url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
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| start
stringlengths 6
10
| end
stringlengths 6
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stringlengths 1
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stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
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https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogDet.lean | Matrix.LogDetAtom.cond_elim | [129, 1] | [137, 28] | have h_Ddet_le_Adet : D.det ≤ A.det := LogDetAtom.optimality_Ddet_le_Adet ht hD hZ hPSD | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_pd : D.PosDef
h_A_psd : A.PosSemidef
⊢ A.PosDef | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_pd : D.PosDef
h_A_psd : A.PosSemidef
h_Ddet_le_Adet : D.det ≤ A.det
⊢ A.PosDef |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogDet.lean | Matrix.LogDetAtom.cond_elim | [129, 1] | [137, 28] | have h_Adet_pos : 0 < A.det := lt_of_lt_of_le h_D_pd.det_pos h_Ddet_le_Adet | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_pd : D.PosDef
h_A_psd : A.PosSemidef
h_Ddet_le_Adet : D.det ≤ A.det
⊢ A.PosDef | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_pd : D.PosDef
h_A_psd : A.PosSemidef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ A.PosDef |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogDet.lean | Matrix.LogDetAtom.cond_elim | [129, 1] | [137, 28] | rw [h_A_psd.PosDef_iff_det_ne_zero] | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_pd : D.PosDef
h_A_psd : A.PosSemidef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ A.PosDef | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_pd : D.PosDef
h_A_psd : A.PosSemidef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ A.det ≠ 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogDet.lean | Matrix.LogDetAtom.cond_elim | [129, 1] | [137, 28] | apply ne_of_gt h_Adet_pos | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_D_pd : D.PosDef
h_A_psd : A.PosSemidef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ A.det ≠ 0 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogDet.lean | Matrix.LogDetAtom.optimality | [139, 1] | [148, 12] | have h_A_pd : A.PosDef := LogDetAtom.cond_elim ht hD hZ hPSD | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
⊢ ∑ i : n, t i ≤ A.det.log | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
⊢ ∑ i : n, t i ≤ A.det.log |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogDet.lean | Matrix.LogDetAtom.optimality | [139, 1] | [148, 12] | have h_Ddet_le_Adet : D.det ≤ A.det := LogDetAtom.optimality_Ddet_le_Adet ht hD hZ hPSD | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
⊢ ∑ i : n, t i ≤ A.det.log | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
⊢ ∑ i : n, t i ≤ A.det.log |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogDet.lean | Matrix.LogDetAtom.optimality | [139, 1] | [148, 12] | have h_Adet_pos: 0 < A.det := h_A_pd.det_pos | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
⊢ ∑ i : n, t i ≤ A.det.log | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ ∑ i : n, t i ≤ A.det.log |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogDet.lean | Matrix.LogDetAtom.optimality | [139, 1] | [148, 12] | rw [← Real.exp_le_exp, Real.exp_sum, Real.exp_log h_Adet_pos] | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ ∑ i : n, t i ≤ A.det.log | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ ∏ x : n, (t x).exp ≤ A.det |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogDet.lean | Matrix.LogDetAtom.optimality | [139, 1] | [148, 12] | apply le_trans (Finset.prod_le_prod (fun i _ => le_of_lt ((t i).exp_pos)) (fun i _ => ht i)) | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ ∏ x : n, (t x).exp ≤ A.det | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ ∏ i : n, Y.diag i ≤ A.det |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogDet.lean | Matrix.LogDetAtom.optimality | [139, 1] | [148, 12] | convert h_Ddet_le_Adet | n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ ∏ i : n, Y.diag i ≤ A.det | case h.e'_3
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ ∏ i : n, Y.diag i = D.det |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogDet.lean | Matrix.LogDetAtom.optimality | [139, 1] | [148, 12] | simp [hD] | case h.e'_3
n : Type
inst✝³ : Fintype n
inst✝² : LinearOrder n
inst✝¹ : LocallyFiniteOrderBot n
𝕜 : Type
inst✝ : RCLike 𝕜
A : Matrix n n ℝ
hA : A.PosDef
t : n → ℝ
Y Z D : Matrix n n ℝ
ht : ∀ (i : n), (t i).exp ≤ Y.diag i
hD : D = diagonal Y.diag
hZ : Z = Y.toUpperTri
