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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
xs'.repr ((toLin' A) (EuclideanSpace.single j 1)) ((Fintype.equivOfCardEq β―).symm i)
case h.e'_3
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ (diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π n)) i j =
β(as' ((Fintype.equivOfCardEq β―).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq β―).symm i) | case h.e'_3
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ (diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π n)) i j =
β(as' ((Fintype.equivOfCardEq β―).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq β―).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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ (diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π n)) i j =
β(as' ((Fintype.equivOfCardEq β―).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq β―).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] | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
β’ β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j) | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
β’ β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) ((βxs β β(Fintype.equivOfCardEq β―)) 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 | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
β’ β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) ((βxs β β(Fintype.equivOfCardEq β―)) j) | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i jβ : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
j : Fin (Fintype.card n)
β’ Module.End.HasEigenvector (toLin' A) (β(as' j)) ((βxs β β(Fintype.equivOfCardEq β―)) 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)) | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i jβ : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
j : Fin (Fintype.card n)
β’ Module.End.HasEigenvector (toLin' A) (β(as' j)) ((βxs β β(Fintype.equivOfCardEq β―)) 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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
xs'.repr ((toLin' A) (EuclideanSpace.single j 1)) ((Fintype.equivOfCardEq β―).symm i) | case h.e'_2
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
xs'.repr (A.mulVec (EuclideanSpace.single j 1)) ((Fintype.equivOfCardEq β―).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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
xs'.repr (A.mulVec (EuclideanSpace.single j 1)) ((Fintype.equivOfCardEq β―).symm i) | case h.e'_2
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ xs.repr (A.transpose j) i =
xs.repr (A.mulVec (EuclideanSpace.single j 1))
((Fintype.equivOfCardEq β―).symm.symm ((Fintype.equivOfCardEq β―).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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ xs.repr (A.transpose j) i =
xs.repr (A.mulVec (EuclideanSpace.single j 1))
((Fintype.equivOfCardEq β―).symm.symm ((Fintype.equivOfCardEq β―).symm i)) | case h.e'_2
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ 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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ EquivLike.coe xs.repr (A.transpose j) i = xs.repr (fun i => A i j * 1) i | case h.e'_2
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ 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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ 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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ (diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π n)) i j =
β(as' ((Fintype.equivOfCardEq β―).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq β―).symm i) | case h.e'_3
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ β(as i) * xs.toBasis.toMatrix (β(Pi.basisFun π n)) i j =
β(as' ((Fintype.equivOfCardEq β―).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq β―).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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ β(as i) * xs.toBasis.toMatrix (β(Pi.basisFun π n)) i j =
β(as' ((Fintype.equivOfCardEq β―).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq β―).symm i) | case h.e'_3
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ β(as i) * EquivLike.coe xs.repr (Pi.single j 1) i =
β(as' ((Fintype.equivOfCardEq β―).symm i)) *
xs.repr (fun a => Decidable.rec (fun h => (fun h => 0 a) h) (fun h => (fun h => β― βΈ 1) h) (instβ 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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ β(as i) * EquivLike.coe xs.repr (Pi.single j 1) i =
