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https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_diagonal | [48, 1] | [62, 10] | rfl | 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 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.conjTranspose_mul_mul | [64, 1] | [69, 81] | refine' ⟨isHermitian_conjTranspose_mul_mul _ hM.1, _⟩ | m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
⊢ (N.conjTranspose * M * N).PosSemidef | m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
⊢ ∀ (x : n → 𝕜), 0 ≤ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.conjTranspose_mul_mul | [64, 1] | [69, 81] | intro x | m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
⊢ ∀ (x : n → 𝕜), 0 ≤ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x | m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.conjTranspose_mul_mul | [64, 1] | [69, 81] | convert hM.2 (N.mulVec x) using 1 | m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x | case h.e'_4
m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
x : n → 𝕜
⊢ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x = star (N.mulVec x) ⬝ᵥ M.mulVec (N.mulVec x) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.conjTranspose_mul_mul | [64, 1] | [69, 81] | rw [mul_assoc, mulVec_mulVec, ← mulVec_mulVec, dotProduct_mulVec, star_mulVec] | case h.e'_4
m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
x : n → 𝕜
⊢ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x = star (N.mulVec x) ⬝ᵥ M.mulVec (N.mulVec x) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.conjTranspose_mul_mul | [71, 1] | [78, 17] | refine' ⟨isHermitian_conjTranspose_mul_mul _ hM.1, _⟩ | m : Type ?u.28307
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 N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
⊢ (N.conjTranspose * M * N).PosDef | m : Type ?u.28307
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 N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
⊢ ∀ (x : n → 𝕜), x ≠ 0 → 0 < star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.conjTranspose_mul_mul | [71, 1] | [78, 17] | intros x hx | m : Type ?u.28307
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 N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
⊢ ∀ (x : n → 𝕜), x ≠ 0 → 0 < star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x | m : Type ?u.28307
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 N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
x : n → 𝕜
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.conjTranspose_mul_mul | [71, 1] | [78, 17] | convert
hM.2 (N.mulVec x) (fun h => hx (eq_zero_of_mulVec_eq_zero hN h)) using 1 | m : Type ?u.28307
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 N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
x : n → 𝕜
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x | case h.e'_4
m : Type ?u.28307
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 N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
x : n → 𝕜
hx : x ≠ 0
⊢ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x = star (N.mulVec x) ⬝ᵥ M.mulVec (N.mulVec x) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.conjTranspose_mul_mul | [71, 1] | [78, 17] | rw [Matrix.mul_assoc, mulVec_mulVec, ← mulVec_mulVec, dotProduct_mulVec,
star_mulVec] | case h.e'_4
m : Type ?u.28307
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 N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
x : n → 𝕜
hx : x ≠ 0
⊢ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x = star (N.mulVec x) ⬝ᵥ M.mulVec (N.mulVec x) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.IsHermitian.nonsingular_inv | [80, 1] | [83, 65] | refine' (Matrix.inv_eq_right_inv _).symm | m : Type ?u.34280
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.IsHermitian
hMdet : IsUnit M.det
⊢ M⁻¹.IsHermitian | m : Type ?u.34280
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.IsHermitian
hMdet : IsUnit M.det
⊢ M * M⁻¹.conjTranspose = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.IsHermitian.nonsingular_inv | [80, 1] | [83, 65] | rw [conjTranspose_nonsing_inv, hM.eq, mul_nonsing_inv _ hMdet] | m : Type ?u.34280
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.IsHermitian
hMdet : IsUnit M.det
⊢ M * M⁻¹.conjTranspose = 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.conj_symm | [85, 1] | [88, 43] | nth_rewrite 1 [star_dotProduct, star_mulVec] | m : Type ?u.36361
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
x : n → 𝕜
M : Matrix n n 𝕜
hM : M.IsHermitian
⊢ star (star x ⬝ᵥ M.mulVec x) = star x ⬝ᵥ M.mulVec x | m : Type ?u.36361
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
x : n → 𝕜
M : Matrix n n 𝕜
hM : M.IsHermitian
⊢ star (star (star x ᵥ* M.conjTranspose ⬝ᵥ x)) = star x ⬝ᵥ M.mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.conj_symm | [85, 1] | [88, 43] | rw [star_star, dotProduct_mulVec, hM.eq] | m : Type ?u.36361
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
x : n → 𝕜
M : Matrix n n 𝕜
hM : M.IsHermitian