hPSD : (D.fromBlocks Z Z.transpose A).PosSemidef
h_A_pd : A.PosDef
h_Ddet_le_Adet : D.det ≤ A.det
h_Adet_pos : 0 < A.det
⊢ ∏ i : n, Y.diag i = D.det | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Eigenspace.lean | Module.End.eigenspace_add | [17, 1] | [24, 73] | rintro x ⟨hf, hg⟩ | K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f g : End R M
a b : R
⊢ f.eigenspace a ⊓ g.eigenspace b ≤ (f + g).eigenspace (a + b) | case intro
K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f g : End R M
a b : R
x : M
hf : x ∈ ↑(f.eigenspace a)
hg : x ∈ ↑(g.eigenspace b)
⊢ x ∈ (f + g).eigenspace (a + b) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Eigenspace.lean | Module.End.eigenspace_add | [17, 1] | [24, 73] | simp only [eigenspace, SetLike.mem_coe, LinearMap.mem_ker,
LinearMap.sub_apply, algebraMap_end_apply] at hf hg | case intro
K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f g : End R M
a b : R
x : M
hf : x ∈ ↑(f.eigenspace a)
hg : x ∈ ↑(g.eigenspace b)
⊢ x ∈ (f + g).eigenspace (a + b) | case intro
K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f g : End R M
a b : R
x : M
hf : f x - a • x = 0
hg : g x - b • x = 0
⊢ x ∈ (f + g).eigenspace (a + b) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Eigenspace.lean | Module.End.eigenspace_add | [17, 1] | [24, 73] | simp only [eigenspace, map_add, LinearMap.mem_ker, LinearMap.sub_apply,
LinearMap.add_apply, algebraMap_end_apply] | case intro
K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f g : End R M
a b : R
x : M
hf : f x - a • x = 0
hg : g x - b • x = 0
⊢ x ∈ (f + g).eigenspace (a + b) | case intro
K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f g : End R M
a b : R
x : M
hf : f x - a • x = 0
hg : g x - b • x = 0
⊢ f x + g x - (a • x + b • x) = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Eigenspace.lean | Module.End.eigenspace_add | [17, 1] | [24, 73] | rw [← add_sub, add_comm (a • x), ← sub_sub, hg, add_sub, add_zero, hf] | case intro
K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f g : End R M
a b : R
x : M
hf : f x - a • x = 0
hg : g x - b • x = 0
⊢ f x + g x - (a • x + b • x) = 0 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Eigenspace.lean | Module.End.eigenspace_one | [26, 1] | [29, 64] | apply eq_top_iff.2 | K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
⊢ eigenspace 1 1 = ⊤ | K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
⊢ ⊤ ≤ eigenspace 1 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Eigenspace.lean | Module.End.eigenspace_one | [26, 1] | [29, 64] | intros x _ | K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
⊢ ⊤ ≤ eigenspace 1 1 | K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x : M
a✝ : x ∈ ⊤
⊢ x ∈ eigenspace 1 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Eigenspace.lean | Module.End.eigenspace_one | [26, 1] | [29, 64] | simp only [mem_eigenspace_iff, LinearMap.one_apply, one_smul] | K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x : M
a✝ : x ∈ ⊤
⊢ x ∈ eigenspace 1 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Eigenspace.lean | Module.End.has_eigenvector_one | [35, 1] | [36, 60] | rw [eigenspace_one] | K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x : M
hx : x ≠ 0
⊢ x ∈ eigenspace 1 1 | K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x : M
hx : x ≠ 0
⊢ x ∈ ⊤ |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Eigenspace.lean | Module.End.has_eigenvector_one | [35, 1] | [36, 60] | apply Submodule.mem_top | K R : Type v
V M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
x : M
hx : x ≠ 0
⊢ x ∈ ⊤ | 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/SchurComplement.lean | Matrix.schur_complement_eq₁₁ | [28, 1] | [37, 7] | simp [Function.star_sum_elim, fromBlocks_mulVec, vecMul_fromBlocks, add_vecMul,
dotProduct_mulVec, vecMul_sub, Matrix.mul_assoc, vecMul_mulVec, hA.eq,
conjTranspose_nonsing_inv, star_mulVec] | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
x : m → 𝕜
y : n → 𝕜
inst✝ : Invertible A
hA : A.IsHermitian
⊢ star (x ⊕ᵥ y) ᵥ* A.fromBlocks B B.conjTranspose D ⬝ᵥ (x ⊕ᵥ y) =
star (x + (A⁻¹ * B).mulVec y) ᵥ* A ⬝ᵥ (x + (A⁻¹ * B).mulVec y) + star y ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ y | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
x : m → 𝕜
y : n → 𝕜
inst✝ : Invertible A
hA : A.IsHermitian
⊢ star x ᵥ* A ⬝ᵥ x + star y ᵥ* B.conjTranspose ⬝ᵥ x + (star x ᵥ* B ⬝ᵥ y + star y ᵥ* D ⬝ᵥ y) =
star x ᵥ* A ⬝ᵥ x + star y ᵥ* B.conjTranspose ⬝ᵥ x +
(star x ᵥ* B ⬝ᵥ y + star y ᵥ* (B.conjTranspose * (A⁻¹ * B)) ⬝ᵥ y) +
