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_diagonal
|
[48, 1]
|
[62, 10]
|
rfl
|
case hs.h
m : Type ?u.8318
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.8330
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
f : n → ℝ
hf : ∀ (i : n), 0 < f i
x : n → ℝ
hx : ∀ (x_1 : n), x x_1 * (diagonal f).mulVec x x_1 ≤ 0
i : n
this✝ : f i * (x i * x i) ≤ 0
this : x i * x i ≤ 0
⊢ 0 = 0 i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemidef.conjTranspose_mul_mul
|
[64, 1]
|
[69, 81]
|
refine' ⟨isHermitian_conjTranspose_mul_mul _ hM.1, _⟩
|
m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
⊢ (N.conjTranspose * M * N).PosSemidef
|
m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
⊢ ∀ (x : n → 𝕜), 0 ≤ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemidef.conjTranspose_mul_mul
|
[64, 1]
|
[69, 81]
|
intro x
|
m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
⊢ ∀ (x : n → 𝕜), 0 ≤ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x
|
m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemidef.conjTranspose_mul_mul
|
[64, 1]
|
[69, 81]
|
convert hM.2 (N.mulVec x) using 1
|
m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x
|
case h.e'_4
m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
x : n → 𝕜
⊢ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x = star (N.mulVec x) ⬝ᵥ M.mulVec (N.mulVec x)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemidef.conjTranspose_mul_mul
|
[64, 1]
|
[69, 81]
|
rw [mul_assoc, mulVec_mulVec, ← mulVec_mulVec, dotProduct_mulVec, star_mulVec]
|
case h.e'_4
m : Type ?u.23145
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
x : n → 𝕜
⊢ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x = star (N.mulVec x) ⬝ᵥ M.mulVec (N.mulVec x)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef.conjTranspose_mul_mul
|
[71, 1]
|
[78, 17]
|
refine' ⟨isHermitian_conjTranspose_mul_mul _ hM.1, _⟩
|
m : Type ?u.28307
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
⊢ (N.conjTranspose * M * N).PosDef
|
m : Type ?u.28307
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
⊢ ∀ (x : n → 𝕜), x ≠ 0 → 0 < star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef.conjTranspose_mul_mul
|
[71, 1]
|
[78, 17]
|
intros x hx
|
m : Type ?u.28307
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
⊢ ∀ (x : n → 𝕜), x ≠ 0 → 0 < star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x
|
m : Type ?u.28307
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
x : n → 𝕜
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef.conjTranspose_mul_mul
|
[71, 1]
|
[78, 17]
|
convert
hM.2 (N.mulVec x) (fun h => hx (eq_zero_of_mulVec_eq_zero hN h)) using 1
|
m : Type ?u.28307
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
x : n → 𝕜
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x
|
case h.e'_4
m : Type ?u.28307
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
x : n → 𝕜
hx : x ≠ 0
⊢ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x = star (N.mulVec x) ⬝ᵥ M.mulVec (N.mulVec x)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef.conjTranspose_mul_mul
|
[71, 1]
|
[78, 17]
|
rw [Matrix.mul_assoc, mulVec_mulVec, ← mulVec_mulVec, dotProduct_mulVec,
star_mulVec]
|
case h.e'_4
m : Type ?u.28307
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M N : Matrix n n 𝕜
hM : M.PosDef
hN : N.det ≠ 0
x : n → 𝕜
hx : x ≠ 0
⊢ star x ⬝ᵥ (N.conjTranspose * M * N).mulVec x = star (N.mulVec x) ⬝ᵥ M.mulVec (N.mulVec x)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.IsHermitian.nonsingular_inv
|
[80, 1]
|
[83, 65]
|
refine' (Matrix.inv_eq_right_inv _).symm
|
m : Type ?u.34280
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.IsHermitian
hMdet : IsUnit M.det
⊢ M⁻¹.IsHermitian
|
m : Type ?u.34280
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.IsHermitian
hMdet : IsUnit M.det
⊢ M * M⁻¹.conjTranspose = 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.IsHermitian.nonsingular_inv
|
[80, 1]
|
[83, 65]
|
rw [conjTranspose_nonsing_inv, hM.eq, mul_nonsing_inv _ hMdet]
|
m : Type ?u.34280
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.IsHermitian
hMdet : IsUnit M.det
⊢ M * M⁻¹.conjTranspose = 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.conj_symm
|
[85, 1]
|
[88, 43]
