url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.BlockTriangular_blockDiagonal | [42, 1] | [46, 10] | rintro ⟨i, i'⟩ ⟨j, j'⟩ h | α : Type u_1
β : Type ?u.2091
m : Type u_2
n : Type ?u.2097
o : Type ?u.2100
m' : α → Type ?u.2105
n' : α → Type ?u.2110
R : Type v
inst✝² : CommRing R
M : Matrix m m R
b : m → α
inst✝¹ : Preorder α
inst✝ : DecidableEq α
d : α → Matrix m m R
⊢ (blockDiagonal d).BlockTriangular Prod.snd | case mk.mk
α : Type u_1
β : Type ?u.2091
m : Type u_2
n : Type ?u.2097
o : Type ?u.2100
m' : α → Type ?u.2105
n' : α → Type ?u.2110
R : Type v
inst✝² : CommRing R
M : Matrix m m R
b : m → α
inst✝¹ : Preorder α
inst✝ : DecidableEq α
d : α → Matrix m m R
i : m
i' : α
j : m
j' : α
h : (j, j').2 < (i, i').2
⊢ blockDiagonal d (i, i') (j, j') = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.BlockTriangular_blockDiagonal | [42, 1] | [46, 10] | rw [blockDiagonal'_eq_blockDiagonal, BlockTriangular_blockDiagonal'] | case mk.mk
α : Type u_1
β : Type ?u.2091
m : Type u_2
n : Type ?u.2097
o : Type ?u.2100
m' : α → Type ?u.2105
n' : α → Type ?u.2110
R : Type v
inst✝² : CommRing R
M : Matrix m m R
b : m → α
inst✝¹ : Preorder α
inst✝ : DecidableEq α
d : α → Matrix m m R
i : m
i' : α
j : m
j' : α
h : (j, j').2 < (i, i').2
⊢ blockDiagonal d (i, i') (j, j') = 0 | case mk.mk.a
α : Type u_1
β : Type ?u.2091
m : Type u_2
n : Type ?u.2097
o : Type ?u.2100
m' : α → Type ?u.2105
n' : α → Type ?u.2110
R : Type v
inst✝² : CommRing R
M : Matrix m m R
b : m → α
inst✝¹ : Preorder α
inst✝ : DecidableEq α
d : α → Matrix m m R
i : m
i' : α
j : m
j' : α
h : (j, j').2 < (i, i').2
⊢ ⟨j', j⟩.fst < ⟨i', i⟩.fst |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.BlockTriangular_blockDiagonal | [42, 1] | [46, 10] | exact h | case mk.mk.a
α : Type u_1
β : Type ?u.2091
m : Type u_2
n : Type ?u.2097
o : Type ?u.2100
m' : α → Type ?u.2105
n' : α → Type ?u.2110
R : Type v
inst✝² : CommRing R
M : Matrix m m R
b : m → α
inst✝¹ : Preorder α
inst✝ : DecidableEq α
d : α → Matrix m m R
i : m
i' : α
j : m
j' : α
h : (j, j').2 < (i, i').2
⊢ ⟨j', j⟩.fst < ⟨i', i⟩.fst | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toBlock_inverse_mul_toBlock_eq_one_of_BlockTriangular | [52, 1] | [63, 29] | let p := (fun i => b i < k) | α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1 | α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toBlock_inverse_mul_toBlock_eq_one_of_BlockTriangular | [52, 1] | [63, 29] | have h_sum : M⁻¹.toBlock p p * M.toBlock p p +
M⁻¹.toBlock p (fun i => ¬ p i) * M.toBlock (fun i => ¬ p i) p = 1 := by
rw [← toBlock_mul_eq_add, inv_mul_of_invertible M, toBlock_one_self] | α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1 | α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toBlock_inverse_mul_toBlock_eq_one_of_BlockTriangular | [52, 1] | [63, 29] | have h_zero : M.toBlock (fun i => ¬ p i) p = 0 := by
{ ext i j
simpa using hM (lt_of_lt_of_le j.2 (le_of_not_lt i.2)) } | α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1 | α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
h_zero : M.toBlock (fun i => ¬p i) p = 0
⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toBlock_inverse_mul_toBlock_eq_one_of_BlockTriangular | [52, 1] | [63, 29] | simpa [h_zero] using h_sum | α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
h_zero : M.toBlock (fun i => ¬p i) p = 0
⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toBlock_inverse_mul_toBlock_eq_one_of_BlockTriangular | [52, 1] | [63, 29] | rw [← toBlock_mul_eq_add, inv_mul_of_invertible M, toBlock_one_self] | α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
⊢ M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toBlock_inverse_mul_toBlock_eq_one_of_BlockTriangular | [52, 1] | [63, 29] | ext i j | α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
⊢ M.toBlock (fun i => ¬p i) p = 0 | case a
α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // ¬p a }
j : { a // p a }
⊢ M.toBlock (fun i => ¬p i) p i j = 0 i j |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toBlock_inverse_mul_toBlock_eq_one_of_BlockTriangular | [52, 1] | [63, 29] | simpa using hM (lt_of_lt_of_le j.2 (le_of_not_lt i.2)) | case a
α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // ¬p a }
j : { a // p a }
⊢ M.toBlock (fun i => ¬p i) p i j = 0 i j | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | let p := (λ i => b i = k) | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
