statement stringlengths 1 615 | proof stringlengths 0 8.26k | type stringclasses 11
values | symbolic_name stringlengths 1 64 | library stringclasses 29
values | filename stringclasses 121
values | imports listlengths 0 15 | deps listlengths 0 20 | docstring stringlengths 0 1.5k | source_url stringclasses 1
value | commit stringclasses 1
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Vec.soCone (t : m → ℝ) (X : Matrix m n ℝ) : Prop | ∀ i, Real.soCone (t i) (X i) | def | Real.Vec.soCone | Lib.Cones | CvxLean/Lib/Cones/SOCone.lean | [
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Math.Data.Vec"
] | [
"Real.soCone"
] | `m` copies of the `n`-dimensional second-order cone `(𝒬ⁿ)ᵐ`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
Vec.rotatedSoCone (v w : m → ℝ) (X : Matrix m n ℝ) : Prop | ∀ i, Real.rotatedSoCone (v i) (w i) (X i) | def | Real.Vec.rotatedSoCone | Lib.Cones | CvxLean/Lib/Cones/SOCone.lean | [
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Math.Data.Vec"
] | [
"Real.rotatedSoCone"
] | `m` copies of the `n`-dimensional rotated second-order cone `(𝒬ᵣⁿ)ᵐ`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
rotateSoCone {n : ℕ} (t : ℝ) (x : Fin n.succ → ℝ) : ℝ × ℝ × (Fin n → ℝ) | ((t + x 0) / sqrt 2, (t - x 0) / sqrt 2, fun i => x i.succ) | def | Real.rotateSoCone | Lib.Cones | CvxLean/Lib/Cones/SOCone.lean | [
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Math.Data.Vec"
] | [] | If `(t, x) ∈ 𝒬ⁿ⁺¹` then `r(t, x) ∈ 𝒬ᵣⁿ⁺²`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
rotateSoCone_rotatedSoCone {n : ℕ} {t : ℝ} {x : Fin n.succ → ℝ} (h : soCone t x) :
let (v, w, x) | rotateSoCone t x; rotatedSoCone v w x := by
simp [rotatedSoCone, rotateSoCone]
have habsx0t : |x 0| ≤ t := by
rw [soCone, Fin.sum_univ_succ] at h
have hS : 0 ≤ ∑ i : Fin n, x (Fin.succ i) ^ 2 :=
Finset.sum_nonneg (fun i _ => (rpow_two (x i.succ)).symm ▸ sq_nonneg (x i.succ))
exact abs_le_of_sqrt_s... | lemma | Real.rotateSoCone_rotatedSoCone | Lib.Cones | CvxLean/Lib/Cones/SOCone.lean | [
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
unrotateSoCone {n : ℕ} (v w : Real) (x : Fin n → ℝ) : ℝ × (Fin n.succ → ℝ) | ((v + w) / sqrt 2, Matrix.vecCons ((v - w) / sqrt 2) x) | def | Real.unrotateSoCone | Lib.Cones | CvxLean/Lib/Cones/SOCone.lean | [
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Math.Data.Vec"
] | [] | If `(v, w, x) ∈ 𝒬ⁿ⁺²` then `u(v, w, x) ∈ 𝒬ᵣⁿ⁺¹`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
unrotateSoCone_soCone {n : ℕ} {v w : ℝ} {x : Fin n → ℝ} (h : rotatedSoCone v w x) :
let (t, x) | unrotateSoCone v w x; soCone t x := by
simp [soCone, unrotateSoCone]
replace ⟨h, hv, hw⟩ := h
rw [sqrt_le_iff]
split_ands
{ simp [le_div_iff]; linarith }
{ rw [Fin.sum_univ_succ]
simp [Matrix.vecCons]
rw [add_comm, ← le_sub_iff_add_le]
field_simp
have hrw : ((v + w) ^ 2 - (v - w) ^ 2) / 2 = ... | lemma | Real.unrotateSoCone_soCone | Lib.Cones | CvxLean/Lib/Cones/SOCone.lean | [
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
soCone_add_sub_two_mul_of_nonneg {x y : ℝ} (z : ℝ)
(hx : 0 ≤ x) (hy : 0 ≤ y) : soCone (x + y) ![x - y, 2 * z] ↔ z ^ (2 : ℝ) ≤ x * y | by
have hxy := add_nonneg hx hy
conv => lhs; unfold soCone; simp [sqrt_le_left hxy, ← le_sub_iff_add_le']
ring_nf; simp | lemma | Real.soCone_add_sub_two_mul_of_nonneg | Lib.Cones | CvxLean/Lib/Cones/SOCone.lean | [
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Math.Data.Vec"
] | [] | To handle powers, a common trick is to use the fact that for
`x, y ≥ 0` and `z ∈ ℝ`,
`((x + y), (x - y, 2z)ᵀ) ∈ 𝒬ⁿ⁺¹ ↔ z ^ 2 ≤ xy`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
soCone_add_sub_two_of_nonneg {x y : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) :
soCone (x + y) ![x - y, 2] ↔ 1 ≤ x * y | by
have h := soCone_add_sub_two_mul_of_nonneg 1 hx hy
rw [mul_one, one_rpow] at h
exact h | lemma | Real.soCone_add_sub_two_of_nonneg | Lib.Cones | CvxLean/Lib/Cones/SOCone.lean | [
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Math.Data.Vec"
] | [] | Same as `soCone_add_sub_two_mul_of_nonneg` with `z = 1`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
soCone_sub_add_two_mul_of_nonneg {x y : ℝ} (z : ℝ) :
soCone (x - y) ![x + y, 2 * z] ↔ y ≤ x ∧ z ^ (2 : ℝ) ≤ -(x * y) | by
conv => lhs; unfold soCone; simp [sqrt_le_iff, ← le_sub_iff_add_le']
apply Iff.and
· rfl
· ring_nf!; rw [← neg_mul, ← div_le_iff (by norm_num)]; simp | lemma | Real.soCone_sub_add_two_mul_of_nonneg | Lib.Cones | CvxLean/Lib/Cones/SOCone.lean | [
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Math.Data.Vec"
] | [] | Same as `soCone_add_sub_two_mul_of_nonneg` replacing `y` by `-y`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
vec_soCone_apply_to_soCone_add_sub_two_mul {n : ℕ} (x y z : Fin n → ℝ) (i : Fin n) :
(soCone ((x + y) i) ((![x - y, 2 • z]ᵀ) i)) ↔ (soCone (x i + y i) ![x i - y i, 2 * z i]) | by
dsimp; convert Iff.rfl; funext j; fin_cases j <;> simp | lemma | Real.vec_soCone_apply_to_soCone_add_sub_two_mul | Lib.Cones | CvxLean/Lib/Cones/SOCone.lean | [
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
vec_soCone_apply_to_soCone_add_sub_two {n : ℕ} (x y : Fin n → ℝ) (i : Fin n) :
(soCone ((x + y) i) ((![x - y, 2]ᵀ) i)) ↔ (soCone (x i + y i) ![x i - y i, 2]) | by
dsimp; convert Iff.rfl; funext j; fin_cases j <;> simp | lemma | Real.vec_soCone_apply_to_soCone_add_sub_two | Lib.Cones | CvxLean/Lib/Cones/SOCone.lean | [
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Analysis.SpecialFunctions.Pow.Real",
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
zeroCone (x : ℝ) : Prop | x = 0 | def | Real.zeroCone | Lib.Cones | CvxLean/Lib/Cones/ZeroCone.lean | [
"Mathlib.Data.Real.Basic",
"CvxLean.Lib.Math.Data.Matrix"
] | [] | The zero cone `{0}`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
Vec.zeroCone {n} [Fintype n] (x : n → ℝ) : Prop | ∀ i, Real.zeroCone (x i) | def | Real.Vec.zeroCone | Lib.Cones | CvxLean/Lib/Cones/ZeroCone.lean | [
