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
|
Finset.one_add_prod_le_prod_one_add
|
[19, 1]
|
[32, 37]
|
convert @Finset.sum_le_univ_sum_of_nonneg (Finset n) ℝ _ _ _ _ _
|
n : Type u_1
inst✝¹ : Fintype n
inst✝ : Nonempty n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ ∑ x ∈ {univ, ∅}, (∏ _a ∈ x, 1) * ∏ a ∈ univ \ x, f a ≤ ∑ t : Finset n, (∏ _a ∈ t, 1) * ∏ a ∈ univ \ t, f a
|
case convert_3
n : Type u_1
inst✝¹ : Fintype n
inst✝ : Nonempty n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ ∀ (x : Finset n), 0 ≤ (∏ _a ∈ x, 1) * ∏ a ∈ univ \ x, f a
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Finset.one_add_prod_le_prod_one_add
|
[19, 1]
|
[32, 37]
|
simp [hf, prod_nonneg]
|
case convert_3
n : Type u_1
inst✝¹ : Fintype n
inst✝ : Nonempty n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ ∀ (x : Finset n), 0 ≤ (∏ _a ∈ x, 1) * ∏ a ∈ univ \ x, f a
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Finset.one_add_prod_le_prod_one_add
|
[19, 1]
|
[32, 37]
|
rw [prod_add, powerset_univ]
|
n : Type u_1
inst✝¹ : Fintype n
inst✝ : Nonempty n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ ∑ t : Finset n, (∏ _a ∈ t, 1) * ∏ a ∈ univ \ t, f a = ∏ i : n, (1 + f i)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.IsHermitian.eigenvectorMatrix_inv_mul
|
[46, 1]
|
[47, 41]
|
apply Basis.toMatrix_mul_toMatrix_flip
|
n : Type u_1
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq n
inst✝³ : LinearOrder n
inst✝² : LocallyFiniteOrderBot n
𝕜 : Type u_2
inst✝¹ : DecidableEq 𝕜
inst✝ : RCLike 𝕜
A : Matrix n n 𝕜
hA : A.IsHermitian
⊢ hA.eigenvectorMatrixInv * hA.eigenvectorMatrix = 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.IsHermitian.spectral_theorem''
|
[50, 1]
|
[54, 67]
|
rw [conjTranspose_eigenvectorMatrix, Matrix.mul_assoc, ← spectral_theorem,
← Matrix.mul_assoc, eigenvectorMatrix_mul_inv, Matrix.one_mul]
|
n : Type u_1
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq n
inst✝³ : LinearOrder n
inst✝² : LocallyFiniteOrderBot n
𝕜 : Type u_2
inst✝¹ : DecidableEq 𝕜
inst✝ : RCLike 𝕜
A : Matrix n n 𝕜
hA : A.IsHermitian
⊢ hA.eigenvectorMatrix * diagonal (RCLike.ofReal ∘ hA.eigenvalues) * hA.eigenvectorMatrix.conjTranspose = A
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosSemidef.sqrt_mul_sqrt
|
[63, 1]
|
[78, 11]
|
simp [IsHermitian.sqrt, Matrix.mul_assoc]
|
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosSemidef
⊢ ⋯.sqrt * ⋯.sqrt =
⋯.eigenvectorMatrix *
((diagonal fun i => (⋯.eigenvalues i).sqrt) * (⋯.eigenvectorMatrix.transpose * ⋯.eigenvectorMatrix) *
diagonal fun i => (⋯.eigenvalues i).sqrt) *
⋯.eigenvectorMatrix.transpose
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosSemidef.sqrt_mul_sqrt
|
[63, 1]
|
[78, 11]
|
rw [← conjTranspose_eq_transpose, hA.1.conjTranspose_eigenvectorMatrix,
hA.1.eigenvectorMatrix_inv_mul, Matrix.mul_one, diagonal_mul_diagonal,
← hA.1.conjTranspose_eigenvectorMatrix]
|
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosSemidef
⊢ ⋯.eigenvectorMatrix *
((diagonal fun i => (⋯.eigenvalues i).sqrt) * (⋯.eigenvectorMatrix.transpose * ⋯.eigenvectorMatrix) *
diagonal fun i => (⋯.eigenvalues i).sqrt) *
⋯.eigenvectorMatrix.transpose =
A
|
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosSemidef
⊢ (⋯.eigenvectorMatrix * diagonal fun i => (⋯.eigenvalues i).sqrt * (⋯.eigenvalues i).sqrt) *
⋯.eigenvectorMatrix.conjTranspose =
A
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosSemidef.sqrt_mul_sqrt
|
[63, 1]
|
[78, 11]
|
convert hA.1.spectral_theorem''
|
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosSemidef
⊢ (⋯.eigenvectorMatrix * diagonal fun i => (⋯.eigenvalues i).sqrt * (⋯.eigenvalues i).sqrt) *
⋯.eigenvectorMatrix.conjTranspose =
A
|
case h.e'_2.h.e'_5.h.e'_6.h.e'_5.h
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosSemidef
x✝ : n
⊢ (⋯.eigenvalues x✝).sqrt * (⋯.eigenvalues x✝).sqrt = (RCLike.ofReal ∘ ⋯.eigenvalues) x✝
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosSemidef.sqrt_mul_sqrt
|
[63, 1]
|
[78, 11]
|
rw [← Real.sqrt_mul (hA.eigenvalues_nonneg _), Real.sqrt_mul_self (hA.eigenvalues_nonneg _)]
|
case h.e'_2.h.e'_5.h.e'_6.h.e'_5.h
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosSemidef
x✝ : n
⊢ (⋯.eigenvalues x✝).sqrt * (⋯.eigenvalues x✝).sqrt = (RCLike.ofReal ∘ ⋯.eigenvalues) x✝
|
case h.e'_2.h.e'_5.h.e'_6.h.e'_5.h
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosSemidef
x✝ : n
⊢ ⋯.eigenvalues x✝ = (RCLike.ofReal ∘ ⋯.eigenvalues) x✝
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosSemidef.sqrt_mul_sqrt
|
[63, 1]
|
[78, 11]
|
simp
|
case h.e'_2.h.e'_5.h.e'_6.h.e'_5.h
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosSemidef
x✝ : n
⊢ ⋯.eigenvalues x✝ = (RCLike.ofReal ∘ ⋯.eigenvalues) x✝
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.IsHermitian.one_add
|
[85, 1]
|
[86, 55]
|
dsimp [IsHermitian]
|
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.IsHermitian
⊢ (1 + A).IsHermitian
|
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.IsHermitian
⊢ (1 + A).conjTranspose = 1 + A
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.IsHermitian.one_add
|
[85, 1]
|
[86, 55]
|
rw [IsHermitian.add _ hA]
|
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.IsHermitian
⊢ (1 + A).conjTranspose = 1 + A
|
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.IsHermitian
⊢ IsHermitian 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.IsHermitian.one_add
|
[85, 1]
|
[86, 55]
|
simp
|
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.IsHermitian
⊢ IsHermitian 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosDef.PosDef_sqrt
|
[95, 1]
|
[102, 68]
|
unfold IsHermitian.sqrt
|
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosDef
⊢ ⋯.sqrt.PosDef
|
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosDef
⊢ ((⋯.eigenvectorMatrix * diagonal fun i => (⋯.eigenvalues i).sqrt) * ⋯.eigenvectorMatrix.transpose).PosDef
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosDef.PosDef_sqrt
|
[95, 1]
|
[102, 68]
|
refine'
PosDef.conjTranspose_mul_mul _ (hA.1.eigenvectorMatrixᵀ)
(PosDef_diagonal (fun i => Real.sqrt_pos.2 (hA.eigenvalues_pos i))) _
|
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosDef
⊢ ((⋯.eigenvectorMatrix * diagonal fun i => (⋯.eigenvalues i).sqrt) * ⋯.eigenvectorMatrix.transpose).PosDef
|
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosDef
⊢ ⋯.eigenvectorMatrix.transpose.det ≠ 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosDef.PosDef_sqrt
|
[95, 1]
|
[102, 68]
|
rw [det_transpose]
|
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosDef
⊢ ⋯.eigenvectorMatrix.transpose.det ≠ 0
|
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosDef
⊢ ⋯.eigenvectorMatrix.det ≠ 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosDef.PosDef_sqrt
|
[95, 1]
|
[102, 68]
|
apply det_ne_zero_of_right_inverse hA.1.eigenvectorMatrix_mul_inv
|
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : LinearOrder n
inst✝ : LocallyFiniteOrderBot n
A : Matrix n n ℝ
hA : A.PosDef
⊢ ⋯.eigenvectorMatrix.det ≠ 0
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosSemidef.PosDef_iff_det_ne_zero
|
[104, 1]
|
[119, 17]
|
refine' ⟨PosDef.det_ne_zero, _⟩
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
⊢ M.PosDef ↔ M.det ≠ 0
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
⊢ M.det ≠ 0 → M.PosDef
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosSemidef.PosDef_iff_det_ne_zero
|
[104, 1]
|
[119, 17]
|
intro hdet
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
⊢ M.det ≠ 0 → M.PosDef
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
