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/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
|
repr_gramSchmidt_diagonal
|
[18, 1]
|
[24, 67]
|
intros j hj
|
𝕜 : Type u_2
E : Type u_1
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
i : ι
b : Basis ι 𝕜 E
⊢ ∀ x ∈ Finset.Iio i, (b.repr ↑((orthogonalProjection (Submodule.span 𝕜 {gramSchmidt 𝕜 (⇑b) x})) (b i))) i = 0
|
𝕜 : Type u_2
E : Type u_1
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
i : ι
b : Basis ι 𝕜 E
j : ι
hj : j ∈ Finset.Iio i
⊢ (b.repr ↑((orthogonalProjection (Submodule.span 𝕜 {gramSchmidt 𝕜 (⇑b) j})) (b i))) i = 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
|
repr_gramSchmidt_diagonal
|
[18, 1]
|
[24, 67]
|
rw [Finset.mem_Iio] at hj
|
𝕜 : Type u_2
E : Type u_1
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
i : ι
b : Basis ι 𝕜 E
j : ι
hj : j ∈ Finset.Iio i
⊢ (b.repr ↑((orthogonalProjection (Submodule.span 𝕜 {gramSchmidt 𝕜 (⇑b) j})) (b i))) i = 0
|
𝕜 : Type u_2
E : Type u_1
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
i : ι
b : Basis ι 𝕜 E
j : ι
hj : j < i
⊢ (b.repr ↑((orthogonalProjection (Submodule.span 𝕜 {gramSchmidt 𝕜 (⇑b) j})) (b i))) i = 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
|
repr_gramSchmidt_diagonal
|
[18, 1]
|
[24, 67]
|
simp [orthogonalProjection_singleton, gramSchmidt_triangular hj]
|
𝕜 : Type u_2
E : Type u_1
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
i : ι
b : Basis ι 𝕜 E
j : ι
hj : j < i
⊢ (b.repr ↑((orthogonalProjection (Submodule.span 𝕜 {gramSchmidt 𝕜 (⇑b) j})) (b i))) i = 0
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Matrix.lean
|
Matrix.vecCons_zero_zero
|
[30, 1]
|
[31, 52]
|
ext i
|
m : ?m.1347
n✝ : ?m.1350
α : Type u_1
n : ℕ
inst✝ : Zero α
⊢ vecCons 0 0 = 0
|
case h
m : ?m.1347
n✝ : ?m.1350
α : Type u_1
n : ℕ
inst✝ : Zero α
i : Fin n.succ
⊢ vecCons 0 0 i = 0 i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Matrix.lean
|
Matrix.vecCons_zero_zero
|
[30, 1]
|
[31, 52]
|
refine' Fin.cases _ _ i <;> simp [vecCons]
|
case h
m : ?m.1347
n✝ : ?m.1350
α : Type u_1
n : ℕ
inst✝ : Zero α
i : Fin n.succ
⊢ vecCons 0 0 i = 0 i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Matrix.lean
|
Matrix.smul_vecCons
|
[33, 1]
|
[35, 52]
|
ext i
|
m : ?m.2988
n✝ : ?m.2991
α : Type u_1
n : ℕ
inst✝¹ : Zero α
inst✝ : SMulZeroClass ℝ α
x : ℝ
y : α
v : Fin n → α
⊢ x • vecCons y v = vecCons (x • y) (x • v)
|
case h
m : ?m.2988
n✝ : ?m.2991
α : Type u_1
n : ℕ
inst✝¹ : Zero α
inst✝ : SMulZeroClass ℝ α
x : ℝ
y : α
v : Fin n → α
i : Fin n.succ
⊢ (x • vecCons y v) i = vecCons (x • y) (x • v) i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Matrix.lean
|
Matrix.smul_vecCons
|
[33, 1]
|
[35, 52]
|
refine' Fin.cases _ _ i <;> simp [vecCons]
|
case h
m : ?m.2988
n✝ : ?m.2991
α : Type u_1
n : ℕ
inst✝¹ : Zero α
inst✝ : SMulZeroClass ℝ α
x : ℝ
y : α
v : Fin n → α
i : Fin n.succ
⊢ (x • vecCons y v) i = vecCons (x • y) (x • v) i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Matrix.lean
|
Matrix.add_vecCons
|
[37, 1]
|
[39, 52]
|
ext i
|
m : ?m.4819
n✝ : ?m.4822
α : Type u_1
n : ℕ
inst✝² : Zero α
inst✝¹ : SMulZeroClass ℝ α
inst✝ : Add α
x : α
v : Fin n → α
y : α
w : Fin n → α
⊢ vecCons x v + vecCons y w = vecCons (x + y) (v + w)
|
case h
m : ?m.4819
n✝ : ?m.4822
α : Type u_1
n : ℕ
inst✝² : Zero α
inst✝¹ : SMulZeroClass ℝ α
inst✝ : Add α
x : α
v : Fin n → α
y : α
w : Fin n → α
i : Fin n.succ
⊢ (vecCons x v + vecCons y w) i = vecCons (x + y) (v + w) i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Matrix.lean
|
Matrix.add_vecCons
|
[37, 1]
|
[39, 52]
|
refine' Fin.cases _ _ i <;> simp [vecCons]
|
case h
m : ?m.4819
n✝ : ?m.4822
α : Type u_1
n : ℕ
inst✝² : Zero α
inst✝¹ : SMulZeroClass ℝ α
inst✝ : Add α
x : α
v : Fin n → α
y : α
w : Fin n → α
i : Fin n.succ
⊢ (vecCons x v + vecCons y w) i = vecCons (x + y) (v + w) i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean
|
LDL.lowerInv_diagonal
|
[23, 1]
|
[29, 24]
|
simp only [lowerInv_eq_gramSchmidtBasis, gramSchmidtBasis]
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
⊢ lowerInv hS i i = 1
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
⊢ ((Pi.basisFun 𝕜 n).toMatrix ⇑(Basis.mk ⋯ ⋯)).transpose i i = 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean
|
LDL.lowerInv_diagonal
|
[23, 1]
|
[29, 24]
|
letI := NormedAddCommGroup.ofMatrix hS.transpose
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
⊢ ((Pi.basisFun 𝕜 n).toMatrix ⇑(Basis.mk ⋯ ⋯)).transpose i i = 1
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
this : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯
⊢ ((Pi.basisFun 𝕜 n).toMatrix ⇑(Basis.mk ⋯ ⋯)).transpose i i = 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean
|
LDL.lowerInv_diagonal
|
[23, 1]
|
[29, 24]
|
letI := InnerProductSpace.ofMatrix hS.transpose
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
this : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯
⊢ ((Pi.basisFun 𝕜 n).toMatrix ⇑(Basis.mk ⋯ ⋯)).transpose i i = 1
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
this✝ : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯
this : InnerProductSpace 𝕜 (n → 𝕜) := InnerProductSpace.ofMatrix ⋯
⊢ ((Pi.basisFun 𝕜 n).toMatrix ⇑(Basis.mk ⋯ ⋯)).transpose i i = 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean
|