|
nth_rewrite 1 [star_dotProduct, star_mulVec]
|
m : Type ?u.36361
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
x : n → 𝕜
M : Matrix n n 𝕜
hM : M.IsHermitian
⊢ star (star x ⬝ᵥ M.mulVec x) = star x ⬝ᵥ M.mulVec x
|
m : Type ?u.36361
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
x : n → 𝕜
M : Matrix n n 𝕜
hM : M.IsHermitian
⊢ star (star (star x ᵥ* M.conjTranspose ⬝ᵥ x)) = star x ⬝ᵥ M.mulVec x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.conj_symm
|
[85, 1]
|
[88, 43]
|
rw [star_star, dotProduct_mulVec, hM.eq]
|
m : Type ?u.36361
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
x : n → 𝕜
M : Matrix n n 𝕜
hM : M.IsHermitian
⊢ star (star (star x ᵥ* M.conjTranspose ⬝ᵥ x)) = star x ⬝ᵥ M.mulVec x
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef.nonsingular_inv
|
[90, 1]
|
[99, 13]
|
refine' ⟨hM.1.nonsingular_inv (isUnit_iff_ne_zero.2 hM.det_ne_zero), _⟩
|
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
⊢ M⁻¹.PosDef
|
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
⊢ ∀ (x : n → 𝕜), x ≠ 0 → 0 < star x ⬝ᵥ M⁻¹.mulVec x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef.nonsingular_inv
|
[90, 1]
|
[99, 13]
|
intros x hx
|
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
⊢ ∀ (x : n → 𝕜), x ≠ 0 → 0 < star x ⬝ᵥ M⁻¹.mulVec x
|
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef.nonsingular_inv
|
[90, 1]
|
[99, 13]
|
have hMMinv := mul_nonsing_inv _ (isUnit_iff_ne_zero.2 hM.det_ne_zero)
|
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x
|
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef.nonsingular_inv
|
[90, 1]
|
[99, 13]
|
have hMinvdet : M⁻¹.det ≠ 0 := det_ne_zero_of_left_inverse hMMinv
|
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x
|
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef.nonsingular_inv
|
[90, 1]
|
[99, 13]
|
have hres :=
hM.2 (M⁻¹.mulVec x) (fun h => hx (eq_zero_of_mulVec_eq_zero hMinvdet h))
|
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x
|
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star (M⁻¹.mulVec x) ⬝ᵥ M.mulVec (M⁻¹.mulVec x)
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef.nonsingular_inv
|
[90, 1]
|
[99, 13]
|
rw [mulVec_mulVec, hMMinv, one_mulVec, star_dotProduct] at hres
|
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star (M⁻¹.mulVec x) ⬝ᵥ M.mulVec (M⁻¹.mulVec x)
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x
|
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star (star x ⬝ᵥ M⁻¹.mulVec x)
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef.nonsingular_inv
|
[90, 1]
|
[99, 13]
|
rw [conj_symm ((@isHermitian_inv _ _ _ _ _ _ M hM.Invertible).2 hM.1)] at hres
|
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star (star x ⬝ᵥ M⁻¹.mulVec x)
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x
|
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star x ⬝ᵥ M⁻¹.mulVec x
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef.nonsingular_inv
|
[90, 1]
|
[99, 13]
|
exact hres
|
m : Type ?u.40080
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type u_2
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n 𝕜
hM : M.PosDef
x : n → 𝕜
hx : x ≠ 0
hMMinv : M * M⁻¹ = 1
hMinvdet : M⁻¹.det ≠ 0
hres : 0 < star x ⬝ᵥ M⁻¹.mulVec x
⊢ 0 < star x ⬝ᵥ M⁻¹.mulVec x
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemidef.mul_mul_of_IsHermitian
|
[101, 1]
|
[103, 54]
|
convert hM.conjTranspose_mul_mul M N
|
m : Type ?u.45085
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.IsHermitian
⊢ (N * M * N).PosSemidef
|
case h.e'_8.h.e'_5.h.e'_5
m : Type ?u.45085
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.IsHermitian
⊢ N = N.conjTranspose
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemidef.mul_mul_of_IsHermitian
|
[101, 1]
|
[103, 54]
|
exact hN.symm
|
case h.e'_8.h.e'_5.h.e'_5
m : Type ?u.45085
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.IsHermitian
⊢ N = N.conjTranspose
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemidef.add
|
[105, 1]
|
[110, 37]
|
refine' ⟨hM.1.add hN.1, _⟩
|
m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
⊢ (M + N).PosSemidef
|
m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
⊢ ∀ (x : n → 𝕜), 0 ≤ star x ⬝ᵥ (M + N).mulVec x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemidef.add