⊢ star (star (star x ᵥ* M.conjTranspose ⬝ᵥ x)) = star x ⬝ᵥ M.mulVec x | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.nonsingular_inv | [90, 1] | [99, 13] | refine' ⟨hM.1.nonsingular_inv (isUnit_iff_ne_zero.2 hM.det_ne_zero), _⟩ | m : Type ?u.40080
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⁻¹.PosDef | m : Type ?u.40080
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
⊢ ∀ (x : n → 𝕜), x ≠ 0 → 0 < star x ⬝ᵥ M⁻¹.mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.nonsingular_inv | [90, 1] | [99, 13] | intros x hx | m : Type ?u.40080
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
⊢ ∀ (x : n → 𝕜), x ≠ 0 → 0 < star x ⬝ᵥ M⁻¹.mulVec x | m : Type ?u.40080
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
x : n → 𝕜
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.nonsingular_inv | [90, 1] | [99, 13] | have hMMinv := mul_nonsing_inv _ (isUnit_iff_ne_zero.2 hM.det_ne_zero) | m : Type ?u.40080
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
x : n → 𝕜
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x | m : Type ?u.40080
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
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.nonsingular_inv | [90, 1] | [99, 13] | have hMinvdet : M⁻¹.det ≠ 0 := det_ne_zero_of_left_inverse hMMinv | m : Type ?u.40080
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
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x | m : Type ?u.40080
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
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.nonsingular_inv | [90, 1] | [99, 13] | have hres :=
hM.2 (M⁻¹.mulVec x) (fun h => hx (eq_zero_of_mulVec_eq_zero hMinvdet h)) | m : Type ?u.40080
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
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x | m : Type ?u.40080
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
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star (M⁻¹.mulVec x) ⬝ᵥ M.mulVec (M⁻¹.mulVec x)
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.nonsingular_inv | [90, 1] | [99, 13] | rw [mulVec_mulVec, hMMinv, one_mulVec, star_dotProduct] at hres | m : Type ?u.40080
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
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star (M⁻¹.mulVec x) ⬝ᵥ M.mulVec (M⁻¹.mulVec x)
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x | m : Type ?u.40080
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
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star (star x ⬝ᵥ M⁻¹.mulVec x)
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.nonsingular_inv | [90, 1] | [99, 13] | rw [conj_symm ((@isHermitian_inv _ _ _ _ _ _ M hM.Invertible).2 hM.1)] at hres | m : Type ?u.40080
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
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star (star x ⬝ᵥ M⁻¹.mulVec x)
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x | m : Type ?u.40080
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
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star x ⬝ᵥ M⁻¹.mulVec x
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef.nonsingular_inv | [90, 1] | [99, 13] | exact hres | m : Type ?u.40080
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
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star x ⬝ᵥ M⁻¹.mulVec x
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.mul_mul_of_IsHermitian | [101, 1] | [103, 54] | convert hM.conjTranspose_mul_mul M N | m : Type ?u.45085
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.IsHermitian
⊢ (N * M * N).PosSemidef | case h.e'_8.h.e'_5.h.e'_5
m : Type ?u.45085
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.IsHermitian
⊢ N = N.conjTranspose |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.mul_mul_of_IsHermitian | [101, 1] | [103, 54] | exact hN.symm | case h.e'_8.h.e'_5.h.e'_5
m : Type ?u.45085
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.IsHermitian
⊢ N = N.conjTranspose | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.add | [105, 1] | [110, 37] | refine' ⟨hM.1.add hN.1, _⟩ | m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
⊢ (M + N).PosSemidef | m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
⊢ ∀ (x : n → 𝕜), 0 ≤ star x ⬝ᵥ (M + N).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.add | [105, 1] | [110, 37] | intros x | m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
⊢ ∀ (x : n → 𝕜), 0 ≤ star x ⬝ᵥ (M + N).mulVec x | m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (M + N).mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.add | [105, 1] | [110, 37] | simp only [add_mulVec, dotProduct_add, map_add] | m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (M + N).mulVec x | m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ M.mulVec x + star x ⬝ᵥ N.mulVec x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemidef.add | [105, 1] | [110, 37] | apply add_nonneg (hM.2 x) (hN.2 x) | m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ M.mulVec x + star x ⬝ᵥ N.mulVec x | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.isUnit_det_of_PosDef_inv | [112, 1] | [117, 28] | apply isUnit_iff_ne_zero.2 | m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
