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/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Vec.lean
|
Vec.le_div_iff
|
[111, 1]
|
[116, 51]
|
exact hi
|
case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a ≤ b / c
i : m
hi : a i * c i ≤ b i
⊢ (a * c) i ≤ b i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Vec.lean
|
Vec.le_div_iff
|
[111, 1]
|
[116, 51]
|
intro h i
|
case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
⊢ a * c ≤ b → a ≤ b / c
|
case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
⊢ a i ≤ (b / c) i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Vec.lean
|
Vec.le_div_iff
|
[111, 1]
|
[116, 51]
|
have hi := h i
|
case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
⊢ a i ≤ (b / c) i
|
case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
hi : (a * c) i ≤ b i
⊢ a i ≤ (b / c) i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Vec.lean
|
Vec.le_div_iff
|
[111, 1]
|
[116, 51]
|
simp at hi
|
case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
hi : (a * c) i ≤ b i
⊢ a i ≤ (b / c) i
|
case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
hi : a i * c i ≤ b i
⊢ a i ≤ (b / c) i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Vec.lean
|
Vec.le_div_iff
|
[111, 1]
|
[116, 51]
|
dsimp
|
case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
hi : a i * c i ≤ b i
⊢ a i ≤ (b / c) i
|
case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
hi : a i * c i ≤ b i
⊢ a i ≤ b i / c i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Vec.lean
|
Vec.le_div_iff
|
[111, 1]
|
[116, 51]
|
rw [_root_.le_div_iff (hc i)]
|
case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
hi : a i * c i ≤ b i
⊢ a i ≤ b i / c i
|
case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
hi : a i * c i ≤ b i
⊢ a i * c i ≤ b i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Vec.lean
|
Vec.le_div_iff
|
[111, 1]
|
[116, 51]
|
exact hi
|
case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
hi : a i * c i ≤ b i
⊢ a i * c i ≤ b i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
simp [rotatedSoCone, rotateSoCone]
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
⊢ match t.rotateSoCone x with
| (v, w, x) => v.rotatedSoCone w x
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
have habsx0t : |x 0| ≤ t := by
rw [soCone, Fin.sum_univ_succ] at h
have hS : 0 ≤ ∑ i : Fin n, x (Fin.succ i) ^ 2 :=
Finset.sum_nonneg (fun i _ => (rpow_two (x i.succ)).symm ▸ sq_nonneg (x i.succ))
exact abs_le_of_sqrt_sq_add_nonneg_le hS h
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
have ht : 0 ≤ t := le_trans (abs_nonneg _) habsx0t
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
replace ⟨hx0t, hnx0t⟩ := abs_le.mp habsx0t
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
split_ands
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2
case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t + x 0) / √2
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
{ field_simp
have hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2 := by norm_cast; ring
rw [hrw, le_sub_iff_add_le, add_comm]
unfold soCone at h; norm_cast at h ⊢
rw [← Fin.sum_univ_succ (f := fun i => (x i) ^ 2)]
rw [← sqrt_le_left ht]
exact h }
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2
case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t + x 0) / √2
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2
|
case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t + x 0) / √2
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
{ simp [le_div_iff]; linarith }
|
case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t + x 0) / √2
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2
|
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
{ simp [le_div_iff]; linarith }
|
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
rw [soCone, Fin.sum_univ_succ] at h
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
⊢ |x 0| ≤ t
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : (x 0 ^ 2 + ∑ i : Fin n, x i.succ ^ 2).sqrt ≤ t
⊢ |x 0| ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
have hS : 0 ≤ ∑ i : Fin n, x (Fin.succ i) ^ 2 :=
Finset.sum_nonneg (fun i _ => (rpow_two (x i.succ)).symm ▸ sq_nonneg (x i.succ))
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : (x 0 ^ 2 + ∑ i : Fin n, x i.succ ^ 2).sqrt ≤ t
⊢ |x 0| ≤ t
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : (x 0 ^ 2 + ∑ i : Fin n, x i.succ ^ 2).sqrt ≤ t
hS : 0 ≤ ∑ i : Fin n, x i.succ ^ 2