⊢ M⁻¹.toSquareBlock b k * M.toSquareBlock b k = 1 | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
⊢ M⁻¹.toSquareBlock b k * M.toSquareBlock b k = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | have h_sum : M⁻¹.toBlock p p * M.toBlock p p +
M⁻¹.toBlock p (λ i => ¬ p i) * M.toBlock (λ i => ¬ p i) p = 1 := by
rw [← toBlock_mul_eq_add, inv_mul_of_invertible M, toBlock_one_self] | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
⊢ M⁻¹.toSquareBlock b k * M.toSquareBlock b k = 1 | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
⊢ M⁻¹.toSquareBlock b k * M.toSquareBlock b k = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | have h_zero : ∀ i j l,
M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 := by
{ intro i j l
by_cases hj : b j.1 ≤ k
{ have hj := lt_of_le_of_ne hj j.2
have hM' := blockTriangular_inv_of_blockTriangular hM
apply mul_eq_zero_of_left
simpa using hM' (lt_of_lt_of_eq hj i.2.symm) }
{ have hj := lt_of_not_ge hj
apply mul_eq_zero_of_right
simpa using hM (lt_of_eq_of_lt l.2 hj) }} | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
⊢ M⁻¹.toSquareBlock b k * M.toSquareBlock b k = 1 | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
h_zero :
∀ (i : { a // p a }) (j : { a // (fun i => ¬p i) a }) (l : { a // p a }),
M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0
⊢ M⁻¹.toSquareBlock b k * M.toSquareBlock b k = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | have h_zero' :
M⁻¹.toBlock p (λ (i : m) => ¬p i) * M.toBlock (λ (i : m) => ¬p i) p = 0 := by
{ ext i l
apply sum_eq_zero (λ j _ => h_zero i j l) } | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
h_zero :
∀ (i : { a // p a }) (j : { a // (fun i => ¬p i) a }) (l : { a // p a }),
M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0
⊢ M⁻¹.toSquareBlock b k * M.toSquareBlock b k = 1 | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
h_zero :
∀ (i : { a // p a }) (j : { a // (fun i => ¬p i) a }) (l : { a // p a }),
M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0
h_zero' : (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 0
⊢ M⁻¹.toSquareBlock b k * M.toSquareBlock b k = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | simpa [h_zero'] using h_sum | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
h_zero :
∀ (i : { a // p a }) (j : { a // (fun i => ¬p i) a }) (l : { a // p a }),
M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0
h_zero' : (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 0
⊢ M⁻¹.toSquareBlock b k * M.toSquareBlock b k = 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | rw [← toBlock_mul_eq_add, inv_mul_of_invertible M, toBlock_one_self] | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
⊢ M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | intro i j l | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
⊢ ∀ (i : { a // p a }) (j : { a // (fun i => ¬p i) a }) (l : { a // p a }),
M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | by_cases hj : b j.1 ≤ k | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 | case pos
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj : b ↑j ≤ k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0
case neg
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj : ¬b ↑j ≤ k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | { have hj := lt_of_le_of_ne hj j.2
have hM' := blockTriangular_inv_of_blockTriangular hM
apply mul_eq_zero_of_left
simpa using hM' (lt_of_lt_of_eq hj i.2.symm) } | case pos
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj : b ↑j ≤ k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0
case neg
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj : ¬b ↑j ≤ k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 | case neg
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj : ¬b ↑j ≤ k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | { have hj := lt_of_not_ge hj
apply mul_eq_zero_of_right
simpa using hM (lt_of_eq_of_lt l.2 hj) } | case neg
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj : ¬b ↑j ≤ k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | have hj := lt_of_le_of_ne hj j.2 | case pos
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj : b ↑j ≤ k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 | case pos
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : b ↑j ≤ k
hj : b ↑j < k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | have hM' := blockTriangular_inv_of_blockTriangular hM | case pos