"Mathlib.Data.Real.Basic",
"CvxLean.Lib.Math.Data.Matrix"
] | [
"Real.zeroCone"
] | The `n`-dimensional zero cone `{0}ⁿ`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
Matrix.zeroCone {n m} [Fintype n] [Fintype m] (M : Matrix n m ℝ) : Prop | ∀ i j, Real.zeroCone (M i j) | def | Real.Matrix.zeroCone | Lib.Cones | CvxLean/Lib/Cones/ZeroCone.lean | [
"Mathlib.Data.Real.Basic",
"CvxLean.Lib.Math.Data.Matrix"
] | [
"Real.zeroCone"
] | The `n×m`-dimensional zero cone `{0}ⁿˣᵐ`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
Matrix.quadForm {n : ℕ} (R : Matrix (Fin n) (Fin n) ℝ) (x : Fin n → ℝ) : ℝ | x ⬝ᵥ R.mulVec x | def | Matrix.quadForm | Lib.Math | CvxLean/Lib/Math/CovarianceEstimation.lean | [
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
gaussianPdf {n : ℕ} (R : Matrix (Fin n) (Fin n) ℝ) (x : Fin n → ℝ) : ℝ | (1 / sqrt (((2 * π) ^ n) * R.det)) * exp (- R⁻¹.quadForm x / 2) | def | gaussianPdf | Lib.Math | CvxLean/Lib/Math/CovarianceEstimation.lean | [
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
covarianceMatrix {N n : ℕ} (y : Fin N → Fin n → ℝ) : Matrix (Fin n) (Fin n) ℝ | fun i j : Fin n => (∑ k : Fin N, y k i * y k j) / N | def | covarianceMatrix | Lib.Math | CvxLean/Lib/Math/CovarianceEstimation.lean | [
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
gaussianPdf_pos {n : ℕ} (R : Matrix (Fin n) (Fin n) ℝ) (y : Fin n → ℝ) (h : R.PosDef) :
0 < gaussianPdf R y | by
refine' mul_pos (div_pos zero_lt_one (sqrt_pos.2 (mul_pos _ h.det_pos))) (exp_pos _)
exact (pow_pos (mul_pos (by positivity) pi_pos) n) | lemma | gaussianPdf_pos | Lib.Math | CvxLean/Lib/Math/CovarianceEstimation.lean | [
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef"
] | [
"gaussianPdf"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
prod_gaussianPdf_pos {N n : ℕ} (R : Matrix (Fin n) (Fin n) ℝ) (y : Fin N → Fin n → ℝ)
(h : R.PosDef) : 0 < ∏ i : Fin N, gaussianPdf R (y i) | Finset.prod_pos (fun _ _ => gaussianPdf_pos _ _ h) | lemma | prod_gaussianPdf_pos | Lib.Math | CvxLean/Lib/Math/CovarianceEstimation.lean | [
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef"
] | [
"gaussianPdf",
"gaussianPdf_pos"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
log_prod_gaussianPdf {N n : ℕ} (y : Fin N → Fin n → ℝ) (R : Matrix (Fin n) (Fin n) ℝ)
(hR : R.PosDef) : log (∏ i : Fin N, gaussianPdf R (y i)) =
∑ i : Fin N, (- (log (sqrt ((2 * π) ^ n)) + log (sqrt R.det)) + - R⁻¹.quadForm (y i) / 2) | by
have : ∀ i,
i ∈ Finset.univ → gaussianPdf R (y i) ≠ 0 := fun i _ => ne_of_gt (gaussianPdf_pos _ _ hR)
have sqrt_2_pi_n_R_det_ne_zero: sqrt ((2 * π) ^ n * R.det) ≠ 0 := by
refine' ne_of_gt (sqrt_pos.2 (mul_pos _ hR.det_pos))
exact (pow_pos (mul_pos (by positivity) pi_pos) _)
rw [log_prod Finset.univ... | lemma | log_prod_gaussianPdf | Lib.Math | CvxLean/Lib/Math/CovarianceEstimation.lean | [
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef"
] | [
"gaussianPdf",
"gaussianPdf_pos"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
sum_quadForm {n : ℕ} (R : Matrix (Fin n) (Fin n) ℝ) {m : ℕ} (y : Fin m → Fin n → ℝ) :
∑ i, R.quadForm (y i) = m * (covarianceMatrix y * Rᵀ).trace | by
by_cases h : m = 0
{ subst h; simp }
simp only [Matrix.quadForm, Matrix.trace, covarianceMatrix, diag, mul_apply, Finset.sum_mul,
Finset.sum_div]
erw [Finset.sum_comm (γ := Fin m)]
simp_rw [Finset.mul_sum]
congr; ext i
have hmnz : (m : ℝ) ≠ 0 := by simp [h]
simp_rw [← mul_assoc, mul_div_cancel₀ _... | lemma | sum_quadForm | Lib.Math | CvxLean/Lib/Math/CovarianceEstimation.lean | [
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef"
] | [
"Matrix.quadForm",
"covarianceMatrix"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
Real.inverse_eq_inv (a : ℝ) : Ring.inverse a = a⁻¹ | by simp | lemma | Real.inverse_eq_inv | Lib.Math | CvxLean/Lib/Math/CovarianceEstimation.lean | [
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
LDL.invertible_diag : Invertible (LDL.diag hA) | by
rw [LDL.diag_eq_lowerInv_conj]
refine @invertibleMul _ _ _ _ (@invertibleMul _ _ _ _ _ hA.Invertible) _ | instance | Matrix.LDL.invertible_diag | Lib.Math | CvxLean/Lib/Math/LogDet.lean | [
"Mathlib.LinearAlgebra.Matrix.LDL",
"CvxLean.Lib.Math.SchurComplement",
"CvxLean.Lib.Math.Subadditivity",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Triangular",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.ToUpperTri",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.LDL"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
PosSemidef_zero : PosSemidef (0 : Matrix n n 𝕜) | by
simp [PosSemidef] | lemma | Matrix.PosSemidef_zero | Lib.Math | CvxLean/Lib/Math/LogDet.lean | [
"Mathlib.LinearAlgebra.Matrix.LDL",
"CvxLean.Lib.Math.SchurComplement",
"CvxLean.Lib.Math.Subadditivity",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Triangular",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.ToUpperTri",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.LDL"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
LogDetAtom.feasibility_PosDef {D Z : Matrix n n ℝ} (hD : D = LDL.diag hA)
(hZ : Z = LDL.diag hA * (LDL.lower hA)ᵀ) : (fromBlocks D Z Zᵀ A).PosSemidef | by
have h_D_eq : D = Z * A⁻¹ * Zᴴ :=
calc D
= D * D⁻¹ * D := by
rw [hD, Matrix.mul_inv_of_invertible, Matrix.one_mul]
_ = D * (LDL.lowerInv hA * A * (LDL.lowerInv hA)ᵀ)⁻¹ * Dᵀ := by
erw [hD, LDL.diag, diagonal_transpose, ← LDL.diag, LDL.diag_eq_lowerInv_conj]
rfl
_ = D ... | lemma | Matrix.LogDetAtom.feasibility_PosDef | Lib.Math | CvxLean/Lib/Math/LogDet.lean | [
"Mathlib.LinearAlgebra.Matrix.LDL",
"CvxLean.Lib.Math.SchurComplement",
"CvxLean.Lib.Math.Subadditivity",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Triangular",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.ToUpperTri",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.LDL"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