⊢ M.PosDef
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosSemidef.PosDef_iff_det_ne_zero
|
[104, 1]
|
[119, 17]
|
refine' ⟨hM.1, _⟩
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
⊢ M.PosDef
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
⊢ ∀ (x : n → ℝ), x ≠ 0 → 0 < star x ⬝ᵥ M.mulVec x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosSemidef.PosDef_iff_det_ne_zero
|
[104, 1]
|
[119, 17]
|
intros x hx
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
⊢ ∀ (x : n → ℝ), x ≠ 0 → 0 < star x ⬝ᵥ M.mulVec x
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ M.mulVec x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosSemidef.PosDef_iff_det_ne_zero
|
[104, 1]
|
[119, 17]
|
apply lt_of_le_of_ne' (hM.2 x)
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ M.mulVec x
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ star x ⬝ᵥ M.mulVec x ≠ 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosSemidef.PosDef_iff_det_ne_zero
|
[104, 1]
|
[119, 17]
|
rw [← hM.sqrt_mul_sqrt, ← mulVec_mulVec, dotProduct_mulVec, ← transpose_transpose hM.1.sqrt,
vecMul_transpose, transpose_transpose, ← conjTranspose_eq_transpose,
hM.PosSemidef_sqrt.1.eq]
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ star x ⬝ᵥ M.mulVec x ≠ 0
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ ⋯.sqrt.mulVec (star x) ⬝ᵥ ⋯.sqrt.mulVec x ≠ 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosSemidef.PosDef_iff_det_ne_zero
|
[104, 1]
|
[119, 17]
|
simp only [RCLike.re_to_real, star, id]
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ ⋯.sqrt.mulVec (star x) ⬝ᵥ ⋯.sqrt.mulVec x ≠ 0
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ (⋯.sqrt.mulVec fun i => x i) ⬝ᵥ ⋯.sqrt.mulVec x ≠ 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosSemidef.PosDef_iff_det_ne_zero
|
[104, 1]
|
[119, 17]
|
change @inner ℝ (EuclideanSpace ℝ _) _ (hM.1.sqrt.mulVec x) (hM.1.sqrt.mulVec x) ≠ 0
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ (⋯.sqrt.mulVec fun i => x i) ⬝ᵥ ⋯.sqrt.mulVec x ≠ 0
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ ≠ 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosSemidef.PosDef_iff_det_ne_zero
|
[104, 1]
|
[119, 17]
|
intro hinner
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
⊢ ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ ≠ 0
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
hinner : ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ = 0
⊢ False
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosSemidef.PosDef_iff_det_ne_zero
|
[104, 1]
|
[119, 17]
|
have sqrtMdet0 : hM.1.sqrt.det = 0 := by
refine' exists_mulVec_eq_zero_iff.1 ⟨x, hx, _⟩
rw [inner_self_eq_zero.1 hinner]
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
hinner : ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ = 0
⊢ False
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
hinner : ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ = 0
sqrtMdet0 : ⋯.sqrt.det = 0
⊢ False
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosSemidef.PosDef_iff_det_ne_zero
|
[104, 1]
|
[119, 17]
|
rw [← hM.sqrt_mul_sqrt, det_mul, sqrtMdet0, mul_zero] at hdet
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
hinner : ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ = 0
sqrtMdet0 : ⋯.sqrt.det = 0
⊢ False
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : 0 ≠ 0
x : n → ℝ
hx : x ≠ 0
hinner : ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ = 0
sqrtMdet0 : ⋯.sqrt.det = 0
⊢ False
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosSemidef.PosDef_iff_det_ne_zero
|
[104, 1]
|
[119, 17]
|
apply hdet rfl
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : 0 ≠ 0
x : n → ℝ
hx : x ≠ 0
hinner : ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ = 0
sqrtMdet0 : ⋯.sqrt.det = 0