β(as' ((Fintype.equivOfCardEq β―).symm i)) *
xs.repr (fun a => Decidable.rec (fun h => (fun h => 0 a) h) (fun h => (fun h => β― βΈ 1) h) (instβ a j)) i | case h.e'_3.e_a.e_a.e_a
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ i = (Fintype.equivOfCardEq β―) ((Fintype.equivOfCardEq β―).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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ i = (Fintype.equivOfCardEq β―) ((Fintype.equivOfCardEq β―).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_cancelβ (det_ne_zero_of_left_inverse
(Basis.toMatrix_mul_toMatrix_flip (Pi.basisFun π n) xs.toBasis)) | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
β’ A.det = β(β i : n, as i) | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
β’ (xs.toBasis.toMatrix β(Pi.basisFun π n)).det * A.det = (xs.toBasis.toMatrix β(Pi.basisFun π n)).det * β(β 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 [β det_mul, spectral_theorem xs as hxs, det_mul, mul_comm, det_diagonal] | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
β’ (xs.toBasis.toMatrix β(Pi.basisFun π n)).det * A.det = (xs.toBasis.toMatrix β(Pi.basisFun π n)).det * β(β i : n, as i) | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
β’ (xs.toBasis.toMatrix β(Pi.basisFun π n)).det * β i : n, (RCLike.ofReal β as) i =
(xs.toBasis.toMatrix β(Pi.basisFun π n)).det * β(β 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 | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
β’ (xs.toBasis.toMatrix β(Pi.basisFun π n)).det * β i : n, (RCLike.ofReal β as) i =
(xs.toBasis.toMatrix β(Pi.basisFun π n)).det * β(β 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] | π : Type u_2
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_1
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
hA : A.IsHermitian
i : n
β’ Module.End.HasEigenvector (toLin' A) (β(hA.eigenvalues i)) (hA.eigenvectorBasis i) | π : Type u_2
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_1
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
hA : A.IsHermitian
i : n
β’ Module.End.HasEigenvector (toLin' A) (β(hA.eigenvalues i))
((β(β―.eigenvectorBasis β―) β β(Fintype.equivOfCardEq β―).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 | π : Type u_2
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_1
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
hA : A.IsHermitian
i : n
β’ Module.End.HasEigenvector (toLin' A) (β(hA.eigenvalues i))
((β(β―.eigenvectorBasis β―) β β(Fintype.equivOfCardEq β―).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] | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
β’ xs.toBasis.toMatrix β(Pi.basisFun π n) * A = diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π n) | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
β’ (of fun i j => (xs.toBasis.repr (A.transpose j)) i) =
diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π 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 | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
β’ (of fun i j => (xs.toBasis.repr (A.transpose j)) i) =
diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π n) | case a
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
β’ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π 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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
β’ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π n)) i j | case a
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
β’ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π 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) β β :=
fun i => as <| (Fintype.equivOfCardEq (Fintype.card_fin _)) i | case a
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
β’ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π n)) i j | case a
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
β’ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π 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' : β 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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
β’ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π n)) i j | case a
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π 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' π _
(PiLp 2 (fun (_ : n) => π)) _ _ (Fintype.card n) (Matrix.toLin' A)
(EuclideanSpace.single j 1)
((Fintype.equivOfCardEq (Fintype.card_fin _)).symm i)
xs' as' hxs' | case a