|
[105, 1]
|
[110, 37]
|
intros x
|
m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
⊢ ∀ (x : n → 𝕜), 0 ≤ star x ⬝ᵥ (M + N).mulVec x
|
m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (M + N).mulVec x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemidef.add
|
[105, 1]
|
[110, 37]
|
simp only [add_mulVec, dotProduct_add, map_add]
|
m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ (M + N).mulVec x
|
m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ M.mulVec x + star x ⬝ᵥ N.mulVec x
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemidef.add
|
[105, 1]
|
[110, 37]
|
apply add_nonneg (hM.2 x) (hN.2 x)
|
m : Type ?u.47670
n : Type u_1
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
𝕜 : Type u_2
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
M N : Matrix n n 𝕜
hM : M.PosSemidef
hN : N.PosSemidef
x : n → 𝕜
⊢ 0 ≤ star x ⬝ᵥ M.mulVec x + star x ⬝ᵥ N.mulVec x
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.isUnit_det_of_PosDef_inv
|
[112, 1]
|
[117, 28]
|
apply isUnit_iff_ne_zero.2
|
m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
h : M⁻¹.PosDef
⊢ IsUnit M.det
|
m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
h : M⁻¹.PosDef
⊢ M.det ≠ 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.isUnit_det_of_PosDef_inv
|
[112, 1]
|
[117, 28]
|
have := h.isUnit_det
|
m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
h : M⁻¹.PosDef
⊢ M.det ≠ 0
|
m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
h : M⁻¹.PosDef
this : IsUnit M⁻¹.det
⊢ M.det ≠ 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.isUnit_det_of_PosDef_inv
|
[112, 1]
|
[117, 28]
|
rw [det_nonsing_inv, isUnit_ring_inverse] at this
|
m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
h : M⁻¹.PosDef
this : IsUnit M⁻¹.det
⊢ M.det ≠ 0
|
m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
h : M⁻¹.PosDef
this : IsUnit M.det
⊢ M.det ≠ 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.isUnit_det_of_PosDef_inv
|
[112, 1]
|
[117, 28]
|
apply IsUnit.ne_zero this
|
m : Type ?u.49238
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.49250
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
h : M⁻¹.PosDef
this : IsUnit M.det
⊢ M.det ≠ 0
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef_inv_iff_PosDef
|
[119, 1]
|
[125, 31]
|
constructor
|
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M⁻¹.PosDef ↔ M.PosDef
|
case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M⁻¹.PosDef → M.PosDef
case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M.PosDef → M⁻¹.PosDef
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef_inv_iff_PosDef
|
[119, 1]
|
[125, 31]
|
{ intros hM
rw [← Matrix.nonsing_inv_nonsing_inv M (isUnit_det_of_PosDef_inv hM)]
apply hM.nonsingular_inv }
|
case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M⁻¹.PosDef → M.PosDef
case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M.PosDef → M⁻¹.PosDef
|
case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M.PosDef → M⁻¹.PosDef
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef_inv_iff_PosDef
|
[119, 1]
|
[125, 31]
|
{ intros hM
exact hM.nonsingular_inv }
|
case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M.PosDef → M⁻¹.PosDef
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef_inv_iff_PosDef
|
[119, 1]
|
[125, 31]
|
intros hM
|
case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M⁻¹.PosDef → M.PosDef
|
case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M⁻¹.PosDef
⊢ M.PosDef
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef_inv_iff_PosDef
|
[119, 1]
|
[125, 31]
|
rw [← Matrix.nonsing_inv_nonsing_inv M (isUnit_det_of_PosDef_inv hM)]
|
case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M⁻¹.PosDef
⊢ M.PosDef
|
case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M⁻¹.PosDef
⊢ M⁻¹⁻¹.PosDef
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef_inv_iff_PosDef
|
[119, 1]
|
[125, 31]
|
apply hM.nonsingular_inv
|
case mp
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M⁻¹.PosDef
⊢ M⁻¹⁻¹.PosDef
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef_inv_iff_PosDef
|
[119, 1]
|
[125, 31]
|
intros hM
|
case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
