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/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | simp only [mem_inter_iff, mem_setOf, mem_image, mem_closedBall, Complex.dist_eq, sub_zero,
Super.ext, j] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c ... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c ... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | constructor | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c ... | case mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.pote... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | intro ⟨zp1, x, xp, xz⟩ | case mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.pote... | case mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x✝ : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.pot... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | rw [← xz, s.ray_potential xp] at zp1 | case mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x✝ : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.pot... | case mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x✝ : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.pot... | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x✝ : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | use x, zp1, xz | case mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x✝ : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.pot... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x✝ : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | intro ⟨x, xp, xz⟩ | case mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.pot... | case mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x✝ : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.po... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | have zp1 := lt_of_le_of_lt xp post | case mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x✝ : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.po... | case mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x✝ : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.po... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x✝ : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | rw [← xz, s.ray_potential zp1] | case mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x✝ : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.po... | case mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x✝ : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.po... | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x✝ : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | use xp, x, zp1 | case mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x✝ : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.po... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x✝ : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | simp only [mem_diff, mem_setOf, u] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | use p01.le | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c z... | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | contrapose i0 | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
p0 : ℝ := s.potential c z0
m0 : s.potential c z0 < s.p c
p1 : ℝ... | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | simp only [not_not, mem_image, mem_setOf, not_forall, exists_prop] at i0 ⊢ | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
p0 : ℝ := s.potential c z0
m0 : s.potential c z0 < s.p c
p1 : ℝ... | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
p0 : ℝ := s.potential c z0
m0 : s.potential c z0 < s.p c
p1 : ℝ... | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
p0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | exact i0 | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
p0 : ℝ := s.potential c z0
m0 : s.potential c z0 < s.p c
p1 : ℝ... | 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 x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
p0... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | have m : z ∈ jᶜ := by rw [compl_inter]; right; exact zu.2 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c ... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c ... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | have lt : s.potential c z < p1 := lt_of_le_of_lt (zm z0u) p01 | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c ... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c ... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | apply (jc.isOpen_compl.eventually_mem m).mp | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c ... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c ... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | apply ((Continuous.potential s).along_snd.continuousAt.eventually_lt continuousAt_const lt).mp | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c ... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c ... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | refine eventually_of_forall fun w lt m ↦ ?_ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c ... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c ... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | rw [compl_inter] at m | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c ... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c ... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | cases' m with m m | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c ... | case inl
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.pot... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | rw [compl_inter] | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c ... | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c ... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | 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 x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.potential c ... | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.poten... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | exact zu.2 | case h
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.poten... | 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 x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | simp only [compl_setOf, mem_setOf, not_le] at m | case inl
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.pot... | case inl
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.pot... | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | linarith | case inl
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.pot... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | apply zm | case inr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.pot... | case inr.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 x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.p... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | simp only [mem_diff, mem_setOf, u] | case inr.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 x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.p... | case inr.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 x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.p... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.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 x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_surj | [291, 1] | [338, 79] | use lt.le, m | case inr.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 x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i0 : ∀ (x : ℂ), (c, x) ∈ s.ext → ¬s.ray c x = z0
p0 : ℝ := s.p... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.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 x : ℂ
a z✝ : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
z0 : S
i... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_bij | [341, 1] | [347, 61] | refine ⟨fun _ m ↦ s.ray_post m, ?_, ?_⟩ | S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ BijOn (fun y => (y.1, s.ray y.1 y.2)) s.ext s.post | case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ InjOn (fun y => (y.1, s.ray y.1 y.2)) s.ext
case refine_2
S : Ty... | Please generate a tactic in lean4 to solve the state.