⊢ |x 0| ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
exact abs_le_of_sqrt_sq_add_nonneg_le hS h
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : (x 0 ^ 2 + ∑ i : Fin n, x i.succ ^ 2).sqrt ≤ t
hS : 0 ≤ ∑ i : Fin n, x i.succ ^ 2
⊢ |x 0| ≤ t
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
field_simp
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) * (t - x 0)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
have hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2 := by norm_cast; ring
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) * (t - x 0)
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) * (t - x 0)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
rw [hrw, le_sub_iff_add_le, add_comm]
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) * (t - x 0)
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
⊢ x 0 ^ 2 + ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ t ^ 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
unfold soCone at h
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
⊢ x 0 ^ 2 + ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ t ^ 2
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : (∑ i : Fin n.succ, x i ^ 2).sqrt ≤ t
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
⊢ x 0 ^ 2 + ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ t ^ 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
norm_cast at h ⊢
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : (∑ i : Fin n.succ, x i ^ 2).sqrt ≤ t
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
⊢ x 0 ^ 2 + ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ t ^ 2
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
h : (∑ x_1 : Fin n.succ, x x_1 ^ 2).sqrt ≤ t
⊢ x 0 ^ 2 + ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ t ^ 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
rw [← Fin.sum_univ_succ (f := fun i => (x i) ^ 2)]
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
h : (∑ x_1 : Fin n.succ, x x_1 ^ 2).sqrt ≤ t
⊢ x 0 ^ 2 + ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ t ^ 2
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
h : (∑ x_1 : Fin n.succ, x x_1 ^ 2).sqrt ≤ t
⊢ ∑ i : Fin (n + 1), x i ^ 2 ≤ t ^ 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
rw [← sqrt_le_left ht]
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
h : (∑ x_1 : Fin n.succ, x x_1 ^ 2).sqrt ≤ t
⊢ ∑ i : Fin (n + 1), x i ^ 2 ≤ t ^ 2
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
h : (∑ x_1 : Fin n.succ, x x_1 ^ 2).sqrt ≤ t
⊢ (∑ i : Fin (n + 1), x i ^ 2).sqrt ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
exact h
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
h : (∑ x_1 : Fin n.succ, x x_1 ^ 2).sqrt ≤ t
⊢ (∑ i : Fin (n + 1), x i ^ 2).sqrt ≤ t
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
ring
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
simp [le_div_iff]
|
case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t + x 0) / √2
|
case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ t + x 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
linarith
|
case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ t + x 0
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
simp [le_div_iff]
|
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2
|
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ x 0 ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
linarith
|
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ x 0 ≤ t
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
simp [soCone, unrotateSoCone]
|
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : v.rotatedSoCone w x
⊢ match v.unrotateSoCone w x with
| (t, x) => t.soCone x
|
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : v.rotatedSoCone w x
⊢ (∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2).sqrt ≤ (v + w) / √2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
replace ⟨h, hv, hw⟩ := h
|
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : v.rotatedSoCone w x
⊢ (∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2).sqrt ≤ (v + w) / √2
|
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ (∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2).sqrt ≤ (v + w) / √2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
rw [sqrt_le_iff]
|
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ (∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2).sqrt ≤ (v + w) / √2
|