⊢ M.PosDef → M⁻¹.PosDef
|
case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosDef
⊢ M⁻¹.PosDef
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosDef_inv_iff_PosDef
|
[119, 1]
|
[125, 31]
|
exact hM.nonsingular_inv
|
case mpr
m : Type ?u.50670
n : Type u_1
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
𝕜 : Type ?u.50682
inst✝⁵ : NormedField 𝕜
inst✝⁴ : PartialOrder 𝕜
inst✝³ : StarRing 𝕜
inst✝² : StarOrderedRing 𝕜
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : M.PosDef
⊢ M⁻¹.PosDef
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemiDef.IsSymm
|
[127, 1]
|
[140, 12]
|
let h_A_IsHermitian := hA.1
|
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
⊢ A.IsSymm
|
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian : A.IsHermitian := hA.left
⊢ A.IsSymm
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemiDef.IsSymm
|
[127, 1]
|
[140, 12]
|
rw [Matrix.isHermitian_iff_isSymmetric] at h_A_IsHermitian
|
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian : A.IsHermitian := hA.left
⊢ A.IsSymm
|
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian : (toEuclideanLin A).IsSymmetric
⊢ A.IsSymm
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemiDef.IsSymm
|
[127, 1]
|
[140, 12]
|
simp [LinearMap.IsSymmetric, toEuclideanLin] at h_A_IsHermitian
|
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian : (toEuclideanLin A).IsSymmetric
⊢ A.IsSymm
|
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
⊢ A.IsSymm
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemiDef.IsSymm
|
[127, 1]
|
[140, 12]
|
apply IsSymm.ext
|
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
⊢ A.IsSymm
|
case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
⊢ ∀ (i j : Fin n), A j i = A i j
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemiDef.IsSymm
|
[127, 1]
|
[140, 12]
|
intros i j
|
case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
⊢ ∀ (i j : Fin n), A j i = A i j
|
case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
⊢ A j i = A i j
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemiDef.IsSymm
|
[127, 1]
|
[140, 12]
|
have hAij := h_A_IsHermitian (fun k => if k = i then 1 else 0) (fun k => if k = j then 1 else 0)
|
case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
⊢ A j i = A i j
|
case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hAij :
(Finset.univ.sum fun x =>
A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = i then 1 else 0) x * if x = j then 1 else 0) =
Finset.univ.sum fun x =>
(if x = i then 1 else 0) * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = j then 1 else 0) x
⊢ A j i = A i j
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemiDef.IsSymm
|
[127, 1]
|
[140, 12]
|
have hi : (Finset.sum Finset.univ fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ)))
(fun k => if k = i then 1 else 0) x) = A j i := by simp
|
case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hAij :
(Finset.univ.sum fun x =>
A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = i then 1 else 0) x * if x = j then 1 else 0) =
Finset.univ.sum fun x =>
(if x = i then 1 else 0) * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = j then 1 else 0) x
⊢ A j i = A i j
|
case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hAij :
(Finset.univ.sum fun x =>
A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = i then 1 else 0) x * if x = j then 1 else 0) =
Finset.univ.sum fun x =>
(if x = i then 1 else 0) * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = j then 1 else 0) x
hi : (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i
⊢ A j i = A i j
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemiDef.IsSymm
|
[127, 1]
|
[140, 12]
|
have hj : (Finset.sum Finset.univ fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ)))
(fun k => if k = j then 1 else 0) x) = A i j := by simp
|
case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hAij :
(Finset.univ.sum fun x =>
A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = i then 1 else 0) x * if x = j then 1 else 0) =
Finset.univ.sum fun x =>
(if x = i then 1 else 0) * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = j then 1 else 0) x
hi : (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i