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : b ↑j ≤ k
hj : b ↑j < k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 | case pos
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : b ↑j ≤ k
hj : b ↑j < k
hM' : M⁻¹.BlockTriangular b
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | apply mul_eq_zero_of_left | case pos
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : b ↑j ≤ k
hj : b ↑j < k
hM' : M⁻¹.BlockTriangular b
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 | case pos.h
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : b ↑j ≤ k
hj : b ↑j < k
hM' : M⁻¹.BlockTriangular b
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | simpa using hM' (lt_of_lt_of_eq hj i.2.symm) | case pos.h
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : b ↑j ≤ k
hj : b ↑j < k
hM' : M⁻¹.BlockTriangular b
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j = 0 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | have hj := lt_of_not_ge hj | case neg
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj : ¬b ↑j ≤ k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 | case neg
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : ¬b ↑j ≤ k
hj : k < b ↑j
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | apply mul_eq_zero_of_right | case neg
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : ¬b ↑j ≤ k
hj : k < b ↑j
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 | case neg.h
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : ¬b ↑j ≤ k
hj : k < b ↑j
⊢ M.toBlock (fun i => ¬p i) p j l = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | simpa using hM (lt_of_eq_of_lt l.2 hj) | case neg.h
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : ¬b ↑j ≤ k
hj : k < b ↑j
⊢ M.toBlock (fun i => ¬p i) p j l = 0 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | ext i l | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
h_zero :
∀ (i : { a // p a }) (j : { a // (fun i => ¬p i) a }) (l : { a // p a }),
M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0
⊢ (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 0 | case a
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
h_zero :
∀ (i : { a // p a }) (j : { a // (fun i => ¬p i) a }) (l : { a // p a }),
M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0
i l : { a // p a }
⊢ ((M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p) i l = 0 i l |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | apply sum_eq_zero (λ j _ => h_zero i j l) | case a
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
h_zero :
∀ (i : { a // p a }) (j : { a // (fun i => ¬p i) a }) (l : { a // p a }),
M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0
i l : { a // p a }
⊢ ((M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p) i l = 0 i l | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.schur_complement_eq₁₁ | [28, 1] | [37, 7] | simp [Function.star_sum_elim, fromBlocks_mulVec, vecMul_fromBlocks, add_vecMul,
dotProduct_mulVec, vecMul_sub, Matrix.mul_assoc, vecMul_mulVec, hA.eq,
conjTranspose_nonsing_inv, star_mulVec] | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
x : m → 𝕜
y : n → 𝕜
inst✝ : Invertible A
hA : A.IsHermitian
⊢ star (x ⊕ᵥ y) ᵥ* A.fromBlocks B B.conjTranspose D ⬝ᵥ (x ⊕ᵥ y) =
star (x + (A⁻¹ * B).mulVec y) ᵥ* A ⬝ᵥ (x + (A⁻¹ * B).mulVec y) + star y ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ y | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
x : m → 𝕜
y : n → 𝕜
inst✝ : Invertible A
hA : A.IsHermitian
⊢ star x ᵥ* A ⬝ᵥ x + star y ᵥ* B.conjTranspose ⬝ᵥ x + (star x ᵥ* B ⬝ᵥ y + star y ᵥ* D ⬝ᵥ y) =
star x ᵥ* A ⬝ᵥ x + star y ᵥ* B.conjTranspose ⬝ᵥ x +
(star x ᵥ* B ⬝ᵥ y + star y ᵥ* (B.conjTranspose * (A⁻¹ * B)) ⬝ᵥ y) +
(star y ᵥ* D ⬝ᵥ y - star y ᵥ* (B.conjTranspose * (A⁻¹ * B)) ⬝ᵥ y) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.schur_complement_eq₁₁ | [28, 1] | [37, 7] | abel | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
x : m → 𝕜
y : n → 𝕜
inst✝ : Invertible A
hA : A.IsHermitian
⊢ star x ᵥ* A ⬝ᵥ x + star y ᵥ* B.conjTranspose ⬝ᵥ x + (star x ᵥ* B ⬝ᵥ y + star y ᵥ* D ⬝ᵥ y) =
star x ᵥ* A ⬝ᵥ x + star y ᵥ* B.conjTranspose ⬝ᵥ x +
(star x ᵥ* B ⬝ᵥ y + star y ᵥ* (B.conjTranspose * (A⁻¹ * B)) ⬝ᵥ y) +
(star y ᵥ* D ⬝ᵥ y - star y ᵥ* (B.conjTranspose * (A⁻¹ * B)) ⬝ᵥ y) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.schur_complement_eq₂₂ | [39, 1] | [48, 7] | simp [Function.star_sum_elim, fromBlocks_mulVec, vecMul_fromBlocks, add_vecMul,
