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/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
{ erw [toLin'_apply]
simp only [xs', OrthonormalBasis.coe_toBasis_repr_apply, of_apply,
OrthonormalBasis.repr_reindex]
erw [Equiv.symm_apply_apply, EuclideanSpace.single, WithLp.equiv_symm_pi_apply 2, mulVec_single]
simp_rw [mul_one]
rfl }
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
xs'.repr ((toLin' A) (EuclideanSpace.single j 1)) ((Fintype.equivOfCardEq ⋯).symm i)
case h.e'_3
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ (diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j =
↑(as' ((Fintype.equivOfCardEq ⋯).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq ⋯).symm i)
|
case h.e'_3
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ (diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j =
↑(as' ((Fintype.equivOfCardEq ⋯).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq ⋯).symm i)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
{ simp only [diagonal_mul, Function.comp]
erw [Basis.toMatrix_apply, OrthonormalBasis.coe_toBasis_repr_apply,
OrthonormalBasis.repr_reindex, Pi.basisFun_apply, LinearMap.coe_stdBasis,
EuclideanSpace.single, WithLp.equiv_symm_pi_apply 2,
Equiv.symm_apply_apply, Equiv.apply_symm_apply]
congr; simp }
|
case h.e'_3
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ (diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j =
↑(as' ((Fintype.equivOfCardEq ⋯).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq ⋯).symm i)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
simp only [xs', OrthonormalBasis.coe_reindex, Equiv.symm_symm]
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
⊢ ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
⊢ ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) ((⇑xs ∘ ⇑(Fintype.equivOfCardEq ⋯)) j)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
intros j
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
⊢ ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) ((⇑xs ∘ ⇑(Fintype.equivOfCardEq ⋯)) j)
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j✝ : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
j : Fin (Fintype.card n)
⊢ Module.End.HasEigenvector (toLin' A) (↑(as' j)) ((⇑xs ∘ ⇑(Fintype.equivOfCardEq ⋯)) j)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
exact (hxs ((Fintype.equivOfCardEq (Fintype.card_fin _)) j))
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j✝ : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
j : Fin (Fintype.card n)
⊢ Module.End.HasEigenvector (toLin' A) (↑(as' j)) ((⇑xs ∘ ⇑(Fintype.equivOfCardEq ⋯)) j)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
erw [toLin'_apply]
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
xs'.repr ((toLin' A) (EuclideanSpace.single j 1)) ((Fintype.equivOfCardEq ⋯).symm i)
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
xs'.repr (A.mulVec (EuclideanSpace.single j 1)) ((Fintype.equivOfCardEq ⋯).symm i)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
simp only [xs', OrthonormalBasis.coe_toBasis_repr_apply, of_apply,
OrthonormalBasis.repr_reindex]
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
xs'.repr (A.mulVec (EuclideanSpace.single j 1)) ((Fintype.equivOfCardEq ⋯).symm i)
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ xs.repr (A.transpose j) i =
xs.repr (A.mulVec (EuclideanSpace.single j 1))
((Fintype.equivOfCardEq ⋯).symm.symm ((Fintype.equivOfCardEq ⋯).symm i))
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
erw [Equiv.symm_apply_apply, EuclideanSpace.single, WithLp.equiv_symm_pi_apply 2, mulVec_single]
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ xs.repr (A.transpose j) i =
xs.repr (A.mulVec (EuclideanSpace.single j 1))
((Fintype.equivOfCardEq ⋯).symm.symm ((Fintype.equivOfCardEq ⋯).symm i))
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ EquivLike.coe xs.repr (A.transpose j) i = xs.repr (fun i => A i j * 1) i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
simp_rw [mul_one]
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ EquivLike.coe xs.repr (A.transpose j) i = xs.repr (fun i => A i j * 1) i
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ EquivLike.coe xs.repr (A.transpose j) i = xs.repr (fun i => A i j) i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
rfl
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ EquivLike.coe xs.repr (A.transpose j) i = xs.repr (fun i => A i j) i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
simp only [diagonal_mul, Function.comp]
|
case h.e'_3
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ (diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j =
↑(as' ((Fintype.equivOfCardEq ⋯).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq ⋯).symm i)
|
case h.e'_3
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ ↑(as i) * xs.toBasis.toMatrix (⇑(Pi.basisFun 𝕜 n)) i j =
