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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/Lib/Math/LogSumExp.lean | Vec.sum_exp_eq_sum_div | [12, 1] | [17, 42] | unfold Vec.sum | n : ℕ
x : Fin n → ℝ
t : ℝ
⊢ sum (exp (x - const n t)) = sum (exp x) / t.exp | n : ℕ
x : Fin n → ℝ
t : ℝ
⊢ (Finset.univ.sum fun i => exp (x - const n t) i) = (Finset.univ.sum fun i => exp x i) / t.exp |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogSumExp.lean | Vec.sum_exp_eq_sum_div | [12, 1] | [17, 42] | rw [Finset.sum_div] | n : ℕ
x : Fin n → ℝ
t : ℝ
⊢ (Finset.univ.sum fun i => exp (x - const n t) i) = (Finset.univ.sum fun i => exp x i) / t.exp | n : ℕ
x : Fin n → ℝ
t : ℝ
⊢ (Finset.univ.sum fun i => exp (x - const n t) i) = Finset.univ.sum fun i => exp x i / t.exp |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogSumExp.lean | Vec.sum_exp_eq_sum_div | [12, 1] | [17, 42] | congr | n : ℕ
x : Fin n → ℝ
t : ℝ
⊢ (Finset.univ.sum fun i => exp (x - const n t) i) = Finset.univ.sum fun i => exp x i / t.exp | case e_f
n : ℕ
x : Fin n → ℝ
t : ℝ
⊢ (fun i => exp (x - const n t) i) = fun i => exp x i / t.exp |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogSumExp.lean | Vec.sum_exp_eq_sum_div | [12, 1] | [17, 42] | ext i | case e_f
n : ℕ
x : Fin n → ℝ
t : ℝ
⊢ (fun i => exp (x - const n t) i) = fun i => exp x i / t.exp | case e_f.h
n : ℕ
x : Fin n → ℝ
t : ℝ
i : Fin n
⊢ exp (x - const n t) i = exp x i / t.exp |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogSumExp.lean | Vec.sum_exp_eq_sum_div | [12, 1] | [17, 42] | simp [Vec.exp, Vec.const, Real.exp_sub] | case e_f.h
n : ℕ
x : Fin n → ℝ
t : ℝ
i : Fin n
⊢ exp (x - const n t) i = exp x i / t.exp | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogSumExp.lean | Vec.sum_exp_pos | [19, 1] | [23, 28] | apply Finset.sum_pos | n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ 0 < sum (exp x) | case h
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ ∀ i ∈ Finset.univ, 0 < exp x i
case hs
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ Finset.univ.Nonempty |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogSumExp.lean | Vec.sum_exp_pos | [19, 1] | [23, 28] | { intros i _; simp [Vec.exp, Real.exp_pos] } | case h
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ ∀ i ∈ Finset.univ, 0 < exp x i
case hs
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ Finset.univ.Nonempty | case hs
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ Finset.univ.Nonempty |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogSumExp.lean | Vec.sum_exp_pos | [19, 1] | [23, 28] | { existsi ⟨0, hn⟩; simp } | case hs
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ Finset.univ.Nonempty | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogSumExp.lean | Vec.sum_exp_pos | [19, 1] | [23, 28] | intros i _ | case h
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ ∀ i ∈ Finset.univ, 0 < exp x i | case h
n : ℕ
hn : 0 < n
x : Fin n → ℝ
i : Fin n
a✝ : i ∈ Finset.univ
⊢ 0 < exp x i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogSumExp.lean | Vec.sum_exp_pos | [19, 1] | [23, 28] | simp [Vec.exp, Real.exp_pos] | case h
n : ℕ
hn : 0 < n
x : Fin n → ℝ
i : Fin n
a✝ : i ∈ Finset.univ
⊢ 0 < exp x i | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogSumExp.lean | Vec.sum_exp_pos | [19, 1] | [23, 28] | existsi ⟨0, hn⟩ | case hs
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ Finset.univ.Nonempty | case hs