dotProduct_mulVec, vecMul_sub, Matrix.mul_assoc, vecMul_mulVec, hD.eq,
conjTranspose_nonsing_inv, star_mulVec] | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
x : m → 𝕜
y : n → 𝕜
inst✝ : Invertible D
hD : D.IsHermitian
⊢ star (x ⊕ᵥ y) ᵥ* A.fromBlocks B B.conjTranspose D ⬝ᵥ (x ⊕ᵥ y) =
star ((D⁻¹ * B.conjTranspose).mulVec x + y) ᵥ* D ⬝ᵥ ((D⁻¹ * B.conjTranspose).mulVec x + y) +
star x ᵥ* (A - B * D⁻¹ * B.conjTranspose) ⬝ᵥ x | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
x : m → 𝕜
y : n → 𝕜
inst✝ : Invertible D
hD : D.IsHermitian
⊢ star x ᵥ* A ⬝ᵥ x + star y ᵥ* B.conjTranspose ⬝ᵥ x + (star x ᵥ* B ⬝ᵥ y + star y ᵥ* D ⬝ᵥ y) =
star x ᵥ* (B * (D⁻¹ * B.conjTranspose)) ⬝ᵥ x + star y ᵥ* B.conjTranspose ⬝ᵥ x +
(star x ᵥ* B ⬝ᵥ y + star y ᵥ* D ⬝ᵥ y) +
(star x ᵥ* A ⬝ᵥ x - star x ᵥ* (B * (D⁻¹ * B.conjTranspose)) ⬝ᵥ x) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.schur_complement_eq₂₂ | [39, 1] | [48, 7] | abel | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
x : m → 𝕜
y : n → 𝕜
inst✝ : Invertible D
hD : D.IsHermitian
⊢ star x ᵥ* A ⬝ᵥ x + star y ᵥ* B.conjTranspose ⬝ᵥ x + (star x ᵥ* B ⬝ᵥ y + star y ᵥ* D ⬝ᵥ y) =
star x ᵥ* (B * (D⁻¹ * B.conjTranspose)) ⬝ᵥ x + star y ᵥ* B.conjTranspose ⬝ᵥ x +
(star x ᵥ* B ⬝ᵥ y + star y ᵥ* D ⬝ᵥ y) +
(star x ᵥ* A ⬝ᵥ x - star x ᵥ* (B * (D⁻¹ * B.conjTranspose)) ⬝ᵥ x) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | have hBAB : (Bᴴ * A⁻¹ * B).IsHermitian := by
apply isHermitian_conjTranspose_mul_mul
apply hA.inv | n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
⊢ (A.fromBlocks B B.conjTranspose D).IsHermitian ↔ (D - B.conjTranspose * A⁻¹ * B).IsHermitian | n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ (A.fromBlocks B B.conjTranspose D).IsHermitian ↔ (D - B.conjTranspose * A⁻¹ * B).IsHermitian |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | rw [isHermitian_fromBlocks_iff] | n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ (A.fromBlocks B B.conjTranspose D).IsHermitian ↔ (D - B.conjTranspose * A⁻¹ * B).IsHermitian | n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian ↔
(D - B.conjTranspose * A⁻¹ * B).IsHermitian |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | constructor | n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian ↔
(D - B.conjTranspose * A⁻¹ * B).IsHermitian | case mp
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian →
(D - B.conjTranspose * A⁻¹ * B).IsHermitian
case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ (D - B.conjTranspose * A⁻¹ * B).IsHermitian →
A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | apply isHermitian_conjTranspose_mul_mul | n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
⊢ (B.conjTranspose * A⁻¹ * B).IsHermitian | case hA
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
⊢ A⁻¹.IsHermitian |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | apply hA.inv | case hA
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
⊢ A⁻¹.IsHermitian | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | intro h | case mp
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian →
(D - B.conjTranspose * A⁻¹ * B).IsHermitian | case mp
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian
⊢ (D - B.conjTranspose * A⁻¹ * B).IsHermitian |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | apply IsHermitian.sub h.2.2.2 hBAB | case mp
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian
⊢ (D - B.conjTranspose * A⁻¹ * B).IsHermitian | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | intro h | case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ (D - B.conjTranspose * A⁻¹ * B).IsHermitian →
A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian | case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : (D - B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | refine' ⟨hA, rfl, conjTranspose_conjTranspose B, _⟩ | case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : (D - B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian | case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : (D - B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ D.IsHermitian |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | rw [← sub_add_cancel D] | case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : (D - B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ D.IsHermitian | case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : (D - B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ (D - ?mpr + ?mpr).IsHermitian