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ 0 ≤ (v + w) / √2 ∧ ∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2 ≤ ((v + w) / √2) ^ 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
split_ands
|
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ 0 ≤ (v + w) / √2 ∧ ∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2 ≤ ((v + w) / √2) ^ 2
|
case refine_1
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ 0 ≤ (v + w) / √2
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ ∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2 ≤ ((v + w) / √2) ^ 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
{ simp [le_div_iff]; linarith }
|
case refine_1
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ 0 ≤ (v + w) / √2
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ ∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2 ≤ ((v + w) / √2) ^ 2
|
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ ∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2 ≤ ((v + w) / √2) ^ 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
{ rw [Fin.sum_univ_succ]
simp [Matrix.vecCons]
rw [add_comm, ← le_sub_iff_add_le]
field_simp
have hrw : ((v + w) ^ 2 - (v - w) ^ 2) / 2 = v * w * 2 := by norm_cast; ring
norm_cast at hrw h
rwa [hrw] }
|
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ ∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2 ≤ ((v + w) / √2) ^ 2
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
simp [le_div_iff]
|
case refine_1
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ 0 ≤ (v + w) / √2
|
case refine_1
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ 0 ≤ v + w
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
linarith
|
case refine_1
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ 0 ≤ v + w
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
rw [Fin.sum_univ_succ]
|
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ ∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2 ≤ ((v + w) / √2) ^ 2
|
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ Matrix.vecCons ((v - w) / √2) x 0 ^ 2 + ∑ i : Fin n, Matrix.vecCons ((v - w) / √2) x i.succ ^ 2 ≤ ((v + w) / √2) ^ 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
simp [Matrix.vecCons]
|
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ Matrix.vecCons ((v - w) / √2) x 0 ^ 2 + ∑ i : Fin n, Matrix.vecCons ((v - w) / √2) x i.succ ^ 2 ≤ ((v + w) / √2) ^ 2
|
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ (v - w) ^ 2 / 2 + ∑ x_1 : Fin n, x x_1 ^ 2 ≤ (v + w) ^ 2 / 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
rw [add_comm, ← le_sub_iff_add_le]
|
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ (v - w) ^ 2 / 2 + ∑ x_1 : Fin n, x x_1 ^ 2 ≤ (v + w) ^ 2 / 2
|
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ ∑ x_1 : Fin n, x x_1 ^ 2 ≤ (v + w) ^ 2 / 2 - (v - w) ^ 2 / 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
field_simp
|
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ ∑ x_1 : Fin n, x x_1 ^ 2 ≤ (v + w) ^ 2 / 2 - (v - w) ^ 2 / 2
|
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ ∑ x_1 : Fin n, x x_1 ^ 2 ≤ ((v + w) ^ 2 - (v - w) ^ 2) / 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
have hrw : ((v + w) ^ 2 - (v - w) ^ 2) / 2 = v * w * 2 := by norm_cast; ring
|
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ ∑ x_1 : Fin n, x x_1 ^ 2 ≤ ((v + w) ^ 2 - (v - w) ^ 2) / 2
|
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
hrw : ((v + w) ^ 2 - (v - w) ^ 2) / 2 = v * w * 2
⊢ ∑ x_1 : Fin n, x x_1 ^ 2 ≤ ((v + w) ^ 2 - (v - w) ^ 2) / 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
norm_cast at hrw h
|
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
hrw : ((v + w) ^ 2 - (v - w) ^ 2) / 2 = v * w * 2
⊢ ∑ x_1 : Fin n, x x_1 ^ 2 ≤ ((v + w) ^ 2 - (v - w) ^ 2) / 2
|
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
hv : 0 ≤ v
hw : 0 ≤ w
hrw : ((v + w) ^ 2 - (v - w) ^ 2) / 2 = v * w * 2
h : ∑ x_1 : Fin n, x x_1 ^ 2 ≤ v * w * 2
⊢ ∑ x_1 : Fin n, x x_1 ^ 2 ≤ ((v + w) ^ 2 - (v - w) ^ 2) / 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
rwa [hrw]
|
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
hv : 0 ≤ v
hw : 0 ≤ w
hrw : ((v + w) ^ 2 - (v - w) ^ 2) / 2 = v * w * 2
h : ∑ x_1 : Fin n, x x_1 ^ 2 ≤ v * w * 2
⊢ ∑ x_1 : Fin n, x x_1 ^ 2 ≤ ((v + w) ^ 2 - (v - w) ^ 2) / 2
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
ring
|
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ ((v + w) ^ 2 - (v - w) ^ 2) / 2 = v * w * 2
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.soCone_add_sub_two_mul_of_nonneg
