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 |
|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | contDiffAt_iff_analytic_at2 | [58, 1] | [66, 45] | intro a | case mpr
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n... | case mpr
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Holomorphic.lean | contDiffAt_iff_analytic_at2 | [58, 1] | [66, 45] | exact a.contDiffAt.of_le le_top | case mpr
E✝ : Type
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace ℂ E✝
inst✝⁶ : CompleteSpace E✝
F : Type
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℂ F
inst✝³ : CompleteSpace F
E : Type
f : ℂ × ℂ → E
x : ℂ × ℂ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
n : ℕ∞
n1 : 1 ≤ n... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ball | [53, 1] | [71, 59] | rcases x with ⟨x, m⟩ | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : ↑(closure s)
⊢ ∃ g r, 0 < r ∧ (∀ z ∈ Metric.ball (↑x) r, p g z) ∧ (𝓝ˢ (s ∩ Metric.ball (↑x) r)).EventuallyEq g f | case mk
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ closure s
⊢ ∃ g r, 0 < r ∧ (∀ z ∈ Metric.ball (↑⟨x, m⟩) r, p g z) ∧ (𝓝ˢ (s ∩ Metric.ball (↑⟨x, m⟩) r)).EventuallyEq g f |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ball | [53, 1] | [71, 59] | simp only [Subtype.coe_mk] | case mk
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ closure s
⊢ ∃ g r, 0 < r ∧ (∀ z ∈ Metric.ball (↑⟨x, m⟩) r, p g z) ∧ (𝓝ˢ (s ∩ Metric.ball (↑⟨x, m⟩) r)).EventuallyEq g f | case mk
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ closure s
⊢ ∃ g r, 0 < r ∧ (∀ z ∈ Metric.ball x r, p g z) ∧ (𝓝ˢ (s ∩ Metric.ball x r)).EventuallyEq g f |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ball | [53, 1] | [71, 59] | rcases b.point m with ⟨g, pg, e⟩ | case mk
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ closure s
⊢ ∃ g r, 0 < r ∧ (∀ z ∈ Metric.ball x r, p g z) ∧ (𝓝ˢ (s ∩ Metric.ball x r)).EventuallyEq g f | case mk.intro.intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg : ∀ᶠ (z : E) in 𝓝 x, p g z
e : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
⊢ ∃ g r, 0 < r ∧ (∀ z ∈ Metric.ball x r, p g z) ∧ (𝓝ˢ (s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ball | [53, 1] | [71, 59] | rcases Metric.eventually_nhds_iff_ball.mp pg with ⟨r, rp, pg⟩ | case mk.intro.intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg : ∀ᶠ (z : E) in 𝓝 x, p g z
e : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
⊢ ∃ g r, 0 < r ∧ (∀ z ∈ Metric.ball x r, p g z) ∧ (𝓝ˢ (s... | case mk.intro.intro.intro.intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y ∈ Metric.ball x r,... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ball | [53, 1] | [71, 59] | rcases Filter.frequently_iff.mp e (Metric.ball_mem_nhds _ rp) with ⟨y, yb, ys, e⟩ | case mk.intro.intro.intro.intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y ∈ Metric.ball x r,... | case mk.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e✝ : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ball | [53, 1] | [71, 59] | use g, r, rp, fun z zr ↦ pg z zr | case mk.intro.intro.intro.intro.intro.intro.intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e✝ : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y... | case right
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e✝ : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y ∈ Metric.ball x r, p g y
y : E
yb : y ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ball | [53, 1] | [71, 59] | simp only [Filter.EventuallyEq, Filter.eventually_iff, mem_nhdsSet_iff_forall] | case right
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e✝ : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y ∈ Metric.ball x r, p g y
y : E
yb : y ... | case right
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e✝ : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y ∈ Metric.ball x r, p g y
y : E
yb : y ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ball | [53, 1] | [71, 59] | intro z ⟨zs, zr⟩ | case right
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e✝ : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y ∈ Metric.ball x r, p g y
y : E
yb : y ... | case right
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e✝ : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y ∈ Metric.ball x r, p g y
y : E
yb : y... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ball | [53, 1] | [71, 59] | simp only [← Filter.eventually_iff] | case right
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e✝ : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y ∈ Metric.ball x r, p g y