STATE:
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ BijOn (fun y => (y... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_bij | [341, 1] | [347, 61] | intro ⟨c0, x0⟩ m0 ⟨c1, x1⟩ m1 e | case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ InjOn (fun y => (y.1, s.ray y.1 y.2)) s.ext | case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c0 x0 : ℂ
m0 : (c0, x0) ∈ s.ext
c1 x1 : ℂ
m1 : (c1, x1) ∈ s.ext
e :... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ InjO... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_bij | [341, 1] | [347, 61] | simp only [Prod.ext_iff] at e ⊢ | case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c0 x0 : ℂ
m0 : (c0, x0) ∈ s.ext
c1 x1 : ℂ
m1 : (c1, x1) ∈ s.ext
e :... | case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c0 x0 : ℂ
m0 : (c0, x0) ∈ s.ext
c1 x1 : ℂ
m1 : (c1, x1) ∈ s.ext
e :... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c0 x0 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_bij | [341, 1] | [347, 61] | rcases e with ⟨ec, ex⟩ | case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c0 x0 : ℂ
m0 : (c0, x0) ∈ s.ext
c1 x1 : ℂ
m1 : (c1, x1) ∈ s.ext
e :... | case refine_1.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c0 x0 : ℂ
m0 : (c0, x0) ∈ s.ext
c1 x1 : ℂ
m1 : (c1, x1) ∈ s.e... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c0 x0 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_bij | [341, 1] | [347, 61] | rw [ec] at m0 ex | case refine_1.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c0 x0 : ℂ
m0 : (c0, x0) ∈ s.ext
c1 x1 : ℂ
m1 : (c1, x1) ∈ s.e... | case refine_1.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c0 x0 c1 : ℂ
m0 : (c1, x0) ∈ s.ext
x1 : ℂ
m1 : (c1, x1) ∈ s.e... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_bij | [341, 1] | [347, 61] | use ec, s.ray_inj m0 m1 ex | case refine_1.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c0 x0 c1 : ℂ
m0 : (c1, x0) ∈ s.ext
x1 : ℂ
m1 : (c1, x1) ∈ s.e... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_bij | [341, 1] | [347, 61] | intro ⟨c, x⟩ m | case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ SurjOn (fun y => (y.1, s.ray y.1 y.2)) s.ext s.post | case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
x : S
m : (c, x) ∈ s.post
⊢ (c, x) ∈ (fun y => (y.1, s.ray ... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c x : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
⊢ Surj... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_bij | [341, 1] | [347, 61] | simp only [mem_image, Prod.ext_iff] | case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
x : S
m : (c, x) ∈ s.post
⊢ (c, x) ∈ (fun y => (y.1, s.ray ... | case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
x : S
m : (c, x) ∈ s.post
⊢ ∃ x_1 ∈ s.ext, x_1.1 = c ∧ s.ra... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_bij | [341, 1] | [347, 61] | rcases s.ray_surj m with ⟨x, m, e⟩ | case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
x : S
m : (c, x) ∈ s.post
⊢ ∃ x_1 ∈ s.ext, x_1.1 = c ∧ s.ra... | case refine_2.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝¹ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
x✝ : S
m✝ : (c, x✝) ∈ s.post
x : ℂ
m : (c, x) ... | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Ray.lean | Super.ray_bij | [341, 1] | [347, 61] | use⟨c, x⟩, m, rfl, e | case refine_2.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝¹ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePreimage s
c : ℂ
x✝ : S
m✝ : (c, x✝) ∈ s.post
x : ℂ
m : (c, x) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.intro
S : Type
inst✝⁵ : TopologicalSpace S
inst✝⁴ : CompactSpace S
inst✝³ : T3Space S
inst✝² : ChartedSpace ℂ S
inst✝¹ : AnalyticManifold I S
f : ℂ → S → S
c✝ x✝¹ : ℂ
a z : S
d n : ℕ
s✝ : Super f d a
y : ℂ × ℂ
s : Super f d a
inst✝ : OnePr... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | spheres_subset_closedBall | [83, 1] | [88, 44] | rw [←closedBall_prod_same, Set.subset_def] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
⊢ sphere c0 r ×ˢ sphere c1 r ⊆ closedBall (c0, c1) r | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
⊢ ∀ x ∈ sphere c0 r ×ˢ sphere c1 r, x ∈ closedBall c0 r ×ˢ closedBall c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
⊢ sphere c0 r ×ˢ sphere c1 r ⊆ closedBall (c0, c1) r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | spheres_subset_closedBall | [83, 1] | [88, 44] | intro z | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