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 ℝ
h : M⁻¹.PosDef
⊢ IsUnit M.det | m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
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 ℝ
h : M⁻¹.PosDef
⊢ M.det ≠ 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.isUnit_det_of_PosDef_inv | [112, 1] | [117, 28] | have := h.isUnit_det | m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
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 ℝ
h : M⁻¹.PosDef
⊢ M.det ≠ 0 | m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
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 ℝ
h : M⁻¹.PosDef
this : IsUnit M⁻¹.det
⊢ M.det ≠ 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.isUnit_det_of_PosDef_inv | [112, 1] | [117, 28] | rw [det_nonsing_inv, isUnit_ring_inverse] at this | m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
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 ℝ
h : M⁻¹.PosDef
this : IsUnit M⁻¹.det
⊢ M.det ≠ 0 | m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
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 ℝ
h : M⁻¹.PosDef
this : IsUnit M.det
⊢ M.det ≠ 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.isUnit_det_of_PosDef_inv | [112, 1] | [117, 28] | apply IsUnit.ne_zero this | m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
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 ℝ
h : M⁻¹.PosDef
this : IsUnit M.det
⊢ M.det ≠ 0 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_inv_iff_PosDef | [119, 1] | [125, 31] | constructor | m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
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 ℝ
⊢ M⁻¹.PosDef ↔ M.PosDef | case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
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 ℝ
⊢ M⁻¹.PosDef → M.PosDef
case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
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 ℝ
⊢ M.PosDef → M⁻¹.PosDef |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_inv_iff_PosDef | [119, 1] | [125, 31] | { intros hM
rw [← Matrix.nonsing_inv_nonsing_inv M (isUnit_det_of_PosDef_inv hM)]
apply hM.nonsingular_inv } | case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
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 ℝ
⊢ M⁻¹.PosDef → M.PosDef
case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
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 ℝ
⊢ M.PosDef → M⁻¹.PosDef | case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
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 ℝ
⊢ M.PosDef → M⁻¹.PosDef |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_inv_iff_PosDef | [119, 1] | [125, 31] | { intros hM
exact hM.nonsingular_inv } | case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
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 ℝ
⊢ M.PosDef → M⁻¹.PosDef | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_inv_iff_PosDef | [119, 1] | [125, 31] | intros hM | case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
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 ℝ
⊢ M⁻¹.PosDef → M.PosDef | case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
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 ℝ
hM : M⁻¹.PosDef
⊢ M.PosDef |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_inv_iff_PosDef | [119, 1] | [125, 31] | rw [← Matrix.nonsing_inv_nonsing_inv M (isUnit_det_of_PosDef_inv hM)] | case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
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 ℝ
hM : M⁻¹.PosDef
⊢ M.PosDef | case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
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 ℝ
hM : M⁻¹.PosDef
⊢ M⁻¹⁻¹.PosDef |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_inv_iff_PosDef | [119, 1] | [125, 31] | apply hM.nonsingular_inv | case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
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 ℝ
hM : M⁻¹.PosDef
⊢ M⁻¹⁻¹.PosDef | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_inv_iff_PosDef | [119, 1] | [125, 31] | intros hM | case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
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 ℝ
⊢ M.PosDef → M⁻¹.PosDef | case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
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 ℝ
hM : M.PosDef
⊢ M⁻¹.PosDef |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosDef_inv_iff_PosDef | [119, 1] | [125, 31] | exact hM.nonsingular_inv | case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
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 ℝ
hM : M.PosDef
⊢ M⁻¹.PosDef | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemiDef.IsSymm | [127, 1] | [140, 12] | let h_A_IsHermitian := hA.1 | m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
⊢ A.IsSymm | m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian : A.IsHermitian := hA.left