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ ⟨0, hn⟩ ∈ Finset.univ |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LogSumExp.lean | Vec.sum_exp_pos | [19, 1] | [23, 28] | simp | case hs
n : ℕ
hn : 0 < n
x : Fin n → ℝ
⊢ ⟨0, hn⟩ ∈ Finset.univ | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.abs_le_of_sqrt_sq_add_nonneg_le | [64, 1] | [69, 50] | rw [sqrt_le_iff] at h | a b c : ℝ
hb : 0 ≤ b
h : (a ^ 2 + b).sqrt ≤ c
⊢ |a| ≤ c | a b c : ℝ
hb : 0 ≤ b
h : 0 ≤ c ∧ a ^ 2 + b ≤ c ^ 2
⊢ |a| ≤ c |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.abs_le_of_sqrt_sq_add_nonneg_le | [64, 1] | [69, 50] | replace ⟨hc, h⟩ := h | a b c : ℝ
hb : 0 ≤ b
h : 0 ≤ c ∧ a ^ 2 + b ≤ c ^ 2
⊢ |a| ≤ c | a b c : ℝ
hb : 0 ≤ b
hc : 0 ≤ c
h : a ^ 2 + b ≤ c ^ 2
⊢ |a| ≤ c |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.abs_le_of_sqrt_sq_add_nonneg_le | [64, 1] | [69, 50] | replace h := le_trans (le_add_of_nonneg_right hb) h | a b c : ℝ
hb : 0 ≤ b
hc : 0 ≤ c
h : a ^ 2 + b ≤ c ^ 2
⊢ |a| ≤ c | a b c : ℝ
hb : 0 ≤ b
hc : 0 ≤ c
h : a ^ 2 ≤ c ^ 2
⊢ |a| ≤ c |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.abs_le_of_sqrt_sq_add_nonneg_le | [64, 1] | [69, 50] | rwa [rpow_two, sq_le_sq, abs_of_nonneg hc] at h | a b c : ℝ
hb : 0 ≤ b
hc : 0 ≤ c
h : a ^ 2 ≤ c ^ 2
⊢ |a| ≤ c | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.log_eq_log | [73, 1] | [83, 13] | rw [h] | x y : ℝ
hx : 0 < x
hy : 0 < y
h : x = y
⊢ x.log = y.log | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.log_eq_log | [73, 1] | [83, 13] | have hxmem := Set.mem_Ioi.2 hx | x y : ℝ
hx : 0 < x
hy : 0 < y
h : x.log = y.log
⊢ x = y | x y : ℝ
hx : 0 < x
hy : 0 < y
h : x.log = y.log
hxmem : x ∈ Set.Ioi 0
⊢ x = y |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.log_eq_log | [73, 1] | [83, 13] | have hymem := Set.mem_Ioi.2 hy | x y : ℝ
hx : 0 < x
hy : 0 < y
h : x.log = y.log
hxmem : x ∈ Set.Ioi 0
⊢ x = y | x y : ℝ
hx : 0 < x
hy : 0 < y
h : x.log = y.log
hxmem : x ∈ Set.Ioi 0
hymem : y ∈ Set.Ioi 0
⊢ x = y |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.log_eq_log | [73, 1] | [83, 13] | have heq : Set.restrict (Set.Ioi 0) log ⟨x, hxmem⟩ =
Set.restrict (Set.Ioi 0) log ⟨y, hymem⟩ := by
simp [h] | x y : ℝ
hx : 0 < x
hy : 0 < y
h : x.log = y.log
hxmem : x ∈ Set.Ioi 0
hymem : y ∈ Set.Ioi 0
⊢ x = y | x y : ℝ
hx : 0 < x
hy : 0 < y
h : x.log = y.log
hxmem : x ∈ Set.Ioi 0
hymem : y ∈ Set.Ioi 0
heq : (Set.Ioi 0).restrict log ⟨x, hxmem⟩ = (Set.Ioi 0).restrict log ⟨y, hymem⟩
⊢ x = y |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.log_eq_log | [73, 1] | [83, 13] | have h := log_injOn_pos.injective heq | x y : ℝ
hx : 0 < x
hy : 0 < y
h : x.log = y.log
hxmem : x ∈ Set.Ioi 0
hymem : y ∈ Set.Ioi 0
heq : (Set.Ioi 0).restrict log ⟨x, hxmem⟩ = (Set.Ioi 0).restrict log ⟨y, hymem⟩
⊢ x = y | x y : ℝ
hx : 0 < x
hy : 0 < y
h✝ : x.log = y.log
hxmem : x ∈ Set.Ioi 0
hymem : y ∈ Set.Ioi 0
heq : (Set.Ioi 0).restrict log ⟨x, hxmem⟩ = (Set.Ioi 0).restrict log ⟨y, hymem⟩
h : ⟨x, hxmem⟩ = ⟨y, hymem⟩
⊢ x = y |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.log_eq_log | [73, 1] | [83, 13] | simp [Subtype.eq] at h | x y : ℝ
hx : 0 < x
hy : 0 < y
h✝ : x.log = y.log
hxmem : x ∈ Set.Ioi 0
hymem : y ∈ Set.Ioi 0
heq : (Set.Ioi 0).restrict log ⟨x, hxmem⟩ = (Set.Ioi 0).restrict log ⟨y, hymem⟩
h : ⟨x, hxmem⟩ = ⟨y, hymem⟩
⊢ x = y | x y : ℝ
hx : 0 < x
hy : 0 < y
h✝ : x.log = y.log
hxmem : x ∈ Set.Ioi 0
hymem : y ∈ Set.Ioi 0
heq : (Set.Ioi 0).restrict log ⟨x, hxmem⟩ = (Set.Ioi 0).restrict log ⟨y, hymem⟩
h : x = y
⊢ x = y |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.log_eq_log | [73, 1] | [83, 13] | exact h | x y : ℝ