⊢ False
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosSemidef.PosDef_iff_det_ne_zero
|
[104, 1]
|
[119, 17]
|
refine' exists_mulVec_eq_zero_iff.1 ⟨x, hx, _⟩
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
hinner : ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ = 0
⊢ ⋯.sqrt.det = 0
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
hinner : ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ = 0
⊢ ⋯.sqrt.mulVec x = 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.PosSemidef.PosDef_iff_det_ne_zero
|
[104, 1]
|
[119, 17]
|
rw [inner_self_eq_zero.1 hinner]
|
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosSemidef
hdet : M.det ≠ 0
x : n → ℝ
hx : x ≠ 0
hinner : ⟪⋯.sqrt.mulVec x, ⋯.sqrt.mulVec x⟫_ℝ = 0
⊢ ⋯.sqrt.mulVec x = 0
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add'
|
[121, 1]
|
[168, 57]
|
let sqrtA := hA.1.sqrt
|
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
⊢ A.det + B.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
⊢ 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'
|
[121, 1]
|
[168, 57]
|
have isUnit_det_sqrtA :=
isUnit_iff_ne_zero.2 hA.PosDef_sqrt.det_ne_zero
|
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
⊢ A.det + B.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
⊢ 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'
|
[121, 1]
|
[168, 57]
|
have : IsUnit sqrtA :=
(isUnit_iff_isUnit_det _).2 isUnit_det_sqrtA
|
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
⊢ A.det + B.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
⊢ 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'
|
[121, 1]
|
[168, 57]
|
have IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian := by
{ apply IsHermitian.nonsingular_inv (hA.posSemidef.PosSemidef_sqrt.1)
exact isUnit_det_sqrtA }
|
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
⊢ A.det + B.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
⊢ 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'
|
[121, 1]
|
[168, 57]
|
have PosSemidef_ABA : (sqrtA⁻¹ * B * sqrtA⁻¹).PosSemidef :=
PosSemidef.mul_mul_of_IsHermitian hB IsHermitian_sqrtA
|
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
⊢ A.det + B.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
⊢ 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'
|
[121, 1]
|
[168, 57]
|
let μ := PosSemidef_ABA.1.eigenvalues
|
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
⊢ A.det + B.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.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'
|
[121, 1]
|
[168, 57]
|
calc A.det + B.det
= A.det * (1 + (sqrtA⁻¹ * B * sqrtA⁻¹).det) := by
rw [det_mul, det_mul, mul_comm _ B.det, mul_assoc, ← det_mul, ← Matrix.mul_inv_rev,
hA.posSemidef.sqrt_mul_sqrt, mul_add, mul_one, mul_comm, mul_assoc, ← det_mul,
nonsing_inv_mul _ (isUnit_iff_ne_zero.2 hA.det_ne_zero), det_one, mul_one]
_ = A.det * (1 + ∏ i, μ i) := by
rw [PosSemidef_ABA.1.det_eq_prod_eigenvalues]
rfl
_ ≤ A.det * ∏ i, (1 + μ i) := by
apply (mul_le_mul_left hA.det_pos).2
apply Finset.one_add_prod_le_prod_one_add μ PosSemidef_ABA.eigenvalues_nonneg
_ = A.det * (1 + sqrtA⁻¹ * B * sqrtA⁻¹).det := by
rw [mul_eq_mul_left_iff]; left; symm
rw [det_eq_prod_eigenvalues PosSemidef_ABA.1.eigenvectorBasis
(fun i => 1 + (PosSemidef_ABA.1.eigenvalues i)) _]
{ simp }
intro i
convert PosSemidef_ABA.1.has_eigenvector_one_add i using 1
simp only [map_add, toLin'_one, toLin'_mul, add_left_inj]
rfl
_ = (A + B).det := by
rw [← det_mul, ← det_conj this (A + B)]
apply congr_arg
rw [← hA.posSemidef.sqrt_mul_sqrt]
change sqrtA * sqrtA * (1 + sqrtA⁻¹ * B * sqrtA⁻¹) = sqrtA * (sqrtA * sqrtA + B) * sqrtA⁻¹
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]
|
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 + 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'
