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/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply eventually_nhdsWithin_of_forall | 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‖ ≤ b
z1 : ℂ
... | 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
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‖ ≤ b
... | 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)
b : ℝ
bh : ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | intro x xs | 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
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‖ ≤ b
... | 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
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‖ ≤ b
... | 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
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply MeasureTheory.ae_of_all _ | 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
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‖ ≤ b
... | 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
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‖ ≤ b
... | 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
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | intro t _ | 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
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‖ ≤ b
... | 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
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‖ ≤ b
... | 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
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp only [deriv_circleMap] | 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
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‖ ≤ b
... | 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
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‖ ≤ b
... | 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
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | rw [norm_smul, Complex.norm_eq_abs] | 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
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‖ ≤ b
... | 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
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‖ ≤ b
... | 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
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp only [map_mul, abs_circleMap_zero, Complex.abs_I, mul_one] | 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
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‖ ≤ b
... | 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
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‖ ≤ b
... | 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
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | have bx := @bh (x, circleMap c1 r t) (Set.mk_mem_prod xs (circleMap_mem_sphere c1
(by linarith) t)) | 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
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‖ ≤ b
... | 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
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‖ ≤ b
... | 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
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp only [uncurry] at bx | 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
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‖ ≤ b
... | 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
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‖ ≤ b
... | 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
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | calc |r| * ‖f x (circleMap c1 r t)‖ ≤ |r| * b := by bound
_ = r * b := by rw [abs_of_pos rp] | 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
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‖ ≤ b
... | 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
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | 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 → 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‖ ≤ b
z1 : 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 → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | bound | 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‖ ≤ b
z1 : 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 → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | rw [abs_of_pos rp] | 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‖ ≤ b
z1 : 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 → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp | 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‖ ≤ b
z1 : 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 → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply eventually_nhdsWithin_of_forall | 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... | case intro.refine_1.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 → 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 → ‖un... | Please generate a tactic in lean4 to solve the state.
STATE:
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... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | intro x xs | case intro.refine_1.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 → 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 → ‖un... | case intro.refine_1.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 → 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 → ‖un... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.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 → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphe... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply ContinuousOn.aestronglyMeasurable | case intro.refine_1.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 → 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 → ‖un... | case intro.refine_1.h.hf
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 → ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.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 → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphe... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply ContinuousOn.smul | case intro.refine_1.h.hf
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 → ... | case intro.refine_1.h.hf.hf
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... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf
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 s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp | case intro.refine_1.h.hf.hf
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... | case intro.refine_1.h.hf.hf
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... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hf
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 ×... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | exact ContinuousOn.mul (Continuous.continuousOn (continuous_circleMap _ _)) continuousOn_const | case intro.refine_1.h.hf.hf
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... | case intro.refine_1.h.hf.hg
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... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hf
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 ×... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | have comp : (fun t ↦ f x (circleMap c1 r t)) = uncurry f ∘ fun t ↦ (x, circleMap c1 r t) := by
apply funext; intro t; simp | case intro.refine_1.h.hf.hg
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... | case intro.refine_1.h.hf.hg
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... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hg
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 ×... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp | case intro.refine_1.h.hf.hg