⊢ A j i = A i j
|
case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hAij :
(Finset.univ.sum fun x =>
A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = i then 1 else 0) x * if x = j then 1 else 0) =
Finset.univ.sum fun x =>
(if x = i then 1 else 0) * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = j then 1 else 0) x
hi : (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i
hj : (Finset.univ.sum fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = j then 1 else 0) x) = A i j
⊢ A j i = A i j
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemiDef.IsSymm
|
[127, 1]
|
[140, 12]
|
simp [WithLp.equiv, mulVec, dotProduct] at hAij
|
case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hAij :
(Finset.univ.sum fun x =>
A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = i then 1 else 0) x * if x = j then 1 else 0) =
Finset.univ.sum fun x =>
(if x = i then 1 else 0) * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = j then 1 else 0) x
hi : (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i
hj : (Finset.univ.sum fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = j then 1 else 0) x) = A i j
⊢ A j i = A i j
|
case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hi : (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i
hj : (Finset.univ.sum fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = j then 1 else 0) x) = A i j
hAij :
(Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) =
Finset.univ.sum fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = j then 1 else 0) x
⊢ A j i = A i j
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemiDef.IsSymm
|
[127, 1]
|
[140, 12]
|
erw [hi, hj] at hAij
|
case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hi : (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i
hj : (Finset.univ.sum fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = j then 1 else 0) x) = A i j
hAij :
(Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) =
Finset.univ.sum fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = j then 1 else 0) x
⊢ A j i = A i j
|
case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hi : (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i
hj : (Finset.univ.sum fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = j then 1 else 0) x) = A i j
hAij : A j i = A i j
⊢ A j i = A i j
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemiDef.IsSymm
|
[127, 1]
|
[140, 12]
|
rw [hAij]
|
case a
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hi : (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i
hj : (Finset.univ.sum fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = j then 1 else 0) x) = A i j
hAij : A j i = A i j
⊢ A j i = A i j
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemiDef.IsSymm
|
[127, 1]
|
[140, 12]
|
simp
|
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hAij :
(Finset.univ.sum fun x =>
A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = i then 1 else 0) x * if x = j then 1 else 0) =
Finset.univ.sum fun x =>
(if x = i then 1 else 0) * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = j then 1 else 0) x
⊢ (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean
|
Matrix.PosSemiDef.IsSymm
|
[127, 1]
|
[140, 12]
|
simp
|
m : Type ?u.51903
n✝ : Type ?u.51906
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
𝕜 : Type ?u.51915
inst✝⁴ : NormedField 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : StarRing 𝕜
inst✝¹ : StarOrderedRing 𝕜
inst✝ : RCLike 𝕜
n : ℕ
A : Matrix (Fin n) (Fin n) ℝ
hA : A.PosSemidef
h_A_IsHermitian :
∀ (x y : EuclideanSpace ℝ (Fin n)),
(Finset.univ.sum fun x_1 => A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) x) x_1 * y x_1) =
Finset.univ.sum fun x_1 => x x_1 * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) y) x_1
i j : Fin n
hAij :
(Finset.univ.sum fun x =>
A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = i then 1 else 0) x * if x = j then 1 else 0) =
Finset.univ.sum fun x =>
(if x = i then 1 else 0) * A.mulVec ((WithLp.equiv 2 (Fin n → ℝ)) fun k => if k = j then 1 else 0) x
hi : (Finset.univ.sum fun x => A j x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = i then 1 else 0) x) = A j i
