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/Subadditivity.lean
|
Matrix.det_add_det_le_det_add'
|
[121, 1]
|
[168, 57]
|
rw [← det_mul, ← det_conj this (A + B)]
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ A.det * (1 + sqrtA⁻¹ * B * sqrtA⁻¹).det = (A + B).det
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ (A * (1 + sqrtA⁻¹ * B * sqrtA⁻¹)).det = (sqrtA * (A + B) * sqrtA⁻¹).det
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add'
|
[121, 1]
|
[168, 57]
|
apply congr_arg
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ (A * (1 + sqrtA⁻¹ * B * sqrtA⁻¹)).det = (sqrtA * (A + B) * sqrtA⁻¹).det
|
case h
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ A * (1 + sqrtA⁻¹ * B * sqrtA⁻¹) = sqrtA * (A + B) * sqrtA⁻¹
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add'
|
[121, 1]
|
[168, 57]
|
rw [← hA.posSemidef.sqrt_mul_sqrt]
|
case h
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ A * (1 + sqrtA⁻¹ * B * sqrtA⁻¹) = sqrtA * (A + B) * sqrtA⁻¹
|
case h
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ ⋯.sqrt * ⋯.sqrt * (1 + sqrtA⁻¹ * B * sqrtA⁻¹) = sqrtA * (⋯.sqrt * ⋯.sqrt + B) * sqrtA⁻¹
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add'
|
[121, 1]
|
[168, 57]
|
change sqrtA * sqrtA * (1 + sqrtA⁻¹ * B * sqrtA⁻¹) = sqrtA * (sqrtA * sqrtA + B) * sqrtA⁻¹
|
case h
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ ⋯.sqrt * ⋯.sqrt * (1 + sqrtA⁻¹ * B * sqrtA⁻¹) = sqrtA * (⋯.sqrt * ⋯.sqrt + B) * sqrtA⁻¹
|
case h
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ sqrtA * sqrtA * (1 + sqrtA⁻¹ * B * sqrtA⁻¹) = sqrtA * (sqrtA * sqrtA + B) * sqrtA⁻¹
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add'
|
[121, 1]
|
[168, 57]
|
rw [Matrix.mul_add, Matrix.mul_one, Matrix.mul_add, Matrix.add_mul,
Matrix.mul_assoc, Matrix.mul_assoc, Matrix.mul_assoc, Matrix.mul_assoc,
← Matrix.mul_assoc _ _ (B * _),
Matrix.mul_nonsing_inv _ isUnit_det_sqrtA, Matrix.one_mul, Matrix.mul_one,
hA.posSemidef.sqrt_mul_sqrt, Matrix.mul_assoc]
|
case h
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosDef
hB : B.PosSemidef
sqrtA : Matrix n n ℝ := ⋯.sqrt
isUnit_det_sqrtA : IsUnit ⋯.sqrt.det
this : IsUnit sqrtA
IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian
PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef
μ : n → ℝ := ⋯.eigenvalues
⊢ sqrtA * sqrtA * (1 + sqrtA⁻¹ * B * sqrtA⁻¹) = sqrtA * (sqrtA * sqrtA + B) * sqrtA⁻¹
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add
|
[170, 1]
|
[179, 77]
|
by_cases hA' : A.det = 0
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
⊢ A.det + B.det ≤ (A + B).det
|
case pos
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : A.det = 0
⊢ A.det + B.det ≤ (A + B).det
case neg
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : ¬A.det = 0
⊢ A.det + B.det ≤ (A + B).det