y : E
yb : y... | case right
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e✝ : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y ∈ Metric.ball x r, p g y
y : E
yb : y... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ball | [53, 1] | [71, 59] | rcases local_preconnected_nhdsSet (b.convex.inter (convex_ball _ _)).isPreconnected n with
⟨u, uo, iu, up, uc⟩ | case right
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e✝ : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y ∈ Metric.ball x r, p g y
y : E
yb : y... | case right.intro.intro.intro.intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e✝ : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y ∈ Metric.ball... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ball | [53, 1] | [71, 59] | have eq := b.unique uo uc (fun _ m ↦ (up m).1) (fun _ m ↦ (up m).2) ⟨y, iu ⟨ys, yb⟩, e⟩ | case right.intro.intro.intro.intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e✝ : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y ∈ Metric.ball... | case right.intro.intro.intro.intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e✝ : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y ∈ Metric.ball... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ball | [53, 1] | [71, 59] | exact eq.eventuallyEq_of_mem (uo.mem_nhds (iu ⟨zs, zr⟩)) | case right.intro.intro.intro.intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e✝ : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y ∈ Metric.ball... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ball | [53, 1] | [71, 59] | refine Filter.inter_mem ?_ ?_ | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e✝ : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y ∈ Metric.ball x r, p g y
y : E
yb : y ∈ Metric.b... | case refine_1
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e✝ : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y ∈ Metric.ball x r, p g y
y : E
yb ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ball | [53, 1] | [71, 59] | exact nhdsSet_mono (inter_subset_right _ _)
(Filter.mem_of_superset isOpen_ball.mem_nhdsSet_self pg) | case refine_1
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e✝ : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y ∈ Metric.ball x r, p g y
y : E
yb ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ball | [53, 1] | [71, 59] | exact nhdsSet_mono (inter_subset_left _ _) b.start | case refine_2
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : E
m : x ∈ closure s
g : E → α
pg✝ : ∀ᶠ (z : E) in 𝓝 x, p g z
e✝ : ∃ᶠ (z : E) in 𝓝 x, z ∈ s ∧ g z = f z
r : ℝ
rp : r > 0
pg : ∀ y ∈ Metric.ball x r, p g y
y : E
yb ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.exists_cover | [95, 1] | [100, 74] | refine b.compact.elim_finite_subcover (fun x : closure s ↦ Metric.ball (x : E) (b.r x))
(fun _ ↦ isOpen_ball) ?_ | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
⊢ ∃ c, closure s ⊆ ⋃ x ∈ c, Metric.ball (↑x) (b.r x) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
⊢ closure s ⊆ ⋃ i, (fun x => Metric.ball (↑x) (b.r x)) i |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.exists_cover | [95, 1] | [100, 74] | intro x m | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
⊢ closure s ⊆ ⋃ i, (fun x => Metric.ball (↑x) (b.r x)) i | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ closure s
⊢ x ∈ ⋃ i, (fun x => Metric.ball (↑x) (b.r x)) i |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.exists_cover | [95, 1] | [100, 74] | exact mem_iUnion_of_mem ⟨x, m⟩ (mem_ball_self (b.rp ⟨x, m⟩)) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ closure s
⊢ x ∈ ⋃ i, (fun x => Metric.ball (↑x) (b.r x)) i | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.yt | [114, 1] | [115, 96] | simp only [Base.t, Base.y, mem_iUnion₂, mem_iUnion] at m ⊢ | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
m : z ∈ b.t
⊢ z ∈ Metric.ball (↑(b.y m)) (b.r (b.y m)) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
m✝ : z ∈ b.t
m : ∃ i, ∃ (_ : i ∈ b.c), z ∈ Metric.ball (↑i) (b.r i)
⊢ z ∈ Metric.ball (↑(choose ⋯)) (b.r (choose ⋯)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.yt | [114, 1] | [115, 96] | exact choose_spec (choose_spec m) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
m✝ : z ∈ b.t
m : ∃ i, ∃ (_ : i ∈ b.c), z ∈ Metric.ball (↑i) (b.r i)
⊢ z ∈ Metric.ball (↑(choose ⋯)) (b.r (choose ⋯)) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | rcases x0 with ⟨x0, m0⟩ | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
x0 x1 : ↑(closure s)