⊢ ∀ x ∈ sphere c0 r ×ˢ sphere c1 r, x ∈ closedBall c0 r ×ˢ closedBall c1 r | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
⊢ z ∈ sphere c0 r ×ˢ sphere c1 r → z ∈ closedBall c0 r ×ˢ closedBall c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
⊢ ∀ x ∈ sphere c0 r ×ˢ sphere c1 r, x ∈ closedBall c0 r ×ˢ closedBall c1 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | spheres_subset_closedBall | [83, 1] | [88, 44] | simp only [Set.mem_prod, mem_sphere_iff_norm, Complex.norm_eq_abs, Metric.mem_closedBall, and_imp] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
⊢ z ∈ sphere c0 r ×ˢ sphere c1 r → z ∈ closedBall c0 r ×ˢ closedBall c1 r | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
⊢ Complex.abs (z.1 - c0) = r → Complex.abs (z.2 - c1) = r → dist z.1 c0 ≤ r ∧ dist z.2 c1 ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
⊢ z ∈ sphere c0 r ×ˢ sphere c1 r → z ∈ closedBall c0 r ×ˢ closedBall c1 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | spheres_subset_closedBall | [83, 1] | [88, 44] | rw [Complex.dist_eq, Complex.dist_eq] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
⊢ Complex.abs (z.1 - c0) = r → Complex.abs (z.2 - c1) = r → dist z.1 c0 ≤ r ∧ dist z.2 c1 ≤ r | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
⊢ Complex.abs (z.1 - c0) = r → Complex.abs (z.2 - c1) = r → Complex.abs (z.1 - c0) ≤ r ∧ Complex.abs (z.2 - c1) ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
⊢ Complex.abs (z.1 - c0) = r → Complex.abs (z.2 - c1) = r → dist z.1 c0 ≤ r ∧ dist z.2 c... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | spheres_subset_closedBall | [83, 1] | [88, 44] | intro a b | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
⊢ Complex.abs (z.1 - c0) = r → Complex.abs (z.2 - c1) = r → Complex.abs (z.1 - c0) ≤ r ∧ Complex.abs (z.2 - c1) ≤ r | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b✝ : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
a : Complex.abs (z.1 - c0) = r
b : Complex.abs (z.2 - c1) = r
⊢ Complex.abs (z.1 - c0) ≤ r ∧ Complex.abs (z.2 - c1) ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
⊢ Complex.abs (z.1 - c0) = r → Complex.abs (z.2 - c1) = r → Complex.abs (z.1 - c0) ≤ r ∧... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | spheres_subset_closedBall | [83, 1] | [88, 44] | exact ⟨le_of_eq a, le_of_eq b⟩ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b✝ : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
a : Complex.abs (z.1 - c0) = r
b : Complex.abs (z.2 - c1) = r
⊢ Complex.abs (z.1 - c0) ≤ r ∧ Complex.abs (z.2 - c1) ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c1✝ w0 w1 : ℂ
r✝ b✝ : ℝ
c0 c1 : ℂ
r : ℝ
z : ℂ × ℂ
a : Complex.abs (z.1 - c0) = r
b : Complex.abs (z.2 - c1) = r
⊢ Complex.abs (z.1 - c0) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | mem_open_closed | [93, 1] | [94, 69] | simp only [Metric.mem_ball, Metric.mem_closedBall] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ z ∈ ball c r → z ∈ closedBall c r | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ dist z c < r → dist z c ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ z ∈ ball c r → z ∈ closedBall c r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | mem_open_closed | [93, 1] | [94, 69] | exact le_of_lt | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ dist z c < r → dist z c ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ dist z c < r → dist z c ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | mem_sphere_closed | [96, 1] | [97, 94] | simp only [mem_sphere_iff_norm, Complex.norm_eq_abs, Metric.mem_closedBall] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ z ∈ sphere c r → z ∈ closedBall c r | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ Complex.abs (z - c) = r → dist z c ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ z ∈ sphere c r → z ∈ closedBall c r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | mem_sphere_closed | [96, 1] | [97, 94] | exact le_of_eq | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ Complex.abs (z - c) = r → dist z c ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
z c : ℂ
r : ℝ
⊢ Complex.abs (z - c) = r → dist z c ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | center_not_in_sphere | [100, 1] | [102, 50] | simp only [mem_sphere_iff_norm, Complex.norm_eq_abs] at zs | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c z : ℂ