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... | case intro.refine_1.h.hf.hg
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... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hg
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 ×... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | rw [comp] | case intro.refine_1.h.hf.hg
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... | case intro.refine_1.h.hf.hg
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... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hg
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 ×... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply ContinuousOn.comp fc | case intro.refine_1.h.hf.hg
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... | case intro.refine_1.h.hf.hg.hf
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 c... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hg
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 ×... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | exact ContinuousOn.prod continuousOn_const (Continuous.continuousOn (continuous_circleMap _ _)) | case intro.refine_1.h.hf.hg.hf
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 c... | case intro.refine_1.h.hf.hg.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 → 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hg.hf
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) (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | intro t _ | case intro.refine_1.h.hf.hg.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 → 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... | case intro.refine_1.h.hf.hg.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 → 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hg.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 → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp | case intro.refine_1.h.hf.hg.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 → 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... | case intro.refine_1.h.hf.hg.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 → 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... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hg.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 → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | exact ⟨xs, by linarith⟩ | case intro.refine_1.h.hf.hg.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 → 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... | case intro.refine_1.h.hs
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 → ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hf.hg.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 → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | exact measurableSet_uIoc | case intro.refine_1.h.hs
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 → ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_1.h.hs
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 s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply funext | 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‖ ≤ b
z1 : ℂ
... | 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
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‖ ≤ b
... | 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)
b : ℝ
bh : ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | intro t | 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
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‖ ≤ b
... | 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
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‖ ≤ b
... | 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
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp | 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
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‖ ≤ b
... | 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
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | 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 → 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‖ ≤ b
z1 : 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 → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply MeasureTheory.ae_of_all _ | case intro.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
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... | case intro.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
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.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
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | intro t _ | case intro.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
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... | case intro.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
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.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
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp | case intro.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
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... | case intro.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
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.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
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply ContinuousOn.smul continuousOn_const | case intro.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
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... | case intro.refine_2.hg
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 → ‖u... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.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
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | have comp : (fun x ↦ f x (circleMap c1 r t)) = uncurry f ∘ fun x ↦ (x, circleMap c1 r t) := by
apply funext; intro t; simp | case intro.refine_2.hg
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 → ‖u... | case intro.refine_2.hg
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 → ‖u... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.hg
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 sph... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | rw [comp] | case intro.refine_2.hg
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 → ‖u... | case intro.refine_2.hg
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 → ‖u... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.hg
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 sph... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply ContinuousOn.comp fc (ContinuousOn.prod continuousOn_id continuousOn_const) | case intro.refine_2.hg
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 → ‖u... | case intro.refine_2.hg
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 → ‖u... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.hg
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 sph... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | intro x xs | case intro.refine_2.hg
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 → ‖u... | case intro.refine_2.hg
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 → ‖u... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.hg
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 sph... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp | case intro.refine_2.hg
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 → ‖u... | case intro.refine_2.hg
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 → ‖u... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.hg