hx : 0 < x
hy : 0 < y
h✝ : x.log = y.log
hxmem : x ∈ Set.Ioi 0
hymem : y ∈ Set.Ioi 0
heq : (Set.Ioi 0).restrict log ⟨x, hxmem⟩ = (Set.Ioi 0).restrict log ⟨y, hymem⟩
h : x = y
⊢ x = y | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.log_eq_log | [73, 1] | [83, 13] | simp [h] | x y : ℝ
hx : 0 < x
hy : 0 < y
h : x.log = y.log
hxmem : x ∈ Set.Ioi 0
hymem : y ∈ Set.Ioi 0
⊢ (Set.Ioi 0).restrict log ⟨x, hxmem⟩ = (Set.Ioi 0).restrict log ⟨y, hymem⟩ | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.div_pow_eq_mul_pow_neg | [85, 1] | [87, 37] | rw [div_eq_mul_inv, ← rpow_neg hb] | a b c : ℝ
hb : 0 ≤ b
⊢ a / b ^ c = a * b ^ (-c) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.one_div_eq_pow_neg_one | [89, 1] | [90, 65] | rw [rpow_neg (le_of_lt ha), rpow_one, div_eq_mul_inv, one_mul] | a : ℝ
ha : 0 < a
⊢ 1 / a = a ^ (-1) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.inv_eq_pow_neg_one | [92, 1] | [93, 49] | rw [inv_eq_one_div, one_div_eq_pow_neg_one ha] | a : ℝ
ha : 0 < a
⊢ a⁻¹ = a ^ (-1) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.pow_half_two | [95, 1] | [98, 11] | show rpow (rpow _ _) _ = _ | x : ℝ
hx : 0 ≤ x
⊢ (x ^ (1 / 2)) ^ 2 = x | x : ℝ
hx : 0 ≤ x
⊢ (x.rpow (1 / 2)).rpow 2 = x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.pow_half_two | [95, 1] | [98, 11] | rw [rpow_eq_pow, rpow_eq_pow, ← rpow_mul hx] | x : ℝ
hx : 0 ≤ x
⊢ (x.rpow (1 / 2)).rpow 2 = x | x : ℝ
hx : 0 ≤ x
⊢ x ^ (1 / 2 * 2) = x |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.pow_half_two | [95, 1] | [98, 11] | norm_num | x : ℝ
hx : 0 ≤ x
⊢ x ^ (1 / 2 * 2) = x | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.pow_two_le_pow_two | [100, 1] | [102, 72] | rw [rpow_two, rpow_two, sq_le_sq, abs_of_nonneg hx, abs_of_nonneg hy] | x y : ℝ
hx : 0 ≤ x
hy : 0 ≤ y
⊢ x ^ 2 ≤ y ^ 2 ↔ x ≤ y | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.binomial_two | [104, 1] | [106, 29] | ring | x y : ℝ
⊢ (x + y) ^ 2 = x ^ 2 + (2 * (x * y) + y ^ 2) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.rpow_eq_mul_rpow_pred | [108, 1] | [110, 76] | conv => left; rw [(by ring : y = (y - 1) + 1), rpow_add_one hx, mul_comm] | x y : ℝ
hx : x ≠ 0
⊢ x ^ y = x * x ^ (y - 1) | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.rpow_eq_mul_rpow_pred | [108, 1] | [110, 76] | ring | x y : ℝ
hx : x ≠ 0
⊢ y = y - 1 + 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Real.lean | Real.exp_neg_eq_one_div | [112, 1] | [113, 31] | rw [exp_neg, inv_eq_one_div] | x : ℝ
⊢ (-x).exp = 1 / x.exp | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.BlockTriangular_blockDiagonal' | [36, 1] | [40, 71] | rintro ⟨i, i'⟩ ⟨j, j'⟩ h | α : Type u_1
β : Type ?u.1255
m : Type ?u.1258
n : Type ?u.1261
o : Type ?u.1264
m' : α → Type u_2
n' : α → Type ?u.1274
R : Type v
inst✝² : CommRing R
M : Matrix m m R
b : m → α
inst✝¹ : Preorder α
inst✝ : DecidableEq α
d : (i : α) → Matrix (m' i) (m' i) R
⊢ (blockDiagonal' d).BlockTriangular Sigma.fst | case mk.mk
α : Type u_1
β : Type ?u.1255
m : Type ?u.1258
n : Type ?u.1261
o : Type ?u.1264
m' : α → Type u_2
n' : α → Type ?u.1274
R : Type v
inst✝² : CommRing R
M : Matrix m m R
b : m → α
inst✝¹ : Preorder α
inst✝ : DecidableEq α
d : (i : α) → Matrix (m' i) (m' i) R
i : α
i' : m' i
j : α
j' : m' j
h : ⟨j, j'⟩.fst < ⟨i, i'⟩.fst
⊢ blockDiagonal' d ⟨i, i'⟩ ⟨j, j'⟩ = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.BlockTriangular_blockDiagonal' | [36, 1] | [40, 71] | apply blockDiagonal'_apply_ne d i' j' (fun h' => ne_of_lt h h'.symm) | case mk.mk
α : Type u_1
