z | case neg.intro.mk.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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.p... | 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) t
e : ℂ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.mk.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
n t : Set (ℂ × S)
b : B... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | constructor | 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) t
e : ℂ... | case h.left
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) t
... | 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | use 1, zero_lt_one | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t = ∅
⊢ ∃ r > 0, ∀ (e : ℂ) (z : S), (e, z) ∈ t → r ≤ s.po... | case right
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t = ∅
⊢ ∀ (e : ℂ) (z : S), (e, z) ∈ t → 1 ≤ s.potential... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c n t
t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | simp only [t0, gt_iff_lt, Set.mem_empty_iff_false, IsEmpty.forall_iff, forall_const,
imp_true_iff, and_true_iff] | case right
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t = ∅
⊢ ∀ (e : ℂ) (z : S), (e, z) ∈ t → 1 ≤ s.potential... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n t... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | refine ContinuousOn.mono ?_ b.near | S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
⊢ ContinuousOn (uncurry s.potential) t | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
⊢ ContinuousOn (uncurry s.potential) s.near | 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Non... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | intro ⟨c, z⟩ m | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
⊢ ContinuousOn (uncurry s.potential) s.near | S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c✝ n t
t0 : t.Nonempty
c : ℂ
z : S
m : (c, z) ∈ s.near
⊢ ContinuousWithinAt... | 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Non... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | apply ContinuousAt.continuousWithinAt | S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c✝ n t
t0 : t.Nonempty
c : ℂ
z : S
m : (c, z) ∈ s.near
⊢ ContinuousWithinAt... | 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
n t : Set (ℂ × S)
b : Barrier s c✝ n t
t0 : t.Nonempty
c : ℂ
z : S
m : (c, z) ∈ s.near
⊢ ContinuousA... | 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
n t : Set (ℂ × S)
b : Barrier s c✝ n t
t0 : t.... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | apply ContinuousAt.potential_of_reaches s | 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
n t : Set (ℂ × S)
b : Barrier s c✝ n t
t0 : t.Nonempty
c : ℂ
z : S
m : (c, z) ∈ s.near
⊢ ContinuousA... | 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
n t : Set (ℂ × S)
b : Barrier s c✝ n t
t0 : t.Nonempty
c : ℂ
z : S
m : (c, z) ∈ s.near
⊢ ∃ n, (c, (f... | 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
n t : Set (ℂ × S)
b : Barrier s c✝ n t
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | use 0 | 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
n t : Set (ℂ × S)
b : Barrier s c✝ n t
t0 : t.Nonempty
c : ℂ
z : S
m : (c, z) ∈ s.near
⊢ ∃ n, (c, (f... | 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
n t : Set (ℂ × S)
b : Barrier s c✝ n t
t0 : t.Nonempty
c : ℂ
z : S
m : (c, z) ∈ s.near
⊢ (c, (f c)^[... | 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
n t : Set (ℂ × S)
b : Barrier s c✝ n t
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | simpa only [Function.iterate_zero_apply] | 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
n t : Set (ℂ × S)
b : Barrier s c✝ n t
t0 : t.Nonempty
c : ℂ
z : S
m : (c, z) ∈ s.near
⊢ (c, (f c)^[... | no goals | 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
n t : Set (ℂ × S)
b : Barrier s c✝ n t
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | have h := b.hole e | case h.left
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) t
... | case h.left
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) t
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | contrapose h | case h.left
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) t
... | case h.left
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) t
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | simp only [not_lt] at h | case h.left
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) t
... | case h.left
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) t
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | have h' := le_antisymm h s.potential_nonneg | case h.left
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) t
... | case h.left
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) t
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | simp only [s.potential_eq_zero, s.preimage_eq, exists_const] at h' | case h.left
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) t
... | case h.left
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) t
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | simp only [not_not, ← h', ps] | case h.left
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) t
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | intro e z m | case h.right
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) t... | case h.right
S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
S : Type
inst✝⁵ : TopologicalSpace 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
n t : Set (ℂ × S)
b : Barrier s c ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | simp only [isMinOn_iff, uncurry] at pm ⊢ | case h.right
S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) ... | case h.right
S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.potential_large | [319, 1] | [336, 72] | exact pm _ m | case h.right