LogDetAtom.feasibility_PosDef' {D Z Y : Matrix n n ℝ} (hY : Y = LDL.diag hA * (LDL.lower hA)ᵀ)
(hD : D = diagonal Y.diag) (hZ : Z = Y.toUpperTri) : (fromBlocks D Z Zᵀ A).PosSemidef | by
have hY_tri : upperTriangular Y := by
{ rw [hY]
apply upperTriangular.mul
apply BlockTriangular_diagonal
apply lowerTriangular.transpose
apply LDL.lowerTriangular_lower }
haveI := hA.Invertible
rw [hZ, hY_tri.toUpperTri_eq]
apply LogDetAtom.feasibility_PosDef _ _ hY
simp [hD, hY, LDL.diag... | lemma | Matrix.LogDetAtom.feasibility_PosDef' | Lib.Math | CvxLean/Lib/Math/LogDet.lean | [
"Mathlib.LinearAlgebra.Matrix.LDL",
"CvxLean.Lib.Math.SchurComplement",
"CvxLean.Lib.Math.Subadditivity",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Triangular",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.ToUpperTri",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.LDL"
] | [
"LDL.lowerTriangular_lower"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
LDL.diagEntries_pos {A : Matrix n n ℝ} (hA: A.PosDef) (i : n) :
0 < LDL.diagEntries hA i | by
have : (LDL.lowerInv hA).det ≠ 0 := by simp [LDL.det_lowerInv hA]
have : LDL.lowerInv hA i ≠ 0 := fun h =>
this (Matrix.det_eq_zero_of_row_eq_zero i (λ j => congr_fun h j))
exact hA.2 (LDL.lowerInv hA i) this | lemma | Matrix.LDL.diagEntries_pos | Lib.Math | CvxLean/Lib/Math/LogDet.lean | [
"Mathlib.LinearAlgebra.Matrix.LDL",
"CvxLean.Lib.Math.SchurComplement",
"CvxLean.Lib.Math.Subadditivity",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Triangular",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.ToUpperTri",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.LDL"
] | [
"LDL.det_lowerInv"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
LogDetAtom.solution_eq_atom {A : Matrix n n ℝ} (hA: A.PosDef) :
∑ i, Real.log (LDL.diagEntries hA i) = Real.log (A.det) | by
conv => rhs; rw [(LDL.lower_conj_diag hA).symm]
have heqsum := Real.log_prod Finset.univ (LDL.diagEntries hA)
(fun i _ => ne_of_gt (LDL.diagEntries_pos hA i))
simp [LDL.diag, heqsum.symm] | lemma | Matrix.LogDetAtom.solution_eq_atom | Lib.Math | CvxLean/Lib/Math/LogDet.lean | [
"Mathlib.LinearAlgebra.Matrix.LDL",
"CvxLean.Lib.Math.SchurComplement",
"CvxLean.Lib.Math.Subadditivity",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Triangular",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.ToUpperTri",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.LDL"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
LogDetAtom.feasibility_exp {A : Matrix n n ℝ} (hA: A.PosDef) (i : n) :
LDL.diagEntries hA i ≤ ((LDL.diag hA) * ((LDL.lower hA)ᵀ)).diag i | by
simp [LDL.diag] | lemma | Matrix.LogDetAtom.feasibility_exp | Lib.Math | CvxLean/Lib/Math/LogDet.lean | [
"Mathlib.LinearAlgebra.Matrix.LDL",
"CvxLean.Lib.Math.SchurComplement",
"CvxLean.Lib.Math.Subadditivity",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Triangular",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.ToUpperTri",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.LDL"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
IsHermitian₁₁_of_IsHermitian_toBlock {A B C D : Matrix n n ℝ}
(h : (fromBlocks A B C D).IsHermitian) : IsHermitian A | by
ext i j; simpa using congr_fun (congr_fun h (Sum.inl i)) (Sum.inl j) | lemma | Matrix.IsHermitian₁₁_of_IsHermitian_toBlock | Lib.Math | CvxLean/Lib/Math/LogDet.lean | [
"Mathlib.LinearAlgebra.Matrix.LDL",
"CvxLean.Lib.Math.SchurComplement",
"CvxLean.Lib.Math.Subadditivity",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Triangular",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.ToUpperTri",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.LDL"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
IsHermitian₂₂_of_IsHermitian_toBlock {A B C D : Matrix n n ℝ}
(h : (fromBlocks A B C D).IsHermitian) : IsHermitian D | by
ext i j; simpa using congr_fun (congr_fun h (Sum.inr i)) (Sum.inr j) | lemma | Matrix.IsHermitian₂₂_of_IsHermitian_toBlock | Lib.Math | CvxLean/Lib/Math/LogDet.lean | [
"Mathlib.LinearAlgebra.Matrix.LDL",
"CvxLean.Lib.Math.SchurComplement",
"CvxLean.Lib.Math.Subadditivity",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Triangular",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.ToUpperTri",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.LDL"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
PosSemidef₁₁_of_PosSemidef_toBlock {A B C D : Matrix n n ℝ}
(h : (fromBlocks A B C D).PosSemidef) : PosSemidef A | ⟨IsHermitian₁₁_of_IsHermitian_toBlock h.1,
fun x => by simpa [Matrix.fromBlocks_mulVec, star] using h.2 (Sum.elim x 0)⟩ | lemma | Matrix.PosSemidef₁₁_of_PosSemidef_toBlock | Lib.Math | CvxLean/Lib/Math/LogDet.lean | [
"Mathlib.LinearAlgebra.Matrix.LDL",
"CvxLean.Lib.Math.SchurComplement",
"CvxLean.Lib.Math.Subadditivity",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Triangular",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.ToUpperTri",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.LDL"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
PosSemidef₂₂_of_PosSemidef_toBlock {A B C D : Matrix n n ℝ}
(h : (fromBlocks A B C D).PosSemidef) :
PosSemidef D | ⟨IsHermitian₂₂_of_IsHermitian_toBlock h.1,
fun x => by simpa [Matrix.fromBlocks_mulVec, star] using h.2 (Sum.elim 0 x)⟩ | lemma | Matrix.PosSemidef₂₂_of_PosSemidef_toBlock | Lib.Math | CvxLean/Lib/Math/LogDet.lean | [
"Mathlib.LinearAlgebra.Matrix.LDL",
"CvxLean.Lib.Math.SchurComplement",
"CvxLean.Lib.Math.Subadditivity",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Triangular",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.ToUpperTri",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.LDL"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
LogDetAtom.optimality_D_posdef {t : n → ℝ} {Y Z D : Matrix n n ℝ}
(ht : ∀ i, (t i).exp ≤ Y.diag i) (hD : D = Matrix.diagonal (Y.diag)) (_hZ : Z = Y.toUpperTri)
(hPSD : (fromBlocks D Z Zᵀ A).PosSemidef) : D.PosDef | by