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
ne : ∃ z, z ∈ ball (↑x0) r0 ∩ ball (↑x1) r1
⊢ ∃ w, w ∈ s ∩ ball (↑x0) r0 ∩ ball (↑x1) r1 | case mk
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
x1 : ↑(closure s)
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑x1) r1
⊢ ∃ w, w ∈ s ∩ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑x... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | rcases x1 with ⟨x1, m1⟩ | case mk
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
x1 : ↑(closure s)
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑x1) r1
⊢ ∃ w, w ∈ s ∩ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑x... | case mk.mk
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
⊢ ∃ w, w ∈ s ∩ ball (↑⟨x0, ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | simp only [Subtype.coe_mk] | case mk.mk
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
⊢ ∃ w, w ∈ s ∩ ball (↑⟨x0, ... | case mk.mk
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
⊢ ∃ w, w ∈ s ∩ ball x0 r0 ∩... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | have x01 : ‖x1 - x0‖ < r0 + r1 := by
rcases ne with ⟨z, m0, m1⟩; simp only [mem_ball, dist_eq_norm] at m0 m1
calc ‖x1 - x0‖
_ = ‖z - x0 - (z - x1)‖ := by abel_nf
_ ≤ ‖z - x0‖ + ‖z - x1‖ := (norm_sub_le _ _)
_ < r0 + r1 := add_lt_add m0 m1 | case mk.mk
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
⊢ ∃ w, w ∈ s ∩ ball x0 r0 ∩... | case mk.mk
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
⊢... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | have sub : ∀ (x : E) {a b : ℝ}, 0 < a → 0 < b → (a / (a + b)) • x - x = -((b / (a + b)) • x) := by
intro x a b ap bp; have rnz := (add_pos ap bp).ne'
calc (a / (a + b)) • x - x
_ = (a / (a + b) - (a + b) / (a + b)) • x := by simp only [one_smul, sub_smul, div_self rnz]
_ = -((b / (a + b)) • x) := by rw [← s... | case mk.mk
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
⊢... | case mk.mk
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | have le : ∀ {a : ℝ}, 0 < a → a / (r0 + r1) * ‖x1 - x0‖ < a := by
intro a ap; apply lt_of_lt_of_le (mul_lt_mul_of_pos_left x01 (div_pos ap (add_pos r0p r1p)))
rw [div_mul_cancel₀ _ (add_pos r0p r1p).ne'] | case mk.mk
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
s... | case mk.mk
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
s... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | have f : ∃ᶠ p : E × E in 𝓝 (x0, x1), p.1 ∈ s ∧ p.2 ∈ s := by
simp only [nhds_prod_eq]; rw [Prod.frequently (p := fun x ↦ x ∈ s) (q := fun x ↦ x ∈ s)]
use mem_closure_iff_frequently.mp m0, mem_closure_iff_frequently.mp m1 | case mk.mk
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
s... | case mk.mk
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f✝ : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | rcases(f.and_eventually e).exists with ⟨⟨z0, z1⟩, ⟨m0, m1⟩, m⟩ | case mk.mk
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f✝ : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
... | case mk.mk.intro.mk.intro.intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f✝ : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0✝ : x0 ∈ closure s
x1 : E
m1✝ : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0✝⟩) r0 ∩ ball (↑⟨x1, m1✝⟩) r1
x... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | refine ⟨_, ⟨?_, m.1⟩, m.2⟩ | case mk.mk.intro.mk.intro.intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f✝ : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0✝ : x0 ∈ closure s
x1 : E
m1✝ : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0✝⟩) r0 ∩ ball (↑⟨x1, m1✝⟩) r1
x... | case mk.mk.intro.mk.intro.intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f✝ : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0✝ : x0 ∈ closure s
x1 : E
m1✝ : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0✝⟩) r0 ∩ ball (↑⟨x1, m1✝⟩) r1
x... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | apply c m0 m1 | case mk.mk.intro.mk.intro.intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f✝ : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0✝ : x0 ∈ closure s
x1 : E
m1✝ : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0✝⟩) r0 ∩ ball (↑⟨x1, m1✝⟩) r1
x... | case mk.mk.intro.mk.intro.intro.a
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f✝ : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0✝ : x0 ∈ closure s
x1 : E
m1✝ : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0✝⟩) r0 ∩ ball (↑⟨x1, m1✝⟩) r1... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | bound | case mk.mk.intro.mk.intro.intro.a