↑(as' ((Fintype.equivOfCardEq ⋯).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq ⋯).symm i)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
erw [Basis.toMatrix_apply, OrthonormalBasis.coe_toBasis_repr_apply,
OrthonormalBasis.repr_reindex, Pi.basisFun_apply, LinearMap.coe_stdBasis,
EuclideanSpace.single, WithLp.equiv_symm_pi_apply 2,
Equiv.symm_apply_apply, Equiv.apply_symm_apply]
|
case h.e'_3
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ ↑(as i) * xs.toBasis.toMatrix (⇑(Pi.basisFun 𝕜 n)) i j =
↑(as' ((Fintype.equivOfCardEq ⋯).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq ⋯).symm i)
|
case h.e'_3
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ ↑(as i) * EquivLike.coe xs.repr (Pi.single j 1) i =
↑(as' ((Fintype.equivOfCardEq ⋯).symm i)) *
xs.repr (fun a => Decidable.rec (fun h => (fun h => 0 a) h) (fun h => (fun h => ⋯ ▸ 1) h) (inst✝ a j)) i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
congr
|
case h.e'_3
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ ↑(as i) * EquivLike.coe xs.repr (Pi.single j 1) i =
↑(as' ((Fintype.equivOfCardEq ⋯).symm i)) *
xs.repr (fun a => Decidable.rec (fun h => (fun h => 0 a) h) (fun h => (fun h => ⋯ ▸ 1) h) (inst✝ a j)) i
|
case h.e'_3.e_a.e_a.e_a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ i = (Fintype.equivOfCardEq ⋯) ((Fintype.equivOfCardEq ⋯).symm i)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
simp
|
case h.e'_3.e_a.e_a.e_a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ i = (Fintype.equivOfCardEq ⋯) ((Fintype.equivOfCardEq ⋯).symm i)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.det_eq_prod_eigenvalues
|
[68, 1]
|
[73, 7]
|
apply mul_left_cancel₀ (det_ne_zero_of_left_inverse
(Basis.toMatrix_mul_toMatrix_flip (Pi.basisFun 𝕜 n) xs.toBasis))
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
⊢ A.det = ↑(∏ i : n, as i)
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
⊢ (xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)).det * A.det = (xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)).det * ↑(∏ i : n, as i)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.det_eq_prod_eigenvalues
|
[68, 1]
|
[73, 7]
|
rw [← det_mul, spectral_theorem xs as hxs, det_mul, mul_comm, det_diagonal]
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
⊢ (xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)).det * A.det = (xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)).det * ↑(∏ i : n, as i)
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
⊢ (xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)).det * ∏ i : n, (RCLike.ofReal ∘ as) i =
(xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)).det * ↑(∏ i : n, as i)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.det_eq_prod_eigenvalues
|
[68, 1]
|
[73, 7]
|
simp
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
⊢ (xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)).det * ∏ i : n, (RCLike.ofReal ∘ as) i =
(xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)).det * ↑(∏ i : n, as i)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.IsHermitian.hasEigenvector_eigenvectorBasis
|
[30, 1]
|
[33, 62]
|
simp only [IsHermitian.eigenvectorBasis, OrthonormalBasis.coe_reindex]
|
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_1
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.IsHermitian
i : n
⊢ Module.End.HasEigenvector (toLin' A) (↑(hA.eigenvalues i)) (hA.eigenvectorBasis i)
|
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_1
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.IsHermitian
i : n
⊢ Module.End.HasEigenvector (toLin' A) (↑(hA.eigenvalues i))
((⇑(⋯.eigenvectorBasis ⋯) ∘ ⇑(Fintype.equivOfCardEq ⋯).symm) i)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.IsHermitian.hasEigenvector_eigenvectorBasis
|
[30, 1]
|
[33, 62]
|
apply LinearMap.IsSymmetric.hasEigenvector_eigenvectorBasis
|
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_1
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.IsHermitian
i : n
⊢ Module.End.HasEigenvector (toLin' A) (↑(hA.eigenvalues i))
((⇑(⋯.eigenvectorBasis ⋯) ∘ ⇑(Fintype.equivOfCardEq ⋯).symm) i)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
rw [basis_toMatrix_basisFun_mul]
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
⊢ xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n) * A = diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
⊢ (of fun i j => (xs.toBasis.repr (A.transpose j)) i) =
diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
ext i j
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
⊢ (of fun i j => (xs.toBasis.repr (A.transpose j)) i) =
diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)
|
case a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
let xs' := xs.reindex (Fintype.equivOfCardEq (Fintype.card_fin _)).symm
|
case a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j
|