(star y ᵥ* D ⬝ᵥ y - star y ᵥ* (B.conjTranspose * (A⁻¹ * B)) ⬝ᵥ y) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.schur_complement_eq₁₁ | [28, 1] | [37, 7] | abel | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
x : m → 𝕜
y : n → 𝕜
inst✝ : Invertible A
hA : A.IsHermitian
⊢ star x ᵥ* A ⬝ᵥ x + star y ᵥ* B.conjTranspose ⬝ᵥ x + (star x ᵥ* B ⬝ᵥ y + star y ᵥ* D ⬝ᵥ y) =
star x ᵥ* A ⬝ᵥ x + star y ᵥ* B.conjTranspose ⬝ᵥ x +
(star x ᵥ* B ⬝ᵥ y + star y ᵥ* (B.conjTranspose * (A⁻¹ * B)) ⬝ᵥ y) +
(star y ᵥ* D ⬝ᵥ y - star y ᵥ* (B.conjTranspose * (A⁻¹ * B)) ⬝ᵥ y) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.schur_complement_eq₂₂ | [39, 1] | [48, 7] | simp [Function.star_sum_elim, fromBlocks_mulVec, vecMul_fromBlocks, add_vecMul,
dotProduct_mulVec, vecMul_sub, Matrix.mul_assoc, vecMul_mulVec, hD.eq,
conjTranspose_nonsing_inv, star_mulVec] | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
x : m → 𝕜
y : n → 𝕜
inst✝ : Invertible D
hD : D.IsHermitian
⊢ star (x ⊕ᵥ y) ᵥ* A.fromBlocks B B.conjTranspose D ⬝ᵥ (x ⊕ᵥ y) =
star ((D⁻¹ * B.conjTranspose).mulVec x + y) ᵥ* D ⬝ᵥ ((D⁻¹ * B.conjTranspose).mulVec x + y) +
star x ᵥ* (A - B * D⁻¹ * B.conjTranspose) ⬝ᵥ x | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
x : m → 𝕜
y : n → 𝕜
inst✝ : Invertible D
hD : D.IsHermitian
⊢ star x ᵥ* A ⬝ᵥ x + star y ᵥ* B.conjTranspose ⬝ᵥ x + (star x ᵥ* B ⬝ᵥ y + star y ᵥ* D ⬝ᵥ y) =
star x ᵥ* (B * (D⁻¹ * B.conjTranspose)) ⬝ᵥ x + star y ᵥ* B.conjTranspose ⬝ᵥ x +
(star x ᵥ* B ⬝ᵥ y + star y ᵥ* D ⬝ᵥ y) +
(star x ᵥ* A ⬝ᵥ x - star x ᵥ* (B * (D⁻¹ * B.conjTranspose)) ⬝ᵥ x) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.schur_complement_eq₂₂ | [39, 1] | [48, 7] | abel | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
x : m → 𝕜
y : n → 𝕜
inst✝ : Invertible D
hD : D.IsHermitian
⊢ star x ᵥ* A ⬝ᵥ x + star y ᵥ* B.conjTranspose ⬝ᵥ x + (star x ᵥ* B ⬝ᵥ y + star y ᵥ* D ⬝ᵥ y) =
star x ᵥ* (B * (D⁻¹ * B.conjTranspose)) ⬝ᵥ x + star y ᵥ* B.conjTranspose ⬝ᵥ x +
(star x ᵥ* B ⬝ᵥ y + star y ᵥ* D ⬝ᵥ y) +
(star x ᵥ* A ⬝ᵥ x - star x ᵥ* (B * (D⁻¹ * B.conjTranspose)) ⬝ᵥ x) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | have hBAB : (Bᴴ * A⁻¹ * B).IsHermitian := by
apply isHermitian_conjTranspose_mul_mul
apply hA.inv | n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
⊢ (A.fromBlocks B B.conjTranspose D).IsHermitian ↔ (D - B.conjTranspose * A⁻¹ * B).IsHermitian | n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ (A.fromBlocks B B.conjTranspose D).IsHermitian ↔ (D - B.conjTranspose * A⁻¹ * B).IsHermitian |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | rw [isHermitian_fromBlocks_iff] | n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ (A.fromBlocks B B.conjTranspose D).IsHermitian ↔ (D - B.conjTranspose * A⁻¹ * B).IsHermitian | n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian ↔
(D - B.conjTranspose * A⁻¹ * B).IsHermitian |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | constructor | n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian ↔
(D - B.conjTranspose * A⁻¹ * B).IsHermitian | case mp
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian →
(D - B.conjTranspose * A⁻¹ * B).IsHermitian
case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ (D - B.conjTranspose * A⁻¹ * B).IsHermitian →
A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | apply isHermitian_conjTranspose_mul_mul | n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
⊢ (B.conjTranspose * A⁻¹ * B).IsHermitian | case hA
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
⊢ A⁻¹.IsHermitian |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | apply hA.inv | case hA
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
⊢ A⁻¹.IsHermitian | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | intro h | case mp
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian →
(D - B.conjTranspose * A⁻¹ * B).IsHermitian | case mp
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian
⊢ (D - B.conjTranspose * A⁻¹ * B).IsHermitian |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | apply IsHermitian.sub h.2.2.2 hBAB | case mp
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian
⊢ (D - B.conjTranspose * A⁻¹ * B).IsHermitian | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | intro h | case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ (D - B.conjTranspose * A⁻¹ * B).IsHermitian →