have h_D_psd : D.PosSemidef := PosSemidef₁₁_of_PosSemidef_toBlock hPSD
show D.PosDef
{ rw [h_D_psd.PosDef_iff_det_ne_zero, hD, det_diagonal, Finset.prod_ne_zero_iff]
exact λ i _ => ne_of_gt (lt_of_lt_of_le ((t i).exp_pos) (ht i)) } | lemma | Matrix.LogDetAtom.optimality_D_posdef | Lib.Math | CvxLean/Lib/Math/LogDet.lean | [
"Mathlib.LinearAlgebra.Matrix.LDL",
"CvxLean.Lib.Math.SchurComplement",
"CvxLean.Lib.Math.Subadditivity",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Triangular",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.ToUpperTri",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.LDL"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
LogDetAtom.optimality_Ddet_le_Adet {t : n → ℝ} {Y Z D : Matrix n n ℝ}
(ht : ∀ i, (t i).exp ≤ Y.diag i) (hD : D = Matrix.diagonal (Y.diag)) (hZ : Z = Y.toUpperTri)
(hPSD : (fromBlocks D Z Zᵀ A).PosSemidef) : D.det ≤ A.det | by
by_cases h_nonempty : Nonempty n
{ have h_D_pd : D.PosDef := LogDetAtom.optimality_D_posdef ht hD hZ hPSD
haveI h_D_invertible : Invertible D := h_D_pd.Invertible
have h_Zdet : Z.det = D.det := by
{ rw [hZ, det_of_upperTriangular (upperTriangular_toUpperTri Y), hD, det_diagonal]
simp [toUpperTr... | lemma | Matrix.LogDetAtom.optimality_Ddet_le_Adet | Lib.Math | CvxLean/Lib/Math/LogDet.lean | [
"Mathlib.LinearAlgebra.Matrix.LDL",
"CvxLean.Lib.Math.SchurComplement",
"CvxLean.Lib.Math.Subadditivity",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Triangular",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.ToUpperTri",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.LDL"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
LogDetAtom.cond_elim {t : n → ℝ} {Y Z D : Matrix n n ℝ} (ht : ∀ i, (t i).exp ≤ Y.diag i)
(hD : D = Matrix.diagonal (Y.diag)) (hZ : Z = Y.toUpperTri)
(hPSD : (fromBlocks D Z Zᵀ A).PosSemidef) : A.PosDef | by
have h_D_pd : D.PosDef := LogDetAtom.optimality_D_posdef ht hD hZ hPSD
have h_A_psd : A.PosSemidef := PosSemidef₂₂_of_PosSemidef_toBlock hPSD
have h_Ddet_le_Adet : D.det ≤ A.det := LogDetAtom.optimality_Ddet_le_Adet ht hD hZ hPSD
have h_Adet_pos : 0 < A.det := lt_of_lt_of_le h_D_pd.det_pos h_Ddet_le_Adet
r... | lemma | Matrix.LogDetAtom.cond_elim | Lib.Math | CvxLean/Lib/Math/LogDet.lean | [
"Mathlib.LinearAlgebra.Matrix.LDL",
"CvxLean.Lib.Math.SchurComplement",
"CvxLean.Lib.Math.Subadditivity",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Triangular",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.ToUpperTri",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.LDL"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
LogDetAtom.optimality {t : n → ℝ} {Y Z D : Matrix n n ℝ} (ht : ∀ i, (t i).exp ≤ Y.diag i)
(hD : D = Matrix.diagonal (Y.diag)) (hZ : Z = Y.toUpperTri)
(hPSD : (fromBlocks D Z Zᵀ A).PosSemidef) : ∑ i, t i ≤ Real.log A.det | by
have h_A_pd : A.PosDef := LogDetAtom.cond_elim ht hD hZ hPSD
have h_Ddet_le_Adet : D.det ≤ A.det := LogDetAtom.optimality_Ddet_le_Adet ht hD hZ hPSD
have h_Adet_pos: 0 < A.det := h_A_pd.det_pos
rw [← Real.exp_le_exp, Real.exp_sum, Real.exp_log h_Adet_pos]
apply le_trans (Finset.prod_le_prod (fun i _ => le_... | lemma | Matrix.LogDetAtom.optimality | Lib.Math | CvxLean/Lib/Math/LogDet.lean | [
"Mathlib.LinearAlgebra.Matrix.LDL",
"CvxLean.Lib.Math.SchurComplement",
"CvxLean.Lib.Math.Subadditivity",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Triangular",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.ToUpperTri",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.LDL"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
sum_exp_eq_sum_div {n} (x : Fin n → ℝ) (t : ℝ) :
Vec.sum (Vec.exp (x - Vec.const n t)) = (Vec.sum (Vec.exp x)) / (Real.exp t) | by
unfold Vec.sum
rw [Finset.sum_div]
congr; ext i
simp [Vec.exp, Vec.const, Real.exp_sub] | lemma | Vec.sum_exp_eq_sum_div | Lib.Math | CvxLean/Lib/Math/LogSumExp.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Math.Data.Vec"
] | [
"Vec.const",
"Vec.exp",
"Vec.sum"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
sum_exp_pos {n} (hn : 0 < n) (x : Fin n → ℝ) :
0 < Vec.sum (Vec.exp x) | by
apply Finset.sum_pos
{ intros i _; simp [Vec.exp, Real.exp_pos] }
{ existsi ⟨0, hn⟩; simp } | lemma | Vec.sum_exp_pos | Lib.Math | CvxLean/Lib/Math/LogSumExp.lean | [
"CvxLean.Lib.Math.Data.Real",
"CvxLean.Lib.Math.Data.Vec"
] | [
"Vec.exp",
"Vec.sum"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
schur_complement_eq₁₁ [Fintype m] [DecidableEq m] [Fintype n] {A : Matrix m m 𝕜}
(B : Matrix m n 𝕜) (D : Matrix n n 𝕜) (x : m → 𝕜) (y : n → 𝕜) [Invertible A]
(hA : A.IsHermitian) :
vecMul (star (x ⊕ᵥ y)) (fromBlocks A B Bᴴ D) ⬝ᵥ (x ⊕ᵥ y) =
vecMul (star (x + (A⁻¹ * B).mulVec y)) A ⬝ᵥ (x + (A⁻¹ * B... | by
simp [Function.star_sum_elim, fromBlocks_mulVec, vecMul_fromBlocks, add_vecMul,
dotProduct_mulVec, vecMul_sub, Matrix.mul_assoc, vecMul_mulVec, hA.eq,
conjTranspose_nonsing_inv, star_mulVec]
abel | lemma | Matrix.schur_complement_eq₁₁ | Lib.Math | CvxLean/Lib/Math/SchurComplement.lean | [
"Mathlib.LinearAlgebra.Matrix.Symmetric",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.PosDef",
"Mathlib.Algebra.Star.Pi"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
schur_complement_eq₂₂ [Fintype m] [Fintype n] [DecidableEq n] (A : Matrix m m 𝕜)
(B : Matrix m n 𝕜) {D : Matrix n n 𝕜} (x : m → 𝕜) (y : n → 𝕜) [Invertible D]
(hD : D.IsHermitian) :
vecMul (star (x ⊕ᵥ y)) (fromBlocks A B Bᴴ D) ⬝ᵥ (x ⊕ᵥ y) =
vecMul (star ((D⁻¹ * Bᴴ).mulVec x + y)) D ⬝ᵥ ((D⁻¹ * Bᴴ).... | by
simp [Function.star_sum_elim, fromBlocks_mulVec, vecMul_fromBlocks, add_vecMul,
dotProduct_mulVec, vecMul_sub, Matrix.mul_assoc, vecMul_mulVec, hD.eq,
conjTranspose_nonsing_inv, star_mulVec]
abel | lemma | Matrix.schur_complement_eq₂₂ | Lib.Math | CvxLean/Lib/Math/SchurComplement.lean | [
"Mathlib.LinearAlgebra.Matrix.Symmetric",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.PosDef",