|
[87, 1]
|
[94, 16]
|
have hxy := add_nonneg hx hy
|
n : Type ?u.52431
m : Type ?u.52428
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
⊢ (x + y).soCone ![x - y, 2 * z] ↔ z ^ 2 ≤ x * y
|
n : Type ?u.52431
m : Type ?u.52428
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
hxy : 0 ≤ x + y
⊢ (x + y).soCone ![x - y, 2 * z] ↔ z ^ 2 ≤ x * y
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.soCone_add_sub_two_mul_of_nonneg
|
[87, 1]
|
[94, 16]
|
conv => lhs; unfold soCone; simp [sqrt_le_left hxy, ← le_sub_iff_add_le']
|
n : Type ?u.52431
m : Type ?u.52428
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
hxy : 0 ≤ x + y
⊢ (x + y).soCone ![x - y, 2 * z] ↔ z ^ 2 ≤ x * y
|
n : Type ?u.52431
m : Type ?u.52428
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
hxy : 0 ≤ x + y
⊢ (2 * z) ^ 2 ≤ (x + y) ^ 2 - (x - y) ^ 2 ↔ z ^ 2 ≤ x * y
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.soCone_add_sub_two_mul_of_nonneg
|
[87, 1]
|
[94, 16]
|
ring_nf
|
n : Type ?u.52431
m : Type ?u.52428
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
hxy : 0 ≤ x + y
⊢ (2 * z) ^ 2 ≤ (x + y) ^ 2 - (x - y) ^ 2 ↔ z ^ 2 ≤ x * y
|
n : Type ?u.52431
m : Type ?u.52428
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
hxy : 0 ≤ x + y
⊢ z ^ 2 * 4 ≤ x * y * 4 ↔ z ^ 2 ≤ x * y
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.soCone_add_sub_two_mul_of_nonneg
|
[87, 1]
|
[94, 16]
|
simp
|
n : Type ?u.52431
m : Type ?u.52428
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
hxy : 0 ≤ x + y
⊢ z ^ 2 * 4 ≤ x * y * 4 ↔ z ^ 2 ≤ x * y
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.soCone_add_sub_two_of_nonneg
|
[96, 1]
|
[101, 10]
|
have h := soCone_add_sub_two_mul_of_nonneg 1 hx hy
|
n : Type ?u.62790
m : Type ?u.62787
inst✝¹ : Fintype m
inst✝ : Fintype n
x y : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
⊢ (x + y).soCone ![x - y, 2] ↔ 1 ≤ x * y
|
n : Type ?u.62790
m : Type ?u.62787
inst✝¹ : Fintype m
inst✝ : Fintype n
x y : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
h : (x + y).soCone ![x - y, 2 * 1] ↔ 1 ^ 2 ≤ x * y
⊢ (x + y).soCone ![x - y, 2] ↔ 1 ≤ x * y
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.soCone_add_sub_two_of_nonneg
|
[96, 1]
|
[101, 10]
|
rw [mul_one, one_rpow] at h
|
n : Type ?u.62790
m : Type ?u.62787
inst✝¹ : Fintype m
inst✝ : Fintype n
x y : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
h : (x + y).soCone ![x - y, 2 * 1] ↔ 1 ^ 2 ≤ x * y
⊢ (x + y).soCone ![x - y, 2] ↔ 1 ≤ x * y
|
n : Type ?u.62790
m : Type ?u.62787
inst✝¹ : Fintype m
inst✝ : Fintype n
x y : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
h : (x + y).soCone ![x - y, 2] ↔ 1 ≤ x * y
⊢ (x + y).soCone ![x - y, 2] ↔ 1 ≤ x * y
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.soCone_add_sub_two_of_nonneg
|
[96, 1]
|
[101, 10]
|
exact h
|
n : Type ?u.62790
m : Type ?u.62787
inst✝¹ : Fintype m
inst✝ : Fintype n
x y : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
h : (x + y).soCone ![x - y, 2] ↔ 1 ≤ x * y
⊢ (x + y).soCone ![x - y, 2] ↔ 1 ≤ x * y
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.soCone_sub_add_two_mul_of_nonneg
|
[103, 1]
|
[109, 63]
|
conv => lhs; unfold soCone; simp [sqrt_le_iff, ← le_sub_iff_add_le']
|
n : Type ?u.63307
m : Type ?u.63304
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
⊢ (x - y).soCone ![x + y, 2 * z] ↔ y ≤ x ∧ z ^ 2 ≤ -(x * y)
|
n : Type ?u.63307
m : Type ?u.63304
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
⊢ y ≤ x ∧ (2 * z) ^ 2 ≤ (x - y) ^ 2 - (x + y) ^ 2 ↔ y ≤ x ∧ z ^ 2 ≤ -(x * y)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.soCone_sub_add_two_mul_of_nonneg
|
[103, 1]
|
[109, 63]
|
apply Iff.and
|
n : Type ?u.63307
m : Type ?u.63304
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
⊢ y ≤ x ∧ (2 * z) ^ 2 ≤ (x - y) ^ 2 - (x + y) ^ 2 ↔ y ≤ x ∧ z ^ 2 ≤ -(x * y)
|
case h₁
n : Type ?u.63307
m : Type ?u.63304
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
⊢ y ≤ x ↔ y ≤ x
case h₂
n : Type ?u.63307
m : Type ?u.63304
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
⊢ (2 * z) ^ 2 ≤ (x - y) ^ 2 - (x + y) ^ 2 ↔ z ^ 2 ≤ -(x * y)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.soCone_sub_add_two_mul_of_nonneg
|
[103, 1]
|
[109, 63]
|
rfl
|
case h₁
n : Type ?u.63307
m : Type ?u.63304
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
⊢ y ≤ x ↔ y ≤ x
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.soCone_sub_add_two_mul_of_nonneg
|
[103, 1]
|
[109, 63]
|
ring_nf!
