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stringclasses 147
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stringlengths 7
101
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stringlengths 1
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stringlengths 6
10
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stringlengths 6
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stringlengths 1
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stringlengths 3
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stringlengths 6
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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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
n : Type ?u.63307
m : Type ?u.63304
inst✝¹ : Fintype m
inst✝ : Fintype n
x y z : ℝ
⊢ 0 < 4
TACTIC:
|
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] | Please generate a tactic in lean4 to solve the state.
STATE:
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]
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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]
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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] | Please generate a tactic in lean4 to solve the state.
STATE:
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]
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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]
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.det_nonneg | [16, 1] | [20, 30] | rw [hM.1.det_eq_prod_eigenvalues] | m : Type ?u.340
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.352
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
M : Matrix n n ℝ
hM : M.PosSemidef
inst✝ : DecidableEq n
⊢ 0 ≤ M.det | m : Type ?u.340
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.352
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
M : Matrix n n ℝ
hM : M.PosSemidef
inst✝ : DecidableEq n
⊢ 0 ≤ Finset.univ.prod fun i => ↑(⋯.eigenvalues i) | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.340
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.352
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
M : Matrix n n ℝ
hM : M.PosSemidef
inst✝ : DecidableEq n
⊢ 0 ≤ M.det
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.det_nonneg | [16, 1] | [20, 30] | apply Finset.prod_nonneg | m : Type ?u.340
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.352
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
M : Matrix n n ℝ
hM : M.PosSemidef
inst✝ : DecidableEq n
⊢ 0 ≤ Finset.univ.prod fun i => ↑(⋯.eigenvalues i) | case h0
m : Type ?u.340
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.352
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
M : Matrix n n ℝ
hM : M.PosSemidef
inst✝ : DecidableEq n
⊢ ∀ i ∈ Finset.univ, 0 ≤ ↑(⋯.eigenvalues i) | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.340
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.352
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
M : Matrix n n ℝ
hM : M.PosSemidef
inst✝ : DecidableEq n
⊢ 0 ≤ Finset.univ.prod fun i => ↑(⋯.eigenvalues i)
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.det_nonneg | [16, 1] | [20, 30] | intros i _hi | case h0
m : Type ?u.340
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.352
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
M : Matrix n n ℝ
hM : M.PosSemidef
inst✝ : DecidableEq n
⊢ ∀ i ∈ Finset.univ, 0 ≤ ↑(⋯.eigenvalues i) | case h0
m : Type ?u.340
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.352
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
M : Matrix n n ℝ
hM : M.PosSemidef
inst✝ : DecidableEq n
i : n
_hi : i ∈ Finset.univ
⊢ 0 ≤ ↑(⋯.eigenvalues i) | Please generate a tactic in lean4 to solve the state.
STATE:
case h0
m : Type ?u.340
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.352
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
M : Matrix n n ℝ
hM : M.PosSemidef
inst✝ : DecidableEq n
⊢ ∀ i ∈ Finset.univ, 0 ≤ ↑(⋯.eigenvalues i)
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.det_nonneg | [16, 1] | [20, 30] | apply eigenvalues_nonneg hM | case h0
m : Type ?u.340
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.352
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
M : Matrix n n ℝ
hM : M.PosSemidef
inst✝ : DecidableEq n
i : n
_hi : i ∈ Finset.univ
⊢ 0 ≤ ↑(⋯.eigenvalues i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h0
m : Type ?u.340
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.352
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
M : Matrix n n ℝ
hM : M.PosSemidef
inst✝ : DecidableEq n
i : n
_hi : i ∈ Finset.univ
⊢ 0 ≤ ↑(⋯.eigenvalues i)
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.det_ne_zero | [22, 1] | [29, 32] | rw [← Matrix.nondegenerate_iff_det_ne_zero] | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
⊢ M.det ≠ 0 | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
⊢ M.Nondegenerate | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
⊢ M.det ≠ 0
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.det_ne_zero | [22, 1] | [29, 32] | intros v hv | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
⊢ M.Nondegenerate | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
⊢ v = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
⊢ M.Nondegenerate
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.det_ne_zero | [22, 1] | [29, 32] | have hv' := hv (star v) | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
⊢ v = 0 | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