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f✝ : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0✝ : x0 ∈ closure s
x1 : E
m1✝ : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0✝⟩) r0 ∩ ball (↑⟨x1, m1✝⟩) r1... | case mk.mk.intro.mk.intro.intro.a
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f✝ : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0✝ : x0 ∈ closure s
x1 : E
m1✝ : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0✝⟩) r0 ∩ ball (↑⟨x1, m1✝⟩) r1... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | bound | case mk.mk.intro.mk.intro.intro.a
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f✝ : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0✝ : x0 ∈ closure s
x1 : E
m1✝ : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0✝⟩) r0 ∩ ball (↑⟨x1, m1✝⟩) r1... | case mk.mk.intro.mk.intro.intro.a
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f✝ : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0✝ : x0 ∈ closure s
x1 : E
m1✝ : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0✝⟩) r0 ∩ ball (↑⟨x1, m1✝⟩) r1... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | simp only [← add_div, add_comm r1 r0, div_self (add_pos r0p r1p).ne'] | case mk.mk.intro.mk.intro.intro.a
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f✝ : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0✝ : x0 ∈ closure s
x1 : E
m1✝ : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0✝⟩) r0 ∩ ball (↑⟨x1, m1✝⟩) r1... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | rcases ne with ⟨z, m0, m1⟩ | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
⊢ ‖x1 - x0‖ < r0 + r1 | case intro.intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0✝ : x0 ∈ closure s
x1 : E
m1✝ : x1 ∈ closure s
z : E
m0 : z ∈ ball (↑⟨x0, m0✝⟩) r0
m1 : z ∈ ball (↑⟨x1, m1✝⟩) r1
⊢ ‖x1 - ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | simp only [mem_ball, dist_eq_norm] at m0 m1 | case intro.intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0✝ : x0 ∈ closure s
x1 : E
m1✝ : x1 ∈ closure s
z : E
m0 : z ∈ ball (↑⟨x0, m0✝⟩) r0
m1 : z ∈ ball (↑⟨x1, m1✝⟩) r1
⊢ ‖x1 - ... | case intro.intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0✝ : x0 ∈ closure s
x1 : E
m1✝ : x1 ∈ closure s
z : E
m0 : ‖z - x0‖ < r0
m1 : ‖z - x1‖ < r1
⊢ ‖x1 - x0‖ < r0 + r1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | calc ‖x1 - x0‖
_ = ‖z - x0 - (z - x1)‖ := by abel_nf
_ ≤ ‖z - x0‖ + ‖z - x1‖ := (norm_sub_le _ _)
_ < r0 + r1 := add_lt_add m0 m1 | case intro.intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0✝ : x0 ∈ closure s
x1 : E
m1✝ : x1 ∈ closure s
z : E
m0 : ‖z - x0‖ < r0
m1 : ‖z - x1‖ < r1
⊢ ‖x1 - x0‖ < r0 + r1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | abel_nf | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0✝ : x0 ∈ closure s
x1 : E
m1✝ : x1 ∈ closure s
z : E
m0 : ‖z - x0‖ < r0
m1 : ‖z - x1‖ < r1
⊢ ‖x1 - x0‖ = ‖z - x0 - (z - x1)‖ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | intro x a b ap bp | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
⊢ ∀ (x : E) ... | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
x : E
a b : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | have rnz := (add_pos ap bp).ne' | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
x : E
a b : ... | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
x : E
a b : ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | calc (a / (a + b)) • x - x
_ = (a / (a + b) - (a + b) / (a + b)) • x := by simp only [one_smul, sub_smul, div_self rnz]
_ = -((b / (a + b)) • x) := by rw [← sub_div, sub_add_cancel_left, neg_div, neg_smul] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
x : E
a b : ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | simp only [one_smul, sub_smul, div_self rnz] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
x : E
a b : ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | rw [← sub_div, sub_add_cancel_left, neg_div, neg_smul] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
x : E
a b : ... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | intro a ap | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
sub : ∀ (x :... | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
sub : ∀ (x :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | apply lt_of_lt_of_le (mul_lt_mul_of_pos_left x01 (div_pos ap (add_pos r0p r1p))) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
sub : ∀ (x :... | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