case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : (D - B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ Matrix n n 𝕜 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | apply IsHermitian.add h hBAB | case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : (D - B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ (D - ?mpr + ?mpr).IsHermitian
case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : (D - B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ Matrix n n 𝕜 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₂₂ | [65, 1] | [70, 79] | rw [← isHermitian_submatrix_equiv (Equiv.sumComm n m), Equiv.sumComm_apply,
fromBlocks_submatrix_sum_swap_sum_swap] | n : Type u_1
m : Type u_2
𝕜 : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hD : D.IsHermitian
⊢ (A.fromBlocks B B.conjTranspose D).IsHermitian ↔ (A - B * D⁻¹ * B.conjTranspose).IsHermitian | n : Type u_1
m : Type u_2
𝕜 : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hD : D.IsHermitian
⊢ (D.fromBlocks B.conjTranspose B A).IsHermitian ↔ (A - B * D⁻¹ * B.conjTranspose).IsHermitian |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₂₂ | [65, 1] | [70, 79] | convert IsHermitian.fromBlocks₁₁ _ _ hD <;> rw [conjTranspose_conjTranspose] | n : Type u_1
m : Type u_2
𝕜 : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hD : D.IsHermitian
⊢ (D.fromBlocks B.conjTranspose B A).IsHermitian ↔ (A - B * D⁻¹ * B.conjTranspose).IsHermitian | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | rw [PosSemidef, IsHermitian.fromBlocks₁₁ _ _ hA.1] | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ (A.fromBlocks B B.conjTranspose D).PosSemidef ↔ (D - B.conjTranspose * A⁻¹ * B).PosSemidef | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ ((D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x) ↔
(D - B.conjTranspose * A⁻¹ * B).PosSemidef |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | constructor | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ ((D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x) ↔
(D - B.conjTranspose * A⁻¹ * B).PosSemidef | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ ((D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x) →
(D - B.conjTranspose * A⁻¹ * B).PosSemidef
case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ (D - B.conjTranspose * A⁻¹ * B).PosSemidef →
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | { intro h; refine' ⟨h.1, _⟩; intro x
have := h.2 (- ((A⁻¹ * B).mulVec x) ⊕ᵥ x)
rw [dotProduct_mulVec, schur_complement_eq₁₁ B D _ _ hA.1, neg_add_self,
dotProduct_zero, zero_add] at this
rw [dotProduct_mulVec]; exact this } | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ ((D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x) →
(D - B.conjTranspose * A⁻¹ * B).PosSemidef
case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ (D - B.conjTranspose * A⁻¹ * B).PosSemidef →
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ (D - B.conjTranspose * A⁻¹ * B).PosSemidef →
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | { intro h; refine' ⟨h.1, _⟩; intro x
rw [dotProduct_mulVec, ← Sum.elim_comp_inl_inr x, schur_complement_eq₁₁ B D _ _ hA.1]
apply le_add_of_nonneg_of_le
{ rw [← dotProduct_mulVec]
apply hA.posSemidef.2 }
{ rw [← dotProduct_mulVec _ _ (x ∘ Sum.inr)]
apply h.2 } } | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ (D - B.conjTranspose * A⁻¹ * B).PosSemidef →
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | intro h | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ ((D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x) →
(D - B.conjTranspose * A⁻¹ * B).PosSemidef | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
⊢ (D - B.conjTranspose * A⁻¹ * B).PosSemidef |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | refine' ⟨h.1, _⟩ | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
⊢ (D - B.conjTranspose * A⁻¹ * B).PosSemidef | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
⊢ ∀ (x : n → 𝕜), 0 ≤ star x ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | intro x | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
⊢ ∀ (x : n → 𝕜), 0 ≤ star x ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec x | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | have := h.2 (- ((A⁻¹ * B).mulVec x) ⊕ᵥ x) | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec x | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
