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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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