sub : ∀ (x :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | rw [div_mul_cancel₀ _ (add_pos r0p r1p).ne'] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
sub : ∀ (x :... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | refine ContinuousAt.eventually_mem ?_ ((isOpen_ball.inter isOpen_ball).mem_nhds ?_) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
sub : ∀ (x :... | case refine_1
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | exact ((continuous_fst.const_smul _).add (continuous_snd.const_smul _)).continuousAt | case refine_1
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | simp only [mem_inter_iff, mem_ball, dist_eq_norm, ← sub_add_eq_add_sub _ x0 _,
add_sub_assoc _ _ x1] | case refine_2
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r... | case refine_2
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | nth_rw 1 [add_comm r0 r1] | case refine_2
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r... | case refine_2
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | simp only [sub _ r0p r1p, sub _ r1p r0p] | case refine_2
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r... | case refine_2
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | simp only [add_comm r1 r0, neg_add_eq_sub, ← sub_eq_add_neg, ← smul_sub, norm_smul,
Real.norm_eq_abs, abs_div, abs_of_pos r0p, abs_of_pos r1p, abs_of_pos (add_pos r0p r1p),
norm_sub_rev (x0 : E) x1] | case refine_2
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r... | case refine_2
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | use le r0p, le r1p | case refine_2
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | simp only [nhds_prod_eq] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
sub : ∀ (x :... | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
sub : ∀ (x :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | rw [Prod.frequently (p := fun x ↦ x ∈ s) (q := fun x ↦ x ∈ s)] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
sub : ∀ (x :... | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
sub : ∀ (x :... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Convex.inter_ball | [124, 1] | [159, 72] | use mem_closure_iff_frequently.mp m0, mem_closure_iff_frequently.mp m1 | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
c : Convex ℝ s
r0 r1 : ℝ
r0p : 0 < r0
r1p : 0 < r1
x0 : E
m0 : x0 ∈ closure s
x1 : E
m1 : x1 ∈ closure s
ne : ∃ z, z ∈ ball (↑⟨x0, m0⟩) r0 ∩ ball (↑⟨x1, m1⟩) r1
x01 : ‖x1 - x0‖ < r0 + r1
sub : ∀ (x :... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ug | [166, 1] | [175, 50] | intro z ⟨zt, m⟩ | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : ↑(closure s)
⊢ EqOn b.u (b.g x) (b.t ∩ Metric.ball (↑x) (b.r x)) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : ↑(closure s)
z : E
zt : z ∈ b.t
m : z ∈ Metric.ball (↑x) (b.r x)
⊢ b.u z = b.g x z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ug | [166, 1] | [175, 50] | simp only [Base.u, zt, dif_pos] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : ↑(closure s)
z : E
zt : z ∈ b.t
m : z ∈ Metric.ball (↑x) (b.r x)
⊢ b.u z = b.g x z | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : ↑(closure s)
z : E
zt : z ∈ b.t
m : z ∈ Metric.ball (↑x) (b.r x)
⊢ b.g (b.y ⋯) z = b.g x z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ug | [166, 1] | [175, 50] | refine b.unique (isOpen_ball.inter isOpen_ball)
((convex_ball _ _).inter (convex_ball _ _)).isPreconnected
(fun _ m ↦ b.gp _ (inter_subset_left _ _ m)) (fun _ m ↦ b.gp _ (inter_subset_right _ _ m))
?_ ⟨b.yt zt, m⟩ | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : ↑(closure s)
z : E
zt : z ∈ b.t
m : z ∈ Metric.ball (↑x) (b.r x)
⊢ b.g (b.y ⋯) z = b.g x z | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : ↑(closure s)
z : E
zt : z ∈ b.t
m : z ∈ Metric.ball (↑x) (b.r x)
⊢ ∃ x_1 ∈ Metric.ball (↑(b.y ⋯)) (b.r (b.y ⋯)) ∩ Metric.ball (↑x) (b.r x), b.g (b.y ⋯) x_1 = b.g x x_1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ug | [166, 1] | [175, 50] | rcases b.convex.inter_ball (b.y zt) x (b.rp _) (b.rp _) ⟨_, ⟨b.yt zt, m⟩⟩ with ⟨w, m⟩ | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : ↑(closure s)
z : E
zt : z ∈ b.t
m : z ∈ Metric.ball (↑x) (b.r x)
⊢ ∃ x_1 ∈ Metric.ball (↑(b.y ⋯)) (b.r (b.y ⋯)) ∩ Metric.ball (↑x) (b.r x), b.g (b.y ⋯) x_1 = b.g x x_1 | case intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : ↑(closure s)
z : E
zt : z ∈ b.t
m✝ : z ∈ Metric.ball (↑x) (b.r x)
w : E
m : w ∈ s ∩ Metric.ball (↑(b.y zt)) (b.r (b.y zt)) ∩ Metric.ball (↑x) (b.r x)