"Mathlib.Algebra.Star.Pi"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
IsHermitian.fromBlocks₁₁ [Fintype m] [DecidableEq m] {A : Matrix m m 𝕜} (B : Matrix m n 𝕜)
(D : Matrix n n 𝕜) (hA : A.IsHermitian) :
(Matrix.fromBlocks A B Bᴴ D).IsHermitian ↔ (D - Bᴴ * A⁻¹ * B).IsHermitian | by
have hBAB : (Bᴴ * A⁻¹ * B).IsHermitian := by
apply isHermitian_conjTranspose_mul_mul
apply hA.inv
rw [isHermitian_fromBlocks_iff]
constructor
· intro h
apply IsHermitian.sub h.2.2.2 hBAB
· intro h
refine' ⟨hA, rfl, conjTranspose_conjTranspose B, _⟩
rw [← sub_add_cancel D]
apply IsHe... | lemma | Matrix.IsHermitian.fromBlocks₁₁ | Lib.Math | CvxLean/Lib/Math/SchurComplement.lean | [
"Mathlib.LinearAlgebra.Matrix.Symmetric",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.PosDef",
"Mathlib.Algebra.Star.Pi"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
IsHermitian.fromBlocks₂₂ [Fintype n] [DecidableEq n] (A : Matrix m m 𝕜) (B : Matrix m n 𝕜)
{D : Matrix n n 𝕜} (hD : D.IsHermitian) :
(Matrix.fromBlocks A B Bᴴ D).IsHermitian ↔ (A - B * D⁻¹ * Bᴴ).IsHermitian | by
rw [← isHermitian_submatrix_equiv (Equiv.sumComm n m), Equiv.sumComm_apply,
fromBlocks_submatrix_sum_swap_sum_swap]
convert IsHermitian.fromBlocks₁₁ _ _ hD <;> rw [conjTranspose_conjTranspose] | lemma | Matrix.IsHermitian.fromBlocks₂₂ | Lib.Math | CvxLean/Lib/Math/SchurComplement.lean | [
"Mathlib.LinearAlgebra.Matrix.Symmetric",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.PosDef",
"Mathlib.Algebra.Star.Pi"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
PosSemidef.fromBlocks₁₁ [Fintype m] [DecidableEq m] [Fintype n] {A : Matrix m m 𝕜}
(B : Matrix m n 𝕜) (D : Matrix n n 𝕜) (hA : A.PosDef) [Invertible A] :
(fromBlocks A B Bᴴ D).PosSemidef ↔ (D - Bᴴ * A⁻¹ * B).PosSemidef | by
rw [PosSemidef, IsHermitian.fromBlocks₁₁ _ _ hA.1]
constructor
{ intro h; refine' ⟨h.1, _⟩; intro x
have := h.2 (- ((A⁻¹ * B).mulVec x) ⊕ᵥ x)
rw [dotProduct_mulVec, schur_complement_eq₁₁ B D _ _ hA.1, neg_add_self,
dotProduct_zero, zero_add] at this
rw [dotProduct_mulVec]; exact this }
{ in... | lemma | Matrix.PosSemidef.fromBlocks₁₁ | Lib.Math | CvxLean/Lib/Math/SchurComplement.lean | [
"Mathlib.LinearAlgebra.Matrix.Symmetric",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.PosDef",
"Mathlib.Algebra.Star.Pi"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
PosSemidef.fromBlocks₂₂ [Fintype m] [Fintype n] [DecidableEq n] (A : Matrix m m 𝕜)
(B : Matrix m n 𝕜) {D : Matrix n n 𝕜} (hD : D.PosDef) [Invertible D] :
(fromBlocks A B Bᴴ D).PosSemidef ↔ (A - B * D⁻¹ * Bᴴ).PosSemidef | by
rw [← posSemidef_submatrix_equiv (Equiv.sumComm n m), Equiv.sumComm_apply,
fromBlocks_submatrix_sum_swap_sum_swap]
convert @PosSemidef.fromBlocks₁₁ m n 𝕜 _ _ _ _ _ _ _ hD _ <;>
rw [conjTranspose_conjTranspose] | lemma | Matrix.PosSemidef.fromBlocks₂₂ | Lib.Math | CvxLean/Lib/Math/SchurComplement.lean | [
"Mathlib.LinearAlgebra.Matrix.Symmetric",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse",
"Mathlib.LinearAlgebra.Matrix.PosDef",
"Mathlib.Algebra.Star.Pi"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
one_add_prod_le_prod_one_add {n : Type _} [Fintype n] [Nonempty n]
(f : n → ℝ) (hf : ∀ i, 0 ≤ f i) : 1 + (∏ i, f i) ≤ ∏ i, (1 + f i) | by
classical
calc 1 + (∏ i, f i)
= (∏ _a : n, 1 : ℝ) * ∏ a : n in univ \ univ, f a
+ (∏ a : n in ∅, 1) * ∏ a : n in univ \ ∅, f a := by
{ simp }
_ = ∑ x in {univ, ∅}, (∏ _a : n in x, 1 : ℝ) * ∏ a : n in univ \ x, f a := by
{ rw [Finset.sum_pair]; simp; exact Finset.univ_nonempty.ne_empty }
_ ≤... | lemma | Finset.one_add_prod_le_prod_one_add | Lib.Math | CvxLean/Lib/Math/Subadditivity.lean | [
"Mathlib.LinearAlgebra.Matrix.PosDef",
"Mathlib.LinearAlgebra.Matrix.Spectrum",
"Mathlib.LinearAlgebra.Eigenspace.Basic",
"Mathlib.LinearAlgebra.Matrix.LDL",
"Mathlib.LinearAlgebra.Matrix.DotProduct",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Spectrum",
"Cv... | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
eigenvectorMatrix_inv_mul : hA.eigenvectorMatrixInv * hA.eigenvectorMatrix = 1 | by
apply Basis.toMatrix_mul_toMatrix_flip | lemma | Matrix.IsHermitian.eigenvectorMatrix_inv_mul | Lib.Math | CvxLean/Lib/Math/Subadditivity.lean | [
"Mathlib.LinearAlgebra.Matrix.PosDef",
"Mathlib.LinearAlgebra.Matrix.Spectrum",
"Mathlib.LinearAlgebra.Eigenspace.Basic",
"Mathlib.LinearAlgebra.Matrix.LDL",
"Mathlib.LinearAlgebra.Matrix.DotProduct",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Spectrum",
"Cv... | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
spectral_theorem'' :
hA.eigenvectorMatrix * diagonal (RCLike.ofReal ∘ hA.eigenvalues) * hA.eigenvectorMatrixᴴ =
A | by
rw [conjTranspose_eigenvectorMatrix, Matrix.mul_assoc, ← spectral_theorem,
← Matrix.mul_assoc, eigenvectorMatrix_mul_inv, Matrix.one_mul] | theorem | Matrix.IsHermitian.spectral_theorem'' | Lib.Math | CvxLean/Lib/Math/Subadditivity.lean | [
"Mathlib.LinearAlgebra.Matrix.PosDef",
"Mathlib.LinearAlgebra.Matrix.Spectrum",
"Mathlib.LinearAlgebra.Eigenspace.Basic",
"Mathlib.LinearAlgebra.Matrix.LDL",
"Mathlib.LinearAlgebra.Matrix.DotProduct",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Spectrum",
"Cv... | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
IsHermitian.sqrt {A : Matrix n n ℝ} (hA : A.IsHermitian) : Matrix n n ℝ | hA.eigenvectorMatrix * Matrix.diagonal (fun i => (hA.eigenvalues i).sqrt) * hA.eigenvectorMatrixᵀ | def | Matrix.IsHermitian.sqrt | Lib.Math | CvxLean/Lib/Math/Subadditivity.lean | [
"Mathlib.LinearAlgebra.Matrix.PosDef",
"Mathlib.LinearAlgebra.Matrix.Spectrum",
"Mathlib.LinearAlgebra.Eigenspace.Basic",
"Mathlib.LinearAlgebra.Matrix.LDL",
"Mathlib.LinearAlgebra.Matrix.DotProduct",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Spectrum",
"Cv... | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
conjTranspose_eq_transpose {m n : Type _} {A : Matrix m n ℝ} : Aᴴ = Aᵀ | rfl | lemma | Matrix.conjTranspose_eq_transpose | Lib.Math | CvxLean/Lib/Math/Subadditivity.lean | [