⊢ v = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
⊢ v = 0
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.det_ne_zero | [22, 1] | [29, 32] | rw [← star_eq_zero] | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
⊢ v = 0 | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
⊢ star v = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
⊢ v = 0
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.det_ne_zero | [22, 1] | [29, 32] | by_contra h | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
⊢ star v = 0 | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
h : ¬star v = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
⊢ star v = 0
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.det_ne_zero | [22, 1] | [29, 32] | have := hM.2 (star v) h | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
h : ¬star v = 0
⊢ False | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
h : ¬star v = 0
this : 0 < star (star v) ⬝ᵥ M.mulVec (star v)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
h : ¬star v = 0
⊢ False
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.det_ne_zero | [22, 1] | [29, 32] | simp [star_star, hv'] at this | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
h : ¬star v = 0
this : 0 < star (star v) ⬝ᵥ M.mulVec (star v)
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
h : ¬star v = 0
this : 0 < star (star v) ⬝ᵥ M.mulVec (star v)
⊢ False
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef_diagonal | [38, 1] | [46, 50] | refine' ⟨isHermitian_diagonal _, _⟩ | m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ (diagonal f).PosSemidef | m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ ∀ (x : n → ℝ), 0 ≤ star x ⬝ᵥ (diagonal f).mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ (diagonal f).PosSemidef
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef_diagonal | [38, 1] | [46, 50] | intro x | m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ ∀ (x : n → ℝ), 0 ≤ star x ⬝ᵥ (diagonal f).mulVec x | m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
⊢ 0 ≤ star x ⬝ᵥ (diagonal f).mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ ∀ (x : n → ℝ), 0 ≤ star x ⬝ᵥ (diagonal f).mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef_diagonal | [38, 1] | [46, 50] | simp only [star, id_def, RCLike.re_to_real] | m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
⊢ 0 ≤ star x ⬝ᵥ (diagonal f).mulVec x | m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
⊢ 0 ≤ (fun i => x i) ⬝ᵥ (diagonal f).mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
⊢ 0 ≤ star x ⬝ᵥ (diagonal f).mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef_diagonal | [38, 1] | [46, 50] | apply Finset.sum_nonneg' | m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
⊢ 0 ≤ (fun i => x i) ⬝ᵥ (diagonal f).mulVec x | case h
m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
⊢ ∀ (i : n), 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
⊢ 0 ≤ (fun i => x i) ⬝ᵥ (diagonal f).mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef_diagonal | [38, 1] | [46, 50] | intro i | case h
m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
⊢ ∀ (i : n), 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i | case h
m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
i : n
⊢ 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
⊢ ∀ (i : n), 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef_diagonal | [38, 1] | [46, 50] | rw [mulVec_diagonal f x i, mul_comm, mul_assoc] | case h
m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
i : n
⊢ 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i | case h
m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
i : n
⊢ 0 ≤ f i * (x i * (fun i => x i) i) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
i : n
⊢ 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef_diagonal | [38, 1] | [46, 50] | exact mul_nonneg (hf i) (mul_self_nonneg (x i)) | case h
m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
i : n
⊢ 0 ≤ f i * (x i * (fun i => x i) i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
i : n
⊢ 0 ≤ f i * (x i * (fun i => x i) i)
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | refine' ⟨isHermitian_diagonal _, _⟩ | m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
⊢ (diagonal f).PosDef | m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
⊢ ∀ (x : n → ℝ), x ≠ 0 → 0 < star x ⬝ᵥ (diagonal f).mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
⊢ (diagonal f).PosDef
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | intros x hx | m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
⊢ ∀ (x : n → ℝ), x ≠ 0 → 0 < star x ⬝ᵥ (diagonal f).mulVec x | m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ (diagonal f).mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
⊢ ∀ (x : n → ℝ), x ≠ 0 → 0 < star x ⬝ᵥ (diagonal f).mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | simp only [star, id_def, RCLike.re_to_real] | m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ (diagonal f).mulVec x | m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ 0 < (fun i => x i) ⬝ᵥ (diagonal f).mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ (diagonal f).mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | apply Finset.sum_pos' | m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ 0 < (fun i => x i) ⬝ᵥ (diagonal f).mulVec x | case h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∀ i ∈ Finset.univ, 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i