⊢ ∃ x_1 ∈ Metric.b... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.ug | [166, 1] | [175, 50] | exact ⟨w, ⟨m.1.2, m.2⟩, _root_.trans ((b.gf _).self_of_nhdsSet ⟨m.1.1, m.1.2⟩)
((b.gf x).self_of_nhdsSet ⟨m.1.1, m.2⟩).symm⟩ | case intro
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
x : ↑(closure s)
z : E
zt : z ∈ b.t
m✝ : z ∈ Metric.ball (↑x) (b.r x)
w : E
m : w ∈ s ∩ Metric.ball (↑(b.y zt)) (b.r (b.y zt)) ∩ Metric.ball (↑x) (b.r x)
⊢ ∃ x_1 ∈ Metric.b... | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.uf | [178, 1] | [185, 66] | simp only [Filter.EventuallyEq, Filter.eventually_iff, mem_nhdsSet_iff_forall] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
⊢ (𝓝ˢ s).EventuallyEq b.u f | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
⊢ ∀ x ∈ s, {x | b.u x = f x} ∈ 𝓝 x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.uf | [178, 1] | [185, 66] | intro z m | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
⊢ ∀ x ∈ s, {x | b.u x = f x} ∈ 𝓝 x | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
z : E
m : z ∈ s
⊢ {x | b.u x = f x} ∈ 𝓝 z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.uf | [178, 1] | [185, 66] | simp only [← Filter.eventually_iff] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
z : E
m : z ∈ s
⊢ {x | b.u x = f x} ∈ 𝓝 z | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
z : E
m : z ∈ s
⊢ ∀ᶠ (x : E) in 𝓝 z, b.u x = f x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.uf | [178, 1] | [185, 66] | set x : closure s := ⟨z, subset_closure m⟩ | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
z : E
m : z ∈ s
⊢ ∀ᶠ (x : E) in 𝓝 z, b.u x = f x | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
z : E
m : z ∈ s
x : ↑(closure s) := ⟨z, ⋯⟩
⊢ ∀ᶠ (x : E) in 𝓝 z, b.u x = f x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.uf | [178, 1] | [185, 66] | have zs : z ∈ Metric.ball (x : E) (b.r x) := mem_ball_self (b.rp x) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
z : E
m : z ∈ s
x : ↑(closure s) := ⟨z, ⋯⟩
⊢ ∀ᶠ (x : E) in 𝓝 z, b.u x = f x | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
z : E
m : z ∈ s
x : ↑(closure s) := ⟨z, ⋯⟩
zs : z ∈ Metric.ball (↑x) (b.r x)
⊢ ∀ᶠ (x : E) in 𝓝 z, b.u x = f x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.uf | [178, 1] | [185, 66] | have ug := (b.ug x).eventuallyEq_of_mem ((b.ot.inter isOpen_ball).mem_nhds
⟨b.cover (subset_closure m), zs⟩) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
z : E
m : z ∈ s
x : ↑(closure s) := ⟨z, ⋯⟩
zs : z ∈ Metric.ball (↑x) (b.r x)
⊢ ∀ᶠ (x : E) in 𝓝 z, b.u x = f x | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
z : E
m : z ∈ s
x : ↑(closure s) := ⟨z, ⋯⟩
zs : z ∈ Metric.ball (↑x) (b.r x)
ug : (𝓝 z).EventuallyEq b.u (b.g x)
⊢ ∀ᶠ (x : E) in 𝓝 z, b.u x = f x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.uf | [178, 1] | [185, 66] | exact ug.trans ((b.gf x).filter_mono (nhds_le_nhdsSet ⟨m, zs⟩)) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z✝ : E
b : Base p s f
z : E
m : z ∈ s
x : ↑(closure s) := ⟨z, ⋯⟩
zs : z ∈ Metric.ball (↑x) (b.r x)
ug : (𝓝 z).EventuallyEq b.u (b.g x)
⊢ ∀ᶠ (x : E) in 𝓝 z, b.u x = f x | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.up | [188, 1] | [191, 92] | apply Filter.eventually_of_mem (b.ot.mem_nhdsSet.mpr b.cover) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
⊢ ∀ᶠ (z : E) in 𝓝ˢ (closure s), p b.u z | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
⊢ ∀ x ∈ b.t, p b.u x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.up | [188, 1] | [191, 92] | intro x m | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
⊢ ∀ x ∈ b.t, p b.u x | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ b.t
⊢ p b.u x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.up | [188, 1] | [191, 92] | refine b.congr (b.gp (b.y m) (b.yt m)) ?_ | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ b.t
⊢ p b.u x | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ b.t
⊢ (𝓝 x).EventuallyEq (b.g (b.y m)) b.u |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/Continuation.lean | Base.up | [188, 1] | [191, 92] | exact ((b.ug _).eventuallyEq_of_mem ((b.ot.inter isOpen_ball).mem_nhds ⟨m, b.yt m⟩)).symm | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
α : Type
p : (E → α) → E → Prop
s : Set E
f : E → α
z : E
b : Base p s f
x : E
m : x ∈ b.t
⊢ (𝓝 x).EventuallyEq (b.g (b.y m)) b.u | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | rw [summable_iff_vanishing_norm] | f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
⊢ Summable fun n => f n z | f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