LDL.lowerInv_diagonal
|
[23, 1]
|
[29, 24]
|
rw [Basis.coe_mk, ← @repr_gramSchmidt_diagonal 𝕜 (n → 𝕜) _ _ _ n _ _ _ i (Pi.basisFun 𝕜 n)]
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
this✝ : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯
this : InnerProductSpace 𝕜 (n → 𝕜) := InnerProductSpace.ofMatrix ⋯
⊢ ((Pi.basisFun 𝕜 n).toMatrix ⇑(Basis.mk ⋯ ⋯)).transpose i i = 1
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
this✝ : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯
this : InnerProductSpace 𝕜 (n → 𝕜) := InnerProductSpace.ofMatrix ⋯
⊢ ((Pi.basisFun 𝕜 n).toMatrix (gramSchmidt 𝕜 ⇑(Pi.basisFun 𝕜 n))).transpose i i =
((Pi.basisFun 𝕜 n).repr (gramSchmidt 𝕜 (⇑(Pi.basisFun 𝕜 n)) i)) i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean
|
LDL.lowerInv_diagonal
|
[23, 1]
|
[29, 24]
|
simp [Basis.toMatrix]
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
this✝ : NormedAddCommGroup (n → 𝕜) := NormedAddCommGroup.ofMatrix ⋯
this : InnerProductSpace 𝕜 (n → 𝕜) := InnerProductSpace.ofMatrix ⋯
⊢ ((Pi.basisFun 𝕜 n).toMatrix (gramSchmidt 𝕜 ⇑(Pi.basisFun 𝕜 n))).transpose i i =
((Pi.basisFun 𝕜 n).repr (gramSchmidt 𝕜 (⇑(Pi.basisFun 𝕜 n)) i)) i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean
|
LDL.lower_eq_to_matrix
|
[31, 1]
|
[39, 72]
|
simp only [lower, lowerInv_eq_gramSchmidtBasis]
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ lower hS = ((gramSchmidtBasis (Pi.basisFun 𝕜 n)).toMatrix ⇑(Pi.basisFun 𝕜 n)).transpose
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n))).transpose⁻¹ =
((gramSchmidtBasis (Pi.basisFun 𝕜 n)).toMatrix ⇑(Pi.basisFun 𝕜 n)).transpose
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean
|
LDL.lower_eq_to_matrix
|
[31, 1]
|
[39, 72]
|
apply Matrix.inv_eq_left_inv
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ ((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n))).transpose⁻¹ =
((gramSchmidtBasis (Pi.basisFun 𝕜 n)).toMatrix ⇑(Pi.basisFun 𝕜 n)).transpose
|
case h
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ ((gramSchmidtBasis (Pi.basisFun 𝕜 n)).toMatrix ⇑(Pi.basisFun 𝕜 n)).transpose *
((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n))).transpose =
1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean
|
LDL.lower_eq_to_matrix
|
[31, 1]
|
[39, 72]
|
rw [← transpose_mul, Basis.toMatrix_mul_toMatrix_flip, transpose_one]
|
case h
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ ((gramSchmidtBasis (Pi.basisFun 𝕜 n)).toMatrix ⇑(Pi.basisFun 𝕜 n)).transpose *
((Pi.basisFun 𝕜 n).toMatrix ⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n))).transpose =
1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean
|
LDL.lowerTriangular_lowerInv
|
[41, 1]
|
[42, 28]
|
apply lowerInv_triangular
|
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
n : Type u_1
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ (lowerInv hS).lowerTriangular
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean
|
LDL.lower_diagonal
|
[54, 1]
|
[57, 77]
|
simpa using
diag_inv_mul_diag_eq_one_of_lowerTriangular (lowerTriangular_lower hS) i
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
i : n
⊢ lower hS i i = 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean
|
LDL.det_lowerInv
|
[59, 1]
|
[64, 25]
|
have h := det_of_lowerTriangular (lowerInv hS) (by apply lowerInv_triangular)
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ (lowerInv hS).det = 1
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
h : (lowerInv hS).det = Finset.univ.prod fun i => lowerInv hS i i
⊢ (lowerInv hS).det = 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean
|
LDL.det_lowerInv
|
[59, 1]
|
[64, 25]
|
rw [h, Finset.prod_eq_one]
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
h : (lowerInv hS).det = Finset.univ.prod fun i => lowerInv hS i i
⊢ (lowerInv hS).det = 1
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
h : (lowerInv hS).det = Finset.univ.prod fun i => lowerInv hS i i
⊢ ∀ x ∈ Finset.univ, lowerInv hS x x = 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean
|
LDL.det_lowerInv
|
[59, 1]
|
[64, 25]
|
intros
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
h : (lowerInv hS).det = Finset.univ.prod fun i => lowerInv hS i i
⊢ ∀ x ∈ Finset.univ, lowerInv hS x x = 1
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
h : (lowerInv hS).det = Finset.univ.prod fun i => lowerInv hS i i
x✝ : n
a✝ : x✝ ∈ Finset.univ
⊢ lowerInv hS x✝ x✝ = 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean
|
LDL.det_lowerInv
|
[59, 1]
|
[64, 25]
|
rw [lowerInv_diagonal]
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
h : (lowerInv hS).det = Finset.univ.prod fun i => lowerInv hS i i
x✝ : n
a✝ : x✝ ∈ Finset.univ
⊢ lowerInv hS x✝ x✝ = 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean
|
LDL.det_lowerInv
|
[59, 1]
|
[64, 25]
|
apply lowerInv_triangular
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ (lowerInv hS).BlockTriangular ⇑OrderDual.toDual
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/LDL.lean
|
LDL.det_lower
|
[66, 1]
|
[68, 15]
|
simp [lower]
|
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : IsWellOrder n fun x x_1 => x < x_1
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
⊢ (lower hS).det = 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.PosSemidef_zero
|