⊢ A.IsSymm |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemiDef.IsSymm | [127, 1] | [140, 12] | rw [Matrix.isHermitian_iff_isSymmetric] at h_A_IsHermitian | m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian : A.IsHermitian := hA.left
⊢ A.IsSymm | m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian : (toEuclideanLin A).IsSymmetric
⊢ A.IsSymm |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemiDef.IsSymm | [127, 1] | [140, 12] | simp [LinearMap.IsSymmetric, toEuclideanLin] at h_A_IsHermitian | m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian : (toEuclideanLin A).IsSymmetric
⊢ A.IsSymm | m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
⊢ A.IsSymm |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemiDef.IsSymm | [127, 1] | [140, 12] | apply IsSymm.ext | m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
⊢ A.IsSymm | case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
⊢ ∀ (i j : Fin n), A j i = A i j |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemiDef.IsSymm | [127, 1] | [140, 12] | intros i j | case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
⊢ ∀ (i j : Fin n), A j i = A i j | case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
⊢ A j i = A i j |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemiDef.IsSymm | [127, 1] | [140, 12] | have hAij := h_A_IsHermitian (fun k => if k = i then 1 else 0) (fun k => if k = j then 1 else 0) | case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
⊢ A j i = A i j | case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hAij :
(Finset.univ.sum fun x =>
A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = i then 1 else 0) x * if x = j then 1 else 0) =
Finset.univ.sum fun x =>
(if x = i then 1 else 0) * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = j then 1 else 0) x
⊢ A j i = A i j |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemiDef.IsSymm | [127, 1] | [140, 12] | have hi : (Finset.sum Finset.univ fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ)))
(fun k => if k = i then 1 else 0) x) = A j i := by simp | case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hAij :
(Finset.univ.sum fun x =>
A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = i then 1 else 0) x * if x = j then 1 else 0) =
Finset.univ.sum fun x =>
(if x = i then 1 else 0) * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = j then 1 else 0) x
⊢ A j i = A i j | case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hAij :
(Finset.univ.sum fun x =>
A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = i then 1 else 0) x * if x = j then 1 else 0) =
Finset.univ.sum fun x =>
(if x = i then 1 else 0) * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = j then 1 else 0) x
hi : (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i
⊢ A j i = A i j |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemiDef.IsSymm | [127, 1] | [140, 12] | have hj : (Finset.sum Finset.univ fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ)))
(fun k => if k = j then 1 else 0) x) = A i j := by simp | case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hAij :
(Finset.univ.sum fun x =>
A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = i then 1 else 0) x * if x = j then 1 else 0) =
Finset.univ.sum fun x =>
(if x = i then 1 else 0) * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = j then 1 else 0) x
hi : (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i
⊢ A j i = A i j | case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hAij :
(Finset.univ.sum fun x =>
A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = i then 1 else 0) x * if x = j then 1 else 0) =
Finset.univ.sum fun x =>
(if x = i then 1 else 0) * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = j then 1 else 0) x
hi : (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i
hj : (Finset.univ.sum fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = j then 1 else 0) x) = A i j
⊢ A j i = A i j |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemiDef.IsSymm | [127, 1] | [140, 12] | simp [WithLp.equiv, mulVec, dotProduct] at hAij | case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hAij :
(Finset.univ.sum fun x =>
A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = i then 1 else 0) x * if x = j then 1 else 0) =
Finset.univ.sum fun x =>
(if x = i then 1 else 0) * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = j then 1 else 0) x
hi : (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i
hj : (Finset.univ.sum fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = j then 1 else 0) x) = A i j
⊢ A j i = A i j | case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hi : (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i
hj : (Finset.univ.sum fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = j then 1 else 0) x) = A i j
hAij :
(Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) =
Finset.univ.sum fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = j then 1 else 0) x