|
case h₂
n : Type ?u.63307
m : Type ?u.63304
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
⊢ (2 * z) ^ 2 ≤ (x - y) ^ 2 - (x + y) ^ 2 ↔ z ^ 2 ≤ -(x * y)
|
case h₂
n : Type ?u.63307
m : Type ?u.63304
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
⊢ z ^ 2 * 4 ≤ -(x * y * 4) ↔ z ^ 2 ≤ -(x * y)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.soCone_sub_add_two_mul_of_nonneg
|
[103, 1]
|
[109, 63]
|
rw [← neg_mul, ← div_le_iff (by norm_num)]
|
case h₂
n : Type ?u.63307
m : Type ?u.63304
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
⊢ z ^ 2 * 4 ≤ -(x * y * 4) ↔ z ^ 2 ≤ -(x * y)
|
case h₂
n : Type ?u.63307
m : Type ?u.63304
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
⊢ z ^ 2 * 4 / 4 ≤ -(x * y) ↔ z ^ 2 ≤ -(x * y)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.soCone_sub_add_two_mul_of_nonneg
|
[103, 1]
|
[109, 63]
|
simp
|
case h₂
n : Type ?u.63307
m : Type ?u.63304
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
⊢ z ^ 2 * 4 / 4 ≤ -(x * y) ↔ z ^ 2 ≤ -(x * y)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.soCone_sub_add_two_mul_of_nonneg
|
[103, 1]
|
[109, 63]
|
norm_num
|
n : Type ?u.63307
m : Type ?u.63304
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
⊢ 0 < 4
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.vec_soCone_apply_to_soCone_add_sub_two_mul
|
[113, 1]
|
[115, 57]
|
dsimp
|
n✝ : Type ?u.71362
m : Type ?u.71359
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
x y z : Fin n → ℝ
i : Fin n
⊢ ((x + y) i).soCone (![x - y, 2 • z]ᵀ i) ↔ (x i + y i).soCone ![x i - y i, 2 * z i]
|
n✝ : Type ?u.71362
m : Type ?u.71359
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
x y z : Fin n → ℝ
i : Fin n
⊢ (x i + y i).soCone (![x - y, 2 • z]ᵀ i) ↔ (x i + y i).soCone ![x i - y i, 2 * z i]
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.vec_soCone_apply_to_soCone_add_sub_two_mul
|
[113, 1]
|
[115, 57]
|
convert Iff.rfl
|
n✝ : Type ?u.71362
m : Type ?u.71359
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
x y z : Fin n → ℝ
i : Fin n
⊢ (x i + y i).soCone (![x - y, 2 • z]ᵀ i) ↔ (x i + y i).soCone ![x i - y i, 2 * z i]
|
case h.e'_2.h.e'_4
n✝ : Type ?u.71362
m : Type ?u.71359
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
x y z : Fin n → ℝ
i : Fin n
⊢ ![x i - y i, 2 * z i] = ![x - y, 2 • z]ᵀ i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.vec_soCone_apply_to_soCone_add_sub_two_mul
|
[113, 1]
|
[115, 57]
|
funext j
|
case h.e'_2.h.e'_4
n✝ : Type ?u.71362
m : Type ?u.71359
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
x y z : Fin n → ℝ
i : Fin n
⊢ ![x i - y i, 2 * z i] = ![x - y, 2 • z]ᵀ i
|
case h.e'_2.h.e'_4.h
n✝ : Type ?u.71362
m : Type ?u.71359
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
x y z : Fin n → ℝ
i : Fin n
j : Fin (Nat.succ 1)
⊢ ![x i - y i, 2 * z i] j = ![x - y, 2 • z]ᵀ i j
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.vec_soCone_apply_to_soCone_add_sub_two_mul
|
[113, 1]
|
[115, 57]
|
fin_cases j <;> simp
|
case h.e'_2.h.e'_4.h
n✝ : Type ?u.71362
m : Type ?u.71359
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
x y z : Fin n → ℝ
i : Fin n
j : Fin (Nat.succ 1)
⊢ ![x i - y i, 2 * z i] j = ![x - y, 2 • z]ᵀ i j
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.vec_soCone_apply_to_soCone_add_sub_two
|
[117, 1]
|
[119, 57]
|
dsimp
|
n✝ : Type ?u.77315
m : Type ?u.77312
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
x y : Fin n → ℝ
i : Fin n
⊢ ((x + y) i).soCone (![x - y, 2]ᵀ i) ↔ (x i + y i).soCone ![x i - y i, 2]
|
n✝ : Type ?u.77315
m : Type ?u.77312