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
(diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π n)) i j | case h.e'_2
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
xs'.repr ((toLin' A) (EuclideanSpace.single j 1)) ((Fintype.equivOfCardEq β―).symm i)
case h.e'_3
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ (diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π n)) i j =
β(as' ((Fintype.equivOfCardEq β―).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq β―).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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
xs'.repr ((toLin' A) (EuclideanSpace.single j 1)) ((Fintype.equivOfCardEq β―).symm i)
case h.e'_3
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ (diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π n)) i j =
β(as' ((Fintype.equivOfCardEq β―).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq β―).symm i) | case h.e'_3
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ (diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π n)) i j =
β(as' ((Fintype.equivOfCardEq β―).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq β―).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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ (diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π n)) i j =
β(as' ((Fintype.equivOfCardEq β―).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq β―).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] | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
β’ β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j) | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
β’ β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) ((βxs β β(Fintype.equivOfCardEq β―)) 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 | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
β’ β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) ((βxs β β(Fintype.equivOfCardEq β―)) j) | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i jβ : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
j : Fin (Fintype.card n)
β’ Module.End.HasEigenvector (toLin' A) (β(as' j)) ((βxs β β(Fintype.equivOfCardEq β―)) 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)) | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i jβ : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
j : Fin (Fintype.card n)
β’ Module.End.HasEigenvector (toLin' A) (β(as' j)) ((βxs β β(Fintype.equivOfCardEq β―)) 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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
xs'.repr ((toLin' A) (EuclideanSpace.single j 1)) ((Fintype.equivOfCardEq β―).symm i) | case h.e'_2
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
xs'.repr (A.mulVec (EuclideanSpace.single j 1)) ((Fintype.equivOfCardEq β―).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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ of (fun i j => (xs.toBasis.repr (A.transpose j)) i) i j =
xs'.repr (A.mulVec (EuclideanSpace.single j 1)) ((Fintype.equivOfCardEq β―).symm i) | case h.e'_2
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ xs.repr (A.transpose j) i =
xs.repr (A.mulVec (EuclideanSpace.single j 1))
((Fintype.equivOfCardEq β―).symm.symm ((Fintype.equivOfCardEq β―).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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ xs.repr (A.transpose j) i =
xs.repr (A.mulVec (EuclideanSpace.single j 1))
((Fintype.equivOfCardEq β―).symm.symm ((Fintype.equivOfCardEq β―).symm i)) | case h.e'_2
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ 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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ EquivLike.coe xs.repr (A.transpose j) i = xs.repr (fun i => A i j * 1) i | case h.e'_2
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ 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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ 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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ (diagonal (RCLike.ofReal β as) * xs.toBasis.toMatrix β(Pi.basisFun π n)) i j =
β(as' ((Fintype.equivOfCardEq β―).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq β―).symm i) | case h.e'_3
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ β(as i) * xs.toBasis.toMatrix (β(Pi.basisFun π n)) i j =
β(as' ((Fintype.equivOfCardEq β―).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq β―).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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ β(as i) * xs.toBasis.toMatrix (β(Pi.basisFun π n)) i j =