β : Type ?u.1255
m : Type ?u.1258
n : Type ?u.1261
o : Type ?u.1264
m' : α → Type u_2
n' : α → Type ?u.1274
R : Type v
inst✝² : CommRing R
M : Matrix m m R
b : m → α
inst✝¹ : Preorder α
inst✝ : DecidableEq α
d : (i : α) → Matrix (m' i) (m' i) R
i : α
i' : m' i
j : α
j' : m' j
h : ⟨j, j'⟩.fst < ⟨i, i'⟩.fst
⊢ blockDiagonal' d ⟨i, i'⟩ ⟨j, j'⟩ = 0 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.BlockTriangular_blockDiagonal | [42, 1] | [46, 10] | rintro ⟨i, i'⟩ ⟨j, j'⟩ h | α : Type u_1
β : Type ?u.2091
m : Type u_2
n : Type ?u.2097
o : Type ?u.2100
m' : α → Type ?u.2105
n' : α → Type ?u.2110
R : Type v
inst✝² : CommRing R
M : Matrix m m R
b : m → α
inst✝¹ : Preorder α
inst✝ : DecidableEq α
d : α → Matrix m m R
⊢ (blockDiagonal d).BlockTriangular Prod.snd | case mk.mk
α : Type u_1
β : Type ?u.2091
m : Type u_2
n : Type ?u.2097
o : Type ?u.2100
m' : α → Type ?u.2105
n' : α → Type ?u.2110
R : Type v
inst✝² : CommRing R
M : Matrix m m R
b : m → α
inst✝¹ : Preorder α
inst✝ : DecidableEq α
d : α → Matrix m m R
i : m
i' : α
j : m
j' : α
h : (j, j').2 < (i, i').2
⊢ blockDiagonal d (i, i') (j, j') = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.BlockTriangular_blockDiagonal | [42, 1] | [46, 10] | rw [blockDiagonal'_eq_blockDiagonal, BlockTriangular_blockDiagonal'] | case mk.mk
α : Type u_1
β : Type ?u.2091
m : Type u_2
n : Type ?u.2097
o : Type ?u.2100
m' : α → Type ?u.2105
n' : α → Type ?u.2110
R : Type v
inst✝² : CommRing R
M : Matrix m m R
b : m → α
inst✝¹ : Preorder α
inst✝ : DecidableEq α
d : α → Matrix m m R
i : m
i' : α
j : m
j' : α
h : (j, j').2 < (i, i').2
⊢ blockDiagonal d (i, i') (j, j') = 0 | case mk.mk.a
α : Type u_1
β : Type ?u.2091
m : Type u_2
n : Type ?u.2097
o : Type ?u.2100
m' : α → Type ?u.2105
n' : α → Type ?u.2110
R : Type v
inst✝² : CommRing R
M : Matrix m m R
b : m → α
inst✝¹ : Preorder α
inst✝ : DecidableEq α
d : α → Matrix m m R
i : m
i' : α
j : m
j' : α
h : (j, j').2 < (i, i').2
⊢ ⟨j', j⟩.fst < ⟨i', i⟩.fst |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.BlockTriangular_blockDiagonal | [42, 1] | [46, 10] | exact h | case mk.mk.a
α : Type u_1
β : Type ?u.2091
m : Type u_2
n : Type ?u.2097
o : Type ?u.2100
m' : α → Type ?u.2105
n' : α → Type ?u.2110
R : Type v
inst✝² : CommRing R
M : Matrix m m R
b : m → α
inst✝¹ : Preorder α
inst✝ : DecidableEq α
d : α → Matrix m m R
i : m
i' : α
j : m
j' : α
h : (j, j').2 < (i, i').2
⊢ ⟨j', j⟩.fst < ⟨i', i⟩.fst | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toBlock_inverse_mul_toBlock_eq_one_of_BlockTriangular | [52, 1] | [63, 29] | let p := (fun i => b i < k) | α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1 | α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toBlock_inverse_mul_toBlock_eq_one_of_BlockTriangular | [52, 1] | [63, 29] | have h_sum : M⁻¹.toBlock p p * M.toBlock p p +
M⁻¹.toBlock p (fun i => ¬ p i) * M.toBlock (fun i => ¬ p i) p = 1 := by
rw [← toBlock_mul_eq_add, inv_mul_of_invertible M, toBlock_one_self] | α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1 | α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toBlock_inverse_mul_toBlock_eq_one_of_BlockTriangular | [52, 1] | [63, 29] | have h_zero : M.toBlock (fun i => ¬ p i) p = 0 := by
{ ext i j
simpa using hM (lt_of_lt_of_le j.2 (le_of_not_lt i.2)) } | α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1 | α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
h_zero : M.toBlock (fun i => ¬p i) p = 0
⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toBlock_inverse_mul_toBlock_eq_one_of_BlockTriangular | [52, 1] | [63, 29] | simpa [h_zero] using h_sum | α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
h_zero : M.toBlock (fun i => ¬p i) p = 0
⊢ ((M⁻¹.toBlock (fun i => b i < k) fun i => b i < k) * M.toBlock (fun i => b i < k) fun i => b i < k) = 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toBlock_inverse_mul_toBlock_eq_one_of_BlockTriangular | [52, 1] | [63, 29] | rw [← toBlock_mul_eq_add, inv_mul_of_invertible M, toBlock_one_self] | α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
⊢ M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toBlock_inverse_mul_toBlock_eq_one_of_BlockTriangular | [52, 1] | [63, 29] | ext i j | α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
⊢ M.toBlock (fun i => ¬p i) p = 0 | case a
α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // ¬p a }
j : { a // p a }
⊢ M.toBlock (fun i => ¬p i) p i j = 0 i j |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toBlock_inverse_mul_toBlock_eq_one_of_BlockTriangular | [52, 1] | [63, 29] | simpa using hM (lt_of_lt_of_le j.2 (le_of_not_lt i.2)) | case a
α : Type u_1
β : Type ?u.3857
m : Type u_2
n : Type ?u.3863
o : Type ?u.3866
m' : α → Type ?u.3871
n' : α → Type ?u.3876
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i < k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // ¬p a }
j : { a // p a }
⊢ M.toBlock (fun i => ¬p i) p i j = 0 i j | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | let p := (λ i => b i = k) | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
⊢ M⁻¹.toSquareBlock b k * M.toSquareBlock b k = 1 | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
⊢ M⁻¹.toSquareBlock b k * M.toSquareBlock b k = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | have h_sum : M⁻¹.toBlock p p * M.toBlock p p +
M⁻¹.toBlock p (λ i => ¬ p i) * M.toBlock (λ i => ¬ p i) p = 1 := by
rw [← toBlock_mul_eq_add, inv_mul_of_invertible M, toBlock_one_self] | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
⊢ M⁻¹.toSquareBlock b k * M.toSquareBlock b k = 1 | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
⊢ M⁻¹.toSquareBlock b k * M.toSquareBlock b k = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | have h_zero : ∀ i j l,
M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 := by
{ intro i j l
by_cases hj : b j.1 ≤ k
{ have hj := lt_of_le_of_ne hj j.2
have hM' := blockTriangular_inv_of_blockTriangular hM
apply mul_eq_zero_of_left
simpa using hM' (lt_of_lt_of_eq hj i.2.symm) }
{ have hj := lt_of_not_ge hj
apply mul_eq_zero_of_right
simpa using hM (lt_of_eq_of_lt l.2 hj) }} | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
⊢ M⁻¹.toSquareBlock b k * M.toSquareBlock b k = 1 | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
h_zero :
∀ (i : { a // p a }) (j : { a // (fun i => ¬p i) a }) (l : { a // p a }),
M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0
⊢ M⁻¹.toSquareBlock b k * M.toSquareBlock b k = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | have h_zero' :
M⁻¹.toBlock p (λ (i : m) => ¬p i) * M.toBlock (λ (i : m) => ¬p i) p = 0 := by
{ ext i l
apply sum_eq_zero (λ j _ => h_zero i j l) } | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
h_zero :
∀ (i : { a // p a }) (j : { a // (fun i => ¬p i) a }) (l : { a // p a }),
M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0
⊢ M⁻¹.toSquareBlock b k * M.toSquareBlock b k = 1 | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
h_zero :
∀ (i : { a // p a }) (j : { a // (fun i => ¬p i) a }) (l : { a // p a }),
M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0