case a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
let as' : Fin (Fintype.card n) → ℝ :=
fun i => as <| (Fintype.equivOfCardEq (Fintype.card_fin _)) i
|
case a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j
|
case a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
have hxs' : ∀ j, Module.End.HasEigenvector (Matrix.toLin' A) (as' j) (xs' j) := by
simp only [xs', OrthonormalBasis.coe_reindex, Equiv.symm_symm]
intros j
exact (hxs ((Fintype.equivOfCardEq (Fintype.card_fin _)) j))
|
case a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j
|
case a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
convert @LinearMap.spectral_theorem' 𝕜 _
(PiLp 2 (fun (_ : n) => 𝕜)) _ _ (Fintype.card n) (Matrix.toLin' A)
(EuclideanSpace.single j 1)
((Fintype.equivOfCardEq (Fintype.card_fin _)).symm i)
xs' as' hxs'
|
case a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
xs'.repr ((toLin' A) (EuclideanSpace.single j 1)) ((Fintype.equivOfCardEq ⋯).symm i)
case h.e'_3
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ (diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j =
↑(as' ((Fintype.equivOfCardEq ⋯).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq ⋯).symm i)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
{ erw [toLin'_apply]
simp only [xs', OrthonormalBasis.coe_toBasis_repr_apply, of_apply,
OrthonormalBasis.repr_reindex]
erw [Equiv.symm_apply_apply, EuclideanSpace.single, WithLp.equiv_symm_pi_apply 2, mulVec_single]
simp_rw [mul_one]
rfl }
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
xs'.repr ((toLin' A) (EuclideanSpace.single j 1)) ((Fintype.equivOfCardEq ⋯).symm i)
case h.e'_3
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ (diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j =
↑(as' ((Fintype.equivOfCardEq ⋯).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq ⋯).symm i)
|
case h.e'_3
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ (diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j =
↑(as' ((Fintype.equivOfCardEq ⋯).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq ⋯).symm i)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
{ simp only [diagonal_mul, Function.comp]
erw [Basis.toMatrix_apply, OrthonormalBasis.coe_toBasis_repr_apply,
OrthonormalBasis.repr_reindex, Pi.basisFun_apply, LinearMap.coe_stdBasis,
EuclideanSpace.single, WithLp.equiv_symm_pi_apply 2,
Equiv.symm_apply_apply, Equiv.apply_symm_apply]
congr; simp }
|
case h.e'_3
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ (diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j =
↑(as' ((Fintype.equivOfCardEq ⋯).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq ⋯).symm i)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
simp only [xs', OrthonormalBasis.coe_reindex, Equiv.symm_symm]
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
⊢ ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
⊢ ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) ((⇑xs ∘ ⇑(Fintype.equivOfCardEq ⋯)) j)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
intros j
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
⊢ ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) ((⇑xs ∘ ⇑(Fintype.equivOfCardEq ⋯)) j)
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j✝ : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
j : Fin (Fintype.card n)
⊢ Module.End.HasEigenvector (toLin' A) (↑(as' j)) ((⇑xs ∘ ⇑(Fintype.equivOfCardEq ⋯)) j)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
exact (hxs ((Fintype.equivOfCardEq (Fintype.card_fin _)) j))
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j✝ : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
j : Fin (Fintype.card n)
⊢ Module.End.HasEigenvector (toLin' A) (↑(as' j)) ((⇑xs ∘ ⇑(Fintype.equivOfCardEq ⋯)) j)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
erw [toLin'_apply]
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
xs'.repr ((toLin' A) (EuclideanSpace.single j 1)) ((Fintype.equivOfCardEq ⋯).symm i)
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
xs'.repr (A.mulVec (EuclideanSpace.single j 1)) ((Fintype.equivOfCardEq ⋯).symm i)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
simp only [xs', OrthonormalBasis.coe_toBasis_repr_apply, of_apply,
OrthonormalBasis.repr_reindex]
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
xs'.repr (A.mulVec (EuclideanSpace.single j 1)) ((Fintype.equivOfCardEq ⋯).symm i)
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ xs.repr (A.transpose j) i =
xs.repr (A.mulVec (EuclideanSpace.single j 1))
((Fintype.equivOfCardEq ⋯).symm.symm ((Fintype.equivOfCardEq ⋯).symm i))
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
erw [Equiv.symm_apply_apply, EuclideanSpace.single, WithLp.equiv_symm_pi_apply 2, mulVec_single]
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ xs.repr (A.transpose j) i =
xs.repr (A.mulVec (EuclideanSpace.single j 1))