"Mathlib.LinearAlgebra.Matrix.PosDef",
"Mathlib.LinearAlgebra.Matrix.Spectrum",
"Mathlib.LinearAlgebra.Eigenspace.Basic",
"Mathlib.LinearAlgebra.Matrix.LDL",
"Mathlib.LinearAlgebra.Matrix.DotProduct",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Spectrum",
"Cv... | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
PosSemidef.sqrt_mul_sqrt {A : Matrix n n ℝ} (hA : A.PosSemidef) :
hA.1.sqrt * hA.1.sqrt = A | calc
hA.1.sqrt * hA.1.sqrt =
hA.1.eigenvectorMatrix * (Matrix.diagonal (fun i => (hA.1.eigenvalues i).sqrt)
* (hA.1.eigenvectorMatrixᵀ * hA.1.eigenvectorMatrix)
* Matrix.diagonal (fun i => (hA.1.eigenvalues i).sqrt)) * hA.1.eigenvectorMatrixᵀ := by
simp [IsHermitian.sqrt, Matrix.mul_assoc]
... | lemma | Matrix.PosSemidef.sqrt_mul_sqrt | Lib.Math | CvxLean/Lib/Math/Subadditivity.lean | [
"Mathlib.LinearAlgebra.Matrix.PosDef",
"Mathlib.LinearAlgebra.Matrix.Spectrum",
"Mathlib.LinearAlgebra.Eigenspace.Basic",
"Mathlib.LinearAlgebra.Matrix.LDL",
"Mathlib.LinearAlgebra.Matrix.DotProduct",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Spectrum",
"Cv... | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
PosSemidef.PosSemidef_sqrt {A : Matrix n n ℝ} (hA : A.PosSemidef) :
hA.1.sqrt.PosSemidef | PosSemidef.conjTranspose_mul_mul _ _
(PosSemidef_diagonal (fun i => Real.sqrt_nonneg (hA.1.eigenvalues i))) | lemma | Matrix.PosSemidef.PosSemidef_sqrt | Lib.Math | CvxLean/Lib/Math/Subadditivity.lean | [
"Mathlib.LinearAlgebra.Matrix.PosDef",
"Mathlib.LinearAlgebra.Matrix.Spectrum",
"Mathlib.LinearAlgebra.Eigenspace.Basic",
"Mathlib.LinearAlgebra.Matrix.LDL",
"Mathlib.LinearAlgebra.Matrix.DotProduct",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Spectrum",
"Cv... | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
IsHermitian.one_add {A : Matrix n n ℝ} (hA : A.IsHermitian) : (1 + A).IsHermitian | by
dsimp [IsHermitian]; rw [IsHermitian.add _ hA]; simp | lemma | Matrix.IsHermitian.one_add | Lib.Math | CvxLean/Lib/Math/Subadditivity.lean | [
"Mathlib.LinearAlgebra.Matrix.PosDef",
"Mathlib.LinearAlgebra.Matrix.Spectrum",
"Mathlib.LinearAlgebra.Eigenspace.Basic",
"Mathlib.LinearAlgebra.Matrix.LDL",
"Mathlib.LinearAlgebra.Matrix.DotProduct",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Spectrum",
"Cv... | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
IsHermitian.has_eigenvector_one_add {A : Matrix n n ℝ} (hA : A.IsHermitian) (i : n) :
Module.End.HasEigenvector
(1 + Matrix.toLin' A) (1 + (hA.eigenvalues i)) ((hA.eigenvectorBasis) i) | Module.End.has_eigenvector_add
(Module.End.has_eigenvector_one (hA.hasEigenvector_eigenvectorBasis i).2)
(hA.hasEigenvector_eigenvectorBasis i) | lemma | Matrix.IsHermitian.has_eigenvector_one_add | Lib.Math | CvxLean/Lib/Math/Subadditivity.lean | [
"Mathlib.LinearAlgebra.Matrix.PosDef",
"Mathlib.LinearAlgebra.Matrix.Spectrum",
"Mathlib.LinearAlgebra.Eigenspace.Basic",
"Mathlib.LinearAlgebra.Matrix.LDL",
"Mathlib.LinearAlgebra.Matrix.DotProduct",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Spectrum",
"Cv... | [
"Module.End.has_eigenvector_add",
"Module.End.has_eigenvector_one"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
PosDef.PosDef_sqrt {A : Matrix n n ℝ} (hA : A.PosDef) : hA.1.sqrt.PosDef | by
unfold IsHermitian.sqrt
refine'
PosDef.conjTranspose_mul_mul _ (hA.1.eigenvectorMatrixᵀ)
(PosDef_diagonal (fun i => Real.sqrt_pos.2 (hA.eigenvalues_pos i))) _
show det hA.1.eigenvectorMatrixᵀ ≠ 0
rw [det_transpose]
apply det_ne_zero_of_right_inverse hA.1.eigenvectorMatrix_mul_inv | lemma | Matrix.PosDef.PosDef_sqrt | Lib.Math | CvxLean/Lib/Math/Subadditivity.lean | [
"Mathlib.LinearAlgebra.Matrix.PosDef",
"Mathlib.LinearAlgebra.Matrix.Spectrum",
"Mathlib.LinearAlgebra.Eigenspace.Basic",
"Mathlib.LinearAlgebra.Matrix.LDL",
"Mathlib.LinearAlgebra.Matrix.DotProduct",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Spectrum",
"Cv... | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
PosSemidef.PosDef_iff_det_ne_zero [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosSemidef) :
M.PosDef ↔ M.det ≠ 0 | by
refine' ⟨PosDef.det_ne_zero, _⟩; intro hdet; refine' ⟨hM.1, _⟩
intros x hx
apply lt_of_le_of_ne' (hM.2 x)
rw [← hM.sqrt_mul_sqrt, ← mulVec_mulVec, dotProduct_mulVec, ← transpose_transpose hM.1.sqrt,
vecMul_transpose, transpose_transpose, ← conjTranspose_eq_transpose,
hM.PosSemidef_sqrt.1.eq]
simp o... | lemma | Matrix.PosSemidef.PosDef_iff_det_ne_zero | Lib.Math | CvxLean/Lib/Math/Subadditivity.lean | [
"Mathlib.LinearAlgebra.Matrix.PosDef",
"Mathlib.LinearAlgebra.Matrix.Spectrum",
"Mathlib.LinearAlgebra.Eigenspace.Basic",
"Mathlib.LinearAlgebra.Matrix.LDL",
"Mathlib.LinearAlgebra.Matrix.DotProduct",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Spectrum",
"Cv... | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
det_add_det_le_det_add' [Nonempty n] (A B : Matrix n n ℝ) (hA : A.PosDef)
(hB : B.PosSemidef) : A.det + B.det ≤ (A + B).det | by
let sqrtA := hA.1.sqrt
have isUnit_det_sqrtA :=
isUnit_iff_ne_zero.2 hA.PosDef_sqrt.det_ne_zero
have : IsUnit sqrtA :=
(isUnit_iff_isUnit_det _).2 isUnit_det_sqrtA
have IsHermitian_sqrtA : sqrtA⁻¹.IsHermitian := by
{ apply IsHermitian.nonsingular_inv (hA.posSemidef.PosSemidef_sqrt.1)
exact isUn... | lemma | Matrix.det_add_det_le_det_add' | Lib.Math | CvxLean/Lib/Math/Subadditivity.lean | [
"Mathlib.LinearAlgebra.Matrix.PosDef",
"Mathlib.LinearAlgebra.Matrix.Spectrum",
"Mathlib.LinearAlgebra.Eigenspace.Basic",
"Mathlib.LinearAlgebra.Matrix.LDL",
"Mathlib.LinearAlgebra.Matrix.DotProduct",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Spectrum",
"Cv... | [
"Finset.one_add_prod_le_prod_one_add"
] | Subadditivity lemma for positive semidefinite matrices. This version assumes that one of the
matrices is positive definite. See `det_add_det_le_det_add` for the more general statement.