β(as' ((Fintype.equivOfCardEq β―).symm i)) * xs'.repr (EuclideanSpace.single j 1) ((Fintype.equivOfCardEq β―).symm i) | case h.e'_3
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ β(as i) * EquivLike.coe xs.repr (Pi.single j 1) i =
β(as' ((Fintype.equivOfCardEq β―).symm i)) *
xs.repr (fun a => Decidable.rec (fun h => (fun h => 0 a) h) (fun h => (fun h => β― βΈ 1) h) (instβ 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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ β(as i) * EquivLike.coe xs.repr (Pi.single j 1) i =
β(as' ((Fintype.equivOfCardEq β―).symm i)) *
xs.repr (fun a => Decidable.rec (fun h => (fun h => 0 a) h) (fun h => (fun h => β― βΈ 1) h) (instβ a j)) i | case h.e'_3.e_a.e_a.e_a
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ i = (Fintype.equivOfCardEq β―) ((Fintype.equivOfCardEq β―).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
π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
i j : n
xs' : OrthonormalBasis (Fin (Fintype.card n)) π (EuclideanSpace π n) := xs.reindex (Fintype.equivOfCardEq β―).symm
as' : Fin (Fintype.card n) β β := fun i => as ((Fintype.equivOfCardEq β―) i)
hxs' : β (j : Fin (Fintype.card n)), Module.End.HasEigenvector (toLin' A) (β(as' j)) (xs' j)
β’ i = (Fintype.equivOfCardEq β―) ((Fintype.equivOfCardEq β―).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_cancelβ (det_ne_zero_of_left_inverse
(Basis.toMatrix_mul_toMatrix_flip (Pi.basisFun π n) xs.toBasis)) | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
β’ A.det = β(β i : n, as i) | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
β’ (xs.toBasis.toMatrix β(Pi.basisFun π n)).det * A.det = (xs.toBasis.toMatrix β(Pi.basisFun π n)).det * β(β 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 [β det_mul, spectral_theorem xs as hxs, det_mul, mul_comm, det_diagonal] | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
β’ (xs.toBasis.toMatrix β(Pi.basisFun π n)).det * A.det = (xs.toBasis.toMatrix β(Pi.basisFun π n)).det * β(β i : n, as i) | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
β’ (xs.toBasis.toMatrix β(Pi.basisFun π n)).det * β i : n, (RCLike.ofReal β as) i =
(xs.toBasis.toMatrix β(Pi.basisFun π n)).det * β(β 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 | π : Type u_1
instβΒ³ : RCLike π
instβΒ² : DecidableEq π
n : Type u_2
instβΒΉ : Fintype n
instβ : DecidableEq n
A : Matrix n n π
xs : OrthonormalBasis n π (EuclideanSpace π n)
as : n β β
hxs : β (j : n), Module.End.HasEigenvector (toLin' A) (β(as j)) (xs j)
β’ (xs.toBasis.toMatrix β(Pi.basisFun π n)).det * β i : n, (RCLike.ofReal β as) i =
(xs.toBasis.toMatrix β(Pi.basisFun π n)).det * β(β 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 : β
β’ x.exp β€ t β x.expCone 1 t | t x : β
β’ x.exp β€ t β 0 < 1 β§ 1 * (x / 1).exp β€ t β¨ 1 = 0 β§ 0 β€ t β§ x β€ 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 : β
β’ x.exp β€ t β 0 < 1 β§ 1 * (x / 1).exp β€ t β¨ 1 = 0 β§ 0 β€ t β§ x β€ 0 | t x : β
β’ (x.exp β€ t β 0 < 1 β§ 1 * (x / 1).exp β€ t β¨ 1 = 0 β§ 0 β€ t β§ x β€ 0) β§
(0 < 1 β§ 1 * (x / 1).exp β€ t β¨ 1 = 0 β§ 0 β€ t β§ x β€ 0 β x.exp β€ 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 : β
β’ (x.exp β€ t β 0 < 1 β§ 1 * (x / 1).exp β€ t β¨ 1 = 0 β§ 0 β€ t β§ x β€ 0) β§
(0 < 1 β§ 1 * (x / 1).exp β€ t β¨ 1 = 0 β§ 0 β€ t β§ x β€ 0 β x.exp β€ t) | case refine_1
t x : β
β’ x.exp β€ t β 0 < 1 β§ 1 * (x / 1).exp β€ t β¨ 1 = 0 β§ 0 β€ t β§ x β€ 0
case refine_2
t x : β
β’ 0 < 1 β§ 1 * (x / 1).exp β€ t β¨ 1 = 0 β§ 0 β€ t β§ x β€ 0 β x.exp β€ 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 : β
β’ x.exp β€ t β 0 < 1 β§ 1 * (x / 1).exp β€ t β¨ 1 = 0 β§ 0 β€ t β§ x β€ 0
case refine_2
t x : β
β’ 0 < 1 β§ 1 * (x / 1).exp β€ t β¨ 1 = 0 β§ 0 β€ t β§ x β€ 0 β x.exp β€ t | case refine_2
t x : β
β’ 0 < 1 β§ 1 * (x / 1).exp β€ t β¨ 1 = 0 β§ 0 β€ t β§ x β€ 0 β x.exp β€ 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) β€ 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 : β
β’ 0 < 1 β§ 1 * (x / 1).exp β€ t β¨ 1 = 0 β§ 0 β€ t β§ x β€ 0 β x.exp β€ 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 : β
β’ x.exp β€ t β 0 < 1 β§ 1 * (x / 1).exp β€ t β¨ 1 = 0 β§ 0 β€ t β§ x β€ 0 | case refine_1
t x : β
hexp : x.exp β€ t
β’ 0 < 1 β§ 1 * (x / 1).exp β€ t β¨ 1 = 0 β§ 0 β€ t β§ x β€ 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 : β
hexp : x.exp β€ t
β’ 0 < 1 β§ 1 * (x / 1).exp β€ t β¨ 1 = 0 β§ 0 β€ t β§ x β€ 0 | case refine_1.h