((Fintype.equivOfCardEq ⋯).symm.symm ((Fintype.equivOfCardEq ⋯).symm i))
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ EquivLike.coe xs.repr (A.transpose j) i = xs.repr (fun i => A i j * 1) i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
simp_rw [mul_one]
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ EquivLike.coe xs.repr (A.transpose j) i = xs.repr (fun i => A i j * 1) i
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ EquivLike.coe xs.repr (A.transpose j) i = xs.repr (fun i => A i j) i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
rfl
|
case h.e'_2
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ EquivLike.coe xs.repr (A.transpose j) i = xs.repr (fun i => A i j) i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
simp only [diagonal_mul, Function.comp]
|
case h.e'_3
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ (diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)) i j =
↑(as' ((Fintype.equivOfCardEq ⋯).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq ⋯).symm i)
|
case h.e'_3
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ ↑(as i) * xs.toBasis.toMatrix (⇑(Pi.basisFun 𝕜 n)) i j =
↑(as' ((Fintype.equivOfCardEq ⋯).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq ⋯).symm i)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
erw [Basis.toMatrix_apply, OrthonormalBasis.coe_toBasis_repr_apply,
OrthonormalBasis.repr_reindex, Pi.basisFun_apply, LinearMap.coe_stdBasis,
EuclideanSpace.single, WithLp.equiv_symm_pi_apply 2,
Equiv.symm_apply_apply, Equiv.apply_symm_apply]
|
case h.e'_3
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ ↑(as i) * xs.toBasis.toMatrix (⇑(Pi.basisFun 𝕜 n)) i j =
↑(as' ((Fintype.equivOfCardEq ⋯).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq ⋯).symm i)
|
case h.e'_3
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ ↑(as i) * EquivLike.coe xs.repr (Pi.single j 1) i =
↑(as' ((Fintype.equivOfCardEq ⋯).symm i)) *
xs.repr (fun a => Decidable.rec (fun h => (fun h => 0 a) h) (fun h => (fun h => ⋯ ▸ 1) h) (inst✝ a j)) i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
congr
|
case h.e'_3
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ ↑(as i) * EquivLike.coe xs.repr (Pi.single j 1) i =
↑(as' ((Fintype.equivOfCardEq ⋯).symm i)) *
xs.repr (fun a => Decidable.rec (fun h => (fun h => 0 a) h) (fun h => (fun h => ⋯ ▸ 1) h) (inst✝ a j)) i
|
case h.e'_3.e_a.e_a.e_a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ i = (Fintype.equivOfCardEq ⋯) ((Fintype.equivOfCardEq ⋯).symm i)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.spectral_theorem
|
[37, 1]
|
[66, 18]
|
simp
|
case h.e'_3.e_a.e_a.e_a
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) 𝕜 (EuclideanSpace 𝕜 n) := xs.reindex (Fintype.equivOfCardEq ⋯).symm
as' : Fin (Fintype.card n) → ℝ := fun i => as ((Fintype.equivOfCardEq ⋯) i)
hxs' : ∀ (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (↑(as' j)) (xs' j)
⊢ i = (Fintype.equivOfCardEq ⋯) ((Fintype.equivOfCardEq ⋯).symm i)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.det_eq_prod_eigenvalues
|
[68, 1]
|
[73, 7]
|
apply mul_left_cancel₀ (det_ne_zero_of_left_inverse
(Basis.toMatrix_mul_toMatrix_flip (Pi.basisFun 𝕜 n) xs.toBasis))
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
⊢ A.det = ↑(∏ i : n, as i)
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
⊢ (xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)).det * A.det = (xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)).det * ↑(∏ i : n, as i)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.det_eq_prod_eigenvalues
|
[68, 1]
|
[73, 7]
|
rw [← det_mul, spectral_theorem xs as hxs, det_mul, mul_comm, det_diagonal]
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
⊢ (xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)).det * A.det = (xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)).det * ↑(∏ i : n, as i)
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
⊢ (xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)).det * ∏ i : n, (RCLike.ofReal ∘ as) i =
(xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)).det * ↑(∏ i : n, as i)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Spectrum.lean
|
Matrix.det_eq_prod_eigenvalues
|
[68, 1]
|
[73, 7]
|
simp
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
inst✝² : DecidableEq 𝕜
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)
as : n → ℝ
hxs : ∀ (j : n), Module.End.HasEigenvector (toLin' A) (↑(as j)) (xs j)
⊢ (xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)).det * ∏ i : n, (RCLike.ofReal ∘ as) i =
(xs.toBasis.toMatrix ⇑(Pi.basisFun 𝕜 n)).det * ↑(∏ i : n, as i)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
unfold expCone
|
t x : ℝ
⊢ x.exp ≤ t ↔ x.expCone 1 t
|
t x : ℝ
⊢ x.exp ≤ t ↔ 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
rw [iff_def]
|
t x : ℝ
⊢ x.exp ≤ t ↔ 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