The argument is taken from Andreas Thom's comment on mathoverflow:
https://mathoverflow.net/questions/65424/determinant-of-sum-of-posi... | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
det_add_det_le_det_add [Nonempty n] (A B : Matrix n n ℝ) (hA : A.PosSemidef)
(hB : B.PosSemidef) : A.det + B.det ≤ (A + B).det | by
by_cases hA' : A.det = 0
{ by_cases hB' : B.det = 0
{ simp [hA', hB']
apply (hA.add hB).det_nonneg }
{ rw [add_comm A B, add_comm A.det B.det]
apply det_add_det_le_det_add' _ _ (hB.PosDef_iff_det_ne_zero.2 hB') hA } }
{ apply det_add_det_le_det_add' _ _ (hA.PosDef_iff_det_ne_zero.2 hA') hB ... | lemma | Matrix.det_add_det_le_det_add | Lib.Math | CvxLean/Lib/Math/Subadditivity.lean | [
"Mathlib.LinearAlgebra.Matrix.PosDef",
"Mathlib.LinearAlgebra.Matrix.Spectrum",
"Mathlib.LinearAlgebra.Eigenspace.Basic",
"Mathlib.LinearAlgebra.Matrix.LDL",
"Mathlib.LinearAlgebra.Matrix.DotProduct",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef",
"CvxLean.Lib.Math.LinearAlgebra.Matrix.Spectrum",
"Cv... | [] | Subadditivity lemma for positive semidefinite matrices. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
repr_gramSchmidt_diagonal {i : ι} (b : Basis ι 𝕜 E) :
b.repr (gramSchmidt 𝕜 b i) i = 1 | by
rw [gramSchmidt_def, map_sub, Finsupp.sub_apply, Basis.repr_self, Finsupp.single_eq_same,
sub_eq_self, map_sum, Finsupp.coe_finset_sum, Finset.sum_apply, Finset.sum_eq_zero]
intros j hj
rw [Finset.mem_Iio] at hj
simp [orthogonalProjection_singleton, gramSchmidt_triangular hj] | lemma | repr_gramSchmidt_diagonal | Lib.Math.Analysis.InnerProductSpace | CvxLean/Lib/Math/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | [
"Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
spectral_theorem' (v : E) (i : Fin n)
(xs : OrthonormalBasis (Fin n) 𝕜 E) (as : Fin n → ℝ)
(hxs : ∀ j, Module.End.HasEigenvector T (as j) (xs j)) :
xs.repr (T v) i = as i * xs.repr v i | by
suffices hsuff : ∀ (w : EuclideanSpace 𝕜 (Fin n)),
T (xs.repr.symm w) = xs.repr.symm (fun i => as i * w i) by
simpa only [LinearIsometryEquiv.symm_apply_apply,
LinearIsometryEquiv.apply_symm_apply]
using congr_arg (fun (v : E) => (xs.repr) v i) (hsuff ((xs.repr) v))
intros w
simp_rw ... | lemma | LinearMap.spectral_theorem' | Lib.Math.Analysis.InnerProductSpace | CvxLean/Lib/Math/Analysis/InnerProductSpace/Spectrum.lean | [
"Mathlib.LinearAlgebra.Matrix.Spectrum"
] | [] | *Diagonalization theorem*, *spectral theorem*; version 3: A self-adjoint operator `T` on a
finite-dimensional inner product space `E` acts diagonally on the identification of `E` with
Euclidean space induced by an orthonormal basis of eigenvectors of `T`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
Array.zeros [Zero α] (n : Nat) : Array α | Array.mk (List.replicate n 0) | def | Array.zeros | Lib.Math.Data | CvxLean/Lib/Math/Data/Array.lean | [
"Mathlib.Data.List.Basic",
"Mathlib.Algebra.Group.Defs"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
DArray.zeros [Zero α] (a b : Nat) : Array (Array α) | Array.mk (List.replicate b (Array.zeros a)) | def | DArray.zeros | Lib.Math.Data | CvxLean/Lib/Math/Data/Array.lean | [
"Mathlib.Data.List.Basic",
"Mathlib.Algebra.Group.Defs"
] | [
"Array.zeros"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
Array.filterIdx (as : Array α) (f : Nat → Bool) | Id.run do
let mut bs := #[]
for h : i in [:as.size] do
if f i then
bs := bs.push (as.get ⟨i, h.2⟩)
return bs | def | Array.filterIdx | Lib.Math.Data | CvxLean/Lib/Math/Data/Array.lean | [
"Mathlib.Data.List.Basic",
"Mathlib.Algebra.Group.Defs"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
Array.zipWithMAux (f : α → β → MetaM γ) (as : Array α) (bs : Array β) (i : Nat) (cs : Array γ) :
MetaM (Array γ) | do
if h : i < as.size then
let a := as.get ⟨i, h⟩;
if h : i < bs.size then
let b := bs.get ⟨i, h⟩;
return ← zipWithMAux f as bs (i+1) <| cs.push <| ← f a b
else
pure cs
else
pure cs
termination_by as.size - i | def | Array.zipWithMAux | Lib.Math.Data | CvxLean/Lib/Math/Data/Array.lean | [
"Mathlib.Data.List.Basic",
"Mathlib.Algebra.Group.Defs"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
Array.zipWithM (f : α → β → MetaM γ) (as : Array α) (bs : Array β) :
MetaM (Array γ) | zipWithMAux f as bs 0 #[] | def | Array.zipWithM | Lib.Math.Data | CvxLean/Lib/Math/Data/Array.lean | [
"Mathlib.Data.List.Basic",
"Mathlib.Algebra.Group.Defs"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
Array.drop (n : ℕ) (as : Array α) : Array α | Array.mk (as.data.drop n) | def | Array.drop | Lib.Math.Data | CvxLean/Lib/Math/Data/Array.lean | [
"Mathlib.Data.List.Basic",
"Mathlib.Algebra.Group.Defs"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
Array.take (n : ℕ) (as : Array α) : Array α | Array.mk (as.data.take n) | def | Array.take | Lib.Math.Data | CvxLean/Lib/Math/Data/Array.lean | [
"Mathlib.Data.List.Basic",
"Mathlib.Algebra.Group.Defs"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
findIdx' (p : α → Prop) [DecidablePred p] : List α → Option ℕ | | [] => none
| (a::l) => if p a then some 0 else (findIdx' p l).map Nat.succ | def | List.findIdx' | Lib.Math.Data | CvxLean/Lib/Math/Data/List.lean | [
"Mathlib.Algebra.Ring.Basic"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
toVec (x : List α) : Fin x.length → α | fun i => List.get x ⟨i, i.2⟩ | def | List.toVec | Lib.Math.Data | CvxLean/Lib/Math/Data/List.lean | [
"Mathlib.Algebra.Ring.Basic"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
finRange' (d n : Nat) : List (Fin n) | if hn : 0 < n then (List.range d).map (Fin.ofNat' · hn) else [] | def | List.finRange' | Lib.Math.Data | CvxLean/Lib/Math/Data/List.lean | [
"Mathlib.Algebra.Ring.Basic"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
sum' [Zero α] [Add α] (L : List α) : α | L.foldl (· + ·) 0 | def | List.sum' | Lib.Math.Data | CvxLean/Lib/Math/Data/List.lean | [
"Mathlib.Algebra.Ring.Basic"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
const (k : α) : Matrix m n α | fun _ _ => k | def | Matrix.const | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
abs (A : Matrix m n ℝ) : Matrix m n ℝ | fun i j => |A i j| | def | Matrix.abs | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
vecCons_zero_zero {n} [Zero α] : vecCons (0 : α) (0 : Fin n → α) = 0 | by
ext i; refine' Fin.cases _ _ i <;> simp [vecCons] | theorem | Matrix.vecCons_zero_zero | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
smul_vecCons {n} [Zero α] [SMulZeroClass ℝ α] (x : ℝ) (y : α) (v : Fin n → α) :
x • vecCons y v = vecCons (x • y) (x • v) | by
ext i; refine' Fin.cases _ _ i <;> simp [vecCons] | theorem | Matrix.smul_vecCons | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
add_vecCons {n} [Zero α] [SMulZeroClass ℝ α] [Add α] (x : α) (v : Fin n → α) (y : α)
