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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