⊢ (Finset.univ.sum fun x => A i x * (Equiv.refl (WithLp 2 (Fin n → ℝ))) (fun k => if k = j then 1 else 0) x) = A i j
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Tactic/DCP/AtomLibrary/Fns/Norm.lean
|
CvxLean.norm2₂_eq_norm2
|
[57, 1]
|
[58, 36]
|
simp [Norm.norm, sqrt_eq_rpow]
|
x y : ℝ
⊢ ‖![x, y]‖ = (x ^ 2 + y ^ 2).sqrt
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.hminₚ_pos
|
[180, 1]
|
[182, 25]
|
unfold hminₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < hminₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.hminₚ_pos
|
[180, 1]
|
[182, 25]
|
norm_num
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.hminₚ_le_hmaxₚ
|
[184, 1]
|
[185, 31]
|
unfold hminₚ hmaxₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ hminₚ ≤ hmaxₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 1 ≤ 100
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.hminₚ_le_hmaxₚ
|
[184, 1]
|
[185, 31]
|
norm_num
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 1 ≤ 100
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.wminₚ_pos
|
[193, 1]
|
[195, 25]
|
unfold wminₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < wminₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.wminₚ_pos
|
[193, 1]
|
[195, 25]
|
norm_num
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.wminₚ_le_wmaxₚ
|
[197, 1]
|
[198, 31]
|
unfold wminₚ wmaxₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ wminₚ ≤ wmaxₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 1 ≤ 100
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.wminₚ_le_wmaxₚ
|
[197, 1]
|
[198, 31]
|
norm_num
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 1 ≤ 100
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.Rmaxₚ_pos
|
[203, 1]
|
[205, 25]
|
unfold Rmaxₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < Rmaxₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 10
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.Rmaxₚ_pos
|
[203, 1]
|
[205, 25]
|
norm_num
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 10
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.σₚ_pos
|
[210, 1]
|
[212, 22]
|
unfold σₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < σₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 0.5
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.σₚ_pos
|
[210, 1]
|
[212, 22]
|
norm_num
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 0.5
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.F₁ₚ_pos
|
[217, 1]
|
[219, 23]
|
unfold F₁ₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < F₁ₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 10
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.F₁ₚ_pos
|
[217, 1]
|
[219, 23]
|
norm_num
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 10
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.F₂ₚ_pos
|
[224, 1]
|
[226, 23]
|
unfold F₂ₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < F₂ₚ
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 20
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/TrussDesign.lean
|
TrussDesign.F₂ₚ_pos
|
[224, 1]
|
[226, 23]
|
norm_num
|
hmin hmax : ℝ
hmin_pos : 0 < hmin
hmin_le_hmax : hmin ≤ hmax
wmin wmax : ℝ
wmin_pos : 0 < wmin
wmin_le_wmax : wmin ≤ wmax
Rmax σ F₁ F₂ : ℝ
⊢ 0 < 20
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
simp [rotatedSoCone, rotateSoCone]
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
⊢ match t.rotateSoCone x with
| (v, w, x) => v.rotatedSoCone w x
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
have habsx0t : |x 0| ≤ t := by
rw [soCone, Fin.sum_univ_succ] at h
have hS : 0 ≤ ∑ i : Fin n, x (Fin.succ i) ^ 2 :=
Finset.sum_nonneg (fun i _ => (rpow_two (x i.succ)).symm ▸ sq_nonneg (x i.succ))
exact abs_le_of_sqrt_sq_add_nonneg_le hS h
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
have ht : 0 ≤ t := le_trans (abs_nonneg _) habsx0t
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
replace ⟨hx0t, hnx0t⟩ := abs_le.mp habsx0t
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
split_ands
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2 ∧ 0 ≤ (t + x 0) / √2 ∧ 0 ≤ (t - x 0) / √2
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2
case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t + x 0) / √2