r : ℝ
rp : r > 0
zs : z ∈ sphere c r
⊢ z - c ≠ 0 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c z : ℂ
r : ℝ
rp : r > 0
zs : Complex.abs (z - c) = r
⊢ z - c ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c z : ℂ
r : ℝ
rp : r > 0
zs : z ∈ sphere c r
⊢ z - c ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | center_not_in_sphere | [100, 1] | [102, 50] | rw [←Complex.abs.ne_zero_iff, zs] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c z : ℂ
r : ℝ
rp : r > 0
zs : Complex.abs (z - c) = r
⊢ z - c ≠ 0 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c z : ℂ
r : ℝ
rp : r > 0
zs : Complex.abs (z - c) = r
⊢ r ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c z : ℂ
r : ℝ
rp : r > 0
zs : Complex.abs (z - c) = r
⊢ z - c ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | center_not_in_sphere | [100, 1] | [102, 50] | exact rp.ne' | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c z : ℂ
r : ℝ
rp : r > 0
zs : Complex.abs (z - c) = r
⊢ r ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
c z : ℂ
r : ℝ
rp : r > 0
zs : Complex.abs (z - c) = r
⊢ r ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc0 | [105, 1] | [110, 83] | refine ContinuousOn.comp h.fc ?_ ?_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
⊢ ContinuousOn (fun z0 => f (z0, w1)) (closedBall c0 r) | case refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
⊢ ContinuousOn (fun z0 => (z0, w1)) (closedBall c0 r)
case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ :... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
⊢ ContinuousOn (fun z0 => f (z0, w1)) (closedBall c0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc0 | [105, 1] | [110, 83] | exact ContinuousOn.prod continuousOn_id continuousOn_const | case refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
⊢ ContinuousOn (fun z0 => (z0, w1)) (closedBall c0 r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
⊢ ContinuousOn (fun z0 => (z0, w1)) (closedBall c0 r)
T... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc0 | [105, 1] | [110, 83] | intro z0 z0m | case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
⊢ Set.MapsTo (fun z0 => (z0, w1)) (closedBall c0 r) s | case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
z0 : ℂ
z0m : z0 ∈ closedBall c0 r
⊢ (fun z0 => (z0, w1)) z0 ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
⊢ Set.MapsTo (fun z0 => (z0, w1)) (closedBall c0 r) s
T... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc0 | [105, 1] | [110, 83] | apply h.rs | case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
z0 : ℂ
z0m : z0 ∈ closedBall c0 r
⊢ (fun z0 => (z0, w1)) z0 ∈ s | case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
z0 : ℂ
z0m : z0 ∈ closedBall c0 r
⊢ (fun z0 => (z0, w1)) z0 ∈ closedBall (c0, c1) r | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
z0 : ℂ
z0m : z0 ∈ closedBall c0 r
⊢ (fun z0 => (z0, w1)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc0 | [105, 1] | [110, 83] | rw [← closedBall_prod_same] | case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
z0 : ℂ
z0m : z0 ∈ closedBall c0 r
⊢ (fun z0 => (z0, w1)) z0 ∈ closedBall (c0, c1) r | case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
z0 : ℂ
z0m : z0 ∈ closedBall c0 r
⊢ (fun z0 => (z0, w1)) z0 ∈ closedBall c0 r ×ˢ closedBall c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
z0 : ℂ
z0m : z0 ∈ closedBall c0 r
⊢ (fun z0 => (z0, w... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc0 | [105, 1] | [110, 83] | exact Set.mem_prod.mpr ⟨z0m, mem_open_closed w1m⟩ | case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
z0 : ℂ
z0m : z0 ∈ closedBall c0 r
⊢ (fun z0 => (z0, w1)) z0 ∈ closedBall c0 r ×ˢ closedBall c1 r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w1m : w1 ∈ ball c1 r
z0 : ℂ
z0m : z0 ∈ closedBall c0 r
⊢ (fun z0 => (z0, w... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc1 | [113, 1] | [118, 67] | refine ContinuousOn.comp h.fc ?_ ?_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
⊢ ContinuousOn (fun z1 => f (w0, z1)) (closedBall c1 r) | case refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
⊢ ContinuousOn (fun z1 => (w0, z1)) (closedBall c1 r)
case refine_2
E : Type
inst✝² : NormedAddCommGroup E
in... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
⊢ ContinuousOn (fun z1 => f (w0, z1)) (closedBall c1 r)