case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∃ i ∈ Finset.univ, 0 < (fun i => x i) i * (diagonal f).mulVec x i | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ 0 < (fun i => x i) ⬝ᵥ (diagonal f).mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | { intros i _
rw [mulVec_diagonal f x i, mul_comm, mul_assoc]
exact mul_nonneg (le_of_lt (hf i)) (mul_self_nonneg (x i)) } | case h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∀ i ∈ Finset.univ, 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i
case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∃ i ∈ Finset.univ, 0 < (fun i => x i) i * (diagonal f).mulVec x i | case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∃ i ∈ Finset.univ, 0 < (fun i => x i) i * (diagonal f).mulVec x i | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∀ i ∈ Finset.univ, 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i
case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∃ i ∈ Finset.univ, 0 < (fun i => x i) i * (diagonal f).mulVec x i
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | { contrapose hx; simp at hx ⊢
ext i
have := hx i
rw [mulVec_diagonal f x i, mul_comm, mul_assoc] at this
have := nonpos_of_mul_nonpos_right this (hf i)
rw [mul_self_eq_zero.1 (le_antisymm this (mul_self_nonneg (x i)))]
rfl } | case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∃ i ∈ Finset.univ, 0 < (fun i => x i) i * (diagonal f).mulVec x i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∃ i ∈ Finset.univ, 0 < (fun i => x i) i * (diagonal f).mulVec x i
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | intros i _ | case h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∀ i ∈ Finset.univ, 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i | case h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
i : n
a✝ : i ∈ Finset.univ
⊢ 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∀ i ∈ Finset.univ, 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | rw [mulVec_diagonal f x i, mul_comm, mul_assoc] | case h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
i : n
a✝ : i ∈ Finset.univ
⊢ 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i | case h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
i : n
a✝ : i ∈ Finset.univ
⊢ 0 ≤ f i * (x i * (fun i => x i) i) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
i : n
a✝ : i ∈ Finset.univ
⊢ 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | exact mul_nonneg (le_of_lt (hf i)) (mul_self_nonneg (x i)) | case h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
i : n
a✝ : i ∈ Finset.univ
⊢ 0 ≤ f i * (x i * (fun i => x i) i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
i : n
a✝ : i ∈ Finset.univ
⊢ 0 ≤ f i * (x i * (fun i => x i) i)
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | contrapose hx | case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∃ i ∈ Finset.univ, 0 < (fun i => x i) i * (diagonal f).mulVec x i | case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ¬∃ i ∈ Finset.univ, 0 < (fun i => x i) i * (diagonal f).mulVec x i
⊢ ¬x ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∃ i ∈ Finset.univ, 0 < (fun i => x i) i * (diagonal f).mulVec x i
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | simp at hx ⊢ | case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ¬∃ i ∈ Finset.univ, 0 < (fun i => x i) i * (diagonal f).mulVec x i
⊢ ¬x ≠ 0 | case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
⊢ x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ¬∃ i ∈ Finset.univ, 0 < (fun i => x i) i * (diagonal f).mulVec x i
⊢ ¬x ≠ 0
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | ext i | case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
⊢ x = 0 | case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
⊢ x i = 0 i | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
⊢ x = 0
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | have := hx i | case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
⊢ x i = 0 i | case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this : x i * (diagonal f).mulVec x i ≤ 0
⊢ x i = 0 i | Please generate a tactic in lean4 to solve the state.
STATE:
case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
⊢ x i = 0 i
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | rw [mulVec_diagonal f x i, mul_comm, mul_assoc] at this | case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this : x i * (diagonal f).mulVec x i ≤ 0
⊢ x i = 0 i | case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this : f i * (x i * x i) ≤ 0
⊢ x i = 0 i | Please generate a tactic in lean4 to solve the state.
STATE:
case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this : x i * (diagonal f).mulVec x i ≤ 0
⊢ x i = 0 i
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | have := nonpos_of_mul_nonpos_right this (hf i) | case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this : f i * (x i * x i) ≤ 0
⊢ x i = 0 i | case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this✝ : f i * (x i * x i) ≤ 0
this : x i * x i ≤ 0
⊢ x i = 0 i | Please generate a tactic in lean4 to solve the state.
STATE:
case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this : f i * (x i * x i) ≤ 0
⊢ x i = 0 i
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | rw [mul_self_eq_zero.1 (le_antisymm this (mul_self_nonneg (x i)))] | case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this✝ : f i * (x i * x i) ≤ 0
this : x i * x i ≤ 0
⊢ x i = 0 i | case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this✝ : f i * (x i * x i) ≤ 0
this : x i * x i ≤ 0
⊢ 0 = 0 i | Please generate a tactic in lean4 to solve the state.