h_zero' : (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 0
⊢ M⁻¹.toSquareBlock b k * M.toSquareBlock b k = 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | simpa [h_zero'] using h_sum | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
h_zero :
∀ (i : { a // p a }) (j : { a // (fun i => ¬p i) a }) (l : { a // p a }),
M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0
h_zero' : (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 0
⊢ M⁻¹.toSquareBlock b k * M.toSquareBlock b k = 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | rw [← toBlock_mul_eq_add, inv_mul_of_invertible M, toBlock_one_self] | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
⊢ M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | intro i j l | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
⊢ ∀ (i : { a // p a }) (j : { a // (fun i => ¬p i) a }) (l : { a // p a }),
M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | by_cases hj : b j.1 ≤ k | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 | case pos
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj : b ↑j ≤ k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0
case neg
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj : ¬b ↑j ≤ k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | { have hj := lt_of_le_of_ne hj j.2
have hM' := blockTriangular_inv_of_blockTriangular hM
apply mul_eq_zero_of_left
simpa using hM' (lt_of_lt_of_eq hj i.2.symm) } | case pos
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj : b ↑j ≤ k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0
case neg
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj : ¬b ↑j ≤ k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 | case neg
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj : ¬b ↑j ≤ k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | { have hj := lt_of_not_ge hj
apply mul_eq_zero_of_right
simpa using hM (lt_of_eq_of_lt l.2 hj) } | case neg
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj : ¬b ↑j ≤ k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | have hj := lt_of_le_of_ne hj j.2 | case pos
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj : b ↑j ≤ k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 | case pos
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : b ↑j ≤ k
hj : b ↑j < k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | have hM' := blockTriangular_inv_of_blockTriangular hM | case pos
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : b ↑j ≤ k
hj : b ↑j < k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 | case pos
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : b ↑j ≤ k
hj : b ↑j < k
hM' : M⁻¹.BlockTriangular b
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | apply mul_eq_zero_of_left | case pos
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : b ↑j ≤ k
hj : b ↑j < k
hM' : M⁻¹.BlockTriangular b
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 | case pos.h
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : b ↑j ≤ k
hj : b ↑j < k
hM' : M⁻¹.BlockTriangular b
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | simpa using hM' (lt_of_lt_of_eq hj i.2.symm) | case pos.h
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : b ↑j ≤ k
hj : b ↑j < k
hM' : M⁻¹.BlockTriangular b
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j = 0 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | have hj := lt_of_not_ge hj | case neg
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj : ¬b ↑j ≤ k