TACTIC:... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc1 | [113, 1] | [118, 67] | exact ContinuousOn.prod continuousOn_const continuousOn_id | case refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
⊢ ContinuousOn (fun z1 => (w0, z1)) (closedBall c1 r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
⊢ ContinuousOn (fun z1 => (w0, z1)) (closedBall c... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc1 | [113, 1] | [118, 67] | intro z1 z1m | case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
⊢ Set.MapsTo (fun z1 => (w0, z1)) (closedBall c1 r) s | case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
z1 : ℂ
z1m : z1 ∈ closedBall c1 r
⊢ (fun z1 => (w0, z1)) z1 ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
⊢ Set.MapsTo (fun z1 => (w0, z1)) (closedBall c1 ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc1 | [113, 1] | [118, 67] | apply h.rs | case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
z1 : ℂ
z1m : z1 ∈ closedBall c1 r
⊢ (fun z1 => (w0, z1)) z1 ∈ s | case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
z1 : ℂ
z1m : z1 ∈ closedBall c1 r
⊢ (fun z1 => (w0, z1)) z1 ∈ closedBall (c0, c1) r | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
z1 : ℂ
z1m : z1 ∈ closedBall c1 r
⊢ (fun z1 => (w... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc1 | [113, 1] | [118, 67] | rw [← closedBall_prod_same] | case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
z1 : ℂ
z1m : z1 ∈ closedBall c1 r
⊢ (fun z1 => (w0, z1)) z1 ∈ closedBall (c0, c1) r | case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
z1 : ℂ
z1m : z1 ∈ closedBall c1 r
⊢ (fun z1 => (w0, z1)) z1 ∈ closedBall c0 r ×ˢ closedBall c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
z1 : ℂ
z1m : z1 ∈ closedBall c1 r
⊢ (fun z1 => ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fc1 | [113, 1] | [118, 67] | exact Set.mem_prod.mpr ⟨w0m, z1m⟩ | case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
z1 : ℂ
z1m : z1 ∈ closedBall c1 r
⊢ (fun z1 => (w0, z1)) z1 ∈ closedBall c0 r ×ˢ closedBall c1 r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
z1 : ℂ
z1m : z1 ∈ closedBall c1 r
⊢ (fun z1 => ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fd0 | [121, 1] | [125, 40] | apply h.rs | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ s | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall (c0, c1) r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fd0 | [121, 1] | [125, 40] | rw [←closedBall_prod_same] | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall (c0, c1) r | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall c0 r ×ˢ closedBall c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall (c0, ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fd0 | [121, 1] | [125, 40] | exact Set.mem_prod.mpr ⟨w0m, w1m⟩ | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall c0 r ×ˢ closedBall c1 r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall c0 r ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fd1 | [128, 1] | [132, 40] | apply h.rs | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ s | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall (c0, c1) r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fd1 | [128, 1] | [132, 40] | rw [←closedBall_prod_same] | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall (c0, c1) r | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall c0 r ×ˢ closedBall c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall (c0, ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | Separate.fd1 | [128, 1] | [132, 40] | exact Set.mem_prod.mpr ⟨w0m, w1m⟩ | case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall c0 r ×ˢ closedBall c1 r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ closedBall c0 r
w1m : w1 ∈ closedBall c1 r
⊢ (w0, w1) ∈ closedBall c0 r ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1 | [135, 1] | [140, 92] | refine Complex.two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable
Set.countable_empty wm fc ?_ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ f z
⊢ ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, r), (z - w)⁻¹ • f z... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ f z
⊢ ∀ x ∈ ball c r \ ∅, DifferentiableAt ℂ (fun z => f z) x | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1 | [135, 1] | [140, 92] | intro z zm | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ f z