S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c n t
t0 : t.Nonempty
pc : ContinuousOn (uncurry s.potential) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
S : Type
inst✝⁵ : TopologicalSpace 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✝¹ : S
d n✝ : ℕ
s : Super f d a
inst✝ : OnePreimage s
n t : Set (ℂ × S)
b : Barrier s c... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.closed_fast | [344, 1] | [349, 98] | apply isClosed_iUnion_of_finite | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : 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 t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
⊢ IsClosed (b.fast m) | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
⊢ ∀ (i : Fin m), IsClosed ((fun p => (p.1, (f p.1)^[↑i] p.2)) ⁻¹' t) | 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 t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
⊢ IsClosed (b.fast m)
TACTI... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.closed_fast | [344, 1] | [349, 98] | intro k | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
⊢ ∀ (i : Fin m), IsClosed ((fun p => (p.1, (f p.1)^[↑i] p.2)) ⁻¹' t) | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
k : Fin m
⊢ IsClosed ((fun p => (p.1, (f p.1)^[↑k] p.2)) ⁻¹' t) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
⊢ ∀ (i : Fin m), IsC... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.closed_fast | [344, 1] | [349, 98] | refine IsClosed.preimage ?_ b.compact.isClosed | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
k : Fin m
⊢ IsClosed ((fun p => (p.1, (f p.1)^[↑k] p.2)) ⁻¹' t) | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
k : Fin m
⊢ Continuous fun p => (p.1, (f p.1)^[↑k] p.2) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
k : Fin m
⊢ IsClosed... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.closed_fast | [344, 1] | [349, 98] | apply continuous_fst.prod_mk | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
k : Fin m
⊢ Continuous fun p => (p.1, (f p.1)^[↑k] p.2) | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
k : Fin m
⊢ Continuous fun x => (f x.1)^[↑k] x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
k : Fin m
⊢ Continuo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.closed_fast | [344, 1] | [349, 98] | generalize hn : (k : ℕ) = n | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
k : Fin m
⊢ Continuous fun x => (f x.1)^[↑k] x.2 | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m : ℕ
k : Fin m
n : ℕ
hn : ↑k = n
⊢ Continuous fun x => (f x.1)^[n] x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝ : ℕ
s : Super f d a
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
k : Fin m
⊢ Continuo... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.closed_fast | [344, 1] | [349, 98] | clear k hn | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m : ℕ
k : Fin m
n : ℕ
hn : ↑k = n
⊢ Continuous fun x => (f x.1)^[n] x.2 | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m n : ℕ
⊢ Continuous fun x => (f x.1)^[n] x.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m : ℕ
k : Fin m
n : ℕ
h... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.closed_fast | [344, 1] | [349, 98] | induction' n with n h | case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m n : ℕ
⊢ Continuous fun x => (f x.1)^[n] x.2 | case h.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
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
⊢ Continuous fun x => (f x.1)^[0] x.2
case h.succ
S : Type
inst✝⁴ : Topolog... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m n : ℕ
⊢ Continuous fu... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.closed_fast | [344, 1] | [349, 98] | simp only [Function.iterate_zero_apply] | case h.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
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
⊢ Continuous fun x => (f x.1)^[0] x.2
case h.succ
S : Type
inst✝⁴ : Topolog... | case h.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
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
⊢ Continuous fun x => x.2
case h.succ
S : Type
inst✝⁴ : TopologicalSpace S
... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
⊢ Continuous fu... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.closed_fast | [344, 1] | [349, 98] | exact continuous_snd | case h.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
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
⊢ Continuous fun x => x.2
case h.succ
S : Type
inst✝⁴ : TopologicalSpace S
... | case h.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✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m n : ℕ
h : Continuous fun x => (f x.1)^[n] x.2
⊢ Continuous fun x => (f x.1)^[... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
⊢ Continuous fu... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.closed_fast | [344, 1] | [349, 98] | simp only [Function.iterate_succ_apply'] | case h.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✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m n : ℕ
h : Continuous fun x => (f x.1)^[n] x.2
⊢ Continuous fun x => (f x.1)^[... | case h.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✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m n : ℕ
h : Continuous fun x => (f x.1)^[n] x.2
⊢ Continuous fun x => f x.1 ((f... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m n : ℕ
h : Contin... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.closed_fast | [344, 1] | [349, 98] | exact s.fa.continuous.comp (continuous_fst.prod_mk h) | case h.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✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m n : ℕ
h : Continuous fun x => (f x.1)^[n] x.2