[30, 1]
|
[32, 20]
|
simp [PosSemidef]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
⊢ PosSemidef 0
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef
|
[34, 1]
|
[50, 16]
|
have h_D_eq : D = Z * A⁻¹ * Zᴴ :=
calc D
= D * D⁻¹ * D := by
rw [hD, Matrix.mul_inv_of_invertible, Matrix.one_mul]
_ = D * (LDL.lowerInv hA * A * (LDL.lowerInv hA)ᵀ)⁻¹ * Dᵀ := by
erw [hD, LDL.diag, diagonal_transpose, ← LDL.diag, LDL.diag_eq_lowerInv_conj]
rfl
_ = D * (LDL.lower hA)ᵀ * A⁻¹ * (D * (LDL.lower hA)ᵀ)ᵀ := by
simp only [hD, LDL.lower, transpose_mul, transpose_transpose, transpose_nonsing_inv,
Matrix.mul_assoc, Matrix.mul_inv_rev]
_ = Z * A⁻¹ * Zᴴ := by
rw [hZ, hD]; rfl
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z : Matrix n n ℝ
hD : D = LDL.diag hA
hZ : Z = LDL.diag hA * (LDL.lower hA).transpose
⊢ (D.fromBlocks Z Z.transpose A).PosSemidef
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z : Matrix n n ℝ
hD : D = LDL.diag hA
hZ : Z = LDL.diag hA * (LDL.lower hA).transpose
h_D_eq : D = Z * A⁻¹ * Z.conjTranspose
⊢ (D.fromBlocks Z Z.transpose A).PosSemidef
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef
|
[34, 1]
|
[50, 16]
|
haveI := hA.Invertible
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z : Matrix n n ℝ
hD : D = LDL.diag hA
hZ : Z = LDL.diag hA * (LDL.lower hA).transpose
h_D_eq : D = Z * A⁻¹ * Z.conjTranspose
⊢ (D.fromBlocks Z Z.transpose A).PosSemidef
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z : Matrix n n ℝ
hD : D = LDL.diag hA
hZ : Z = LDL.diag hA * (LDL.lower hA).transpose
h_D_eq : D = Z * A⁻¹ * Z.conjTranspose
this : Invertible A
⊢ (D.fromBlocks Z Z.transpose A).PosSemidef
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef
|
[34, 1]
|
[50, 16]
|
erw [PosSemidef.fromBlocks₂₂ _ _ hA]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z : Matrix n n ℝ
hD : D = LDL.diag hA
hZ : Z = LDL.diag hA * (LDL.lower hA).transpose
h_D_eq : D = Z * A⁻¹ * Z.conjTranspose
this : Invertible A
⊢ (D.fromBlocks Z Z.transpose A).PosSemidef
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z : Matrix n n ℝ
hD : D = LDL.diag hA
hZ : Z = LDL.diag hA * (LDL.lower hA).transpose
h_D_eq : D = Z * A⁻¹ * Z.conjTranspose
this : Invertible A
⊢ (D - Z * A⁻¹ * Z.conjTranspose).PosSemidef
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef
|
[34, 1]
|
[50, 16]
|
simp [h_D_eq]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z : Matrix n n ℝ
hD : D = LDL.diag hA
hZ : Z = LDL.diag hA * (LDL.lower hA).transpose
h_D_eq : D = Z * A⁻¹ * Z.conjTranspose
this : Invertible A
⊢ (D - Z * A⁻¹ * Z.conjTranspose).PosSemidef
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef
|
[34, 1]
|
[50, 16]
|
rw [hD, Matrix.mul_inv_of_invertible, Matrix.one_mul]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z : Matrix n n ℝ
hD : D = LDL.diag hA
hZ : Z = LDL.diag hA * (LDL.lower hA).transpose
⊢ D = D * D⁻¹ * D
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef
|
[34, 1]
|
[50, 16]
|
erw [hD, LDL.diag, diagonal_transpose, ← LDL.diag, LDL.diag_eq_lowerInv_conj]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z : Matrix n n ℝ
hD : D = LDL.diag hA
hZ : Z = LDL.diag hA * (LDL.lower hA).transpose
⊢ D * D⁻¹ * D = D * (LDL.lowerInv hA * A * (LDL.lowerInv hA).transpose)⁻¹ * D.transpose
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z : Matrix n n ℝ
hD : D = LDL.diag hA
hZ : Z = LDL.diag hA * (LDL.lower hA).transpose
⊢ LDL.lowerInv hA * A * (LDL.lowerInv hA).conjTranspose * (LDL.lowerInv hA * A * (LDL.lowerInv hA).conjTranspose)⁻¹ *
(LDL.lowerInv hA * A * (LDL.lowerInv hA).conjTranspose) =
LDL.lowerInv hA * A * (LDL.lowerInv hA).conjTranspose * (LDL.lowerInv hA * A * (LDL.lowerInv hA).transpose)⁻¹ *
(LDL.lowerInv hA * A * (LDL.lowerInv hA).conjTranspose)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef
|
[34, 1]
|
[50, 16]
|
rfl
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z : Matrix n n ℝ
hD : D = LDL.diag hA
hZ : Z = LDL.diag hA * (LDL.lower hA).transpose
⊢ LDL.lowerInv hA * A * (LDL.lowerInv hA).conjTranspose * (LDL.lowerInv hA * A * (LDL.lowerInv hA).conjTranspose)⁻¹ *
(LDL.lowerInv hA * A * (LDL.lowerInv hA).conjTranspose) =
LDL.lowerInv hA * A * (LDL.lowerInv hA).conjTranspose * (LDL.lowerInv hA * A * (LDL.lowerInv hA).transpose)⁻¹ *
(LDL.lowerInv hA * A * (LDL.lowerInv hA).conjTranspose)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef
|
[34, 1]
|
[50, 16]
|
simp only [hD, LDL.lower, transpose_mul, transpose_transpose, transpose_nonsing_inv,
Matrix.mul_assoc, Matrix.mul_inv_rev]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z : Matrix n n ℝ
hD : D = LDL.diag hA
hZ : Z = LDL.diag hA * (LDL.lower hA).transpose
⊢ D * (LDL.lowerInv hA * A * (LDL.lowerInv hA).transpose)⁻¹ * D.transpose =
D * (LDL.lower hA).transpose * A⁻¹ * (D * (LDL.lower hA).transpose).transpose
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef
|
[34, 1]
|
[50, 16]
|
rw [hZ, hD]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z : Matrix n n ℝ
hD : D = LDL.diag hA
hZ : Z = LDL.diag hA * (LDL.lower hA).transpose
⊢ D * (LDL.lower hA).transpose * A⁻¹ * (D * (LDL.lower hA).transpose).transpose = Z * A⁻¹ * Z.conjTranspose
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z : Matrix n n ℝ
hD : D = LDL.diag hA
hZ : Z = LDL.diag hA * (LDL.lower hA).transpose