⊢ ∀ x ∈ ball c r \ ∅, DifferentiableAt ℂ (fun z => f z) x | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ f z
z : ℂ
zm : z ∈ ball c r \ ∅
⊢ DifferentiableAt ℂ (fun z =... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1 | [135, 1] | [140, 92] | apply fd z _ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ f z
z : ℂ
zm : z ∈ ball c r \ ∅
⊢ DifferentiableAt ℂ (fun z =... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ f z
z : ℂ
zm : z ∈ ball c r \ ∅
⊢ z ∈ ball c r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1 | [135, 1] | [140, 92] | simp only [Metric.mem_ball, Set.diff_empty] at zm ⊢ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ f z
z : ℂ
zm : z ∈ ball c r \ ∅
⊢ z ∈ ball c r | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ f z
z : ℂ
zm : dist z c < r
⊢ dist z c < r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1 | [135, 1] | [140, 92] | assumption | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ f z
z : ℂ
zm : dist z c < r
⊢ dist z c < r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b r : ℝ
c w : ℂ
f : ℂ → E
wm : w ∈ ball c r
fc : ContinuousOn f (closedBall c r)
fd : ∀ z ∈ ball c r, DifferentiableAt ℂ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2 | [143, 1] | [160, 55] | have h1 := fun z0 (z0m : z0 ∈ closedBall c0 r) ↦
cauchy1 w1m (h.fc1 z0m) fun z1 z1m ↦ h.fd1 z0m (mem_open_closed z1m) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
⊢ ((2 * ↑π * I)⁻¹ •
∮ (z0 : ℂ) in C(c0, r), (z0 - w0)⁻¹ • (2 * ↑π * I)⁻¹ • ∮ (z1 : ℂ) in C(c1, r), (z1 ... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
⊢ ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
⊢ ((2 * ↑π * I)⁻¹ •
∮ (z0 : ℂ) in C(c0, r)... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2 | [143, 1] | [160, 55] | have ic1 : ContinuousOn (fun z0 ↦ (2 * π * I : ℂ)⁻¹ • ∮ z1 in C(c1, r), (z1 - w1)⁻¹ • f (z0, z1))
(closedBall c0 r) :=
(h.fc0 w1m).congr h1 | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
⊢ ... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2 | [143, 1] | [160, 55] | have id1 : DifferentiableOn ℂ (fun z0 ↦ (2 * π * I : ℂ)⁻¹ • ∮ z1 in C(c1, r), (z1 - w1)⁻¹
• f (z0, z1)) (ball c0 r) := by
rw [differentiableOn_congr fun z zs ↦ h1 z (mem_open_closed zs)]
intro z0 z0m; apply DifferentiableAt.differentiableWithinAt
exact h.fd0 (mem_open_closed z0m) (mem_open_closed w1m) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2 | [143, 1] | [160, 55] | have h01 :=
cauchy1 w0m ic1 fun z0 z0m ↦
DifferentiableOn.differentiableAt id1 (IsOpen.mem_nhds isOpen_ball z0m) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2 | [143, 1] | [160, 55] | exact _root_.trans h01 (h1 w0 (mem_open_closed w0m)) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2 | [143, 1] | [160, 55] | rw [differentiableOn_congr fun z zs ↦ h1 z (mem_open_closed zs)] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2 | [143, 1] | [160, 55] | intro z0 z0m | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2 | [143, 1] | [160, 55] | apply DifferentiableAt.differentiableWithinAt | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0, w1)
ic... | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0,... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy2 | [143, 1] | [160, 55] | exact h.fd0 (mem_open_closed z0m) (mem_open_closed w1m) | case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c1, r), (z - w1)⁻¹ • f (z0, z)) = f (z0,... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
h : Separate f c0 c1 r b s
w0m : w0 ∈ ball c0 r
w1m : w1 ∈ ball c1 r
h1 : ∀ z0 ∈ closedBall c0 r, ((2 * ↑π * I... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | simp at wm | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : w ∈ ball 0 r
⊢ HasSum (fun n => w ^ n • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z - c)⁻... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
⊢ HasSum (fun n => w ^ n • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : w ∈ ball 0 r
⊢ HasSum (fun n => w ^ n • (2... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | have ci : CircleIntegrable f c r := ContinuousOn.circleIntegrable (by linarith) fc | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
⊢ HasSum (fun n => w ^ n • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, r), (z - c)⁻¹ ^ n • (z ... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
⊢ HasSum (fun n => w ^ n • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