⊢ Continuous fun x => f x.1 ((f... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.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✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m n : ℕ
h : Contin... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.mem_fast | [351, 1] | [355, 51] | simp only [Barrier.fast, Set.mem_iUnion, Set.mem_preimage] | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : 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 t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
⊢ (e, z) ∈ b.fast m ↔ ∃ n < m, (e, (f e)^[n] z) ∈ t | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : 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 t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
⊢ (∃ i, (e, (f e)^[↑i] z) ∈ t) ↔ ∃ n < m, (e, (f e)^[n] z) ∈ t | 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 t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
⊢ (e, z) ∈ b.f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.mem_fast | [351, 1] | [355, 51] | 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
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
⊢ (∃ i, (e, (f e)^[↑i] z) ∈ t) ↔ ∃ n < m, (e, (f e)^[n] z) ∈ t | 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
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
⊢ (∃ i, (e, (f e)^[↑i] z) ∈ t) → ∃ n < m, (e, (f e)^[n] z) ∈ t
cas... | 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 t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
⊢ (∃ i, (e, (f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.mem_fast | [351, 1] | [355, 51] | 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
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
⊢ (∃ i, (e, (f e)^[↑i] z) ∈ t) → ∃ n < m, (e, (f e)^[n] z) ∈ t
cas... | 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
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
h : ∃ i, (e, (f e)^[↑i] z) ∈ t
⊢ ∃ n < m, (e, (f e)^[n] z) ∈ t
cas... | 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
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
⊢ (∃ i... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.mem_fast | [351, 1] | [355, 51] | rcases h with ⟨n, 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
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
h : ∃ i, (e, (f e)^[↑i] z) ∈ t
⊢ ∃ n < m, (e, (f e)^[n] z) ∈ t
cas... | case mp.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✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m : ℕ
e : ℂ
z : S
n : Fin m
h : (e, (f e)^[↑n] z) ∈ t
⊢ ∃ n < m, (e, (f e)^[... | 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
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
h : ∃ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.mem_fast | [351, 1] | [355, 51] | use n, Fin.is_lt _, h | case mp.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✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m : ℕ
e : ℂ
z : S
n : Fin m
h : (e, (f e)^[↑n] z) ∈ t
⊢ ∃ n < m, (e, (f e)^[... | 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 t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
⊢ (∃ n < m, (e, (f e)^[n] z) ∈ t) → ∃ i, (e, (f e)^[↑i] z) ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.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✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m : ℕ
e : ℂ
z :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.mem_fast | [351, 1] | [355, 51] | intro h | 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 t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
⊢ (∃ n < m, (e, (f e)^[n] z) ∈ t) → ∃ i, (e, (f e)^[↑i] z) ∈ t | 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 t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
h : ∃ n < m, (e, (f e)^[n] z) ∈ t
⊢ ∃ i, (e, (f e)^[↑i] z) ∈ t | 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 t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
⊢ (∃ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.mem_fast | [351, 1] | [355, 51] | rcases h with ⟨n, nm, h⟩ | 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 t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
h : ∃ n < m, (e, (f e)^[n] z) ∈ t
⊢ ∃ i, (e, (f e)^[↑i] z) ∈ t | case mpr.intro.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✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m : ℕ
e : ℂ
z : S
n : ℕ
nm : n < m
h : (e, (f e)^[n] z) ∈ t
⊢ ∃ i, (e... | 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 t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
h : ∃... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.mem_fast | [351, 1] | [355, 51] | use⟨n, nm⟩, h | case mpr.intro.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✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m : ℕ
e : ℂ
z : S
n : ℕ
nm : n < m
h : (e, (f e)^[n] z) ∈ t
⊢ ∃ i, (e... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.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✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m : ℕ
e ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.fast_reaches | [357, 1] | [359, 69] | rw [b.mem_fast] at q | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : 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 t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
q : (e, z) ∈ b.fast m
⊢ ∃ n, (e, (f e)^[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
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
q : ∃ n < m, (e, (f e)^[n] z) ∈ t
⊢ ∃ n, (e, (f e)^[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
n t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
q : (e, z) ∈ b... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.fast_reaches | [357, 1] | [359, 69] | rcases q with ⟨n, _, q⟩ | S : Type
inst✝⁴ : TopologicalSpace S
inst✝³ : CompactSpace S
inst✝² : T3Space S
inst✝¹ : 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 t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