⊢ LDL.diag hA * (LDL.lower hA).transpose * A⁻¹ * (LDL.diag hA * (LDL.lower hA).transpose).transpose =
LDL.diag hA * (LDL.lower hA).transpose * A⁻¹ * (LDL.diag hA * (LDL.lower hA).transpose).conjTranspose
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef
|
[34, 1]
|
[50, 16]
|
rfl
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z : Matrix n n ℝ
hD : D = LDL.diag hA
hZ : Z = LDL.diag hA * (LDL.lower hA).transpose
⊢ LDL.diag hA * (LDL.lower hA).transpose * A⁻¹ * (LDL.diag hA * (LDL.lower hA).transpose).transpose =
LDL.diag hA * (LDL.lower hA).transpose * A⁻¹ * (LDL.diag hA * (LDL.lower hA).transpose).conjTranspose
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef'
|
[52, 1]
|
[63, 26]
|
have hY_tri : upperTriangular Y := by
{ rw [hY]
apply upperTriangular.mul
apply BlockTriangular_diagonal
apply lowerTriangular.transpose
apply LDL.lowerTriangular_lower }
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
⊢ (D.fromBlocks Z Z.transpose A).PosSemidef
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hY_tri : Y.upperTriangular
⊢ (D.fromBlocks Z Z.transpose A).PosSemidef
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef'
|
[52, 1]
|
[63, 26]
|
haveI := hA.Invertible
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hY_tri : Y.upperTriangular
⊢ (D.fromBlocks Z Z.transpose A).PosSemidef
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hY_tri : Y.upperTriangular
this : Invertible A
⊢ (D.fromBlocks Z Z.transpose A).PosSemidef
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef'
|
[52, 1]
|
[63, 26]
|
rw [hZ, hY_tri.toUpperTri_eq]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hY_tri : Y.upperTriangular
this : Invertible A
⊢ (D.fromBlocks Z Z.transpose A).PosSemidef
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hY_tri : Y.upperTriangular
this : Invertible A
⊢ (D.fromBlocks Y Y.transpose A).PosSemidef
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef'
|
[52, 1]
|
[63, 26]
|
apply LogDetAtom.feasibility_PosDef _ _ hY
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hY_tri : Y.upperTriangular
this : Invertible A
⊢ (D.fromBlocks Y Y.transpose A).PosSemidef
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hY_tri : Y.upperTriangular
this : Invertible A
⊢ D = LDL.diag hA
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef'
|
[52, 1]
|
[63, 26]
|
simp [hD, hY, LDL.diag]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hY_tri : Y.upperTriangular
this : Invertible A
⊢ D = LDL.diag hA
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef'
|
[52, 1]
|
[63, 26]
|
rw [hY]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
⊢ Y.upperTriangular
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
⊢ (LDL.diag hA * (LDL.lower hA).transpose).upperTriangular
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef'
|
[52, 1]
|
[63, 26]
|
apply upperTriangular.mul
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
⊢ (LDL.diag hA * (LDL.lower hA).transpose).upperTriangular
|
case hM
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
⊢ (LDL.diag hA).upperTriangular
case hN
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
⊢ (LDL.lower hA).transpose.upperTriangular
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef'
|
[52, 1]
|
[63, 26]
|
apply BlockTriangular_diagonal
|
case hM
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
⊢ (LDL.diag hA).upperTriangular
case hN
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
⊢ (LDL.lower hA).transpose.upperTriangular
|
case hN
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
⊢ (LDL.lower hA).transpose.upperTriangular
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef'
|
[52, 1]
|
[63, 26]
|
apply lowerTriangular.transpose
|
case hN
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
⊢ (LDL.lower hA).transpose.upperTriangular
|
case hN.hM
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
⊢ (LDL.lower hA).lowerTriangular
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_PosDef'
|
[52, 1]
|
[63, 26]
|
apply LDL.lowerTriangular_lower
|
case hN.hM
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
D Z Y : Matrix n n ℝ
hY : Y = LDL.diag hA * (LDL.lower hA).transpose
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
⊢ (LDL.lower hA).lowerTriangular
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LDL.diagEntries_pos
|
[65, 1]
|
[70, 38]
|
have : (LDL.lowerInv hA).det ≠ 0 := by simp [LDL.det_lowerInv hA]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA✝ : A✝.PosDef
A : Matrix n n ℝ
hA : A.PosDef
i : n
⊢ 0 < LDL.diagEntries hA i
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA✝ : A✝.PosDef
A : Matrix n n ℝ
hA : A.PosDef
i : n
this : (LDL.lowerInv hA).det ≠ 0
⊢ 0 < LDL.diagEntries hA i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LDL.diagEntries_pos
|
[65, 1]
|
[70, 38]
|
have : LDL.lowerInv hA i ≠ 0 := fun h =>
this (Matrix.det_eq_zero_of_row_eq_zero i (λ j => congr_fun h j))
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA✝ : A✝.PosDef
A : Matrix n n ℝ
hA : A.PosDef
i : n
this : (LDL.lowerInv hA).det ≠ 0
⊢ 0 < LDL.diagEntries hA i
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA✝ : A✝.PosDef
A : Matrix n n ℝ
hA : A.PosDef
i : n
this✝ : (LDL.lowerInv hA).det ≠ 0
this : LDL.lowerInv hA i ≠ 0