A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian | case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : (D - B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | refine' ⟨hA, rfl, conjTranspose_conjTranspose B, _⟩ | case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : (D - B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ A.IsHermitian ∧ B.conjTranspose = B.conjTranspose ∧ B.conjTranspose.conjTranspose = B ∧ D.IsHermitian | case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : (D - B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ D.IsHermitian |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | rw [← sub_add_cancel D] | case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : (D - B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ D.IsHermitian | case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : (D - B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ (D - ?mpr + ?mpr).IsHermitian
case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : (D - B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ Matrix n n 𝕜 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₁₁ | [50, 1] | [63, 33] | apply IsHermitian.add h hBAB | case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : (D - B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ (D - ?mpr + ?mpr).IsHermitian
case mpr
n : Type u_3
m : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype m
inst✝ : DecidableEq m
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.IsHermitian
hBAB : (B.conjTranspose * A⁻¹ * B).IsHermitian
h : (D - B.conjTranspose * A⁻¹ * B).IsHermitian
⊢ Matrix n n 𝕜 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₂₂ | [65, 1] | [70, 79] | rw [← isHermitian_submatrix_equiv (Equiv.sumComm n m), Equiv.sumComm_apply,
fromBlocks_submatrix_sum_swap_sum_swap] | n : Type u_1
m : Type u_2
𝕜 : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hD : D.IsHermitian
⊢ (A.fromBlocks B B.conjTranspose D).IsHermitian ↔ (A - B * D⁻¹ * B.conjTranspose).IsHermitian | n : Type u_1
m : Type u_2
𝕜 : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hD : D.IsHermitian
⊢ (D.fromBlocks B.conjTranspose B A).IsHermitian ↔ (A - B * D⁻¹ * B.conjTranspose).IsHermitian |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.IsHermitian.fromBlocks₂₂ | [65, 1] | [70, 79] | convert IsHermitian.fromBlocks₁₁ _ _ hD <;> rw [conjTranspose_conjTranspose] | n : Type u_1
m : Type u_2
𝕜 : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hD : D.IsHermitian
⊢ (D.fromBlocks B.conjTranspose B A).IsHermitian ↔ (A - B * D⁻¹ * B.conjTranspose).IsHermitian | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | rw [PosSemidef, IsHermitian.fromBlocks₁₁ _ _ hA.1] | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ (A.fromBlocks B B.conjTranspose D).PosSemidef ↔ (D - B.conjTranspose * A⁻¹ * B).PosSemidef | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ ((D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x) ↔
(D - B.conjTranspose * A⁻¹ * B).PosSemidef |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | constructor | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ ((D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x) ↔
(D - B.conjTranspose * A⁻¹ * B).PosSemidef | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ ((D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x) →
(D - B.conjTranspose * A⁻¹ * B).PosSemidef
case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ (D - B.conjTranspose * A⁻¹ * B).PosSemidef →
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | { intro h; refine' ⟨h.1, _⟩; intro x
have := h.2 (- ((A⁻¹ * B).mulVec x) ⊕ᵥ x)
rw [dotProduct_mulVec, schur_complement_eq₁₁ B D _ _ hA.1, neg_add_self,
dotProduct_zero, zero_add] at this
rw [dotProduct_mulVec]; exact this } | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ ((D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x) →
(D - B.conjTranspose * A⁻¹ * B).PosSemidef
case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ (D - B.conjTranspose * A⁻¹ * B).PosSemidef →
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ (D - B.conjTranspose * A⁻¹ * B).PosSemidef →
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | { intro h; refine' ⟨h.1, _⟩; intro x
rw [dotProduct_mulVec, ← Sum.elim_comp_inl_inr x, schur_complement_eq₁₁ B D _ _ hA.1]
apply le_add_of_nonneg_of_le
{ rw [← dotProduct_mulVec]
apply hA.posSemidef.2 }
{ rw [← dotProduct_mulVec _ _ (x ∘ Sum.inr)]
apply h.2 } } | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ (D - B.conjTranspose * A⁻¹ * B).PosSemidef →
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | intro h | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ ((D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x) →
(D - B.conjTranspose * A⁻¹ * B).PosSemidef | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