STATE:
case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this✝ : f i * (x i * x i) ≤ 0
this : x i * x i ≤ 0
⊢ x i = 0 i
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | rfl | case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this✝ : f i * (x i * x i) ≤ 0
this : x i * x i ≤ 0
⊢ 0 = 0 i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this✝ : f i * (x i * x i) ≤ 0
this : x i * x i ≤ 0
⊢ 0 = 0 i
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.conjTranspose_mul_mul | [64, 1] | [69, 81] | refine' ⟨isHermitian_conjTranspose_mul_mul _ hM.1, _⟩ | m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
⊢ (N.conjTranspose * M * N).PosSemidef | m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
⊢ ∀ (x : n → 𝕜), 0 ≤ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
⊢ (N.conjTranspose * M * N).PosSemidef
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.conjTranspose_mul_mul | [64, 1] | [69, 81] | intro x | m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
⊢ ∀ (x : n → 𝕜), 0 ≤ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x | m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
⊢ ∀ (x : n → 𝕜), 0 ≤ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.conjTranspose_mul_mul | [64, 1] | [69, 81] | convert hM.2 (N.mulVec x) using 1 | m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x | case h.e'_4
m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
x : n → 𝕜
⊢ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x = star (N.mulVec x) ⬝ᵥ M.mulVec (N.mulVec x) | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.conjTranspose_mul_mul | [64, 1] | [69, 81] | rw [mul_assoc, mulVec_mulVec, ← mulVec_mulVec, dotProduct_mulVec, star_mulVec] | case h.e'_4
m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
x : n → 𝕜
⊢ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x = star (N.mulVec x) ⬝ᵥ M.mulVec (N.mulVec x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
x : n → 𝕜
⊢ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x = star (N.mulVec x) ⬝ᵥ M.mulVec (N.mulVec x)
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.conjTranspose_mul_mul | [71, 1] | [78, 17] | refine' ⟨isHermitian_conjTranspose_mul_mul _ hM.1, _⟩ | m : Type ?u.28307
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
⊢ (N.conjTranspose * M * N).PosDef | m : Type ?u.28307
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
⊢ ∀ (x : n → 𝕜), x ≠ 0 → 0 < star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.28307
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
⊢ (N.conjTranspose * M * N).PosDef
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.conjTranspose_mul_mul | [71, 1] | [78, 17] | intros x hx | m : Type ?u.28307
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
⊢ ∀ (x : n → 𝕜), x ≠ 0 → 0 < star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x | m : Type ?u.28307
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
x : n → 𝕜
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.28307
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
⊢ ∀ (x : n → 𝕜), x ≠ 0 → 0 < star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.conjTranspose_mul_mul | [71, 1] | [78, 17] | convert
hM.2 (N.mulVec x) (fun h => hx (eq_zero_of_mulVec_eq_zero hN h)) using 1 | m : Type ?u.28307
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
x : n → 𝕜
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x | case h.e'_4
m : Type ?u.28307
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
x : n → 𝕜
hx : x ≠ 0
⊢ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x = star (N.mulVec x) ⬝ᵥ M.mulVec (N.mulVec x) | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.28307
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
x : n → 𝕜
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.conjTranspose_mul_mul | [71, 1] | [78, 17] | rw [Matrix.mul_assoc, mulVec_mulVec, ← mulVec_mulVec, dotProduct_mulVec,
star_mulVec] | case h.e'_4
m : Type ?u.28307
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
x : n → 𝕜
hx : x ≠ 0
⊢ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x = star (N.mulVec x) ⬝ᵥ M.mulVec (N.mulVec x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
m : Type ?u.28307
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
x : n → 𝕜
hx : x ≠ 0
⊢ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x = star (N.mulVec x) ⬝ᵥ M.mulVec (N.mulVec x)
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.IsHermitian.nonsingular_inv | [80, 1] | [83, 65] | refine' (Matrix.inv_eq_right_inv _).symm | m : Type ?u.34280
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.IsHermitian
hMdet : IsUnit M.det
⊢ M⁻¹.IsHermitian | m : Type ?u.34280
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.IsHermitian
hMdet : IsUnit M.det
⊢ M * M⁻¹.conjTranspose = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.34280
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.IsHermitian
hMdet : IsUnit M.det
⊢ M⁻¹.IsHermitian
