url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/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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