⊢ (D - B.conjTranspose * A⁻¹ * B).PosSemidef |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | refine' ⟨h.1, _⟩ | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
⊢ (D - B.conjTranspose * A⁻¹ * B).PosSemidef | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
⊢ ∀ (x : n → 𝕜), 0 ≤ star x ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | intro x | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
⊢ ∀ (x : n → 𝕜), 0 ≤ star x ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec x | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | have := h.2 (- ((A⁻¹ * B).mulVec x) ⊕ᵥ x) | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec x | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
x : n → 𝕜
this : 0 ≤ star (-(A⁻¹ * B).mulVec x ⊕ᵥ x) ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec (-(A⁻¹ * B).mulVec x ⊕ᵥ x)
⊢ 0 ≤ star x ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | rw [dotProduct_mulVec, schur_complement_eq₁₁ B D _ _ hA.1, neg_add_self,
dotProduct_zero, zero_add] at this | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
x : n → 𝕜
this : 0 ≤ star (-(A⁻¹ * B).mulVec x ⊕ᵥ x) ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec (-(A⁻¹ * B).mulVec x ⊕ᵥ x)
⊢ 0 ≤ star x ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec x | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
x : n → 𝕜
this : 0 ≤ star x ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x
⊢ 0 ≤ star x ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | rw [dotProduct_mulVec] | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
x : n → 𝕜
this : 0 ≤ star x ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x
⊢ 0 ≤ star x ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec x | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
x : n → 𝕜
this : 0 ≤ star x ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x
⊢ 0 ≤ star x ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | exact this | case mp
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h :
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x
x : n → 𝕜
this : 0 ≤ star x ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x
⊢ 0 ≤ star x ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | intro h | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
⊢ (D - B.conjTranspose * A⁻¹ * B).PosSemidef →
(D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
⊢ (D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | refine' ⟨h.1, _⟩ | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
⊢ (D - B.conjTranspose * A⁻¹ * B).IsHermitian ∧
∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
⊢ ∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | intro x | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
⊢ ∀ (x : m ⊕ n → 𝕜), 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | rw [dotProduct_mulVec, ← Sum.elim_comp_inl_inr x, schur_complement_eq₁₁ B D _ _ hA.1] | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (A.fromBlocks B B.conjTranspose D).mulVec x | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤
star (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) ᵥ* A ⬝ᵥ (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) +
star (x ∘ Sum.inr) ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x ∘ Sum.inr |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | apply le_add_of_nonneg_of_le | case mpr
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤
star (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) ᵥ* A ⬝ᵥ (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) +
star (x ∘ Sum.inr) ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x ∘ Sum.inr | case mpr.ha
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) ᵥ* A ⬝ᵥ (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr))
case mpr.hbc
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inr) ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x ∘ Sum.inr |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | { rw [← dotProduct_mulVec]
apply hA.posSemidef.2 } | case mpr.ha
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) ᵥ* A ⬝ᵥ (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr))
case mpr.hbc
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inr) ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x ∘ Sum.inr | case mpr.hbc
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inr) ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x ∘ Sum.inr |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | { rw [← dotProduct_mulVec _ _ (x ∘ Sum.inr)]
apply h.2 } | case mpr.hbc
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inr) ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x ∘ Sum.inr | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | rw [← dotProduct_mulVec] | case mpr.ha