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 sph... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | exact ⟨xs, by linarith⟩ | case intro.refine_2.hg
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 → ‖u... | case intro.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
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 → ‖un... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.refine_2.hg
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 sph... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | exact z1s | case intro.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
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 → ‖un... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.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
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphe... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | apply funext | 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‖ ≤ b
z1 : ℂ
... | 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
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‖ ≤ b
... | 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)
b : ℝ
bh : ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | intro t | 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
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‖ ≤ b
... | 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
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‖ ≤ b
... | 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
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | simp | 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
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‖ ≤ b
... | 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
f : ℂ → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.circleIntegral | [191, 1] | [225, 14] | 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 → 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‖ ≤ b
z1 : 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 → ℂ → E
s : Set ℂ
rp : r > 0
cs : IsCompact s
fc : ContinuousOn (uncurry f) (s ×ˢ sphere c1 r)
b : ℝ
bh : ∀ ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | refine ContinuousOn.inv₀ (ContinuousOn.sub continuousOn_id continuousOn_const) fun z 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 w : ℂ
r : ℝ
wr : w ∈ ball 0 r
⊢ ContinuousOn (fun z => (z - (c + w))⁻¹) (sphere 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
c w : ℂ
r : ℝ
wr : w ∈ ball 0 r
z : ℂ
zs : z ∈ sphere c r
⊢ z - (c + w) ≠ 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 w : ℂ
r : ℝ
wr : w ∈ ball 0 r
⊢ ContinuousOn (fun z => (z - (c + w))⁻¹) (sphere c r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | rw [←Complex.abs.ne_zero_iff] | 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 w : ℂ
r : ℝ
wr : w ∈ ball 0 r
z : ℂ
zs : z ∈ sphere c r
⊢ z - (c + w) ≠ 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 w : ℂ
r : ℝ
wr : w ∈ ball 0 r
z : ℂ
zs : z ∈ sphere c r
⊢ Complex.abs (z - (c + w)) ≠ 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 w : ℂ
r : ℝ
wr : w ∈ ball 0 r
z : ℂ
zs : z ∈ sphere c r
⊢ z - (c + w) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | simp only [mem_ball_zero_iff, Complex.norm_eq_abs, mem_sphere_iff_norm] at zs wr | 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 w : ℂ
r : ℝ
wr : w ∈ ball 0 r
z : ℂ
zs : z ∈ sphere c r
⊢ Complex.abs (z - (c + w)) ≠ 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 w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ Complex.abs (z - (c + w)) ≠ 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 w : ℂ
r : ℝ
wr : w ∈ ball 0 r
z : ℂ
zs : z ∈ sphere c r
⊢ Complex.abs (z - (c + w)) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | apply ne_of_gt | 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 w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ Complex.abs (z - (c + w)) ≠ 0 | 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
c w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ 0 < Complex.abs (z - (c + w)) | 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 w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ Complex.abs (z - (c + w)) ≠ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | calc abs (z - (c + w))
_ = abs (z - c + -w) := by ring_nf
_ ≥ abs (z - c) - abs (-w) := by bound
_ = r - abs (-w) := by rw [zs]
_ = r - abs w := by rw [Complex.abs.map_neg]
_ > r - r := (sub_lt_sub_left wr _)
_ = 0 := by ring | 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
c w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ 0 < Complex.abs (z - (c + w)) | 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
c w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ 0 < Complex.abs (z - (c + w))
TACTI... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | ring_nf | 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 w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ Complex.abs (z - (c + w)) = Complex.abs (z - c + -w) | 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 w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ Complex.abs (z - (c + w)) = Complex.abs (z... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | bound | 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 w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ Complex.abs (z - c + -w) ≥ Complex.abs (z - c) - Complex.abs (-w) | 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 w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ Complex.abs (z - c + -w) ≥ Complex.abs (z ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | rw [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 w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ Complex.abs (z - c) - Complex.abs (-w) = r - Complex.abs (-w) | 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 w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ Complex.abs (z - c) - Complex.abs (-w) = r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | rw [Complex.abs.map_neg] | 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 w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ r - Complex.abs (-w) = r - Complex.abs w | 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 w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ r - Complex.abs (-w) = r - Complex.abs w
T... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.inv_sphere_ball | [234, 1] | [246, 21] | ring | 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 w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ 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 w : ℂ
r : ℝ
z : ℂ
zs : Complex.abs (z - c) = r
wr : Complex.abs w < r
⊢ r - r = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.cauchy1 | [249, 1] | [262, 17] | apply ContinuousOn.circleIntegral rp (isCompact_sphere _ _) | 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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun z0 => ∮ (z1 : ℂ) in C(c1, r), (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)) (sphere c0 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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (uncurry fun z0 z1 => (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)) (sphere c0 r ×ˢ sphere 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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun z0 => ∮ (z1 : ℂ) in C(c1, r), ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.cauchy1 | [249, 1] | [262, 17] | apply ContinuousOn.smul | 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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (uncurry fun z0 z1 => (z1 - c1)⁻¹ ^ n1 • (z1 - c1)⁻¹ • f (z0, z1)) (sphere c0 r ×ˢ sphere c1 r) | case hf