|
[121, 1]
|
[168, 57]
|
apply IsHermitian.nonsingular_inv (hA.posSemidef.PosSemidef_sqrt.1)
|
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
⊢ sqrtA⁻¹.IsHermitian
|
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
⊢ IsUnit ⋯.sqrt.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]
|
exact isUnit_det_sqrtA
|
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
⊢ IsUnit ⋯.sqrt.det
|
no goals
|
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_mul, mul_comm _ B.det, mul_assoc, ← det_mul, ← Matrix.mul_inv_rev,
hA.posSemidef.sqrt_mul_sqrt, mul_add, mul_one, mul_comm, mul_assoc, ← det_mul,
nonsing_inv_mul _ (isUnit_iff_ne_zero.2 hA.det_ne_zero), det_one, mul_one]
|
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 + B.det = A.det * (1 + (sqrtA⁻¹ * B * sqrtA⁻¹).det)
|
no goals
|
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 [PosSemidef_ABA.1.det_eq_prod_eigenvalues]
|
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.det * (1 + ∏ i : n, μ i)
|
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 + ∏ i : n, ↑(⋯.eigenvalues i)) = A.det * (1 + ∏ i : n, μ i)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add'
|
[121, 1]
|
[168, 57]
|
rfl
|
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 + ∏ i : n, ↑(⋯.eigenvalues i)) = A.det * (1 + ∏ i : n, μ i)
|
no goals
|
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 (mul_le_mul_left hA.det_pos).2
|
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 + ∏ i : n, μ i) ≤ A.det * ∏ i : n, (1 + μ i)
|
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
⊢ 1 + ∏ i : n, μ i ≤ ∏ i : n, (1 + μ i)
|
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 Finset.one_add_prod_le_prod_one_add μ PosSemidef_ABA.eigenvalues_nonneg
|
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
⊢ 1 + ∏ i : n, μ i ≤ ∏ i : n, (1 + μ i)
|
no goals
|
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 [mul_eq_mul_left_iff]
|
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 * ∏ i : n, (1 + μ i) = A.det * (1 + sqrtA⁻¹ * B * sqrtA⁻¹).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
⊢ ∏ i : n, (1 + μ i) = (1 + sqrtA⁻¹ * B * sqrtA⁻¹).det ∨ A.det = 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add'
|
[121, 1]
|
[168, 57]
|
left
|
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
⊢ ∏ i : n, (1 + μ i) = (1 + sqrtA⁻¹ * B * sqrtA⁻¹).det ∨ A.det = 0
|
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
⊢ ∏ i : n, (1 + μ i) = (1 + sqrtA⁻¹ * 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]
|
symm
|
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
⊢ ∏ i : n, (1 + μ i) = (1 + sqrtA⁻¹ * 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
⊢ (1 + sqrtA⁻¹ * B * sqrtA⁻¹).det = ∏ i : n, (1 + μ i)
|
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_eq_prod_eigenvalues PosSemidef_ABA.1.eigenvectorBasis
(fun i => 1 + (PosSemidef_ABA.1.eigenvalues i)) _]
|
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
⊢ (1 + sqrtA⁻¹ * B * sqrtA⁻¹).det = ∏ i : n, (1 + μ i)
|
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
⊢ ↑(∏ i : n, (1 + ⋯.eigenvalues i)) = ∏ i : n, (1 + μ i)
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
⊢ ∀ (j : n),
Module.End.HasEigenvector (toLin' (1 + sqrtA⁻¹ * B * sqrtA⁻¹)) (↑((fun i => 1 + ⋯.eigenvalues i) j))
(⋯.eigenvectorBasis j)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add'
|
[121, 1]
|
[168, 57]
|
{ simp }
|
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
⊢ ↑(∏ i : n, (1 + ⋯.eigenvalues i)) = ∏ i : n, (1 + μ i)
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
⊢ ∀ (j : n),
Module.End.HasEigenvector (toLin' (1 + sqrtA⁻¹ * B * sqrtA⁻¹)) (↑((fun i => 1 + ⋯.eigenvalues i) j))
(⋯.eigenvectorBasis j)
|
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
⊢ ∀ (j : n),