⊢ A j i = A i j |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemiDef.IsSymm | [127, 1] | [140, 12] | erw [hi, hj] at hAij | case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hi : (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i
hj : (Finset.univ.sum fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = j then 1 else 0) x) = A i j
hAij :
(Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) =
Finset.univ.sum fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = j then 1 else 0) x
⊢ A j i = A i j | case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hi : (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i
hj : (Finset.univ.sum fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = j then 1 else 0) x) = A i j
hAij : A j i = A i j
⊢ A j i = A i j |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemiDef.IsSymm | [127, 1] | [140, 12] | rw [hAij] | case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hi : (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i
hj : (Finset.univ.sum fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = j then 1 else 0) x) = A i j
hAij : A j i = A i j
⊢ A j i = A i j | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemiDef.IsSymm | [127, 1] | [140, 12] | simp | m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hAij :
(Finset.univ.sum fun x =>
A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = i then 1 else 0) x * if x = j then 1 else 0) =
Finset.univ.sum fun x =>
(if x = i then 1 else 0) * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = j then 1 else 0) x
⊢ (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean | Matrix.PosSemiDef.IsSymm | [127, 1] | [140, 12] | simp | m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hAij :
(Finset.univ.sum fun x =>
A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = i then 1 else 0) x * if x = j then 1 else 0) =
Finset.univ.sum fun x =>
(if x = i then 1 else 0) * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = j then 1 else 0) x
hi : (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i
⊢ (Finset.univ.sum fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = j then 1 else 0) x) = A i j | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Tactic/DCP/AtomLibrary/Fns/Norm.lean | CvxLean.norm2₂_eq_norm2 | [57, 1] | [58, 36] | simp [Norm.norm, sqrt_eq_rpow] | x y : ℝ
⊢ ‖![x, y]‖ = (x ^ 2 + y ^ 2).sqrt | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/TrussDesign.lean | TrussDesign.hminₚ_pos | [180, 1] | [182, 25] | unfold hminₚ | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < hminₚ | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/TrussDesign.lean | TrussDesign.hminₚ_pos | [180, 1] | [182, 25] | norm_num | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/TrussDesign.lean | TrussDesign.hminₚ_le_hmaxₚ | [184, 1] | [185, 31] | unfold hminₚ hmaxₚ | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ hminₚ ≤ hmaxₚ | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 1 ≤ 100 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/TrussDesign.lean | TrussDesign.hminₚ_le_hmaxₚ | [184, 1] | [185, 31] | norm_num | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 1 ≤ 100 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/TrussDesign.lean | TrussDesign.wminₚ_pos | [193, 1] | [195, 25] | unfold wminₚ | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < wminₚ | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/TrussDesign.lean | TrussDesign.wminₚ_pos | [193, 1] | [195, 25] | norm_num | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/TrussDesign.lean | TrussDesign.wminₚ_le_wmaxₚ | [197, 1] | [198, 31] | unfold wminₚ wmaxₚ | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ wminₚ ≤ wmaxₚ | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 1 ≤ 100 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/TrussDesign.lean | TrussDesign.wminₚ_le_wmaxₚ | [197, 1] | [198, 31] | norm_num | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 1 ≤ 100 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/TrussDesign.lean | TrussDesign.Rmaxₚ_pos | [203, 1] | [205, 25] | unfold Rmaxₚ | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < Rmaxₚ | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 10 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/TrussDesign.lean | TrussDesign.Rmaxₚ_pos | [203, 1] | [205, 25] | norm_num | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 10 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/TrussDesign.lean | TrussDesign.σₚ_pos | [210, 1] | [212, 22] | unfold σₚ | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < σₚ | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 0.5 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/TrussDesign.lean | TrussDesign.σₚ_pos | [210, 1] | [212, 22] | norm_num | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 0.5 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/TrussDesign.lean | TrussDesign.F₁ₚ_pos | [217, 1] | [219, 23] | unfold F₁ₚ | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < F₁ₚ | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 10 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/TrussDesign.lean | TrussDesign.F₁ₚ_pos | [217, 1] | [219, 23] | norm_num | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 10 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/TrussDesign.lean | TrussDesign.F₂ₚ_pos | [224, 1] | [226, 23] | unfold F₂ₚ | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < F₂ₚ | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 20 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/TrussDesign.lean | TrussDesign.F₂ₚ_pos | [224, 1] | [226, 23] | norm_num | hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 20 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | simp [rotatedSoCone, rotateSoCone] | n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