⊢ 0 < LDL.diagEntries hA i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LDL.diagEntries_pos
|
[65, 1]
|
[70, 38]
|
exact hA.2 (LDL.lowerInv hA i) this
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA✝ : A✝.PosDef
A : Matrix n n ℝ
hA : A.PosDef
i : n
this✝ : (LDL.lowerInv hA).det ≠ 0
this : LDL.lowerInv hA i ≠ 0
⊢ 0 < LDL.diagEntries hA i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LDL.diagEntries_pos
|
[65, 1]
|
[70, 38]
|
simp [LDL.det_lowerInv hA]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA✝ : A✝.PosDef
A : Matrix n n ℝ
hA : A.PosDef
i : n
⊢ (LDL.lowerInv hA).det ≠ 0
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.solution_eq_atom
|
[72, 1]
|
[77, 31]
|
conv => rhs; rw [(LDL.lower_conj_diag hA).symm]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA✝ : A✝.PosDef
A : Matrix n n ℝ
hA : A.PosDef
⊢ ∑ i : n, (LDL.diagEntries hA i).log = A.det.log
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA✝ : A✝.PosDef
A : Matrix n n ℝ
hA : A.PosDef
⊢ ∑ i : n, (LDL.diagEntries hA i).log = (LDL.lower hA * LDL.diag hA * (LDL.lower hA).conjTranspose).det.log
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.solution_eq_atom
|
[72, 1]
|
[77, 31]
|
have heqsum := Real.log_prod Finset.univ (LDL.diagEntries hA)
(fun i _ => ne_of_gt (LDL.diagEntries_pos hA i))
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA✝ : A✝.PosDef
A : Matrix n n ℝ
hA : A.PosDef
⊢ ∑ i : n, (LDL.diagEntries hA i).log = (LDL.lower hA * LDL.diag hA * (LDL.lower hA).conjTranspose).det.log
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA✝ : A✝.PosDef
A : Matrix n n ℝ
hA : A.PosDef
heqsum : (∏ i : n, LDL.diagEntries hA i).log = ∑ i : n, (LDL.diagEntries hA i).log
⊢ ∑ i : n, (LDL.diagEntries hA i).log = (LDL.lower hA * LDL.diag hA * (LDL.lower hA).conjTranspose).det.log
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.solution_eq_atom
|
[72, 1]
|
[77, 31]
|
simp [LDL.diag, heqsum.symm]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA✝ : A✝.PosDef
A : Matrix n n ℝ
hA : A.PosDef
heqsum : (∏ i : n, LDL.diagEntries hA i).log = ∑ i : n, (LDL.diagEntries hA i).log
⊢ ∑ i : n, (LDL.diagEntries hA i).log = (LDL.lower hA * LDL.diag hA * (LDL.lower hA).conjTranspose).det.log
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.feasibility_exp
|
[79, 1]
|
[81, 18]
|
simp [LDL.diag]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA✝ : A✝.PosDef
A : Matrix n n ℝ
hA : A.PosDef
i : n
⊢ LDL.diagEntries hA i ≤ (LDL.diag hA * (LDL.lower hA).transpose).diag i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.IsHermitian₁₁_of_IsHermitian_toBlock
|
[83, 1]
|
[85, 71]
|
ext i j
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA : A✝.PosDef
A B C D : Matrix n n ℝ
h : (A.fromBlocks B C D).IsHermitian
⊢ A.IsHermitian
|
case a
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA : A✝.PosDef
A B C D : Matrix n n ℝ
h : (A.fromBlocks B C D).IsHermitian
i j : n
⊢ A.conjTranspose i j = A i j
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.IsHermitian₁₁_of_IsHermitian_toBlock
|
[83, 1]
|
[85, 71]
|
simpa using congr_fun (congr_fun h (Sum.inl i)) (Sum.inl j)
|
case a
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA : A✝.PosDef
A B C D : Matrix n n ℝ
h : (A.fromBlocks B C D).IsHermitian
i j : n
⊢ A.conjTranspose i j = A i j
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.IsHermitian₂₂_of_IsHermitian_toBlock
|
[87, 1]
|
[89, 71]
|
ext i j
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA : A✝.PosDef
A B C D : Matrix n n ℝ
h : (A.fromBlocks B C D).IsHermitian
⊢ D.IsHermitian
|
case a
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA : A✝.PosDef
A B C D : Matrix n n ℝ
h : (A.fromBlocks B C D).IsHermitian
i j : n
⊢ D.conjTranspose i j = D i j
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.IsHermitian₂₂_of_IsHermitian_toBlock
|
[87, 1]
|
[89, 71]
|
simpa using congr_fun (congr_fun h (Sum.inr i)) (Sum.inr j)
|
case a
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA : A✝.PosDef
A B C D : Matrix n n ℝ
h : (A.fromBlocks B C D).IsHermitian
i j : n
⊢ D.conjTranspose i j = D i j
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.PosSemidef₁₁_of_PosSemidef_toBlock
|
[91, 1]
|
[94, 80]
|
simpa [Matrix.fromBlocks_mulVec, star] using h.2 (Sum.elim x 0)
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA : A✝.PosDef
A B C D : Matrix n n ℝ
h : (A.fromBlocks B C D).PosSemidef
x : n → ℝ
⊢ 0 ≤ star x ⬝ᵥ A.mulVec x
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.PosSemidef₂₂_of_PosSemidef_toBlock
|
[96, 1]
|
[100, 80]
|
simpa [Matrix.fromBlocks_mulVec, star] using h.2 (Sum.elim 0 x)
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A✝ : Matrix n n ℝ
hA : A✝.PosDef
A B C D : Matrix n n ℝ
h : (A.fromBlocks B C D).PosSemidef
x : n → ℝ
⊢ 0 ≤ star x ⬝ᵥ D.mulVec x
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality_D_posdef
|
[102, 1]
|
[108, 70]
|
have h_D_psd : D.PosSemidef := PosSemidef₁₁_of_PosSemidef_toBlock hPSD
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
_hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
⊢ D.PosDef
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
_hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_psd : D.PosSemidef
⊢ D.PosDef
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality_D_posdef
|
[102, 1]
|
[108, 70]
|