Module.End.HasEigenvector (toLin' (1 + sqrtA⁻¹ * B * sqrtA⁻¹)) (↑((fun i => 1 + ⋯.eigenvalues i) j))
(⋯.eigenvectorBasis j)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add'
|
[121, 1]
|
[168, 57]
|
intro i
|
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
⊢ ∀ (j : n),
Module.End.HasEigenvector (toLin' (1 + sqrtA⁻¹ * B * sqrtA⁻¹)) (↑((fun i => 1 + ⋯.eigenvalues i) j))
(⋯.eigenvectorBasis j)
|
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
i : n
⊢ Module.End.HasEigenvector (toLin' (1 + sqrtA⁻¹ * B * sqrtA⁻¹)) (↑((fun i => 1 + ⋯.eigenvalues i) i))
(⋯.eigenvectorBasis i)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add'
|
[121, 1]
|
[168, 57]
|
convert PosSemidef_ABA.1.has_eigenvector_one_add i using 1
|
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
i : n
⊢ Module.End.HasEigenvector (toLin' (1 + sqrtA⁻¹ * B * sqrtA⁻¹)) (↑((fun i => 1 + ⋯.eigenvalues i) i))
(⋯.eigenvectorBasis i)
|
case h.e'_6.h.h.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
i : n
e_2✝ : (n → ℝ) = EuclideanSpace ℝ n
e_3✝ : EuclideanDomain.toCommRing = Real.commRing
he✝¹ : Pi.addCommGroup = WithLp.instAddCommGroup 2 ((i : n) → (fun x => ℝ) i)
he✝ : Pi.Function.module n ℝ ℝ = WithLp.instModule 2 ℝ ((i : n) → (fun x => ℝ) i)
⊢ toLin' (1 + sqrtA⁻¹ * B * sqrtA⁻¹) = 1 + toLin' (sqrtA⁻¹ * 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]
|
simp only [map_add, toLin'_one, toLin'_mul, add_left_inj]
|
case h.e'_6.h.h.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
i : n
e_2✝ : (n → ℝ) = EuclideanSpace ℝ n
e_3✝ : EuclideanDomain.toCommRing = Real.commRing
he✝¹ : Pi.addCommGroup = WithLp.instAddCommGroup 2 ((i : n) → (fun x => ℝ) i)
he✝ : Pi.Function.module n ℝ ℝ = WithLp.instModule 2 ℝ ((i : n) → (fun x => ℝ) i)
⊢ toLin' (1 + sqrtA⁻¹ * B * sqrtA⁻¹) = 1 + toLin' (sqrtA⁻¹ * B * sqrtA⁻¹)
|
case h.e'_6.h.h.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
i : n
e_2✝ : (n → ℝ) = EuclideanSpace ℝ n
e_3✝ : EuclideanDomain.toCommRing = Real.commRing
he✝¹ : Pi.addCommGroup = WithLp.instAddCommGroup 2 ((i : n) → (fun x => ℝ) i)
he✝ : Pi.Function.module n ℝ ℝ = WithLp.instModule 2 ℝ ((i : n) → (fun x => ℝ) i)
⊢ LinearMap.id = 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]
|
rfl
|
case h.e'_6.h.h.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
i : n
e_2✝ : (n → ℝ) = EuclideanSpace ℝ n
e_3✝ : EuclideanDomain.toCommRing = Real.commRing
he✝¹ : Pi.addCommGroup = WithLp.instAddCommGroup 2 ((i : n) → (fun x => ℝ) i)
he✝ : Pi.Function.module n ℝ ℝ = WithLp.instModule 2 ℝ ((i : n) → (fun x => ℝ) i)
⊢ LinearMap.id = 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Subadditivity.lean
|
Matrix.det_add_det_le_det_add'
|
[121, 1]
|
[168, 57]
|
simp
|
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
⊢ ↑(∏ i : n, (1 + ⋯.eigenvalues i)) = ∏ i : n, (1 + μ i)
|
no goals
|
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/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/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
unfold expCone
|
t x : ℝ
⊢ x.exp ≤ t ↔ x.expCone 1 t
|
t x : ℝ
⊢ x.exp ≤ t ↔ 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
rw [iff_def]
|
t x : ℝ
⊢ x.exp ≤ t ↔ 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
|
t x : ℝ
⊢ (x.exp ≤ t → 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0) ∧
(0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0 → x.exp ≤ t)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
split_ands
|
t x : ℝ
⊢ (x.exp ≤ t → 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0) ∧
(0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0 → x.exp ≤ t)
|
case refine_1
t x : ℝ
⊢ x.exp ≤ t → 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
case refine_2
t x : ℝ
⊢ 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0 → x.exp ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