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add
|
[170, 1]
|
[179, 77]
|
{ by_cases hB' : B.det = 0
{ simp [hA', hB']
apply (hA.add hB).det_nonneg }
{ rw [add_comm A B, add_comm A.det B.det]
apply det_add_det_le_det_add' _ _ (hB.PosDef_iff_det_ne_zero.2 hB') hA } }
|
case pos
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : A.det = 0
⊢ A.det + B.det ≤ (A + B).det
case neg
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : ¬A.det = 0
⊢ A.det + B.det ≤ (A + B).det
|
case neg
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : ¬A.det = 0
⊢ A.det + B.det ≤ (A + B).det
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add
|
[170, 1]
|
[179, 77]
|
{ apply det_add_det_le_det_add' _ _ (hA.PosDef_iff_det_ne_zero.2 hA') hB }
|
case neg
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : ¬A.det = 0
⊢ A.det + B.det ≤ (A + B).det
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add
|
[170, 1]
|
[179, 77]
|
by_cases hB' : B.det = 0
|
case pos
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : A.det = 0
⊢ A.det + B.det ≤ (A + B).det
|
case pos
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : A.det = 0
hB' : B.det = 0
⊢ A.det + B.det ≤ (A + B).det
case neg
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : A.det = 0
hB' : ¬B.det = 0
⊢ A.det + B.det ≤ (A + B).det
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add
|
[170, 1]
|
[179, 77]
|
{ simp [hA', hB']
apply (hA.add hB).det_nonneg }
|
case pos
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : A.det = 0
hB' : B.det = 0
⊢ A.det + B.det ≤ (A + B).det
case neg
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : A.det = 0
hB' : ¬B.det = 0
⊢ A.det + B.det ≤ (A + B).det
|
case neg
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : A.det = 0
hB' : ¬B.det = 0
⊢ A.det + B.det ≤ (A + B).det
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add
|
[170, 1]
|
[179, 77]
|
{ rw [add_comm A B, add_comm A.det B.det]
apply det_add_det_le_det_add' _ _ (hB.PosDef_iff_det_ne_zero.2 hB') hA }
|
case neg
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : A.det = 0
hB' : ¬B.det = 0
⊢ A.det + B.det ≤ (A + B).det
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add
|
[170, 1]
|
[179, 77]
|
simp [hA', hB']
|
case pos
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : A.det = 0
hB' : B.det = 0
⊢ A.det + B.det ≤ (A + B).det
|
case pos
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : A.det = 0
hB' : B.det = 0
⊢ 0 ≤ (A + B).det
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add
|
[170, 1]
|
[179, 77]
|
apply (hA.add hB).det_nonneg
|
case pos
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : A.det = 0
hB' : B.det = 0
⊢ 0 ≤ (A + B).det
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add
|
[170, 1]
|
[179, 77]
|
rw [add_comm A B, add_comm A.det B.det]
|
case neg