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.IsHermitian.nonsingular_inv | [80, 1] | [83, 65] | rw [conjTranspose_nonsing_inv, hM.eq, mul_nonsing_inv _ hMdet] | m : Type ?u.34280
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.IsHermitian
hMdet : IsUnit M.det
⊢ M * M⁻¹.conjTranspose = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.34280
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.IsHermitian
hMdet : IsUnit M.det
⊢ M * M⁻¹.conjTranspose = 1
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.conj_symm | [85, 1] | [88, 43] | nth_rewrite 1 [star_dotProduct, star_mulVec] | m : Type ?u.36361
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
x : n → 𝕜
M : Matrix n n 𝕜
hM : M.IsHermitian
⊢ star (star x ⬝ᵥ M.mulVec x) = star x ⬝ᵥ M.mulVec x | m : Type ?u.36361
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
x : n → 𝕜
M : Matrix n n 𝕜
hM : M.IsHermitian
⊢ star (star (star x ᵥ* M.conjTranspose ⬝ᵥ x)) = star x ⬝ᵥ M.mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.36361
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
x : n → 𝕜
M : Matrix n n 𝕜
hM : M.IsHermitian
⊢ star (star x ⬝ᵥ M.mulVec x) = star x ⬝ᵥ M.mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.conj_symm | [85, 1] | [88, 43] | rw [star_star, dotProduct_mulVec, hM.eq] | m : Type ?u.36361
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
x : n → 𝕜
M : Matrix n n 𝕜
hM : M.IsHermitian
⊢ star (star (star x ᵥ* M.conjTranspose ⬝ᵥ x)) = star x ⬝ᵥ M.mulVec x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.36361
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
x : n → 𝕜
M : Matrix n n 𝕜
hM : M.IsHermitian
⊢ star (star (star x ᵥ* M.conjTranspose ⬝ᵥ x)) = star x ⬝ᵥ M.mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.nonsingular_inv | [90, 1] | [99, 13] | refine' ⟨hM.1.nonsingular_inv (isUnit_iff_ne_zero.2 hM.det_ne_zero), _⟩ | m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
⊢ M⁻¹.PosDef | m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
⊢ ∀ (x : n → 𝕜), x ≠ 0 → 0 < star x ⬝ᵥ M⁻¹.mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
⊢ M⁻¹.PosDef
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.nonsingular_inv | [90, 1] | [99, 13] | intros x hx | m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
⊢ ∀ (x : n → 𝕜), x ≠ 0 → 0 < star x ⬝ᵥ M⁻¹.mulVec x | m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
⊢ ∀ (x : n → 𝕜), x ≠ 0 → 0 < star x ⬝ᵥ M⁻¹.mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.nonsingular_inv | [90, 1] | [99, 13] | have hMMinv := mul_nonsing_inv _ (isUnit_iff_ne_zero.2 hM.det_ne_zero) | m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x | m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.nonsingular_inv | [90, 1] | [99, 13] | have hMinvdet : M⁻¹.det ≠ 0 := det_ne_zero_of_left_inverse hMMinv | m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x | m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.nonsingular_inv | [90, 1] | [99, 13] | have hres :=
hM.2 (M⁻¹.mulVec x) (fun h => hx (eq_zero_of_mulVec_eq_zero hMinvdet h)) | m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x | m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star (M⁻¹.mulVec x) ⬝ᵥ M.mulVec (M⁻¹.mulVec x)
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.nonsingular_inv | [90, 1] | [99, 13] | rw [mulVec_mulVec, hMMinv, one_mulVec, star_dotProduct] at hres | m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star (M⁻¹.mulVec x) ⬝ᵥ M.mulVec (M⁻¹.mulVec x)
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x | m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star (star x ⬝ᵥ M⁻¹.mulVec x)
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star (M⁻¹.mulVec x) ⬝ᵥ M.mulVec (M⁻¹.mulVec x)
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.nonsingular_inv | [90, 1] | [99, 13] | rw [conj_symm ((@isHermitian_inv _ _ _ _ _ _ M hM.Invertible).2 hM.1)] at hres | m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star (star x ⬝ᵥ M⁻¹.mulVec x)
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x | m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star x ⬝ᵥ M⁻¹.mulVec x
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star (star x ⬝ᵥ M⁻¹.mulVec x)
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.nonsingular_inv | [90, 1] | [99, 13] | exact hres | m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star x ⬝ᵥ M⁻¹.mulVec x
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star x ⬝ᵥ M⁻¹.mulVec x
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.mul_mul_of_IsHermitian | [101, 1] | [103, 54] | convert hM.conjTranspose_mul_mul M N | m : Type ?u.45085
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.IsHermitian
⊢ (N * M * N).PosSemidef | case h.e'_8.h.e'_5.h.e'_5
m : Type ?u.45085
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.IsHermitian
⊢ N = N.conjTranspose | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.45085
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.IsHermitian
⊢ (N * M * N).PosSemidef
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.mul_mul_of_IsHermitian | [101, 1] | [103, 54] | exact hN.symm | case h.e'_8.h.e'_5.h.e'_5