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) ᵥ* A ⬝ᵥ (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) | case mpr.ha
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) ⬝ᵥ A.mulVec (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | apply hA.posSemidef.2 | case mpr.ha
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) ⬝ᵥ A.mulVec (x ∘ Sum.inl + (A⁻¹ * B).mulVec (x ∘ Sum.inr)) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | rw [← dotProduct_mulVec _ _ (x ∘ Sum.inr)] | case mpr.hbc
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inr) ᵥ* (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ x ∘ Sum.inr | case mpr.hbc
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inr) ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec (x ∘ Sum.inr) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₁₁ | [72, 1] | [88, 20] | apply h.2 | case mpr.hbc
n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Fintype n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hA : A.PosDef
inst✝ : Invertible A
h : (D - B.conjTranspose * A⁻¹ * B).PosSemidef
x : m ⊕ n → 𝕜
⊢ 0 ≤ star (x ∘ Sum.inr) ⬝ᵥ (D - B.conjTranspose * A⁻¹ * B).mulVec (x ∘ Sum.inr) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₂₂ | [90, 1] | [96, 35] | rw [← posSemidef_submatrix_equiv (Equiv.sumComm n m), Equiv.sumComm_apply,
fromBlocks_submatrix_sum_swap_sum_swap] | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hD : D.PosDef
inst✝ : Invertible D
⊢ (A.fromBlocks B B.conjTranspose D).PosSemidef ↔ (A - B * D⁻¹ * B.conjTranspose).PosSemidef | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hD : D.PosDef
inst✝ : Invertible D
⊢ (D.fromBlocks B.conjTranspose B A).PosSemidef ↔ (A - B * D⁻¹ * B.conjTranspose).PosSemidef |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/SchurComplement.lean | Matrix.PosSemidef.fromBlocks₂₂ | [90, 1] | [96, 35] | convert @PosSemidef.fromBlocks₁₁ m n 𝕜 _ _ _ _ _ _ _ hD _ <;>
rw [conjTranspose_conjTranspose] | n : Type u_2
m : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq n
A : Matrix m m 𝕜
B : Matrix m n 𝕜
D : Matrix n n 𝕜
hD : D.PosDef
inst✝ : Invertible D
⊢ (D.fromBlocks B.conjTranspose B A).PosSemidef ↔ (A - B * D⁻¹ * B.conjTranspose).PosSemidef | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.det_nonneg | [16, 1] | [20, 30] | rw [hM.1.det_eq_prod_eigenvalues] | m : Type ?u.340
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.352
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
M : Matrix n n ℝ
hM : M.PosSemidef
inst✝ : DecidableEq n
⊢ 0 ≤ M.det | m : Type ?u.340
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.352
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
M : Matrix n n ℝ
hM : M.PosSemidef
inst✝ : DecidableEq n
⊢ 0 ≤ Finset.univ.prod fun i => ↑(⋯.eigenvalues i) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.det_nonneg | [16, 1] | [20, 30] | apply Finset.prod_nonneg | m : Type ?u.340
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.352
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
M : Matrix n n ℝ
hM : M.PosSemidef
inst✝ : DecidableEq n
⊢ 0 ≤ Finset.univ.prod fun i => ↑(⋯.eigenvalues i) | case h0
m : Type ?u.340
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.352
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
M : Matrix n n ℝ
hM : M.PosSemidef
inst✝ : DecidableEq n
⊢ ∀ i ∈ Finset.univ, 0 ≤ ↑(⋯.eigenvalues i) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.det_nonneg | [16, 1] | [20, 30] | intros i _hi | case h0
m : Type ?u.340
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.352
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
M : Matrix n n ℝ
hM : M.PosSemidef
inst✝ : DecidableEq n
⊢ ∀ i ∈ Finset.univ, 0 ≤ ↑(⋯.eigenvalues i) | case h0
m : Type ?u.340
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.352
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
M : Matrix n n ℝ
hM : M.PosSemidef
inst✝ : DecidableEq n
i : n
_hi : i ∈ Finset.univ
⊢ 0 ≤ ↑(⋯.eigenvalues i) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.det_nonneg | [16, 1] | [20, 30] | apply eigenvalues_nonneg hM | case h0
m : Type ?u.340
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.352
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
M : Matrix n n ℝ
hM : M.PosSemidef
inst✝ : DecidableEq n
i : n
_hi : i ∈ Finset.univ
⊢ 0 ≤ ↑(⋯.eigenvalues i) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.det_ne_zero | [22, 1] | [29, 32] | rw [← Matrix.nondegenerate_iff_det_ne_zero] | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