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : A.det = 0
hB' : ¬B.det = 0
⊢ A.det + B.det ≤ (A + B).det
|
case neg
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : A.det = 0
hB' : ¬B.det = 0
⊢ B.det + A.det ≤ (B + A).det
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add
|
[170, 1]
|
[179, 77]
|
apply det_add_det_le_det_add' _ _ (hB.PosDef_iff_det_ne_zero.2 hB') hA
|
case neg
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : A.det = 0
hB' : ¬B.det = 0
⊢ B.det + A.det ≤ (B + A).det
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add
|
[170, 1]
|
[179, 77]
|
apply det_add_det_le_det_add' _ _ (hA.PosDef_iff_det_ne_zero.2 hA') hB
|
case neg
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : Nonempty n
A B : Matrix n n ℝ
hA : A.PosSemidef
hB : B.PosSemidef
hA' : ¬A.det = 0
⊢ A.det + B.det ≤ (A + B).det
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.abs_le_of_sqrt_sq_add_nonneg_le
|
[64, 1]
|
[69, 50]
|
rw [sqrt_le_iff] at h
|
a b c : ℝ
hb : 0 ≤ b
h : (a ^ 2 + b).sqrt ≤ c
⊢ |a| ≤ c
|
a b c : ℝ
hb : 0 ≤ b
h : 0 ≤ c ∧ a ^ 2 + b ≤ c ^ 2
⊢ |a| ≤ c
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.abs_le_of_sqrt_sq_add_nonneg_le
|
[64, 1]
|
[69, 50]
|
replace ⟨hc, h⟩ := h
|
a b c : ℝ
hb : 0 ≤ b
h : 0 ≤ c ∧ a ^ 2 + b ≤ c ^ 2
⊢ |a| ≤ c
|
a b c : ℝ
hb : 0 ≤ b
hc : 0 ≤ c
h : a ^ 2 + b ≤ c ^ 2
⊢ |a| ≤ c
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.abs_le_of_sqrt_sq_add_nonneg_le
|
[64, 1]
|
[69, 50]
|
replace h := le_trans (le_add_of_nonneg_right hb) h
|
a b c : ℝ
hb : 0 ≤ b
hc : 0 ≤ c
h : a ^ 2 + b ≤ c ^ 2
⊢ |a| ≤ c
|
a b c : ℝ
hb : 0 ≤ b
hc : 0 ≤ c
h : a ^ 2 ≤ c ^ 2
⊢ |a| ≤ c
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.abs_le_of_sqrt_sq_add_nonneg_le
|
[64, 1]
|
[69, 50]
|
rwa [rpow_two, sq_le_sq, abs_of_nonneg hc] at h
|
a b c : ℝ
hb : 0 ≤ b
hc : 0 ≤ c
h : a ^ 2 ≤ c ^ 2
⊢ |a| ≤ c
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.log_eq_log
|
[73, 1]
|
[83, 13]
|
rw [h]
|
x y : ℝ
hx : 0 < x
hy : 0 < y
h : x = y
⊢ x.log = y.log
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.log_eq_log
|
[73, 1]
|
[83, 13]
|
have hxmem := Set.mem_Ioi.2 hx
|
x y : ℝ
hx : 0 < x
hy : 0 < y
h : x.log = y.log
⊢ x = y
|
x y : ℝ
hx : 0 < x
hy : 0 < y
h : x.log = y.log
hxmem : x ∈ Set.Ioi 0
⊢ x = y
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.log_eq_log
|
[73, 1]
|
[83, 13]
|
have hymem := Set.mem_Ioi.2 hy
|
x y : ℝ
hx : 0 < x
hy : 0 < y
h : x.log = y.log
hxmem : x ∈ Set.Ioi 0
⊢ x = y
|
x y : ℝ
hx : 0 < x
hy : 0 < y
h : x.log = y.log
hxmem : x ∈ Set.Ioi 0
hymem : y ∈ Set.Ioi 0
⊢ x = y
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.log_eq_log
|
[73, 1]
|
[83, 13]
|
have heq : Set.restrict (Set.Ioi 0) log ⟨x, hxmem⟩ =
Set.restrict (Set.Ioi 0) log ⟨y, hymem⟩ := by
simp [h]
|
x y : ℝ
hx : 0 < x
hy : 0 < y
h : x.log = y.log