t x : β
hexp : x.exp β€ t
β’ 0 < 1 β§ 1 * (x / 1).exp β€ 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 : β
hexp : x.exp β€ t
β’ 0 < 1 β§ 1 * (x / 1).exp β€ t | case refine_1.h.refine_1
t x : β
hexp : x.exp β€ t
β’ 0 < 1
case refine_1.h.refine_2
t x : β
hexp : x.exp β€ t
β’ 1 * (x / 1).exp β€ 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 : β
hexp : x.exp β€ t
β’ 0 < 1
case refine_1.h.refine_2
t x : β
hexp : x.exp β€ t
β’ 1 * (x / 1).exp β€ t | case refine_1.h.refine_2
t x : β
hexp : x.exp β€ t
β’ 1 * (x / 1).exp β€ 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 : β
hexp : x.exp β€ t
β’ 1 * (x / 1).exp β€ 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 : β
hexp : x.exp β€ t
β’ 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 : β
hexp : x.exp β€ t
β’ 1 * (x / 1).exp β€ 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 : β
β’ 0 < 1 β§ 1 * (x / 1).exp β€ t β¨ 1 = 0 β§ 0 β€ t β§ x β€ 0 β x.exp β€ t | case refine_2
t x : β
h : 0 < 1 β§ 1 * (x / 1).exp β€ t β¨ 1 = 0 β§ 0 β€ t β§ x β€ 0
β’ x.exp β€ 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) β€ 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 : β
h : 0 < 1 β§ 1 * (x / 1).exp β€ t β¨ 1 = 0 β§ 0 β€ t β§ x β€ 0
β’ x.exp β€ 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) β€ t := h.2 | case refine_2.inl
t x : β
h : 0 < 1 β§ 1 * (x / 1).exp β€ t
β’ x.exp β€ t | case refine_2.inl
t x : β
hβ : 0 < 1 β§ 1 * (x / 1).exp β€ t
h : 1 * (x / 1).exp β€ t
β’ x.exp β€ 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 : β
hβ : 0 < 1 β§ 1 * (x / 1).exp β€ t
h : 1 * (x / 1).exp β€ t
β’ x.exp β€ 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 : β
h : 1 = 0 β§ 0 β€ t β§ x β€ 0
β’ x.exp β€ t | case refine_2.inr
t x : β
h : 1 = 0 β§ 0 β€ t β§ x β€ 0
β’ 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 : β
h : 1 = 0 β§ 0 β€ t β§ x β€ 0
β’ 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} := β¨β¨β¨k, rflβ©β©, fun j => Subtype.ext j.propertyβ© | Ξ± : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.upperTriangular
k : m
β’ Mβ»ΒΉ k k * M k k = 1 | Ξ± : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := β¨k, β―β©, uniq := β― }
β’ Mβ»ΒΉ 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) β¨k, rflβ©) β¨k, rflβ© | Ξ± : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := β¨k, β―β©, uniq := β― }
β’ Mβ»ΒΉ k k * M k k = 1 | Ξ± : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := β¨k, β―β©, uniq := β― }
h : (Mβ»ΒΉ.toSquareBlock id k * M.toSquareBlock id k) β¨k, β―β© β¨k, β―β© = 1 β¨k, β―β© β¨k, β―β©
β’ Mβ»ΒΉ 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 | Ξ± : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := β¨k, β―β©, uniq := β― }
h : (Mβ»ΒΉ.toSquareBlock id k * M.toSquareBlock id k) β¨k, β―β© β¨k, β―β© = 1 β¨k, β―β© β¨k, β―β©
β’ Mβ»ΒΉ k k * M k k = 1 | Ξ± : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := β¨k, β―β©, uniq := β― }
h :
β i : { a // id a = k }, Mul.mul (Mβ»ΒΉ.toSquareBlock id k β¨k, β―β© i) (M.toSquareBlock id k i β¨k, β―β©) =
OfNat.ofNat 1 β¨k, β―β© β¨k, β―β©
β’ Mβ»ΒΉ 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 | Ξ± : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := β¨k, β―β©, uniq := β― }
h :
β i : { a // id a = k }, Mul.mul (Mβ»ΒΉ.toSquareBlock id k β¨k, β―β© i) (M.toSquareBlock id k i β¨k, β―β©) =
OfNat.ofNat 1 β¨k, β―β© β¨k, β―β©
β’ Mβ»ΒΉ k k * M k k = 1 | Ξ± : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := β¨k, β―β©, uniq := β― }
h : Mul.mul (Mβ»ΒΉ.toSquareBlock id k β¨k, β―β© default) (M.toSquareBlock id k default β¨k, β―β©) = OfNat.ofNat 1 β¨k, β―β© β¨k, β―β©
β’ Mβ»ΒΉ 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 | Ξ± : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := β¨k, β―β©, uniq := β― }
h : Mul.mul (Mβ»ΒΉ.toSquareBlock id k β¨k, β―β© default) (M.toSquareBlock id k default β¨k, β―β©) = OfNat.ofNat 1 β¨k, β―β© β¨k, β―β©
β’ Mβ»ΒΉ k k * M k k = 1 | Ξ± : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := β¨k, β―β©, uniq := β― }
h : Mul.mul (Mβ»ΒΉ.toSquareBlock id k β¨k, β―β© β¨k, β―β©) (M.toSquareBlock id k β¨k, β―β© β¨k, β―β©) = 1
β’ Mβ»ΒΉ 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 [β h] | Ξ± : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := β¨k, β―β©, uniq := β― }
h : Mul.mul (Mβ»ΒΉ.toSquareBlock id k β¨k, β―β© β¨k, β―β©) (M.toSquareBlock id k β¨k, β―β© β¨k, β―β©) = 1
β’ Mβ»ΒΉ k k * M k k = 1 | Ξ± : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := β¨k, β―β©, uniq := β― }