⊢ ∀ ε > 0, ∃ s, ∀ (t : Finset ℕ), Disjoint t s → ‖t.sum fun i => f i z‖ < ε |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | intro e ep | f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
⊢ ∀ ε > 0, ∃ s, ∀ (t : Finset ℕ), Disjoint t s → ‖t.sum fun i => f i z‖ < ε | f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
e : ℝ
ep : e > 0
⊢ ∃ s, ∀ (t : Finset ℕ), Disjoint t s → ‖t.sum fun i => f i z‖ < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | rcases h e ep with ⟨m, hm⟩ | f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
e : ℝ
ep : e > 0
⊢ ∃ s, ∀ (t : Finset ℕ), Disjoint t s → ‖t.sum fun i => f i z‖ < e | case intro
f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
⊢ ∃ s, ∀ (t : Finset ℕ), Disjoint t s → ‖t.sum fun i => f i z‖ < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | use Finset.range m | case intro
f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
⊢ ∃ s, ∀ (t : Finset ℕ), Disjoint t s → ‖t.sum fun i => f i z‖ < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
⊢ ∀ (t : Finset ℕ), Disjoint t (Finset.range m) → ‖t.sum fun i => f i z‖ < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | intro A d | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
⊢ ∀ (t : Finset ℕ), Disjoint t (Finset.range m) → ‖t.sum fun i => f i z‖ < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
A : Finset ℕ
d : Disjoint A (Finset.range m)
⊢ ‖A.sum fun i => f i z‖ < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | calc ‖A.sum (fun n ↦ f n z)‖
_ ≤ A.sum (fun n ↦ ‖f n z‖) := by bound
_ < e := hm _ _ (late_iff_disjoint_range.mpr d) zs | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
A : Finset ℕ
d : Disjoint A (Finset.range m)
⊢ ‖A.sum fun i => f i z‖ < e | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_summable | [47, 1] | [54, 55] | bound | f : ℕ → ℂ → ℂ
s : Set ℂ
z : ℂ
zs : z ∈ s
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
A : Finset ℕ
d : Disjoint A (Finset.range m)
⊢ ‖A.sum fun n => f n z‖ ≤ A.sum fun n => ‖f n z‖ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_uniform_cauchy_series | [57, 1] | [67, 51] | rw [Metric.uniformCauchySeqOn_iff] | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
⊢ UniformCauchySeqOn (fun N z => N.sum fun n => f n z) atTop s | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < ε |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_uniform_cauchy_series | [57, 1] | [67, 51] | intro e ep | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
⊢ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < ε | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
⊢ ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_uniform_cauchy_series | [57, 1] | [67, 51] | rcases h e ep with ⟨m, hm⟩ | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
⊢ ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < e | case intro
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
⊢ ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_uniform_cauchy_series | [57, 1] | [67, 51] | use Finset.range m | case intro
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
⊢ ∃ N, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (m.sum fun n => f n x) (n.sum fun n => f n x) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
⊢ ∀ m_1 ≥ Finset.range m, ∀ n ≥ Finset.range m, ∀ x ∈ s, dist (m_1.sum fun n => f n x) (n.sum fun n => f n x) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_uniform_cauchy_series | [57, 1] | [67, 51] | intro A HA B HB z zs | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
⊢ ∀ m_1 ≥ Finset.range m, ∀ n ≥ Finset.range m, ∀ x ∈ s, dist (m_1.sum fun n => f n x) (n.sum fun n => f n x) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
A : Finset ℕ
HA : A ≥ Finset.range m
B : Finset ℕ
HB : B ≥ Finset.range m
z : ℂ
zs : z ∈ s
⊢ dist (A.sum fun n => f n z) (B.sum fun n => f n z) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_uniform_cauchy_series | [57, 1] | [67, 51] | calc dist (A.sum fun n ↦ f n z) (B.sum fun n ↦ f n z)
_ ≤ (A ∆ B).sum fun n ↦ abs (f n z) := symmDiff_bound _ _ _
_ < e := hm (A ∆ B) z (symmDiff_late HA HB) zs | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e
A : Finset ℕ
HA : A ≥ Finset.range m
B : Finset ℕ
HB : B ≥ Finset.range m
z : ℂ
zs : z ∈ s
⊢ dist (A.sum fun n => f n z) (B.sum fun n => f n z) < e | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [HasUniformSum, Metric.tendstoUniformlyOn_iff] | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
⊢ HasUniformSum f (tsumOn f) s | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