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
x y : Fin n → ℝ
i : Fin n
⊢ (x i + y i).soCone (![x - y, 2]ᵀ i) ↔ (x i + y i).soCone ![x i - y i, 2]
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.vec_soCone_apply_to_soCone_add_sub_two
|
[117, 1]
|
[119, 57]
|
convert Iff.rfl
|
n✝ : Type ?u.77315
m : Type ?u.77312
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
x y : Fin n → ℝ
i : Fin n
⊢ (x i + y i).soCone (![x - y, 2]ᵀ i) ↔ (x i + y i).soCone ![x i - y i, 2]
|
case h.e'_2.h.e'_4
n✝ : Type ?u.77315
m : Type ?u.77312
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
x y : Fin n → ℝ
i : Fin n
⊢ ![x i - y i, 2] = ![x - y, 2]ᵀ i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.vec_soCone_apply_to_soCone_add_sub_two
|
[117, 1]
|
[119, 57]
|
funext j
|
case h.e'_2.h.e'_4
n✝ : Type ?u.77315
m : Type ?u.77312
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
x y : Fin n → ℝ
i : Fin n
⊢ ![x i - y i, 2] = ![x - y, 2]ᵀ i
|
case h.e'_2.h.e'_4.h
n✝ : Type ?u.77315
m : Type ?u.77312
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
x y : Fin n → ℝ
i : Fin n
j : Fin (Nat.succ 1)
⊢ ![x i - y i, 2] j = ![x - y, 2]ᵀ i j
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.vec_soCone_apply_to_soCone_add_sub_two
|
[117, 1]
|
[119, 57]
|
fin_cases j <;> simp
|
case h.e'_2.h.e'_4.h
n✝ : Type ?u.77315
m : Type ?u.77312
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
x y : Fin n → ℝ
i : Fin n
j : Fin (Nat.succ 1)
⊢ ![x i - y i, 2] j = ![x - y, 2]ᵀ i j
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogSumExp.lean
|
Vec.sum_exp_eq_sum_div
|
[12, 1]
|
[17, 42]
|
unfold Vec.sum
|
n : ℕ
x : Fin n → ℝ
t : ℝ
⊢ sum (exp (x - const n t)) = sum (exp x) / t.exp
|
n : ℕ
x : Fin n → ℝ
t : ℝ
⊢ (Finset.univ.sum fun i => exp (x - const n t) i) = (Finset.univ.sum fun i => exp x i) / t.exp
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogSumExp.lean
|
Vec.sum_exp_eq_sum_div
|
[12, 1]
|
[17, 42]
|
rw [Finset.sum_div]
|
n : ℕ
x : Fin n → ℝ
t : ℝ
⊢ (Finset.univ.sum fun i => exp (x - const n t) i) = (Finset.univ.sum fun i => exp x i) / t.exp
|
n : ℕ
x : Fin n → ℝ
t : ℝ
⊢ (Finset.univ.sum fun i => exp (x - const n t) i) = Finset.univ.sum fun i => exp x i / t.exp
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogSumExp.lean
|
Vec.sum_exp_eq_sum_div
|
[12, 1]
|
[17, 42]
|
congr
|
n : ℕ
x : Fin n → ℝ
t : ℝ
⊢ (Finset.univ.sum fun i => exp (x - const n t) i) = Finset.univ.sum fun i => exp x i / t.exp
|
case e_f
n : ℕ
x : Fin n → ℝ
t : ℝ
⊢ (fun i => exp (x - const n t) i) = fun i => exp x i / t.exp
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogSumExp.lean
|
Vec.sum_exp_eq_sum_div
|
[12, 1]
|
[17, 42]
|
ext i
|
case e_f
n : ℕ
x : Fin n → ℝ
t : ℝ
⊢ (fun i => exp (x - const n t) i) = fun i => exp x i / t.exp
|
case e_f.h
n : ℕ
x : Fin n → ℝ
t : ℝ
i : Fin n
⊢ exp (x - const n t) i = exp x i / t.exp
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogSumExp.lean
|
Vec.sum_exp_eq_sum_div
|
[12, 1]
|
[17, 42]
|
simp [Vec.exp, Vec.const, Real.exp_sub]
|
case e_f.h
n : ℕ
x : Fin n → ℝ
t : ℝ
i : Fin n
⊢ exp (x - const n t) i = exp x i / t.exp
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogSumExp.lean
|
Vec.sum_exp_pos
|
[19, 1]
|
[23, 28]
|
apply Finset.sum_pos
|
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ 0 < sum (exp x)
|
case h
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ ∀ i ∈ Finset.univ, 0 < exp x i
case hs
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ Finset.univ.Nonempty
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogSumExp.lean
|
Vec.sum_exp_pos
|