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
{ field_simp
have hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2 := by norm_cast; ring
rw [hrw, le_sub_iff_add_le, add_comm]
unfold soCone at h; norm_cast at h ⊢
rw [← Fin.sum_univ_succ (f := fun i => (x i) ^ 2)]
rw [← sqrt_le_left ht]
exact h }
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2
case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t + x 0) / √2
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2
|
case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t + x 0) / √2
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
{ simp [le_div_iff]; linarith }
|
case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t + x 0) / √2
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2
|
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
{ simp [le_div_iff]; linarith }
|
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
rw [soCone, Fin.sum_univ_succ] at h
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
⊢ |x 0| ≤ t
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : (x 0 ^ 2 + ∑ i : Fin n, x i.succ ^ 2).sqrt ≤ t
⊢ |x 0| ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
have hS : 0 ≤ ∑ i : Fin n, x (Fin.succ i) ^ 2 :=
Finset.sum_nonneg (fun i _ => (rpow_two (x i.succ)).symm ▸ sq_nonneg (x i.succ))
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : (x 0 ^ 2 + ∑ i : Fin n, x i.succ ^ 2).sqrt ≤ t
⊢ |x 0| ≤ t
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : (x 0 ^ 2 + ∑ i : Fin n, x i.succ ^ 2).sqrt ≤ t
hS : 0 ≤ ∑ i : Fin n, x i.succ ^ 2
⊢ |x 0| ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
exact abs_le_of_sqrt_sq_add_nonneg_le hS h
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : (x 0 ^ 2 + ∑ i : Fin n, x i.succ ^ 2).sqrt ≤ t
hS : 0 ≤ ∑ i : Fin n, x i.succ ^ 2
⊢ |x 0| ≤ t
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
field_simp
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) / √2 * ((t - x 0) / √2) * 2
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) * (t - x 0)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
have hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2 := by norm_cast; ring
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) * (t - x 0)
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) * (t - x 0)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
rw [hrw, le_sub_iff_add_le, add_comm]
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
⊢ ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ (t + x 0) * (t - x 0)
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
⊢ x 0 ^ 2 + ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ t ^ 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
unfold soCone at h
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
⊢ x 0 ^ 2 + ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ t ^ 2
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : (∑ i : Fin n.succ, x i ^ 2).sqrt ≤ t
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
⊢ x 0 ^ 2 + ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ t ^ 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
norm_cast at h ⊢
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : (∑ i : Fin n.succ, x i ^ 2).sqrt ≤ t
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
⊢ x 0 ^ 2 + ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ t ^ 2
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
h : (∑ x_1 : Fin n.succ, x x_1 ^ 2).sqrt ≤ t
⊢ x 0 ^ 2 + ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ t ^ 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
rw [← Fin.sum_univ_succ (f := fun i => (x i) ^ 2)]
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
h : (∑ x_1 : Fin n.succ, x x_1 ^ 2).sqrt ≤ t
⊢ x 0 ^ 2 + ∑ x_1 : Fin n, x x_1.succ ^ 2 ≤ t ^ 2
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
h : (∑ x_1 : Fin n.succ, x x_1 ^ 2).sqrt ≤ t
⊢ ∑ i : Fin (n + 1), x i ^ 2 ≤ t ^ 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
rw [← sqrt_le_left ht]
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
h : (∑ x_1 : Fin n.succ, x x_1 ^ 2).sqrt ≤ t
⊢ ∑ i : Fin (n + 1), x i ^ 2 ≤ t ^ 2
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
h : (∑ x_1 : Fin n.succ, x x_1 ^ 2).sqrt ≤ t