⊢ ∀ ε > 0, ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < ε |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | intro e ep | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
⊢ ∀ ε > 0, ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < ε | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
⊢ ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rcases h (e / 4) (by linarith) with ⟨m, hm⟩ | f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
⊢ ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < e | case intro
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
⊢ ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [Filter.eventually_atTop] | case intro
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
⊢ ∀ᶠ (n : Finset ℕ) in atTop, ∀ x ∈ s, dist (tsumOn f x) (n.sum fun n => f n x) < e | case intro
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
⊢ ∃ a, ∀ b ≥ a, ∀ x ∈ s, dist (tsumOn f x) (b.sum fun n => f n x) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | use Finset.range m | case intro
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
⊢ ∃ a, ∀ b ≥ a, ∀ x ∈ s, dist (tsumOn f x) (b.sum fun n => f n x) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
⊢ ∀ b ≥ Finset.range m, ∀ x ∈ s, dist (tsumOn f x) (b.sum fun n => f n x) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | intro N Nm z zs | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
⊢ ∀ b ≥ Finset.range m, ∀ x ∈ s, dist (tsumOn f x) (b.sum fun n => f n x) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
⊢ dist (tsumOn f z) (N.sum fun n => f n z) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [tsumOn] | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
⊢ dist (tsumOn f z) (N.sum fun n => f n z) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
⊢ dist (∑' (n : ℕ), f n z) (N.sum fun n => f n z) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | generalize G : tsum (fun n ↦ f n z) = g | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
⊢ dist (∑' (n : ℕ), f n z) (N.sum fun n => f n z) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
⊢ dist g (N.sum fun n => f n z) < e |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | have S : Summable (fun n ↦ f n z) := uniformVanishing_to_summable zs h | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
⊢ dist g (N.sum fun n => f n z) < e | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
S : Summable fun n => f n z
⊢ dist g (N.sum fun n => f n z) ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | have GS : HasSum (fun n ↦ f n z) g := by rw [← G]; exact Summable.hasSum S | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
S : Summable fun n => f n z
⊢ dist g (N.sum fun n => f n z) ... | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
S : Summable fun n => f n z
GS : HasSum (fun n => f n z) g
⊢... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | clear S | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
S : Summable fun n => f n z
GS : HasSum (fun n => f n z) g
⊢... | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
GS : HasSum (fun n => f n z) g
⊢ dist g (N.sum fun n => f n ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [HasSum] at GS | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
GS : HasSum (fun n => f n z) g
⊢ dist g (N.sum fun n => f n ... | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
GS : Filter.Tendsto (fun s => s.sum fun b => f b z) atTop (�... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rw [Metric.tendsto_atTop] at GS | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
GS : Filter.Tendsto (fun s => s.sum fun b => f b z) atTop (�... | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
GS : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (n.sum fun b => f b z) g < ... |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Series.lean | uniformVanishing_to_tendsto_uniformly_on | [70, 1] | [103, 25] | rcases GS (e / 4) (by linarith) with ⟨M, HM⟩ | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
GS : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (n.sum fun b => f b z) g < ... | case h.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
h : UniformVanishing f s
e : ℝ
ep : e > 0
m : ℕ
hm : ∀ (N : Finset ℕ) (z : ℂ), Late N m → z ∈ s → (N.sum fun n => Complex.abs (f n z)) < e / 4
N : Finset ℕ
Nm : N ≥ Finset.range m
z : ℂ
zs : z ∈ s
g : ℂ
G : ∑' (n : ℕ), f n z = g
GS : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (n.sum fun b => f b z... |
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