[19, 1]
|
[23, 28]
|
{ intros i _; simp [Vec.exp, Real.exp_pos] }
|
case h
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ ∀ i ∈ Finset.univ, 0 < exp x i
case hs
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ Finset.univ.Nonempty
|
case hs
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ Finset.univ.Nonempty
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogSumExp.lean
|
Vec.sum_exp_pos
|
[19, 1]
|
[23, 28]
|
{ existsi ⟨0, hn⟩; simp }
|
case hs
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ Finset.univ.Nonempty
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogSumExp.lean
|
Vec.sum_exp_pos
|
[19, 1]
|
[23, 28]
|
intros i _
|
case h
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ ∀ i ∈ Finset.univ, 0 < exp x i
|
case h
n : ℕ
hn : 0 < n
x : Fin n → ℝ
i : Fin n
a✝ : i ∈ Finset.univ
⊢ 0 < exp x i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogSumExp.lean
|
Vec.sum_exp_pos
|
[19, 1]
|
[23, 28]
|
simp [Vec.exp, Real.exp_pos]
|
case h
n : ℕ
hn : 0 < n
x : Fin n → ℝ
i : Fin n
a✝ : i ∈ Finset.univ
⊢ 0 < exp x i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogSumExp.lean
|
Vec.sum_exp_pos
|
[19, 1]
|
[23, 28]
|
existsi ⟨0, hn⟩
|
case hs
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ Finset.univ.Nonempty
|
case hs
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ ⟨0, hn⟩ ∈ Finset.univ
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogSumExp.lean
|
Vec.sum_exp_pos
|
[19, 1]
|
[23, 28]
|
simp
|
case hs
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ ⟨0, hn⟩ ∈ Finset.univ
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/Arith/PositivityExt.lean
|
Mathlib.Meta.Positivity.Real.log_nonneg_of_ge_one
|
[17, 1]
|
[18, 41]
|
linarith
|
x : ℝ
h : 0 ≤ x - 1
⊢ 1 ≤ x
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/Arith/PositivityExt.lean
|
Mathlib.Meta.Positivity.Real.log_pos_of_gt_one
|
[20, 1]
|
[21, 38]
|
linarith
|
x : ℝ
h : 0 < x - 1
⊢ 1 < x
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/Arith/PositivityExt.lean
|
Mathlib.Meta.Positivity.Real.one_sub_one_div_sq_nonneg_of_le_one
|
[38, 1]
|
[45, 32]
|
have hx1 : x ≤ 1 := by linarith
|
x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
⊢ 0 ≤ 1 / x ^ 2 - 1
|
x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
⊢ 0 ≤ 1 / x ^ 2 - 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/Arith/PositivityExt.lean
|
Mathlib.Meta.Positivity.Real.one_sub_one_div_sq_nonneg_of_le_one
|
[38, 1]
|
[45, 32]
|
have h0x : 0 ≤ x := by positivity
|
x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
⊢ 0 ≤ 1 / x ^ 2 - 1
|
x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
h0x : 0 ≤ x
⊢ 0 ≤ 1 / x ^ 2 - 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/Arith/PositivityExt.lean
|
Mathlib.Meta.Positivity.Real.one_sub_one_div_sq_nonneg_of_le_one
|
[38, 1]
|
[45, 32]
|
have h0x2 : 0 < x ^ (2 : ℝ) := by positivity
|
x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
h0x : 0 ≤ x
⊢ 0 ≤ 1 / x ^ 2 - 1
|
x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
h0x : 0 ≤ x
h0x2 : 0 < x ^ 2
⊢ 0 ≤ 1 / x ^ 2 - 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/Arith/PositivityExt.lean
|
Mathlib.Meta.Positivity.Real.one_sub_one_div_sq_nonneg_of_le_one
|
[38, 1]
|
[45, 32]
|
rw [le_sub_iff_add_le, zero_add, le_div_iff h0x2, one_mul]
|
x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
h0x : 0 ≤ x
h0x2 : 0 < x ^ 2
⊢ 0 ≤ 1 / x ^ 2 - 1
|
x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
h0x : 0 ≤ x
h0x2 : 0 < x ^ 2
⊢ x ^ 2 ≤ 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/Arith/PositivityExt.lean