{ rw [h_D_psd.PosDef_iff_det_ne_zero, hD, det_diagonal, Finset.prod_ne_zero_iff]
exact λ i _ => ne_of_gt (lt_of_lt_of_le ((t i).exp_pos) (ht i)) }
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
_hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_psd : D.PosSemidef
⊢ D.PosDef
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality_D_posdef
|
[102, 1]
|
[108, 70]
|
rw [h_D_psd.PosDef_iff_det_ne_zero, hD, det_diagonal, Finset.prod_ne_zero_iff]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
_hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_psd : D.PosSemidef
⊢ D.PosDef
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
_hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_psd : D.PosSemidef
⊢ ∀ a ∈ Finset.univ, Y.diag a ≠ 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality_D_posdef
|
[102, 1]
|
[108, 70]
|
exact λ i _ => ne_of_gt (lt_of_lt_of_le ((t i).exp_pos) (ht i))
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
_hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_psd : D.PosSemidef
⊢ ∀ a ∈ Finset.univ, Y.diag a ≠ 0
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality_Ddet_le_Adet
|
[110, 1]
|
[127, 32]
|
by_cases h_nonempty : Nonempty n
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
⊢ D.det ≤ A.det
|
case pos
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : Nonempty n
⊢ D.det ≤ A.det
case neg
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : ¬Nonempty n
⊢ D.det ≤ A.det
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality_Ddet_le_Adet
|
[110, 1]
|
[127, 32]
|
{ have h_D_pd : D.PosDef := LogDetAtom.optimality_D_posdef ht hD hZ hPSD
haveI h_D_invertible : Invertible D := h_D_pd.Invertible
have h_Zdet : Z.det = D.det := by
{ rw [hZ, det_of_upperTriangular (upperTriangular_toUpperTri Y), hD, det_diagonal]
simp [toUpperTri] }
have h_ZDZ_semidef : (Zᴴ * D⁻¹ * Z).PosSemidef :=
PosSemidef.conjTranspose_mul_mul D⁻¹ Z h_D_pd.nonsingular_inv.posSemidef
have h_AZDZ_semidef : (A - Zᴴ * D⁻¹ * Z).PosSemidef :=
(PosSemidef.fromBlocks₁₁ Z A h_D_pd).1 hPSD
show D.det ≤ A.det
{ apply le_of_add_le_of_nonneg_left _ h_AZDZ_semidef.det_nonneg
simpa [h_Zdet] using det_add_det_le_det_add _ _ h_ZDZ_semidef h_AZDZ_semidef } }
|
case pos
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : Nonempty n
⊢ D.det ≤ A.det
case neg
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : ¬Nonempty n
⊢ D.det ≤ A.det
|
case neg
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : ¬Nonempty n
⊢ D.det ≤ A.det
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality_Ddet_le_Adet
|
[110, 1]
|
[127, 32]
|
{ haveI h_empty := not_nonempty_iff.1 h_nonempty
simp [Matrix.det_isEmpty] }
|
case neg
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : ¬Nonempty n
⊢ D.det ≤ A.det
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality_Ddet_le_Adet
|
[110, 1]
|
[127, 32]
|
have h_D_pd : D.PosDef := LogDetAtom.optimality_D_posdef ht hD hZ hPSD
|
case pos
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : Nonempty n
⊢ D.det ≤ A.det
|
case pos
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : Nonempty n
h_D_pd : D.PosDef
⊢ D.det ≤ A.det
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality_Ddet_le_Adet
|
[110, 1]
|
[127, 32]
|
haveI h_D_invertible : Invertible D := h_D_pd.Invertible
|
case pos
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : Nonempty n
h_D_pd : D.PosDef
⊢ D.det ≤ A.det
|
case pos
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : Nonempty n
h_D_pd : D.PosDef
h_D_invertible : Invertible D
⊢ D.det ≤ A.det
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality_Ddet_le_Adet
|
[110, 1]
|
[127, 32]
|
have h_Zdet : Z.det = D.det := by
{ rw [hZ, det_of_upperTriangular (upperTriangular_toUpperTri Y), hD, det_diagonal]
simp [toUpperTri] }
|
case pos
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : Nonempty n
h_D_pd : D.PosDef
h_D_invertible : Invertible D
⊢ D.det ≤ A.det
|
case pos
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : Nonempty n
h_D_pd : D.PosDef
h_D_invertible : Invertible D
h_Zdet : Z.det = D.det
⊢ D.det ≤ A.det
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality_Ddet_le_Adet
|
[110, 1]
|
[127, 32]
|
have h_ZDZ_semidef : (Zᴴ * D⁻¹ * Z).PosSemidef :=
PosSemidef.conjTranspose_mul_mul D⁻¹ Z h_D_pd.nonsingular_inv.posSemidef
|
case pos
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : Nonempty n
h_D_pd : D.PosDef
h_D_invertible : Invertible D
h_Zdet : Z.det = D.det
⊢ D.det ≤ A.det
|
case pos
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : Nonempty n
h_D_pd : D.PosDef
h_D_invertible : Invertible D
h_Zdet : Z.det = D.det
h_ZDZ_semidef : (Z.conjTranspose * D⁻¹ * Z).PosSemidef
⊢ D.det ≤ A.det
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality_Ddet_le_Adet
|
[110, 1]
|
[127, 32]
|
have h_AZDZ_semidef : (A - Zᴴ * D⁻¹ * Z).PosSemidef :=
(PosSemidef.fromBlocks₁₁ Z A h_D_pd).1 hPSD
|
case pos
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : Nonempty n
h_D_pd : D.PosDef
h_D_invertible : Invertible D
h_Zdet : Z.det = D.det
h_ZDZ_semidef : (Z.conjTranspose * D⁻¹ * Z).PosSemidef
⊢ D.det ≤ A.det
|
case pos
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : Nonempty n
h_D_pd : D.PosDef
h_D_invertible : Invertible D
h_Zdet : Z.det = D.det
h_ZDZ_semidef : (Z.conjTranspose * D⁻¹ * Z).PosSemidef