⊢ M.det ≠ 0 | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
⊢ M.Nondegenerate |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.det_ne_zero | [22, 1] | [29, 32] | intros v hv | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
⊢ M.Nondegenerate | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
⊢ v = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.det_ne_zero | [22, 1] | [29, 32] | have hv' := hv (star v) | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
⊢ v = 0 | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
⊢ v = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.det_ne_zero | [22, 1] | [29, 32] | rw [← star_eq_zero] | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
⊢ v = 0 | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
⊢ star v = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.det_ne_zero | [22, 1] | [29, 32] | by_contra h | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
⊢ star v = 0 | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
h : ¬star v = 0
⊢ False |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.det_ne_zero | [22, 1] | [29, 32] | have := hM.2 (star v) h | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
h : ¬star v = 0
⊢ False | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
h : ¬star v = 0
this : 0 < star (star v) ⬝ᵥ M.mulVec (star v)
⊢ False |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.det_ne_zero | [22, 1] | [29, 32] | simp [star_star, hv'] at this | m : Type ?u.1418
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
v : n → 𝕜
hv : ∀ (w : n → 𝕜), v ⬝ᵥ M.mulVec w = 0
hv' : v ⬝ᵥ M.mulVec (star v) = 0
h : ¬star v = 0
this : 0 < star (star v) ⬝ᵥ M.mulVec (star v)
⊢ False | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef_diagonal | [38, 1] | [46, 50] | refine' ⟨isHermitian_diagonal _, _⟩ | m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ (diagonal f).PosSemidef | m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ ∀ (x : n → ℝ), 0 ≤ star x ⬝ᵥ (diagonal f).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef_diagonal | [38, 1] | [46, 50] | intro x | m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
⊢ ∀ (x : n → ℝ), 0 ≤ star x ⬝ᵥ (diagonal f).mulVec x | m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
⊢ 0 ≤ star x ⬝ᵥ (diagonal f).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef_diagonal | [38, 1] | [46, 50] | simp only [star, id_def, RCLike.re_to_real] | m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
⊢ 0 ≤ star x ⬝ᵥ (diagonal f).mulVec x | m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
⊢ 0 ≤ (fun i => x i) ⬝ᵥ (diagonal f).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef_diagonal | [38, 1] | [46, 50] | apply Finset.sum_nonneg' | m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
⊢ 0 ≤ (fun i => x i) ⬝ᵥ (diagonal f).mulVec x | case h
m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
⊢ ∀ (i : n), 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef_diagonal | [38, 1] | [46, 50] | intro i | case h
m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
⊢ ∀ (i : n), 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i | case h
m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
i : n
⊢ 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef_diagonal | [38, 1] | [46, 50] | rw [mulVec_diagonal f x i, mul_comm, mul_assoc] | case h
m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
i : n
⊢ 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i | case h
m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
i : n
⊢ 0 ≤ f i * (x i * (fun i => x i) i) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef_diagonal | [38, 1] | [46, 50] | exact mul_nonneg (hf i) (mul_self_nonneg (x i)) | case h
m : Type ?u.6640
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.6652
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 ≤ f i
x : n → ℝ
i : n
⊢ 0 ≤ f i * (x i * (fun i => x i) i) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | refine' ⟨isHermitian_diagonal _, _⟩ | m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
⊢ (diagonal f).PosDef | m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
⊢ ∀ (x : n → ℝ), x ≠ 0 → 0 < star x ⬝ᵥ (diagonal f).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | intros x hx | m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