{ intro hexp
apply Or.intro_left
split_ands
{ apply Real.zero_lt_one }
{ rwa [div_one, one_mul] } }
|
case refine_1
t x : ℝ
⊢ x.exp ≤ t → 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
case refine_2
t x : ℝ
⊢ 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0 → x.exp ≤ t
|
case refine_2
t x : ℝ
⊢ 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0 → x.exp ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
{ intro h
cases h with
| inl h =>
have h : 1 * exp (x / 1) ≤ t := h.2
rwa [div_one, one_mul] at h
| inr h =>
exfalso
exact zero_ne_one h.1.symm }
|
case refine_2
t x : ℝ
⊢ 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0 → x.exp ≤ t
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
intro hexp
|
case refine_1
t x : ℝ
⊢ x.exp ≤ t → 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
|
case refine_1
t x : ℝ
hexp : x.exp ≤ t
⊢ 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
apply Or.intro_left
|
case refine_1
t x : ℝ
hexp : x.exp ≤ t
⊢ 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
|
case refine_1.h
t x : ℝ
hexp : x.exp ≤ t
⊢ 0 < 1 ∧ 1 * (x / 1).exp ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
split_ands
|
case refine_1.h
t x : ℝ
hexp : x.exp ≤ t
⊢ 0 < 1 ∧ 1 * (x / 1).exp ≤ t
|
case refine_1.h.refine_1
t x : ℝ
hexp : x.exp ≤ t
⊢ 0 < 1
case refine_1.h.refine_2
t x : ℝ
hexp : x.exp ≤ t
⊢ 1 * (x / 1).exp ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
{ apply Real.zero_lt_one }
|
case refine_1.h.refine_1
t x : ℝ
hexp : x.exp ≤ t
⊢ 0 < 1
case refine_1.h.refine_2
t x : ℝ
hexp : x.exp ≤ t
⊢ 1 * (x / 1).exp ≤ t
|
case refine_1.h.refine_2
t x : ℝ
hexp : x.exp ≤ t
⊢ 1 * (x / 1).exp ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
{ rwa [div_one, one_mul] }
|
case refine_1.h.refine_2
t x : ℝ
hexp : x.exp ≤ t
⊢ 1 * (x / 1).exp ≤ t
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
apply Real.zero_lt_one
|
case refine_1.h.refine_1
t x : ℝ
hexp : x.exp ≤ t
⊢ 0 < 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
rwa [div_one, one_mul]
|
case refine_1.h.refine_2
t x : ℝ
hexp : x.exp ≤ t
⊢ 1 * (x / 1).exp ≤ t
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
intro h
|
case refine_2
t x : ℝ
⊢ 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0 → x.exp ≤ t
|
case refine_2
t x : ℝ
h : 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
⊢ x.exp ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
cases h with
| inl h =>
have h : 1 * exp (x / 1) ≤ t := h.2
rwa [div_one, one_mul] at h
| inr h =>
exfalso
exact zero_ne_one h.1.symm
|
case refine_2
t x : ℝ
h : 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
⊢ x.exp ≤ t
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
have h : 1 * exp (x / 1) ≤ t := h.2
|
case refine_2.inl
t x : ℝ
h : 0 < 1 ∧ 1 * (x / 1).exp ≤ t
⊢ x.exp ≤ t
|
case refine_2.inl
t x : ℝ
h✝ : 0 < 1 ∧ 1 * (x / 1).exp ≤ t
h : 1 * (x / 1).exp ≤ t
⊢ x.exp ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
rwa [div_one, one_mul] at h
|
case refine_2.inl
t x : ℝ
h✝ : 0 < 1 ∧ 1 * (x / 1).exp ≤ t
h : 1 * (x / 1).exp ≤ t
⊢ x.exp ≤ t
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
exfalso
|
case refine_2.inr
t x : ℝ
h : 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
⊢ x.exp ≤ t
|
case refine_2.inr
t x : ℝ
h : 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
⊢ False
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
exact zero_ne_one h.1.symm
|
case refine_2.inr
t x : ℝ
h : 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
⊢ False
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Analysis/InnerProductSpace/Spectrum.lean
|
LinearMap.spectral_theorem'
|
[15, 1]
|
[29, 78]
|
suffices hsuff : ∀ (w : EuclideanSpace 𝕜 (Fin n)),
T (xs.repr.symm w) = xs.repr.symm (fun i => as i * w i) by