|
t x : ℝ
⊢ (x.exp ≤ t → 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0) ∧
(0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0 → x.exp ≤ t)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
split_ands
|
t x : ℝ
⊢ (x.exp ≤ t → 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0) ∧
(0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0 → x.exp ≤ t)
|
case refine_1
t x : ℝ
⊢ x.exp ≤ t → 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
case refine_2
t x : ℝ
⊢ 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0 → x.exp ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
{ intro hexp
apply Or.intro_left
split_ands
{ apply Real.zero_lt_one }
{ rwa [div_one, one_mul] } }
|
case refine_1
t x : ℝ
⊢ x.exp ≤ t → 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
case refine_2
t x : ℝ
⊢ 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0 → x.exp ≤ t
|
case refine_2
t x : ℝ
⊢ 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0 → x.exp ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
{ intro h
cases h with
| inl h =>
have h : 1 * exp (x / 1) ≤ t := h.2
rwa [div_one, one_mul] at h
| inr h =>
exfalso
exact zero_ne_one h.1.symm }
|
case refine_2
t x : ℝ
⊢ 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0 → x.exp ≤ t
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
intro hexp
|
case refine_1
t x : ℝ
⊢ x.exp ≤ t → 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
|
case refine_1
t x : ℝ
hexp : x.exp ≤ t
⊢ 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
apply Or.intro_left
|
case refine_1
t x : ℝ
hexp : x.exp ≤ t
⊢ 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
|
case refine_1.h
t x : ℝ
hexp : x.exp ≤ t
⊢ 0 < 1 ∧ 1 * (x / 1).exp ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
split_ands
|
case refine_1.h
t x : ℝ
hexp : x.exp ≤ t
⊢ 0 < 1 ∧ 1 * (x / 1).exp ≤ t
|
case refine_1.h.refine_1
t x : ℝ
hexp : x.exp ≤ t
⊢ 0 < 1
case refine_1.h.refine_2
t x : ℝ
hexp : x.exp ≤ t
⊢ 1 * (x / 1).exp ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
{ apply Real.zero_lt_one }
|
case refine_1.h.refine_1
t x : ℝ
hexp : x.exp ≤ t
⊢ 0 < 1
case refine_1.h.refine_2
t x : ℝ
hexp : x.exp ≤ t
⊢ 1 * (x / 1).exp ≤ t
|
case refine_1.h.refine_2
t x : ℝ
hexp : x.exp ≤ t
⊢ 1 * (x / 1).exp ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
{ rwa [div_one, one_mul] }
|
case refine_1.h.refine_2
t x : ℝ
hexp : x.exp ≤ t
⊢ 1 * (x / 1).exp ≤ t
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
apply Real.zero_lt_one
|
case refine_1.h.refine_1
t x : ℝ
hexp : x.exp ≤ t
⊢ 0 < 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
rwa [div_one, one_mul]
|
case refine_1.h.refine_2
t x : ℝ
hexp : x.exp ≤ t
⊢ 1 * (x / 1).exp ≤ t
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
intro h
|
case refine_2
t x : ℝ
⊢ 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0 → x.exp ≤ t
|
case refine_2
t x : ℝ
h : 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
⊢ x.exp ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
cases h with
| inl h =>
have h : 1 * exp (x / 1) ≤ t := h.2
rwa [div_one, one_mul] at h
| inr h =>
exfalso
exact zero_ne_one h.1.symm
|
case refine_2
t x : ℝ
h : 0 < 1 ∧ 1 * (x / 1).exp ≤ t ∨ 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
⊢ x.exp ≤ t
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
have h : 1 * exp (x / 1) ≤ t := h.2
|
case refine_2.inl
t x : ℝ
h : 0 < 1 ∧ 1 * (x / 1).exp ≤ t
⊢ x.exp ≤ t
|
case refine_2.inl
t x : ℝ
h✝ : 0 < 1 ∧ 1 * (x / 1).exp ≤ t
h : 1 * (x / 1).exp ≤ t
⊢ x.exp ≤ t
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
rwa [div_one, one_mul] at h
|
case refine_2.inl
t x : ℝ
h✝ : 0 < 1 ∧ 1 * (x / 1).exp ≤ t
h : 1 * (x / 1).exp ≤ t
⊢ x.exp ≤ t
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
exfalso
|
case refine_2.inr
t x : ℝ
h : 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
⊢ x.exp ≤ t
|
case refine_2.inr
t x : ℝ
h : 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
⊢ False
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Cones/ExpCone.lean
|
Real.exp_iff_expCone
|
[23, 1]
|
[39, 37]
|
exact zero_ne_one h.1.symm
|
case refine_2.inr
t x : ℝ
h : 1 = 0 ∧ 0 ≤ t ∧ x ≤ 0
⊢ False
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean
|
Matrix.diag_inv_mul_diag_eq_one_of_upperTriangular
|
[57, 1]
|
[65, 68]
|
letI : Unique {a // id a = k} := ⟨⟨⟨k, rfl⟩⟩, fun j => Subtype.ext j.property⟩
|
α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
⊢ M⁻¹ k k * M k k = 1
|
α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
⊢ M⁻¹ k k * M k k = 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean
|
Matrix.diag_inv_mul_diag_eq_one_of_upperTriangular
|
[57, 1]
|
[65, 68]
|
have h := congr_fun (congr_fun
(toSquareBlock_inv_mul_toSquareBlock_eq_one hM k) ⟨k, rfl⟩) ⟨k, rfl⟩
|
α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
⊢ M⁻¹ k k * M k k = 1
|