h : Mul.mul (Mβ»ΒΉ.toSquareBlock id k β¨k, β―β© β¨k, β―β©) (M.toSquareBlock id k β¨k, β―β© β¨k, β―β©) = 1
β’ Mβ»ΒΉ k k * M k k = Mul.mul (Mβ»ΒΉ.toSquareBlock id k β¨k, β―β© β¨k, β―β©) (M.toSquareBlock id k β¨k, β―β© β¨k, β―β©) |
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] | Ξ± : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := β¨k, β―β©, uniq := β― }
h : Mul.mul (Mβ»ΒΉ.toSquareBlock id k β¨k, β―β© β¨k, β―β©) (M.toSquareBlock id k β¨k, β―β© β¨k, β―β©) = 1
β’ Mβ»ΒΉ k k * M k k = Mul.mul (Mβ»ΒΉ.toSquareBlock id k β¨k, β―β© β¨k, β―β©) (M.toSquareBlock id k β¨k, β―β© β¨k, β―β©) | Ξ± : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := β¨k, β―β©, uniq := β― }
h : Mul.mul (Mβ»ΒΉ.toSquareBlock id k β¨k, β―β© β¨k, β―β©) (M.toSquareBlock id k β¨k, β―β© β¨k, β―β©) = 1
β’ Mβ»ΒΉ k k * M k k = Mul.mul (Mβ»ΒΉ 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 | Ξ± : Type ?u.10355
m : Type u_1
n : Type ?u.10361
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.upperTriangular
k : m
this : Unique { a // id a = k } := { default := β¨k, β―β©, uniq := β― }
h : Mul.mul (Mβ»ΒΉ.toSquareBlock id k β¨k, β―β© β¨k, β―β©) (M.toSquareBlock id k β¨k, β―β© β¨k, β―β©) = 1
β’ Mβ»ΒΉ k k * M k k = Mul.mul (Mβ»ΒΉ 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} :=
β¨β¨β¨k, rflβ©β©, fun j => Subtype.ext j.propertyβ© | Ξ± : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.lowerTriangular
k : m
β’ Mβ»ΒΉ k k * M k k = 1 | Ξ± : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := β¨k, β―β©, uniq := β― }
β’ Mβ»ΒΉ 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) β¨k, rflβ©) β¨k, rflβ© | Ξ± : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := β¨k, β―β©, uniq := β― }
β’ Mβ»ΒΉ k k * M k k = 1 | Ξ± : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := β¨k, β―β©, uniq := β― }
h : (Mβ»ΒΉ.toSquareBlock (βOrderDual.toDual) k * M.toSquareBlock (βOrderDual.toDual) k) β¨k, β―β© β¨k, β―β© = 1 β¨k, β―β© β¨k, β―β©
β’ Mβ»ΒΉ 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 | Ξ± : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := β¨k, β―β©, uniq := β― }
h : (Mβ»ΒΉ.toSquareBlock (βOrderDual.toDual) k * M.toSquareBlock (βOrderDual.toDual) k) β¨k, β―β© β¨k, β―β© = 1 β¨k, β―β© β¨k, β―β©
β’ Mβ»ΒΉ k k * M k k = 1 | Ξ± : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := β¨k, β―β©, uniq := β― }
h :
β i : { a // OrderDual.toDual a = k },
Mul.mul (Mβ»ΒΉ.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© i) (M.toSquareBlock (βOrderDual.toDual) k i β¨k, β―β©) =
OfNat.ofNat 1 β¨k, β―β© β¨k, β―β©
β’ Mβ»ΒΉ 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 | Ξ± : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := β¨k, β―β©, uniq := β― }
h :
β i : { a // OrderDual.toDual a = k },
Mul.mul (Mβ»ΒΉ.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© i) (M.toSquareBlock (βOrderDual.toDual) k i β¨k, β―β©) =
OfNat.ofNat 1 β¨k, β―β© β¨k, β―β©
β’ Mβ»ΒΉ k k * M k k = 1 | Ξ± : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := β¨k, β―β©, uniq := β― }
h :
Mul.mul (Mβ»ΒΉ.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© default)
(M.toSquareBlock (βOrderDual.toDual) k default β¨k, β―β©) =
OfNat.ofNat 1 β¨k, β―β© β¨k, β―β©
β’ Mβ»ΒΉ 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 | Ξ± : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := β¨k, β―β©, uniq := β― }
h :
Mul.mul (Mβ»ΒΉ.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© default)
(M.toSquareBlock (βOrderDual.toDual) k default β¨k, β―β©) =
OfNat.ofNat 1 β¨k, β―β© β¨k, β―β©
β’ Mβ»ΒΉ k k * M k k = 1 | Ξ± : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := β¨k, β―β©, uniq := β― }
h :
Mul.mul (Mβ»ΒΉ.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© β¨k, β―β©)
(M.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© β¨k, β―β©) =
1
β’ Mβ»ΒΉ 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 [β h] | Ξ± : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := β¨k, β―β©, uniq := β― }
h :
Mul.mul (Mβ»ΒΉ.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© β¨k, β―β©)
(M.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© β¨k, β―β©) =
1
β’ Mβ»ΒΉ k k * M k k = 1 | Ξ± : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := β¨k, β―β©, uniq := β― }
h :
Mul.mul (Mβ»ΒΉ.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© β¨k, β―β©)
(M.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© β¨k, β―β©) =
1
β’ Mβ»ΒΉ k k * M k k =
Mul.mul (Mβ»ΒΉ.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© β¨k, β―β©)