m : Type ?u.45085
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.IsHermitian
⊢ N = N.conjTranspose | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_8.h.e'_5.h.e'_5
m : Type ?u.45085
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.IsHermitian
⊢ N = N.conjTranspose
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.add | [105, 1] | [110, 37] | refine' ⟨hM.1.add hN.1, _⟩ | m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
⊢ (M + N).PosSemidef | m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
⊢ ∀ (x : n → 𝕜), 0 ≤ star x ⬝ᵥ (M + N).mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
⊢ (M + N).PosSemidef
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.add | [105, 1] | [110, 37] | intros x | m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
⊢ ∀ (x : n → 𝕜), 0 ≤ star x ⬝ᵥ (M + N).mulVec x | m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (M + N).mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
⊢ ∀ (x : n → 𝕜), 0 ≤ star x ⬝ᵥ (M + N).mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.add | [105, 1] | [110, 37] | simp only [add_mulVec, dotProduct_add, map_add] | m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (M + N).mulVec x | m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ M.mulVec x + star x ⬝ᵥ N.mulVec x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (M + N).mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.add | [105, 1] | [110, 37] | apply add_nonneg (hM.2 x) (hN.2 x) | m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ M.mulVec x + star x ⬝ᵥ N.mulVec x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ M.mulVec x + star x ⬝ᵥ N.mulVec x
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.isUnit_det_of_PosDef_inv | [112, 1] | [117, 28] | apply isUnit_iff_ne_zero.2 | m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
h : M⁻¹.PosDef
⊢ IsUnit M.det | m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
h : M⁻¹.PosDef
⊢ M.det ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
h : M⁻¹.PosDef
⊢ IsUnit M.det
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.isUnit_det_of_PosDef_inv | [112, 1] | [117, 28] | have := h.isUnit_det | m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
h : M⁻¹.PosDef
⊢ M.det ≠ 0 | m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
h : M⁻¹.PosDef
this : IsUnit M⁻¹.det
⊢ M.det ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
h : M⁻¹.PosDef
⊢ M.det ≠ 0
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.isUnit_det_of_PosDef_inv | [112, 1] | [117, 28] | rw [det_nonsing_inv, isUnit_ring_inverse] at this | m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
h : M⁻¹.PosDef
this : IsUnit M⁻¹.det
⊢ M.det ≠ 0 | m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
h : M⁻¹.PosDef
this : IsUnit M.det
⊢ M.det ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
h : M⁻¹.PosDef
this : IsUnit M⁻¹.det
⊢ M.det ≠ 0
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.isUnit_det_of_PosDef_inv | [112, 1] | [117, 28] | apply IsUnit.ne_zero this | m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
h : M⁻¹.PosDef
this : IsUnit M.det
⊢ M.det ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
h : M⁻¹.PosDef
this : IsUnit M.det
⊢ M.det ≠ 0
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_inv_iff_PosDef | [119, 1] | [125, 31] | constructor | m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M⁻¹.PosDef ↔ M.PosDef | case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M⁻¹.PosDef → M.PosDef
case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M.PosDef → M⁻¹.PosDef | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M⁻¹.PosDef ↔ M.PosDef
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_inv_iff_PosDef | [119, 1] | [125, 31] | { intros hM
rw [← Matrix.nonsing_inv_nonsing_inv M (isUnit_det_of_PosDef_inv hM)]
apply hM.nonsingular_inv } | case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M⁻¹.PosDef → M.PosDef
case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M.PosDef → M⁻¹.PosDef | case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M.PosDef → M⁻¹.PosDef | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M⁻¹.PosDef → M.PosDef
case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M.PosDef → M⁻¹.PosDef
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_inv_iff_PosDef | [119, 1] | [125, 31] | { intros hM
exact hM.nonsingular_inv } | case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M.PosDef → M⁻¹.PosDef | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M.PosDef → M⁻¹.PosDef
TACTIC:
|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_inv_iff_PosDef | [119, 1] | [125, 31] | intros hM | case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M⁻¹.PosDef → M.PosDef | case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M⁻¹.PosDef
⊢ M.PosDef | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M⁻¹.PosDef → M.PosDef
TACTIC:
|
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