α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : (M⁻¹.toSquareBlock id k * M.toSquareBlock id k) ⟨k, ⋯⟩ ⟨k, ⋯⟩ = 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean
|
Matrix.diag_inv_mul_diag_eq_one_of_upperTriangular
|
[57, 1]
|
[65, 68]
|
dsimp only [HMul.hMul, dotProduct] at h
|
α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : (M⁻¹.toSquareBlock id k * M.toSquareBlock id k) ⟨k, ⋯⟩ ⟨k, ⋯⟩ = 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1
|
α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
∑ i : { a // id a = k }, Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ i) (M.toSquareBlock id k i ⟨k, ⋯⟩) =
OfNat.ofNat 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean
|
Matrix.diag_inv_mul_diag_eq_one_of_upperTriangular
|
[57, 1]
|
[65, 68]
|
rw [@Fintype.sum_unique _ _ _ _] at h
|
α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
∑ i : { a // id a = k }, Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ i) (M.toSquareBlock id k i ⟨k, ⋯⟩) =
OfNat.ofNat 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1
|
α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ default) (M.toSquareBlock id k default ⟨k, ⋯⟩) = OfNat.ofNat 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean
|
Matrix.diag_inv_mul_diag_eq_one_of_upperTriangular
|
[57, 1]
|
[65, 68]
|
simp at h
|
α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ default) (M.toSquareBlock id k default ⟨k, ⋯⟩) = OfNat.ofNat 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1
|
α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) (M.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) = 1
⊢ M⁻¹ k k * M k k = 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean
|
Matrix.diag_inv_mul_diag_eq_one_of_upperTriangular
|
[57, 1]
|
[65, 68]
|
rw [← h]
|
α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) (M.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) = 1
⊢ M⁻¹ k k * M k k = 1
|
α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) (M.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) = 1
⊢ M⁻¹ k k * M k k = Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) (M.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean
|
Matrix.diag_inv_mul_diag_eq_one_of_upperTriangular
|
[57, 1]
|
[65, 68]
|
simp [toSquareBlock, toSquareBlockProp]
|
α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) (M.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) = 1
⊢ M⁻¹ k k * M k k = Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) (M.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
|
α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) (M.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) = 1
⊢ M⁻¹ k k * M k k = Mul.mul (M⁻¹ k k) (M k k)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean
|
Matrix.diag_inv_mul_diag_eq_one_of_upperTriangular
|
[57, 1]
|
[65, 68]
|
rfl
|
α : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : Mul.mul (M⁻¹.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) (M.toSquareBlock id k ⟨k, ⋯⟩ ⟨k, ⋯⟩) = 1
⊢ M⁻¹ k k * M k k = Mul.mul (M⁻¹ k k) (M k k)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean
|
Matrix.diag_inv_mul_diag_eq_one_of_lowerTriangular
|
[67, 1]
|
[75, 68]
|
letI : Unique {a // OrderDual.toDual a = k} :=
⟨⟨⟨k, rfl⟩⟩, fun j => Subtype.ext j.property⟩
|
α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
⊢ M⁻¹ k k * M k k = 1
|
α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
⊢ M⁻¹ k k * M k k = 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean
|
Matrix.diag_inv_mul_diag_eq_one_of_lowerTriangular
|
[67, 1]
|
[75, 68]
|
have h := congr_fun (congr_fun
(toSquareBlock_inv_mul_toSquareBlock_eq_one hM k) ⟨k, rfl⟩) ⟨k, rfl⟩
|
α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
⊢ M⁻¹ k k * M k k = 1
|
α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k * M.toSquareBlock (⇑OrderDual.toDual) k) ⟨k, ⋯⟩ ⟨k, ⋯⟩ = 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean
|
Matrix.diag_inv_mul_diag_eq_one_of_lowerTriangular
|
[67, 1]
|
[75, 68]
|
dsimp [HMul.hMul, dotProduct] at h
|
α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h : (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k * M.toSquareBlock (⇑OrderDual.toDual) k) ⟨k, ⋯⟩ ⟨k, ⋯⟩ = 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1
|
α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
∑ i : { a // OrderDual.toDual a = k },
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ i) (M.toSquareBlock (⇑OrderDual.toDual) k i ⟨k, ⋯⟩) =
OfNat.ofNat 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean
|
Matrix.diag_inv_mul_diag_eq_one_of_lowerTriangular
|
[67, 1]
|
[75, 68]
|
rw [@Fintype.sum_unique _ _ _ this] at h
|
α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
∑ i : { a // OrderDual.toDual a = k },
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ i) (M.toSquareBlock (⇑OrderDual.toDual) k i ⟨k, ⋯⟩) =
OfNat.ofNat 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1