hxmem : x ∈ Set.Ioi 0
hymem : y ∈ Set.Ioi 0
⊢ x = y
|
x y : ℝ
hx : 0 < x
hy : 0 < y
h : x.log = y.log
hxmem : x ∈ Set.Ioi 0
hymem : y ∈ Set.Ioi 0
heq : (Set.Ioi 0).restrict log ⟨x, hxmem⟩ = (Set.Ioi 0).restrict log ⟨y, hymem⟩
⊢ x = y
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.log_eq_log
|
[73, 1]
|
[83, 13]
|
have h := log_injOn_pos.injective heq
|
x y : ℝ
hx : 0 < x
hy : 0 < y
h : x.log = y.log
hxmem : x ∈ Set.Ioi 0
hymem : y ∈ Set.Ioi 0
heq : (Set.Ioi 0).restrict log ⟨x, hxmem⟩ = (Set.Ioi 0).restrict log ⟨y, hymem⟩
⊢ x = y
|
x y : ℝ
hx : 0 < x
hy : 0 < y
h✝ : x.log = y.log
hxmem : x ∈ Set.Ioi 0
hymem : y ∈ Set.Ioi 0
heq : (Set.Ioi 0).restrict log ⟨x, hxmem⟩ = (Set.Ioi 0).restrict log ⟨y, hymem⟩
h : ⟨x, hxmem⟩ = ⟨y, hymem⟩
⊢ x = y
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.log_eq_log
|
[73, 1]
|
[83, 13]
|
simp [Subtype.eq] at h
|
x y : ℝ
hx : 0 < x
hy : 0 < y
h✝ : x.log = y.log
hxmem : x ∈ Set.Ioi 0
hymem : y ∈ Set.Ioi 0
heq : (Set.Ioi 0).restrict log ⟨x, hxmem⟩ = (Set.Ioi 0).restrict log ⟨y, hymem⟩
h : ⟨x, hxmem⟩ = ⟨y, hymem⟩
⊢ x = y
|
x y : ℝ
hx : 0 < x
hy : 0 < y
h✝ : x.log = y.log
hxmem : x ∈ Set.Ioi 0
hymem : y ∈ Set.Ioi 0
heq : (Set.Ioi 0).restrict log ⟨x, hxmem⟩ = (Set.Ioi 0).restrict log ⟨y, hymem⟩
h : x = y
⊢ x = y
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.log_eq_log
|
[73, 1]
|
[83, 13]
|
exact h
|
x y : ℝ
hx : 0 < x
hy : 0 < y
h✝ : x.log = y.log
hxmem : x ∈ Set.Ioi 0
hymem : y ∈ Set.Ioi 0
heq : (Set.Ioi 0).restrict log ⟨x, hxmem⟩ = (Set.Ioi 0).restrict log ⟨y, hymem⟩
h : x = y
⊢ x = y
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.log_eq_log
|
[73, 1]
|
[83, 13]
|
simp [h]
|
x y : ℝ
hx : 0 < x
hy : 0 < y
h : x.log = y.log
hxmem : x ∈ Set.Ioi 0
hymem : y ∈ Set.Ioi 0
⊢ (Set.Ioi 0).restrict log ⟨x, hxmem⟩ = (Set.Ioi 0).restrict log ⟨y, hymem⟩
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.div_pow_eq_mul_pow_neg
|
[85, 1]
|
[87, 37]
|
rw [div_eq_mul_inv, ← rpow_neg hb]
|
a b c : ℝ
hb : 0 ≤ b
⊢ a / b ^ c = a * b ^ (-c)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.one_div_eq_pow_neg_one
|
[89, 1]
|
[90, 65]
|
rw [rpow_neg (le_of_lt ha), rpow_one, div_eq_mul_inv, one_mul]
|
a : ℝ
ha : 0 < a
⊢ 1 / a = a ^ (-1)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.inv_eq_pow_neg_one
|
[92, 1]
|
[93, 49]
|
rw [inv_eq_one_div, one_div_eq_pow_neg_one ha]
|
a : ℝ
ha : 0 < a
⊢ a⁻¹ = a ^ (-1)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.pow_half_two
|
[95, 1]
|
[98, 11]
|
show rpow (rpow _ _) _ = _
|
x : ℝ
hx : 0 ≤ x
⊢ (x ^ (1 / 2)) ^ 2 = x
|
x : ℝ
hx : 0 ≤ x
⊢ (x.rpow (1 / 2)).rpow 2 = x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.pow_half_two
|
[95, 1]
|
[98, 11]
|
rw [rpow_eq_pow, rpow_eq_pow, ← rpow_mul hx]
|
x : ℝ
hx : 0 ≤ x
⊢ (x.rpow (1 / 2)).rpow 2 = x
|
x : ℝ
hx : 0 ≤ x