q : ∃ n < m, (e, (f e)^[n] z) ∈ t
⊢ ∃ n, (e, (f e)^[n] z) ∈ s.near | case intro.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✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m : ℕ
e : ℂ
z : S
n : ℕ
left✝ : n < m
q : (e, (f e)^[n] z) ∈ t
⊢ ∃ n, (e,... | 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 t : Set (ℂ × S)
b : Barrier s c n t
m : ℕ
e : ℂ
z : S
q : ∃ n < m, (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Barrier.fast_reaches | [357, 1] | [359, 69] | exact ⟨n, b.near q⟩ | case intro.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✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m : ℕ
e : ℂ
z : S
n : ℕ
left✝ : n < m
q : (e, (f e)^[n] z) ∈ t
⊢ ∃ n, (e,... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.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✝¹ : ℕ
s : Super f d a
n✝ t : Set (ℂ × S)
b : Barrier s c n✝ t
m : ℕ
e : ℂ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Continuous.potential | [362, 1] | [422, 22] | refine continuous_iff_lower_upperSemicontinuous.mpr ⟨?_, UpperSemicontinuous.potential s⟩ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c : ℂ
a z : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
⊢ Continuous (uncurry s.potential) | S : Type
inst✝⁵ : TopologicalSpace 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
⊢ LowerSemicontinuous (uncurry s.potential) | 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
⊢ Continuous (uncurry s.potential)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Continuous.potential | [362, 1] | [422, 22] | intro ⟨c, z⟩ | S : Type
inst✝⁵ : TopologicalSpace 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
⊢ LowerSemicontinuous (uncurry s.potential) | S : Type
inst✝⁵ : TopologicalSpace 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 : S
⊢ LowerSemicontinuousAt (uncurry s.potential) (c, z) | 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
⊢ LowerSemicontinuous (uncurry s.potential)
TACTI... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Continuous.potential | [362, 1] | [422, 22] | by_cases re : ∃ n, (c, (f c)^[n] z) ∈ s.near | S : Type
inst✝⁵ : TopologicalSpace 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 : S
⊢ LowerSemicontinuousAt (uncurry s.potential) (c, z) | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
re : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ LowerSemicontinuousAt (uncurry s.potential) (c, z... | 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
z : S
⊢ LowerSemicontinuousAt (uncurry s.... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Continuous.potential | [362, 1] | [422, 22] | exact (ContinuousAt.potential_of_reaches s re).lowerSemicontinuousAt | case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
re : ∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ LowerSemicontinuousAt (uncurry s.potential) (c, z... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
re : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ LowerSemicontinuousAt (uncurry s.potential) (c, ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
re : ∃ n, (c, (f c)^[n] z)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Continuous.potential | [362, 1] | [422, 22] | simp only [not_exists] at re | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
re : ¬∃ n, (c, (f c)^[n] z) ∈ s.near
⊢ LowerSemicontinuousAt (uncurry s.potential) (c, ... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
re : ∀ (x : ℕ), (c, (f c)^[x] z) ∉ s.near
⊢ LowerSemicontinuousAt (uncurry s.potential)... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
re : ¬∃ n, (c, (f c)^[n] z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Continuous.potential | [362, 1] | [422, 22] | intro y y1 | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
re : ∀ (x : ℕ), (c, (f c)^[x] z) ∉ s.near
⊢ LowerSemicontinuousAt (uncurry s.potential)... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
re : ∀ (x : ℕ), (c, (f c)^[x] z) ∉ s.near
y : ℝ
y1 : y < uncurry s.potential (c, z)
⊢ ∀... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
re : ∀ (x : ℕ), (c, (f c)^... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Continuous.potential | [362, 1] | [422, 22] | simp only [ContinuousAt, uncurry, s.potential_eq_one re] at y1 ⊢ | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
re : ∀ (x : ℕ), (c, (f c)^[x] z) ∉ s.near
y : ℝ
y1 : y < uncurry s.potential (c, z)
⊢ ∀... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
re : ∀ (x : ℕ), (c, (f c)^[x] z) ∉ s.near
y : ℝ
y1 : y < 1
⊢ ∀ᶠ (x' : ℂ × S) in 𝓝 (c, ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
re : ∀ (x : ℕ), (c, (f c)^... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Continuous.potential | [362, 1] | [422, 22] | contrapose re | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
re : ∀ (x : ℕ), (c, (f c)^[x] z) ∉ s.near
y : ℝ
y1 : y < 1
⊢ ∀ᶠ (x' : ℂ × S) in 𝓝 (c, ... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
y : ℝ
y1 : y < 1
re : ¬∀ᶠ (x' : ℂ × S) in 𝓝 (c, z), y < s.potential x'.1 x'.2
⊢ ¬∀ (x ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
re : ∀ (x : ℕ), (c, (f c)^... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Potential.lean | Continuous.potential | [362, 1] | [422, 22] | simp only [Filter.not_eventually, not_lt] at re | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
y : ℝ
y1 : y < 1
re : ¬∀ᶠ (x' : ℂ × S) in 𝓝 (c, z), y < s.potential x'.1 x'.2
⊢ ¬∀ (x ... | case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
y : ℝ
y1 : y < 1
re : ∃ᶠ (x : ℂ × S) in 𝓝 (c, z), s.potential x.1 x.2 ≤ y
⊢ ¬∀ (x : ℕ)... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ : ℂ
a z✝ : S
d n : ℕ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
z : S
y : ℝ
y1 : y < 1
re : ¬∀ᶠ ... |
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