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => (x.2 - c1)⁻¹ ^ n1) (sphere c0 r ×ˢ sphere c1 r)
case hg
E : Type
inst✝² : Nor... | 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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (uncurry fun z0 z1 => (z1 - c1)⁻¹ ^... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.cauchy1 | [249, 1] | [262, 17] | apply ContinuousOn.pow | case hf
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => (x.2 - c1)⁻¹ ^ n1) (sphere c0 r ×ˢ sphere c1 r)
case hg
E : Type
inst✝² : Nor... | case hf.hf
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => (x.2 - c1)⁻¹) (sphere c0 r ×ˢ sphere c1 r)
case hg
E : Type
inst✝² : Norme... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => (x.2 - c1)⁻¹ ^ n1... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.cauchy1 | [249, 1] | [262, 17] | apply ContinuousOn.inv₀ | case hf.hf
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => (x.2 - c1)⁻¹) (sphere c0 r ×ˢ sphere c1 r)
case hg
E : Type
inst✝² : Norme... | case hf.hf.hf
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => x.2 - c1) (sphere c0 r ×ˢ sphere c1 r)
case hf.hf.h0
E : Type
inst✝² : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => (x.2 - c1)⁻¹) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.cauchy1 | [249, 1] | [262, 17] | apply Continuous.continuousOn | case hf.hf.hf
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => x.2 - c1) (sphere c0 r ×ˢ sphere c1 r)
case hf.hf.h0
E : Type
inst✝² : ... | case hf.hf.hf.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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ Continuous fun x => x.2 - c1
case hf.hf.h0
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : N... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => x.2 - c1) (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.cauchy1 | [249, 1] | [262, 17] | exact Continuous.sub (Continuous.snd continuous_id) continuous_const | case hf.hf.hf.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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ Continuous fun x => x.2 - c1
case hf.hf.h0
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : N... | case hf.hf.h0
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ∀ x ∈ sphere c0 r ×ˢ sphere c1 r, x.2 - c1 ≠ 0
case hg
E : Type
inst✝² : NormedAddCommGroup E
... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.hf.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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ Continuous fun x => x.2 - c1
ca... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.cauchy1 | [249, 1] | [262, 17] | intro x xp | case hf.hf.h0
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ∀ x ∈ sphere c0 r ×ˢ sphere c1 r, x.2 - c1 ≠ 0
case hg
E : Type
inst✝² : NormedAddCommGroup E
... | case hf.hf.h0
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
x : ℂ × ℂ
xp : x ∈ sphere c0 r ×ˢ sphere c1 r
⊢ x.2 - c1 ≠ 0
case hg
E : Type
inst✝² : NormedAdd... | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.h0
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ∀ x ∈ sphere c0 r ×ˢ sphere c1 r, ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.cauchy1 | [249, 1] | [262, 17] | exact center_not_in_sphere rp (Set.mem_prod.mp xp).right | case hf.hf.h0
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
x : ℂ × ℂ
xp : x ∈ sphere c0 r ×ˢ sphere c1 r
⊢ x.2 - c1 ≠ 0
case hg
E : Type
inst✝² : NormedAdd... | case hg
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => (x.2 - c1)⁻¹ • f (x.1, x.2)) (sphere c0 r ×ˢ sphere c1 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf.hf.h0
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
x : ℂ × ℂ
xp : x ∈ sphere c0 r ×ˢ sp... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.cauchy1 | [249, 1] | [262, 17] | apply ContinuousOn.smul | case hg
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => (x.2 - c1)⁻¹ • f (x.1, x.2)) (sphere c0 r ×ˢ sphere c1 r) | case hg.hf
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => (x.2 - c1)⁻¹) (sphere c0 r ×ˢ sphere c1 r)
case hg.hg
E : Type
inst✝² : No... | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => (x.2 - c1)⁻¹ • f ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.cauchy1 | [249, 1] | [262, 17] | apply ContinuousOn.inv₀ | case hg.hf
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => (x.2 - c1)⁻¹) (sphere c0 r ×ˢ sphere c1 r)
case hg.hg
E : Type
inst✝² : No... | case hg.hf.hf
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => x.2 - c1) (sphere c0 r ×ˢ sphere c1 r)
case hg.hf.h0
E : Type
inst✝² : ... | Please generate a tactic in lean4 to solve the state.
STATE:
case hg.hf
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => (x.2 - c1)⁻¹) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.cauchy1 | [249, 1] | [262, 17] | apply Continuous.continuousOn | case hg.hf.hf
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => x.2 - c1) (sphere c0 r ×ˢ sphere c1 r)
case hg.hf.h0
E : Type
inst✝² : ... | case hg.hf.hf.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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ Continuous fun x => x.2 - c1
case hg.hf.h0
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : N... | Please generate a tactic in lean4 to solve the state.
STATE:
case hg.hf.hf
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => x.2 - c1) (... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.cauchy1 | [249, 1] | [262, 17] | exact Continuous.sub (Continuous.snd continuous_id) continuous_const | case hg.hf.hf.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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ Continuous fun x => x.2 - c1
case hg.hf.h0
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : N... | case hg.hf.h0
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ∀ x ∈ sphere c0 r ×ˢ sphere c1 r, x.2 - c1 ≠ 0
case hg.hg
E : Type
inst✝² : NormedAddCommGroup... | Please generate a tactic in lean4 to solve the state.
STATE:
case hg.hf.hf.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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ Continuous fun x => x.2 - c1
ca... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.cauchy1 | [249, 1] | [262, 17] | intro x xp | case hg.hf.h0
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ∀ x ∈ sphere c0 r ×ˢ sphere c1 r, x.2 - c1 ≠ 0
case hg.hg
E : Type
inst✝² : NormedAddCommGroup... | case hg.hf.h0
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
x : ℂ × ℂ
xp : x ∈ sphere c0 r ×ˢ sphere c1 r
⊢ x.2 - c1 ≠ 0
case hg.hg
E : Type
inst✝² : Normed... | Please generate a tactic in lean4 to solve the state.
STATE:
case hg.hf.h0
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ∀ x ∈ sphere c0 r ×ˢ sphere c1 r, ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.cauchy1 | [249, 1] | [262, 17] | exact center_not_in_sphere rp (Set.mem_prod.mp xp).right | case hg.hf.h0
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
x : ℂ × ℂ
xp : x ∈ sphere c0 r ×ˢ sphere c1 r
⊢ x.2 - c1 ≠ 0
case hg.hg
E : Type
inst✝² : Normed... | case hg.hg
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => f (x.1, x.2)) (sphere c0 r ×ˢ sphere c1 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case hg.hf.h0
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
x : ℂ × ℂ
xp : x ∈ sphere c0 r ×ˢ sp... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.cauchy1 | [249, 1] | [262, 17] | simp | case hg.hg
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => f (x.1, x.2)) (sphere c0 r ×ˢ sphere c1 r) | case hg.hg
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => f x) (sphere c0 r ×ˢ sphere c1 r) | Please generate a tactic in lean4 to solve the state.