h_AZDZ_semidef : (A - Z.conjTranspose * D⁻¹ * Z).PosSemidef
⊢ D.det ≤ A.det
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality_Ddet_le_Adet
|
[110, 1]
|
[127, 32]
|
{ apply le_of_add_le_of_nonneg_left _ h_AZDZ_semidef.det_nonneg
simpa [h_Zdet] using det_add_det_le_det_add _ _ h_ZDZ_semidef h_AZDZ_semidef }
|
case pos
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : Nonempty n
h_D_pd : D.PosDef
h_D_invertible : Invertible D
h_Zdet : Z.det = D.det
h_ZDZ_semidef : (Z.conjTranspose * D⁻¹ * Z).PosSemidef
h_AZDZ_semidef : (A - Z.conjTranspose * D⁻¹ * Z).PosSemidef
⊢ D.det ≤ A.det
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality_Ddet_le_Adet
|
[110, 1]
|
[127, 32]
|
rw [hZ, det_of_upperTriangular (upperTriangular_toUpperTri Y), hD, det_diagonal]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : Nonempty n
h_D_pd : D.PosDef
h_D_invertible : Invertible D
⊢ Z.det = D.det
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : Nonempty n
h_D_pd : D.PosDef
h_D_invertible : Invertible D
⊢ ∏ i : n, Y.toUpperTri i i = ∏ i : n, Y.diag i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality_Ddet_le_Adet
|
[110, 1]
|
[127, 32]
|
simp [toUpperTri]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : Nonempty n
h_D_pd : D.PosDef
h_D_invertible : Invertible D
⊢ ∏ i : n, Y.toUpperTri i i = ∏ i : n, Y.diag i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality_Ddet_le_Adet
|
[110, 1]
|
[127, 32]
|
apply le_of_add_le_of_nonneg_left _ h_AZDZ_semidef.det_nonneg
|
case pos
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : Nonempty n
h_D_pd : D.PosDef
h_D_invertible : Invertible D
h_Zdet : Z.det = D.det
h_ZDZ_semidef : (Z.conjTranspose * D⁻¹ * Z).PosSemidef
h_AZDZ_semidef : (A - Z.conjTranspose * D⁻¹ * Z).PosSemidef
⊢ D.det ≤ A.det
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : Nonempty n
h_D_pd : D.PosDef
h_D_invertible : Invertible D
h_Zdet : Z.det = D.det
h_ZDZ_semidef : (Z.conjTranspose * D⁻¹ * Z).PosSemidef
h_AZDZ_semidef : (A - Z.conjTranspose * D⁻¹ * Z).PosSemidef
⊢ D.det + (A - Z.conjTranspose * D⁻¹ * Z).det ≤ A.det
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality_Ddet_le_Adet
|
[110, 1]
|
[127, 32]
|
simpa [h_Zdet] using det_add_det_le_det_add _ _ h_ZDZ_semidef h_AZDZ_semidef
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : Nonempty n
h_D_pd : D.PosDef
h_D_invertible : Invertible D
h_Zdet : Z.det = D.det
h_ZDZ_semidef : (Z.conjTranspose * D⁻¹ * Z).PosSemidef
h_AZDZ_semidef : (A - Z.conjTranspose * D⁻¹ * Z).PosSemidef
⊢ D.det + (A - Z.conjTranspose * D⁻¹ * Z).det ≤ A.det
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality_Ddet_le_Adet
|
[110, 1]
|
[127, 32]
|
haveI h_empty := not_nonempty_iff.1 h_nonempty
|
case neg
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : ¬Nonempty n
⊢ D.det ≤ A.det
|
case neg
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : ¬Nonempty n
h_empty : IsEmpty n
⊢ D.det ≤ A.det
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality_Ddet_le_Adet
|
[110, 1]
|
[127, 32]
|
simp [Matrix.det_isEmpty]
|
case neg
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_nonempty : ¬Nonempty n
h_empty : IsEmpty n
⊢ D.det ≤ A.det
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.cond_elim
|
[129, 1]
|
[137, 28]
|
have h_D_pd : D.PosDef := LogDetAtom.optimality_D_posdef ht hD hZ hPSD
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
⊢ A.PosDef
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_pd : D.PosDef
⊢ A.PosDef
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.cond_elim
|
[129, 1]
|
[137, 28]
|
have h_A_psd : A.PosSemidef := PosSemidef₂₂_of_PosSemidef_toBlock hPSD
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_pd : D.PosDef
⊢ A.PosDef
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_pd : D.PosDef
h_A_psd : A.PosSemidef
⊢ A.PosDef
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.cond_elim
|
[129, 1]
|
[137, 28]
|
have h_Ddet_le_Adet : D.det ≤ A.det := LogDetAtom.optimality_Ddet_le_Adet ht hD hZ hPSD
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_pd : D.PosDef
h_A_psd : A.PosSemidef
⊢ A.PosDef
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_pd : D.PosDef
h_A_psd : A.PosSemidef
h_Ddet_le_Adet : D.det ≤ A.det
⊢ A.PosDef
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.cond_elim
|
[129, 1]
|
[137, 28]
|
have h_Adet_pos : 0 < A.det := lt_of_lt_of_le h_D_pd.det_pos h_Ddet_le_Adet
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_pd : D.PosDef
h_A_psd : A.PosSemidef
h_Ddet_le_Adet : D.det ≤ A.det
⊢ A.PosDef
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_pd : D.PosDef
h_A_psd : A.PosSemidef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ A.PosDef
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.cond_elim
|
[129, 1]
|
[137, 28]
|
rw [h_A_psd.PosDef_iff_det_ne_zero]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_pd : D.PosDef
h_A_psd : A.PosSemidef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ A.PosDef
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_pd : D.PosDef
h_A_psd : A.PosSemidef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ A.det ≠ 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.cond_elim
|
[129, 1]
|
[137, 28]
|
apply ne_of_gt h_Adet_pos
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_pd : D.PosDef