⊢ match t.rotateSoCone x with
| (v, w, x) => v.rotatedSoCone w x | n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | have habsx0t : |x 0| ≤ t := by
rw [soCone, Fin.sum_univ_succ] at h
have hS : 0 ≤ ∑ i : Fin n, x (Fin.succ i) ^ 2 :=
Finset.sum_nonneg (fun i _ => (rpow_two (x i.succ)).symm ▸ sq_nonneg (x i.succ))
exact abs_le_of_sqrt_sq_add_nonneg_le hS h | n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2 | n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | have ht : 0 ≤ t := le_trans (abs_nonneg _) habsx0t | n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2 | n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | replace ⟨hx0t, hnx0t⟩ := abs_le.mp habsx0t | n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2 | n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | split_ands | n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2 | case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2
case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t + x 0) / √2
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | { field_simp
have hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2 := by norm_cast; ring
rw [hrw, le_sub_iff_add_le, add_comm]
unfold soCone at h; norm_cast at h ⊢
rw [← Fin.sum_univ_succ (f := fun i => (x i) ^ 2)]
rw [← sqrt_le_left ht]
exact h } | case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2
case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t + x 0) / √2
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2 | case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t + x 0) / √2
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | { simp [le_div_iff]; linarith } | case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t + x 0) / √2
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2 | case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | { simp [le_div_iff]; linarith } | case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | rw [soCone, Fin.sum_univ_succ] at h | n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
⊢ |x 0| ≤ t | n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : (x 0 ^ 2 + ∑ i : Fin n, x i.succ ^ 2).sqrt ≤ t
⊢ |x 0| ≤ t |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | have hS : 0 ≤ ∑ i : Fin n, x (Fin.succ i) ^ 2 :=
Finset.sum_nonneg (fun i _ => (rpow_two (x i.succ)).symm ▸ sq_nonneg (x i.succ)) | n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : (x 0 ^ 2 + ∑ i : Fin n, x i.succ ^ 2).sqrt ≤ t
⊢ |x 0| ≤ t | n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : (x 0 ^ 2 + ∑ i : Fin n, x i.succ ^ 2).sqrt ≤ t
hS : 0 ≤ ∑ i : Fin n, x i.succ ^ 2
⊢ |x 0| ≤ t |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | exact abs_le_of_sqrt_sq_add_nonneg_le hS h | n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : (x 0 ^ 2 + ∑ i : Fin n, x i.succ ^ 2).sqrt ≤ t
hS : 0 ≤ ∑ i : Fin n, x i.succ ^ 2
⊢ |x 0| ≤ t | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | field_simp | case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 | case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) * (t - x 0) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | have hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2 := by norm_cast; ring | case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) * (t - x 0) | case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) * (t - x 0) |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | rw [hrw, le_sub_iff_add_le, add_comm] | case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) * (t - x 0) | case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
⊢ x 0 ^ 2 + ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ t ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | unfold soCone at h | case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
⊢ x 0 ^ 2 + ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ t ^ 2 | case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : (∑ i : Fin n.succ, x i ^ 2).sqrt ≤ t
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
⊢ x 0 ^ 2 + ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ t ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | norm_cast at h ⊢ | case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : (∑ i : Fin n.succ, x i ^ 2).sqrt ≤ t
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