⊢ x ^ (1 / 2 * 2) = x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.pow_half_two
|
[95, 1]
|
[98, 11]
|
norm_num
|
x : ℝ
hx : 0 ≤ x
⊢ x ^ (1 / 2 * 2) = x
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.pow_two_le_pow_two
|
[100, 1]
|
[102, 72]
|
rw [rpow_two, rpow_two, sq_le_sq, abs_of_nonneg hx, abs_of_nonneg hy]
|
x y : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
⊢ x ^ 2 ≤ y ^ 2 ↔ x ≤ y
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.binomial_two
|
[104, 1]
|
[106, 29]
|
ring
|
x y : ℝ
⊢ (x + y) ^ 2 = x ^ 2 + (2 * (x * y) + y ^ 2)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.rpow_eq_mul_rpow_pred
|
[108, 1]
|
[110, 76]
|
conv => left; rw [(by ring : y = (y - 1) + 1), rpow_add_one hx, mul_comm]
|
x y : ℝ
hx : x ≠ 0
⊢ x ^ y = x * x ^ (y - 1)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.rpow_eq_mul_rpow_pred
|
[108, 1]
|
[110, 76]
|
ring
|
x y : ℝ
hx : x ≠ 0
⊢ y = y - 1 + 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Data/Real.lean
|
Real.exp_neg_eq_one_div
|
[112, 1]
|
[113, 31]
|
rw [exp_neg, inv_eq_one_div]
|
x : ℝ
⊢ (-x).exp = 1 / x.exp
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Relaxation.lean
|
Minimization.Relaxation.induces_original_problem_optimality
|
[57, 1]
|
[66, 41]
|
refine ⟨h_feas_y, ?_⟩
|
D E F R : Type
inst✝ : Preorder R
p : Minimization D R
q : Minimization E R
r : Minimization F R
Rx : p.Relaxation q
phi_inv : E → D
phi_left_inv : Function.LeftInverse Rx.phi phi_inv
h_objFun : ∀ (x : D), p.feasible x → p.objFun x = q.objFun (Rx.phi x)
y : E
h_opt_y : q.optimal y
h_feas_y : p.feasible (phi_inv y)
⊢ p.optimal (phi_inv y)
|
D E F R : Type
inst✝ : Preorder R
p : Minimization D R
q : Minimization E R
r : Minimization F R
Rx : p.Relaxation q
phi_inv : E → D
phi_left_inv : Function.LeftInverse Rx.phi phi_inv
h_objFun : ∀ (x : D), p.feasible x → p.objFun x = q.objFun (Rx.phi x)
y : E
h_opt_y : q.optimal y
h_feas_y : p.feasible (phi_inv y)
⊢ ∀ (y_1 : D), p.feasible y_1 → p.objFun (phi_inv y) ≤ p.objFun y_1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Relaxation.lean
|
Minimization.Relaxation.induces_original_problem_optimality
|
[57, 1]
|
[66, 41]
|
intros x h_feas_x
|
D E F R : Type
inst✝ : Preorder R
p : Minimization D R
q : Minimization E R
r : Minimization F R
Rx : p.Relaxation q
phi_inv : E → D
phi_left_inv : Function.LeftInverse Rx.phi phi_inv
h_objFun : ∀ (x : D), p.feasible x → p.objFun x = q.objFun (Rx.phi x)
y : E
h_opt_y : q.optimal y
h_feas_y : p.feasible (phi_inv y)
⊢ ∀ (y_1 : D), p.feasible y_1 → p.objFun (phi_inv y) ≤ p.objFun y_1
|
D E F R : Type
inst✝ : Preorder R
p : Minimization D R
q : Minimization E R
r : Minimization F R
Rx : p.Relaxation q
phi_inv : E → D
phi_left_inv : Function.LeftInverse Rx.phi phi_inv
h_objFun : ∀ (x : D), p.feasible x → p.objFun x = q.objFun (Rx.phi x)