h_A_psd : A.PosSemidef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ A.det ≠ 0
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality
|
[139, 1]
|
[148, 12]
|
have h_A_pd : A.PosDef := LogDetAtom.cond_elim ht hD hZ hPSD
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
⊢ ∑ i : n, t i ≤ A.det.log
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
⊢ ∑ i : n, t i ≤ A.det.log
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality
|
[139, 1]
|
[148, 12]
|
have h_Ddet_le_Adet : D.det ≤ A.det := LogDetAtom.optimality_Ddet_le_Adet ht hD hZ hPSD
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
⊢ ∑ i : n, t i ≤ A.det.log
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
⊢ ∑ i : n, t i ≤ A.det.log
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality
|
[139, 1]
|
[148, 12]
|
have h_Adet_pos: 0 < A.det := h_A_pd.det_pos
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
⊢ ∑ i : n, t i ≤ A.det.log
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ ∑ i : n, t i ≤ A.det.log
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality
|
[139, 1]
|
[148, 12]
|
rw [← Real.exp_le_exp, Real.exp_sum, Real.exp_log h_Adet_pos]
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ ∑ i : n, t i ≤ A.det.log
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ ∏ x : n, (t x).exp ≤ A.det
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality
|
[139, 1]
|
[148, 12]
|
apply le_trans (Finset.prod_le_prod (fun i _ => le_of_lt ((t i).exp_pos)) (fun i _ => ht i))
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ ∏ x : n, (t x).exp ≤ A.det
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ ∏ i : n, Y.diag i ≤ A.det
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality
|
[139, 1]
|
[148, 12]
|
convert h_Ddet_le_Adet
|
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ ∏ i : n, Y.diag i ≤ A.det
|
case h.e'_3
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ ∏ i : n, Y.diag i = D.det
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LogDet.lean
|
Matrix.LogDetAtom.optimality
|
[139, 1]
|
[148, 12]
|
simp [hD]
|
case h.e'_3
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ ∏ i : n, Y.diag i = D.det
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.IsHermitian.hasEigenvector_eigenvectorBasis
|
[30, 1]
|
[33, 62]
|
simp only [IsHermitian.eigenvectorBasis, OrthonormalBasis.coe_reindex]
|
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_1
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.IsHermitian
i : n
⊢ Module.End.HasEigenvector (toLin' A) (↑(hA.eigenvalues i)) (hA.eigenvectorBasis i)
|
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_1
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.IsHermitian
i : n
⊢ Module.End.HasEigenvector (toLin' A) (↑(hA.eigenvalues i))
((⇑(⋯.eigenvectorBasis ⋯) ∘ ⇑(Fintype.equivOfCardEq ⋯).symm) i)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.IsHermitian.hasEigenvector_eigenvectorBasis
|
[30, 1]
|
[33, 62]
|
apply LinearMap.IsSymmetric.hasEigenvector_eigenvectorBasis
|
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_1
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.IsHermitian
i : n
⊢ Module.End.HasEigenvector (toLin' A) (↑(hA.eigenvalues i))
((⇑(⋯.eigenvectorBasis ⋯) ∘ ⇑(Fintype.equivOfCardEq ⋯).symm) i)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
rw [basis_toMatrix_basisFun_mul]
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
⊢ xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n) * A = diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
⊢ (of fun i j => (xs.toBasis.repr (A.transpose j)) i) =
diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
ext i j
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
⊢ (of fun i j => (xs.toBasis.repr (A.transpose j)) i) =
diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)
|
case a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
let xs' := xs.reindex (Fintype.equivOfCardEq (Fintype.card_fin _)).symm
|
case a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j
|
case a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
let as' : Fin (Fintype.card n) → ℝ :=
fun i => as <| (Fintype.equivOfCardEq (Fintype.card_fin _)) i
|
case a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j
|
case a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
have hxs' : ∀ j, Module.End.HasEigenvector (Matrix.toLin' A) (as' j) (xs' j) := by
simp only [xs', OrthonormalBasis.coe_reindex, Equiv.symm_symm]
intros j
exact (hxs ((Fintype.equivOfCardEq (Fintype.card_fin _)) j))
|
case a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j
|
case a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
convert @LinearMap.spectral_theorem' 𝕜 _
(PiLp 2 (fun (_ : n) => 𝕜)) _ _ (Fintype.card n) (Matrix.toLin' A)
(EuclideanSpace.single j 1)
((Fintype.equivOfCardEq (Fintype.card_fin _)).symm i)
xs' as' hxs'
|
case a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
xs'.repr ((toLin' A) (EuclideanSpace.single j 1)) ((Fintype.equivOfCardEq ⋯).symm i)
case h.e'_3
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ (diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j =
↑(as' ((Fintype.equivOfCardEq ⋯).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq ⋯).symm i)
|
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