⊢ ∀ (x : n → ℝ), x ≠ 0 → 0 < star x ⬝ᵥ (diagonal f).mulVec x | m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ (diagonal f).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | simp only [star, id_def, RCLike.re_to_real] | m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ (diagonal f).mulVec x | m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ 0 < (fun i => x i) ⬝ᵥ (diagonal f).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | apply Finset.sum_pos' | m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ 0 < (fun i => x i) ⬝ᵥ (diagonal f).mulVec x | case h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∀ i ∈ Finset.univ, 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i
case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∃ i ∈ Finset.univ, 0 < (fun i => x i) i * (diagonal f).mulVec x i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | { intros i _
rw [mulVec_diagonal f x i, mul_comm, mul_assoc]
exact mul_nonneg (le_of_lt (hf i)) (mul_self_nonneg (x i)) } | case h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∀ i ∈ Finset.univ, 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i
case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∃ i ∈ Finset.univ, 0 < (fun i => x i) i * (diagonal f).mulVec x i | case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∃ i ∈ Finset.univ, 0 < (fun i => x i) i * (diagonal f).mulVec x i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | { contrapose hx; simp at hx ⊢
ext i
have := hx i
rw [mulVec_diagonal f x i, mul_comm, mul_assoc] at this
have := nonpos_of_mul_nonpos_right this (hf i)
rw [mul_self_eq_zero.1 (le_antisymm this (mul_self_nonneg (x i)))]
rfl } | case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∃ i ∈ Finset.univ, 0 < (fun i => x i) i * (diagonal f).mulVec x i | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | intros i _ | case h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∀ i ∈ Finset.univ, 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i | case h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
i : n
a✝ : i ∈ Finset.univ
⊢ 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | rw [mulVec_diagonal f x i, mul_comm, mul_assoc] | case h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
i : n
a✝ : i ∈ Finset.univ
⊢ 0 ≤ (fun i => x i) i * (diagonal f).mulVec x i | case h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
i : n
a✝ : i ∈ Finset.univ
⊢ 0 ≤ f i * (x i * (fun i => x i) i) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | exact mul_nonneg (le_of_lt (hf i)) (mul_self_nonneg (x i)) | case h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
i : n
a✝ : i ∈ Finset.univ
⊢ 0 ≤ f i * (x i * (fun i => x i) i) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | contrapose hx | case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : x ≠ 0
⊢ ∃ i ∈ Finset.univ, 0 < (fun i => x i) i * (diagonal f).mulVec x i | case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ¬∃ i ∈ Finset.univ, 0 < (fun i => x i) i * (diagonal f).mulVec x i
⊢ ¬x ≠ 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | simp at hx ⊢ | case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ¬∃ i ∈ Finset.univ, 0 < (fun i => x i) i * (diagonal f).mulVec x i
⊢ ¬x ≠ 0 | case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
⊢ x = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | ext i | case hs
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
⊢ x = 0 | case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
⊢ x i = 0 i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | have := hx i | case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
⊢ x i = 0 i | case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this : x i * (diagonal f).mulVec x i ≤ 0
⊢ x i = 0 i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | rw [mulVec_diagonal f x i, mul_comm, mul_assoc] at this | case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this : x i * (diagonal f).mulVec x i ≤ 0
⊢ x i = 0 i | case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this : f i * (x i * x i) ≤ 0
⊢ x i = 0 i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | have := nonpos_of_mul_nonpos_right this (hf i) | case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this : f i * (x i * x i) ≤ 0
⊢ x i = 0 i | case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this✝ : f i * (x i * x i) ≤ 0
this : x i * x i ≤ 0
⊢ x i = 0 i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | rw [mul_self_eq_zero.1 (le_antisymm this (mul_self_nonneg (x i)))] | case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this✝ : f i * (x i * x i) ≤ 0
this : x i * x i ≤ 0
⊢ x i = 0 i | case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this✝ : f i * (x i * x i) ≤ 0
this : x i * x i ≤ 0
⊢ 0 = 0 i |
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