⊢ HasSum (fun n => w ^ n... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | have h := hasSum_cauchyPowerSeries_integral ci wm | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
⊢ HasSum (fun n => w ^ n • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in ... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
h :
HasSum (fun n => (cauchyPowerSeries f c r n) fun x ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | simp_rw [cauchyPowerSeries_apply] at h | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
h :
HasSum (fun n => (cauchyPowerSeries f c r n) fun x ... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
h :
HasSum (fun n => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c,... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | generalize hs : (2*π*I : ℂ)⁻¹ = s | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
h :
HasSum (fun n => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c,... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
h :
HasSum (fun n => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | simp_rw [hs] at h | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
h :
HasSum (fun n => (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
h :
HasSum (fun n => s •... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | generalize hg : (s • ∮ z : ℂ in C(c, r), (z - (c + w))⁻¹ • f z) = g | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
h :
HasSum (fun n => s •... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
h :
HasSum (fun n => s •... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | rw [hg] at h | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
h :
HasSum (fun n => s •... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
g : E
h : HasSum (fun n =>... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | simp_rw [div_eq_mul_inv, mul_pow, ← smul_smul, circleIntegral.integral_smul, smul_comm s _] at h | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
g : E
h : HasSum (fun n =>... | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
g : E
hg : (s • ∮ (z : ℂ) ... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | assumption | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f c r
s : ℂ
hs : (2 * ↑π * I)⁻¹ = s
g : E
hg : (s • ∮ (z : ℂ) ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
ci : CircleIntegrable f... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | cauchy1_hasSum | [176, 1] | [188, 13] | linarith | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
⊢ 0 ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r✝ b : ℝ
f : ℂ → E
c w : ℂ
r : ℝ
rp : r > 0
fc : ContinuousOn f (sphere c r)
wm : Complex.abs w < r
⊢ 0 ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | rcases (IsCompact.prod cs (isCompact_sphere _ _)).bddAbove_image fc.norm with ⟨b, bh⟩ | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
⊢ ContinuousOn (fun z0 => ∮ (z1 : ℂ) in C(c1, r), f z0 z1) s | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : b ∈ upperBounds ((fun x => ‖uncurry f x‖) '' s ×ˢ sp... | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
⊢ ContinuousOn... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp only [mem_upperBounds, Set.forall_mem_image] at bh | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : b ∈ upperBounds ((fun x => ‖uncurry f x‖) '' s ×ˢ sp... | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | intro z1 z1s | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ... | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | have fb : ∀ᶠ x : ℂ in 𝓝[s] z1, ∀ᵐ t : ℝ, t ∈ Set.uIoc 0 (2 * π) →
‖deriv (circleMap c1 r) t • (fun z1 : ℂ ↦ f x z1) (circleMap c1 r t)‖ ≤ r * b := by
apply eventually_nhdsWithin_of_forall; intro x xs
apply MeasureTheory.ae_of_all _; intro t _; simp only [deriv_circleMap]
rw [norm_smul, Complex.norm_eq_abs]
... | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ... | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | refine intervalIntegral.continuousWithinAt_of_dominated_interval ?_ fb (by simp) ?_ | case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncurry f x‖ ... | case intro.refine_1
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ⦃x : ℂ × ℂ⦄, x ∈ s ×ˢ sphere c1 r → ‖uncu... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f✝ : ℂ × ℂ → E
s✝ : Set (ℂ × ℂ)
c0 c1 w0 w1 : ℂ
r b✝ : ℝ
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b ... |
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