STATE:
case hg.hg
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => f (x.1, x.2)) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | ContinuousOn.cauchy1 | [249, 1] | [262, 17] | exact fc | case hg.hg
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => f x) (sphere c0 r ×ˢ sphere c1 r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg.hg
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
n1 : ℕ
rp : r > 0
fc : ContinuousOn f (sphere c0 r ×ˢ sphere c1 r)
⊢ ContinuousOn (fun x => f x) (sphere c... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | rw [circleIntegral] | 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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ ... | 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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | simp_rw [circleIntegral] | 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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ ... | 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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | simp | 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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ ... | 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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | apply intervalIntegral.hasSum_integral_of_dominated_convergence fun n _ ↦ r * b 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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ ... | case hF_meas
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summ... | 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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | intro n | case hF_meas
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summ... | case hF_meas
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summ... | Please generate a tactic in lean4 to solve the state.
STATE:
case hF_meas
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | apply ContinuousOn.aestronglyMeasurable | case hF_meas
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summ... | case hF_meas.hf
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : S... | Please generate a tactic in lean4 to solve the state.
STATE:
case hF_meas
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | apply ContinuousOn.smul | case hF_meas.hf
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : S... | case hF_meas.hf.hf
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs ... | Please generate a tactic in lean4 to solve the state.
STATE:
case hF_meas.hf
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (spher... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | apply ContinuousOn.mul (Continuous.continuousOn (continuous_circleMap _ _)) continuousOn_const | case hF_meas.hf.hf
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs ... | case hF_meas.hf.hg
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs ... | Please generate a tactic in lean4 to solve the state.
STATE:
case hF_meas.hf.hf
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sp... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | apply ContinuousOn.comp (fc n) (Continuous.continuousOn (continuous_circleMap _ _)) | case hF_meas.hf.hg
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs ... | case hF_meas.hf.hg
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs ... | Please generate a tactic in lean4 to solve the state.
STATE:
case hF_meas.hf.hg
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sp... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | intro t _ | case hF_meas.hf.hg
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs ... | case hF_meas.hf.hg
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs ... | Please generate a tactic in lean4 to solve the state.
STATE:
case hF_meas.hf.hg
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sp... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | exact circleMap_mem_sphere _ (by linarith) _ | case hF_meas.hf.hg
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs ... | case hF_meas.hs
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : S... | Please generate a tactic in lean4 to solve the state.
STATE:
case hF_meas.hf.hg
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sp... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | exact measurableSet_uIoc | case hF_meas.hs
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : S... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hF_meas.hs
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (spher... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | 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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summable b
h : ∀ ... | 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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | intro n | case h_bound
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summ... | case h_bound
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_bound
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | apply MeasureTheory.ae_of_all | case h_bound
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summ... | case h_bound.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Su... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_bound
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | intro t _ | case h_bound.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Su... | case h_bound.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Su... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_bound.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | rw [norm_smul, Complex.norm_eq_abs] | case h_bound.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Su... | case h_bound.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Su... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_bound.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | simp | case h_bound.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Su... | case h_bound.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Su... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_bound.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | rw [abs_of_pos rp] | case h_bound.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Su... | case h_bound.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Su... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_bound.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | refine mul_le_mul_of_nonneg_left ?_ rp.le | case h_bound.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Su... | case h_bound.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Su... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_bound.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | exact fb n (circleMap c r t) (circleMap_mem_sphere _ (by linarith) _) | case h_bound.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Su... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h_bound.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | apply MeasureTheory.ae_of_all | case bound_summable
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs... | case bound_summable.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
... | Please generate a tactic in lean4 to solve the state.
STATE:
case bound_summable
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | intro t _ | case bound_summable.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
... | case bound_summable.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
... | Please generate a tactic in lean4 to solve the state.
STATE:
case bound_summable.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | exact Summable.mul_left _ bs | case bound_summable.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bound_summable.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | simp only [_root_.intervalIntegrable_const] | case bound_integrable
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bound_integrable
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Osgood.lean | sum_integral_commute | [274, 1] | [294, 71] | apply MeasureTheory.ae_of_all | case h_lim
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summab... | case h_lim.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
f : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r)
fb : ∀ (n : ℕ), ∀ z ∈ sphere c r, ‖f n z‖ ≤ b n
bs : Summ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h_lim
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 : ℕ → ℂ → E
g : ℂ → E
c : ℂ
r : ℝ
b : ℕ → ℝ
rp : r > 0
fc : ∀ (n : ℕ), ContinuousOn (f n) (sphere c r... |
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