x : n → 𝕜
this : 0 ≤ star (-(A⁻¹ * B).mulVec x ⊕ᵥ x) ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec (-(A⁻¹ * B).mulVec x ⊕ᵥ x)
⊢ 0 ≤ star x ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | rw [dotProduct_mulVec, schur_complement_eq₁₁ B D _ _ hA.1, neg_add_self,
dotProduct_zero, zero_add] at this | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
x : n → 𝕜
this : 0 ≤ star (-(A⁻¹ * B).mulVec x ⊕ᵥ x) ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec (-(A⁻¹ * B).mulVec x ⊕ᵥ x)
⊢ 0 ≤ star x ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec x | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
x : n → 𝕜
this : 0 ≤ star x ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x
⊢ 0 ≤ star x ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | rw [dotProduct_mulVec] | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
x : n → 𝕜
this : 0 ≤ star x ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x
⊢ 0 ≤ star x ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec x | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
x : n → 𝕜
this : 0 ≤ star x ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x
⊢ 0 ≤ star x ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | exact this | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
x : n → 𝕜
this : 0 ≤ star x ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x
⊢ 0 ≤ star x ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | intro h | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ (D - B.conjTranspose * A⁻¹ * B).PosSemidef →
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
⊢ (D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | refine' ⟨h.1, _⟩ | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
⊢ (D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
⊢ ∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | intro x | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
⊢ ∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | rw [dotProduct_mulVec, ← Sum.elim_comp_inl_inr x, schur_complement_eq₁₁ B D _ _ hA.1] | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤
star (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) ᵥ* A ⬝ᵥ (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) +
star (x ∘ Sum.inr) ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x ∘ Sum.inr |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | apply le_add_of_nonneg_of_le | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤
star (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) ᵥ* A ⬝ᵥ (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) +
star (x ∘ Sum.inr) ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x ∘ Sum.inr | case mpr.ha
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) ᵥ* A ⬝ᵥ (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr))
case mpr.hbc
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inr) ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x ∘ Sum.inr |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | { rw [← dotProduct_mulVec]
apply hA.posSemidef.2 } | case mpr.ha
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) ᵥ* A ⬝ᵥ (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr))
case mpr.hbc
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inr) ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x ∘ Sum.inr | case mpr.hbc
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inr) ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x ∘ Sum.inr |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | { rw [← dotProduct_mulVec _ _ (x ∘ Sum.inr)]
apply h.2 } | case mpr.hbc
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inr) ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x ∘ Sum.inr | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | rw [← dotProduct_mulVec] | case mpr.ha
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) ᵥ* A ⬝ᵥ (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) | case mpr.ha
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) ⬝ᵥ A.mulVec (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | apply hA.posSemidef.2 | case mpr.ha
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) ⬝ᵥ A.mulVec (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | rw [← dotProduct_mulVec _ _ (x ∘ Sum.inr)] | case mpr.hbc
