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
| input
stringlengths 73
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
t x : β
β’ x.exp β€ t β x.expCone 1 t
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.h
t x : β
hexp : x.exp β€ t
β’ 0 < 1 β§ 1 * (x / 1).exp β€ t
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.h.refine_2
t x : β
hexp : x.exp β€ t
β’ 1 * (x / 1).exp β€ t
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.h.refine_1
t x : β
hexp : x.exp β€ t
β’ 0 < 1
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.h.refine_2
t x : β
hexp : x.exp β€ t
β’ 1 * (x / 1).exp β€ t
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inl
t x : β
h : 0 < 1 β§ 1 * (x / 1).exp β€ t
β’ x.exp β€ t
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr
t x : β
h : 1 = 0 β§ 0 β€ t β§ x β€ 0
β’ x.exp β€ t
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr
t x : β
h : 1 = 0 β§ 0 β€ t β§ x β€ 0
β’ False
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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©)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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©)
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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©)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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©)
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ 0 < nβ
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ 0 < 10
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ StrongLT 0 dβ
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
i : Fin nβ
β’ 0 i < dβ i
TACTIC:
|
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©
|
Please generate a tactic in lean4 to solve the state.
STATE:
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©
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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©
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ StrongLT 0 sminβ
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
i : Fin nβ
β’ 0 i < sminβ i
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ sminβ β€ smaxβ
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
i : Fin nβ
β’ sminβ i β€ smaxβ i
TACTIC:
|
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©
|
Please generate a tactic in lean4 to solve the state.
STATE:
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©
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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©
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ 0 β€ aβ
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ 0 β€ 1
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
n : β
d Οmin Οmax smin smax : Fin n β β
F : β β β
β’ 0 β€ aβ β’ dβ ^ 2
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.