|
α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ default)
(M.toSquareBlock (⇑OrderDual.toDual) k default ⟨k, ⋯⟩) =
OfNat.ofNat 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean
|
Matrix.diag_inv_mul_diag_eq_one_of_lowerTriangular
|
[67, 1]
|
[75, 68]
|
simp at h
|
α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ default)
(M.toSquareBlock (⇑OrderDual.toDual) k default ⟨k, ⋯⟩) =
OfNat.ofNat 1 ⟨k, ⋯⟩ ⟨k, ⋯⟩
⊢ M⁻¹ k k * M k k = 1
|
α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
(M.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩) =
1
⊢ M⁻¹ k k * M k k = 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean
|
Matrix.diag_inv_mul_diag_eq_one_of_lowerTriangular
|
[67, 1]
|
[75, 68]
|
rw [← h]
|
α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
(M.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩) =
1
⊢ M⁻¹ k k * M k k = 1
|
α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
(M.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩) =
1
⊢ M⁻¹ k k * M k k =
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
(M.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean
|
Matrix.diag_inv_mul_diag_eq_one_of_lowerTriangular
|
[67, 1]
|
[75, 68]
|
simp [toSquareBlock, toSquareBlockProp]
|
α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
(M.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩) =
1
⊢ M⁻¹ k k * M k k =
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
(M.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
|
α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
(M.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩) =
1
⊢ M⁻¹ k k * M k k = Mul.mul (M⁻¹ k k) (M k k)
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Lib/Math/LinearAlgebra/Matrix/Triangular.lean
|
Matrix.diag_inv_mul_diag_eq_one_of_lowerTriangular
|
[67, 1]
|
[75, 68]
|
rfl
|
α : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
inst✝³ : CommRing R
M N : Matrix m m R
inst✝² : Fintype m
inst✝¹ : LinearOrder m
inst✝ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := ⟨k, ⋯⟩, uniq := ⋯ }
h :
Mul.mul (M⁻¹.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩)
(M.toSquareBlock (⇑OrderDual.toDual) k ⟨k, ⋯⟩ ⟨k, ⋯⟩) =
1
⊢ M⁻¹ k k * M k k = Mul.mul (M⁻¹ k k) (M k k)
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.simp_vec_fraction
|
[43, 1]
|
[47, 49]
|
have h_di_pos := h_d_pos i
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
h_d_pos : StrongLT 0 d
s : Fin n → ℝ
i : Fin n
⊢ d i / (d i / s i) = s i
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
h_d_pos : StrongLT 0 d
s : Fin n → ℝ
i : Fin n
h_di_pos : 0 i < d i
⊢ d i / (d i / s i) = s i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.simp_vec_fraction
|
[43, 1]
|
[47, 49]
|
simp at h_di_pos
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
h_d_pos : StrongLT 0 d
s : Fin n → ℝ
i : Fin n
h_di_pos : 0 i < d i
⊢ d i / (d i / s i) = s i
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
h_d_pos : StrongLT 0 d
s : Fin n → ℝ
i : Fin n
h_di_pos : 0 < d i
⊢ d i / (d i / s i) = s i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.simp_vec_fraction
|
[43, 1]
|
[47, 49]
|
have h_di_nonzero : d i ≠ 0 := by linarith
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
h_d_pos : StrongLT 0 d
s : Fin n → ℝ
i : Fin n
h_di_pos : 0 < d i
⊢ d i / (d i / s i) = s i
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
h_d_pos : StrongLT 0 d
s : Fin n → ℝ
i : Fin n
h_di_pos : 0 < d i
h_di_nonzero : d i ≠ 0
⊢ d i / (d i / s i) = s i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.simp_vec_fraction
|
[43, 1]
|
[47, 49]
|
rw [← div_mul, div_self h_di_nonzero, one_mul]
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
h_d_pos : StrongLT 0 d
s : Fin n → ℝ
i : Fin n
h_di_pos : 0 < d i
h_di_nonzero : d i ≠ 0
⊢ d i / (d i / s i) = s i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.simp_vec_fraction
|
[43, 1]
|
[47, 49]
|
linarith
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
h_d_pos : StrongLT 0 d
s : Fin n → ℝ
i : Fin n
h_di_pos : 0 < d i
⊢ d i ≠ 0
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.fold_partial_sum
|
[49, 1]
|
[53, 22]
|
simp [Vec.cumsum]
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
hn : Fact (0 < n)
t : Fin n → ℝ
i : Fin n
⊢ ∑ j ∈ [[0, i]], t j = Vec.cumsum t i
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
hn : Fact (0 < n)
t : Fin n → ℝ
i : Fin n
⊢ ∑ j ∈ [[0, i]], t j = if h : 0 < n then ∑ j ∈ [[⟨0, h⟩, i]], t j else 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.fold_partial_sum
|
[49, 1]
|
[53, 22]
|
split_ifs
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
hn : Fact (0 < n)
t : Fin n → ℝ
i : Fin n
⊢ ∑ j ∈ [[0, i]], t j = if h : 0 < n then ∑ j ∈ [[⟨0, h⟩, i]], t j else 0
|
case pos
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
hn : Fact (0 < n)
t : Fin n → ℝ
i : Fin n
h✝ : 0 < n
⊢ ∑ j ∈ [[0, i]], t j = ∑ j ∈ [[⟨0, h✝⟩, i]], t j