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inr) ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x ∘ Sum.inr | case mpr.hbc
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inr) ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec (x ∘ Sum.inr) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | apply h.2 | case mpr.hbc
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inr) ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec (x ∘ Sum.inr) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₂₂ | [90, 1] | [96, 35] | rw [← posSemidef_submatrix_equiv (Equiv.sumComm n m), Equiv.sumComm_apply,
fromBlocks_submatrix_sum_swap_sum_swap] | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hD : D.PosDef
inst✝ : Invertible D
⊢ (A.fromBlocks B B.conjTranspose D).PosSemidef ↔ (A - B * D⁻¹ * B.conjTranspose).PosSemidef | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hD : D.PosDef
inst✝ : Invertible D
⊢ (D.fromBlocks B.conjTranspose B A).PosSemidef ↔ (A - B * D⁻¹ * B.conjTranspose).PosSemidef |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₂₂ | [90, 1] | [96, 35] | convert @PosSemidef.fromBlocks₁₁ m n 𝕜 _ _ _ _ _ _ _ hD _ <;>
rw [conjTranspose_conjTranspose] | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hD : D.PosDef
inst✝ : Invertible D
⊢ (D.fromBlocks B.conjTranspose B A).PosSemidef ↔ (A - B * D⁻¹ * B.conjTranspose).PosSemidef | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | constructor | m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
⊢ a / b ≤ c ↔ a ≤ c * b | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
⊢ a / b ≤ c → a ≤ c * b
case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
⊢ a ≤ c * b → a / b ≤ c |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | intro h i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
⊢ a / b ≤ c → a ≤ c * b | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a / b ≤ c
i : m
⊢ a i ≤ (c * b) i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | have hi := h i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a / b ≤ c
i : m
⊢ a i ≤ (c * b) i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a / b ≤ c
i : m
hi : (a / b) i ≤ c i
⊢ a i ≤ (c * b) i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | simp at hi | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a / b ≤ c
i : m
hi : (a / b) i ≤ c i
⊢ a i ≤ (c * b) i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a / b ≤ c
i : m
hi : a i / b i ≤ c i
⊢ a i ≤ (c * b) i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | rw [_root_.div_le_iff (hb i)] at hi | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a / b ≤ c
i : m
hi : a i / b i ≤ c i
⊢ a i ≤ (c * b) i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a / b ≤ c
i : m
hi : a i ≤ c i * b i
⊢ a i ≤ (c * b) i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | exact hi | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a / b ≤ c
i : m
hi : a i ≤ c i * b i
⊢ a i ≤ (c * b) i | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | intro h i | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
⊢ a ≤ c * b → a / b ≤ c | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
⊢ (a / b) i ≤ c i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | have hi := h i | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
⊢ (a / b) i ≤ c i | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
hi : a i ≤ (c * b) i
⊢ (a / b) i ≤ c i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | simp at hi | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
hi : a i ≤ (c * b) i
⊢ (a / b) i ≤ c i | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
hi : a i ≤ c i * b i
⊢ (a / b) i ≤ c i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | dsimp | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
hi : a i ≤ c i * b i
⊢ (a / b) i ≤ c i | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
hi : a i ≤ c i * b i
⊢ a i / b i ≤ c i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | rw [_root_.div_le_iff (hb i)] | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
hi : a i ≤ c i * b i
⊢ a i / b i ≤ c i | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
hi : a i ≤ c i * b i
⊢ a i ≤ c i * b i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | exact hi | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
hi : a i ≤ c i * b i
⊢ a i ≤ c i * b i | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.le_div_iff | [111, 1] | [116, 51] | constructor | m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