y : E
h_opt_y : q.optimal y
h_feas_y : p.feasible (phi_inv y)
x : D
h_feas_x : p.feasible x
⊢ p.objFun (phi_inv y) ≤ p.objFun x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Relaxation.lean
|
Minimization.Relaxation.induces_original_problem_optimality
|
[57, 1]
|
[66, 41]
|
rw [h_objFun _ h_feas_y, phi_left_inv]
|
D E F R : Type
inst✝ : Preorder R
p : Minimization D R
q : Minimization E R
r : Minimization F R
Rx : p.Relaxation q
phi_inv : E → D
phi_left_inv : Function.LeftInverse Rx.phi phi_inv
h_objFun : ∀ (x : D), p.feasible x → p.objFun x = q.objFun (Rx.phi x)
y : E
h_opt_y : q.optimal y
h_feas_y : p.feasible (phi_inv y)
x : D
h_feas_x : p.feasible x
⊢ p.objFun (phi_inv y) ≤ p.objFun x
|
D E F R : Type
inst✝ : Preorder R
p : Minimization D R
q : Minimization E R
r : Minimization F R
Rx : p.Relaxation q
phi_inv : E → D
phi_left_inv : Function.LeftInverse Rx.phi phi_inv
h_objFun : ∀ (x : D), p.feasible x → p.objFun x = q.objFun (Rx.phi x)
y : E
h_opt_y : q.optimal y
h_feas_y : p.feasible (phi_inv y)
x : D
h_feas_x : p.feasible x
⊢ q.objFun y ≤ p.objFun x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Relaxation.lean
|
Minimization.Relaxation.induces_original_problem_optimality
|
[57, 1]
|
[66, 41]
|
have h_bound := h_opt_y.2 (Rx.phi x) (Rx.phi_feasibility x h_feas_x)
|
D E F R : Type
inst✝ : Preorder R
p : Minimization D R
q : Minimization E R
r : Minimization F R
Rx : p.Relaxation q
phi_inv : E → D
phi_left_inv : Function.LeftInverse Rx.phi phi_inv
h_objFun : ∀ (x : D), p.feasible x → p.objFun x = q.objFun (Rx.phi x)
y : E
h_opt_y : q.optimal y
h_feas_y : p.feasible (phi_inv y)
x : D
h_feas_x : p.feasible x
⊢ q.objFun y ≤ p.objFun x
|
D E F R : Type
inst✝ : Preorder R
p : Minimization D R
q : Minimization E R
r : Minimization F R
Rx : p.Relaxation q
phi_inv : E → D
phi_left_inv : Function.LeftInverse Rx.phi phi_inv
h_objFun : ∀ (x : D), p.feasible x → p.objFun x = q.objFun (Rx.phi x)
y : E
h_opt_y : q.optimal y
h_feas_y : p.feasible (phi_inv y)
x : D
h_feas_x : p.feasible x
h_bound : q.objFun y ≤ q.objFun (Rx.phi x)
⊢ q.objFun y ≤ p.objFun x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Relaxation.lean
|
Minimization.Relaxation.induces_original_problem_optimality
|
[57, 1]
|
[66, 41]
|
rwa [← h_objFun _ h_feas_x] at h_bound
|
D E F R : Type
inst✝ : Preorder R
p : Minimization D R
q : Minimization E R
r : Minimization F R
Rx : p.Relaxation q
phi_inv : E → D
phi_left_inv : Function.LeftInverse Rx.phi phi_inv
h_objFun : ∀ (x : D), p.feasible x → p.objFun x = q.objFun (Rx.phi x)
y : E
h_opt_y : q.optimal y
h_feas_y : p.feasible (phi_inv y)
x : D
h_feas_x : p.feasible x
h_bound : q.objFun y ≤ q.objFun (Rx.phi x)
⊢ q.objFun y ≤ p.objFun x
|
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
|
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