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 | case neg
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : ¬b ↑j ≤ k
hj : k < b ↑j
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | apply mul_eq_zero_of_right | case neg
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : ¬b ↑j ≤ k
hj : k < b ↑j
⊢ M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0 | case neg.h
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : ¬b ↑j ≤ k
hj : k < b ↑j
⊢ M.toBlock (fun i => ¬p i) p j l = 0 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | simpa using hM (lt_of_eq_of_lt l.2 hj) | case neg.h
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
i : { a // p a }
j : { a // (fun i => ¬p i) a }
l : { a // p a }
hj✝ : ¬b ↑j ≤ k
hj : k < b ↑j
⊢ M.toBlock (fun i => ¬p i) p j l = 0 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | ext i l | α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
h_zero :
∀ (i : { a // p a }) (j : { a // (fun i => ¬p i) a }) (l : { a // p a }),
M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0
⊢ (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 0 | case a
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
h_zero :
∀ (i : { a // p a }) (j : { a // (fun i => ¬p i) a }) (l : { a // p a }),
M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0
i l : { a // p a }
⊢ ((M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p) i l = 0 i l |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/LinearAlgebra/Matrix/Block.lean | Matrix.toSquareBlock_inv_mul_toSquareBlock_eq_one | [82, 1] | [104, 30] | apply sum_eq_zero (λ j _ => h_zero i j l) | case a
α : Type u_1
β : Type ?u.27607
m : Type u_2
n : Type ?u.27613
o : Type ?u.27616
m' : α → Type ?u.27621
n' : α → Type ?u.27626
R : Type v
inst✝⁶ : CommRing R
M : Matrix m m R
b : m → α
inst✝⁵ : DecidableEq m
inst✝⁴ : Fintype m
inst✝³ : DecidableEq n
inst✝² : Fintype n
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i => b i = k
h_sum : M⁻¹.toBlock p p * M.toBlock p p + (M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p = 1
h_zero :
∀ (i : { a // p a }) (j : { a // (fun i => ¬p i) a }) (l : { a // p a }),
M⁻¹.toBlock p (fun i => ¬p i) i j * M.toBlock (fun i => ¬p i) p j l = 0
i l : { a // p a }
⊢ ((M⁻¹.toBlock p fun i => ¬p i) * M.toBlock (fun i => ¬p i) p) i l = 0 i l | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | constructor | m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
⊢ a / b ≤ c ↔ a ≤ c * b | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
⊢ a / b ≤ c → a ≤ c * b
case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
⊢ a ≤ c * b → a / b ≤ c |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | intro h i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
⊢ a / b ≤ c → a ≤ c * b | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a / b ≤ c
i : m
⊢ a i ≤ (c * b) i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | have hi := h i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a / b ≤ c
i : m
⊢ a i ≤ (c * b) i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a / b ≤ c
i : m
hi : (a / b) i ≤ c i
⊢ a i ≤ (c * b) i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | simp at hi | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a / b ≤ c
i : m
hi : (a / b) i ≤ c i
⊢ a i ≤ (c * b) i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a / b ≤ c
i : m
hi : a i / b i ≤ c i
⊢ a i ≤ (c * b) i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | rw [_root_.div_le_iff (hb i)] at hi | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a / b ≤ c