simpa only [LinearIsometryEquiv.symm_apply_apply,
LinearIsometryEquiv.apply_symm_apply]
using congr_arg (fun (v : E) => (xs.repr) v i) (hsuff ((xs.repr) v))
|
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : DecidableEq 𝕜
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hn : FiniteDimensional.finrank 𝕜 E = n
T : E →ₗ[𝕜] E
v : E
i : Fin n
xs : OrthonormalBasis (Fin n) 𝕜 E
as : Fin n → ℝ
hxs : ∀ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j)
⊢ xs.repr (T v) i = ↑(as i) * xs.repr v i
|
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : DecidableEq 𝕜
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hn : FiniteDimensional.finrank 𝕜 E = n
T : E →ₗ[𝕜] E
v : E
i : Fin n
xs : OrthonormalBasis (Fin n) 𝕜 E
as : Fin n → ℝ
hxs : ∀ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j)
⊢ ∀ (w : EuclideanSpace 𝕜 (Fin n)), T (xs.repr.symm w) = xs.repr.symm fun i => ↑(as i) * w i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Analysis/InnerProductSpace/Spectrum.lean
|
LinearMap.spectral_theorem'
|
[15, 1]
|
[29, 78]
|
intros w
|
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : DecidableEq 𝕜
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hn : FiniteDimensional.finrank 𝕜 E = n
T : E →ₗ[𝕜] E
v : E
i : Fin n
xs : OrthonormalBasis (Fin n) 𝕜 E
as : Fin n → ℝ
hxs : ∀ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j)
⊢ ∀ (w : EuclideanSpace 𝕜 (Fin n)), T (xs.repr.symm w) = xs.repr.symm fun i => ↑(as i) * w i
|
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : DecidableEq 𝕜
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hn : FiniteDimensional.finrank 𝕜 E = n
T : E →ₗ[𝕜] E
v : E
i : Fin n
xs : OrthonormalBasis (Fin n) 𝕜 E
as : Fin n → ℝ
hxs : ∀ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j)
w : EuclideanSpace 𝕜 (Fin n)
⊢ T (xs.repr.symm w) = xs.repr.symm fun i => ↑(as i) * w i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Analysis/InnerProductSpace/Spectrum.lean
|
LinearMap.spectral_theorem'
|
[15, 1]
|
[29, 78]
|
simp_rw [← OrthonormalBasis.sum_repr_symm, map_sum, LinearMap.map_smul,
fun j => Module.End.mem_eigenspace_iff.mp (hxs j).1, smul_smul, mul_comm]
|
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : DecidableEq 𝕜
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hn : FiniteDimensional.finrank 𝕜 E = n
T : E →ₗ[𝕜] E
v : E
i : Fin n
xs : OrthonormalBasis (Fin n) 𝕜 E
as : Fin n → ℝ
hxs : ∀ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j)
w : EuclideanSpace 𝕜 (Fin n)
⊢ T (xs.repr.symm w) = xs.repr.symm fun i => ↑(as i) * w i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Analysis/InnerProductSpace/Spectrum.lean
|
LinearMap.spectral_theorem'
|
[15, 1]
|
[29, 78]
|
simpa only [LinearIsometryEquiv.symm_apply_apply,
LinearIsometryEquiv.apply_symm_apply]
using congr_arg (fun (v : E) => (xs.repr) v i) (hsuff ((xs.repr) v))
|
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : DecidableEq 𝕜
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hn : FiniteDimensional.finrank 𝕜 E = n
T : E →ₗ[𝕜] E
v : E
i : Fin n
xs : OrthonormalBasis (Fin n) 𝕜 E
as : Fin n → ℝ
hxs : ∀ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j)
hsuff : ∀ (w : EuclideanSpace 𝕜 (Fin n)), T (xs.repr.symm w) = xs.repr.symm fun i => ↑(as i) * w i
⊢ xs.repr (T v) i = ↑(as i) * xs.repr v i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
|
repr_gramSchmidt_diagonal
|
[18, 1]
|
[24, 67]
|
rw [gramSchmidt_def, map_sub, Finsupp.sub_apply, Basis.repr_self, Finsupp.single_eq_same,
sub_eq_self, map_sum, Finsupp.coe_finset_sum, Finset.sum_apply, Finset.sum_eq_zero]
|
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
⊢ (b.repr (gramSchmidt 𝕜 (⇑b) i)) i = 1
|
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
|
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