case neg
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
hn : Fact (0 < n)
t : Fin n → ℝ
i : Fin n
h✝ : ¬0 < n
⊢ ∑ j ∈ [[0, i]], t j = 0
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.fold_partial_sum
|
[49, 1]
|
[53, 22]
|
rfl
|
case pos
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
hn : Fact (0 < n)
t : Fin n → ℝ
i : Fin n
h✝ : 0 < n
⊢ ∑ j ∈ [[0, i]], t j = ∑ j ∈ [[⟨0, h✝⟩, i]], t j
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.fold_partial_sum
|
[49, 1]
|
[53, 22]
|
linarith [hn.out]
|
case neg
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
hn : Fact (0 < n)
t : Fin n → ℝ
i : Fin n
h✝ : ¬0 < n
⊢ ∑ j ∈ [[0, i]], t j = 0
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.nₚ_pos
|
[148, 1]
|
[148, 48]
|
unfold nₚ
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
⊢ 0 < nₚ
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
⊢ 0 < 10
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.nₚ_pos
|
[148, 1]
|
[148, 48]
|
norm_num
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
⊢ 0 < 10
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.dₚ_pos
|
[154, 1]
|
[155, 50]
|
intro i
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
⊢ StrongLT 0 dₚ
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
i : Fin nₚ
⊢ 0 i < dₚ i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.dₚ_pos
|
[154, 1]
|
[155, 50]
|
fin_cases i <;> (dsimp [dₚ]; norm_num)
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
i : Fin nₚ
⊢ 0 i < dₚ i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.dₚ_pos
|
[154, 1]
|
[155, 50]
|
dsimp [dₚ]
|
case tail.tail.tail.tail.tail.tail.tail.tail.tail.head
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
⊢ 0 ⟨9, ⋯⟩ < dₚ ⟨9, ⋯⟩
|
case tail.tail.tail.tail.tail.tail.tail.tail.tail.head
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
⊢ 0 < ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, ⋯⟩
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.dₚ_pos
|
[154, 1]
|
[155, 50]
|
norm_num
|
case tail.tail.tail.tail.tail.tail.tail.tail.tail.head
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
⊢ 0 < ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, ⋯⟩
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.sminₚ_pos
|
[173, 1]
|
[174, 36]
|
intro i
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
⊢ StrongLT 0 sminₚ
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
i : Fin nₚ
⊢ 0 i < sminₚ i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.sminₚ_pos
|
[173, 1]
|
[174, 36]
|
fin_cases i <;> norm_num
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
i : Fin nₚ
⊢ 0 i < sminₚ i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.sminₚ_le_smaxₚ
|
[179, 1]
|
[180, 60]
|
intro i
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
⊢ sminₚ ≤ smaxₚ
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
i : Fin nₚ
⊢ sminₚ i ≤ smaxₚ i
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.sminₚ_le_smaxₚ
|
[179, 1]
|
[180, 60]
|
fin_cases i <;> (dsimp [sminₚ, smaxₚ]; norm_num)
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
i : Fin nₚ
⊢ sminₚ i ≤ smaxₚ i
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.sminₚ_le_smaxₚ
|
[179, 1]
|
[180, 60]
|
dsimp [sminₚ, smaxₚ]
|
case tail.tail.tail.tail.tail.tail.tail.tail.tail.head
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
⊢ sminₚ ⟨9, ⋯⟩ ≤ smaxₚ ⟨9, ⋯⟩
|
case tail.tail.tail.tail.tail.tail.tail.tail.tail.head
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
⊢ ![0.7828, 0.6235, 0.7155, 0.5340, 0.6329, 0.4259, 0.7798, 0.9604, 0.7298, 0.8405] ⟨9, ⋯⟩ ≤
![1.9624, 1.6036, 1.6439, 1.5641, 1.7194, 1.9090, 1.3193, 1.3366, 1.9470, 2.8803] ⟨9, ⋯⟩
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.sminₚ_le_smaxₚ
|
[179, 1]
|
[180, 60]
|
norm_num
|
case tail.tail.tail.tail.tail.tail.tail.tail.tail.head
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
⊢ ![0.7828, 0.6235, 0.7155, 0.5340, 0.6329, 0.4259, 0.7798, 0.9604, 0.7298, 0.8405] ⟨9, ⋯⟩ ≤
![1.9624, 1.6036, 1.6439, 1.5641, 1.7194, 1.9090, 1.3193, 1.3366, 1.9470, 2.8803] ⟨9, ⋯⟩
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.aₚ_nonneg
|
[188, 1]
|
[189, 51]
|
unfold aₚ
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
⊢ 0 ≤ aₚ
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
⊢ 0 ≤ 1
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.aₚ_nonneg
|
[188, 1]
|
[189, 51]
|
norm_num
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
⊢ 0 ≤ 1
|
no goals
|
https://github.com/verified-optimization/CvxLean.git
|
c62c2f292c6420f31a12e738ebebdfed50f6f840
|
CvxLean/Examples/VehicleSpeedScheduling.lean
|
VehicleSpeedSched.aₚdₚ2_nonneg
|
[191, 1]
|
[193, 55]
|
intros i
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
⊢ 0 ≤ aₚ • dₚ ^ 2
|
n : ℕ
d τmin τmax smin smax : Fin n → ℝ
F : ℝ → ℝ
i : Fin nₚ
⊢ 0 i ≤ (aₚ • dₚ ^ 2) i
|
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