(w : Fin n → α) : vecCons x v + vecCons y w = vecCons (x + y) (v + w) | by
ext i; refine' Fin.cases _ _ i <;> simp [vecCons] | theorem | Matrix.add_vecCons | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
sum [Fintype m] [AddCommMonoid α] (X : Matrix m m α) : α | ∑ i, (∑ j, X i j) | def | Matrix.sum | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
toArray (A : Matrix (Fin n) (Fin n) Float) : Array (Array Float) | (Array.range n).map <| fun i =>
if h : i < n then Vec.Computable.toArray (A ⟨i, h⟩) else Array.mk <| List.replicate n 0 | def | Matrix.Computable.toArray | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [
"Vec.Computable.toArray"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
dotProduct (v w : Fin n → Float) : Float | (Array.zipWith (Vec.Computable.toArray v) (Vec.Computable.toArray w) Float.mul).foldl Float.add 0 | def | Matrix.Computable.dotProduct | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [
"Vec.Computable.toArray"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
mulVec (M : Matrix (Fin n) (Fin m) Float) (v : (Fin m) → Float) : Fin n → Float | fun i => (fun j => M i j) ⬝ᵥᶜ v | def | Matrix.Computable.mulVec | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
vecMul (x : Fin n → Float) (M : Matrix (Fin n) (Fin m) Float) : Fin m → Float | fun j => x ⬝ᵥᶜ fun i => M i j | def | Matrix.Computable.vecMul | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
transpose {m n} {α} (M : Matrix m n α) : Matrix n m α | fun i j => M j i | def | Matrix.Computable.transpose | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
diag {n} {α} (M : Matrix n n α) : n → α | fun i => M i i | def | Matrix.Computable.diag | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
mul (M : Matrix (Fin l) (Fin m) Float) (N : Matrix (Fin m) (Fin n) Float) :
Matrix (Fin l) (Fin n) Float | fun i k => (fun j => M i j) ⬝ᵥᶜ (fun j => N j k) | def | Matrix.Computable.mul | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
sum (A : Matrix (Fin n) (Fin n) Float) : Float | Vec.Computable.sum (fun i => Vec.Computable.sum (A i)) | def | Matrix.Computable.sum | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [
"Vec.Computable.sum"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
trace (A : Matrix (Fin n) (Fin n) Float) : Float | (Vec.Computable.toArray (fun i => A i i)).foldl (· + ·) 0 | def | Matrix.Computable.trace | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [
"Vec.Computable.toArray"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
covarianceMatrix {N n : ℕ} (Y : Matrix (Fin N) (Fin n) Float) (i j : Fin n) : Float | Vec.Computable.sum (fun k => (Y k i) * (Y k j)) / (OfNat.ofNat N) | def | Matrix.Computable.covarianceMatrix | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [
"Vec.Computable.sum",
"covarianceMatrix"
] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
diagonal (x : Fin n → Float) : Matrix (Fin n) (Fin n) Float | fun i j => (if i = j then x i else 0) | def | Matrix.Computable.diagonal | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
fromBlocks {l : Type} {m : Type} {n : Type} {o : Type} {α : Type} :
Matrix n l α → Matrix n m α → Matrix o l α → Matrix o m α → Matrix (n ⊕ o) (l ⊕ m) α | fun A B C D i j =>
match i with
| Sum.inl i =>
match j with
| Sum.inl j => A i j
| Sum.inr j => B i j
| Sum.inr i =>
match j with
| Sum.inl j => C i j
| Sum.inr j => D i j | def | Matrix.Computable.fromBlocks | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
toUpperTri (A : Matrix (Fin n) (Fin n) Float) : Matrix (Fin n) (Fin n) Float | fun i j => if i ≤ j then A i j else 0 | def | Matrix.Computable.toUpperTri | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
minorAux (A : Matrix (Fin n.succ) (Fin n.succ) Float) (a b : Fin n.succ) :
Matrix (Fin n) (Fin n) Float | fun i j =>
let i' : Fin n.succ := if i.val < a.val then i else i.succ;
let j' : Fin n.succ := if j.val < b.val then j else j.succ;
A i' j' | def | Matrix.Computable.minorAux | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
minor (A : Matrix (Fin n) (Fin n) Float) (a b : Fin n) :
Matrix (Fin n.pred) (Fin n.pred) Float | match n with
| 0 => fun _ => Fin.elim0
| _ + 1 => minorAux A a b | def | Matrix.Computable.minor | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
det {n : ℕ} (A : Matrix (Fin n) (Fin n) Float) : Float | if h : 0 < n then
if n == 1 then A ⟨0, h⟩ ⟨0, h⟩ else
(List.finRange n).foldl (fun s i =>
s + (-1) ^ (Float.ofNat i.val) * A i ⟨0, h⟩ * det (minor A i ⟨0, h⟩)) 0
else 0 | def | Matrix.Computable.det | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
cofactor (A : Matrix (Fin n) (Fin n) Float) : Matrix (Fin n) (Fin n) Float | fun i j => (-1) ^ (Float.ofNat (i.val + j.val)) * (A i j) | def | Matrix.Computable.cofactor | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
adjugate (A : Matrix (Fin n) (Fin n) Float) : Matrix (Fin n) (Fin n) Float | transpose (cofactor A) | def | Matrix.Computable.adjugate | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
inv (A : Matrix (Fin n) (Fin n) Float) : Matrix (Fin n) (Fin n) Float | (1 / det A) • adjugate A | def | Matrix.Computable.inv | Lib.Math.Data | CvxLean/Lib/Math/Data/Matrix.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Matrix.Basic",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.Data.Array.Defs",
"CvxLean.Lib.Math.Data.List",
"CvxLean.Lib.Math.Data.Vec"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 | |
entr (x : Real) | -(x * log x) | def | Real.entr | Lib.Math.Data | CvxLean/Lib/Math/Data/Real.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Exponential",
"Mathlib.Analysis.SpecialFunctions.Pow.Real"
] | [] | See `CvxLean.Tactic.DCP.AtomLibrary.Fns.Entr`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
huber (x : Real) | if abs x ≤ 1 then x ^ 2 else 2 * abs x - 1 | def | Real.huber | Lib.Math.Data | CvxLean/Lib/Math/Data/Real.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Exponential",
"Mathlib.Analysis.SpecialFunctions.Pow.Real"
] | [] | See `CvxLean.Tactic.DCP.AtomLibrary.Fns.Huber`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
klDiv (x y : Real) | x * log (x / y) - x + y | def | Real.klDiv | Lib.Math.Data | CvxLean/Lib/Math/Data/Real.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Exponential",
"Mathlib.Analysis.SpecialFunctions.Pow.Real"
] | [] | See `CvxLean.Tactic.DCP.AtomLibrary.Fns.KLDiv`. | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
instDivReal : Div ℝ | by
infer_instance | instance | Real.instDivReal | Lib.Math.Data | CvxLean/Lib/Math/Data/Real.lean | [
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Exponential",
"Mathlib.Analysis.SpecialFunctions.Pow.Real"
] | [] | https://github.com/verified-optimization/CvxLean | c62c2f292c6420f31a12e738ebebdfed50f6f840 |
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