⊢ a ≤ b / c ↔ a * c ≤ b | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
⊢ a ≤ b / c → a * c ≤ b
case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
⊢ a * c ≤ b → a ≤ b / c |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.le_div_iff | [111, 1] | [116, 51] | intro h i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
⊢ a ≤ b / c → a * c ≤ b | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a ≤ b / c
i : m
⊢ (a * c) i ≤ b i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.le_div_iff | [111, 1] | [116, 51] | have hi := h i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a ≤ b / c
i : m
⊢ (a * c) i ≤ b i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a ≤ b / c
i : m
hi : a i ≤ (b / c) i
⊢ (a * c) i ≤ b i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.le_div_iff | [111, 1] | [116, 51] | simp at hi | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a ≤ b / c
i : m
hi : a i ≤ (b / c) i
⊢ (a * c) i ≤ b i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a ≤ b / c
i : m
hi : a i ≤ b i / c i
⊢ (a * c) i ≤ b i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.le_div_iff | [111, 1] | [116, 51] | rw [_root_.le_div_iff (hc i)] at hi | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a ≤ b / c
i : m
hi : a i ≤ b i / c i
⊢ (a * c) i ≤ b i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a ≤ b / c
i : m
hi : a i * c i ≤ b i
⊢ (a * c) i ≤ b i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.le_div_iff | [111, 1] | [116, 51] | exact hi | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a ≤ b / c
i : m
hi : a i * c i ≤ b i
⊢ (a * c) i ≤ b i | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.le_div_iff | [111, 1] | [116, 51] | intro h i | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
⊢ a * c ≤ b → a ≤ b / c | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
⊢ a i ≤ (b / c) i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.le_div_iff | [111, 1] | [116, 51] | have hi := h i | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
⊢ a i ≤ (b / c) i | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
hi : (a * c) i ≤ b i
⊢ a i ≤ (b / c) i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.le_div_iff | [111, 1] | [116, 51] | simp at hi | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
hi : (a * c) i ≤ b i
⊢ a i ≤ (b / c) i | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
hi : a i * c i ≤ b i
⊢ a i ≤ (b / c) i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.le_div_iff | [111, 1] | [116, 51] | dsimp | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
hi : a i * c i ≤ b i
⊢ a i ≤ (b / c) i | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
hi : a i * c i ≤ b i
⊢ a i ≤ b i / c i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.le_div_iff | [111, 1] | [116, 51] | rw [_root_.le_div_iff (hc i)] | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
hi : a i * c i ≤ b i
⊢ a i ≤ b i / c i | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
hi : a i * c i ≤ b i
⊢ a i * c i ≤ b i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.le_div_iff | [111, 1] | [116, 51] | exact hi | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a * c ≤ b
i : m
hi : a i * c i ≤ b i
⊢ a i * c i ≤ b i | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/HypersonicShapeDesign.lean | HypersonicShapeDesign.aₚ_nonneg | [45, 1] | [47, 22] | unfold aₚ | a b : ℝ
⊢ 0 ≤ aₚ | a b : ℝ
⊢ 0 ≤ 5e-2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/HypersonicShapeDesign.lean | HypersonicShapeDesign.aₚ_nonneg | [45, 1] | [47, 22] | norm_num | a b : ℝ
⊢ 0 ≤ 5e-2 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/HypersonicShapeDesign.lean | HypersonicShapeDesign.bₚ_nonneg | [52, 1] | [53, 22] | unfold bₚ | a b : ℝ
⊢ 0 ≤ bₚ | a b : ℝ
⊢ 0 ≤ 0.65 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/HypersonicShapeDesign.lean | HypersonicShapeDesign.bₚ_nonneg | [52, 1] | [53, 22] | norm_num | a b : ℝ
⊢ 0 ≤ 0.65 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/HypersonicShapeDesign.lean | HypersonicShapeDesign.bₚ_lt_one | [55, 1] | [56, 22] | unfold bₚ | a b : ℝ
⊢ bₚ < 1 | a b : ℝ
⊢ 0.65 < 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/HypersonicShapeDesign.lean | HypersonicShapeDesign.bₚ_lt_one | [55, 1] | [56, 22] | norm_num | a b : ℝ
⊢ 0.65 < 1 | no goals |
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