⊢ x 0 ^ 2 + ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ t ^ 2 | case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
h : (∑ x_1 : Fin n.succ, x x_1 ^ 2).sqrt ≤ t
⊢ x 0 ^ 2 + ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ t ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | rw [← Fin.sum_univ_succ (f := fun i => (x i) ^ 2)] | case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
h : (∑ x_1 : Fin n.succ, x x_1 ^ 2).sqrt ≤ t
⊢ x 0 ^ 2 + ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ t ^ 2 | case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
h : (∑ x_1 : Fin n.succ, x x_1 ^ 2).sqrt ≤ t
⊢ ∑ i : Fin (n + 1), x i ^ 2 ≤ t ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | rw [← sqrt_le_left ht] | case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
h : (∑ x_1 : Fin n.succ, x x_1 ^ 2).sqrt ≤ t
⊢ ∑ i : Fin (n + 1), x i ^ 2 ≤ t ^ 2 | case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
h : (∑ x_1 : Fin n.succ, x x_1 ^ 2).sqrt ≤ t
⊢ (∑ i : Fin (n + 1), x i ^ 2).sqrt ≤ t |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | exact h | case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
h : (∑ x_1 : Fin n.succ, x x_1 ^ 2).sqrt ≤ t
⊢ (∑ i : Fin (n + 1), x i ^ 2).sqrt ≤ t | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | ring | n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | simp [le_div_iff] | case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t + x 0) / √2 | case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ t + x 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | linarith | case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ t + x 0 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | simp [le_div_iff] | case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2 | case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ x 0 ≤ t |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.rotateSoCone_rotatedSoCone | [43, 1] | [62, 34] | linarith | case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ x 0 ≤ t | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.unrotateSoCone_soCone | [68, 1] | [81, 16] | simp [soCone, unrotateSoCone] | n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : v.rotatedSoCone w x
⊢ match v.unrotateSoCone w x with
| (t, x) => t.soCone x | n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : v.rotatedSoCone w x
⊢ (∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2).sqrt ≤ (v + w) / √2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.unrotateSoCone_soCone | [68, 1] | [81, 16] | replace ⟨h, hv, hw⟩ := h | n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : v.rotatedSoCone w x
⊢ (∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2).sqrt ≤ (v + w) / √2 | n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ (∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2).sqrt ≤ (v + w) / √2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.unrotateSoCone_soCone | [68, 1] | [81, 16] | rw [sqrt_le_iff] | n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ (∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2).sqrt ≤ (v + w) / √2 | n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ 0 ≤ (v + w) / √2 ∧ ∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2 ≤ ((v + w) / √2) ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.unrotateSoCone_soCone | [68, 1] | [81, 16] | split_ands | n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ 0 ≤ (v + w) / √2 ∧ ∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2 ≤ ((v + w) / √2) ^ 2 | case refine_1
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ 0 ≤ (v + w) / √2
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ ∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2 ≤ ((v + w) / √2) ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.unrotateSoCone_soCone | [68, 1] | [81, 16] | { simp [le_div_iff]; linarith } | case refine_1
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ 0 ≤ (v + w) / √2
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ ∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2 ≤ ((v + w) / √2) ^ 2 | case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ ∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2 ≤ ((v + w) / √2) ^ 2 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.unrotateSoCone_soCone | [68, 1] | [81, 16] | { rw [Fin.sum_univ_succ]
simp [Matrix.vecCons]
rw [add_comm, ← le_sub_iff_add_le]
field_simp
have hrw : ((v + w) ^ 2 - (v - w) ^ 2) / 2 = v * w * 2 := by norm_cast; ring
norm_cast at hrw h
rwa [hrw] } | case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ ∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2 ≤ ((v + w) / √2) ^ 2 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Cones/SOCone.lean | Real.unrotateSoCone_soCone | [68, 1] | [81, 16] | simp [le_div_iff] | case refine_1
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ 0 ≤ (v + w) / √2 | case refine_1
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ 0 ≤ v + w |
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