i : m
hi : a i / b i ≤ c i
⊢ a i ≤ (c * b) i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a / b ≤ c
i : m
hi : a i ≤ c i * b i
⊢ a i ≤ (c * b) i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | exact hi | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a / b ≤ c
i : m
hi : a i ≤ c i * b i
⊢ a i ≤ (c * b) i | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | intro h i | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
⊢ a ≤ c * b → a / b ≤ c | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
⊢ (a / b) i ≤ c i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | have hi := h i | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
⊢ (a / b) i ≤ c i | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
hi : a i ≤ (c * b) i
⊢ (a / b) i ≤ c i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | simp at hi | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
hi : a i ≤ (c * b) i
⊢ (a / b) i ≤ c i | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
hi : a i ≤ c i * b i
⊢ (a / b) i ≤ c i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | dsimp | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
hi : a i ≤ c i * b i
⊢ (a / b) i ≤ c i | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
hi : a i ≤ c i * b i
⊢ a i / b i ≤ c i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | rw [_root_.div_le_iff (hb i)] | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
hi : a i ≤ c i * b i
⊢ a i / b i ≤ c i | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
hi : a i ≤ c i * b i
⊢ a i ≤ c i * b i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.div_le_iff | [104, 1] | [109, 51] | exact hi | case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hb : StrongLT 0 b
h : a ≤ c * b
i : m
hi : a i ≤ c i * b i
⊢ a i ≤ c i * b i | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.le_div_iff | [111, 1] | [116, 51] | constructor | m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
⊢ a ≤ b / c ↔ a * c ≤ b | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
⊢ a ≤ b / c → a * c ≤ b
case mpr
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
⊢ a * c ≤ b → a ≤ b / c |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.le_div_iff | [111, 1] | [116, 51] | intro h i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
⊢ a ≤ b / c → a * c ≤ b | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a ≤ b / c
i : m
⊢ (a * c) i ≤ b i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.le_div_iff | [111, 1] | [116, 51] | have hi := h i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a ≤ b / c
i : m
⊢ (a * c) i ≤ b i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a ≤ b / c
i : m
hi : a i ≤ (b / c) i
⊢ (a * c) i ≤ b i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.le_div_iff | [111, 1] | [116, 51] | simp at hi | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a ≤ b / c
i : m
hi : a i ≤ (b / c) i
⊢ (a * c) i ≤ b i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a ≤ b / c
i : m
hi : a i ≤ b i / c i
⊢ (a * c) i ≤ b i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Lib/Math/Data/Vec.lean | Vec.le_div_iff | [111, 1] | [116, 51] | rw [_root_.le_div_iff (hc i)] at hi | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a ≤ b / c
i : m
hi : a i ≤ b i / c i
⊢ (a * c) i ≤ b i | case mp
m : Type u
n : Type v
inst✝¹ : Fintype m
inst✝ : Fintype n
α : Type w
a b c : m → ℝ
hc : StrongLT 0 c
h : a ≤ b / c
i : m
hi : a i * c i ≤ b i
⊢ (a * c) i ≤ b i |
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