|
Mathlib.Meta.Positivity.Real.one_sub_one_div_sq_nonneg_of_le_one
|
[38, 1]
|
[45, 32]
|
simpa [abs_eq_self.mpr h0x]
|
x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
h0x : 0 ≤ x
h0x2 : 0 < x ^ 2
⊢ x ^ 2 ≤ 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/Arith/PositivityExt.lean
|
Mathlib.Meta.Positivity.Real.one_sub_one_div_sq_nonneg_of_le_one
|
[38, 1]
|
[45, 32]
|
linarith
|
x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
⊢ x ≤ 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/Arith/PositivityExt.lean
|
Mathlib.Meta.Positivity.Real.one_sub_one_div_sq_nonneg_of_le_one
|
[38, 1]
|
[45, 32]
|
positivity
|
x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
⊢ 0 ≤ x
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/Arith/PositivityExt.lean
|
Mathlib.Meta.Positivity.Real.one_sub_one_div_sq_nonneg_of_le_one
|
[38, 1]
|
[45, 32]
|
positivity
|
x : ℝ
h1 : 0 < x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
h0x : 0 ≤ x
⊢ 0 < x ^ 2
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/Arith/PositivityExt.lean
|
Mathlib.Meta.Positivity.Real.one_sub_sq_nonneg_of_le_one
|
[73, 1]
|
[77, 31]
|
have hx1 : x ≤ 1 := by linarith
|
x : ℝ
h1 : 0 ≤ x
h2 : 0 ≤ 1 - x
⊢ 0 ≤ 1 - x ^ 2
|
x : ℝ
h1 : 0 ≤ x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
⊢ 0 ≤ 1 - x ^ 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/Arith/PositivityExt.lean
|
Mathlib.Meta.Positivity.Real.one_sub_sq_nonneg_of_le_one
|
[73, 1]
|
[77, 31]
|
rw [le_sub_iff_add_le, zero_add]
|
x : ℝ
h1 : 0 ≤ x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
⊢ 0 ≤ 1 - x ^ 2
|
x : ℝ
h1 : 0 ≤ x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
⊢ x ^ 2 ≤ 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/Arith/PositivityExt.lean
|
Mathlib.Meta.Positivity.Real.one_sub_sq_nonneg_of_le_one
|
[73, 1]
|
[77, 31]
|
simpa [abs_eq_self.mpr h1]
|
x : ℝ
h1 : 0 ≤ x
h2 : 0 ≤ 1 - x
hx1 : x ≤ 1
⊢ x ^ 2 ≤ 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/Arith/PositivityExt.lean
|
Mathlib.Meta.Positivity.Real.one_sub_sq_nonneg_of_le_one
|
[73, 1]
|
[77, 31]
|
linarith
|
x : ℝ
h1 : 0 ≤ x
h2 : 0 ≤ 1 - x
⊢ x ≤ 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/Arith/PositivityExt.lean
|
Mathlib.Meta.Positivity.Real.exp_sub_one_pos_of_pos
|
[107, 1]
|
[108, 23]
|
simp [h]
|
x : ℝ
h : 0 < x
⊢ 0 < x.exp - 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/Arith/PositivityExt.lean
|
Mathlib.Meta.Positivity.Real.one_sub_div_exp_pos_of_pos
|
[121, 1]
|
[122, 37]
|
field_simp
|
x : ℝ
h : 0 < x
⊢ 0 < 1 - 1 / x.exp
|
x : ℝ
h : 0 < x
⊢ 0 < x.exp
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/Arith/PositivityExt.lean
|
Mathlib.Meta.Positivity.Real.one_sub_div_exp_pos_of_pos
|
[121, 1]
|
[122, 37]
|
positivity
|
x : ℝ
h : 0 < x
⊢ 0 < x.exp
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/Arith/PositivityExt.lean
|
Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos
|
[135, 1]
|
[143, 35]
|
have ha : 0 ≤ a := by linarith
|
a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
⊢ 0 < (a * x) ^ 2 - x ^ 2
|
a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
⊢ 0 < (a * x) ^ 2 - x ^ 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/Arith/PositivityExt.lean
|
Mathlib.Meta.Positivity.Real.scaled_sq_diff_pos_of_pos
|
[135, 1]
|
[143, 35]
|
have ha1 : 1 < a := by linarith
|
a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
⊢ 0 < (a * x) ^ 2 - x ^ 2
|
a x : ℝ
h1 : 0 < a - 1
h2 : 0 < x
ha : 0 ≤ a
ha1 : 1 < a
⊢ 0 < (a * x) ^ 2 - x ^ 2
|
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