⊢ (∑ i : Fin (n + 1), x i ^ 2).sqrt ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
exact h
|
case refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
hrw : (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
h : (∑ x_1 : Fin n.succ, x x_1 ^ 2).sqrt ≤ t
⊢ (∑ i : Fin (n + 1), x i ^ 2).sqrt ≤ t
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
ring
|
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ (t + x 0) * (t - x 0) = t ^ 2 - x 0 ^ 2
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
simp [le_div_iff]
|
case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t + x 0) / √2
|
case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ t + x 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
linarith
|
case refine_2.refine_1
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ t + x 0
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
simp [le_div_iff]
|
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ 0 ≤ (t - x 0) / √2
|
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ x 0 ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.rotateSoCone_rotatedSoCone
|
[43, 1]
|
[62, 34]
|
linarith
|
case refine_2.refine_2
n✝ : Type ?u.2341
m : Type ?u.2338
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
t : ℝ
x : Fin n.succ → ℝ
h : t.soCone x
habsx0t : |x 0| ≤ t
ht : 0 ≤ t
hx0t : -t ≤ x 0
hnx0t : x 0 ≤ t
⊢ x 0 ≤ t
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
simp [soCone, unrotateSoCone]
|
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : v.rotatedSoCone w x
⊢ match v.unrotateSoCone w x with
| (t, x) => t.soCone x
|
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : v.rotatedSoCone w x
⊢ (∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2).sqrt ≤ (v + w) / √2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
replace ⟨h, hv, hw⟩ := h
|
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : v.rotatedSoCone w x
⊢ (∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2).sqrt ≤ (v + w) / √2
|
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ (∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2).sqrt ≤ (v + w) / √2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
rw [sqrt_le_iff]
|
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ (∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2).sqrt ≤ (v + w) / √2
|
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ 0 ≤ (v + w) / √2 ∧ ∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2 ≤ ((v + w) / √2) ^ 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
split_ands
|
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ 0 ≤ (v + w) / √2 ∧ ∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2 ≤ ((v + w) / √2) ^ 2
|
case refine_1
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ 0 ≤ (v + w) / √2
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ ∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2 ≤ ((v + w) / √2) ^ 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
{ simp [le_div_iff]; linarith }
|
case refine_1
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ 0 ≤ (v + w) / √2
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ ∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2 ≤ ((v + w) / √2) ^ 2
|
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ ∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2 ≤ ((v + w) / √2) ^ 2
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
{ rw [Fin.sum_univ_succ]
simp [Matrix.vecCons]
rw [add_comm, ← le_sub_iff_add_le]
field_simp
have hrw : ((v + w) ^ 2 - (v - w) ^ 2) / 2 = v * w * 2 := by norm_cast; ring
norm_cast at hrw h
rwa [hrw] }
|
case refine_2
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ ∑ x_1 : Fin (n + 1), Matrix.vecCons ((v - w) / √2) x x_1 ^ 2 ≤ ((v + w) / √2) ^ 2
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/SOCone.lean
|
Real.unrotateSoCone_soCone
|
[68, 1]
|
[81, 16]
|
simp [le_div_iff]
|
case refine_1
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ 0 ≤ (v + w) / √2
|
case refine_1
n✝ : Type ?u.32056
m : Type ?u.32053
inst✝¹ : Fintype m
inst✝ : Fintype n✝
n : ℕ
v w : ℝ
x : Fin n → ℝ
h : ∑ i : Fin n, x i ^ 2 ≤ v * w * 2
hv : 0 ≤ v
hw : 0 ≤ w
⊢ 0 ≤ v + w
|
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