(M.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© β¨k, β―β©) |
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] | Ξ± : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := β¨k, β―β©, uniq := β― }
h :
Mul.mul (Mβ»ΒΉ.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© β¨k, β―β©)
(M.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© β¨k, β―β©) =
1
β’ Mβ»ΒΉ k k * M k k =
Mul.mul (Mβ»ΒΉ.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© β¨k, β―β©)
(M.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© β¨k, β―β©) | Ξ± : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := β¨k, β―β©, uniq := β― }
h :
Mul.mul (Mβ»ΒΉ.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© β¨k, β―β©)
(M.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© β¨k, β―β©) =
1
β’ Mβ»ΒΉ k k * M k k = Mul.mul (Mβ»ΒΉ 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 | Ξ± : Type ?u.15206
m : Type u_1
n : Type ?u.15212
R : Type u_2
instβΒ³ : CommRing R
M N : Matrix m m R
instβΒ² : Fintype m
instβΒΉ : LinearOrder m
instβ : Invertible M
hM : M.lowerTriangular
k : m
this : Unique { a // OrderDual.toDual a = k } := { default := β¨k, β―β©, uniq := β― }
h :
Mul.mul (Mβ»ΒΉ.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© β¨k, β―β©)
(M.toSquareBlock (βOrderDual.toDual) k β¨k, β―β© β¨k, β―β©) =
1
β’ Mβ»ΒΉ k k * M k k = Mul.mul (Mβ»ΒΉ 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 : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
h_d_pos : StrongLT 0 d
s : Fin n β β
i : Fin n
β’ d i / (d i / s i) = s i | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
h_d_pos : StrongLT 0 d
s : Fin n β β
i : Fin n
h_di_pos : 0 i < d i
β’ 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 : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
h_d_pos : StrongLT 0 d
s : Fin n β β
i : Fin n
h_di_pos : 0 i < d i
β’ d i / (d i / s i) = s i | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
h_d_pos : StrongLT 0 d
s : Fin n β β
i : Fin n
h_di_pos : 0 < d i
β’ 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 β 0 := by linarith | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
h_d_pos : StrongLT 0 d
s : Fin n β β
i : Fin n
h_di_pos : 0 < d i
β’ d i / (d i / s i) = s i | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
h_d_pos : StrongLT 0 d
s : Fin n β β
i : Fin n
h_di_pos : 0 < d i
h_di_nonzero : d i β 0
β’ 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 [β div_mul, div_self h_di_nonzero, one_mul] | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
h_d_pos : StrongLT 0 d
s : Fin n β β
i : Fin n
h_di_pos : 0 < d i
h_di_nonzero : d i β 0
β’ 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 : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
h_d_pos : StrongLT 0 d
s : Fin n β β
i : Fin n
h_di_pos : 0 < d i
β’ d i β 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 : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
hn : Fact (0 < n)
t : Fin n β β
i : Fin n
β’ β j β [[0, i]], t j = Vec.cumsum t i | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
hn : Fact (0 < n)
t : Fin n β β
i : Fin n
β’ β j β [[0, i]], t j = if h : 0 < n then β j β [[β¨0, hβ©, 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 : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
hn : Fact (0 < n)
t : Fin n β β
i : Fin n
β’ β j β [[0, i]], t j = if h : 0 < n then β j β [[β¨0, hβ©, i]], t j else 0 | case pos
n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
hn : Fact (0 < n)
t : Fin n β β
i : Fin n
hβ : 0 < n
β’ β j β [[0, i]], t j = β j β [[β¨0, hββ©, i]], t j
case neg
n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
hn : Fact (0 < n)
t : Fin n β β
i : Fin n
hβ : Β¬0 < n
β’ β j β [[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 : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
hn : Fact (0 < n)
t : Fin n β β
i : Fin n
hβ : 0 < n
β’ β j β [[0, i]], t j = β j β [[β¨0, hββ©, 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 : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
hn : Fact (0 < n)
t : Fin n β β
i : Fin n
hβ : Β¬0 < n
β’ β j β [[0, i]], t j = 0 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/VehicleSpeedScheduling.lean | VehicleSpeedSched.nβ_pos | [148, 1] | [148, 48] | unfold nβ | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ 0 < nβ | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ 0 < 10 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/VehicleSpeedScheduling.lean | VehicleSpeedSched.nβ_pos | [148, 1] | [148, 48] | norm_num | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ 0 < 10 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/VehicleSpeedScheduling.lean | VehicleSpeedSched.dβ_pos | [154, 1] | [155, 50] | intro i | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ StrongLT 0 dβ | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
i : Fin nβ
β’ 0 i < dβ i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/VehicleSpeedScheduling.lean | VehicleSpeedSched.dβ_pos | [154, 1] | [155, 50] | fin_cases i <;> (dsimp [dβ]; norm_num) | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
i : Fin nβ
β’ 0 i < dβ i | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/VehicleSpeedScheduling.lean | VehicleSpeedSched.dβ_pos | [154, 1] | [155, 50] | dsimp [dβ] | case tail.tail.tail.tail.tail.tail.tail.tail.tail.head
n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ 0 β¨9, β―β© < dβ β¨9, β―β© | case tail.tail.tail.tail.tail.tail.tail.tail.tail.head
n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ 0 < ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] β¨9, β―β© |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/VehicleSpeedScheduling.lean | VehicleSpeedSched.dβ_pos | [154, 1] | [155, 50] | norm_num | case tail.tail.tail.tail.tail.tail.tail.tail.tail.head
n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ 0 < ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] β¨9, β―β© | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/VehicleSpeedScheduling.lean | VehicleSpeedSched.sminβ_pos | [173, 1] | [174, 36] | intro i | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ StrongLT 0 sminβ | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
i : Fin nβ
β’ 0 i < sminβ i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/VehicleSpeedScheduling.lean | VehicleSpeedSched.sminβ_pos | [173, 1] | [174, 36] | fin_cases i <;> norm_num | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
i : Fin nβ
β’ 0 i < sminβ i | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/VehicleSpeedScheduling.lean | VehicleSpeedSched.sminβ_le_smaxβ | [179, 1] | [180, 60] | intro i | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ sminβ β€ smaxβ | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
i : Fin nβ
β’ sminβ i β€ smaxβ i |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/VehicleSpeedScheduling.lean | VehicleSpeedSched.sminβ_le_smaxβ | [179, 1] | [180, 60] | fin_cases i <;> (dsimp [sminβ, smaxβ]; norm_num) | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
i : Fin nβ
β’ sminβ i β€ smaxβ i | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/VehicleSpeedScheduling.lean | VehicleSpeedSched.sminβ_le_smaxβ | [179, 1] | [180, 60] | dsimp [sminβ, smaxβ] | case tail.tail.tail.tail.tail.tail.tail.tail.tail.head
n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ sminβ β¨9, β―β© β€ smaxβ β¨9, β―β© | case tail.tail.tail.tail.tail.tail.tail.tail.tail.head
n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ ![0.7828, 0.6235, 0.7155, 0.5340, 0.6329, 0.4259, 0.7798, 0.9604, 0.7298, 0.8405] β¨9, β―β© β€
![1.9624, 1.6036, 1.6439, 1.5641, 1.7194, 1.9090, 1.3193, 1.3366, 1.9470, 2.8803] β¨9, β―β© |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/VehicleSpeedScheduling.lean | VehicleSpeedSched.sminβ_le_smaxβ | [179, 1] | [180, 60] | norm_num | case tail.tail.tail.tail.tail.tail.tail.tail.tail.head
n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ ![0.7828, 0.6235, 0.7155, 0.5340, 0.6329, 0.4259, 0.7798, 0.9604, 0.7298, 0.8405] β¨9, β―β© β€
![1.9624, 1.6036, 1.6439, 1.5641, 1.7194, 1.9090, 1.3193, 1.3366, 1.9470, 2.8803] β¨9, β―β© | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/VehicleSpeedScheduling.lean | VehicleSpeedSched.aβ_nonneg | [188, 1] | [189, 51] | unfold aβ | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ 0 β€ aβ | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ 0 β€ 1 |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/VehicleSpeedScheduling.lean | VehicleSpeedSched.aβ_nonneg | [188, 1] | [189, 51] | norm_num | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ 0 β€ 1 | no goals |
https://github.com/verified-optimization/CvxLean.git | c62c2f292c6420f31a12e738ebebdfed50f6f840 | CvxLean/Examples/VehicleSpeedScheduling.lean | VehicleSpeedSched.aβdβ2_nonneg | [191, 1] | [193, 55] | intros i | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ 0 β€ aβ β’ dβ ^ 2 | n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
i : Fin nβ
β’ 0 i β€ (aβ β’ dβ ^ 2) i |
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